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14162307
|
Vomilenine reductase
|
In enzymology, a vomilenine reductase (EC 1.5.1.32) is an enzyme that catalyzes the chemical reaction
1,2-dihydrovomilenine + NADP+ formula_0 vomilenine + NADPH + H+
Thus, the two substrates of this enzyme are 1,2-dihydrovomilenine and NADP+, whereas its 3 products are vomilenine, NADPH, and H+.
This enzyme belongs to the family of oxidoreductases, specifically those acting on the CH-NH group of donors with NAD+ or NADP+ as acceptor. The systematic name of this enzyme class is 1,2-dihydrovomilenine:NADP+ oxidoreductase. This enzyme participates in indole and ipecac alkaloid biosynthesis.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "\\rightleftharpoons"
}
] |
https://en.wikipedia.org/wiki?curid=14162307
|
14162739
|
Oscillator strength
|
In spectroscopy, oscillator strength is a dimensionless quantity that expresses the probability of absorption or emission of electromagnetic radiation in transitions between energy levels of an atom or molecule. For example, if an emissive state has a small oscillator strength, will outpace . Conversely, "bright" transitions will have large oscillator strengths. The oscillator strength can be thought of as the ratio between the quantum mechanical transition rate and the classical absorption/emission rate of a single electron oscillator with the same frequency as the transition.
Theory.
An atom or a molecule can absorb light and undergo a transition from
one quantum state to another.
The oscillator strength formula_0 of a transition from a lower state
formula_1 to an upper state formula_2 may be defined by
formula_3
where formula_4 is the mass of an electron and formula_5 is
the reduced Planck constant. The quantum states formula_6 1,2, are assumed to have several
degenerate sub-states, which are labeled by formula_7. "Degenerate" means
that they all have the same energy formula_8.
The operator formula_9 is the sum of the x-coordinates formula_10
of all formula_11 electrons in the system, i.e.
formula_12
The oscillator strength is the same for each sub-state formula_13.
The definition can be recast by inserting the Rydberg energy formula_14 and Bohr radius formula_15
formula_16
In case the matrix elements of formula_17 are the same, we can get rid of the sum and of the 1/3 factor
formula_18
Thomas–Reiche–Kuhn sum rule.
To make equations of the previous section applicable to the states belonging to the continuum spectrum, they should be rewritten in terms of matrix elements of the momentum formula_19. In absence of magnetic field, the Hamiltonian can be written as formula_20, and calculating a commutator formula_21 in the basis of eigenfunctions of formula_22 results in the relation between matrix elements
formula_23.
Next, calculating matrix elements of a commutator formula_24 in the same basis and eliminating matrix elements of formula_25, we arrive at
formula_26
Because formula_27, the above expression results in a sum rule
formula_28
where formula_29 are oscillator strengths for quantum transitions between the states formula_30 and formula_31. This is the Thomas-Reiche-Kuhn sum rule, and the term with formula_32 has been omitted because in confined systems such as atoms or molecules the diagonal matrix element formula_33 due to the time inversion symmetry of the Hamiltonian formula_22. Excluding this term eliminates divergency because of the vanishing denominator.
Sum rule and electron effective mass in crystals.
In crystals, the electronic energy spectrum has a band structure formula_34. Near the minimum of an isotropic energy band, electron energy can be expanded in powers of formula_19 as formula_35 where formula_36 is the electron effective mass. It can be shown that it satisfies the equation
formula_37
Here the sum runs over all bands with formula_38. Therefore, the ratio formula_39 of the free electron mass formula_40 to its effective mass formula_36 in a crystal can be considered as the oscillator strength for the transition of an electron from the quantum state at the bottom of the formula_30 band into the same state.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "f_{12}"
},
{
"math_id": 1,
"text": "|1\\rangle"
},
{
"math_id": 2,
"text": "|2\\rangle"
},
{
"math_id": 3,
"text": "\n f_{12} = \\frac{2 }{3}\\frac{m_e}{\\hbar^2}(E_2 - E_1) \\sum_{\\alpha=x,y,z}\n | \\langle 1 m_1 | R_\\alpha | 2 m_2 \\rangle |^2,\n"
},
{
"math_id": 4,
"text": "m_e"
},
{
"math_id": 5,
"text": "\\hbar"
},
{
"math_id": 6,
"text": "|n\\rangle, n="
},
{
"math_id": 7,
"text": "m_n"
},
{
"math_id": 8,
"text": "E_n"
},
{
"math_id": 9,
"text": "R_x"
},
{
"math_id": 10,
"text": "r_{i,x}"
},
{
"math_id": 11,
"text": "N"
},
{
"math_id": 12,
"text": "\n R_\\alpha = \\sum_{i=1}^N r_{i,\\alpha}.\n"
},
{
"math_id": 13,
"text": "|n m_n\\rangle"
},
{
"math_id": 14,
"text": "\\text{Ry}"
},
{
"math_id": 15,
"text": "a_0"
},
{
"math_id": 16,
"text": "\n f_{12} = \\frac{E_2 - E_1}{3\\, \\text{Ry}} \\frac{\\sum_{\\alpha=x,y,z}\n | \\langle 1 m_1 | R_\\alpha | 2 m_2 \\rangle |^2}{a_0^2}.\n"
},
{
"math_id": 17,
"text": "R_x, R_y, R_z"
},
{
"math_id": 18,
"text": "\n f_{12} = 2\\frac{m_e}{\\hbar^2}(E_2 - E_1) \\, | \\langle 1 m_1 | R_x | 2 m_2 \\rangle |^2.\n"
},
{
"math_id": 19,
"text": "\\boldsymbol{p}"
},
{
"math_id": 20,
"text": "H=\\frac{1}{2m}\\boldsymbol{p}^2+V(\\boldsymbol{r})"
},
{
"math_id": 21,
"text": "[H,x]"
},
{
"math_id": 22,
"text": "H"
},
{
"math_id": 23,
"text": "\n x_{nk}=-\\frac{i\\hbar/m}{E_n-E_k}(p_x)_{nk}.\n"
},
{
"math_id": 24,
"text": "[p_x,x]"
},
{
"math_id": 25,
"text": "x"
},
{
"math_id": 26,
"text": "\n \\langle n|[p_x,x]|n\\rangle=\\frac{2i\\hbar}{m}\\sum_{k\\neq n} \\frac{|\\langle n|p_x|k\\rangle|^2}{E_n-E_k}.\n"
},
{
"math_id": 27,
"text": "[p_x,x]=-i\\hbar"
},
{
"math_id": 28,
"text": "\n \\sum_{k\\neq n}f_{nk}=1,\\,\\,\\,\\,\\,f_{nk}=-\\frac{2}{m}\\frac{|\\langle n|p_x|k\\rangle|^2}{E_n-E_k},\n"
},
{
"math_id": 29,
"text": "f_{nk}"
},
{
"math_id": 30,
"text": "n"
},
{
"math_id": 31,
"text": "k"
},
{
"math_id": 32,
"text": "k=n"
},
{
"math_id": 33,
"text": "\\langle n|p_x|n\\rangle=0"
},
{
"math_id": 34,
"text": "E_n(\\boldsymbol{p})"
},
{
"math_id": 35,
"text": "E_n(\\boldsymbol{p})=\\boldsymbol{p}^2/2m^*"
},
{
"math_id": 36,
"text": "m^*"
},
{
"math_id": 37,
"text": "\n \\frac{2}{m}\\sum_{k\\neq n}\\frac{|\\langle n|p_x|k\\rangle|^2}{E_k-E_n}+\\frac{m}{m^*}=1.\n"
},
{
"math_id": 38,
"text": "k\\neq n"
},
{
"math_id": 39,
"text": "m/m^*"
},
{
"math_id": 40,
"text": "m"
}
] |
https://en.wikipedia.org/wiki?curid=14162739
|
14164479
|
HSD17B1
|
Protein-coding gene in the species Homo sapiens
17β-Hydroxysteroid dehydrogenase 1 (17β-HSD1) is an enzyme that in humans is encoded by the "HSD17B1" gene. This enzyme oxidizes or reduces the C17 hydroxy/keto group of androgens and estrogens and hence is able to regulate the potency of these sex steroids
Function.
This enzyme is responsible for the interconversion of estrone (E1) and estradiol (E2) and for the interconversion of androstenedione and testosterone:
17β-estradiol + NADP+ + formula_0 estrone + NADPH + H+
testosterone + NADP+ + formula_0 androstenedione + NADPH + H+
The human 17β-HSD1 isozyme is highly specific for estrogens over androgens whereas the rodent isozyme is less specific.
Discovery.
Human 17β-HSD1 was the first enzyme of the 17β-HSD family to be cloned and to have its sequence identified. Its three-dimensional structure is also the first example of any human steroid-converting enzyme.
Structure.
This enzyme contains a short-chain dehydrogenase domain that contains a characteristic 3-layer (αβα) sandwich known as a Rossmann fold. The human enzyme contains 327 amino acids and exists as a homodimer with two identical subunits of 34.5 kDa The N-terminal short-chain dehydrogenase domain contains binding site for the NADP+/NADPH cofactor. A narrow, hydrophobic C-terminal domain contains a binding pocket for the steroid substrate.
Clinical significance.
Estradiol stimulates while dihydrotestosterone (DHT) inhibits breast cancer growth. Furthermore 17β-HSD1 levels positively correlate with estradiol and negatively correlate with DHT levels in breast cancer cells. Hence 17β-HSD1 represents a possible drug target for breast cancer treatment.
References.
<templatestyles src="Reflist/styles.css" />
Further reading.
<templatestyles src="Refbegin/styles.css" />
|
[
{
"math_id": 0,
"text": "\\rightleftharpoons"
}
] |
https://en.wikipedia.org/wiki?curid=14164479
|
14166896
|
Phronima sedentaria
|
Species of crustacean
<templatestyles src="Template:Taxobox/core/styles.css" />
Phronima sedentaria is a species of amphipod crustaceans found in oceans at a depth of up to . They are large in size relative to other members of the family Phronimidae. Individuals may be found inside barrel-like homes, created most commonly from the tunics of select species of tunicate, where they rear their young. "P. sedentaria" is known to employ multiple feeding strategies and other interesting behaviors, including daily vertical migration. The species is also known by the more common names “pram bug” and “barrel shrimp.”
Description.
"Phronima sedentaria" is the largest and most abundant species in the family Phronimidae. Sexual dimorphism is reflected between male and female members in the more extended and prominent antennae of the males relative to the short, reduced ones of females. More obviously, the size discrepancy between males and females distinguishes them further. Females measure up to 42 mm (1.7 in) long, while males are only 15 mm (0.6 in) long. This species also possesses a complex optical system which involves the use of two sets of compound eyes. Both sets use bundles of crystalline cones to process visual information: one set ("medial eyes") faces dorsally and one set faces laterally ("lateral eyes").
The medial eyes have a very small retina, but very large compound eyes. The compound eyes occupy the entire dorsal surface of the head, collecting light, and guides the light down crystalline cones like optical fibers, until they hit the medial retina. This is hypothesized to be camouflage against predators. While most of its body is transparent, retinas are necessarily opaque. By using optical fibers, a large light-collecting surface can be combined with a small retina.
The medial compound eyes are extremely fine, with angle formula_0 between ommatidia. However, this was at the price of restricted visual field. Each eye can only see an angular width of formula_1, and both visual fields are largely overlapping, meaning that they have fine binocular vision in a narrow beam aimed directly above.
Distribution.
"Phronima sedentaria" is found in temperate, subtropical, and tropical waters of all the world's oceans, including the Mediterranean Sea. It is usually found in midwater pelagic habitats, but can be found migrating all the way to the surface.
Ecology.
"Phronima sedentaria" most commonly exhibits a symbiotic relationship with tunicates of the genera "Pyrosoma spp.", "Doliolum spp.", and "Salpa" "spp". It is not entirely certain how to classify the symbiotic relationship between "P. sedentaria" and its hosts, as instances of commensalism, parasitism, and predation have all been observed. However, some research has suggested that the majority of suborder Hyperiidea members exhibit parasitic behavior. Females of "P. sedentaria" live in the barrel-like bodies of salps, pyrosomes, and cnidarians, and use their strong pleopods to propel their homes through the water. They can somersault rapidly in their barrels, thus quickly changing direction. However, speed is reduced by a factor of three to four when swimming while inside a barrel compared to swimming without one. The shape of the barrel is generally asymmetrical, with one opening three times larger than that of the other. In order to construct their barrels, females first locate a suitable salp, pyrosome, or cnidarian host and either cut into the host or enter through an existing opening. Once inside, the female consumes the organism within and carves out the gelatinous inside, leaving nothing but the tunic. The cells on the tunic layer may serve various functions for the "P. sedentaria," including protection from UV light, storage of acid, and defense against microorganisms, like bacteria. In laboratory experiments with restricted access to potential hosts, females have additionally displayed competition for barrels.
While certain prey (salps, pyrosomes, and cnidaria) have additional uses for "P. sedentaria" in hosting their young and providing feeding platforms, the species is also carnivorous on zooplankton, krill, arrowworms, and other crustaceans. "P. sedentaria" use different feeding techniques depending on the food source, but the leading sets of pereiopods (front legs) are primarily used in all cases. Mouth pieces, such as the mandible, maxillipeds, and maxillae, manipulate the food into small pieces which are then able to fit through the esophagus. Feeding preferentially occurs at nighttime when members of this species undergo a vertical migration of around 200–350 meters to the ocean's surface. Research has shown this species is susceptible to temperature fluctuations outside of the range 8-25 degrees Celsius (46-77 degrees Fahrenheit), which explains the desire for cooler deep water (300–600 meters deep) throughout the day and warmer shallow waters (0–25 meters deep) at night. "P. sedentaria" typically migrate to hypoxic areas (such as the Oxygen Minimum Zone) during the day, causing low metabolic rates and physical activity.
Known predators of "P. sedentaria" include the longnose lancetfish, European flying squid, Pacific pomfret, albacore, and skipjack tuna.
Life cycle & development.
Female "Phronima sedentaria" are capable of producing up to 600 eggs at a time. Juveniles spend their early development within the mother in a specialized pouch called the marsupium. After finding a suitable host, the female begins to transform the barrel into a nursery for its young. She uses her pleopods and anterior pereiopods to remove offspring from the marsupium, while her posterior pereiopods maintain stability and grasp onto the barrel. Once inside, the young organize themselves into a medial ring around the interior of the barrel. This shape is maintained until the mother delivers food, which the offspring then feed on and return to formation afterwards. Young "P. sedentaria" use the barrel as another food source. The offspring of "P. sedentaria" develop within their barrel homes until reaching prematurity, after which they are able to feed and survive independently. Development is characterized by growth stages in which molting occurs. Each molt adds a new segment to the sets of pleopods in the rear. The emergence of sexual dimorphism occurs soon after prematurity.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "0.25^\\circ"
},
{
"math_id": 1,
"text": "15^\\circ"
}
] |
https://en.wikipedia.org/wiki?curid=14166896
|
14167645
|
Delta-aminolevulinic acid dehydratase
|
Protein-coding gene in the species Homo sapiens
Aminolevulinic acid dehydratase (porphobilinogen synthase, or ALA dehydratase, or aminolevulinate dehydratase) is an enzyme (EC 4.2.1.24) that in humans is encoded by the "ALAD" gene. Porphobilinogen synthase (or ALA dehydratase, or aminolevulinate dehydratase) synthesizes porphobilinogen through the asymmetric condensation of two molecules of aminolevulinic acid. All natural tetrapyrroles, including hemes, chlorophylls and vitamin B12, share porphobilinogen as a common precursor. Porphobilinogen synthase is the prototype morpheein.
Function.
It catalyzes the following reaction, the second step of the biosynthesis of porphyrin:
2 5-Aminolevulinic acid formula_0 porphobilinogen + 2 H2O
It therefore catalyzes the condensation of 2 molecules of 5-aminolevulinate to form porphobilinogen (a precursor of heme, cytochromes and other hemoproteins). This reaction is the first common step in the biosynthesis of all biological tetrapyrroles. Zinc is essential for enzymatic activity.
Structure.
The structural basis for allosteric regulation of Porphobilinogen synthase (PBGS) is modulation of a quaternary structure equilibrium between octamer and hexamer (via dimers), which is represented schematically as 6mer* ↔ 2mer* ↔ 2mer ↔ 8mer. The * represents a reorientation between two domains of each subunit that occurs in the dissociated state because it is sterically forbidden in the larger multimers.
PBGS is encoded by a single gene and each PBGS multimer is composed of multiple copies of the same protein. Each PBGS subunit consists of a ~300 residue αβ-barrel domain, which houses the enzyme's active site in its center, and a >25 residue N-terminal arm domain. Allosteric regulation of PBGS can be described in terms of the orientation of the αβ-barrel domain with respect to the N-terminal arm domain.
Each "N"-terminal arm has up to two interactions with other subunits in a PBGS multimer. One of these interactions helps to stabilize a "closed" conformation of the active site lid. The other interaction restricts solvent access from the other end of the αβ-barrel.
In the inactive multimeric state, the N-terminal arm domain is not involved in the lid-stabilizing interaction, and in the crystal structure of the inactive assembly, the active site lid is disordered.
Allosteric regulators.
As a nearly universal enzyme with a highly conserved active site, PBGS would not be a prime target for the development of antimicrobials and/or herbicides. To the contrary, allosteric sites can be much more phylogenetically variable than active sites, thus presenting more drug development opportunities.
Phylogenetic variation in PBGS allostery leads to the framing of discussion of PBGS allosteric regulation in terms of intrinsic and extrinsic factors.
Intrinsic allosteric regulators.
Magnesium.
The allosteric magnesium ion lies at the highly hydrated interface of two pro-octamer dimers. It appears to be easily dissociable, and it has been shown that hexamers accumulate when magnesium is removed "in vitro".
pH.
Though it is not common to consider hydronium ions as allosteric regulators, in the case of PBGS, side chain protonation at locations other than the active site has been shown to affect the quaternary structure equilibrium, and thus to affect the rate of its catalyzed reaction as well.
Extrinsic allosteric regulators.
Small molecule hexamer stabilization.
Inspection of the PBGS 6mer* reveals a surface cavity that is not present in the 8mer. Small molecule binding to this phylogenetically variable cavity has been proposed to stabilize 6mer* of the targeted PBGS and consequently inhibit activity.
Such allosteric regulators are known as "morphlocks" because they lock PBGS in a specific morpheein form (6mer*).
Lead poisoning.
ALAD enzymatic activity is inhibited by lead, beginning at blood lead levels that were once considered to be safe (<10 μg/dL) and continuing to correlate negatively across the range from 5 to 95 μg/dL. Inhibition of ALAD by lead leads to anemia primarily because it both inhibits heme synthesis and shortens the lifespan of circulating red blood cells, but also by stimulating the excessive production of the hormone erythropoietin, leading to inadequate maturation of red cells from their progenitors. A defect in the ALAD structural gene can cause increased sensitivity to lead poisoning and acute hepatic porphyria. Alternatively spliced transcript variants encoding different isoforms have been identified.
Deficiency.
A deficiency of porphobilinogen synthase is usually acquired (rather than hereditary) and can be caused by heavy metal poisoning, especially lead poisoning, as the enzyme is very susceptible to inhibition by heavy metals.
Hereditary insufficiency of porphobilinogen synthase is called porphobilinogen synthase (or ALA dehydratase) deficiency porphyria. It is an extremely rare cause of porphyria, with less than 10 cases ever reported. All disease associated protein variants favor hexamer formation relative to the wild type human enzyme.
PBGS as the prototype morpheein.
The morpheein model of allostery exemplified by PBGS adds an additional layer of understanding to potential mechanisms for regulation of protein function and complements the increased focus that the protein science community is placing on protein structure dynamics.
This model illustrates how the dynamics of phenomena such as alternate protein conformations, alternate oligomeric states, and transient protein-protein interactions can be harnessed for allosteric regulation of catalytic activity.
References.
<templatestyles src="Reflist/styles.css" />
Further reading.
<templatestyles src="Refbegin/styles.css" />
|
[
{
"math_id": 0,
"text": "\\rightleftharpoons"
}
] |
https://en.wikipedia.org/wiki?curid=14167645
|
1416915
|
Tangent cone
|
Generalization of the tangent space to a manifold to the case of certain spaces
In geometry, the tangent cone is a generalization of the notion of the tangent space to a manifold to the case of certain spaces with singularities.
Definitions in nonlinear analysis.
In nonlinear analysis, there are many definitions for a tangent cone, including the adjacent cone, Bouligand's contingent cone, and the Clarke tangent cone. These three cones coincide for a convex set, but they can differ on more general sets.
Clarke tangent cone.
Let formula_0 be a nonempty closed subset of the Banach space formula_1. The Clarke's tangent cone to formula_0 at formula_2, denoted by formula_3 consists of all vectors formula_4, such that for any sequence formula_5 tending to zero, and any sequence formula_6 tending to formula_7, there exists a sequence formula_8 tending to formula_9, such that for all formula_10 holds formula_11
Clarke's tangent cone is always subset of the corresponding contingent cone (and coincides with it, when the set in question is convex). It has the important property of being a closed convex cone.
Definition in convex geometry.
Let "K" be a closed convex subset of a real vector space "V" and ∂"K" be the boundary of "K". The solid tangent cone to "K" at a point "x" ∈ ∂"K" is the closure of the cone formed by all half-lines (or rays) emanating from "x" and intersecting "K" in at least one point "y" distinct from "x". It is a convex cone in "V" and can also be defined as the intersection of the closed half-spaces of "V" containing "K" and bounded by the supporting hyperplanes of "K" at "x". The boundary "T""K" of the solid tangent cone is the tangent cone to "K" and ∂"K" at "x". If this is an affine subspace of "V" then the point "x" is called a smooth point of ∂"K" and ∂"K" is said to be differentiable at "x" and "T""K" is the ordinary tangent space to ∂"K" at "x".
Definition in algebraic geometry.
Let "X" be an affine algebraic variety embedded into the affine space formula_12, with defining ideal formula_13. For any polynomial "f", let formula_14 be the homogeneous component of "f" of the lowest degree, the "initial term" of "f", and let
formula_15
be the homogeneous ideal which is formed by the initial terms formula_14 for all formula_16, the "initial ideal" of "I". The tangent cone to "X" at the origin is the Zariski closed subset of formula_12 defined by the ideal formula_17. By shifting the coordinate system, this definition extends to an arbitrary point of formula_12 in place of the origin. The tangent cone serves as the extension of the notion of the tangent space to "X" at a regular point, where "X" most closely resembles a differentiable manifold, to all of "X". (The tangent cone at a point of formula_12 that is not contained in "X" is empty.)
For example, the nodal curve
formula_18
is singular at the origin, because both partial derivatives of "f"("x", "y") = "y"2 − "x"3 − "x"2 vanish at (0, 0). Thus the Zariski tangent space to "C" at the origin is the whole plane, and has higher dimension than the curve itself (two versus one). On the other hand, the tangent cone is the union of the tangent lines to the two branches of "C" at the origin,
formula_19
Its defining ideal is the principal ideal of "k"["x"] generated by the initial term of "f", namely "y"2 − "x"2 = 0.
The definition of the tangent cone can be extended to abstract algebraic varieties, and even to general Noetherian schemes. Let "X" be an algebraic variety, "x" a point of "X", and ("O""X","x", "m") be the local ring of "X" at "x". Then the tangent cone to "X" at "x" is the spectrum of the associated graded ring of "O""X","x" with respect to the "m"-adic filtration:
formula_20
If we look at our previous example, then we can see that graded pieces contain the same information. So let
formula_21
then if we expand out the associated graded ring
formula_22
we can see that the polynomial defining our variety
formula_23 in formula_24
|
[
{
"math_id": 0,
"text": "A"
},
{
"math_id": 1,
"text": "X"
},
{
"math_id": 2,
"text": "x_0\\in A"
},
{
"math_id": 3,
"text": "\\widehat{T}_A(x_0)"
},
{
"math_id": 4,
"text": "v\\in X"
},
{
"math_id": 5,
"text": "\\{t_n\\}_{n\\ge 1}\\subset\\mathbb{R}"
},
{
"math_id": 6,
"text": "\\{x_n\\}_{n\\ge 1}\\subset A"
},
{
"math_id": 7,
"text": "x_0"
},
{
"math_id": 8,
"text": "\\{v_n\\}_{n\\ge 1}\\subset X"
},
{
"math_id": 9,
"text": "v"
},
{
"math_id": 10,
"text": "n\\ge 1"
},
{
"math_id": 11,
"text": "x_n+t_nv_n\\in A"
},
{
"math_id": 12,
"text": "k^n"
},
{
"math_id": 13,
"text": "I\\subset k[x_1,\\ldots ,x_n]"
},
{
"math_id": 14,
"text": "\\operatorname{in}(f)"
},
{
"math_id": 15,
"text": "\\operatorname{in}(I)\\subset k[x_1,\\ldots ,x_n]"
},
{
"math_id": 16,
"text": "f \\in I"
},
{
"math_id": 17,
"text": "\\operatorname{in}(I)"
},
{
"math_id": 18,
"text": "C: y^2=x^3+x^2 "
},
{
"math_id": 19,
"text": " x=y,\\quad x=-y. "
},
{
"math_id": 20,
"text": "\\operatorname{gr}_m O_{X,x}=\\bigoplus_{i\\geq 0} m^i / m^{i+1}."
},
{
"math_id": 21,
"text": "\n(\\mathcal{O}_{X,x},\\mathfrak{m}) = \\left(\\left(\\frac{k[x,y]}{(y^2 - x^3 - x^2)}\\right)_{(x,y)}, (x,y)\\right)\n"
},
{
"math_id": 22,
"text": "\n\\begin{align}\n\\operatorname{gr}_m O_{X,x} &= \\frac{\\mathcal{O}_{X,x}}{(x,y)} \\oplus \\frac{(x,y)}{(x,y)^2} \\oplus \\frac{(x,y)^2}{(x,y)^3} \\oplus \\cdots \\\\\n&= k \\oplus \\frac{(x,y)}{(x,y)^2} \\oplus \\frac{(x,y)^2}{(x,y)^3} \\oplus \\cdots\n\\end{align}\n"
},
{
"math_id": 23,
"text": "\ny^2 - x^3 - x^2 \\equiv y^2 - x^2\n"
},
{
"math_id": 24,
"text": "\\frac{(x,y)^2}{(x,y)^3}"
}
] |
https://en.wikipedia.org/wiki?curid=1416915
|
1416975
|
Test statistic
|
Statistic used in statistical hypothesis testing
Test statistic is a quantity derived from the sample for statistical hypothesis testing. A hypothesis test is typically specified in terms of a test statistic, considered as a numerical summary of a data-set that reduces the data to one value that can be used to perform the hypothesis test. In general, a test statistic is selected or defined in such a way as to quantify, within observed data, behaviours that would distinguish the null from the alternative hypothesis, where such an alternative is prescribed, or that would characterize the null hypothesis if there is no explicitly stated alternative hypothesis.
An important property of a test statistic is that its sampling distribution under the null hypothesis must be calculable, either exactly or approximately, which allows "p"-values to be calculated. A "test statistic" shares some of the same qualities of a descriptive statistic, and many statistics can be used as both test statistics and descriptive statistics. However, a test statistic is specifically intended for use in statistical testing, whereas the main quality of a descriptive statistic is that it is easily interpretable. Some informative descriptive statistics, such as the sample range, do not make good test statistics since it is difficult to determine their sampling distribution.
Two widely used test statistics are the t-statistic and the F-statistic.
Example.
Suppose the task is to test whether a coin is fair (i.e. has equal probabilities of producing a head or a tail). If the coin is flipped 100 times and the results are recorded, the raw data can be represented as a sequence of 100 heads and tails. If there is interest in the marginal probability of obtaining a tail, only the number "T" out of the 100 flips that produced a tail needs to be recorded. But "T" can also be used as a test statistic in one of two ways:
Using one of these sampling distributions, it is possible to compute either a one-tailed or two-tailed p-value for the null hypothesis that the coin is fair. The test statistic in this case reduces a set of 100 numbers to a single numerical summary that can be used for testing.
Common test statistics.
One-sample tests are appropriate when a sample is being compared to the population from a hypothesis. The population characteristics are known from theory or are calculated from the population.
Two-sample tests are appropriate for comparing two samples, typically experimental and control samples from a scientifically controlled experiment.
Paired tests are appropriate for comparing two samples where it is impossible to control important variables. Rather than comparing two sets, members are paired between samples so the difference between the members becomes the sample. Typically the mean of the differences is then compared to zero. The common example scenario for when a paired difference test is appropriate is when a single set of test subjects has something applied to them and the test is intended to check for an effect.
Z-tests are appropriate for comparing means under stringent conditions regarding normality and a known standard deviation.
A "t"-test is appropriate for comparing means under relaxed conditions (less is assumed).
Tests of proportions are analogous to tests of means (the 50% proportion).
Chi-squared tests use the same calculations and the same probability distribution for different applications:
F-tests (analysis of variance, ANOVA) are commonly used when deciding whether groupings of data by category are meaningful. If the variance of test scores of the left-handed in a class is much smaller than the variance of the whole class, then it may be useful to study lefties as a group. The null hypothesis is that two variances are the same – so the proposed grouping is not meaningful.
In the table below, the symbols used are defined at the bottom of the table. Many other tests can be found in . Proofs exist that the test statistics are appropriate.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "R^2"
}
] |
https://en.wikipedia.org/wiki?curid=1416975
|
1417038
|
Energy charge
|
Measure of energy in cells
The adenylate energy charge is an index used to measure the energy status of biological cells.
ATP or Mg-ATP is the principal molecule for storing and transferring energy in the cell : it is used for biosynthetic pathways, maintenance of transmembrane gradients, movement, cell division, etc... More than 90% of the ATP is produced by phosphorylation of ADP by the ATP synthase. ATP can also be produced by “substrate level phosphorylation” reactions (ADP phosphorylation by (1,3)-bisphosphoglycerate, phosphoenolpyruvate, phosphocreatine), by the succinate-CoA ligase and phosphoenolpyruvate carboxylkinase, and by adenylate kinase, an enzyme that maintains the three adenine nucleotides in equilibrium (<chem>ATP + AMP <=> 2 ADP</chem>).
The energy charge is related to ATP, ADP and AMP concentrations. It was first defined by Atkinson and Walton who found that it was necessary to take into account the concentration of all three nucleotides, rather than just ATP and ADP, to account for the energy status in metabolism. Since the adenylate kinase maintains two ADP molecules in equilibrium with one ATP (<chem>2 ADP <=> ATP + AMP</chem>), Atkinson defined the adenylate energy charge as:
<templatestyles src="Template:Blockquote/styles.css" />formula_0The energy charge of most cells varies between 0.7 and 0.95 - oscillations in this range are quite frequent. Daniel Atkinson showed that when the energy charge increases from 0.6 to 1.0, the citrate lyase and phosphoribosyl pyrophosphate synthetase, two enzymes controlling anabolic (ATP-demanding) pathways are activated, while the phosphofructokinase and the pyruvate dehydrogenase, two enzymes controlling amphibolic pathways (supplying ATP as well as important biosynthetic intermediates) are inhibited He concluded that control of these pathways has evolved to maintain the energy charge within rather narrow limits - in other words, that the energy charge, like the pH of a cell, must be buffered at all times. We now know that most if not all anabolic and catabolic pathways are indeed controlled, directly and indirectly, by the energy charge. In addition to direct regulation of several enzymes by adenyl nucleotides, an AMP-activated protein kinase known as AMP-K phosphorylates and thereby regulates key enzymes when the energy charge decreases. This results in switching off anabolic pathways while switching on catabolic pathways when AMP increases.
Life depends on an adequate energy charge. If ATP synthesis is momentarily insufficient to maintain an adequate energy charge, AMP can be converted by two different pathways to hypoxanthine and ribose-5P, followed by irreversible oxidation of hypoxanthine to uric acid. This helps to buffer the adenylate energy charge by decreasing the total {ATP+ADP+AMP} concentration.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "\\mbox{Energy charge} = \\frac{[\\mbox{ATP}] + \\frac{1}{2} [\\mbox{ADP}]} {[\\mbox{ATP}] + [\\mbox{ADP}] + [\\mbox{AMP}]}"
}
] |
https://en.wikipedia.org/wiki?curid=1417038
|
1417252
|
Fuchsian model
|
Group representation of a Riemann surface
In mathematics, a Fuchsian model is a representation of a hyperbolic Riemann surface "R" as a quotient of the upper half-plane H by a Fuchsian group. Every hyperbolic Riemann surface admits such a representation. The concept is named after Lazarus Fuchs.
A more precise definition.
By the uniformization theorem, every Riemann surface is either elliptic, parabolic or hyperbolic. More precisely this theorem states that a Riemann surface formula_0 which is not isomorphic to either the Riemann sphere (the elliptic case) or a quotient of the complex plane by a discrete subgroup (the parabolic case) must be a quotient of the hyperbolic plane formula_1 by a subgroup formula_2 acting properly discontinuously and freely.
In the Poincaré half-plane model for the hyperbolic plane the group of biholomorphic transformations is the group formula_3 acting by homographies, and the uniformization theorem means that there exists a discrete, torsion-free subgroup formula_4 such that the Riemann surface formula_5 is isomorphic to formula_0. Such a group is called a Fuchsian group, and the isomorphism formula_6 is called a Fuchsian model for formula_0.
Fuchsian models and Teichmüller space.
Let formula_0 be a closed hyperbolic surface and let formula_2 be a Fuchsian group so that formula_5 is a Fuchsian model for formula_0. Let
formula_7
and endow this set with the topology of pointwise convergence (sometimes called "algebraic convergence"). In this particular case this topology can most easily be defined as follows: the group formula_2 is finitely generated since it is isomorphic to the fundamental group of formula_0. Let formula_8 be a generating set: then any formula_9 is determined by the elements formula_10 and so we can identify formula_11 with a subset of formula_12 by the map formula_13. Then we give it the subspace topology.
The Nielsen isomorphism theorem (this is not standard terminology and this result is not directly related to the Dehn–Nielsen theorem) then has the following statement:
<templatestyles src="Block indent/styles.css"/> "For any formula_14 there exists a self-homeomorphism (in fact a quasiconformal map) formula_15 of the upper half-plane formula_1 such that formula_16 for all formula_17."
The proof is very simple: choose an homeomorphism formula_18 and lift it to the hyperbolic plane. Taking a diffeomorphism yields quasi-conformal map since formula_0 is compact.
This result can be seen as the equivalence between two models for Teichmüller space of formula_0: the set of discrete faithful representations of the fundamental group formula_19 into formula_3 modulo conjugacy and the set of marked Riemann surfaces formula_20 where formula_21 is a quasiconformal homeomorphism modulo a natural equivalence relation.
References.
Matsuzaki, K.; Taniguchi, M.: Hyperbolic manifolds and Kleinian groups. Oxford (1998).
|
[
{
"math_id": 0,
"text": "R"
},
{
"math_id": 1,
"text": "\\mathbb H"
},
{
"math_id": 2,
"text": "\\Gamma"
},
{
"math_id": 3,
"text": "\\mathrm{PSL}_2(\\mathbb R)"
},
{
"math_id": 4,
"text": "\\Gamma \\subset \\mathrm{PSL}_2(\\mathbb R)"
},
{
"math_id": 5,
"text": "\\Gamma \\backslash \\mathbb H"
},
{
"math_id": 6,
"text": "R \\cong \\Gamma \\backslash \\mathbb H"
},
{
"math_id": 7,
"text": "A(\\Gamma) = \\{ \\rho \\colon \\Gamma \\to \\mathrm{PSL}_2(\\Reals)\\colon \\rho \\text{ is faithful and discrete }\\}"
},
{
"math_id": 8,
"text": "g_1, \\ldots, g_r"
},
{
"math_id": 9,
"text": "\\rho \\in A(\\Gamma)"
},
{
"math_id": 10,
"text": "\\rho(g_1), \\ldots, \\rho(g_r)"
},
{
"math_id": 11,
"text": "A(\\Gamma)"
},
{
"math_id": 12,
"text": "\\mathrm{PSL}_2(\\mathbb R)^r"
},
{
"math_id": 13,
"text": "\\rho \\mapsto (\\rho(g_1), \\ldots, \\rho(g_r))"
},
{
"math_id": 14,
"text": "\\rho\\in A(\\Gamma)"
},
{
"math_id": 15,
"text": "h"
},
{
"math_id": 16,
"text": "h \\circ \\gamma \\circ h^{-1} = \\rho(\\gamma)"
},
{
"math_id": 17,
"text": "\\gamma \\in \\Gamma"
},
{
"math_id": 18,
"text": "R \\to \\rho(\\Gamma) \\backslash \\mathbb H"
},
{
"math_id": 19,
"text": "\\pi_1(R)"
},
{
"math_id": 20,
"text": "(X, f)"
},
{
"math_id": 21,
"text": "f\\colon R \\to X"
}
] |
https://en.wikipedia.org/wiki?curid=1417252
|
14173190
|
Multiplier (economics)
|
A concept in economics
In macroeconomics, a multiplier is a factor of proportionality that measures how much an endogenous variable changes in response to a change in some exogenous variable.
For example, suppose variable "x"
changes by "k" units, which causes another variable "y" to change by "M" × "k" units. Then the multiplier is "M".
Common uses.
Two multipliers are commonly discussed in introductory macroeconomics.
Commercial banks create money, especially under the fractional-reserve banking system used throughout the world. In this system, money is created whenever a bank gives out a new loan. This is because the loan, when drawn on and spent, mostly finishes up as a deposit back in the banking system and is counted as part of money supply. After putting aside a part of these deposits as mandated bank reserves, the balance is available for the making of further loans by the bank. This process continues multiple times, and is called the multiplier effect.
The multiplier may vary across countries, and will also vary depending on what measures of money are being considered. For example, consider M2 as a measure of the U.S. money supply, and M0 as a measure of the U.S. monetary base. If a $1 increase in M0 by the Federal Reserve causes M2 to increase by $10, then the money multiplier is 10.
Fiscal multipliers.
Multipliers can be calculated to analyze the effects of fiscal policy, or other exogenous changes in spending, on aggregate output.
For example, if an increase in German government spending by €100, with no change in tax rates, causes German GDP to increase by €150, then the "spending multiplier" is 1.5. Other types of fiscal multipliers can also be calculated, like multipliers that describe the effects of changing taxes (such as lump-sum taxes or proportional taxes).
Keynesian and Hansen–Samuelson multipliers.
Keynesian economists often calculate multipliers that measure the effect on aggregate demand only. (To be precise, the usual "Keynesian multiplier" formulas measure how much the IS curve shifts left or right in response to an exogenous change in spending.)
American Economist Paul Samuelson credited Alvin Hansen for the inspiration behind his seminal 1939 contribution. The original Samuelson multiplier-accelerator model (or, as he belatedly baptised it, the "Hansen-Samuelson" model) relies on a multiplier mechanism that is based on a simple Keynesian consumption function with a Robertsonian lag:
formula_0
formula_1
so present consumption is a function of past income (with c as the marginal propensity to consume). Here, t is the tax rate and m is the ratio of imports to GDP. Investment, in turn, is assumed to be composed of three parts:
formula_2
The first part is autonomous investment, the second is investment induced by interest rates and the final part is investment induced by changes in consumption demand (the "acceleration" principle). It is assumed that b > 0. As we are concentrating on the income-expenditure side, let us assume I(r) = 0 (or alternatively, constant interest), so that:
formula_3
Now, assuming away government and foreign sector, aggregate demand at time t is:
formula_4
assuming goods market equilibrium (so formula_5), then in equilibrium:
formula_6
But we know the values of formula_7 and formula_8 are merely formula_0 and formula_9 respectively, then substituting these in:
formula_10
or, rearranging and rewriting as a second order linear difference equation:
formula_11
The solution to this system then becomes elementary. The equilibrium level of Y (call it formula_12, the particular solution) is easily solved by letting formula_13, or:
formula_14
so:
formula_15
The complementary function, formula_16 is also easy to determine. Namely, we know that it will have the form formula_17 where formula_18 and formula_19 are arbitrary constants to be defined and where formula_20 and formula_21 are the two eigenvalues (characteristic roots) of the following characteristic equation:
formula_22
Thus, the entire solution is written as formula_23
Opponents of Keynesianism have sometimes argued that Keynesian multiplier calculations are misleading; for example, according to the theory of Ricardian equivalence, it is impossible to calculate the effect of deficit-financed government spending on demand without specifying how people expect the deficit to be paid off in the future.
Multiplier formula in an open economy.
The three most known multiplier formula are as depicted, where:
General method.
The general method for calculating short-run multipliers is called comparative statics. That is, comparative statics calculates how much one or more endogenous variables change in the short run, given a change in one or more exogenous variables. The comparative statics method is an application of the implicit function theorem.
Dynamic multipliers can also be calculated. That is, one can ask how a change in some exogenous variable in year "t" affects endogenous variables in year "t", in year "t"+1, in year "t"+2, and so forth. A graph showing the impact on some endogenous variable, over time (that is, the multipliers for times "t", "t"+1, "t"+2, etc.), is called an impulse-response function. The general method for calculating impulse response functions is sometimes called comparative dynamics.
History.
The Tableau économique (Economic Table) of François Quesnay (1758), which laid the foundation of the Physiocrat school of economics is credited as the "first precise formulation" of interdependent systems in economics and the origin of multiplier theory. In the tableau économique, one sees variables in one period (time "t") feeding into variables in the next period (time "t"+1), and a constant rate of flow yields geometric series, which computes a multiplier.
The modern theory of the multiplier was developed in the 1930s, by Kahn, Keynes, Giblin, and others, following earlier work in the 1890s by the Australian economist Alfred De Lissa, the Danish economist Julius Wulff, and the German-American economist N. A. J. L. Johannsen.
|
[
{
"math_id": 0,
"text": "C_{t} = C_{0} + cY_{t-1}"
},
{
"math_id": 1,
"text": "1/(1-c(1-t)+m)"
},
{
"math_id": 2,
"text": "I_{t} = I_{0} + I(r) + b (C_{t} - C_{t-1})"
},
{
"math_id": 3,
"text": "I_{t} = I_{0} + b (C_{t} - C_{t-1})"
},
{
"math_id": 4,
"text": "Ytd = C_{t} + I_{t} = C_{0} + I_{0} + cY_{t-1} + b (C_{t} - C_{t-1})"
},
{
"math_id": 5,
"text": "Y_{t} = Ytd"
},
{
"math_id": 6,
"text": "Y_{t} = C_{0} + I_{0} + cY_{t-1} + b (C_{t} - C_{t-1})"
},
{
"math_id": 7,
"text": "C_{t}"
},
{
"math_id": 8,
"text": "C_{t-1}"
},
{
"math_id": 9,
"text": "C_{t-1} = C_{0} + cY_{t-2}"
},
{
"math_id": 10,
"text": "Y_{t} = C_{0} + I_{0} + cY_{t-1} + b (C_{0} + cY_{t-1} - C_{0} - cY_{t-2})"
},
{
"math_id": 11,
"text": "Y_{t} - (1 + b )cY_{t-1} + b cY_{t-2} = (C_{0} + I_{0})"
},
{
"math_id": 12,
"text": "Y_{p}"
},
{
"math_id": 13,
"text": "Y_{t} = Y_{t-1} = Y_{t-2} = Y_{p}"
},
{
"math_id": 14,
"text": "(1 - c - b c + b c)Y_{p} = (C_{0} + I_{0})"
},
{
"math_id": 15,
"text": "Y_{p} = (C_{0} + I_{0})/(1-c)"
},
{
"math_id": 16,
"text": "Y_{c}"
},
{
"math_id": 17,
"text": "Y_{c} = A_{1}r_{1}t + A_{2}r_{2}t"
},
{
"math_id": 18,
"text": "A_{1}"
},
{
"math_id": 19,
"text": "A_{2}"
},
{
"math_id": 20,
"text": "r_{1}"
},
{
"math_id": 21,
"text": "r_{2}"
},
{
"math_id": 22,
"text": "r^{2} - (1+b )cr + b c = 0"
},
{
"math_id": 23,
"text": "Y = Y_{c} + Y_{p}"
}
] |
https://en.wikipedia.org/wiki?curid=14173190
|
14175633
|
Georg Nöbeling
|
German mathematician
Georg August Nöbeling (12 November 1907 – 16 February 2008) was a German mathematician.
Education and career.
Born and raised in Lüdenscheid, Nöbeling studied mathematics and physics at University of Göttingen between 1927 and 1929 and University of Vienna, where he was a student of Karl Menger and received his PhD in 1931 on a generalization of the embedding theorem, which for one special case can be visualized by the Menger sponge. Nöbeling worked and researched in Menger's Mathematical Colloquium with Kurt Gödel, Franz Alt, Abraham Wald, Olga Taussky-Todd and others.
In 1933, he moved to the University of Erlangen, where he habilitated in 1935 under Otto Haupt and obtained a professorship at the same place in 1940. His work focused on analysis, topology, and geometry. 1968/1969 he solved Specker's theorem on abelian groups.
As Rector (1962–1963) of the University of Erlangen he oversaw the merge with the business college in Nuremberg. He also served twice as the chairman of the German Mathematical Society and is a member of the Bavarian Academy of Sciences and Humanities. He celebrated his 100th birthday in 2007.
|
[
{
"math_id": 0,
"text": "\\mathbb{R}^{2n+1}"
}
] |
https://en.wikipedia.org/wiki?curid=14175633
|
14177213
|
Index of industrial production
|
Economic indicator
The Index of Industrial Production (IIP) is an index for India which details out the growth of various sectors in an economy such as mineral mining, electricity and manufacturing. The all India IIP is a composite indicator that measures the short-term changes in the volume of production of a basket of industrial products during a given period with respect to that in a chosen base period. It is compiled and published monthly by the National Statistics Office (NSO), Ministry of Statistical and Programme Implementation six weeks after the reference month ends.
The level of the Index of Industrial Production (IIP) is an abstract number, the magnitude of which represents the status of production in the industrial sector for a given period of time as compared to a reference period of time. The base year was at one time fixed at 1993–94 so that year was assigned an index level of 100. The current base year is 2011-2012.
The Eight Core Industries comprise nearly 40.27% of the weight of items included in the Index of Industrial Production (IIP). These are Refinery products, Electricity, Steel, Coal, Crude oil, Natural gas, Cement and Fertilisers.(Arranged in descending order of their share).
The beginning.
The first official attempt to compute the IIP was made much earlier than even the recommendations on the subject at the international level. The Office of the Economic Advisor, Ministry of Commerce and Industry made the first attempt of compilation and release of IIP with base year 1937, covering 15 important industries, accounting for more than 90 percent of the total production of the selected industries. The all-India IIP is being released as a monthly series since 1950. With the inception of the Central Statistical Organization in 1951, the responsibility for compilation and publication of IIP was vested with this office.
Successive revisions.
As the structure of the industrial sector changes over time, it became necessary to revise the base year of the IIP periodically to capture the changing composition of industrial production and emergence of new products and services so as to measure the real growth of industrial sector (UNSO recommends quinquennial revision of the base year of IIP). After 1937, the successive revised base years were 1946, 1951, 1956, 1960, 1970, 1980–81 and 1993–94. Initially it was covering 15 industries comprising three broad categories: mining, manufacturing and electricity. The scope of the index was restricted to mining and manufacturing sectors consisting of 20 industries with 35 items, when the base year was shifted to 1946 by Economic Adviser, Ministry of Commerce & Industry and it was called Interim Index of Industrial Production. This index was discontinued in April 1956 due to certain shortcomings and was replaced by the revised index with 1951 as the base year covering 88 items, broadly categorised as mining & quarrying (2), manufacturing (17) and electricity (1) compiled by CSO. The items in this index were classified according to the International Standard Industrial Classification (ISIC) 1948 of all economic activities.
The index was revised in July 1962 to the base year 1956 as per the recommendations of a working group constituted by the CSO for the purpose and it had covered 201 items, classified according to the Standard Industrial and Occupational Classification of All Economic Activities published by the CSO in 1962. The index with 1960 as the base year was based on regular monthly series for 312 items and annual series for 436 items. Hence, though the published index was based on regular monthly series for 312 items, weights had been assigned for 436 items with a view to using the same set of weights for the regular monthly index as well as the annual index covering the additional items. However, the mineral index prepared by the IBM excluded gold, salt, petroleum and natural gas.
The next revised series of index numbers with 1970 as the base year, had taken into account of the structural changes occurred in industrial activity of the country since 1960 and this index was released in March 1975 covering 352 items comprising mining (61), manufacturing (290) and electricity (1). The working group (set up in 1978) under the Chairmanship of the then Director General of CSO, decided to shift the base to 1980–81, to reflect the changes that had taken place in the industrial structure and to accommodate the items from small-scale sector.
A notable feature of the revised 1980 index number series was the inclusion of 18 items from the SSI sector, for which the office of the Development Commissioner of Small-Scale Industries (DCSSI) could ensure regular supply of data. The production data for the small-scale sector were included only from the month of July 1984 onwards; prior to this the production data from the directorate general of technical development (DGTD) for large and medium industries alone had been used. For the period April 1981 to June 1984 in respect of these 18 items, average base year (1980–81) production as obtained from DGTD was used. From July 1984 onwards, combined average base year production both for DGTD and DCSSI products was used. The weights for these items were based on ASI 1980–1981 results and no separate weights for DGTD and DCSSI items were allocated in the 1980–81 series.
The next revision of IIP with 1993–94 as the base year containing 543 items (with the addition of 3 items for mining sector and 188 for the manufacturing sector) has come into existence on 27 May 1998 and ever since, the quick estimates of IIP are being released as per the norms set out for the IMF’s SDDS2, with a time lag of six weeks from the reference month. These quick estimates for a given month are revised twice in the subsequent months. To retain the distinctive character and enable the collection of data, the source agencies proposed clubbing of 478 items of the manufacturing sector into 285 item groups and thus making a total of 287 item groups together with one each of electricity and mining & quarrying. The revised series has followed the National Industrial Classification NIC-1987. Another important feature of the latest series is the inclusion of unorganised manufacturing sector (That is, the same 18 SSI products) along with organised sector for the first time in the weighting diagram.
Recent revision of IIP released by CSO with 2004–05 as the base year comprises 682 items. As per chief statistician T C A Anant, this index shall give a better picture of growth in various sectors of the economy, because it is broader and includes technologically advanced goods such as cell phones and iPods. The previous base year (1993–94) was not usable as the list contained an array of outdated items such as typewriters and tape recorders.
Weighted arithmetic mean of quantity relatives with weights being allotted to various items in proportion to value added by manufacture in the base year by using Laspeyres' formula:
formula_0
where formula_1 is the index, formula_2 is the production relative of the ith item for the month in question and formula_3 is the weight allotted to it.
|
[
{
"math_id": 0,
"text": "I = \\frac{\\sum (W_{i}R_{i})}{\\sum W_{i}}"
},
{
"math_id": 1,
"text": "I"
},
{
"math_id": 2,
"text": "R_{i}"
},
{
"math_id": 3,
"text": "W_{i}"
}
] |
https://en.wikipedia.org/wiki?curid=14177213
|
14179284
|
METAP2
|
Protein-coding gene in humans
Methionine aminopeptidase 2 is an enzyme that in humans is encoded by the "METAP2" gene.
Methionine aminopeptidase 2, a member of the dimetallohydrolase family, is a cytosolic metalloenzyme that catalyzes the hydrolytic removal of N-terminal methionine residues from nascent proteins.
MetAP2 is found in all organisms and is especially important because of its critical role in tissue repair and protein degradation. Furthermore, MetAP2 is of particular interest because the enzyme plays a key role in angiogenesis, the growth of new blood vessels, which is necessary for the progression of diseases including solid tumor cancers and rheumatoid arthritis. MetAP2 is also the target of two groups of anti-angiogenic natural products, ovalicin and fumagillin, and their analogs such as beloranib.
Structure.
In living organisms, the start codon that initiates protein synthesis codes for either methionine (eukaryotes) or formylmethionine (prokaryotes). In E. coli (prokaryote), an enzyme called formylmethionine deformylase can cleave the formyl group, leaving just the N-terminal methionine residue. For proteins with small, uncharged penultimate N-terminal residues, a methionine aminopeptidase can cleave the methionine residue.
The number of genes encoding for a methionine aminopeptidase varies between organisms. In E. coli, there is only one known MetAP, a 29,333 Da monomeric enzyme coded for by a gene consisting of 264 codons. The knockout of this gene in E. coli leads to cell inviability. In humans, there are two genes encoding MetAP, MetAP1 and MetAP2. MetAP1 codes for a 42 kDa enzyme, while MetAP2 codes for a 67 kDa enzyme. Yeast MetAP1 is 40 percent homologous to E. coli MetAP; within S. cerevisiae, MetAP2 is 22 percent homologous with the sequence of MetAP1; MetAP2 is highly conserved between "S. cerevisiae" and humans. In contrast to prokaryotes, eukaryotic S. cerevisiae strains lacking the gene for either MetAP1 or MetAP2 are viable, but exhibit a slower growth rate than a control strain expressing both genes.
Active site.
The active site of MetAP2 has a structural motif characteristic of many metalloenzymes—including the dioxygen carrier protein, hemerythrin; the dinuclear non-heme iron protein, ribonucleotide reductase; leucine aminopeptidase; urease; arginase; several phosphatases and phosphoesterases—that includes two bridging carboxylate ligands and a bridging water or hydroxide ligand. Specifically in human MetAP2 (PDB: 1BOA), one of the catalytic metal ions is bound to His331, Glu364, Glu459, Asp263, and a bridging water or hydroxide, while the other metal ion is bound to Asp251 (bidentate), App262 (bidentate), Glu459, and the same bridging water or hydroxide. Here, the two bridging carboxylates are Asp262 and Glu459.
Dimetal center.
The identity of the active site metal ions under physiological conditions has not been successfully established, and remains a controversial issue. MetAP2 shows activity in the presence of Zn(II), Co(II), Mn(II), and Fe(II) ions, and various authors have argued any given metal ion is the physiological one: some in the presence of iron, others in cobalt, others in manganese, and yet others in the presence of zinc. Nonetheless, the majority of crystallographers have crystallized MetAP2 either in the presence of Zn(II) or Co(II) (see PDB database).
Mechanism.
The bridging water or hydroxide ligand acts as a nucleophile during the hydrolysis reaction, but the exact mechanism of catalysis is not yet known. The catalytic mechanisms of hydrolase enzymes depend greatly on the identity of the bridging ligand, which can be challenging to determine due to the difficulty of studying hydrogen atoms via x-ray crystallography.
The histidine residues shown in the mechanism to the right, H178 and H79, are conserved in all MetAPs (MetAP1s and MetAP2s) sequenced to date, suggesting their presence is important to catalytic activity. Based upon X-ray crystallographic data, histidine 79 (H79) has been proposed to help position the methionine residue in the active site and transfer a proton to the newly exposed N-terminal amine. Lowther and Colleagues have proposed two possible mechanisms for MetAP2 in E. coli, shown at the right.
Function.
While previous studies have indicated MetAP2 catalyzes the removal of N-terminal methionine residues in vitro, the function of this enzyme in vivo may be more complex. For example, a significant correlation exists between the inhibition of the enzymatic activity of MetAP2 and inhibition of cell growth, thus implicating the enzyme in endothelial cell proliferation. For this reason, cancer researchers have singled out MetAP2 as a potential target for the inhibition of angiogenesis. Moreover, studies have demonstrated that MetAP2 copurifies and interacts with the α subunit of eukaryotic initiation factor 2 (eIF2), a protein that is necessary for protein synthesis in vivo. Specifically, MetAP2 protects eIF-2α from inhibitory phosphorylation from the enzyme eIF-2α kinase, inhibits RNA-dependent protein kinase (PKR)-catalyzed eIF-2 R-subunit phosphorylation, and also reverses PKR-mediated inhibition of protein synthesis in intact cells.
Clinical significance.
Numerous studies implicate MetAP2 in angiogenesis. Specifically, the covalent binding of either the ovalicin or fumagillin epoxide moiety to the active site histidine residue of MetAP2 has been shown to inactivate the enzyme, thereby inhibiting angiogenesis. The way in which MetAP2 regulates angiogenesis has yet to be established, however, such that further study is required to validate that antiangiogenic activity results directly from MetAP2 inhibition. Nevertheless, with both the growth and metastasis of solid tumors depending heavily on angiogenesis, fumagillin and its analogs—including evexomostat, TNP-470, caplostatin, and beloranib—as well as ovalicin represent potential anticancer agents.
Moreover, the ability of MetAP2 to decrease cell viability in prokaryotic and small eukaryotic organisms has made it a target for antibacterial agents. Thus far, both fumagillin and TNP-470 have been shown to possess antimalarial activity both in vitro and in vivo, and fumarranol, another fumagillin analog, represents a promising lead.
The fumagillin-derived METAP2 inhibitor beloranib (ZGN-433, CDK-732) has shown efficacy in reducing weight in severely obese subjects. MetAP2 inhibitors work by re-establishing insulin sensitivity and balance to the ways the body metabolizes fat, leading to substantial loss of body weight. Development of beloranib was halted in 2016 after two deaths during clinical trials for patients with Praeder-Willi Syndrome.
Evexomostat (SDX-7320) a polymer–drug conjugate of SDX-7539, a MetAP2 inhibitor, is undergoing phase 2 clinical studies
Interactions.
METAP2 has been shown to interact with Protein kinase R.
References.
<templatestyles src="Reflist/styles.css" />
Further reading.
<templatestyles src="Refbegin/styles.css" />
|
[
{
"math_id": 0,
"text": "\\rightleftharpoons"
}
] |
https://en.wikipedia.org/wiki?curid=14179284
|
14179835
|
Activation function
|
Artificial neural network node function
<templatestyles src="Machine learning/styles.css"/>
The activation function of a node in an artificial neural network is a function that calculates the output of the node based on its individual inputs and their weights. Nontrivial problems can be solved using only a few nodes if the activation function is "nonlinear". Modern activation functions include the smooth version of the ReLU, the GELU, which was used in the 2018 BERT model, the logistic (sigmoid) function used in the 2012 speech recognition model developed by Hinton et al, the ReLU used in the 2012 AlexNet computer vision model and in the 2015 ResNet model.
Comparison of activation functions.
Aside from their empirical performance, activation functions also have different mathematical properties:
These properties do not decisively influence performance, nor are they the only mathematical properties that may be useful. For instance, the strictly positive range of the softplus makes it suitable for predicting variances in variational autoencoders.
Mathematical details.
The most common activation functions can be divided into three categories: ridge functions, radial functions and fold functions.
An activation function formula_0 is saturating if formula_1. It is nonsaturating if it is formula_2. Non-saturating activation functions, such as ReLU, may be better than saturating activation functions, because they are less likely to suffer from the vanishing gradient problem.
Ridge activation functions.
Ridge functions are multivariate functions acting on a linear combination of the input variables. Often used examples include:
In biologically inspired neural networks, the activation function is usually an abstraction representing the rate of action potential firing in the cell. In its simplest form, this function is binary—that is, either the neuron is firing or not. Neurons also cannot fire faster than a certain rate, motivating sigmoid activation functions whose range is a finite interval.
The function looks like formula_7, where formula_8 is the Heaviside step function.
If a line has a positive slope, on the other hand, it may reflect the increase in firing rate that occurs as input current increases. Such a function would be of the form formula_9.
Radial activation functions.
A special class of activation functions known as radial basis functions (RBFs) are used in RBF networks, which are extremely efficient as universal function approximators. These activation functions can take many forms, but they are usually found as one of the following functions:
where formula_13 is the vector representing the function "center" and formula_14 and formula_15 are parameters affecting the spread of the radius.
Folding activation functions.
Folding activation functions are extensively used in the pooling layers in convolutional neural networks, and in output layers of multiclass classification networks. These activations perform aggregation over the inputs, such as taking the mean, minimum or maximum. In multiclass classification the softmax activation is often used.
Table of activation functions.
The following table compares the properties of several activation functions that are functions of one fold x from the previous layer or layers:
The following table lists activation functions that are not functions of a single fold x from the previous layer or layers:
<templatestyles src="Citation/styles.css"/>^ Here, formula_16 is the Kronecker delta.
<templatestyles src="Citation/styles.css"/>^ For instance, formula_17 could be iterating through the number of kernels of the previous neural network layer while formula_18 iterates through the number of kernels of the current layer.
Quantum activation functions.
In quantum neural networks programmed on gate-model quantum computers, based on quantum perceptrons instead of variational quantum circuits, the non-linearity of the activation function can be implemented with no need of measuring the output of each perceptron at each layer. The quantum properties loaded within the circuit such as superposition can be preserved by creating the Taylor series of the argument computed by the perceptron itself, with suitable quantum circuits computing the powers up to a wanted approximation degree. Because of the flexibility of such quantum circuits, they can be designed in order to approximate any arbitrary classical activation function.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "f"
},
{
"math_id": 1,
"text": "\\lim_{|v|\\to \\infty} |\\nabla f(v)| = 0"
},
{
"math_id": 2,
"text": "\\lim_{|v|\\to \\infty} |\\nabla f(v)| \\neq 0"
},
{
"math_id": 3,
"text": "\\phi (\\mathbf v)=a +\\mathbf v'\\mathbf b"
},
{
"math_id": 4,
"text": "\\phi (\\mathbf v)=\\max(0,a +\\mathbf v'\\mathbf b)"
},
{
"math_id": 5,
"text": "\\phi (\\mathbf v)=1_{a +\\mathbf v'\\mathbf b>0}"
},
{
"math_id": 6,
"text": "\\phi(\\mathbf v) = (1+\\exp(-a-\\mathbf v'\\mathbf b))^{-1}"
},
{
"math_id": 7,
"text": "\\phi(\\mathbf v)=U(a + \\mathbf v'\\mathbf b)"
},
{
"math_id": 8,
"text": "U"
},
{
"math_id": 9,
"text": "\\phi(\\mathbf v)=a+\\mathbf v'\\mathbf b"
},
{
"math_id": 10,
"text": "\\,\\phi(\\mathbf v)=\\exp\\left(-\\frac{\\|\\mathbf v - \\mathbf c\\|^2}{2\\sigma^2}\\right)"
},
{
"math_id": 11,
"text": "\\,\\phi(\\mathbf v) = \\sqrt{\\|\\mathbf v - \\mathbf c\\|^2 + a^2}"
},
{
"math_id": 12,
"text": "\\,\\phi(\\mathbf v) = \\left(\\|\\mathbf v-\\mathbf c\\|^2 + a^2\\right)^{-\\frac{1}{2}}"
},
{
"math_id": 13,
"text": "\\mathbf c"
},
{
"math_id": 14,
"text": "a"
},
{
"math_id": 15,
"text": "\\sigma"
},
{
"math_id": 16,
"text": "\\delta_{ij}"
},
{
"math_id": 17,
"text": "j"
},
{
"math_id": 18,
"text": "i"
}
] |
https://en.wikipedia.org/wiki?curid=14179835
|
1418
|
Absolute zero
|
Lowest theoretical temperature
Absolute zero is the lowest limit of the thermodynamic temperature scale; a state at which the enthalpy and entropy of a cooled ideal gas reach their minimum value. The fundamental particles of nature have minimum vibrational motion, retaining only quantum mechanical, zero-point energy-induced particle motion. The theoretical temperature is determined by extrapolating the ideal gas law; by international agreement, absolute zero is taken as 0 kelvin (International System of Units), which is −273.15 degrees on the Celsius scale, and equals −459.67 degrees on the Fahrenheit scale (United States customary units or imperial units). The Kelvin and Rankine temperature scales set their zero points at absolute zero by definition.
It is commonly thought of as the lowest temperature possible, but it is not the lowest "enthalpy" state possible, because all real substances begin to depart from the ideal gas when cooled as they approach the change of state to liquid, and then to solid; and the sum of the enthalpy of vaporization (gas to liquid) and enthalpy of fusion (liquid to solid) exceeds the ideal gas's change in enthalpy to absolute zero. In the quantum-mechanical description, matter at absolute zero is in its ground state, the point of lowest internal energy.
The laws of thermodynamics indicate that absolute zero cannot be reached using only thermodynamic means, because the temperature of the substance being cooled approaches the temperature of the cooling agent asymptotically. Even a system at absolute zero, if it could somehow be achieved, would still possess quantum mechanical zero-point energy, the energy of its ground state at absolute zero; the kinetic energy of the ground state cannot be removed.
Scientists and technologists routinely achieve temperatures close to absolute zero, where matter exhibits quantum effects such as
superconductivity, superfluidity, and Bose–Einstein condensation.
Thermodynamics near absolute zero.
At temperatures near , nearly all molecular motion ceases and Δ"S" = 0 for any adiabatic process, where "S" is the entropy. In such a circumstance, pure substances can (ideally) form perfect crystals with no structural imperfections as "T" → 0. Max Planck's strong form of the third law of thermodynamics states the entropy of a perfect crystal vanishes at absolute zero. The original Nernst "heat theorem" makes the weaker and less controversial claim that the entropy change for any isothermal process approaches zero as "T" → 0:
formula_0
The implication is that the entropy of a perfect crystal approaches a constant value. An adiabat is a state with constant entropy, typically represented on a graph as a curve in a manner similar to isotherms and isobars.
The Nernst postulate identifies the isotherm T = 0 as coincident with the adiabat S = 0, although other isotherms and adiabats are distinct. As no two adiabats intersect, no other adiabat can intersect the T = 0 isotherm. Consequently no adiabatic process initiated at nonzero temperature can lead to zero temperature. (≈ Callen, pp. 189–190)
A perfect crystal is one in which the internal lattice structure extends uninterrupted in all directions. The perfect order can be represented by translational symmetry along three (not usually orthogonal) axes. Every lattice element of the structure is in its proper place, whether it is a single atom or a molecular grouping. For substances that exist in two (or more) stable crystalline forms, such as diamond and graphite for carbon, there is a kind of "chemical degeneracy". The question remains whether both can have zero entropy at "T" = 0 even though each is perfectly ordered.
Perfect crystals never occur in practice; imperfections, and even entire amorphous material inclusions, can and do get "frozen in" at low temperatures, so transitions to more stable states do not occur.
Using the Debye model, the specific heat and entropy of a pure crystal are proportional to "T" 3, while the enthalpy and chemical potential are proportional to "T" 4. (Guggenheim, p. 111) These quantities drop toward their "T" = 0 limiting values and approach with "zero" slopes. For the specific heats at least, the limiting value itself is definitely zero, as borne out by experiments to below 10 K. Even the less detailed Einstein model shows this curious drop in specific heats. In fact, all specific heats vanish at absolute zero, not just those of crystals. Likewise for the coefficient of thermal expansion. Maxwell's relations show that various other quantities also vanish. These phenomena were unanticipated.
Since the relation between changes in Gibbs free energy ("G"), the enthalpy ("H") and the entropy is
formula_1
thus, as "T" decreases, Δ"G" and Δ"H" approach each other (so long as Δ"S" is bounded). Experimentally, it is found that all spontaneous processes (including chemical reactions) result in a decrease in "G" as they proceed toward equilibrium. If Δ"S" and/or "T" are small, the condition Δ"G" < 0 may imply that Δ"H" < 0, which would indicate an exothermic reaction. However, this is not required; endothermic reactions can proceed spontaneously if the "T"Δ"S" term is large enough.
Moreover, the slopes of the derivatives of Δ"G" and Δ"H" converge and are equal to zero at "T" = 0. This ensures that Δ"G" and Δ"H" are nearly the same over a considerable range of temperatures and justifies the approximate empirical Principle of Thomsen and Berthelot, which states that "the equilibrium state to which a system proceeds is the one that evolves the greatest amount of heat", i.e., an actual process is the "most exothermic one". (Callen, pp. 186–187)
One model that estimates the properties of an electron gas at absolute zero in metals is the Fermi gas. The electrons, being fermions, must be in different quantum states, which leads the electrons to get very high typical velocities, even at absolute zero. The maximum energy that electrons can have at absolute zero is called the Fermi energy. The Fermi temperature is defined as this maximum energy divided by the Boltzmann constant, and is on the order of 80,000 K for typical electron densities found in metals. For temperatures significantly below the Fermi temperature, the electrons behave in almost the same way as at absolute zero. This explains the failure of the classical equipartition theorem for metals that eluded classical physicists in the late 19th century.
Relation with Bose–Einstein condensate.
A Bose–Einstein condensate (BEC) is a state of matter of a dilute gas of weakly interacting bosons confined in an external potential and cooled to temperatures very near absolute zero. Under such conditions, a large fraction of the bosons occupy the lowest quantum state of the external potential, at which point quantum effects become apparent on a macroscopic scale.
This state of matter was first predicted by Satyendra Nath Bose and Albert Einstein in 1924–25. Bose first sent a paper to Einstein on the quantum statistics of light quanta (now called photons). Einstein was impressed, translated the paper from English to German and submitted it for Bose to the "Zeitschrift für Physik", which published it. Einstein then extended Bose's ideas to material particles (or matter) in two other papers.
Seventy years later, in 1995, the first gaseous condensate was produced by Eric Cornell and Carl Wieman at the University of Colorado at Boulder NIST-JILA lab, using a gas of rubidium atoms cooled to 170 nanokelvin (nK) ().
In 2003, researchers at the Massachusetts Institute of Technology (MIT) achieved a temperature of 450 ± 80 picokelvin (pK) () in a BEC of sodium atoms. The associated black-body (peak emittance) wavelength of 6.4 megameters is roughly the radius of Earth.
In 2021, University of Bremen physicists achieved a BEC with a temperature of only 38 pK, the current coldest temperature record.
Absolute temperature scales.
Absolute, or thermodynamic, temperature is conventionally measured in kelvin (Celsius-scaled increments) and in the Rankine scale (Fahrenheit-scaled increments) with increasing rarity. Absolute temperature measurement is uniquely determined by a multiplicative constant which specifies the size of the "degree", so the "ratios" of two absolute temperatures, "T"2/"T"1, are the same in all scales. The most transparent definition of this standard comes from the Maxwell–Boltzmann distribution. It can also be found in Fermi–Dirac statistics (for particles of half-integer spin) and Bose–Einstein statistics (for particles of integer spin). All of these define the relative numbers of particles in a system as decreasing exponential functions of energy (at the particle level) over "kT", with "k" representing the Boltzmann constant and "T" representing the temperature observed at the macroscopic level.
Negative temperatures.
Temperatures that are expressed as negative numbers on the familiar Celsius or Fahrenheit scales are simply colder than the zero points of those scales. Certain systems can achieve truly negative temperatures; that is, their thermodynamic temperature (expressed in kelvins) can be of a negative quantity. A system with a truly negative temperature is not colder than absolute zero. Rather, a system with a negative temperature is hotter than "any" system with a positive temperature, in the sense that if a negative-temperature system and a positive-temperature system come in contact, heat flows from the negative to the positive-temperature system.
Most familiar systems cannot achieve negative temperatures because adding energy always increases their entropy. However, some systems have a maximum amount of energy that they can hold, and as they approach that maximum energy their entropy actually begins to decrease. Because temperature is defined by the relationship between energy and entropy, such a system's temperature becomes negative, even though energy is being added. As a result, the Boltzmann factor for states of systems at negative temperature increases rather than decreases with increasing state energy. Therefore, no complete system, i.e. including the electromagnetic modes, can have negative temperatures, since there is no highest energy state, so that the sum of the probabilities of the states would diverge for negative temperatures. However, for quasi-equilibrium systems (e.g. spins out of equilibrium with the electromagnetic field) this argument does not apply, and negative effective temperatures are attainable.
On 3 January 2013, physicists announced that for the first time they had created a quantum gas made up of potassium atoms with a negative temperature in motional degrees of freedom.
History.
One of the first to discuss the possibility of an absolute minimal temperature was Robert Boyle. His 1665 "New Experiments and Observations touching Cold", articulated the dispute known as the "primum frigidum". The concept was well known among naturalists of the time. Some contended an absolute minimum temperature occurred within earth (as one of the four classical elements), others within water, others air, and some more recently within nitre. But all of them seemed to agree that, "There is some body or other that is of its own nature supremely cold and by participation of which all other bodies obtain that quality."
Limit to the "degree of cold".
The question of whether there is a limit to the degree of coldness possible, and, if so, where the zero must be placed, was first addressed by the French physicist Guillaume Amontons in 1703, in connection with his improvements in the air thermometer. His instrument indicated temperatures by the height at which a certain mass of air sustained a column of mercury—the pressure, or "spring" of the air varying with temperature. Amontons therefore argued that the zero of his thermometer would be that temperature at which the spring of the air was reduced to nothing. He used a scale that marked the boiling point of water at +73 and the melting point of ice at +<templatestyles src="Fraction/styles.css" />51+1⁄2, so that the zero was equivalent to about −240 on the Celsius scale. Amontons held that the absolute zero cannot be reached, so never attempted to compute it explicitly.
The value of −240 °C, or "431 divisions [in Fahrenheit's thermometer] below the cold of freezing water" was published by George Martine in 1740.
This close approximation to the modern value of −273.15 °C for the zero of the air thermometer was further improved upon in 1779 by Johann Heinrich Lambert, who observed that might be regarded as absolute cold.
Values of this order for the absolute zero were not, however, universally accepted about this period. Pierre-Simon Laplace and Antoine Lavoisier, in their 1780 treatise on heat, arrived at values ranging from 1,500 to 3,000 below the freezing point of water, and thought that in any case it must be at least 600 below. John Dalton in his "Chemical Philosophy" gave ten calculations of this value, and finally adopted −3,000 °C as the natural zero of temperature.
Charles's law.
From 1787 to 1802, it was determined by Jacques Charles (unpublished), John Dalton, and Joseph Louis Gay-Lussac that, at constant pressure, ideal gases expanded or contracted their volume linearly (Charles's law) by about 1/273 parts per degree Celsius of temperature's change up or down, between 0° and 100° C. This suggested that the volume of a gas cooled at about −273 °C would reach zero.
Lord Kelvin's work.
After James Prescott Joule had determined the mechanical equivalent of heat, Lord Kelvin approached the question from an entirely different point of view, and in 1848 devised a scale of absolute temperature that was independent of the properties of any particular substance and was based on Carnot's theory of the Motive Power of Heat and data published by Henri Victor Regnault. It followed from the principles on which this scale was constructed that its zero was placed at −273 °C, at almost precisely the same point as the zero of the air thermometer, where the air volume would reach "nothing". This value was not immediately accepted; values ranging from to , derived from laboratory measurements and observations of astronomical refraction, remained in use in the early 20th century.
The race to absolute zero.
With a better theoretical understanding of absolute zero, scientists were eager to reach this temperature in the lab. By 1845, Michael Faraday had managed to liquefy most gases then known to exist, and reached a new record for lowest temperatures by reaching . Faraday believed that certain gases, such as oxygen, nitrogen, and hydrogen, were permanent gases and could not be liquefied. Decades later, in 1873 Dutch theoretical scientist Johannes Diderik van der Waals demonstrated that these gases could be liquefied, but only under conditions of very high pressure and very low temperatures. In 1877, Louis Paul Cailletet in France and Raoul Pictet in Switzerland succeeded in producing the first droplets of liquid air at . This was followed in 1883 by the production of liquid oxygen by the Polish professors Zygmunt Wróblewski and Karol Olszewski.
Scottish chemist and physicist James Dewar and Dutch physicist Heike Kamerlingh Onnes took on the challenge to liquefy the remaining gases, hydrogen and helium. In 1898, after 20 years of effort, Dewar was the first to liquefy hydrogen, reaching a new low-temperature record of . However, Kamerlingh Onnes, his rival, was the first to liquefy helium, in 1908, using several precooling stages and the Hampson–Linde cycle. He lowered the temperature to the boiling point of helium . By reducing the pressure of the liquid helium, he achieved an even lower temperature, near 1.5 K. These were the coldest temperatures achieved on Earth at the time and his achievement earned him the Nobel Prize in 1913. Kamerlingh Onnes would continue to study the properties of materials at temperatures near absolute zero, describing superconductivity and superfluids for the first time.
Very low temperatures.
The average temperature of the universe today is approximately , or about −270.42 °C, based on measurements of cosmic microwave background radiation. Standard models of the future expansion of the universe predict that the average temperature of the universe is decreasing over time. This temperature is calculated as the mean density of energy in space; it should not be confused with the mean electron temperature (total energy divided by particle count) which has increased over time.
Absolute zero cannot be achieved, although it is possible to reach temperatures close to it through the use of evaporative cooling, cryocoolers, dilution refrigerators, and nuclear adiabatic demagnetization. The use of laser cooling has produced temperatures of less than a billionth of a kelvin. At very low temperatures in the vicinity of absolute zero, matter exhibits many unusual properties, including superconductivity, superfluidity, and Bose–Einstein condensation. To study such phenomena, scientists have worked to obtain even lower temperatures.
See also.
<templatestyles src="Div col/styles.css"/>
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": " \\lim_{T \\to 0} \\Delta S = 0 "
},
{
"math_id": 1,
"text": " \\Delta G = \\Delta H - T \\Delta S \\,"
}
] |
https://en.wikipedia.org/wiki?curid=1418
|
1418203
|
Output elasticity
|
Economic term
In economics, output elasticity is the percentage change of output (GDP or production of a single firm) divided by the percentage change of an input. It is sometimes called "partial output elasticity" to clarify that it refers to the change of only one input.
As with every elasticity, this measure is defined locally, i.e. defined at a point.
If the production function contains only one input, then the output elasticity is also an indicator of the degree of returns to scale. If the coefficient of output elasticity is greater than 1, then production is experiencing increasing returns to scale. If the coefficient is less than 1, then production is experiencing decreasing returns to scale. If the coefficient is 1, then production is experiencing constant returns to scale. Note that returns to scale may change as the level of production changes.
A different usage of the term "output elasticity" is defined as the percentage change in output per one percent change in "all" the inputs. The coefficient of output elasticity can be used to estimate returns to scale.
The mathematical formula is formula_0 where x represents the inputs and Q, the output. Multi-input-multi-output generalisations also exist in the literature.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": " E_Q = \\frac{\\partial Q}{\\partial \\textbf{x}} \\cdot \\frac{\\textbf{x}}{Q} "
}
] |
https://en.wikipedia.org/wiki?curid=1418203
|
14182105
|
Power-added efficiency
|
Power-added efficiency (PAE) is a metric for rating the efficiency of a power amplifier that takes into account the effect of the gain of the amplifier. It is calculated (in percent) as:
formula_0
It differs from most power efficiency descriptions calculated (in percent) as:
formula_1
PAE will be very similar to efficiency when the gain of the amplifier is sufficiently high. But if the amplifier gain is relatively low the amount of power that is needed to drive the input of the amplifier should be considered in a metric that measures the efficiency of said amplifier.
|
[
{
"math_id": 0,
"text": "\n\\mathrm{PAE} = 100 \\times \\frac{P_{OUT}^{RF} - P_{IN}^{RF}}{P_{DC}^{TOTAL}}\n"
},
{
"math_id": 1,
"text": "\n\\eta = 100 \\times {P_{OUT}^{RF}} / {P_{IN}^{DC}}\n"
}
] |
https://en.wikipedia.org/wiki?curid=14182105
|
14182874
|
Schuette–Nesbitt formula
|
In mathematics, the Schuette–Nesbitt formula is a generalization of the inclusion–exclusion principle. It is named after Donald R. Schuette and Cecil J. Nesbitt.
The probabilistic version of the Schuette–Nesbitt formula has practical applications in actuarial science, where it is used to calculate the net single premium for life annuities and life insurances based on the general symmetric status.
Combinatorial versions.
Consider a set Ω and subsets "A"1, ..., "Am". Let
denote the number of subsets to which "ω" ∈ Ω belongs, where we use the indicator functions of the sets "A"1, ..., "Am". Furthermore, for each "k" ∈ {0, 1, ..., "m"}, let
denote the number of intersections of exactly k sets out of "A"1, ..., "Am", to which ω belongs, where the intersection over the empty index set is defined as Ω, hence "N"0
1Ω. Let V denote a vector space over a field R such as the real or complex numbers (or more generally a module over a ring R with multiplicative identity). Then, for every choice of "c"0, ..., "cm" ∈ "V",
where 1{"N"
"n"} denotes the indicator function of the set of all "ω" ∈ Ω with "N"("ω")
"n", and formula_0 is a binomial coefficient. Equality (3) says that the two V-valued functions defined on Ω are the same.
Proof of (3).
We prove that (3) holds pointwise. Take "ω" ∈ Ω and define "n"
"N"("ω").
Then the left-hand side of (3) equals "cn".
Let I denote the set of all those indices "i" ∈ {1, ..., "m"} such that "ω" ∈ "Ai", hence I contains exactly n indices.
Given "J" ⊂ {1, ..., "m"} with k elements, then ω belongs to the intersection ∩"j"∈"J""Aj" if and only if J is a subset of I.
By the combinatorial interpretation of the binomial coefficient, there are "Nk"
formula_1 such subsets (the binomial coefficient is zero for "k" > "n").
Therefore the right-hand side of (3) evaluated at ω equals
formula_2
where we used that the first binomial coefficient is zero for "k" > "n".
Note that the sum (*) is empty and therefore defined as zero for "n" < "l".
Using the factorial formula for the binomial coefficients, it follows that
formula_3
Rewriting (**) with the summation index "j"
"k" − "l" und using the binomial formula for the third equality shows that
formula_4
which is the Kronecker delta. Substituting this result into the above formula and noting that n choose l equals 1 for "l"
"n", it follows that the right-hand side of (3) evaluated at ω also reduces to "cn".
Representation in the polynomial ring.
As a special case, take for V the polynomial ring "R"["x"] with the indeterminate x. Then (3) can be rewritten in a more compact way as
This is an identity for two polynomials whose coefficients depend on ω, which is implicit in the notation.
Proof of (4) using (3): Substituting "cn"
"xn" for "n" ∈ {0, ..., "m"} into (3) and using the binomial formula shows that
formula_5
which proves (4).
Representation with shift and difference operators.
Consider the linear shift operator E and the linear difference operator Δ, which we define here on the sequence space of V by
formula_6
and
formula_7
Substituting "x"
"E" in (4) shows that
where we used that Δ
"E" – "I" with I denoting the identity operator. Note that "E"0 and Δ0 equal the identity operator I on the sequence space, "Ek" and Δ"k" denote the k-fold composition.
<templatestyles src="Math_proof/styles.css" />Direct proof of (5) by the operator method
To prove (5), we first want to verify the equation
involving indicator functions of the sets "A"1, ..., "Am" and their complements with respect to Ω. Suppose an ω from Ω belongs to exactly k sets out of "A"1, ..., "Am", where "k" ∈ {0, ..., "m"}, for simplicity of notation say that ω only belongs to "A"1, ..., "Ak". Then the left-hand side of (✳) is "Ek". On the right-hand side of (✳), the first k factors equal E, the remaining ones equal I, their product is also "Ek", hence the formula (✳) is true.
Note that
formula_8
Inserting this result into equation (✳) and expanding the product gives
formula_9
because the product of indicator functions is the indicator function of the intersection. Using the definition (2), the result (5) follows.
Let (Δ"k""c")0 denote the 0th component of the k-fold composition Δ"k" applied to "c"
("c"0, "c"1, ..., "cm", ...), where Δ0 denotes the identity. Then (3) can be rewritten in a more compact way as
Probabilistic versions.
Consider arbitrary events "A"1, ..., "Am" in a probability space and let E denote the expectation operator. Then N from (1) is the random number of these events which occur simultaneously. Using "Nk" from (2), define
where the intersection over the empty index set is again defined as Ω, hence "S"0
1. If the ring R is also an algebra over the real or complex numbers, then taking the expectation of the coefficients in (4) and using the notation from (7),
in "R"["x"]. If R is the field of real numbers, then this is the probability-generating function of the probability distribution of N.
Similarly, (5) and (6) yield
and, for every sequence "c"
("c"0, "c"1, "c"2, "c"3, ..., "cm", ...),
The quantity on the left-hand side of (6') is the expected value of "c""N".
Remarks.
(0, 1, 1, ...): the left-hand side reduces to the probability of the event {"N" ≥ 1}, which is the union of "A"1, ..., "Am", and the right-hand side is "S"1 – "S"2 + "S"3 – ... – (–1)"m""Sm", because (Δ0"c")0
0 and (Δ"k""c")0
–(–1)"k" for "k" ∈ {1, ..., "m"}.
formula_10
and let I denote the ("m" + 1)-dimensional identity matrix. Then (6) and (6') hold for every vector "c"
("c"0, "c"1, ..., "cm")T in ("m" + 1)-dimensional Euclidean space, where the exponent T in the definition of c denotes the transpose.
For textbook presentations of the probabilistic Schuette–Nesbitt formula (6') and their applications to actuarial science, cf. . Chapter 8, or , Chapter 18 and the Appendix, pp. 577–578.
History.
For independent events, the formula (6') appeared in a discussion of Robert P. White and T.N.E. Greville's paper by Donald R. Schuette and Cecil J. Nesbitt, see . In the two-page note , Hans U. Gerber, called it Schuette–Nesbitt formula and generalized it to arbitrary events. Christian Buchta, see , noticed the combinatorial nature of the formula and published the elementary combinatorial proof of (3).
Cecil J. Nesbitt, PhD, F.S.A., M.A.A.A., received his mathematical education at the University of Toronto and the Institute for Advanced Study in Princeton. He taught actuarial mathematics at the University of Michigan from 1938 to 1980. He served the Society of Actuaries from 1985 to 1987 as Vice-President for Research and Studies. Professor Nesbitt died in 2001. (Short CV taken from , page xv.)
Donald Richard Schuette was a PhD student of C. Nesbitt, he later became professor at the University of Wisconsin–Madison.
The probabilistic version of the Schuette–Nesbitt formula (6') generalizes much older formulae of Waring, which express the probability of the events {"N"
"n"} and {"N" ≥ "n"} in terms of "S"1, "S"2, ..., "Sm". More precisely, with formula_11 denoting the binomial coefficient,
and
see , Sections IV.3 and IV.5, respectively.
To see that these formulae are special cases of the probabilistic version of the Schuette–Nesbitt formula, note that by the binomial theorem
formula_12
Applying this operator identity to the sequence "c"
(0, ..., 0, 1, 0, 0, ...) with n leading zeros and noting that ("E jc")0
1 if "j"
"n" and ("E jc")0
0 otherwise, the formula (8) for {"N"
"n"} follows from (6').
Applying the identity to "c"
(0, ..., 0, 1, 1, 1, ...) with n leading zeros and noting that ("E jc")0
1 if "j" ≥ "n" and ("E jc")0
0 otherwise, equation (6') implies that
formula_13
Expanding (1 – 1)"k" using the binomial theorem and using equation (11) of the formulas involving binomial coefficients, we obtain
formula_14
Hence, we have the formula (9) for {"N" ≥ "n"}.
Applications.
In actuarial science.
Problem: Suppose there are m persons aged "x"1, ..., "xm" with remaining random (but independent) lifetimes "T"1, ..., "Tm". Suppose the group signs a life insurance contract which pays them after t years the amount "cn" if exactly n persons out of m are still alive after t years. How high is the expected payout of this insurance contract in t years?
Solution: Let "Aj" denote the event that person j survives t years, which means that "Aj"
{"Tj" > "t"}. In actuarial notation the probability of this event is denoted by "t pxj" and can be taken from a life table. Use independence to calculate the probability of intersections. Calculate "S"1, ..., "Sm" and use the probabilistic version of the Schuette–Nesbitt formula (6') to calculate the expected value of "cN".
In probability theory.
Let σ be a random permutation of the set {1, ..., "m"} and let "Aj" denote the event that "j" is a fixed point of σ, meaning that "Aj"
{"σ"("j")
"j"}. When the numbers in J, which is a subset of {1, ..., "m"}, are fixed points, then there are ("m" – |"J"|)! ways to permute the remaining "m" – |"J"| numbers, hence
formula_15
By the combinatorical interpretation of the binomial coefficient, there are formula_16 different choices of a subset J of {1, ..., "m"} with k elements, hence (7) simplifies to
formula_17
Therefore, using (4'), the probability-generating function of the number N of fixed points is given by
formula_18
This is the partial sum of the infinite series giving the exponential function at "x" – 1, which in turn is the probability-generating function of the Poisson distribution with parameter 1. Therefore, as m tends to infinity, the distribution of N converges to the Poisson distribution with parameter 1.
|
[
{
"math_id": 0,
"text": "\\textstyle\\binom kl"
},
{
"math_id": 1,
"text": "\\textstyle\\binom nk"
},
{
"math_id": 2,
"text": "\\sum_{k=0}^m \\binom nk\\sum_{l=0}^k (-1)^{k-l}\\binom klc_l\n=\\sum_{l=0}^m\\underbrace{\\sum_{k=l}^n (-1)^{k-l}\\binom nk \\binom kl}_{=:\\,(*)} c_l,"
},
{
"math_id": 3,
"text": "\n\\begin{align}\n(*)\n&=\\sum_{k=l}^n (-1)^{k-l}\\frac{n!}{k!\\,(n-k)!}\\,\\frac{k!}{l!\\,(k-l)!}\\\\\n&=\\underbrace{\\frac{n!}{l!\\,(n-l)!}}_{=\\binom nl}\\underbrace{\\sum_{k=l}^n (-1)^{k-l}\\frac{(n-l)!}{(n-k)!\\,(k-l)!}}_{=:\\,(**)}\\\\\n\\end{align}\n"
},
{
"math_id": 4,
"text": "\n\\begin{align}\n(**)\n&=\\sum_{j=0}^{n-l} (-1)^{j}\\frac{(n-l)!}{(n-l-j)!\\,j!}\\\\\n&=\\sum_{j=0}^{n-l} (-1)^{j}\\binom{n-l}{j}\n=(1-1)^{n-l}\n=\\delta_{ln},\n\\end{align}\n"
},
{
"math_id": 5,
"text": "\n\\sum_{n=0}^m 1_{\\{N=n\\}}x^n\n=\\sum_{k=0}^m N_k\\underbrace{\\sum_{l=0}^k \\binom kl(-1)^{k-l}x^l}_{=\\,(x-1)^k},\n"
},
{
"math_id": 6,
"text": "\\begin{align}\nE:V^{\\mathbb{N}_0}&\\to V^{\\mathbb{N}_0},\\\\\nE(c_0,c_1,c_2,c_3,\\ldots)&\\mapsto(c_1,c_2,c_3,\\ldots),\\\\\n\\end{align}"
},
{
"math_id": 7,
"text": "\\begin{align}\n\\Delta:V^{\\mathbb{N}_0}&\\to V^{\\mathbb{N}_0},\\\\\n\\Delta(c_0,c_1,c_2,c_3\\ldots)&\\mapsto(c_1-c_0,c_2-c_1,c_3-c_2,\\ldots).\\\\\n\\end{align}"
},
{
"math_id": 8,
"text": "\\begin{align}\n1_{A_j^{\\mathrm c}}I+1_{A_j}E\n&=I-1_{A_j}I+1_{A_j}E\\\\\n&=I+1_{A_j}(E-I)=I+1_{A_j}\\Delta,\\qquad j\\in\\{0,\\ldots,m\\}.\n\\end{align}"
},
{
"math_id": 9,
"text": "\\sum_{n=0}^m 1_{\\{N=n\\}}E^n\n=\\sum_{k=0}^m\\sum_{\\scriptstyle J\\subset\\{1,\\ldots,m\\}\\atop\\scriptstyle|J|=k}\n1_{\\cap_{j\\in J}A_j}\\Delta^k,\n"
},
{
"math_id": 10,
"text": "E=\\begin{pmatrix}\n0&1&0&\\cdots&0\\\\\n0&0&1&\\ddots&\\vdots\\\\\n\\vdots&\\ddots&\\ddots&\\ddots&0\\\\\n0&\\cdots&0&0&1\\\\\n0&\\cdots&0&0&0\n\\end{pmatrix},\n\\qquad\n\\Delta=\\begin{pmatrix}\n-1&1&0&\\cdots&0\\\\\n0&-1&1&\\ddots&\\vdots\\\\\n\\vdots&\\ddots&\\ddots&\\ddots&0\\\\\n0&\\cdots&0&-1&1\\\\\n0&\\cdots&0&0&-1\n\\end{pmatrix},\n"
},
{
"math_id": 11,
"text": "\\textstyle\\binom kn"
},
{
"math_id": 12,
"text": "\\Delta^k=(E-I)^k=\\sum_{j=0}^k\\binom kj (-1)^{k-j}E^j,\\qquad k\\in\\mathbb{N}_0."
},
{
"math_id": 13,
"text": "\\mathbb{P}(N\\ge n)=\\sum_{k=n}^m S_k\\sum_{j=n}^k\\binom kj(-1)^{k-j}."
},
{
"math_id": 14,
"text": "\\sum_{j=n}^k\\binom kj(-1)^{k-j}\n=-\\sum_{j=0}^{n-1}\\binom kj(-1)^{k-j}\n=(-1)^{k-n}\\binom{k-1}{n-1}."
},
{
"math_id": 15,
"text": "\\mathbb{P}\\biggl(\\bigcap_{j\\in J}A_j\\biggr)=\\frac{(m-|J|)!}{m!}."
},
{
"math_id": 16,
"text": "\\textstyle\\binom mk"
},
{
"math_id": 17,
"text": "S_k=\\binom mk \\frac{(m-k)!}{m!}=\\frac1{k!}."
},
{
"math_id": 18,
"text": "\\mathbb{E}[x^N]=\\sum_{k=0}^m\\frac{(x-1)^k}{k!},\\qquad x\\in\\mathbb{R}."
}
] |
https://en.wikipedia.org/wiki?curid=14182874
|
14183897
|
Benesi–Hildebrand method
|
Technique in physical chemistry
The Benesi–Hildebrand method is a mathematical approach used in physical chemistry for the determination of the equilibrium constant "K" and stoichiometry of non-bonding interactions. This method has been typically applied to reaction equilibria that form one-to-one complexes, such as charge-transfer complexes and host–guest molecular complexation.
<chem>{H} + G <=> HG</chem>
The theoretical foundation of this method is the assumption that when either one of the reactants is present in excess amounts over the other reactant, the characteristic electronic absorption spectra of the other reactant are transparent in the collective absorption/emission range of the reaction system. Therefore, by measuring the absorption spectra of the reaction before and after the formation of the product and its equilibrium, the association constant of the reaction can be determined.
History.
This method was first developed by Benesi and Hildebrand in 1949, as a means to explain a phenomenon where iodine changes color in various aromatic solvents. This was attributed to the formation of an iodine-solvent complex through acid-base interactions, leading to the observed shifts in the absorption spectrum. Following this development, the Benesi–Hildebrand method has become one of the most common strategies for determining association constants based on absorbance spectra.
Derivation.
To observe one-to-one binding between a single host (H) and guest (G) using UV/Vis absorbance, the Benesi–Hildebrand method can be employed. The basis behind this method is that the acquired absorbance should be a mixture of the host, guest, and the host–guest complex.
formula_0
With the assumption that the initial concentration of the guest (G0) is much larger than the initial concentration of the host (H0), then the absorbance from H0 should be negligible.
formula_1
The absorbance can be collected before and following the formation of the HG complex. This change in absorbance (Δ"A") is what is experimentally acquired, with "A"0 being the initial absorbance before the interaction of HG and A being the absorbance taken at any point of the reaction.
formula_2
Using the Beer–Lambert law, the equation can be rewritten with the absorption coefficients and concentrations of each component.
formula_3
Due to the previous assumption that <chem>[G]_0 \gg [H]_0</chem>, one can expect that [G] = [G]0. Δ"ε" represents the change in value between "ε"HG and "ε"G.
formula_4
A binding isotherm can be described as "the theoretical change in the concentration of one component as a function of the concentration of another component at constant temperature." This can be described by the following equation:
formula_5
By substituting the binding isotherm equation into the previous equation, the equilibrium constant "K"a can now be correlated to the change in absorbance due to the formation of the HG complex.
formula_6
Further modifications results in an equation where a double reciprocal plot can be made with 1/Δ"A" as a function of 1/[G]0. Δ"ε" can be derived from the intercept while "K"a can be calculated from the slope.
formula_7
Limitations and alternatives.
In many cases, the Benesi–Hildebrand method provides excellent linear plots, and reasonable values for "K" and "ε". However, various problems arising from experimental data have been noted from time to time. Some of these issues include: different values of "ε" with different concentration scales, lack of consistency between the Benesi–Hildebrand values and those obtained from other methods (e.g. equilibrium constants from partition measurements), and zero and negative intercepts. Concerns have also surfaced over the accuracy of the Benesi–Hildebrand method as certain conditions cause these calculations to become invalid. For instance, the reactant concentrations must always obey the assumption that the initial concentration of the guest ([G]0) is much larger than the initial concentration of the host ([H]0). In the case when this breaks down, the Benesi–Hildebrand plot deviates from its linear nature and exhibits scatter plot characteristics. Also, in the case of determining the equilibrium constants for weakly bound complexes, it is common for the formation of 2:1 complexes to occur in solution. It has been observed that the existence of these 2:1 complexes generate inappropriate parameters that significantly interfere with the accurate determination of association constants. Due to this fact, one of the criticisms of this method is the inflexibility of only being able to study reactions with 1:1 product complexes.
These limitations can be overcome by using a computational method which is more generally applicable, a non-linear least-squares minimization method. The two parameters, "K" or "ε" are determined by using the Solver module a spreadsheet, by minimizing a sum of squared differences between observed and calculated quantities with respect to the equilibrium constant and molar absorbance or chemical shift values of the individual chemical species involved. The use of this and more sophisticated methods have the additional advantage that they are not limited to systems where a single complex is formed.
Modifications.
Although initially used in conjunction with UV/Vis spectroscopy, many modifications have been made that allow the B–H method to be applied to other spectroscopic techniques involving fluorescence, infrared, and NMR.
Modifications have also been done to further improve the accuracy in the determination of "K" and "ε" based on the Benesi–Hildebrand equations. One such modification was done by Rose and Drago. The equation that they developed is as follows:
formula_8
Their method relied on a set of chosen values of "ε" and the collection of absorbance data and initial concentrations of the host and guest. This would thus allow the calculation of "K"−1. By plotting a graph of "ε"HG versus "K"−1, the result would be a linear relationship. When the procedure is repeated for a series of concentrations and plotted on the same graph, the lines intersect at a point giving the optimum value of "ε"HG and "K"−1. However, some problems have surfaced with this modified method as some examples displayed an imprecise point of intersection or no intersection at all.
More recently, another graphical procedure has been developed in order to evaluate "K" and "ε" independently of each other. This approach relies on a more complex mathematical rearrangement of the Benesi–Hildebrand method but has proven to be quite accurate when compared to standard values.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "A=A^\\ce{HG}+A^\\ce{G}+A^\\ce{H}\\,"
},
{
"math_id": 1,
"text": "A=A^\\ce{HG}+A^\\ce{G}\\,"
},
{
"math_id": 2,
"text": "{\\Delta}A=A-A_0\\,"
},
{
"math_id": 3,
"text": "{\\Delta}A=\\varepsilon^\\ce{HG}[\\ce{HG}]b+\\varepsilon^\\ce{G}[\\ce G]b-\\varepsilon^\\ce{G}[\\ce G]_0b\\,"
},
{
"math_id": 4,
"text": "{\\Delta}A={\\Delta}\\varepsilon[\\ce{HG}]b\\,"
},
{
"math_id": 5,
"text": "[\\ce{HG}] = \\frac{[\\ce H]_0K_{\\rm a}[\\ce G]}{1+K_{\\rm a}[\\ce G]}"
},
{
"math_id": 6,
"text": "{\\Delta}A=b{\\Delta}\\varepsilon{\\frac{[\\ce H]_0K_{\\rm a}[\\ce G]_0}{1+K_{\\rm a}[\\ce G]_0}}"
},
{
"math_id": 7,
"text": "\\frac{1}{{\\Delta}A}=\\frac{1}{b{\\Delta}\\varepsilon[\\ce G]_0[\\ce H]_0K_{\\rm a}} +\\frac{1}{b{\\Delta}\\varepsilon[\\ce H]_0}"
},
{
"math_id": 8,
"text": "K^{-1}=\\frac{A}{\\varepsilon_\\ce{HG}} - [\\ce H]_0 - [\\ce G]_0 + \\frac{C_\\ce{H}C_\\ce{G}}{A}\\varepsilon_\\ce{HG}"
}
] |
https://en.wikipedia.org/wiki?curid=14183897
|
1418498
|
Left recursion
|
In the formal language theory of computer science, left recursion is a special case of recursion where a string is recognized as part of a language by the fact that it decomposes into a string from that same language (on the left) and a suffix (on the right). For instance, formula_0 can be recognized as a sum because it can be broken into formula_1, also a sum, and formula_2, a suitable suffix.
In terms of context-free grammar, a nonterminal is left-recursive if the leftmost symbol in one of its productions is itself (in the case of direct left recursion) or can be made itself by some sequence of substitutions (in the case of indirect left recursion).
Definition.
A grammar is left-recursive if and only if there exists a nonterminal symbol formula_3 that can derive to a sentential form with itself as the leftmost symbol. Symbolically,
formula_4,
where formula_5 indicates the operation of making one or more substitutions, and formula_6 is any sequence of terminal and nonterminal symbols.
Direct left recursion.
Direct left recursion occurs when the definition can be satisfied with only one substitution. It requires a rule of the form
formula_7
where formula_6 is a sequence of nonterminals and terminals . For example, the rule
formula_8
is directly left-recursive. A left-to-right recursive descent parser for this rule might look like
void Expression() {
Expression();
match('+');
Term();
and such code would fall into infinite recursion when executed.
Indirect left recursion.
Indirect left recursion occurs when the definition of left recursion is satisfied via several substitutions. It entails a set of rules following the pattern
formula_9
formula_10
formula_11
formula_12
where formula_13 are sequences that can each yield the empty string, while formula_14 may be any sequences of terminal and nonterminal symbols at all. Note that these sequences may be empty. The derivation
formula_15
then gives formula_16 as leftmost in its final sentential form.
Uses.
Left recursion is commonly used as an idiom for making operations left-associative: that an expression codice_0 is evaluated as codice_1. In this case, that evaluation order could be achieved as a matter of syntax via the three grammatical rules
formula_17
formula_18
formula_19
These only allow parsing the formula_20 codice_0 as consisting of the formula_20 codice_3 and formula_21 codice_4, where codice_3 in turn consists of the formula_20 codice_6 and formula_21 codice_7, while codice_6 consists of the formula_20 codice_9 and formula_21 codice_10, etc.
Removing left recursion.
Left recursion often poses problems for parsers, either because it leads them into infinite recursion (as in the case of most top-down parsers) or because they expect rules in a normal form that forbids it (as in the case of many bottom-up parsers). Therefore, a grammar is often preprocessed to eliminate the left recursion.
Removing direct left recursion.
The general algorithm to remove direct left recursion follows. Several improvements to this method have been made.
For a left-recursive nonterminal formula_3, discard any rules of the form formula_22 and consider those that remain:
formula_23
where:
Replace these with two sets of productions, one set for formula_3:
formula_25
and another set for the fresh nonterminal formula_26 (often called the "tail" or the "rest"):
formula_27
Repeat this process until no direct left recursion remains.
As an example, consider the rule set
formula_28
This could be rewritten to avoid left recursion as
formula_29
formula_30
Removing all left recursion.
The above process can be extended to eliminate all left recursion, by first converting indirect left recursion to direct left recursion on the highest numbered nonterminal in a cycle.
Inputs "A grammar: a set of nonterminals formula_31 and their productions"
Output "A modified grammar generating the same language but without left recursion"
# "For each nonterminal formula_32:"
## "Repeat until an iteration leaves the grammar unchanged:"
### "For each rule formula_33, formula_34 being a sequence of terminals and nonterminals:"
#### "If formula_34 begins with a nonterminal formula_35 and formula_36:"
##### "Let formula_37 be formula_34 without its leading formula_35."
##### "Remove the rule formula_33."
##### "For each rule formula_38:"
###### "Add the rule formula_39."
## "Remove direct left recursion for formula_32 as described above."
Step 1.1.1 amounts to expanding the initial nonterminal formula_35 in the right hand side of some rule formula_40, but only if formula_36. If formula_40 was one step in a cycle of productions giving rise to a left recursion, then this has shortened that cycle by one step, but often at the price of increasing the number of rules.
The algorithm may be viewed as establishing a topological ordering on nonterminals: afterwards there can only be a rule formula_40 if formula_41.
Note that this algorithm is highly sensitive to the nonterminal ordering; optimizations often focus on choosing this ordering well.
Pitfalls.
Although the above transformations preserve the language generated by a grammar, they may change the parse trees that witness strings' recognition. With suitable bookkeeping, tree rewriting can recover the originals, but if this step is omitted, the differences may change the semantics of a parse.
Associativity is particularly vulnerable; left-associative operators typically appear in right-associative-like arrangements under the new grammar. For example, starting with this grammar:
formula_42
formula_43
formula_44
the standard transformations to remove left recursion yield the following:
formula_45
formula_46
formula_47
formula_48
formula_44
Parsing the string "1 - 2 - 3" with the first grammar in an LALR parser (which can handle left-recursive grammars) would have resulted in the parse tree:
This parse tree groups the terms on the left, giving the correct semantics "(1 - 2) - 3".
Parsing with the second grammar gives
which, properly interpreted, signifies "1 + (-2 + (-3))", also correct, but less faithful to the input and much harder to implement for some operators. Notice how terms to the right appear deeper in the tree, much as a right-recursive grammar would arrange them for "1 - (2 - 3)".
Accommodating left recursion in top-down parsing.
A formal grammar that contains left recursion cannot be parsed by a LL(k)-parser or other naive recursive descent parser unless it is converted to a weakly equivalent right-recursive form. In contrast, left recursion is preferred for LALR parsers because it results in lower stack usage than right recursion. However, more sophisticated top-down parsers can implement general context-free grammars by use of curtailment. In 2006, Frost and Hafiz described an algorithm which accommodates ambiguous grammars with direct left-recursive production rules. That algorithm was extended to a complete parsing algorithm to accommodate indirect as well as direct left recursion in polynomial time, and to generate compact polynomial-size representations of the potentially exponential number of parse trees for highly ambiguous grammars by Frost, Hafiz and Callaghan in 2007. The authors then implemented the algorithm as a set of parser combinators written in the Haskell programming language.
|
[
{
"math_id": 0,
"text": "1+2+3"
},
{
"math_id": 1,
"text": "1+2"
},
{
"math_id": 2,
"text": "{}+3"
},
{
"math_id": 3,
"text": "A"
},
{
"math_id": 4,
"text": " A \\Rightarrow^+ A\\alpha"
},
{
"math_id": 5,
"text": "\\Rightarrow^+"
},
{
"math_id": 6,
"text": "\\alpha"
},
{
"math_id": 7,
"text": "A \\to A\\alpha"
},
{
"math_id": 8,
"text": "\\mathit{Expression} \\to \\mathit{Expression} + \\mathit{Term}"
},
{
"math_id": 9,
"text": "A_0 \\to \\beta_0A_1\\alpha_0"
},
{
"math_id": 10,
"text": "A_1 \\to \\beta_1A_2\\alpha_1"
},
{
"math_id": 11,
"text": "\\cdots"
},
{
"math_id": 12,
"text": "A_n \\to \\beta_nA_0\\alpha_n"
},
{
"math_id": 13,
"text": "\\beta_0, \\beta_1, \\ldots, \\beta_n"
},
{
"math_id": 14,
"text": "\\alpha_0, \\alpha_1, \\ldots, \\alpha_n"
},
{
"math_id": 15,
"text": "A_0\\Rightarrow\\beta_0A_1\\alpha_0\\Rightarrow^+ A_1\\alpha_0\\Rightarrow\\beta_1A_2\\alpha_1\\alpha_0\\Rightarrow^+\\cdots\\Rightarrow^+ A_0\\alpha_n\\dots\\alpha_1\\alpha_0"
},
{
"math_id": 16,
"text": "A_0"
},
{
"math_id": 17,
"text": " \\mathit{Expression} \\to \\mathit{Term} "
},
{
"math_id": 18,
"text": " \\mathit{Expression} \\to \\mathit{Expression} + \\mathit{Term} "
},
{
"math_id": 19,
"text": " \\mathit{Expression} \\to \\mathit{Expression} - \\mathit{Term} "
},
{
"math_id": 20,
"text": "\\mathit{Expression}"
},
{
"math_id": 21,
"text": "\\mathit{Term}"
},
{
"math_id": 22,
"text": "A\\rightarrow A"
},
{
"math_id": 23,
"text": "A \\rightarrow A\\alpha_1 \\mid \\ldots \\mid A\\alpha_n \\mid \\beta_1 \\mid \\ldots \\mid \\beta_m "
},
{
"math_id": 24,
"text": "\\beta"
},
{
"math_id": 25,
"text": "A \\rightarrow \\beta_1A^\\prime \\mid \\ldots \\mid \\beta_mA^\\prime"
},
{
"math_id": 26,
"text": "A'"
},
{
"math_id": 27,
"text": "A^\\prime \\rightarrow \\alpha_1A^\\prime \\mid \\ldots \\mid \\alpha_nA^\\prime \\mid \\epsilon"
},
{
"math_id": 28,
"text": "\\mathit{Expression} \\rightarrow \\mathit{Expression}+\\mathit{Expression} \\mid \\mathit{Integer} \\mid \\mathit{String}"
},
{
"math_id": 29,
"text": "\\mathit{Expression} \\rightarrow \\mathit{Integer}\\,\\mathit{Expression}' \\mid \\mathit{String}\\,\\mathit{Expression}'"
},
{
"math_id": 30,
"text": "\\mathit{Expression}' \\rightarrow {}+\\mathit{Expression} \\,\\mathit{Expression}'\\mid \\epsilon"
},
{
"math_id": 31,
"text": "A_1,\\ldots,A_n"
},
{
"math_id": 32,
"text": "A_i"
},
{
"math_id": 33,
"text": "A_i\\rightarrow\\alpha_i"
},
{
"math_id": 34,
"text": "\\alpha_i"
},
{
"math_id": 35,
"text": "A_j"
},
{
"math_id": 36,
"text": "j<i"
},
{
"math_id": 37,
"text": "\\beta_i"
},
{
"math_id": 38,
"text": "A_j\\rightarrow\\alpha_j"
},
{
"math_id": 39,
"text": "A_i\\rightarrow\\alpha_j\\beta_i"
},
{
"math_id": 40,
"text": "A_i \\to A_j \\beta"
},
{
"math_id": 41,
"text": "j>i"
},
{
"math_id": 42,
"text": "\\mathit{Expression} \\rightarrow \\mathit{Expression}\\,-\\,\\mathit{Term} \\mid \\mathit{Term}"
},
{
"math_id": 43,
"text": "\\mathit{Term} \\rightarrow \\mathit{Term}\\,*\\,\\mathit{Factor} \\mid \\mathit{Factor}"
},
{
"math_id": 44,
"text": "\\mathit{Factor} \\rightarrow (\\mathit{Expression}) \\mid \\mathit{Integer}"
},
{
"math_id": 45,
"text": "\\mathit{Expression} \\rightarrow \\mathit{Term}\\ \\mathit{Expression}'"
},
{
"math_id": 46,
"text": "\\mathit{Expression}' \\rightarrow {} - \\mathit{Term}\\ \\mathit{Expression}' \\mid \\epsilon"
},
{
"math_id": 47,
"text": "\\mathit{Term} \\rightarrow \\mathit{Factor}\\ \\mathit{Term}'"
},
{
"math_id": 48,
"text": "\\mathit{Term}' \\rightarrow {} * \\mathit{Factor}\\ \\mathit{Term}' \\mid \\epsilon"
}
] |
https://en.wikipedia.org/wiki?curid=1418498
|
14187697
|
Semicircular potential well
|
Elementary example of quantum phenomena and the applications of quantum mechanics
In quantum mechanics, the case of a particle in a one-dimensional ring is similar to the particle in a box. The particle follows the path of a semicircle from formula_0 to formula_1 where it cannot escape, because the potential from formula_1 to formula_2 is infinite. Instead there is total reflection, meaning the particle bounces back and forth between formula_0 to formula_1. The Schrödinger equation for a free particle which is restricted to a semicircle (technically, whose configuration space is the circle formula_3) is
Wave function.
Using cylindrical coordinates on the 1-dimensional semicircle, the wave function depends only on the angular coordinate, and so
Substituting the Laplacian in cylindrical coordinates, the wave function is therefore expressed as
The moment of inertia for a semicircle, best expressed in cylindrical coordinates, is formula_4. Solving the integral, one finds that the moment of inertia of a semicircle is formula_5, exactly the same for a hoop of the same radius. The wave function can now be expressed as formula_6, which is easily solvable.
Since the particle cannot escape the region from formula_0 to formula_1, the general solution to this differential equation is
Defining formula_7, we can calculate the energy as formula_8. We then apply the boundary conditions, where formula_9 and formula_10 are continuous and the wave function is normalizable:
Like the infinite square well, the first boundary condition demands that the wave function equals 0 at both formula_11 and formula_12. Basically
Since the wave function formula_13, the coefficient A must equal 0 because formula_14. The wave function also equals 0 at formula_15 so we must apply this boundary condition. Discarding the trivial solution where "B"=0, the wave function formula_16 only when "m" is an integer since formula_17. This boundary condition quantizes the energy where the energy equals formula_8 where "m" is any integer. The condition "m"=0 is ruled out because formula_18 everywhere, meaning that the particle is not in the potential at all. Negative integers are also ruled out since they can easily be absorbed in the normalization condition.
We then normalize the wave function, yielding a result where formula_19. The normalized wave function is
The ground state energy of the system is formula_20. Like the particle in a box, there exists nodes in the excited states of the system where both formula_21 and formula_22 are both 0, which means that the probability of finding the particle at these nodes are 0.
Analysis.
Since the wave function is only dependent on the azimuthal angle formula_23, the measurable quantities of the system are the angular position and angular momentum, expressed with the operators formula_23 and formula_24 respectively.
Using cylindrical coordinates, the operators formula_23 and formula_24 are expressed as formula_23 and formula_25 respectively, where these observables play a role similar to position and momentum for the particle in a box. The commutation and uncertainty relations for angular position and angular momentum are given as follows:
Boundary conditions.
As with all quantum mechanics problems, if the boundary conditions are changed so does the wave function. If a particle is confined to the motion of an entire ring ranging from 0 to formula_2, the particle is subject only to a periodic boundary condition (see particle in a ring). If a particle is confined to the motion of formula_26 to formula_27, the issue of even and odd parity becomes important.
The wave equation for such a potential is given as:
where formula_28 and formula_29 are for odd and even "m" respectively.
Similarly, if the semicircular potential well is a finite well, the solution will resemble that of the finite potential well where the angular operators formula_23 and formula_24 replace the linear operators "x" and "p".
|
[
{
"math_id": 0,
"text": " 0 "
},
{
"math_id": 1,
"text": " \\pi "
},
{
"math_id": 2,
"text": " 2 \\pi "
},
{
"math_id": 3,
"text": "S^1"
},
{
"math_id": 4,
"text": "I \\ \\stackrel{\\mathrm{def}}{=}\\ \\iiint_V r^2 \\,\\rho(r,\\phi,z)\\,r dr\\,d\\phi\\,dz \\!"
},
{
"math_id": 5,
"text": "I=m s^2 "
},
{
"math_id": 6,
"text": " -\\frac{\\hbar^2}{2I} \\frac{d^2\\psi}{d\\phi^2} = E\\psi "
},
{
"math_id": 7,
"text": " m=\\sqrt {\\frac{2 I E}{\\hbar^2}} "
},
{
"math_id": 8,
"text": " E= \\frac{m^2 \\hbar ^2}{2I} "
},
{
"math_id": 9,
"text": " \\psi "
},
{
"math_id": 10,
"text": " \\frac{d\\psi}{d\\phi} "
},
{
"math_id": 11,
"text": " \\phi = 0 "
},
{
"math_id": 12,
"text": " \\phi = \\pi "
},
{
"math_id": 13,
"text": " \\psi(0) = 0 "
},
{
"math_id": 14,
"text": " \\cos (0) = 1 "
},
{
"math_id": 15,
"text": " \\phi= \\pi "
},
{
"math_id": 16,
"text": " \\psi (\\pi) = 0 = B \\sin (m \\pi) "
},
{
"math_id": 17,
"text": " \\sin (n \\pi) = 0 "
},
{
"math_id": 18,
"text": " \\psi = 0 "
},
{
"math_id": 19,
"text": " B= \\sqrt{\\frac{2}{\\pi}} "
},
{
"math_id": 20,
"text": " E= \\frac{\\hbar ^2}{2I} "
},
{
"math_id": 21,
"text": " \\psi (\\phi) "
},
{
"math_id": 22,
"text": " \\psi (\\phi) ^2 "
},
{
"math_id": 23,
"text": " \\phi "
},
{
"math_id": 24,
"text": " L_z "
},
{
"math_id": 25,
"text": " -i \\hbar \\frac{d}{d\\phi} "
},
{
"math_id": 26,
"text": "- \\frac{\\pi}{2} "
},
{
"math_id": 27,
"text": " \\frac{\\pi}{2} "
},
{
"math_id": 28,
"text": " \\psi_{\\rm o} (\\phi) "
},
{
"math_id": 29,
"text": " \\psi_{\\rm e} (\\phi) "
}
] |
https://en.wikipedia.org/wiki?curid=14187697
|
14188480
|
Malgrange preparation theorem
|
Theorem about smooth complex functions
In mathematics, the Malgrange preparation theorem is an analogue of the Weierstrass preparation theorem for smooth functions. It was conjectured by René Thom and proved by B. Malgrange (1962–1963, 1964, 1967).
Statement of Malgrange preparation theorem.
Suppose that "f"("t","x") is a smooth complex function of "t"∈R and "x"∈R"n" near the origin, and let "k" be the smallest integer such that
formula_0
Then one form of the preparation theorem states that near the origin "f" can be written as the product of a smooth function "c" that is nonzero at the origin and a smooth function that as a function of "t" is a polynomial of degree "k". In other words,
formula_1
where the functions "c" and "a" are smooth and "c" is nonzero at the origin.
A second form of the theorem, occasionally called the Mather division theorem, is a sort of "division with remainder" theorem: it says that if "f" and "k" satisfy the conditions above and "g" is a smooth function near the origin, then we can write
formula_2
where "q" and "r" are smooth, and as a function of "t", "r" is a polynomial of degree less than "k". This means that
formula_3
for some smooth functions "r""j"("x").
The two forms of the theorem easily imply each other: the first form is the special case of the "division with remainder" form where "g" is "t""k", and the division with remainder form follows from the first form of the theorem as we may assume that "f" as a function of "t" is a polynomial of degree "k".
If the functions "f" and "g" are real, then the functions "c", "a", "q", and "r" can also be taken to be real. In the case of the Weierstrass preparation theorem these functions are uniquely determined by "f" and "g", but uniqueness no longer holds for the Malgrange preparation theorem.
Proof of Malgrange preparation theorem.
The Malgrange preparation theorem can be deduced from the Weierstrass preparation theorem. The obvious way of doing this does not work: although smooth functions have a formal power series expansion at the origin, and the Weierstrass preparation theorem applies to formal power series, the formal power series will not usually converge to smooth functions near the origin. Instead one can use the idea of decomposing a smooth function as a sum of analytic functions by applying a partition of unity to its Fourier transform.
For a proof along these lines see or
Algebraic version of the Malgrange preparation theorem.
The Malgrange preparation theorem can be restated as a theorem about modules over rings of smooth, real-valued germs. If "X" is a manifold, with "p"∈"X", let "C"∞"p"("X") denote the ring of real-valued germs of smooth functions at "p" on "X". Let "M""p"("X") denote the unique maximal ideal of "C"∞"p"("X"), consisting of germs which vanish at p. Let "A" be a "C"∞"p"("X")-module, and let "f":"X" → "Y" be a smooth function between manifolds. Let "q" = "f"("p"). "f" induces a ring homomorphism "f"*:"C"∞"q"(Y) → "C"∞"p"("X") by composition on the right with "f". Thus we can view "A" as a "C"∞"q"("Y")-module. Then the Malgrange preparation theorem says that if "A" is a finitely-generated "C"∞"p"("X")-module, then "A" is a finitely-generated "C"∞"q"("Y")-module if and only if "A"/"M""q"("Y")A is a finite-dimensional real vector space.
|
[
{
"math_id": 0,
"text": "f(0,0)=0, {\\partial f\\over \\partial t}(0,0)=0, \\dots, {\\partial^{k-1} f\\over \\partial t^{k-1}}(0,0)=0, {\\partial^{k} f\\over \\partial t^{k}}(0,0)\\ne0."
},
{
"math_id": 1,
"text": "f(t,x) = c(t,x)\\left(t^k+a_{k-1}(x)t^{k-1}+\\cdots+a_0(x) \\right)"
},
{
"math_id": 2,
"text": "g=qf+r"
},
{
"math_id": 3,
"text": "r(x)=\\sum_{0\\le j<k}t^jr_j(x)"
}
] |
https://en.wikipedia.org/wiki?curid=14188480
|
14189688
|
Cardinal function
|
Function that returns cardinal numbers
In mathematics, a cardinal function (or cardinal invariant) is a function that returns cardinal numbers.
formula_0
The "additivity" of "I" is the smallest number of sets from "I" whose union is not in "I" any more. As any ideal is closed under finite unions, this number is always at least formula_1; if "I" is a σ-ideal, then formula_2
formula_3
The "covering number" of "I" is the smallest number of sets from "I" whose union is all of "X". As "X" itself is not in "I", we must have add("I"&hairsp;) ≤ cov("I"&hairsp;).
formula_4
The "uniformity number" of "I" (sometimes also written formula_5) is the size of the smallest set not in "I". Assuming "I" contains all singletons, add("I"&hairsp;) ≤ non("I"&hairsp;).
formula_6
The "cofinality" of "I" is the cofinality of the partial order ("I", ⊆). It is easy to see that we must have non("I"&hairsp;) ≤ cof("I"&hairsp;) and cov("I"&hairsp;) ≤ cof("I"&hairsp;).
In the case that formula_7 is an ideal closely related to the structure of the reals, such as the ideal of Lebesgue null sets or the ideal of meagre sets, these cardinal invariants are referred to as cardinal characteristics of the continuum.
formula_11
formula_12
Cardinal functions in topology.
Cardinal functions are widely used in topology as a tool for describing various topological properties. Below are some examples. (Note: some authors, arguing that "there are no finite cardinal numbers in general topology", prefer to define the cardinal functions listed below so that they never taken on finite cardinal numbers as values; this requires modifying some of the definitions given below, for example by adding "formula_14" to the right-hand side of the definitions, etc.)
formula_37
* The hereditary cellularity (sometimes called spread) is the least upper bound of cellularities of its subsets: formula_38 or formula_39 where "discrete" means that it is a discrete topological space.
Basic inequalities.
formula_57
formula_58
formula_59
formula_60
Cardinal functions in Boolean algebras.
Cardinal functions are often used in the study of Boolean algebras. We can mention, for example, the following functions:
formula_64
formula_66.
formula_68.
formula_70
Cardinal functions in algebra.
Examples of cardinal functions in algebra are:
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "{\\rm add}(I) = \\min\\{|\\mathcal{A}| : \\mathcal{A}\\subseteq I \\wedge \\bigcup \\mathcal{A} \\notin I\\}."
},
{
"math_id": 1,
"text": "\\aleph_0"
},
{
"math_id": 2,
"text": "\\operatorname{add}(I) \\ge \\aleph_1."
},
{
"math_id": 3,
"text": "\\operatorname{cov}(I) = \\min\\{|\\mathcal{A}| : \\mathcal{A} \\subseteq I \\wedge \\bigcup \\mathcal{A} = X\\}."
},
{
"math_id": 4,
"text": "\\operatorname{non}(I) = \\min\\{|A| : A \\subseteq X\\ \\wedge\\ A \\notin I\\},"
},
{
"math_id": 5,
"text": "{\\rm unif}(I)"
},
{
"math_id": 6,
"text": "{\\rm cof}(I) = \\min\\{|\\mathcal{B}| : \\mathcal{B} \\subseteq I \\wedge \\forall A \\in I(\\exists B \\in \\mathcal{B})(A\\subseteq B)\\}."
},
{
"math_id": 7,
"text": "I"
},
{
"math_id": 8,
"text": "(\\mathbb{P},\\sqsubseteq)"
},
{
"math_id": 9,
"text": "{\\mathfrak b}(\\mathbb{P})"
},
{
"math_id": 10,
"text": "{\\mathfrak d}(\\mathbb{P})"
},
{
"math_id": 11,
"text": "{\\mathfrak b}(\\mathbb{P}) = \\min\\big\\{|Y| : Y \\subseteq \\mathbb{P}\\ \\wedge\\ (\\forall x\\in \\mathbb{P})(\\exists y\\in Y)(y\\not\\sqsubseteq x)\\big\\},"
},
{
"math_id": 12,
"text": "{\\mathfrak d}(\\mathbb{P}) = \\min\\big\\{|Y| : Y \\subseteq \\mathbb{P}\\ \\wedge\\ (\\forall x\\in \\mathbb{P})(\\exists y\\in Y)(x\\sqsubseteq y)\\big\\}."
},
{
"math_id": 13,
"text": "pp_\\kappa(\\lambda)"
},
{
"math_id": 14,
"text": "\\;\\; + \\;\\aleph_0"
},
{
"math_id": 15,
"text": "X"
},
{
"math_id": 16,
"text": "|X|"
},
{
"math_id": 17,
"text": "o(X)."
},
{
"math_id": 18,
"text": "\\operatorname{w}(X)"
},
{
"math_id": 19,
"text": "X."
},
{
"math_id": 20,
"text": "\\operatorname{w}(X) = \\aleph_0"
},
{
"math_id": 21,
"text": "\\pi"
},
{
"math_id": 22,
"text": "\\operatorname{nw}(X)"
},
{
"math_id": 23,
"text": "\\mathcal{N}"
},
{
"math_id": 24,
"text": "x"
},
{
"math_id": 25,
"text": "U"
},
{
"math_id": 26,
"text": "x,"
},
{
"math_id": 27,
"text": "B"
},
{
"math_id": 28,
"text": "x \\in B \\subseteq U."
},
{
"math_id": 29,
"text": "x."
},
{
"math_id": 30,
"text": "\\chi(X) = \\sup \\; \\{\\chi(x,X) : x\\in X\\}."
},
{
"math_id": 31,
"text": "\\chi(X) = \\aleph_0"
},
{
"math_id": 32,
"text": "\\operatorname{d}(X)"
},
{
"math_id": 33,
"text": "\\rm{d}(X) = \\aleph_0"
},
{
"math_id": 34,
"text": "\\operatorname{L}(X)"
},
{
"math_id": 35,
"text": "\\operatorname{L}(X)."
},
{
"math_id": 36,
"text": "\\rm{L}(X) = \\aleph_0"
},
{
"math_id": 37,
"text": "\\operatorname{c}(X) = \\sup\\{|\\mathcal{U}| : \\mathcal{U} \\text{ is a family of mutually disjoint non-empty open subsets of } X\\}."
},
{
"math_id": 38,
"text": "s(X) = {\\rm hc}(X) = \\sup\\{ {\\rm c} (Y) : Y \\subseteq X \\}"
},
{
"math_id": 39,
"text": "s(X) = \\sup\\{|Y|:Y \\subseteq X \\text{ with the subspace topology is discrete} \\}"
},
{
"math_id": 40,
"text": "e(X) = \\sup\\{|Y| : Y \\subseteq X \\text{ is closed and discrete}\\}."
},
{
"math_id": 41,
"text": "t(x, X)"
},
{
"math_id": 42,
"text": "x \\in X"
},
{
"math_id": 43,
"text": "\\alpha"
},
{
"math_id": 44,
"text": "x\\in{\\rm cl}_X(Y)"
},
{
"math_id": 45,
"text": "Y"
},
{
"math_id": 46,
"text": "X,"
},
{
"math_id": 47,
"text": "Z"
},
{
"math_id": 48,
"text": "|Z| \\leq \\alpha,"
},
{
"math_id": 49,
"text": "x\\in \\operatorname{cl}_X(Z)."
},
{
"math_id": 50,
"text": "t(x, X) = \\sup \\left\\{ \\min \\{|Z| : Z\\subseteq Y\\ \\wedge\\ x\\in {\\rm cl}_X(Z)\\} : Y \\subseteq X\\ \\wedge\\ x \\in {\\rm cl}_X(Y)\\right\\}."
},
{
"math_id": 51,
"text": "t(X) = \\sup\\{t(x, X) : x \\in X\\}."
},
{
"math_id": 52,
"text": "t(X) = \\aleph_0"
},
{
"math_id": 53,
"text": "t^+(X)"
},
{
"math_id": 54,
"text": "Y \\subseteq X,"
},
{
"math_id": 55,
"text": "\\alpha,"
},
{
"math_id": 56,
"text": "x\\in{\\rm cl}_X(Z)."
},
{
"math_id": 57,
"text": "c(X) \\leq d(X) \\leq w(X) \\leq o(X) \\leq 2^{|X|}"
},
{
"math_id": 58,
"text": "e(X) \\leq s(X)"
},
{
"math_id": 59,
"text": "\\chi(X) \\leq w(X)"
},
{
"math_id": 60,
"text": "\\operatorname{nw}(X) \\leq w(X) \\text{ and } o(X) \\leq 2^{\\operatorname{nw}(X)}"
},
{
"math_id": 61,
"text": "c(\\mathbb{B})"
},
{
"math_id": 62,
"text": "\\mathbb{B}"
},
{
"math_id": 63,
"text": "{\\rm length}(\\mathbb{B})"
},
{
"math_id": 64,
"text": "{\\rm length}(\\mathbb{B}) = \\sup\\big\\{|A| : A \\subseteq \\mathbb{B} \\text{ is a chain} \\big\\}"
},
{
"math_id": 65,
"text": "{\\rm depth}(\\mathbb{B})"
},
{
"math_id": 66,
"text": "{\\rm depth}(\\mathbb{B}) = \\sup\\big\\{|A| : A \\subseteq \\mathbb{B} \\text{ is a well-ordered subset} \\big\\}"
},
{
"math_id": 67,
"text": "{\\rm Inc}(\\mathbb{B})"
},
{
"math_id": 68,
"text": "{\\rm Inc}({\\mathbb B}) = \\sup\\big\\{|A| : A \\subseteq \\mathbb{B} \\text{ such that } \\forall a,b \\in A \\big(a \\neq b\\ \\Rightarrow \\neg (a\\leq b\\ \\vee \\ b \\leq a)\\big)\\big\\}"
},
{
"math_id": 69,
"text": "\\pi(\\mathbb{B})"
},
{
"math_id": 70,
"text": "\\pi(\\mathbb{B}) = \\min\\big\\{|A| : A \\subseteq \\mathbb{B}\\setminus \\{0\\} \\text{ such that } \\forall b \\in B\\setminus\\{0\\} \\big(\\exists a \\in A\\big)\\big(a \\leq b\\big)\\big\\}."
},
{
"math_id": 71,
"text": "{\\rm rank}(M)"
}
] |
https://en.wikipedia.org/wiki?curid=14189688
|
14189709
|
Rutherford backscattering spectrometry
|
Analytical technique to determine structure and composition of materials
Rutherford backscattering spectrometry (RBS) is an analytical technique used in materials science. Sometimes referred to as high-energy ion scattering (HEIS) spectrometry, RBS is used to determine the structure and composition of materials by measuring the backscattering of a beam of high energy ions (typically protons or alpha particles) impinging on a sample.
Geiger–Marsden experiment.
Rutherford backscattering spectrometry is named after Lord Rutherford, a physicist sometimes referred to as the father of nuclear physics. Rutherford supervised a series of experiments carried out by Hans Geiger and Ernest Marsden between 1909 and 1914 studying the scattering of alpha particles through metal foils. While attempting to eliminate "stray particles" they believed to be caused by an imperfection in their alpha source, Rutherford suggested that Marsden attempt to measure backscattering from a gold foil sample. According to the then-dominant plum-pudding model of the atom, in which small negative electrons were spread through a diffuse positive region, backscattering of the high-energy positive alpha particles should have been nonexistent. At most small deflections should occur as the alpha particles passed almost unhindered through the foil. Instead, when Marsden positioned the detector on the same side of the foil as the alpha particle source, he immediately detected a noticeable backscattered signal. According to Rutherford, "It was quite the most incredible event that has ever happened to me in my life. It was almost as incredible as if you fired a 15-inch shell at a piece of tissue paper and it came back and hit you."
Rutherford interpreted the result of the Geiger–Marsden experiment as an indication of a Coulomb collision with a single massive positive particle. This led him to the conclusion that the atom's positive charge could not be diffuse but instead must be concentrated in a single massive core: the atomic nucleus. Calculations indicated that the charge necessary to accomplish this deflection was approximately 100 times the charge of the electron, close to the atomic number of gold. This led to the development of the Rutherford model of the atom in which a positive nucleus made up of "N"e positive particles, or protons, was surrounded by "N" orbiting electrons of charge -e to balance the nuclear charge. This model was eventually superseded by the Bohr atom, incorporating some early results from quantum mechanics.
If the energy of the incident particle is increased sufficiently, the Coulomb barrier is exceeded and the wavefunctions of the incident and struck particles overlap. This may result in nuclear reactions in certain cases, but frequently the interaction remains elastic, although the scattering cross-sections may fluctuate wildly as a function of energy and no longer be calculable analytically. This case is known as "Elastic (non-Rutherford) Backscattering Spectrometry" (EBS). There has recently been great progress in determining EBS scattering cross-sections, by solving Schrödinger's equation for each interaction. However, for the EBS analysis of matrices containing light elements, the utilization of experimentally measured scattering cross-section data is also considered to be a very credible option.
Basic principles.
We describe Rutherford backscattering as an elastic, hard-sphere collision between a high kinetic energy particle from the incident beam (the "projectile") and a stationary particle located in the sample (the "target"). "Elastic" in this context means that no energy is transferred between the incident particle and the stationary particle during the collision, and the state of the stationary particle is not changed. (Except that for a small amount of momentum, which is ignored.)
Nuclear interactions are generally not elastic, since a collision may result in a nuclear reaction, with the release of considerable quantities of energy. Nuclear reaction analysis (NRA) is useful for detecting light elements. However, this is not Rutherford scattering.
Considering the kinematics of the collision (that is, the conservation of momentum and kinetic energy), the energy E1 of the scattered projectile is reduced from the initial energy E0:
formula_0
where k is known as the "kinematical factor", and
formula_1
where particle 1 is the projectile, particle 2 is the target nucleus, and formula_2 is the scattering angle of the projectile in the laboratory frame of reference (that is, relative to the observer). The plus sign is taken when the mass of the projectile is less than that of the target, otherwise the minus sign is taken.
While this equation correctly determines the energy of the scattered projectile for any particular scattering angle (relative to the observer), it does not describe the probability of observing such an event. For that we need the "differential cross-section" of the backscattering event:
formula_3
where formula_4 and formula_5 are the atomic numbers of the incident and target nuclei. This equation is written in the centre of mass frame of reference and is therefore not a function of the mass of either the projectile or the target nucleus.
The scattering angle in the laboratory frame of reference formula_2 is "not" the same as the scattering angle in the centre of mass frame of reference formula_6 (although for RBS experiments they are usually very similar). However, heavy ion projectiles can easily recoil lighter ions which, if the geometry is right, can be ejected from the target and detected. This is the basis of the Elastic Recoil Detection (ERD, with synonyms ERDA, FRS, HFS) technique. RBS often uses a He beam which readily recoils H, so simultaneous RBS/ERD is frequently done to probe the hydrogen isotope content of samples (although H ERD with a He beam above 1 MeV is not Rutherford: see http://www-nds.iaea.org/sigmacalc). For ERD the scattering angle in the lab frame of reference is quite different from that in the centre of mass frame of reference.
Heavy ions cannot "back"scatter from light ones: it is kinematically prohibited. The kinematical factor must remain real, and this limits the permitted scattering angle in the laboratory frame of reference. In ERD it is often convenient to place the recoil detector at recoil angles large enough to prohibit signal from the scattered beam. The scattered ion intensity is always very large compared to the recoil intensity (the Rutherford scattering cross-section formula goes to infinity as the scattering angle goes to zero), and for ERD the scattered beam usually has to be excluded from the measurement somehow.
The singularity in the Rutherford scattering cross-section formula is unphysical of course. If the scattering cross-section is zero it implies that the projectile never comes close to the target, but in this case it also never penetrates the electron cloud surrounding the nucleus either. The pure Coulomb formula for the scattering cross-section shown above must be corrected for this screening effect, which becomes more important as the energy of the projectile decreases (or, equivalently, its mass increases).
While large-angle scattering only occurs for ions which scatter off target nuclei, inelastic small-angle scattering can also occur off the sample electrons. This results in a gradual decrease in the kinetic energy of incident ions as they penetrate into the sample, so that backscattering off interior nuclei occurs with a lower "effective" incident energy. Similarly backscattered ions lose energy to electrons as they exit the sample. The amount by which the ion energy is lowered after passing through a given distance is referred to as the stopping power of the material and is dependent on the electron distribution. This energy loss varies continuously with respect to distance traversed, so that stopping power is expressed as
formula_7
For high energy ions stopping power is usually proportional to formula_8; however, precise calculation of stopping power is difficult to carry out with any accuracy.
Stopping power (properly, "stopping force") has units of energy per unit length. It is generally given in thin film units, that is eV /(atom/cm2) since it is measured experimentally on thin films whose thickness is always measured absolutely as mass per unit area, avoiding the problem of determining the density of the material which may vary as a function of thickness. Stopping power is now known for all materials at around 2%, see http://www.srim.org.
Instrumentation.
An RBS instrument generally includes three essential components:
Two common source/acceleration arrangements are used in commercial RBS systems, working in either one or two stages. One-stage systems consist of a He+ source connected to an acceleration tube with a high positive potential applied to the ion source, and the ground at the end of the acceleration tube. This arrangement is simple and convenient, but it can be difficult to achieve energies of much more than 1 MeV due to the difficulty of applying very high voltages to the system.
Two-stage systems, or "tandem accelerators", start with a source of He− ions and position the positive terminal at the center of the acceleration tube. A stripper element included in the positive terminal removes electrons from ions which pass through, converting He− ions to He++ ions. The ions thus start out being attracted to the terminal, pass through and become positive, and are repelled until they exit the tube at ground. This arrangement, though more complex, has the advantage of achieving higher accelerations with lower applied voltages: a typical tandem accelerator with an applied voltage of 750 kV can achieve ion energies of over 2 MeV.
Detectors to measure backscattered energy are usually silicon surface barrier detectors, a very thin layer (100 nm) of P-type silicon on an N-type substrate forming a p-n junction. Ions which reach the detector lose some of their energy to inelastic scattering from the electrons, and some of these electrons gain enough energy to overcome the band gap between the semiconductor valence and conduction bands. This means that each ion incident on the detector will produce some number of electron-hole pairs which is dependent on the energy of the ion. These pairs can be detected by applying a voltage across the detector and measuring the current, providing an effective measurement of the ion energy. The relationship between ion energy and the number of electron-hole pairs produced will be dependent on the detector materials, the type of ion and the efficiency of the current measurement; energy resolution is dependent on thermal fluctuations. After one ion is incident on the detector, there will be some "dead time" before the electron-hole pairs recombine in which a second incident ion cannot be distinguished from the first.
Angular dependence of detection can be achieved by using a movable detector, or more practically by separating the surface barrier detector into many independent cells which can be measured independently, covering some range of angles around direct (180 degrees) back-scattering. Angular dependence of the incident beam is controlled by using a tiltable sample stage.
Composition and depth measurement.
The energy loss of a backscattered ion is dependent on two processes: the energy lost in scattering events with sample nuclei, and the energy lost to small-angle scattering from the sample electrons. The first process is dependent on the scattering cross-section of the nucleus and thus on its mass and atomic number. For a given measurement angle, nuclei of two different elements will therefore scatter incident ions to different degrees and with different energies, producing separate peaks on an N(E) plot of measurement count versus energy. These peaks are characteristic of the elements contained in the material, providing a means of analyzing the composition of a sample by matching scattered energies to known scattering cross-sections. Relative concentrations can be determined by measuring the heights of the peaks.
The second energy loss process, the stopping power of the sample electrons, does not result in large discrete losses such as those produced by nuclear collisions. Instead it creates a gradual energy loss dependent on the electron density and the distance traversed in the sample. This energy loss will lower the measured energy of ions which backscatter from nuclei inside the sample in a continuous manner dependent on the depth of the nuclei. The result is that instead of the sharp backscattered peaks one would expect on an N(E) plot, with the width determined by energy and angular resolution, the peaks observed trail off gradually towards lower energy as the ions pass through the depth occupied by that element. Elements which only appear at some depth inside the sample will also have their peak positions shifted by some amount which represents the distance an ion had to traverse to reach those nuclei.
In practice, then, a compositional depth profile can be determined from an RBS N(E) measurement. The elements contained by a sample can be determined from the positions of peaks in the energy spectrum. Depth can be determined from the width and shifted position of these peaks, and relative concentration from the peak heights. This is especially useful for the analysis of a multilayer sample, for example, or for a sample with a composition which varies more continuously with depth.
This kind of measurement can only be used to determine elemental composition; the chemical structure of the sample cannot be determined from the N(E) profile. However, it is possible to learn something about this through RBS by examining the crystal structure. This kind of spatial information can be investigated by taking advantage of blocking and channeling.
Structural measurements: blocking and channeling.
To fully understand the interaction of an incident beam of nuclei with a crystalline structure, it is necessary to comprehend two more key concepts: "blocking" and "channeling".
When a beam of ions with parallel trajectories is incident on a target atom, scattering off that atom will prevent collisions in a cone-shaped region "behind" the target relative to the beam. This occurs because the repulsive potential of the target atom bends close ion trajectories away from their original path, and is referred to as "blocking". The radius of this blocked region, at a distance L from the original atom, is given by
formula_9
When an ion is scattered from deep inside a sample, it can then re-scatter off a second atom, creating a second blocked cone in the direction of the scattered trajectory. This can be detected by carefully varying the detection angle relative to the incident angle.
"Channeling" is observed when the incident beam is aligned with a major symmetry axis of the crystal. Incident nuclei which avoid collisions with surface atoms are excluded from collisions with all atoms deeper in the sample, due to blocking by the first layer of atoms. When the interatomic distance is large compared to the radius of the blocked cone, the incident ions can penetrate many times the interatomic distance without being backscattered. This can result in a drastic reduction of the observed backscattered signal when the incident beam is oriented along one of the symmetry directions, allowing determination of a sample's regular crystal structure. Channeling works best for very small blocking radii, i.e. for high-energy, low-atomic-number incident ions such as He+.
The tolerance for the deviation of the ion beam angle of incidence relative to the symmetry direction depends on the blocking radius, making the allowable deviation angle proportional to
formula_10
While the intensity of an RBS peak is observed to decrease across most of its width when the beam is channeled, a narrow peak at the high-energy end of larger peak will often be observed, representing surface scattering from the first layer of atoms. The presence of this peak opens the possibility of surface sensitivity for RBS measurements.
Profiling of displaced atoms.
In addition, channeling of ions can also be used to analyze a crystalline sample for lattice damage. If atoms within the target are displaced from their crystalline lattice site, this will result in a higher backscattering yield in relation to a perfect crystal. By comparing the spectrum from a sample being analyzed to that from a perfect crystal, and that obtained at a random (non-channeling) orientation (representative of a spectrum from an amorphous sample), it is possible to determine the extent of crystalline damage in terms of a fraction of displaced atoms. Multiplying this fraction by the density of the material when amorphous then also gives an estimate for the concentration of displaced atoms. The energy at which the increased backscattering occurs can also be used to determine the depth at which the displaced atoms are and a defect depth profile can be built up as a result.
Surface sensitivity.
While RBS is generally used to measure the bulk composition and structure of a sample, it is possible to obtain some information about the structure and composition of the sample surface. When the signal is channeled to remove the bulk signal, careful manipulation of the incident and detection angles can be used to determine the relative positions of the first few layers of atoms, taking advantage of blocking effects.
The surface structure of a sample can be changed from the ideal in a number of ways. The first layer of atoms can change its distance from subsequent layers (relaxation); it can assume a different two-dimensional structure than the bulk (reconstruction); or another material can be adsorbed onto the surface. Each of these cases can be detected by RBS. For example, surface reconstruction can be detected by aligning the beam in such a way that channeling should occur, so that only a surface peak of known intensity should be detected. A higher-than-usual intensity or a wider peak will indicate that the first layers of atoms are failing to block the layers beneath, i.e. that the surface has been reconstructed. Relaxations can be detected by a similar procedure with the sample tilted so the ion beam is incident at an angle selected so that first-layer atoms should block backscattering at a diagonal; that is, from atoms which are below and displaced from the blocking atom. A higher-than-expected backscattered yield will indicate that the first layer has been displaced relative to the second layer, or relaxed. Adsorbate materials will be detected by their different composition, changing the position of the surface peak relative to the expected position.
RBS has also been used to measure processes which affect the surface differently from the bulk by analyzing changes in the channeled surface peak. A well-known example of this is the RBS analysis of the premelting of lead surfaces by Frenken, Maree and van der Veen. In an RBS measurement of the Pb(110) surface, a well-defined surface peak which is stable at low temperatures was found to become wider and more intense as temperature increase past two-thirds of the bulk melting temperature. The peak reached the bulk height and width as temperature reached the melting temperature. This increase in the disorder of the surface, making deeper atoms visible to the incident beam, was interpreted as pre-melting of the surface, and computer simulations of the RBS process produced similar results when compared with theoretical pre-melting predictions.
RBS has also been combined with nuclear microscopy, in which a focused ion beam is scanned across a surface in a manner similar to a scanning electron microscope. The energetic analysis of backscattered signals in this kind of application provides compositional information about the surface, while the microprobe itself can be used to examine features such as periodic surface structures.
See also.
<templatestyles src="Div col/styles.css"/>
References.
Citations.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "E_1 = k \\cdot E_0, "
},
{
"math_id": 1,
"text": "k = \\left(\\frac{m_1 \\cos{\\theta_1} \\pm \\sqrt{m_2^2 - m_1^2(\\sin{\\theta_1})^2}}{m_1 + m_2}\\right)^2,"
},
{
"math_id": 2,
"text": "\\theta_1"
},
{
"math_id": 3,
"text": "\\frac{d\\omega}{d\\Omega} = \\left(\\frac{Z_1Z_2e^2}{4E_0}\\right)^2 \n\\frac{1}{\\left(\\sin{\\theta/2}\\right)^4},"
},
{
"math_id": 4,
"text": "Z_1"
},
{
"math_id": 5,
"text": "Z_2"
},
{
"math_id": 6,
"text": "\\theta"
},
{
"math_id": 7,
"text": "S(E) = -{dE \\over dx}. "
},
{
"math_id": 8,
"text": "\\frac{Z_2}{E}"
},
{
"math_id": 9,
"text": "R = 2\\sqrt{\\frac{Z_1Z_2e^2L}{E_0}}"
},
{
"math_id": 10,
"text": "\\sqrt{\\frac{Z_1Z_2}{E_0d}}"
}
] |
https://en.wikipedia.org/wiki?curid=14189709
|
1419
|
Adiabatic process
|
Thermodynamic process in which no mass or heat is exchanged with surroundings
An adiabatic process ("adiabatic" from grc " "" ()" 'impassable') is a type of thermodynamic process
that occurs without transferring heat or mass between the thermodynamic system and its environment. Unlike an isothermal process, an adiabatic process transfers energy to the surroundings only as work. As a key concept in thermodynamics, the adiabatic process supports the theory that explains the first law of thermodynamics. The opposite term to "adiabatic" is "diabatic".
Some chemical and physical processes occur too rapidly for energy to enter or leave the system as heat, allowing a convenient "adiabatic approximation". For example, the adiabatic flame temperature uses this approximation to calculate the upper limit of flame temperature by assuming combustion loses no heat to its surroundings.
In meteorology, adiabatic expansion and cooling of moist air, which can be triggered by winds flowing up and over a mountain for example, can cause the water vapor pressure to exceed the saturation vapor pressure. Expansion and cooling beyond the saturation vapor pressure is often idealized as a "pseudo-adiabatic process" whereby excess vapor instantaneously precipitates into water droplets. The change in temperature of an air undergoing pseudo-adiabatic expansion differs from air undergoing adiabatic expansion because latent heat is released by precipitation.
Description.
A process without transfer of heat to or from a system, so that "Q" = 0, is called adiabatic, and such a system is said to be adiabatically isolated. The simplifying assumption frequently made is that a process is adiabatic. For example, the compression of a gas within a cylinder of an engine is assumed to occur so rapidly that on the time scale of the compression process, little of the system's energy can be transferred out as heat to the surroundings. Even though the cylinders are not insulated and are quite conductive, that process is idealized to be adiabatic. The same can be said to be true for the expansion process of such a system.
The assumption of adiabatic isolation is useful and often combined with other such idealizations to calculate a good first approximation of a system's behaviour. For example, according to Laplace, when sound travels in a gas, there is no time for heat conduction in the medium, and so the propagation of sound is adiabatic. For such an adiabatic process, the modulus of elasticity (Young's modulus) can be expressed as "E" = "γP", where "γ" is the ratio of specific heats at constant pressure and at constant volume ("γ" =) and "P" is the pressure of the gas.
Various applications of the adiabatic assumption.
For a closed system, one may write the first law of thermodynamics as Δ"U" = "Q" − "W", where Δ"U" denotes the change of the system's internal energy, "Q" the quantity of energy added to it as heat, and "W" the work done by the system on its surroundings.
Naturally occurring adiabatic processes are irreversible (entropy is produced).
The transfer of energy as work into an adiabatically isolated system can be imagined as being of two idealized extreme kinds. In one such kind, no entropy is produced within the system (no friction, viscous dissipation, etc.), and the work is only pressure-volume work (denoted by "P" d"V"). In nature, this ideal kind occurs only approximately because it demands an infinitely slow process and no sources of dissipation.
The other extreme kind of work is isochoric work (d"V" = 0), for which energy is added as work solely through friction or viscous dissipation within the system. A stirrer that transfers energy to a viscous fluid of an adiabatically isolated system with rigid walls, without phase change, will cause a rise in temperature of the fluid, but that work is not recoverable. Isochoric work is irreversible. The second law of thermodynamics observes that a natural process, of transfer of energy as work, always consists at least of isochoric work and often both of these extreme kinds of work. Every natural process, adiabatic or not, is irreversible, with Δ"S" > 0, as friction or viscosity are always present to some extent.
Adiabatic compression and expansion.
The adiabatic compression of a gas causes a rise in temperature of the gas. Adiabatic expansion against pressure, or a spring, causes a drop in temperature. In contrast, free expansion is an isothermal process for an ideal gas.
Adiabatic compression occurs when the pressure of a gas is increased by work done on it by its surroundings, e.g., a piston compressing a gas contained within a cylinder and raising the temperature where in many practical situations heat conduction through walls can be slow compared with the compression time. This finds practical application in diesel engines which rely on the lack of heat dissipation during the compression stroke to elevate the fuel vapor temperature sufficiently to ignite it.
Adiabatic compression occurs in the Earth's atmosphere when an air mass descends, for example, in a Katabatic wind, Foehn wind, or Chinook wind flowing downhill over a mountain range. When a parcel of air descends, the pressure on the parcel increases. Because of this increase in pressure, the parcel's volume decreases and its temperature increases as work is done on the parcel of air, thus increasing its internal energy, which manifests itself by a rise in the temperature of that mass of air. The parcel of air can only slowly dissipate the energy by conduction or radiation (heat), and to a first approximation it can be considered adiabatically isolated and the process an adiabatic process.
Adiabatic expansion occurs when the pressure on an adiabatically isolated system is decreased, allowing it to expand in size, thus causing it to do work on its surroundings. When the pressure applied on a parcel of gas is reduced, the gas in the parcel is allowed to expand; as the volume increases, the temperature falls as its internal energy decreases. Adiabatic expansion occurs in the Earth's atmosphere with orographic lifting and lee waves, and this can form pilei or lenticular clouds.
Due in part to adiabatic expansion in mountainous areas, snowfall infrequently occurs in some parts of the Sahara desert.
Adiabatic expansion does not have to involve a fluid. One technique used to reach very low temperatures (thousandths and even millionths of a degree above absolute zero) is via adiabatic demagnetisation, where the change in magnetic field on a magnetic material is used to provide adiabatic expansion. Also, the contents of an expanding universe can be described (to first order) as an adiabatically expanding fluid. (See heat death of the universe.)
Rising magma also undergoes adiabatic expansion before eruption, particularly significant in the case of magmas that rise quickly from great depths such as kimberlites.
In the Earth's convecting mantle (the asthenosphere) beneath the lithosphere, the mantle temperature is approximately an adiabat. The slight decrease in temperature with shallowing depth is due to the decrease in pressure the shallower the material is in the Earth.
Such temperature changes can be quantified using the ideal gas law, or the hydrostatic equation for atmospheric processes.
In practice, no process is truly adiabatic. Many processes rely on a large difference in time scales of the process of interest and the rate of heat dissipation across a system boundary, and thus are approximated by using an adiabatic assumption. There is always some heat loss, as no perfect insulators exist.
Ideal gas (reversible process).
The mathematical equation for an ideal gas undergoing a reversible (i.e., no entropy generation) adiabatic process can be represented by the polytropic process equation
formula_0
where "P" is pressure, "V" is volume, and "γ" is the adiabatic index or heat capacity ratio defined as
formula_1
Here "CP" is the specific heat for constant pressure, "CV" is the specific heat for constant volume, and "f" is the number of degrees of freedom (3 for a monatomic gas, 5 for a diatomic gas or a gas of linear molecules such as carbon dioxide).
For a monatomic ideal gas, "γ" =, and for a diatomic gas (such as nitrogen and oxygen, the main components of air), "γ" =. Note that the above formula is only applicable to classical ideal gases (that is, gases far above absolute zero temperature) and not Bose–Einstein or Fermi gases.
One can also use the ideal gas law to rewrite the above relationship between "P" and "V" as
formula_2
formula_3
where "T" is the absolute or thermodynamic temperature.
Example of adiabatic compression.
The compression stroke in a gasoline engine can be used as an example of adiabatic compression. The model assumptions are: the uncompressed volume of the cylinder is one litre (1 L = 1000 cm3 = 0.001 m3); the gas within is the air consisting of molecular nitrogen and oxygen only (thus a diatomic gas with 5 degrees of freedom, and so "γ" =); the compression ratio of the engine is 10:1 (that is, the 1 L volume of uncompressed gas is reduced to 0.1 L by the piston); and the uncompressed gas is at approximately room temperature and pressure (a warm room temperature of ~27 °C, or 300 K, and a pressure of 1 bar = 100 kPa, i.e. typical sea-level atmospheric pressure).
formula_4
so the adiabatic constant for this example is about 6.31 Pa m4.2.
The gas is now compressed to a 0.1 L (0.0001 m3) volume, which we assume happens quickly enough that no heat enters or leaves the gas through the walls. The adiabatic constant remains the same, but with the resulting pressure unknown
formula_5
We can now solve for the final pressure
formula_6
or 25.1 bar. This pressure increase is more than a simple 10:1 compression ratio would indicate; this is because the gas is not only compressed, but the work done to compress the gas also increases its internal energy, which manifests itself by a rise in the gas temperature and an additional rise in pressure above what would result from a simplistic calculation of 10 times the original pressure.
We can solve for the temperature of the compressed gas in the engine cylinder as well, using the ideal gas law, "PV" = "nRT" ("n" is amount of gas in moles and "R" the gas constant for that gas). Our initial conditions being 100 kPa of pressure, 1 L volume, and 300 K of temperature, our experimental constant ("nR") is:
formula_7
We know the compressed gas has V = 0.1 L and P = , so we can solve for temperature:
formula_8
That is a final temperature of 753 K, or 479 °C, or 896 °F, well above the ignition point of many fuels. This is why a high-compression engine requires fuels specially formulated to not self-ignite (which would cause engine knocking when operated under these conditions of temperature and pressure), or that a supercharger with an intercooler to provide a pressure boost but with a lower temperature rise would be advantageous. A diesel engine operates under even more extreme conditions, with compression ratios of 16:1 or more being typical, in order to provide a very high gas pressure, which ensures immediate ignition of the injected fuel.
Adiabatic free expansion of a gas.
For an adiabatic free expansion of an ideal gas, the gas is contained in an insulated container and then allowed to expand in a vacuum. Because there is no external pressure for the gas to expand against, the work done by or on the system is zero. Since this process does not involve any heat transfer or work, the first law of thermodynamics then implies that the net internal energy change of the system is zero. For an ideal gas, the temperature remains constant because the internal energy only depends on temperature in that case. Since at constant temperature, the entropy is proportional to the volume, the entropy increases in this case, therefore this process is irreversible.
Derivation of "P"–"V" relation for adiabatic compression and expansion.
The definition of an adiabatic process is that heat transfer to the system is zero, "δQ" = 0. Then, according to the first law of thermodynamics,
where "dU" is the change in the internal energy of the system and "δW" is work done "by" the system. Any work ("δW") done must be done at the expense of internal energy "U", since no heat "δQ" is being supplied from the surroundings. Pressure–volume work "δW" done "by" the system is defined as
However, "P" does not remain constant during an adiabatic process but instead changes along with "V".
It is desired to know how the values of "dP" and "dV" relate to each other as the adiabatic process proceeds. For an ideal gas (recall ideal gas law "PV" = "nRT") the internal energy is given by
where "α" is the number of degrees of freedom divided by 2, "R" is the universal gas constant and "n" is the number of moles in the system (a constant).
Differentiating equation (a3) yields
Equation (a4) is often expressed as "dU" = "nCV dT" because "CV" = "αR".
Now substitute equations (a2) and (a4) into equation (a1) to obtain
formula_9
factorize −"P dV":
formula_10
and divide both sides by "PV":
formula_11
After integrating the left and right sides from "V"0 to "V" and from "P"0 to "P" and changing the sides respectively,
formula_12
Exponentiate both sides, substitute with "γ", the heat capacity ratio
formula_13
and eliminate the negative sign to obtain
formula_14
Therefore,
formula_15
and
formula_16
At the same time, the work done by the pressure–volume changes as a result from this process, is equal to
Since we require the process to be adiabatic, the following equation needs to be true
By the previous derivation,
Rearranging (b4) gives
formula_17
Substituting this into (b2) gives
formula_18
Integrating we obtain the expression for work,
formula_19
Substituting "γ" = in second term,
formula_20
Rearranging,
formula_21
Using the ideal gas law and assuming a constant molar quantity (as often happens in practical cases),
formula_22
By the continuous formula,
formula_23
or
formula_24
Substituting into the previous expression for "W",
formula_25
Substituting this expression and (b1) in (b3) gives
formula_26
Simplifying,
formula_27
formula_28
formula_29
Derivation of discrete formula and work expression.
The change in internal energy of a system, measured from state 1 to state 2, is equal to
At the same time, the work done by the pressure–volume changes as a result from this process, is equal to
Since we require the process to be adiabatic, the following equation needs to be true
By the previous derivation,
Rearranging (c4) gives
formula_17
Substituting this into (c2) gives
formula_18
Integrating we obtain the expression for work,
formula_19
Substituting "γ" = in second term,
formula_20
Rearranging,
formula_21
Using the ideal gas law and assuming a constant molar quantity (as often happens in practical cases),
formula_22
By the continuous formula,
formula_23
or
formula_24
Substituting into the previous expression for "W",
formula_25
Substituting this expression and (c1) in (c3) gives
formula_26
Simplifying,
formula_27
formula_28
formula_29
Graphing adiabats.
An adiabat is a curve of constant entropy in a diagram. Some properties of adiabats on a "P"–"V" diagram are indicated. These properties may be read from the classical behaviour of ideal gases, except in the region where "PV" becomes small (low temperature), where quantum effects become important.
The right diagram is a "P"–"V" diagram with a superposition of adiabats and isotherms:
The isotherms are the red curves and the adiabats are the black curves.
The adiabats are isentropic.
Volume is the horizontal axis and pressure is the vertical axis.
Etymology.
The term "adiabatic" () is an anglicization of the Greek term ἀδιάβατος "impassable" (used by Xenophon of rivers). It is used in the thermodynamic sense by Rankine (1866), and adopted by Maxwell in 1871 (explicitly attributing the term to Rankine).
The etymological origin corresponds here to an impossibility of transfer of energy as heat and of transfer of matter across the wall.
The Greek word ἀδιάβατος is formed from privative ἀ- ("not") and διαβατός, "passable", in turn deriving from διά ("through"), and βαῖνειν ("to walk, go, come").
Conceptual significance in thermodynamic theory.
The adiabatic process has been important for thermodynamics since its early days. It was important in the work of Joule because it provided a way of nearly directly relating quantities of heat and work.
Energy can enter or leave a thermodynamic system enclosed by walls that prevent mass transfer only as heat or work. Therefore, a quantity of work in such a system can be related almost directly to an equivalent quantity of heat in a cycle of two limbs. The first limb is an isochoric adiabatic work process increasing the system's internal energy; the second, an isochoric and workless heat transfer returning the system to its original state. Accordingly, Rankine measured quantity of heat in units of work, rather than as a calorimetric quantity. In 1854, Rankine used a quantity that he called "the thermodynamic function" that later was called entropy, and at that time he wrote also of the "curve of no transmission of heat", which he later called an adiabatic curve. Besides its two isothermal limbs, Carnot's cycle has two adiabatic limbs.
For the foundations of thermodynamics, the conceptual importance of this was emphasized by Bryan, by Carathéodory, and by Born. The reason is that calorimetry presupposes a type of temperature as already defined before the statement of the first law of thermodynamics, such as one based on empirical scales. Such a presupposition involves making the distinction between empirical temperature and absolute temperature. Rather, the definition of absolute thermodynamic temperature is best left till the second law is available as a conceptual basis.
In the eighteenth century, the law of conservation of energy was not yet fully formulated or established, and the nature of heat was debated. One approach to these problems was to regard heat, measured by calorimetry, as a primary substance that is conserved in quantity. By the middle of the nineteenth century, it was recognized as a form of energy, and the law of conservation of energy was thereby also recognized. The view that eventually established itself, and is currently regarded as right, is that the law of conservation of energy is a primary axiom, and that heat is to be analyzed as consequential. In this light, heat cannot be a component of the total energy of a single body because it is not a state variable but, rather, a variable that describes a transfer between two bodies. The adiabatic process is important because it is a logical ingredient of this current view.
Divergent usages of the word "adiabatic".
This present article is written from the viewpoint of macroscopic thermodynamics, and the word "adiabatic" is used in this article in the traditional way of thermodynamics, introduced by Rankine. It is pointed out in the present article that, for example, if a compression of a gas is rapid, then there is little time for heat transfer to occur, even when the gas is not adiabatically isolated by a definite wall. In this sense, a rapid compression of a gas is sometimes approximately or loosely said to be "adiabatic", though often far from isentropic, even when the gas is not adiabatically isolated by a definite wall.
Some authors, like Pippard, recommend using "adiathermal" to refer to processes where no heat-exchange occurs (such as Joule expansion), and "adiabatic" to reversible quasi-static adiathermal processes (so that rapid compression of a gas is "not" "adiabatic"). And Laidler has summarized the complicated etymology of "adiabatic".
Quantum mechanics and quantum statistical mechanics, however, use the word "adiabatic" in a very different sense, one that can at times seem almost opposite to the classical thermodynamic sense. In quantum theory, the word "adiabatic" can mean something perhaps near isentropic, or perhaps near quasi-static, but the usage of the word is very different between the two disciplines.
On the one hand, in quantum theory, if a perturbative element of compressive work is done almost infinitely slowly (that is to say quasi-statically), it is said to have been done "adiabatically". The idea is that the shapes of the eigenfunctions change slowly and continuously, so that no quantum jump is triggered, and the change is virtually reversible. While the occupation numbers are unchanged, nevertheless there is change in the energy levels of one-to-one corresponding, pre- and post-compression, eigenstates. Thus a perturbative element of work has been done without heat transfer and without introduction of random change within the system. For example, Max Born writes "Actually, it is usually the 'adiabatic' case with which we have to do: i.e. the limiting case where the external force (or the reaction of the parts of the system on each other) acts very slowly. In this case, to a very high approximation
formula_30
that is, there is no probability for a transition, and the system is in the initial state after cessation of the perturbation. Such a slow perturbation is therefore reversible, as it is classically."
On the other hand, in quantum theory, if a perturbative element of compressive work is done rapidly, it changes the occupation numbers and energies of the eigenstates in proportion to the transition moment integral and in accordance with time-dependent perturbation theory, as well as perturbing the functional form of the eigenstates themselves. In that theory, such a rapid change is said not to be "adiabatic", and the contrary word "diabatic" is applied to it.
Recent research suggests that the power absorbed from the perturbation corresponds to the rate of these non-adiabatic transitions. This corresponds to the classical process of energy transfer in the form of heat, but with the relative time scales reversed in the quantum case. Quantum adiabatic processes occur over relatively long time scales, while classical adiabatic processes occur over relatively short time scales. It should also be noted that the concept of 'heat' (in reference to the quantity of thermal energy transferred) breaks down at the quantum level, and the specific form of energy (typically electromagnetic) must be considered instead. The small or negligible absorption of energy from the perturbation in a quantum adiabatic process provides a good justification for identifying it as the quantum analogue of adiabatic processes in classical thermodynamics, and for the reuse of the term.
Furthermore, in atmospheric thermodynamics, a diabatic process is one in which heat is exchanged.
In classical thermodynamics, such a rapid change would still be called adiabatic because the system is adiabatically isolated, and there is no transfer of energy as heat. The strong irreversibility of the change, due to viscosity or other entropy production, does not impinge on this classical usage.
Thus for a mass of gas, in macroscopic thermodynamics, words are so used that a compression is sometimes loosely or approximately said to be adiabatic if it is rapid enough to avoid significant heat transfer, even if the system is not adiabatically isolated. But in quantum statistical theory, a compression is not called adiabatic if it is rapid, even if the system is adiabatically isolated in the classical thermodynamic sense of the term. The words are used differently in the two disciplines, as stated just above.
References.
<templatestyles src="Reflist/styles.css" />
External links.
Media related to at Wikimedia Commons
|
[
{
"math_id": 0,
"text": " P V^\\gamma = \\text{constant}, "
},
{
"math_id": 1,
"text": " \\gamma = \\frac{C_P}{C_V} = \\frac{f + 2}{f}. "
},
{
"math_id": 2,
"text": " P^{1-\\gamma} T^\\gamma = \\text{constant},"
},
{
"math_id": 3,
"text": " TV^{\\gamma - 1} = \\text{constant}."
},
{
"math_id": 4,
"text": "\\begin{align}\n& P_1 V_1^\\gamma = \\mathrm{constant}_1 = 100\\,000~\\text{Pa} \\times (0.001~\\text{m}^3)^\\frac75 \\\\\n& = 10^5 \\times 6.31 \\times 10^{-5}~\\text{Pa}\\,\\text{m}^{21/5} = 6.31~\\text{Pa}\\,\\text{m}^{21/5},\n\\end{align}"
},
{
"math_id": 5,
"text": " P_2 V_2^\\gamma = \\mathrm{constant}_1 = 6.31~\\text{Pa}\\,\\text{m}^{21/5} = P \\times (0.0001~\\text{m}^3)^\\frac75, "
},
{
"math_id": 6,
"text": " P_2 = P_1\\left (\\frac{V_1}{V_2}\\right)^\\gamma = 100\\,000~\\text{Pa} \\times \\text{10}^{7/5} = 2.51 \\times 10^6~\\text{Pa}"
},
{
"math_id": 7,
"text": " \\frac{PV}{T} = \\mathrm{constant}_2 = \\frac{10^5~\\text{Pa} \\times 10^{-3}~\\text{m}^3}{300~\\text{K}} = 0.333~\\text{Pa}\\,\\text{m}^3\\text{K}^{-1}."
},
{
"math_id": 8,
"text": " T = \\frac{P V}{\\mathrm{constant}_2} = \\frac{2.51 \\times 10^6~\\text{Pa} \\times 10^{-4}~\\text{m}^3}{0.333~\\text{Pa}\\,\\text{m}^3\\text{K}^{-1}} = 753~\\text{K}. "
},
{
"math_id": 9,
"text": " -P \\, dV = \\alpha P \\, dV + \\alpha V \\, dP,"
},
{
"math_id": 10,
"text": " -(\\alpha + 1) P \\, dV = \\alpha V \\, dP,"
},
{
"math_id": 11,
"text": " -(\\alpha + 1) \\frac{dV}{V} = \\alpha \\frac{dP}{P}. "
},
{
"math_id": 12,
"text": " \\ln \\left( \\frac{P}{P_0} \\right) = -\\frac{\\alpha + 1}{\\alpha} \\ln \\left( \\frac{V}{V_0} \\right). "
},
{
"math_id": 13,
"text": " \\left( \\frac{P}{P_0} \\right) = \\left( \\frac{V}{V_0} \\right)^{-\\gamma}, "
},
{
"math_id": 14,
"text": " \\left( \\frac{P}{P_0} \\right) = \\left( \\frac{V_0}{V} \\right)^\\gamma. "
},
{
"math_id": 15,
"text": " \\left( \\frac{P}{P_0} \\right) \\left( \\frac{V}{V_0} \\right)^\\gamma = 1,"
},
{
"math_id": 16,
"text": " P_0 V_0^\\gamma = P V^\\gamma = \\mathrm{constant}. "
},
{
"math_id": 17,
"text": " P = P_1 \\left(\\frac{V_1}{V} \\right)^\\gamma. "
},
{
"math_id": 18,
"text": " W = \\int_{V_1}^{V_2} P_1 \\left(\\frac{V_1}{V} \\right)^\\gamma \\,dV. "
},
{
"math_id": 19,
"text": " W = P_1 V_1^\\gamma \\frac{V_2^{1-\\gamma} - V_1^{1-\\gamma}}{1 - \\gamma} = \\frac{P_2 V_2 - P_1 V_1}{1 - \\gamma}. "
},
{
"math_id": 20,
"text": " W = -\\alpha P_1 V_1^\\gamma \\left( V_2^{1-\\gamma} - V_1^{1-\\gamma} \\right). "
},
{
"math_id": 21,
"text": " W = -\\alpha P_1 V_1 \\left( \\left( \\frac{V_2}{V_1} \\right)^{1-\\gamma} - 1 \\right). "
},
{
"math_id": 22,
"text": " W = -\\alpha n R T_1 \\left( \\left( \\frac{V_2}{V_1} \\right)^{1-\\gamma} - 1 \\right). "
},
{
"math_id": 23,
"text": " \\frac{P_2}{P_1} = \\left(\\frac{V_2}{V_1}\\right)^{-\\gamma}, "
},
{
"math_id": 24,
"text": " \\left(\\frac{P_2}{P_1}\\right)^{-\\frac{1}{\\gamma}} = \\frac{V_2}{V_1}. "
},
{
"math_id": 25,
"text": " W = -\\alpha n R T_1 \\left( \\left( \\frac{P_2}{P_1} \\right)^{\\frac{\\gamma-1}{\\gamma}} - 1 \\right). "
},
{
"math_id": 26,
"text": " \\alpha n R (T_2 - T_1) = \\alpha n R T_1 \\left( \\left( \\frac{P_2}{P_1} \\right)^{\\frac{\\gamma-1}{\\gamma}} - 1 \\right). "
},
{
"math_id": 27,
"text": " T_2 - T_1 = T_1 \\left( \\left( \\frac{P_2}{P_1} \\right)^{\\frac{\\gamma-1}{\\gamma}} - 1 \\right), "
},
{
"math_id": 28,
"text": " \\frac{T_2}{T_1} - 1 = \\left( \\frac{P_2}{P_1} \\right)^{\\frac{\\gamma-1}{\\gamma}} - 1, "
},
{
"math_id": 29,
"text": " T_2 = T_1 \\left( \\frac{P_2}{P_1} \\right)^{\\frac{\\gamma-1}{\\gamma}}. "
},
{
"math_id": 30,
"text": "c_1^2=1,\\,\\,c_2^2=0,\\,\\,c_3^2=0,\\,...\\,,"
}
] |
https://en.wikipedia.org/wiki?curid=1419
|
141916
|
Magma (algebra)
|
Algebraic structure with a binary operation
In abstract algebra, a magma, binar, or, rarely, groupoid is a basic kind of algebraic structure. Specifically, a magma consists of a set equipped with a single binary operation that must be closed by definition. No other properties are imposed.
History and terminology.
The term "groupoid" was introduced in 1927 by Heinrich Brandt describing his Brandt groupoid. The term was then appropriated by B. A. Hausmann and Øystein Ore (1937) in the sense (of a set with a binary operation) used in this article. In a couple of reviews of subsequent papers in Zentralblatt, Brandt strongly disagreed with this overloading of terminology. The Brandt groupoid is a groupoid in the sense used in category theory, but not in the sense used by Hausmann and Ore. Nevertheless, influential books in semigroup theory, including Clifford and Preston (1961) and Howie (1995) use groupoid in the sense of Hausmann and Ore. Hollings (2014) writes that the term "groupoid" is "perhaps most often used in modern mathematics" in the sense given to it in category theory.
According to Bergman and Hausknecht (1996): "There is no generally accepted word for a set with a not necessarily associative binary operation. The word "groupoid" is used by many universal algebraists, but workers in category theory and related areas object strongly to this usage because they use the same word to mean 'category in which all morphisms are invertible'. The term "magma" was used by Serre [Lie Algebras and Lie Groups, 1965]." It also appears in Bourbaki's .
Definition.
A magma is a set "M" matched with an operation • that sends any two elements "a", "b" ∈ "M" to another element, "a" • "b" ∈ "M". The symbol • is a general placeholder for a properly defined operation. To qualify as a magma, the set and operation ("M", •) must satisfy the following requirement (known as the "magma" or closure property):
For all "a", "b" in "M", the result of the operation "a" • "b" is also in "M".
And in mathematical notation:
formula_0
If • is instead a partial operation, then ("M", •) is called a partial magma or, more often, a partial groupoid.
Morphism of magmas.
A morphism of magmas is a function "f" : "M" → "N" that maps magma ("M", •) to magma ("N", ∗) that preserves the binary operation:
"f" ("x" • "y") = "f"("x") ∗ "f"("y").
For example, with "M" equal to the positive real numbers and * as the geometric mean, "N" equal to the real number line, and • as the arithmetic mean, a logarithm "f" is a morphism of the magma ("M", *) to ("N", •).
proof: formula_1
Note that these commutative magmas are not associative; nor do they have an identity element. This morphism of magmas has been used in economics since 1863 when W. Stanley Jevons calculated the rate of inflation in 39 commodities in England in his "A Serious Fall in the Value of Gold Ascertained", page 7.
Notation and combinatorics.
The magma operation may be applied repeatedly, and in the general, non-associative case, the order matters, which is notated with parentheses. Also, the operation • is often omitted and notated by juxtaposition:
("a" • ("b" • "c")) • "d" ≡ ("a"("bc"))"d".
A shorthand is often used to reduce the number of parentheses, in which the innermost operations and pairs of parentheses are omitted, being replaced just with juxtaposition: "xy" • "z" ≡ ("x" • "y") • "z". For example, the above is abbreviated to the following expression, still containing parentheses:
("a" • "bc")"d".
A way to avoid completely the use of parentheses is prefix notation, in which the same expression would be written ••"a"•"bcd". Another way, familiar to programmers, is postfix notation (reverse Polish notation), in which the same expression would be written "abc"••"d"•, in which the order of execution is simply left-to-right (no currying).
The set of all possible strings consisting of symbols denoting elements of the magma, and sets of balanced parentheses is called the Dyck language. The total number of different ways of writing "n" applications of the magma operator is given by the Catalan number "Cn". Thus, for example, "C"2 = 2, which is just the statement that ("ab")"c" and "a"("bc") are the only two ways of pairing three elements of a magma with two operations. Less trivially, "C"3 = 5: (("ab")"c")"d", ("a"("bc"))"d", ("ab")("cd"), "a"(("bc")"d"), and "a"("b"("cd")).
There are "n""n"2 magmas with "n" elements, so there are 1, 1, 16, 19683, , ... (sequence in the OEIS) magmas with 0, 1, 2, 3, 4, ... elements. The corresponding numbers of non-isomorphic magmas are 1, 1, 10, 3330, , ... (sequence in the OEIS) and the numbers of simultaneously non-isomorphic and non-antiisomorphic magmas are 1, 1, 7, 1734, , ... (sequence in the OEIS).
Free magma.
A free magma "MX" on a set "X" is the "most general possible" magma generated by "X" (i.e., there are no relations or axioms imposed on the generators; see free object). The binary operation on "MX" is formed by wrapping each of the two operands in parentheses and juxtaposing them in the same order. For example:
"a" • "b" = ("a")("b"),
"a" • ("a" • "b") = ("a")(("a")("b")),
("a" • "a") • "b" = (("a")("a"))("b").
"MX" can be described as the set of non-associative words on "X" with parentheses retained.
It can also be viewed, in terms familiar in computer science, as the magma of full binary trees with leaves labelled by elements of "X". The operation is that of joining trees at the root. It therefore has a foundational role in syntax.
A free magma has the universal property such that if "f" : "X" → "N" is a function from "X" to any magma "N", then there is a unique extension of "f" to a morphism of magmas "f"′
"f"′ : "MX" → "N".
Types of magma.
Magmas are not often studied as such; instead there are several different kinds of magma, depending on what axioms the operation is required to satisfy. Commonly studied types of magma include:
Note that each of divisibility and invertibility imply the cancellation property.
Classification by properties.
A magma ("S", •), with "x", "y", "u", "z" ∈ "S", is called
"xz" implies "y"
"z"
"zx" implies "y"
"z"
Category of magmas.
The category of magmas, denoted Mag, is the category whose objects are magmas and whose morphisms are magma homomorphisms. The category Mag has direct products, and there is an inclusion functor: Set → Med ↪ Mag as trivial magmas, with operations given by projection "x" T "y" = "y".
An important property is that an injective endomorphism can be extended to an automorphism of a magma extension, just the colimit of the (constant sequence of the) endomorphism.
Because the singleton ({*}, *) is the terminal object of Mag, and because Mag is algebraic, Mag is pointed and complete.
References.
<templatestyles src="Reflist/styles.css" />
<templatestyles src="Refbegin/styles.css" />
|
[
{
"math_id": 0,
"text": "a, b \\in M \\implies a \\cdot b \\in M."
},
{
"math_id": 1,
"text": "\\log{\\sqrt{x y} } \\ = \\ \\frac{\\log x + \\log y}{2} "
}
] |
https://en.wikipedia.org/wiki?curid=141916
|
14194283
|
Non-Hausdorff manifold
|
In geometry and topology, it is a usual axiom of a manifold to be a Hausdorff space. In general topology, this axiom is relaxed, and one studies non-Hausdorff manifolds: spaces locally homeomorphic to Euclidean space, but not necessarily Hausdorff.
Examples.
Line with two origins.
The most familiar non-Hausdorff manifold is the line with two origins, or bug-eyed line. This is the quotient space of two copies of the real line,
formula_0 and formula_1 (with formula_2), obtained by identifying points formula_3 and formula_4 whenever formula_5
An equivalent description of the space is to take the real line formula_6 and replace the origin formula_7 with two origins formula_8 and formula_9 The subspace formula_10 retains its usual Euclidean topology. And a local base of open neighborhoods at each origin formula_11 is formed by the sets formula_12 with formula_13 an open neighborhood of formula_7 in formula_14
For each origin formula_11 the subspace obtained from formula_6 by replacing formula_7 with formula_11 is an open neighborhood of formula_11 homeomorphic to formula_14 Since every point has a neighborhood homeomorphic to the Euclidean line, the space is locally Euclidean. In particular, it is locally Hausdorff, in the sense that each point has a Hausdorff neighborhood. But the space is not Hausdorff, as every neighborhood of formula_8 intersects every neighbourhood of formula_9 It is however a T1 space.
The space is second countable.
The space exhibits several phenomena that do not happen in Hausdorff spaces:
The space does not have the homotopy type of a CW-complex, or of any Hausdorff space.
Line with many origins.
The line with many origins is similar to the line with two origins, but with an arbitrary number of origins. It is constructed by taking an arbitrary set formula_20 with the discrete topology and taking the quotient space of formula_21 that identifies points formula_22 and formula_23 whenever formula_24 Equivalently, it can be obtained from formula_6 by replacing the origin formula_7 with many origins formula_25 one for each formula_26 The neighborhoods of each origin are described as in the two origin case.
If there are infinitely many origins, the space illustrates that the closure of a compact set need not be compact in general. For example, the closure of the compact set formula_27 is the set formula_28 obtained by adding all the origins to formula_29, and that closure is not compact. From being locally Euclidean, such a space is locally compact in the sense that every point has a local base of compact neighborhoods. But the origin points do not have any closed compact neighborhood.
Branching line.
Similar to the line with two origins is the branching line.
This is the quotient space of two copies of the real line
formula_30
with the equivalence relation
formula_31
This space has a single point for each negative real number formula_32 and two points formula_33 for every non-negative number: it has a "fork" at zero.
Etale space.
The etale space of a sheaf, such as the sheaf of continuous real functions over a manifold, is a manifold that is often non-Hausdorff. (The etale space is Hausdorff if it is a sheaf of functions with some sort of analytic continuation property.)
Properties.
Because non-Hausdorff manifolds are locally homeomorphic to Euclidean space, they are locally metrizable (but not metrizable in general) and locally Hausdorff (but not Hausdorff in general).
Notes.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "\\R \\times \\{a\\}"
},
{
"math_id": 1,
"text": "\\R \\times \\{b\\}"
},
{
"math_id": 2,
"text": "a \\neq b"
},
{
"math_id": 3,
"text": "(x,a)"
},
{
"math_id": 4,
"text": "(x,b)"
},
{
"math_id": 5,
"text": "x \\neq 0."
},
{
"math_id": 6,
"text": "\\R"
},
{
"math_id": 7,
"text": "0"
},
{
"math_id": 8,
"text": "0_a"
},
{
"math_id": 9,
"text": "0_b."
},
{
"math_id": 10,
"text": "\\R\\setminus\\{0\\}"
},
{
"math_id": 11,
"text": "0_i"
},
{
"math_id": 12,
"text": "(U\\setminus\\{0\\})\\cup\\{0_i\\}"
},
{
"math_id": 13,
"text": "U"
},
{
"math_id": 14,
"text": "\\R."
},
{
"math_id": 15,
"text": "-1"
},
{
"math_id": 16,
"text": "0_b"
},
{
"math_id": 17,
"text": "[-1,0)\\cup\\{0_a\\}"
},
{
"math_id": 18,
"text": "[-1,0)\\cup\\{0_b\\}"
},
{
"math_id": 19,
"text": "[-1,0)"
},
{
"math_id": 20,
"text": "S"
},
{
"math_id": 21,
"text": "\\R\\times S"
},
{
"math_id": 22,
"text": "(x,\\alpha)"
},
{
"math_id": 23,
"text": "(x,\\beta)"
},
{
"math_id": 24,
"text": "x\\ne 0."
},
{
"math_id": 25,
"text": "0_\\alpha,"
},
{
"math_id": 26,
"text": "\\alpha\\in S."
},
{
"math_id": 27,
"text": "A=[-1,0)\\cup\\{0_\\alpha\\}\\cup(0,1]"
},
{
"math_id": 28,
"text": "A\\cup\\{0_\\beta:\\beta\\in S\\}"
},
{
"math_id": 29,
"text": "A"
},
{
"math_id": 30,
"text": "\\R \\times \\{a\\} \\quad \\text{ and } \\quad \\R \\times \\{b\\}"
},
{
"math_id": 31,
"text": "(x, a) \\sim (x, b) \\quad \\text{ if } \\; x < 0."
},
{
"math_id": 32,
"text": "r"
},
{
"math_id": 33,
"text": "x_a, x_b"
}
] |
https://en.wikipedia.org/wiki?curid=14194283
|
14200011
|
BCM theory
|
Neuroscience model of learning
BCM theory, BCM synaptic modification, or the BCM rule, named for Elie Bienenstock, Leon Cooper, and Paul Munro, is a physical theory of learning in the visual cortex developed in 1981.
The BCM model proposes a sliding threshold for long-term potentiation (LTP) or long-term depression (LTD) induction, and states that synaptic plasticity is stabilized by a dynamic adaptation of the time-averaged postsynaptic activity. According to the BCM model, when a pre-synaptic neuron fires, the post-synaptic neurons will tend to undergo LTP if it is in a high-activity state (e.g., is firing at high frequency, and/or has high internal calcium concentrations), or LTD if it is in a lower-activity state (e.g., firing in low frequency, low internal calcium concentrations). This theory is often used to explain how cortical neurons can undergo both LTP or LTD depending on different conditioning stimulus protocols applied to pre-synaptic neurons (usually high-frequency stimulation, or HFS, for LTP, or low-frequency stimulation, LFS, for LTD).
Development.
In 1949, Donald Hebb proposed a working mechanism for memory and computational adaption in the brain now called Hebbian learning, or the maxim that "cells that fire together, wire together". This notion is foundational in the modern understanding of the brain as a neural network, and though not universally true, remains a good first approximation supported by decades of evidence.
However, Hebb's rule has problems, namely that it has no mechanism for connections to get weaker and no upper bound for how strong they can get. In other words, the model is unstable, both theoretically and computationally. Later modifications gradually improved Hebb's rule, normalizing it and allowing for decay of synapses, where no activity or unsynchronized activity between neurons results in a loss of connection strength. New biological evidence brought this activity to a peak in the 1970s, where theorists formalized various approximations in the theory, such as the use of firing frequency instead of potential in determining neuron excitation, and the assumption of ideal and, more importantly, linear synaptic integration of signals. That is, there is no unexpected behavior in the adding of input currents to determine whether or not a cell will fire.
These approximations resulted in the basic form of BCM below in 1979, but the final step came in the form of mathematical analysis to prove stability and computational analysis to prove applicability, culminating in Bienenstock, Cooper, and Munro's 1982 paper.
Since then, experiments have shown evidence for BCM behavior in both the visual cortex and the hippocampus, the latter of which plays an important role in the formation and storage of memories. Both of these areas are well-studied experimentally, but both theory and experiment have yet to establish conclusive synaptic behavior in other areas of the brain. It has been proposed that in the cerebellum, the parallel-fiber to Purkinje cell synapse follows an "inverse BCM rule", meaning that at the time of parallel fiber activation, a high calcium concentration in the Purkinje cell results in LTD, while a lower concentration results in LTP. Furthermore, the biological implementation for synaptic plasticity in BCM has yet to be established.
Theory.
The basic BCM rule takes the form
formula_0
where:
This model is a modified form of the Hebbian learning rule, formula_10, and requires a suitable choice of function formula_11 to avoid the Hebbian problems of instability.
Bienenstock at al. rewrite formula_5 as a function formula_12 where formula_13 is the time average of formula_14. With this modification and discarding the uniform decay the rule takes the vectorial form:
formula_15
The conditions for stable learning are derived rigorously in BCM noting that with formula_16 and with the approximation of the average output formula_17, it is sufficient that
formula_18
formula_19
or equivalently, that the threshold formula_20, where formula_21 and formula_22 are fixed positive constants.
When implemented, the theory is often taken such that
formula_23
where formula_24 is a time constant of selectivity.
The model has drawbacks, as it requires both long-term potentiation and long-term depression, or increases and decreases in synaptic strength, something which has not been observed in all cortical systems. Further, it requires a variable activation threshold and depends strongly on stability of the selected fixed points formula_22 and formula_21. However, the model's strength is that it incorporates all these requirements from independently derived rules of stability, such as normalizability and a decay function with time proportional to the square of the output.
Example.
This example is a particular case of the one at chapter "Mathematical results" of Bienenstock at al. work, assuming formula_25 and formula_26. With these values formula_27 and we decide formula_28 that fulfills the stability conditions said in previous chapter.
Assume two presynaptic neurons that provides inputs formula_29 and formula_30, its activity a repetitive cycle with half of time formula_31 and remainder time formula_32 . formula_13 time average will be the average of formula_14 value in first and second half of a cycle.
Let initial value of weights formula_33. In the first half of time formula_34 and formula_33, the weighted sum formula_14 is equal to 0.095 and we use same value as initial average formula_13. That means formula_35 , formula_36, formula_37. Adding 10% of the derivative to the weights we obtain new ones formula_38.
In next half of time, inputs are formula_32 and weights formula_38. That means formula_39, formula_13 of full cycle is 0.075, formula_40 , formula_41, formula_42. Adding 10% of the derivative to the weights we obtain new ones formula_43.
Repeating previous cycle we obtain, after several hundred of iterations, that stability is reached with formula_44, formula_45 (first half) and formula_46 (remainder time), formula_47, formula_48 , formula_49 and formula_50.
Note how, as predicted, the final weight vector formula_51 has become orthogonal to one of the input patterns, being the final values of formula_14 in both intervals zeros of the function formula_11.
Experiment.
The first major experimental confirmation of BCM came in 1992 in investigating LTP and LTD in the hippocampus. Serena Dudek's experimental work showed qualitative agreement with the final form of the BCM activation function. This experiment was later replicated in the visual cortex, which BCM was originally designed to model. This work provided further evidence of the necessity for a variable threshold function for stability in Hebbian-type learning (BCM or others).
Experimental evidence has been non-specific to BCM until Rittenhouse "et al." confirmed BCM's prediction of synapse modification in the visual cortex when one eye is selectively closed. Specifically,
formula_52
where formula_53 describes the variance in spontaneous activity or noise in the closed eye and formula_54 is time since closure. Experiment agreed with the general shape of this prediction and provided an explanation for the dynamics of monocular eye closure (monocular deprivation) versus binocular eye closure. The experimental results are far from conclusive, but so far have favored BCM over competing theories of plasticity.
Applications.
While the algorithm of BCM is too complicated for large-scale parallel distributed processing, it has been put to use in lateral networks with some success. Furthermore, some existing computational network learning algorithms have been made to correspond to BCM learning.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "\\,\\frac{d m_j(t)}{d t} = \\phi(\\textbf{c}(t))d_j(t)-\\epsilon m_j(t),"
},
{
"math_id": 1,
"text": "m_j"
},
{
"math_id": 2,
"text": "j"
},
{
"math_id": 3,
"text": "d_j"
},
{
"math_id": 4,
"text": "c(t) = \\textbf{w}(t)\\textbf{d}(t) = \\sum_j w_j(t)d_j(t)"
},
{
"math_id": 5,
"text": "\\phi(c)"
},
{
"math_id": 6,
"text": "\\theta_M"
},
{
"math_id": 7,
"text": "\\phi(c)<0 \n"
},
{
"math_id": 8,
"text": "c < \\theta_M"
},
{
"math_id": 9,
"text": "\\epsilon"
},
{
"math_id": 10,
"text": "\\dot{m_j}=c d_j"
},
{
"math_id": 11,
"text": "\\phi"
},
{
"math_id": 12,
"text": "\\phi(c,\\bar{c})"
},
{
"math_id": 13,
"text": "\\bar{c}"
},
{
"math_id": 14,
"text": "c"
},
{
"math_id": 15,
"text": "\\dot{\\mathbf{m}}(t) = \\phi(c(t),\\bar{c}(t))\\mathbf{d}(t)"
},
{
"math_id": 16,
"text": "c(t)=\\textbf{m}(t)\\cdot\\textbf{d}(t)"
},
{
"math_id": 17,
"text": "\\bar{c}(t) \\approx \\textbf{m}(t)\\bar{\\mathbf{d}}"
},
{
"math_id": 18,
"text": "\\,\\sgn\\phi(c,\\bar{c}) = \\sgn\\left(c-\\left(\\frac{\\bar{c}}{c_0}\\right)^p\\bar{c}\\right) ~~ \\textrm{for} ~ c>0, ~ \\textrm{and}"
},
{
"math_id": 19,
"text": "\\,\\phi(0,\\bar{c}) = 0 ~~ \\textrm{for} ~ \\textrm{all} ~ \\bar{c},"
},
{
"math_id": 20,
"text": "\\theta_M(\\bar{c}) = (\\bar{c}/c_0)^p\\bar{c}"
},
{
"math_id": 21,
"text": "p"
},
{
"math_id": 22,
"text": "c_0"
},
{
"math_id": 23,
"text": "\\,\\phi(c,\\bar{c}) = c(c-\\theta_M) ~~~ \\textrm{and} ~~~ \\theta_M = \\bar{c}^2 = \\frac{1}{\\tau}\\int_{-\\infty}^t c^2(t^\\prime)e^{-(t-t^\\prime)/\\tau}d t^\\prime,"
},
{
"math_id": 24,
"text": "\\tau"
},
{
"math_id": 25,
"text": "p=2\n"
},
{
"math_id": 26,
"text": "c_0 = 1"
},
{
"math_id": 27,
"text": "\\theta_M=(\\bar{c}/c_0)^p\\bar{c}=\\bar{c}^3"
},
{
"math_id": 28,
"text": "\\phi(c,\\bar{c}) = c (c - \\theta_M)"
},
{
"math_id": 29,
"text": "d_1"
},
{
"math_id": 30,
"text": "d_2"
},
{
"math_id": 31,
"text": "\\mathbf{d}=(d_1,d_2)=(0.9,0.1)"
},
{
"math_id": 32,
"text": "\\mathbf{d}=(0.2,0.7\n)"
},
{
"math_id": 33,
"text": "\\mathbf{m}=(0.1,0.05)"
},
{
"math_id": 34,
"text": "\\mathbf{d}=(0.9,0.1)"
},
{
"math_id": 35,
"text": "\\theta_M=0.001"
},
{
"math_id": 36,
"text": "\\phi=0.009"
},
{
"math_id": 37,
"text": "\\dot{m}=(0.008,0.001)"
},
{
"math_id": 38,
"text": "\\mathbf{m}=(0.101,0.051)"
},
{
"math_id": 39,
"text": "c=0.055\n"
},
{
"math_id": 40,
"text": "\\theta_M=0.000\n"
},
{
"math_id": 41,
"text": "\\phi=0.003"
},
{
"math_id": 42,
"text": "\\dot{m}=(0.001,0.002)"
},
{
"math_id": 43,
"text": "\\mathbf{m}=(0.110,0.055)"
},
{
"math_id": 44,
"text": "\\mathbf{m}=(3.246,-0.927)"
},
{
"math_id": 45,
"text": "c=\\sqrt{8}=2.828\n"
},
{
"math_id": 46,
"text": "c=0.000\n"
},
{
"math_id": 47,
"text": "\\bar{c}=\\sqrt{8}/2=1.414"
},
{
"math_id": 48,
"text": "\\theta_M = \\sqrt{8} = 2.828\n"
},
{
"math_id": 49,
"text": "\\phi=0.000"
},
{
"math_id": 50,
"text": "\\dot{m}=(0.000,0.000)"
},
{
"math_id": 51,
"text": "m"
},
{
"math_id": 52,
"text": "\\log\\left(\\frac{m_{\\rm closed}(t)}{m_{\\rm closed}(0)}\\right) \\sim -\\overline{n^2}t,"
},
{
"math_id": 53,
"text": "\\overline{n^2}"
},
{
"math_id": 54,
"text": "t"
}
] |
https://en.wikipedia.org/wiki?curid=14200011
|
14204445
|
Mazur manifold
|
Concept in differential topology
In differential topology, a branch of mathematics, a Mazur manifold is a contractible, compact, smooth four-dimensional manifold-with-boundary which is not diffeomorphic to the standard 4-ball. Usually these manifolds are further required to have a handle decomposition with a single formula_0-handle, and a single formula_1-handle; otherwise, they would simply be called contractible manifolds. The boundary of a Mazur manifold is necessarily a homology 3-sphere.
History.
Barry Mazur and Valentin Poenaru discovered these manifolds simultaneously. Akbulut and Kirby showed that the Brieskorn homology spheres formula_2, formula_3 and formula_4 are boundaries of Mazur manifolds, effectively coining the term `Mazur Manifold.' These results were later generalized to other contractible manifolds by Casson, Harer and Stern. One of the Mazur manifolds is also an example of an Akbulut cork which can be used to construct exotic 4-manifolds.
Mazur manifolds have been used by Fintushel and Stern to construct exotic actions of a group of order 2 on the 4-sphere.
Mazur's discovery was surprising for several reasons:
* Every smooth homology sphere in dimension formula_5 is homeomorphic to the boundary of a compact contractible smooth manifold. This follows from the work of Kervaire and the h-cobordism theorem. Slightly more strongly, every smooth homology 4-sphere is diffeomorphic to the boundary of a compact contractible smooth 5-manifold (also by the work of Kervaire). But not every homology 3-sphere is diffeomorphic to the boundary of a contractible compact smooth 4-manifold. For example, the Poincaré homology sphere does not bound such a 4-manifold because the Rochlin invariant provides an obstruction.
* The h-cobordism Theorem implies that, at least in dimensions formula_6 there is a unique contractible formula_7-manifold with simply-connected boundary, where uniqueness is up to diffeomorphism. This manifold is the unit ball formula_8. It's an open problem as to whether or not formula_9 admits an exotic smooth structure, but by the h-cobordism theorem, such an exotic smooth structure, if it exists, must restrict to an exotic smooth structure on formula_10. Whether or not formula_10 admits an exotic smooth structure is equivalent to another open problem, the smooth Poincaré conjecture in dimension four. Whether or not formula_11 admits an exotic smooth structure is another open problem, closely linked to the Schoenflies problem in dimension four.
Mazur's observation.
Let formula_12 be a Mazur manifold that is constructed as formula_13 union a 2-handle. Here is a sketch of Mazur's argument that the double of such a Mazur manifold is formula_10. formula_14 is a contractible 5-manifold constructed as formula_15 union a 2-handle. The 2-handle can be unknotted since the attaching map is a framed knot in the 4-manifold formula_16. So formula_15 union the 2-handle is diffeomorphic to formula_9. The boundary of formula_9 is formula_10. But the boundary of formula_14 is the double of formula_12.
|
[
{
"math_id": 0,
"text": "1"
},
{
"math_id": 1,
"text": "2"
},
{
"math_id": 2,
"text": "\\Sigma(2,5,7) "
},
{
"math_id": 3,
"text": " \\Sigma(3,4,5)"
},
{
"math_id": 4,
"text": "\\Sigma(2,3,13)"
},
{
"math_id": 5,
"text": "n \\geq 5"
},
{
"math_id": 6,
"text": "n \\geq 6"
},
{
"math_id": 7,
"text": "n"
},
{
"math_id": 8,
"text": "D^n"
},
{
"math_id": 9,
"text": "D^5"
},
{
"math_id": 10,
"text": "S^4"
},
{
"math_id": 11,
"text": "D^4"
},
{
"math_id": 12,
"text": "M"
},
{
"math_id": 13,
"text": "S^1 \\times D^3"
},
{
"math_id": 14,
"text": "M \\times [0,1]"
},
{
"math_id": 15,
"text": "S^1 \\times D^4"
},
{
"math_id": 16,
"text": "S^1 \\times S^3"
}
] |
https://en.wikipedia.org/wiki?curid=14204445
|
14204502
|
(a,b,0) class of distributions
|
Term in probability theory
In probability theory, a member of the ("a", "b", 0) class of distributions is any distribution of a discrete random variable "N" whose values are nonnegative integers whose probability mass function satisfies the recurrence formula
formula_0
for some real numbers "a" and "b", where formula_1.
The (a,b,0) class of distributions is also known as the Panjer, the Poisson-type or the Katz family of distributions, and may be retrieved through the Conway–Maxwell–Poisson distribution.
Only the Poisson, binomial and negative binomial distributions satisfy the full form of this relationship. These are also the three discrete distributions among the six members of the natural exponential family with quadratic variance functions (NEF–QVF).
More general distributions can be defined by fixing some initial values of "pj" and applying the recursion to define subsequent values. This can be of use in fitting distributions to empirical data. However, some further well-known distributions are available if the recursion above need only hold for a restricted range of values of "k": for example the logarithmic distribution and the discrete uniform distribution.
The ("a", "b", 0) class of distributions has important applications in actuarial science in the context of loss models.
Properties.
Sundt proved that only the binomial distribution, the Poisson distribution and the negative binomial distribution belong to this class of distributions, with each distribution being represented by a different sign of "a". Furthermore, it was shown by Fackler that there is a universal formula for all three distributions, called the (united) Panjer distribution.
The more usual parameters of these distributions are determined by both "a" and "b". The properties of these distributions in relation to the present class of distributions are summarised in the following table. Note that formula_2 denotes the probability generating function.
Note that the Panjer distribution reduces to the Poisson distribution in the limit case formula_3; it coincides with the negative binomial distribution for positive, finite real numbers formula_4, and it equals the binomial distribution for negative integers formula_5.
Plotting.
An easy way to quickly determine whether a given sample was taken from a distribution from the ("a","b",0) class is by graphing the ratio of two consecutive observed data (multiplied by a constant) against the "x"-axis.
By multiplying both sides of the recursive formula by formula_6, you get
formula_7
which shows that the left side is obviously a linear function of formula_6. When using a sample of formula_8 data, an approximation of the formula_9's need to be done. If formula_10 represents the number of observations having the value formula_6, then formula_11 is an unbiased estimator of the true formula_9.
Therefore, if a linear trend is seen, then it can be assumed that the data is taken from an ("a","b",0) distribution. Moreover, the slope of the function would be the parameter formula_12, while the ordinate at the origin would be formula_13.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": " \\frac{p_k}{p_{k-1}} = a + \\frac{b}{k}, \\qquad k = 1, 2, 3, \\dots "
},
{
"math_id": 1,
"text": "p_k = P(N = k)"
},
{
"math_id": 2,
"text": "W_N(x)\\,"
},
{
"math_id": 3,
"text": "\\alpha \\rightarrow \\pm\\infty"
},
{
"math_id": 4,
"text": "\\alpha\\in \\mathbb{R}_{>0}"
},
{
"math_id": 5,
"text": " -\\alpha \\in \\mathbb{Z}"
},
{
"math_id": 6,
"text": "k"
},
{
"math_id": 7,
"text": "k \\, \\frac{p_k}{p_{k-1}} = ak + b,"
},
{
"math_id": 8,
"text": "n"
},
{
"math_id": 9,
"text": "p_k"
},
{
"math_id": 10,
"text": "n_k"
},
{
"math_id": 11,
"text": "\\hat{p}_k = \\frac{n_k}{n}"
},
{
"math_id": 12,
"text": "a"
},
{
"math_id": 13,
"text": "b"
}
] |
https://en.wikipedia.org/wiki?curid=14204502
|
14212420
|
16-hydroxysteroid epimerase
|
Class of enzymes
In enzymology, a 16-hydroxysteroid epimerase (EC 5.1.99.2) is an enzyme that catalyzes the chemical reaction
16alpha-hydroxysteroid formula_0 16beta-hydroxysteroid
Hence, this enzyme has one substrate, 16alpha-hydroxysteroid, and one product, 16beta-hydroxysteroid.
This enzyme belongs to the family of isomerases, specifically those racemases and epimerases acting on other compounds. The systematic name of this enzyme class is 16-hydroxysteroid 16-epimerase.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "\\rightleftharpoons"
}
] |
https://en.wikipedia.org/wiki?curid=14212420
|
14212440
|
2-acetolactate mutase
|
Class of enzymes
In enzymology, a 2-acetolactate mutase (EC 5.4.99.3) is an enzyme that catalyzes the chemical reaction
2-acetolactate formula_0 3-hydroxy-3-methyl-2-oxobutanoate
Hence, this enzyme has one substrate, 2-acetolactate, and one product, 3-hydroxy-3-methyl-2-oxobutanoate.
This enzyme belongs to the family of isomerases, specifically those intramolecular transferases transferring other groups. The systematic name of this enzyme class is 2-acetolactate methylmutase. Other names in common use include acetolactate mutase, and acetohydroxy acid isomerase. This enzyme participates in valine, leucine and isoleucine biosynthesis.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "\\rightleftharpoons"
}
] |
https://en.wikipedia.org/wiki?curid=14212440
|
14212456
|
2-aminohexano-6-lactam racemase
|
Class of enzymes
In enzymology, a 2-aminohexano-6-lactam racemase (EC 5.1.1.15) is an enzyme that catalyzes the chemical reaction
L-2-aminohexano-6-lactam formula_0 D-2-aminohexano-6-lactam
Hence, this enzyme has one substrate, L-2-aminohexano-6-lactam, and one product, D-2-aminohexano-6-lactam.
This enzyme belongs to the family of isomerases, specifically those racemases and epimerases acting on amino acids and derivatives. The systematic name of this enzyme class is 2-aminohexano-6-lactam racemase. This enzyme is also called alpha-amino-epsilon-caprolactam racemase.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "\\rightleftharpoons"
}
] |
https://en.wikipedia.org/wiki?curid=14212456
|
14212472
|
2-chloro-4-carboxymethylenebut-2-en-1,4-olide isomerase
|
Class of enzymes
In enzymology, a 2-chloro-4-carboxymethylenebut-2-en-1,4-olide isomerase (EC 5.2.1.10) is an enzyme that catalyzes the chemical reaction
cis-2-chloro-4-carboxymethylenebut-2-en-1,4-olide formula_0 trans-2-chloro-4-carboxymethylenebut-2-en-1,4-olide
Hence, this enzyme has one substrate, cis-2-chloro-4-carboxymethylenebut-2-en-1,4-olide, and one product, trans-2-chloro-4-carboxymethylenebut-2-en-1,4-olide.
This enzyme belongs to the family of isomerases, specifically cis-trans isomerases. The systematic name of this enzyme class is 2-chloro-4-carboxymethylenebut-2-en-1,4-olide cis-trans-isomerase. Other names in common use include 2-chlorocarboxymethylenebutenolide isomerase, and chlorodienelactone isomerase. This enzyme participates in 1,4-dichlorobenzene degradation.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "\\rightleftharpoons"
}
] |
https://en.wikipedia.org/wiki?curid=14212472
|
14212496
|
2-methyleneglutarate mutase
|
Class of enzymes
In enzymology, a 2-methyleneglutarate mutase (EC 5.4.99.4) is an enzyme that catalyzes the chemical reaction
2-methyleneglutarate formula_0 2-methylene-3-methylsuccinate
Hence, this enzyme has one substrate, 2-methyleneglutarate, and one product, 2-methylene-3-methylsuccinate.
This enzyme belongs to the family of isomerases, specifically those intramolecular transferases transferring other groups. The systematic name of this enzyme class is 2-methyleneglutarate carboxy-methylenemethylmutase. This enzyme is also called alpha-methyleneglutarate mutase. This enzyme participates in c5-branched dibasic acid metabolism. It employs one cofactor, cobamide.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "\\rightleftharpoons"
}
] |
https://en.wikipedia.org/wiki?curid=14212496
|
1421250
|
Roger Apéry
|
French mathematician (1916-1994)
Roger Apéry (; 14 November 1916, Rouen – 18 December 1994, Caen) was a French mathematician most remembered for Apéry's theorem, which states that "ζ"(3) is an irrational number. Here, "ζ"("s") denotes the Riemann zeta function.
Biography.
Apéry was born in Rouen in 1916 to a French mother and Greek father. His childhood was spent in Lille until 1926, when the family moved to Paris, where he studied at the Lycée Ledru-Rollin and the Lycée Louis-le-Grand. He was admitted at the École normale supérieure in 1935. His studies were interrupted at the start of World War II; he was mobilized in September 1939, taken prisoner of war in June 1940, repatriated with pleurisy in June 1941, and hospitalized until August 1941. He wrote his doctoral thesis in algebraic geometry under the direction of Paul Dubreil and René Garnier in 1947.
In 1947 Apéry was appointed Maître de conférences (lecturer) at the University of Rennes. In 1949 he was appointed Professor at the University of Caen, where he remained until his retirement.
In 1979 he published an unexpected proof of the irrationality of "ζ"(3), which is the sum of the inverses of the cubes of the positive integers. An indication of the difficulty is that the corresponding problem for other odd powers remains unsolved. Nevertheless, many mathematicians have since worked on the so-called Apéry sequences to seek alternative proofs that might apply to other odd powers (Frits Beukers, Alfred van der Poorten, Marc Prévost, Keith Ball, Tanguy Rivoal, Wadim Zudilin, and others).
Apéry was active in politics and for a few years in the 1960s was president of the Calvados Radical Party of the Left. He abandoned politics after the reforms instituted by Edgar Faure after the 1968 revolt, when he realised that university life was running against the tradition he had always upheld.
Personal life.
Apéry married in 1947 and had three sons, including mathematician François Apéry. His first marriage ended in divorce in 1971. He then remarried in 1972 and divorced in 1977.
In 1994, Apéry died from Parkinson's disease after a long illness in Caen. He was buried next to his parents at the Père Lachaise Cemetery in Paris. His tombstone has a mathematical inscription stating his theorem.
formula_0
|
[
{
"math_id": 0,
"text": " 1 + \\frac{1}{8} + \\frac{1}{27} + \\frac{1}{64} + \\cdots \\neq \\frac{p}{q} "
}
] |
https://en.wikipedia.org/wiki?curid=1421250
|
14212518
|
3-carboxy-cis,cis-muconate cycloisomerase
|
InterPro Family
In enzymology, a 3-carboxy-cis,cis-muconate cycloisomerase (EC 5.5.1.2) is an enzyme that catalyzes the chemical reaction
2-carboxy-2,5-dihydro-5-oxofuran-2-acetate formula_0 cis,cis-butadiene-1,2,4-tricarboxylate
Hence, this enzyme has one substrate, 2-carboxy-2,5-dihydro-5-oxofuran-2-acetate, and one product, cis,cis-butadiene-1,2,4-tricarboxylate.
This enzyme belongs to the family of isomerases, specifically intramolecular lyases. The systematic name of this enzyme class is 2-carboxy-2,5-dihydro-5-oxofuran-2-acetate lyase (decyclizing). Other names in common use include beta-carboxymuconate lactonizing enzyme, and 3-carboxymuconolactone hydrolase. This enzyme participates in benzoate degradation via hydroxylation.
Structural studies.
As of late 2007, only one structure has been solved for this class of enzymes, with the PDB accession code 1Q5N.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "\\rightleftharpoons"
}
] |
https://en.wikipedia.org/wiki?curid=14212518
|
14212533
|
3-(hydroxyamino)phenol mutase
|
Class of enzymes
In enzymology, a 3-(hydroxyamino)phenol mutase (EC 5.4.4.3) is an enzyme that catalyzes the chemical reaction
3-hydroxyaminophenol formula_0 aminohydroquinone
Hence, this enzyme has one substrate, 3-hydroxyaminophenol, and one product, aminohydroquinone.
This enzyme belongs to the family of isomerases, specifically those intramolecular transferases transferring hydroxy groups. The systematic name of this enzyme class is 3-(hydroxyamino)phenol hydroxymutase. Other names in common use include 3-hydroxylaminophenol mutase, and 3HAP mutase.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "\\rightleftharpoons"
}
] |
https://en.wikipedia.org/wiki?curid=14212533
|
14212559
|
3-hydroxybutyryl-CoA epimerase
|
Class of enzymes
In enzymology, a 3-hydroxybutyryl-CoA epimerase (EC 5.1.2.3) is an enzyme that catalyzes the chemical reaction
(S)-3-hydroxybutanoyl-CoA formula_0 (R)-3-hydroxybutanoyl-CoA
Hence, this enzyme has one substrate, (S)-3-hydroxybutanoyl-CoA, and one product, (R)-3-hydroxybutanoyl-CoA.
This enzyme belongs to the family of isomerases, specifically those racemases and epimerases acting on hydroxy acids and derivatives. The systematic name of this enzyme class is 3-hydroxybutanoyl-CoA 3-epimerase. Other names in common use include 3-hydroxybutyryl coenzyme A epimerase, and 3-hydroxyacyl-CoA epimerase. This enzyme participates in fatty acid metabolism and butanoate metabolism.
Structural studies.
As of late 2007, four structures have been solved for this class of enzymes, with PDB accession codes 1WDK, 1WDL, 1WDM, and 2D3T.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "\\rightleftharpoons"
}
] |
https://en.wikipedia.org/wiki?curid=14212559
|
14212575
|
4-carboxymethyl-4-methylbutenolide mutase
|
Class of enzymes
In enzymology, a 4-carboxymethyl-4-methylbutenolide mutase (EC 5.4.99.14) is an enzyme that catalyzes the chemical reaction
4-carboxymethyl-4-methylbut-2-en-1,4-olide formula_0 4-carboxymethyl-3-methylbut-2-en-1,4-olide
Hence, this enzyme has one substrate, 4-carboxymethyl-4-methylbut-2-en-1,4-olide, and one product, 4-carboxymethyl-3-methylbut-2-en-1,4-olide.
This enzyme belongs to the family of isomerases, specifically those intramolecular transferases transferring other groups. The systematic name of this enzyme class is 4-carboxymethyl-4-methylbut-2-en-1,4-olide methylmutase. Other names in common use include 4-methyl-2-enelactone isomerase, 4-methylmuconolactone methylisomerase, and 4-methyl-3-enelactone methyl isomerase.
References.
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{
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14212596
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4-deoxy-L-threo-5-hexosulose-uronate ketol-isomerase
|
Enzyme
In enzymology, a 4-deoxy-L-threo-5-hexosulose-uronate ketol-isomerase (EC 5.3.1.17) is an enzyme that catalyzes the chemical reaction
4-deoxy-L-threo-5-hexosulose uronate formula_0 3-deoxy-D-glycero-2,5-hexodiulosonate
Hence, this enzyme has one substrate, 4-deoxy-L-threo-5-hexosulose uronate, and one product, 3-deoxy-D-glycero-2,5-hexodiulosonate.
This enzyme belongs to the family of isomerases, specifically those intramolecular oxidoreductases interconverting aldoses and ketoses. The systematic name of this enzyme class is 4-deoxy-L-threo-5-hexosulose-uronate aldose-ketose-isomerase. This enzyme is also called 4-deoxy-L-threo-5-hexulose uronate isomerase. This enzyme participates in pentose and glucuronate interconversions.
Structural studies.
As of late 2007, 3 structures have been solved for this class of enzymes, with PDB accession codes 1X8M, 1XRU, and 1YWK.
References.
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[
{
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https://en.wikipedia.org/wiki?curid=14212596
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14212622
|
4-hydroxyproline epimerase
|
Class of enzymes
In enzymology, a 4-hydroxyproline epimerase (EC 5.1.1.8) is an enzyme that catalyzes the chemical reaction
"trans"-4-hydroxy-L-proline formula_0 "cis"-4-hydroxy-D-proline
Hence, this enzyme has one substrate, trans-4-hydroxy-L-proline, and one product, cis-4-hydroxy-D-proline.
This enzyme belongs to the family of isomerases, specifically those racemases and epimerases acting on amino acids and derivatives. The systematic name of this enzyme class is 4-hydroxyproline 2-epimerase. Other names in common use include hydroxyproline epimerase, hydroxyproline 2-epimerase, and L-hydroxyproline epimerase. This enzyme participates in arginine and proline metabolism.
References.
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14212642
|
5-(carboxyamino)imidazole ribonucleotide mutase
|
Class of enzymes
In enzymology, a 5-(carboxyamino)imidazole ribonucleotide mutase (EC 5.4.99.18) is an enzyme that catalyzes the chemical reaction
5-carboxyamino-1-(5-phospho-D-ribosyl)imidazole formula_0 5-amino-1-(5-phospho-D-ribosyl)imidazole-4-carboxylate
Hence, this enzyme has one substrate, 5-carboxyamino-1-(5-phospho-D-ribosyl)imidazole, and one product, 5-amino-1-(5-phospho-D-ribosyl)imidazole-4-carboxylate.
This enzyme belongs to the family of isomerases, specifically those intramolecular transferases transferring other groups. The systematic name of this enzyme class is 5-carboxyamino-1-(5-phospho-D-ribosyl)imidazole carboxymutase. Other names in common use include N5-CAIR mutase, PurE, N5-carboxyaminoimidazole ribonucleotide mutase, and class I PurE.
References.
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https://en.wikipedia.org/wiki?curid=14212642
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14212666
|
5-carboxymethyl-2-hydroxymuconate Delta-isomerase
|
InterPro Family
In enzymology, a 5-carboxymethyl-2-hydroxymuconate Delta-isomerase (EC 5.3.3.10) is an enzyme that catalyzes the chemical reaction
5-carboxymethyl-2-hydroxymuconate formula_0 5-carboxy-2-oxohept-3-enedioate
Hence, this enzyme has one substrate, 5-carboxymethyl-2-hydroxymuconate, and one product, 5-carboxy-2-oxohept-3-enedioate.
This enzyme belongs to the family of isomerases, specifically those intramolecular oxidoreductases transposing C=C bonds. The systematic name of this enzyme class is 5-carboxymethyl-2-hydroxymuconate Delta2,Delta4-2-oxo,Delta3-isomerase. This enzyme participates in tyrosine metabolism and benzoate degradation via hydroxylation.
Structural studies.
As of late 2007, 4 structures have been solved for this class of enzymes, with PDB accession codes 1GTT, 1I7O, 1WZO, and 2DFU.
References.
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{
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https://en.wikipedia.org/wiki?curid=14212666
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14212682
|
Acetoin racemase
|
Class of enzymes
In enzymology, an acetoin racemase (EC 5.1.2.4) is an enzyme that catalyzes the chemical reaction
(S)-acetoin formula_0 (R)-acetoin
This enzyme belongs to the family of isomerases, specifically those racemases and epimerases acting on hydroxy acids and derivatives. The systematic name of this enzyme class is acetoin racemase. This enzyme is also called acetylmethylcarbinol racemase. This enzyme participates in butanoate metabolism.
References.
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14212697
|
Aconitate Delta-isomerase
|
Class of enzymes
In enzymology, an aconitate Δ-isomerase (EC 5.3.3.7) is an enzyme that catalyzes the chemical reaction
trans-aconitate formula_0 cis-aconitate
Hence, this enzyme has one substrate, trans-aconitate, and one product, cis-aconitate.
This enzyme belongs to the family of isomerases, specifically those intramolecular oxidoreductases transposing C=C bonds. The systematic name of this enzyme class is aconitate Delta2-Delta3-isomerase. This enzyme is also called aconitate isomerase.
References.
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14212722
|
ADP-glyceromanno-heptose 6-epimerase
|
In enzymology, an ADP-L-glycero-D-manno-heptose 6-epimerase (EC 5.1.3.20) is an enzyme that catalyzes the chemical reaction
ADP-D-glycero-D-manno-heptose formula_0 ADP-L-glycero-D-manno-heptose
Hence, this enzyme has one substrate, ADP-D-glycero-D-manno-heptose, and one product, ADP-L-glycero-D-manno-heptose.
This enzyme belongs to the family of isomerases, specifically those racemases and epimerases acting on carbohydrates and derivatives. The systematic name of this enzyme class is ADP-L-glycero-D-manno-heptose 6-epimerase. This enzyme participates in lipopolysaccharide biosynthesis. It employs one cofactor, NADP+ in a direct oxidation mechanism.
Structural studies.
As of 2007, only one structure has been solved for this class of enzymes, with the PDB accession code 1EQ2.
References.
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14212744
|
Alanine racemase
|
In enzymology, an alanine racemase (EC 5.1.1.1) is an enzyme that catalyzes the chemical reaction
L-alanine formula_0 D-alanine
Hence, this enzyme has one substrate, L-alanine, and one product, D-alanine.
This enzyme belongs to the family of isomerases, specifically those racemases and epimerases acting on amino acids and derivatives. The systematic name of this enzyme class is alanine racemase. This enzyme is also called L-alanine racemase. This enzyme participates in alanine and aspartate metabolism and D-alanine metabolism. It employs one cofactor, pyridoxal phosphate. At least two compounds, 3-Fluoro-D-alanine and D-Cycloserine are known to inhibit this enzyme.
The D-alanine produced by alanine racemase is used for peptidoglycan biosynthesis. Peptidoglycan is found in the cell walls of all bacteria, including many which are harmful to humans. The enzyme is absent in higher eukaryotes but found everywhere in prokaryotes, making alanine racemase a great target for antimicrobial drug development. Alanine racemase can be found in some invertebrates.
Bacteria can have one (alr gene) or two alanine racemase genes. Bacterial species with two genes for alanine racemase have one that is continually expressed and one that is inducible, which makes it difficult to target both genes for drug studies. However, knockout studies have shown that without the alr gene being expressed, the bacteria would need an external source of D-alanine in order to survive. Therefore, the alr gene is a feasible target for antimicrobial drugs.
Alanine racemase is the only known protein, as of 2002, to contain a left-handed α-helix of 5 amino acids, the longest left-handed α-helix found up until at that point.
Structural studies.
To catalyze the interconversion of D and L alanine, Alanine racemase must position residues capable of exchanging protons on either side of the alpha carbon of alanine. Structural studies of enzyme-inhibitor complexes suggest that Tyrosine 265 and Lysine 39 are these residues. The alpha-proton of the L-enantiomer is oriented toward Tyr265 and the alpha proton of the D-enantiomer is oriented toward Lys39 (Figure 1). The distance between the enzyme residues and the enantiomers is 3.5 Å and 3.6 Å respectively. Structural studies of enzyme complexes with a synthetic L-alanine analog, a tight binding inhibitor and propionate further validate that Tyr265 and Lys39 are catalytic bases for the reaction.
The PLP-L-Ala and PLP-D-Ala complexes are almost superimposability. The regions that do not overlap are the arms connected the pyridine ring of PLP and the alpha carbon of alanine. An interaction between both the phosphate oxygen and pyridine nitrogen atoms to the 5’phosphopyridoxyl region of PLP-Ala probably creates tight binding to the enzyme.
The structure of alanine racemase from "Bacillus stearothermophilus" (Geobacillus stearothermophilus) was determined by X-ray crystallography to a resolution of 1.9 A. The alanine racemase monomer is composed of two domains, an eight-stranded alpha/beta barrel at the N terminus, and a C-terminal domain essentially composed of beta-strand. A model of the two domain structure is shown in Figure 2. The N-terminal domain is also found in the PROSC (proline synthetase co-transcribed bacterial homolog) family of proteins, which are not known to have alanine racemase activity. The pyridoxal 5'-phosphate (PLP) cofactor lies in and above the mouth of the alpha/beta barrel and is covalently linked via an aldimine linkage to a lysine residue, which is at the C terminus of the first beta-strand of the alpha/beta barrel.
Proposed mechanism.
Reaction mechanisms are difficult to fully prove by experiment. The traditional mechanism attributed to an alanine racemase reaction is that of a two-base mechanism with a PLP-stabilized carbanion intermediate. PLP is used as an electron sink stabilize the negative charge resulting from the deprotonation of the alpha carbon. The two based mechanism favors reaction specificity compared to a one base mechanism. The second catalytic residue is pre-positioned to donate a proton quickly after a carbanionic intermediate is formed and thus reduces the chance of alternative reactions occurring. There are two potential conflicts with this traditional mechanism, as identified by Watanabe et al. First, Arg219 forms a hydrogen bond with pyridine nitrogen of PLP. The arginine group has a pKa of about 12.6 and is therefore unlikely to protonate the pyridine. Normally in PLP reactions an acidic amino acid residue such as a carboxylic acid group, with a pKa of about 5, protonates the pyridine ring. The protonation of the pyridine nitrogen allows the nitrogen to accept additional negative charge. Therefore, due to the Arg219, the PLP-stabilized carbanion intermediate is less likely to form. Another problem identified was the need for another basic residue to return Lys39 and Tyr265 back to their protonated and unprotonated forms for L-alanine and vice versa for D-alanine. Watanabe et al. found no amino acid residues or water molecules, other than the carboxylate group of PLP-Ala, to be close enough (within 4.5A) to protonate or deprotonate Lys or Tyr. This is shown in Figure 3.
Based on the crystal structures of N-(5’-phosphopyridoxyl) L- alanine (PKP-L-Ala ( and N-(5’-phosphopyridoxyl) D-alanine (PLP-D-Ala)
Watanabe et al. proposed an alternative mechanism in 2002, as seen in the figure 4. In this mechanism the carboxylate oxygen atoms of PLP-Ala directly participates in catalysis by mediating proton transfer between Lys39 and Tyr265. The crystallization structure identified that the carboxylate oxygen of PLP-L-Ala to the OH of Tyr265 was only 3.6A and the carboxylate oxygen of PLP-L-Ala to the nitrogen of Lys39 was only 3.5A. Therefore, both were close enough to cause a reaction.
This mechanism is supported by mutations of Arg219. Mutations changing Arg219 to a carboxylate result in a quinonoid intermediate being detected whereas none was detected with arginine. The arginine intermediate has much more free energy, is more unstable, than the acidic residue mutants. The destabilization of the intermediate promotes specificity of the reaction.
References.
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14212770
|
Aldose 1-epimerase
|
In enzymology, an aldose 1-epimerase (EC 5.1.3.3) is an enzyme that catalyzes the chemical reaction
alpha-D-glucose formula_0 beta-D-glucose
Hence, this enzyme has one substrate, alpha-D-glucose, and one product, beta-D-glucose.
This enzyme belongs to the family of isomerases, specifically those racemases and epimerases acting on carbohydrates and derivatives. The systematic name of this enzyme class is aldose 1-epimerase. Other names in common use include mutarotase, and aldose mutarotase. This enzyme participates in glycolysis and gluconeogenesis.
Structural studies.
As of late 2007, 23 structures have been solved for this class of enzymes, with PDB accession codes 1L7J, 1L7K, 1LUR, 1MMU, 1MMX, 1MMY, 1MMZ, 1MN0, 1NS0, 1NS2, 1NS4, 1NS7, 1NS8, 1NSM, 1NSR, 1NSS, 1NSU, 1NSV, 1NSX, 1NSZ, 1SNZ, 1SO0, and 1YGA.
References.
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https://en.wikipedia.org/wiki?curid=14212770
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14212794
|
Allantoin racemase
|
In enzymology, an allantoin racemase (EC 5.1.99.3) is an enzyme that catalyzes the chemical reaction
(S)(+)-allantoin formula_0 (R)(−)-allantoin
Hence, this enzyme has one substrate, (S)(+)-allantoin, and one product, (R)(−)-allantoin.
This enzyme belongs to the family of isomerases, specifically those racemases and epimerases acting on other compounds. The systematic name of this enzyme class is allantoin racemase. This enzyme participates in purine metabolism.
References.
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14212831
|
Alpha-methylacyl-CoA racemase
|
Protein-coding gene in the species Homo sapiens
α-Methylacyl-CoA racemase (AMACR, EC 5.1.99.4) is an enzyme that in humans is encoded by the "AMACR" gene. AMACR catalyzes the following chemical reaction:
(2"R")-2-methylacyl-CoA formula_0 (2"S")-2-methylacyl-CoA
In mammalian cells, the enzyme is responsible for converting (2"R")-methylacyl-CoA esters to their (2"S")-methylacyl-CoA epimers and known substrates, including coenzyme A esters of pristanic acid (mostly derived from phytanic acid, a 3-methyl branched-chain fatty acid that is abundant in the diet) and bile acids derived from cholesterol. This transformation is required in order to degrade (2"R")-methylacyl-CoA esters by β-oxidation, which process requires the (2"S")-epimer. The enzyme is known to be localised in peroxisomes and mitochondria, both of which are known to β-oxidize 2-methylacyl-CoA esters.
Nomenclature.
This enzyme belongs to the family of isomerases, specifically the racemases and epimerases which act on other compounds. The systematic name of this enzyme class is 2-methylacyl-CoA 2-epimerase. In vitro experiments with the human enzyme AMACR 1A show that both (2"S")- and (2"R")-methyldecanoyl-CoA esters are substrates and are converted by the enzyme with very similar efficiency. Prolonged incubation of either substrate with the enzyme establishes an equilibrium with both substrates or products present in a near 1:1 ratio. The mechanism of the enzyme requires removal of the α-proton of the 2-methylacyl-CoA to form a deprotonated intermediate (which is probably the enol or enolate) followed by non-sterespecific reprotonation. Thus either epimer is converted into a near 1:1 mixture of both isomers upon full conversion of the substrate.
Clinical significance.
Both decreased and increased levels of the enzyme in humans are linked with diseases.
Neurological diseases.
Reduction of the protein level or activity results in the accumulation of (2R)-methyl fatty acids such as bile acids which causes neurological symptoms. The symptoms are similar to those of adult Refsum disease and usually appear in the late teens or early twenties.
The first documented cases of AMACR deficiency in adults were reported in 2000. This deficiency falls within a class of disorders called peroxisome biogenesis disorders (PBDs), although it is quite different from other peroxisomal disorders and does not share classic Refsum disorder symptoms. The deficiency causes an accumulation of pristanic acid, dihydroxycholestanoic acid (DHCA) and trihydroxycholestanoic acid (THCA) and to a lesser extent phytanic acid. This phenomenon was verified in 2002, when researchers reported of a certain case, "His condition would have been missed if they hadn't measured the pristanic acid concentration."
AMACR deficiency can cause mental impairment, confusion, learning difficulties, and liver damage. It can be treated by dietary elimination of pristanic and phytanic acid through reduced intake of dairy products and meats such as beef, lamb, and chicken. Compliance to the diet is low, however, because of eating habits and loss of weight.
Cancer.
Increased levels of AMACR protein concentration and activity are associated with prostate cancer, and the enzyme is used widely as a biomarker (known in cancer literature as P504S) in biopsy tissues. Around 10 different variants of human AMACR have been identified from prostate cancer tissues, which variants arise from alternative mRNA splicing. Some of these splice variants lack catalytic residues in the active site or have changes in the C-terminus, which is required for dimerisation. Increased levels of AMACR are also associated with some breast, colon, and other cancers, but it is unclear exactly what the role of AMACR is in these cancers.
Antibodies to AMACR are used in immunohistochemistry to demonstrate prostate carcinoma, since the enzyme is greatly overexpressed in this type of tumour.
Ibuprofen metabolism.
The enzyme is also involved in a chiral inversion pathway which converts ibuprofen, a member of the 2-arylpropionic acid (2-APA) non-steroidal anti-inflammatory drug family (NSAIDs), from the "R"-enantiomer to the "S"-enantiomer. The pathway is uni-directional because only "R"-ibuprofen can be converted into ibuprofenoyl-CoA, which is then epimerized by AMACR. Conversion of "S"-ibuprofenoyl-CoA to "S"-ibuprofen is assumed to be performed by one of the many human acyl-CoA thioesterase enzymes (ACOTs). The reaction is of pharmacological importance because ibuprofen is typically used as a racemic mixture, and the drug is converted to the "S"-isomer upon uptake, which inhibits the activity of the cyclo-oxygenase enzymes and induces an anti-inflammatory effect. Human AMACR 1A has been demonstrated to epimerise other 2-APA-CoA esters, suggesting a common chiral inversion pathway for this class of drugs.
References.
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Further reading.
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14212847
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Alpha-pinene-oxide decyclase
|
In enzymology, an α-pinene-oxide decyclase (EC 5.5.1.10) is an enzyme that catalyzes the chemical reaction
α-pinene oxide formula_0 (Z)-2-methyl-5-isopropylhexa-2,5-dienal
Hence, this enzyme has one substrate, α-pinene oxide, and one product, (Z)-2-methyl-5-isopropylhexa-2,5-dienal.
This enzyme belongs to the family of isomerases, specifically the class of intramolecular lyases. The systematic name of this enzyme class is α-pinene-oxide lyase (decyclizing). This enzyme is also called α-pinene oxide lyase. This enzyme participates in limonene and pinene degradation.
References.
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14212870
|
Amino-acid racemase
|
In enzymology, an amino-acid racemase (EC 5.1.1.10) is an enzyme that catalyzes the chemical reaction
an L-amino acid formula_0 a D-amino acid
Hence, this enzyme has one substrate, L-amino acid, and one product, D-amino acid.
This enzyme belongs to the family of isomerases, specifically those racemases and epimerases acting on amino acids and derivatives. The systematic name of this enzyme class is amino-acid racemase. This enzyme is also called L-amino acid racemase. This enzyme participates in 4 metabolic pathways: glycine, serine and threonine metabolism, cysteine metabolism, D-glutamine and D-glutamate metabolism, and D-arginine and D-ornithine metabolism. It employs one cofactor, pyridoxal phosphate.
Structural studies.
As of late 2007, 5 structures have been solved for this class of enzymes, with PDB accession codes 2FKP, 2GGG, 2GGH, 2GGI, and 2GGJ.
References.
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14212898
|
Arabinose-5-phosphate isomerase
|
In enzymology, an arabinose-5-phosphate isomerase (EC 5.3.1.13) is an enzyme that catalyzes the chemical reaction
D-arabinose 5-phosphate formula_0 D-ribulose 5-phosphate
Hence, this enzyme has one substrate, D-arabinose 5-phosphate, and one product, D-ribulose 5-phosphate.
This enzyme belongs to the family of isomerases, specifically those intramolecular oxidoreductases interconverting aldoses and ketoses. The systematic name of this enzyme class is D-arabinose-5-phosphate aldose-ketose-isomerase. Other names in common use include arabinose phosphate isomerase, phosphoarabinoisomerase, and D-arabinose-5-phosphate ketol-isomerase.
References.
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14212916
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Arabinose isomerase
|
In enzymology, an arabinose isomerase (EC 5.3.1.3) is an enzyme that catalyzes the chemical reaction
D-arabinose formula_0 D-ribulose
Hence, this enzyme has one substrate, D-arabinose, and one product, D-ribulose.
This enzyme belongs to the family of isomerases, specifically those intramolecular oxidoreductases interconverting aldoses and ketoses. The systematic name of this enzyme class is D-arabinose aldose-ketose-isomerase. Other names in common use include D-arabinose(L-fucose) isomerase, D-arabinose isomerase, L-fucose isomerase, and D-arabinose ketol-isomerase.
Structural studies.
As of late 2007, only one structure has been solved for this class of enzymes, with the PDB accession code 1FUI.
References.
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14212938
|
Arginine racemase
|
In enzymology, an arginine racemase (EC 5.1.1.9) is an enzyme that catalyzes the chemical reaction
L-arginine formula_0 D-arginine
Hence, this enzyme has one substrate, L-arginine, and one product, D-arginine.
This enzyme belongs to the family of isomerases, specifically those racemases and epimerases acting on amino acids and derivatives. The systematic name of this enzyme class is arginine racemase. This enzyme participates in 3 metabolic pathways: lysine degradation, D-glutamine and D-glutamate metabolism, and D-arginine and D-ornithine metabolism. It employs one cofactor, pyridoxal phosphate.
References.
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14212965
|
Ascopyrone tautomerase
|
In enzymology, an ascopyrone tautomerase (EC 5.3.2.7) is an enzyme that catalyzes the chemical reaction
1,5-anhydro-4-deoxy-D-glycero-hex-3-en-2-ulose formula_0 1,5-anhydro-4-deoxy-D-glycero-hex-1-en-3-ulose
Hence, this enzyme has one substrate, 1,5-anhydro-4-deoxy-D-glycero-hex-3-en-2-ulose, and one product, 1,5-anhydro-4-deoxy-D-glycero-hex-1-en-3-ulose.
The enzyme is involved with the anhydrofructose pathway.
This enzyme belongs to the family of isomerases, specifically those intramolecular oxidoreductases interconverting keto- and enol-groups. The systematic name of this enzyme class is 1,5-anhydro-4-deoxy-D-glycero-hex-3-en-2-ulose Delta3-Delta1-isomerase. Other names in common use include ascopyrone isomerase, ascopyrone intramolecular oxidoreductase, 1,5-anhydro-D-glycero-hex-3-en-2-ulose tautomerase, APM tautomerase, ascopyrone P tautomerase, and APTM.
References.
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14212988
|
Aspartate racemase
|
In enzymology, an aspartate racemase (EC 5.1.1.13) is an enzyme that catalyzes the following chemical reaction:
L-aspartate formula_0 D-aspartate
This enzyme belongs to the family of isomerases, specifically those racemases and epimerases acting on amino acids and amino acid derivatives, including glutamate racemase, proline racemase, and diaminopimelate epimerase.
The systematic name of this enzyme class is aspartate racemase. Other names in common use include D-aspartate racemase, and McyF.
Discovery.
Aspartate racemase was first discovered in the gram-positive bacteria "Streptococcus faecalis" by Lamont "et al". in 1972. It was then determined that aspartate racemase also racemizes L-alanine around half as quickly as it does L-aspartate, but does not show racemase activity in the presence of L-glutamate.
Structure.
The crystallographic structure of bacterial aspartate racemase has been solved in "Pyrococcus horikoshii OT3", "Escherichia coli", "Microcystis aeruginosa", and "Picrophilus torridus DSM 9790".
Homodimer.
In most bacteria for which the structure is known, aspartate racemase exists as a homodimer, where each subunit has a molecular weight of approximately 25 kDa. The complex consists primarily of alpha helices, and additionally features a Rossmann fold in the center of the dimer. The catalytic pocket lies at the cleft formed by the intersection of the two domains. A citrate molecule can fit inside the binding pocket, leading to a contraction of the cleft to make the "closed form" of aspartate racemase.
Two highly conserved cysteine residues are suggested to be responsible for the interconversion of L-aspartate and D-aspartate. These cysteine residues lie 3–4 angstroms away from the α-carbon of aspartate. Site-directed mutagenesis studies showed that the mutation of the upstream cysteine residue to serine resulted in complete loss of racemization activity, while the same mutation in the downstream cysteine residue resulted in retention of 10–20% racemization activity. However, mutation of the acid residue glutamate, which stabilizes the downstream cysteine residue, resulted in complete loss of racemization activity. Up to 9 other residues are known to interact with and stabilize the isomers of aspartate through hydrogen bonding or hydrophobic interactions.
In "E. coli", one of the active cysteine residues is substituted for a threonine residue, allowing for much greater substrate promiscuity. Notably, aspartate racemase in "E. coli" is also able to catalyze the racemization of glutamate.
Monomer.
In 2004, an aspartate racemase was discovered in "Bifidobacterium bifidum" as a 27 kDa monomer. This protein shares nearly identical enzymological properties with homodimeric aspartate racemase isolated from "Streptococcus thermophilus", but has the added characteristic that its thermal stability increases significantly in the presence of aspartate.
Reaction mechanism.
Aspartate racemase catalyzes the following reaction:
Aspartate racemase can accept either L-aspartate or D-aspartate as substrates.
Amino acid racemization is carried out by two dominant mechanisms: one-base mechanisms and two-base mechanisms. In one-base mechanisms, a proton acceptor abstracts the α-hydrogen from the substrate amino acid to form a carbanion intermediate until reprotonated at the other face of the α-carbon. Racemases dependent on pyridoxal-5-phosphate (PLP) typically leverage one-base mechanisms. In the two-base mechanism, an alpha hydrogen is abstracted by a base on one face of the amino acid while another protonated base concertedly donates its hydrogen onto the other face of the amino acid.
PLP-independent mechanism.
Aspartate racemases in bacteria function in the absence of PLP, suggesting a PLP-independent mechanism. A two-base mechanism is supported in the literature, carried out by two thiol groups:
Other PLP-independent isomerases in bacteria include glutamate racemase, proline racemase, and hydroxyproline-2-epimerase.
PLP-dependent mechanism.
Mammalian aspartate racemase, in contrast with bacterial aspartate racemase, is a PLP-dependent enzyme. The exact mechanism is unknown, but it is hypothesized to proceed similarly to mammalian serine racemase as below:
Inhibition.
General inhibitors for cysteine residues have shown to be effective agents against monomeric aspartate racemase. "N"-ethylmaleimide and 5,5'-dithiobis(2-nitrobenzoate) both inhibit monomeric aspartate racemase at 1mM.
Function.
Metabolism of D-aspartate.
One of the primary functions of aspartate racemase in bacteria is the metabolism of D-aspartate. The beginning of D-aspartate metabolism is its conversion to L-alanine. First, D-aspartate is isomerized to L-aspartate by aspartate racemase, followed by decarboxylation to form L-alanine.
Peptidoglycan synthesis.
D-amino acids are common within the peptidoglycan of bacteria. In "Bifidobacterium bifidum", D-aspartate is formed from L-aspartate via aspartate racemase and used as a cross-linker moiety in the peptidoglycan.
Mammalian neurogenesis.
Aspartate racemase is highly expressed in the brain, the heart, and the testes of mammals, all tissues in which D-aspartate is present. D-aspartate is abundant in the embryonic brain, but falls during postnatal development. Retrovirus-mediated expression of short hairpin RNA complementary to aspartate racemase in newborn neurons of the adult hippocampus led to defects in dendritic development and empaired survival of the newborn neurons, suggesting that aspartate racemase may modulate adult neurogenesis in mammals.
Evolution.
Phylogenetic analysis shows that PLP-dependent animal aspartate racemases are in the same family as PLP-dependent animal serine racemases, and the genes encoding them share a common ancestor. Aspartate racemases in animals have independently evolved from serine racemases through amino acid substitutions, namely, the introduction of three consecutive serine residues. Serine racemases isolated from "Saccoglossus kowalevskii" also show both high aspartate and glutamate racemization activity.
References.
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|
[
{
"math_id": 0,
"text": "\\rightleftharpoons"
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] |
https://en.wikipedia.org/wiki?curid=14212988
|
14213010
|
Beta-lysine 5,6-aminomutase
|
Class of enzymes
In enzymology, a beta-lysine 5,6-aminomutase (EC 5.4.3.3) is an enzyme that catalyzes the chemical reaction
(3S)-3,6-diaminohexanoate formula_0 (3S,5S)-3,5-diaminohexanoate
Hence, this enzyme has one substrate, (3S)-3,6-diaminohexanoate, and one product, (3S,5S)-3,5-diaminohexanoate.
This enzyme belongs to the family of isomerases, specifically those intramolecular transferases transferring amino groups. The systematic name of this enzyme class is (3S)-3,6-diaminohexanoate 5,6-aminomutase. Other names in common use include beta-lysine mutase, and L-beta-lysine 5,6-aminomutase. This enzyme participates in lysine degradation. It employs one cofactor, cobamide.
Structural studies.
As of late 2007, only one structure has been solved for this class of enzymes, with the PDB accession code 1XRS.
References.
<templatestyles src="Reflist/styles.css" />
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[
{
"math_id": 0,
"text": "\\rightleftharpoons"
}
] |
https://en.wikipedia.org/wiki?curid=14213010
|
14213034
|
Beta-phosphoglucomutase
|
Enzyme
In enzymology, a β-phosphoglucomutase (EC 5.4.2.6) is an enzyme that catalyzes the chemical reaction
β-D-glucose 1-phosphate formula_0 β-D-glucose 6-phosphate
Hence, this enzyme has one substrate, β-D-glucose 1-phosphate, and one product, β-D-glucose 6-phosphate.
This enzyme belongs to the family of isomerases, specifically the phosphotransferases (phosphomutases), which transfer phosphate groups within a molecule. The systematic name of this enzyme class is beta-D-glucose 1,6-phosphomutase. This enzyme participates in starch and sucrose metabolism.
Structural studies.
20 structures have been solved for this enzyme PDB. Some of the accession codes are 1LVH, 1O03, 1O08, 1Z4N, 1Z4O, and 1ZOL. Most of these structures detail metal fluoride analogue complexes which are used to mimic different states along the reaction coordinate.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "\\rightleftharpoons"
}
] |
https://en.wikipedia.org/wiki?curid=14213034
|
14213050
|
Bornyl diphosphate synthase
|
In enzymology, bornyl diphosphate synthase (BPPS) (EC 5.5.1.8) is an enzyme that catalyzes the chemical reaction
geranyl diphosphate formula_0 (+)-bornyl diphosphate
Bornyl diphosphate synthase is involved in the biosynthesis of the cyclic monoterpenoid bornyl diphosphate. As seen from the reaction above, BPPS takes geranyl diphosphate as its only substrate and isomerizes into the product, (+)- bornyl diphosphate. This reaction comes from a general class of enzymes called terpene synthases that cyclize a universal precursor, geranyl diphosphate, to form varying monocyclic and bicyclic monoterpenes. The biochemical transformation of geranyl diphosphate to cyclic products occurs in a variety of aromatic plants, including both angiosperms and gymnosperms, and is used for various purposes described in sections below. Terpene synthases like BPPS are the primary enzymes in the formation of low-molecular-weight terpene metabolites. The organization of terpene synthases, their characteristic ability to form multiple products, and regulation in response to biotic and abiotic factors contribute to the formation of a diverse group of terpene metabolites. The structural diversity and complexity of terpenes generates an enormous potential for mediating plant–environment interactions.
The systematic name of this enzyme class is (+)-bornyl-diphosphate lyase (decyclizing). Other names in common use include bornyl pyrophosphate synthase, bornyl pyrophosphate synthetase, (+)-bornylpyrophosphate cyclase, and geranyl-diphosphate cyclase (ambiguous). This enzyme participates in monoterpenoid biosynthesis and belongs to the family of isomerases, specifically the class of intramolecular lyases.
Mechanism.
As seen in the mechanism above, bornyl diphosphate synthase catalyzes the cyclization cascade of GPP into (+)- bornyl diphosphate. Following the initial metal-activated diphosphate departure from GPP, the molecule isomerizes to linalyl diphosphate (LPP), which then allows for the rotation around the carbon-carbon bond, and consequent reattachment of the PPi group. The pyrophosphate then stabilizes the cyclization into the terpinyl cation, and another final cyclization yields the 2-bornyl cation. This cation is then neutralized by the stereo-specific C–O bond formation with the final re-attachment of pyrophosphate to create the final product, BPP. Careful consideration of the BPPS structure shows that the active site, discussed in further detail below, guides the positions and conformations of the isoprenoid functionality of the substrate, while the diphosphate position remains essentially anchored in a single location and conformation. Overall, the pyrophosphate plays an important role in stabilizing the carbocations formed throughout the cyclization in the active site of the enzyme. These interactions and the strategic positioning of pyrophosphate is what is believed to lead to its endo-specific recapture in the final step by the bornyl cation.
Enzyme Structure.
Bornyl diphosphate synthase is a homodimeric isomerase, with each of the two monomers containing two α-helical domains. In the case of BPPS, the C-terminal domain directly catalyzes the cyclization of geranyl diphosphate, as seen in the above reaction mechanism, while the N-terminal domain acts as a scaffolding to the active site of the C-terminal during the reaction. The N-terminal domain forms similar α-barrels to that of other terpene cyclases such as epiaristolochene synthase and farnesyltransferase. In ligand complexes, such as with GPP, bornyl diphosphate synthase stabilizes the complex through multiple hydrogen bond interactions, specifically with aspartate-rich motifs. Additionally, arginines in the N terminus may play a stabilizing role in the initial isomerization step of the reaction cascade discussed in the section above. The C-terminal domain, on the other hand, contains 12 α-helices, which define the hydrophobic active site where the cyclization occurs. Critical amino acid segments found in the C-terminal domain are also what allow the required magnesium metal ions to bind and allow the first pyrophosphate release. Specifically, this is accomplished by an aspartate-rich domain DDIYD beginning at D351, in with the boldface represents the residues directly interacting with the magnesium ion, elucidated on the adjacent image.
As of late 2007, 7 structures have been solved for this class of enzymes, with PDB accession codes 1N1B, 1N1Z, 1N20, 1N21, 1N22, 1N23, and 1N24.
Biological Function.
Many properties of plants derive almost exclusively from monoterpene natural products: plants generate these compounds for molecular functions in regulation, communication, and defense. For examples, terpenes often have a strong odor and may protect the plants that produce them from herbivores by deterring them and by attracting predators of said herbivores. The monoterpenes characterized to-date reveal a vast array of structural and functional variations coming from different monocyclic or bicyclic skeletons. Despite this structural and stereochemical diversity, all monoterpenes derive from the same substrate, geranyl diphosphate (GPP). The cyclization of this C10-isoprenoid precursor through sequential carbocation intermediates, as seen in the above sections, and is catalyzed by metal-dependent enzymes: in this case, BPPS cyclizes GPP into bornyl diphosphate. However, the multitude of products coming from only a single substrate helps conclude that this diversity is a consequence of the evolution of variations in the enzyme. Each different enzyme holds an active site that chaperones intermediates through different cyclization pathways, and thus forms myriad monoterpenoids.
Industrial Relevance.
Historically, aromatic plants have been used for their pleasing fragrances, culinary applications, and therapeutic potential. Because bornyl diphosphate synthase is crucial in forming aromatic monoterpenoids within plants, this enzyme is of key industrial relevance. Specifically, while most studies focus on BPPS from "Salvia officinalis", there has been a recent interest in studying LaBPPS, bornyl diphosphate synthase from lavender. This interest arises from the fact that lavender essential oils (EOs) of higher quality produced by a few "Lavandula angustifolia" variations are heavily sought after in the perfume industry. Compared to the BPPS of "Salvia officinalis", LaBPPS showed several differences in amino acid sequence, and the products it catalyzes: in detail, the carbocation intermediates are more stable in LaBPPS than in regular BPPS, leading to a different efficiency of converting GPP into BPP. Given the novelty of LaBPP discovery, further research on this will most likely be of significant use to the perfume and fragrance industry.
References.
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[
{
"math_id": 0,
"text": "\\rightleftharpoons"
}
] |
https://en.wikipedia.org/wiki?curid=14213050
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14213076
|
Carboxy-cis,cis-muconate cyclase
|
Class of enzymes
In enzymology, a carboxy-cis,cis-muconate cyclase (EC 5.5.1.5) is an enzyme that catalyzes the chemical reaction
3-carboxy-2,5-dihydro-5-oxofuran-2-acetate formula_0 3-carboxy-cis,cis-muconate
Hence, this enzyme has one substrate, 3-carboxy-2,5-dihydro-5-oxofuran-2-acetate, and one product, 3-carboxy-cis,cis-muconate.
This enzyme belongs to the family of isomerases, specifically the class of intramolecular lyases. The systematic name of this enzyme class is 3-carboxy-2,5-dihydro-5-oxofuran-2-acetate lyase (decyclizing). This enzyme is also called 3-carboxymuconate cyclase. This enzyme participates in benzoate degradation via hydroxylation.
Structural studies.
As of late 2007, only one structure has been solved for this class of enzymes, with the PDB accession code 1JOF.
References.
<templatestyles src="Reflist/styles.css" />
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[
{
"math_id": 0,
"text": "\\rightleftharpoons"
}
] |
https://en.wikipedia.org/wiki?curid=14213076
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14213091
|
CDP-paratose 2-epimerase
|
In enzymology, a CDP-paratose 2-epimerase (EC 5.1.3.10) is an enzyme that catalyzes the chemical reaction
CDP-3,6-dideoxy-D-glucose formula_0 CDP-3,6-dideoxy-D-mannose
Hence, this enzyme has one substrate, CDP-3,6-dideoxy-D-glucose, and one product, CDP-3,6-dideoxy-D-mannose.
This enzyme belongs to the family of isomerases, specifically those racemases and epimerases acting on carbohydrates and derivatives. The systematic name of this enzyme class is CDP-3,6-dideoxy-D-glucose 2-epimerase. Other names in common use include CDP-paratose epimerase, cytidine diphosphoabequose epimerase, cytidine diphosphodideoxyglucose epimerase, cytidine diphosphoparatose epimerase, and cytidine diphosphate paratose-2-epimerase. It is also incorrectly known as "CDP-abequose epimerase", and "CDP-D-abequose 2-epimerase". This enzyme participates in starch and sucrose metabolism. It employs one cofactor, NAD+.
References.
<templatestyles src="Reflist/styles.css" />
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[
{
"math_id": 0,
"text": "\\rightleftharpoons"
}
] |
https://en.wikipedia.org/wiki?curid=14213091
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14213114
|
Cellobiose epimerase
|
In enzymology a cellobiose epimerase (EC 5.1.3.11) is an enzyme that catalyzes the chemical reaction
cellobiose formula_0 D-glucosyl-D-mannose
Hence, this enzyme has one substrate, cellobiose, and one product, D-glucosyl-D-mannose.
This enzyme belongs to the family of isomerases, specifically those racemases and epimerases acting on carbohydrates and their derivatives. The systematic name of this enzyme class is cellobiose 2-epimerase. Enzymes like these can produce a more rapid syndrome that can speed up the process of many life-threatening diseases such as Necrotizing Fasciitis.
References.
<templatestyles src="Reflist/styles.css" />
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[
{
"math_id": 0,
"text": "\\rightleftharpoons"
}
] |
https://en.wikipedia.org/wiki?curid=14213114
|
14213135
|
Chalcone isomerase
|
In enzymology, a chalcone isomerase (EC 5.5.1.6) is an enzyme that catalyzes the chemical reaction
a chalcone formula_0 a flavanone
Hence, this enzyme has one substrate, a chalcone, and one product, a flavanone.
This enzyme belongs to the family of isomerases, specifically the class of intramolecular lyases. The systematic name of this enzyme class is flavanone lyase (decyclizing). This enzyme is also called chalcone-flavanone isomerase. This enzyme participates in flavonoid biosynthesis.
The "Petunia hybrida" (Petunia) genome contains two genes coding for very similar enzymes, ChiA and ChiB, but only the first seems to encode a functional chalcone isomerase.
Structural studies.
As of late 2007, 7 structures have been solved for this class of enzymes, with PDB accession codes 1EYP, 1EYQ, 1FM7, 1FM8, 1JEP, 1JX0, and 1JX1.
Chalcone isomerase has a core 2-layer alpha/beta structure consisting of beta(3)-alpha(2)-beta-alpha(2)-beta(3).
References.
<templatestyles src="Reflist/styles.css" />
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[
{
"math_id": 0,
"text": "\\rightleftharpoons"
}
] |
https://en.wikipedia.org/wiki?curid=14213135
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14213157
|
Chloromuconate cycloisomerase
|
In enzymology, a chloromuconate cycloisomerase (EC 5.5.1.7) is an enzyme that catalyzes the chemical reaction
2-chloro-2,5-dihydro-5-oxofuran-2-acetate formula_0 3-chloro-cis,cis-muconate
Hence, this enzyme has one substrate, 2-chloro-2,5-dihydro-5-oxofuran-2-acetate, and one product, 3-chloro-cis,cis-muconate.
This enzyme belongs to the family of isomerases, specifically the class of intramolecular lyases. The systematic name of this enzyme class is 2-chloro-2,5-dihydro-5-oxofuran-2-acetate lyase (decyclizing). This enzyme is also called muconate cycloisomerase II. This enzyme participates in gamma-hexachlorocyclohexane degradation and 1,4-dichlorobenzene degradation. It employs one cofactor, manganese.
Structural studies.
As of late 2007, 3 structures have been solved for this class of enzymes, with PDB accession codes 1CHR, 1NU5, and 2CHR.
References.
<templatestyles src="Reflist/styles.css" />
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[
{
"math_id": 0,
"text": "\\rightleftharpoons"
}
] |
https://en.wikipedia.org/wiki?curid=14213157
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14213186
|
Cholestenol Delta-isomerase
|
In enzymology, a cholestenol Δ-isomerase (EC 5.3.3.5) is an enzyme that catalyzes the chemical reaction
5alpha-cholest-7-en-3beta-ol formula_0 5alpha-cholest-8-en-3beta-ol
Hence, this enzyme has one substrate, 5alpha-cholest-7-en-3beta-ol, and one product, 5alpha-cholest-8-en-3beta-ol.
This enzyme belongs to the family of isomerases, specifically those intramolecular oxidoreductases transposing C=C bonds. The systematic name of this enzyme class is Delta7-cholestenol Delta7-Delta8-isomerase. This enzyme participates in biosynthesis of steroids.
References.
<templatestyles src="Reflist/styles.css" />
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[
{
"math_id": 0,
"text": "\\rightleftharpoons"
}
] |
https://en.wikipedia.org/wiki?curid=14213186
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1421466
|
Mediant (mathematics)
|
In mathematics, the mediant of two fractions, generally made up of four positive integers
formula_0 and formula_1 is defined as formula_2
That is to say, the numerator and denominator of the mediant are the sums of the numerators and denominators of the given fractions, respectively. It is sometimes called the freshman sum, as it is a common mistake in the early stages of learning about addition of fractions.
Technically, this is a binary operation on valid fractions (nonzero denominator), considered as ordered pairs of appropriate integers, a priori disregarding the perspective on rational numbers as equivalence classes of fractions. For example, the mediant of the fractions 1/1 and 1/2 is 2/3. However, if the fraction 1/1 is replaced by the fraction 2/2, which is an equivalent fraction denoting the same rational number 1, the mediant of the fractions 2/2 and 1/2 is 3/4. For a stronger connection to rational numbers the fractions may be required to be reduced to lowest terms, thereby selecting unique representatives from the respective equivalence classes.
The Stern–Brocot tree provides an enumeration of all positive rational numbers via mediants in lowest terms, obtained purely by iterative computation of the mediant according to a simple algorithm.
* Componendo:
formula_11
* Dividendo:
formula_12
Graphical determination of mediants.
A positive rational number is one in the form formula_37 where formula_38 are positive natural numbers; "i.e." formula_39. The set of positive rational numbers formula_40 is, therefore, the Cartesian product of formula_41 by itself; "i.e." formula_42. A point with coordinates formula_43 represents the rational number formula_37, and the slope of a segment connecting the origin of coordinates to this point is formula_37. Since formula_38 are not required to be coprime, point formula_43 represents one and only one rational number, but a rational number is represented by more than one point; "e.g." formula_44 are all representations of the rational number formula_45. This is a slight modification of the formal definition of rational numbers, restricting them to positive values, and flipping the order of the terms in the ordered pair formula_43 so that the slope of the segment becomes equal to the rational number.
Two points formula_46 where formula_47 are two representations of (possibly equivalent) rational numbers formula_37 and formula_48. The line segments connecting the origin of coordinates to formula_43 and formula_49 form two adjacent sides in a parallelogram. The vertex of the parallelogram opposite to the origin of coordinates is the point formula_50, which is the mediant of formula_37 and formula_48.
The area of the parallelogram is formula_51, which is also the magnitude of the cross product of vectors formula_52 and formula_53. It follows from the formal definition of rational number equivalence that the area is zero if formula_37 and formula_48 are equivalent. In this case, one segment coincides with the other, since their slopes are equal. The area of the parallelogram formed by two consecutive rational numbers in the Stern–Brocot tree is always 1.
Generalization.
The notion of mediant can be generalized to "n" fractions, and a generalized mediant inequality holds, a fact that seems to have been first noticed by Cauchy. More precisely, the weighted mediant formula_54 of "n" fractions formula_55 is defined by formula_56 (with formula_57). It can be shown that formula_54 lies somewhere between the smallest and the largest fraction among the formula_58.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": " \\frac{a}{c} \\quad"
},
{
"math_id": 1,
"text": "\\quad \\frac{b}{d} \\quad"
},
{
"math_id": 2,
"text": "\\quad \\frac{a+b}{c+d}. "
},
{
"math_id": 3,
"text": "a/c < b/d "
},
{
"math_id": 4,
"text": "c\\cdot d> 0"
},
{
"math_id": 5,
"text": "\\frac a c < \\frac{a+b}{c+d} < \\frac b d. "
},
{
"math_id": 6,
"text": "\\frac{a+b}{c+d}-\\frac a c={{bc-ad}\\over{c(c+d)}} ={d\\over{c+d}}\\left( \\frac{b}{d}-\\frac a c \\right)"
},
{
"math_id": 7,
"text": "\\frac b d-\\frac{a+b}{c+d}={{bc-ad}\\over{d(c+d)}} ={c\\over{c+d}}\\left( \\frac{b}{d}-\\frac a c \\right). "
},
{
"math_id": 8,
"text": "a/c = b/d"
},
{
"math_id": 9,
"text": "c \\ne 0,\\ d \\ne 0"
},
{
"math_id": 10,
"text": "\\frac a c = \\frac b d = \\frac{a+b}{c+d}"
},
{
"math_id": 11,
"text": "\\frac{a+c}{c} = \\frac{b+d}{d}"
},
{
"math_id": 12,
"text": "\\frac{a-c}{c} = \\frac{b-d}{d}"
},
{
"math_id": 13,
"text": "bc-ad=1"
},
{
"math_id": 14,
"text": " a'/c' "
},
{
"math_id": 15,
"text": " a'=\\lambda_1 a + \\lambda_2 b "
},
{
"math_id": 16,
"text": " c' = \\lambda_1 c + \\lambda_2 d "
},
{
"math_id": 17,
"text": " \\lambda_1,\\,\\lambda_2 "
},
{
"math_id": 18,
"text": " \\lambda_i "
},
{
"math_id": 19,
"text": "\\frac{\\lambda_1 a+\\lambda_2 b}{\\lambda_1 c+\\lambda_2 d }-\\frac a c=\\lambda_2 {{bc-ad} \\over {c(\\lambda_1 c+\\lambda_2 d)}}\n"
},
{
"math_id": 20,
"text": "\\frac b d - \\frac{\\lambda_1 a+\\lambda_2 b}{\\lambda_1 c+\\lambda_2 d }=\\lambda_1 {{bc-ad}\\over{d(\\lambda_1 c+\\lambda_2 d )}} "
},
{
"math_id": 21,
"text": "bc-ad=1 \\, "
},
{
"math_id": 22,
"text": " a'=\\lambda_1 a+ \\lambda_2 b "
},
{
"math_id": 23,
"text": " c' = \\lambda_1 c+ \\lambda_2 d "
},
{
"math_id": 24,
"text": " \\lambda_1,\\lambda_2 "
},
{
"math_id": 25,
"text": " c'\\ge c+d. "
},
{
"math_id": 26,
"text": " \\Delta(v_1,v_2,v_3)"
},
{
"math_id": 27,
"text": " \\text{area}(\\Delta) = {{bc-ad}\\over 2} \\, ."
},
{
"math_id": 28,
"text": " p=(p_1,p_2) "
},
{
"math_id": 29,
"text": " p_1=\\lambda_1 a+\\lambda_2 b,\\; p_2=\\lambda_1 c+\\lambda_2 d, "
},
{
"math_id": 30,
"text": " \\lambda_1\\ge 0,\\,\\lambda_2 \\ge 0, \\,\\lambda_1+\\lambda_2 \\le 1. "
},
{
"math_id": 31,
"text": " \\text{area}(\\Delta)=v_\\mathrm{interior} + {v_\\mathrm{boundary}\\over 2} - 1 "
},
{
"math_id": 32,
"text": " \\ge 1 "
},
{
"math_id": 33,
"text": " q_2 = \\lambda_1 c+ \\lambda_2 d \\le \\max(c,d)<c+d "
},
{
"math_id": 34,
"text": " \\lambda_1+\\lambda_2 \\le 1. "
},
{
"math_id": 35,
"text": "?\\left(\\frac{p+r}{q+s}\\right) = \\frac1 2 \\left(?\\left(\\frac p q\\right) + {}?\\left(\\frac r s\\right)\\right)"
},
{
"math_id": 36,
"text": " bc-ad=1"
},
{
"math_id": 37,
"text": "a/b"
},
{
"math_id": 38,
"text": "a,b"
},
{
"math_id": 39,
"text": "a,b\\in\\mathbb{N}^{+}"
},
{
"math_id": 40,
"text": "\\mathbb{Q}^{+}"
},
{
"math_id": 41,
"text": "\\mathbb{N}^{+}"
},
{
"math_id": 42,
"text": "\\mathbb{Q}^{+}=(\\mathbb{N}^{+})^2"
},
{
"math_id": 43,
"text": "(b,a)"
},
{
"math_id": 44,
"text": "(4,2),(60,30),(48,24)"
},
{
"math_id": 45,
"text": "1/2"
},
{
"math_id": 46,
"text": "(b,a)\\neq(d,c)"
},
{
"math_id": 47,
"text": "a,b,c,d\\in\\mathbb{N}^{+}"
},
{
"math_id": 48,
"text": "c/d"
},
{
"math_id": 49,
"text": "(d,c)"
},
{
"math_id": 50,
"text": "(b+d,a+c)"
},
{
"math_id": 51,
"text": "bc-ad"
},
{
"math_id": 52,
"text": "\\langle b,a\\rangle"
},
{
"math_id": 53,
"text": "\\langle d,c\\rangle"
},
{
"math_id": 54,
"text": "m_w"
},
{
"math_id": 55,
"text": "a_1/b_1,\\ldots,a_n/b_n"
},
{
"math_id": 56,
"text": "\\frac{\\sum_i w_i a_i}{\\sum_i w_i b_i}"
},
{
"math_id": 57,
"text": "w_i>0"
},
{
"math_id": 58,
"text": "a_i/b_i"
}
] |
https://en.wikipedia.org/wiki?curid=1421466
|
14214764
|
Chondroitin-glucuronate 5-epimerase
|
In enzymology, a chondroitin-glucuronate 5-epimerase (EC 5.1.3.19) is an enzyme that catalyzes the chemical reaction
chondroitin D-glucuronate formula_0 dermatan L-iduronate
Hence, this enzyme has one substrate, chondroitin D-glucuronate, and one product, dermatan L-iduronate.
This enzyme belongs to the family of isomerases, specifically those racemases and epimerases acting on carbohydrates and derivatives. The systematic name of this enzyme class is chondroitin-D-glucuronate 5-epimerase. Other names in common use include polyglucuronate 5-epimerase, dermatan-sulfate 5-epimerase, urunosyl C-5 epimerase, and chondroitin D-glucuronosyl 5-epimerase. This enzyme participates in chondroitin sulfate biosynthesis.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "\\rightleftharpoons"
}
] |
https://en.wikipedia.org/wiki?curid=14214764
|
14214817
|
Copalyl diphosphate synthase
|
In enzymology, a copalyl diphosphate synthase (EC 5.5.1.12) is an enzyme that catalyzes the chemical reaction
geranylgeranyl diphosphate formula_0 (+)-copalyl diphosphate
Hence, this enzyme has one substrate, geranylgeranyl diphosphate, and one product, (+)-copalyl diphosphate.
This enzyme belongs to the family of isomerases, specifically the class of intramolecular lyases. The systematic name of this enzyme class is (+)-copalyl-diphosphate lyase (decyclizing). This enzyme participates in diterpenoid biosynthesis.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "\\rightleftharpoons"
}
] |
https://en.wikipedia.org/wiki?curid=14214817
|
14214866
|
Corticosteroid side-chain-isomerase
|
In enzymology, a corticosteroid side-chain-isomerase (EC 5.3.1.21) is an enzyme that catalyzes the chemical reaction
11-deoxycorticosterone formula_0 20-hydroxy-3-oxopregn-4-en-21-al
Hence, this enzyme has one substrate, 11-deoxycorticosterone, and one product, 20-hydroxy-3-oxopregn-4-en-21-al.
This enzyme belongs to the family of isomerases, specifically those intramolecular oxidoreductases interconverting aldoses and ketoses. The systematic name of this enzyme class is 11-deoxycorticosterone aldose-ketose-isomerase.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "\\rightleftharpoons"
}
] |
https://en.wikipedia.org/wiki?curid=14214866
|
14214937
|
Cycloartenol synthase
|
In enzymology, a cycloartenol synthase (EC 5.4.99.8) is an enzyme that catalyzes the chemical reaction
("S")-2,3-epoxysqualene formula_0 cycloartenol
Hence, this enzyme has one substrate, ("S")-2,3-epoxysqualene, and one product, cycloartenol.
This enzyme is an oxidosqualene cyclase and belongs to the family of isomerases, specifically those intramolecular transferases transferring other groups. This enzyme participates in biosynthesis of steroids.
Nomenclature.
The systematic name of this enzyme class is:
Other names in common use include:
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "\\rightleftharpoons"
}
] |
https://en.wikipedia.org/wiki?curid=14214937
|
14215091
|
Cycloeucalenol cycloisomerase
|
In enzymology, a cycloeucalenol cycloisomerase (EC 5.5.1.9) is an enzyme that catalyzes the chemical reaction
cycloeucalenol formula_0 obtusifoliol
Hence, this enzyme has one substrate, cycloeucalenol, and one product, obtusifoliol.
This enzyme belongs to the family of isomerases, specifically the class of intramolecular lyases. The systematic name of this enzyme class is cycloeucalenol lyase (cyclopropane-decyclizing). This enzyme is also called cycloeucalenol---obtusifoliol isomerase. This enzyme participates in biosynthesis of steroids.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "\\rightleftharpoons"
}
] |
https://en.wikipedia.org/wiki?curid=14215091
|
14215118
|
Diaminopimelate epimerase
|
In enzymology, a diaminopimelate epimerase (EC 5.1.1.7) is an enzyme that catalyzes the chemical reaction
LL-2,6-diaminoheptanedioate formula_0 meso-diaminoheptanedioate
Hence, this enzyme has one substrate, LL-2,6-diaminoheptanedioate, and one product, meso-diaminoheptanedioate.
This enzyme belongs to the family of isomerases, specifically those racemases and epimerases acting on amino acids and derivatives. The systematic name of this enzyme class is LL-2,6-diaminoheptanedioate 2-epimerase. This enzyme participates in lysine biosynthesis.
Background.
Bacteria, plants and fungi metabolise aspartic acid to produce four amino acids - lysine, threonine, methionine and isoleucine - in a series of reactions known as the aspartate pathway. Additionally, several important metabolic intermediates are produced by these reactions, such as diaminopimelic acid, an essential component of bacterial cell wall biosynthesis, and dipicolinic acid, which is involved in sporulation (spore production) in Gram-positive bacteria. Members of the animal kingdom do not possess this pathway and must therefore acquire these essential amino acids through their diet. Research into improving the metabolic flux through this pathway has the potential to increase the yield of the essential amino acids in important crops, thus improving their nutritional value. Additionally, since the enzymes are not present in animals, inhibitors of them are promising targets for the development of novel antibiotics and herbicides.
The lysine/diaminopimelic acid branch of the aspartate pathway produces the essential amino acid lysine via the intermediate meso-diaminopimelic acid (meso-DAP), which is also a vital cell wall component in Gram-negative bacteria. The production of dihydropicolinate from aspartate-semialdehyde controls flux into the lysine/diaminopimelic acid pathway. Three variants of this pathway exist, differing in how tetrahydropicolinate (formed by reduction of dihydropicolinate) is metabolised to meso-DAP. One variant, the most commonly found one in archaea and bacteria, uses primarily succinyl intermediates, while a second variant, found only in Bacillus, utilizes primarily acetyl intermediates. In the third variant, found in some Gram-positive bacteria, a dehydrogenase converts tetrahydropicolinate directly to meso-DAP. In all variants meso-DAP is subsequently converted to lysine by a decarboxylase, or, in Gram-negative bacteria, assimilated into the cell wall. Evidence exists that a fourth, currently unknown, variant of this pathway may function in plants.
Diaminopimelate epimerase (EC 5.1.1.7), which catalyses the isomerisation of L,L-dimaminopimelate to meso-DAP in the biosynthetic pathway leading from aspartate to lysine, is a member of the broader family of PLP-independent amino acid racemases. This enzyme is a monomeric protein of about 30 kDa consisting of two domains which are similar in structure though they share little Sequence alignment. Each domain consists of mixed beta-sheets which fold into a barrel around the central helix. The active site cleft is formed from both domains and contains two conserved cysteines thought to function as the acid and base in the Catalysis. Other PLP-independent racemases such as glutamate racemase have been shown to share a similar structure and mechanism of catalysis.
Structural studies.
As of late 2007, 4 structures have been solved for this class of enzymes, with PDB accession codes 1BWZ, 1GQZ, 2GKE, and 2GKJ.
References.
<templatestyles src="Reflist/styles.css" />
Further reading.
<templatestyles src="Refbegin/styles.css" />
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[
{
"math_id": 0,
"text": "\\rightleftharpoons"
}
] |
https://en.wikipedia.org/wiki?curid=14215118
|
14215146
|
Dichloromuconate cycloisomerase
|
In enzymology, a dichloromuconate cycloisomerase (EC 5.5.1.11) is an enzyme that catalyzes the chemical reaction
2,4-dichloro-2,5-dihydro-5-oxofuran-2-acetate formula_0 2,4-dichloro-cis,cis-muconate
Hence, this enzyme has one substrate, 2,4-dichloro-2,5-dihydro-5-oxofuran-2-acetate, and one product, 2,4-dichloro-cis,cis-muconate.
This enzyme belongs to the family of isomerases, specifically the class of intramolecular lyases. The systematic name of this enzyme class is 2,4-dichloro-2,5-dihydro-5-oxofuran-2-acetate lyase (decyclizing). This enzyme participates in 1,4-dichlorobenzene degradation. It employs one cofactor, manganese.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "\\rightleftharpoons"
}
] |
https://en.wikipedia.org/wiki?curid=14215146
|
14215182
|
D-lysine 5,6-aminomutase
|
In enzymology, D-lysine 5,6-aminomutase (EC 5.4.3.4) is an enzyme that catalyzes the chemical reaction
D-lysine formula_0 2,5-diaminohexanoate
Hence, this enzyme has one substrate, D-lysine, and one product, 2,5-diaminohexanoate.
This enzyme participates in lysine degradation. It employs one cofactor, cobamide.
Background.
D-lysine 5,6-aminomutase belongs to the isomerase family of enzymes, specifically intramolecular transferases, which transfers amino groups. Its systematic name is D-2,6-diaminohexanoate 5,6-aminomutase. Other names in common use include D-α-lysine mutase and adenosylcobalamin-dependent D-lysine 5,6-aminomutase, which can be abbreviated as 5,6-LAM.
5,6-LAM is capable of reversibly catalyzing the migration of an amino group from ε-carbon to δ-carbon in both D-lysine and L-β-lysine, and catalyzing the migration of hydrogen atoms from δ-carbon to ε-carbon at the same time. It demonstrates greatest catalytic activity in 20mM Tris•HCl at pH 9.0-9.2.
In the early 1950s, 5,6-LAM was discovered in the amino-acid-fermenting bacteria "Clostridium sticklandii," in which lysine undergoes degradation under anaerobic conditions to equimolar amounts of acetate and butyrate"."
Later, isotopic studies uncovered two possible pathways. In pathway A, both acetate and butyrate are generated from C2-C3 cleavage of D-lysine. Unlike pathway A, pathway B involves C5-C4 degradation, producing the same products.
D-lysine 5,6-aminomutase (5,6-LAM) is responsible for the first conversion in pathway B to convert D-α-lysine into 2,5-diaminohexanoate. Unlike other members of the family of aminomutases (like 2,3-LAM), which are peculiar to a single substrate, 5,6-LAM can reversibly catalyze both the reaction of D-lysine to 2,5-diaminohexanoic acid and the reaction of L-β-lysine to 3,5-diaminohexanoic acid.
Structure.
Subunits.
5,6-LAM is an α2β2 tetramer. The structure of the alpha subunit is predominantly a PLP-binding TIM barrel domain, with several additional alpha-helices and beta-strands at the N and C termini. These helices and strands form an intertwined accessory clamp structure that wraps around the sides of the TIM barrel and extends up toward the Ado ligand of the Cbl cofactor, which is the beta subunit providing most of the interactions observed between the protein and the Ado ligand of the Cbl, suggesting that its role is mainly in stabilizing AdoCbl in the precatalytic resting state. The β subunit binds AdoCbl while the PLP directly binds to α subunit. PLP also directly binds to Lys144 of the β subunit to form an internal aldimine. PLP and AdoCbl are separated by a distance of 24Å.
Mechanism.
Catalytic cycle.
The catalytic cycle starts with Ado-CH2• (5'-deoxyadenosyl radical derived from adenosylcobalamine) abstracting a hydrogen atom from PLP-D-lysine adduct (substrate-related precursor SH) to generate a substrate-related radical (S•), with the radical located at carbon 5 of the lysine residue. The latter undergoes an internal cyclization/addition to the imine nitrogen producing an aziridinecarbinyl radical (I•) — a more thermodynamically stable intermediate with the radical being at a benzylic position. Rearrangement of I• produces a product-related radical (P•), which then participates in the final step of hydrogen transfer from AdoH to afford the PLP-product complex (PH).
Structure-based catalysis.
Further understanding of the catalytic mechanism can be derived from the X-ray structure.
First, an evident conformational change is observed after the substrate is added to the system. With a substrate-free enzyme, the distance between AdoCbl and PLP is about 24 Å. PLP participates in multiple non-covalent interactions with the enzyme with 5,6-LAM presenting an “open” state.
The first step of the catalytic cycle involves the enzyme accepting the substrate by forming an external aldimine with PLP replacing the PLP-Lys144β internal aldimine. With the cleavage of the internal aldimine, the β unit is able to swing towards to the top of the α unit and block the empty site. Therefore, generation of the Ado-CH2• radical leads to a change in the structure of the active domain, bringing the AdoCbl and PLP-substrate complex closer to each other, thus locking the enzyme in a “closed” state. The closed state exists until the radical transfer occurs when the product is released and AdoCbl is reformed. At the same time, the closed state is transformed to the open state again to wait for the next substrate.
Also worth mentioning is the locking mechanism to prevent the radical reaction without the presence of substrate discovered by Catherine Drennan's group. Lys144 of the β subunit is located at a short G-rich loop highly conserved across all 5,6-LAMs, which blocks the AdoCbl from the reaction site. Based on X-ray structure analysis, when the open structure is applied, the axes of the TIM barrel and Rossmann domains are in different directions. With the addition of the substrate, the subunits rearrange to turn the axes into each other to facilitate the catalysis. For example, in wild type 5,6-LAM, the phenol ring of Tyr263α is oriented in a slipped geometry with pyridine ring of PLP, generating a π-π stacking interaction, which is capable of modulating the electron distribution of the high-energetic radical intermediate.
History.
Early insights into the mechanism of the catalytic reaction mainly focused on isotopic methods. Both pathways of lysine degradation and the role of 5,6-LAM were discovered in early work by Stadtman's group during 1950s-1960s. In 1971, having a tritiated α-lysine, 2,5-diaminohexanoate, and coenzyme in hand, Colin Morley and T. Stadtman discovered the role of 5'-deoxyadenosylcobalamin (AdoCbl) as a source for hydrogen migration. Recently, much progress has been made toward detecting the intermediates of the reaction, especially towards I•. Based on quantum-mechanical calculations, it was proposed that with 5-fluorolysine as a substitute for D-lysine the 5-FS• species can be captured and analyzed. A similar approach was applied towards PLP modification, when it was modified to 4’-cyanoPLP or PLP-NO. The radical intermediate I• analogue is hypothesized to be easily detected to support the proposed mechanism. Other simulations can also provide some insights into the catalytic reaction.
References.
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[
{
"math_id": 0,
"text": "\\rightleftharpoons"
}
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https://en.wikipedia.org/wiki?curid=14215182
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14215211
|
D-lyxose ketol-isomerase
|
Class of enzymes
In enzymology, a D-lyxose ketol-isomerase (EC 5.3.1.15) is an enzyme that catalyzes the chemical reaction
D-lyxose formula_0 D-xylulose
Hence, this enzyme has one substrate, D-lyxose, and one product, D-xylulose.
This enzyme belongs to the family of isomerases, specifically those intramolecular oxidoreductases interconverting aldoses and ketoses. The systematic name of this enzyme class is D-lyxose aldose-ketose-isomerase. Other names in common use include D-lyxose isomerase, and D-lyxose ketol-isomerase. This enzyme participates in pentose and glucuronate interconversions.
Structural studies.
As of late 2007, two structures have been solved for this class of enzymes, with PDB accession codes 1QO2 and 1VZW.
References.
<templatestyles src="Reflist/styles.css" />
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[
{
"math_id": 0,
"text": "\\rightleftharpoons"
}
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https://en.wikipedia.org/wiki?curid=14215211
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14215259
|
D-ornithine 4,5-aminomutase
|
In enzymology, a D-ornithine 4,5-aminomutase (EC 5.4.3.5) is an enzyme that catalyzes the chemical reaction
D-ornithine formula_0 (2R,4S)-2,4-diaminopentanoate
Hence, this enzyme has one substrate, D-ornithine, and one product, (2R,4S)-2,4-diaminopentanoate.
This enzyme belongs to the family of isomerases, specifically those intramolecular transferases transferring amino groups. The systematic name of this enzyme class is D-ornithine 4,5-aminomutase. Other names in common use include D-alpha-ornithine 5,4-aminomutase, and D-ornithine aminomutase. This enzyme participates in d-arginine and d-ornithine metabolism. It has 3 cofactors: pyridoxal phosphate, Cobamide coenzyme, and Dithiothreitol.
References.
<templatestyles src="Reflist/styles.css" />
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[
{
"math_id": 0,
"text": "\\rightleftharpoons"
}
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https://en.wikipedia.org/wiki?curid=14215259
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14215282
|
DTDP-4-dehydrorhamnose 3,5-epimerase
|
Enzyme
In enzymology, a dTDP-4-dehydrorhamnose 3,5-epimerase (EC 5.1.3.13) is an enzyme that catalyzes the chemical reaction
dTDP-4-dehydro-6-deoxy-D-glucose formula_0 dTDP-4-dehydro-6-deoxy-L-mannose
Hence, this enzyme has one substrate, dTDP-4-dehydro-6-deoxy-D-glucose, and one product, dTDP-4-dehydro-6-deoxy-L-mannose.
This enzyme belongs to the family of isomerases, specifically those racemases and epimerases acting on carbohydrates and derivatives. The systematic name of this enzyme class is dTDP-4-dehydro-6-deoxy-D-glucose 3,5-epimerase. Other names in common use include dTDP-L-rhamnose synthetase, dTDP-L-rhamnose synthetase, thymidine diphospho-4-ketorhamnose 3,5-epimerase, TDP-4-ketorhamnose 3,5-epimerase, dTDP-4-dehydro-6-deoxy-D-glucose 3,5-epimerase, and TDP-4-keto-L-rhamnose-3,5-epimerase. This enzyme participates in 3 metabolic pathways: nucleotide sugars metabolism, streptomycin biosynthesis, and polyketide sugar unit biosynthesis.
Structural studies.
The crystal structure of RmlC from "Methanobacterium thermoautotrophicum" was determined in the presence and absence of a substrate analogue. RmlC is a homodimer comprising a central jelly roll motif, which extends in two directions into longer beta-sheets. Binding of dTDP is stabilised by ionic interactions to the phosphate group and by a combination of ionic and hydrophobic interactions with the base. The active site, which is located in the centre of the jelly roll, is formed by residues that are conserved in all known RmlC sequence homologues. The active site is lined with a number of charged residues and a number of residues with hydrogen-bonding potentials, which together comprise a potential network for substrate binding and catalysis. The active site is also lined with aromatic residues which provide favourable environments for the base moiety of dTDP and potentially for the sugar moiety of the substrate.
As of late 2007, 14 structures have been solved for this class of enzymes, with PDB accession codes 1DZR, 1DZT, 1EP0, 1EPZ, 1NXM, 1NYW, 1NZC, 1PM7, 1RTV, 1UPI, 1WLT, 2B9U, 2IXC, and 2IXL.
References.
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[
{
"math_id": 0,
"text": "\\rightleftharpoons"
}
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https://en.wikipedia.org/wiki?curid=14215282
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14215330
|
Farnesol 2-isomerase
|
In enzymology, a farnesol 2-isomerase (EC 5.2.1.9) is an enzyme that catalyzes the chemical reaction
2-trans,6-trans-farnesol formula_0 2-cis,6-trans-farnesol
Hence, this enzyme has one substrate, 2-trans,6-trans-farnesol, and one product, 2-cis,6-trans-farnesol.
This enzyme belongs to the family of isomerases, specifically cis-trans isomerases. The systematic name of this enzyme class is 2-trans,6-trans-farnesol 2-cis-trans-isomerase. This enzyme is also called farnesol isomerase.
References.
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[
{
"math_id": 0,
"text": "\\rightleftharpoons"
}
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https://en.wikipedia.org/wiki?curid=14215330
|
14215355
|
Furylfuramide isomerase
|
In enzymology, a furylfuramide isomerase (EC 5.2.1.6) is an enzyme that catalyzes the chemical reaction
(E)-2-(2-furyl)-3-(5-nitro-2-furyl)acrylamide formula_0 (Z)-2-(2-furyl)-3-(5-nitro-2-furyl)acrylamide
Hence, this enzyme has one substrate, (E)-2-(2-furyl)-3-(5-nitro-2-furyl)acrylamide, and one product, (Z)-2-(2-furyl)-3-(5-nitro-2-furyl)acrylamide.
This enzyme belongs to the family of isomerases, specifically cis-trans isomerases. The systematic name of this enzyme class is 2-(2-furyl)-3-(5-nitro-2-furyl)acrylamide cis-trans-isomerase. It has 2 cofactors: NAD+, and NADH.
References.
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[
{
"math_id": 0,
"text": "\\rightleftharpoons"
}
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https://en.wikipedia.org/wiki?curid=14215355
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14215382
|
Galactose-6-phosphate isomerase
|
In enzymology, a galactose-6-phosphate isomerase (EC 5.3.1.26) is an enzyme that catalyzes the chemical reaction
D-galactose 6-phosphate formula_0 D-tagatose 6-phosphate
Hence, this enzyme has one substrate, D-galactose 6-phosphate, and one product, D-tagatose 6-phosphate.
This enzyme belongs to the family of isomerases, specifically those intramolecular oxidoreductases interconverting aldoses and ketoses. The systematic name of this enzyme class is D-galactose-6-phosphate aldose-ketose-isomerase. This enzyme participates in galactose metabolism.
References.
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[
{
"math_id": 0,
"text": "\\rightleftharpoons"
}
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https://en.wikipedia.org/wiki?curid=14215382
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14215411
|
GDP-mannose 3,5-epimerase
|
In enzymology, a GDP-mannose 3,5-epimerase (EC 5.1.3.18) is an enzyme that catalyzes the chemical reaction
GDP-mannose formula_0 GDP-L-galactose + GDP-L-gulose - initial reaction overview (updated in 2020, see below)
Hence, this enzyme has one substrate, GDP-mannose, and two products, GDP-L-galactose and GDP-L-gulose
Since only GDP-L-gulose (the C5-epimer of GDP-D-mannose) was found in the reaction mixture, it was postulated that the enzyme performs the C5-epimerization prior to the C3-epimerization. However, GDP-D-altrose was recently found as a reaction product, which means that both reaction routes can occur: C5-prior-to-C3 and C3-prior-to-C5.
This also means that the GDP-mannose 3,5-epimerase has three reaction products, namely the main product GDP-L-galactose (C3,5-epimer) and two sideproducts GDP-L-gulose (C5-epimer) + GDP-D-altrose (C3-epimer).
GDP-D-mannose formula_0 GDP-L-galactose (C3,5-epimer) + GDP-L-gulose (C5-epimer) + GDP-D-altrose (C3-epimer) - reaction overview update in 2020
This enzyme belongs to the family of isomerases, specifically those racemases and epimerases acting on carbohydrates and derivatives. The systematic name of this enzyme class is GDP-mannose 3,5-epimerase. Other names in common use include GDP-D-mannose:GDP-L-galactose epimerase, guanosine 5'-diphosphate D-mannose:guanosine 5'-diphosphate, GM35E, and L-galactose epimerase. This enzyme participates in ascorbate and aldarate metabolism.
Structural studies.
As of late 2007, 4 structures have been solved for this class of enzymes, with PDB accession codes 2C54, 2C59, 2C5A, and 2C5E.
References.
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[
{
"math_id": 0,
"text": "\\rightleftharpoons"
}
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https://en.wikipedia.org/wiki?curid=14215411
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14215438
|
Glucose-6-phosphate 1-epimerase
|
In enzymology, a glucose-6-phosphate 1-epimerase (EC 5.1.3.15) is an enzyme that catalyzes the chemical reaction
alpha-D-glucose 6-phosphate formula_0 beta-D-glucose 6-phosphate
Hence, this enzyme has one substrate, alpha-D-glucose 6-phosphate, and one product, beta-D-glucose 6-phosphate.
This enzyme belongs to the family of isomerases, specifically those racemases and epimerases acting on carbohydrates and derivatives. The systematic name of this enzyme class is D-glucose-6-phosphate 1-epimerase. This enzyme participates in glycolysis / gluconeogenesis.
Structural studies.
As of late 2013, 3 structures have been solved for this class of enzymes, with PDB accession codes 2CIQ, 2CIR, and 2CIS.
References.
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[
{
"math_id": 0,
"text": "\\rightleftharpoons"
}
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https://en.wikipedia.org/wiki?curid=14215438
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14215470
|
Glucuronate isomerase
|
In enzymology, a glucuronate isomerase (EC 5.3.1.12) is an enzyme that catalyzes the chemical reaction
D-glucuronate formula_0 D-fructuronate
Hence, this enzyme has one substrate, D-glucuronate, and one product, D-fructuronate.
This enzyme belongs to the family of isomerases, specifically those intramolecular oxidoreductases interconverting aldoses and ketoses. The systematic name of this enzyme class is D-glucuronate aldose-ketose-isomerase. Other names in common use include uronic isomerase, uronate isomerase, D-glucuronate isomerase, uronic acid isomerase, and D-glucuronate ketol-isomerase. This enzyme participates in pentose and glucuronate interconversions.
Structural studies.
As of late 2007, two structures have been solved for this class of enzymes, with PDB accession codes 1J5S and 2Q01.
References.
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[
{
"math_id": 0,
"text": "\\rightleftharpoons"
}
] |
https://en.wikipedia.org/wiki?curid=14215470
|
14215507
|
Glutamate-1-semialdehyde 2,1-aminomutase
|
In enzymology, a glutamate-1-semialdehyde 2,1-aminomutase (EC 5.4.3.8) is an enzyme that catalyzes the chemical reaction
L-glutamate 1-semialdehyde formula_0 5-aminolevulinate
Hence, this enzyme has one substrate, L-glutamate-1-semialdehyde, and one product, 5-aminolevulinate.
This enzyme belongs to the family of isomerases, specifically those intramolecular transferases transferring amino groups. The systematic name of this enzyme class is (S)-4-amino-5-oxopentanoate 4,5-aminomutase. This enzyme is also called glutamate-1-semialdehyde aminotransferase. This enzyme participates in porphyrin and chlorophyll biosynthesis. It employs one cofactor, pyridoxal phosphate.
Structural studies.
As of late 2007, 10 structures have been solved for this class of enzymes, with PDB accession codes 2CFB, 2E7U, 2EPJ, 2GSA, 2HOY, 2HOZ, 2HP1, 2HP2, 3GSB, and 4GSA.
References.
<templatestyles src="Reflist/styles.css" />
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[
{
"math_id": 0,
"text": "\\rightleftharpoons"
}
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https://en.wikipedia.org/wiki?curid=14215507
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14215535
|
Glutamate racemase
|
In enzymology, glutamate racemase (MurI with a capital "i") (EC 5.1.1.3) is an enzyme that catalyzes the chemical reaction
L-glutamate formula_0 D-glutamate
Hence, this enzyme RacE has one substrate, L-glutamate, and one product, D-glutamate.
This enzyme belongs to the family of isomerases, specifically those racemases and epimerases acting on amino acids and derivatives, including proline racemase, aspartate racemase, and diaminopimelate epimerase. This enzyme participates in glutamate metabolism that is essential for cell wall biosynthesis in bacteria. Glutamate racemase performs the additional function of gyrase inhibition, preventing gyrase from binding to DNA.
Glutamate racemase (MurI) serves two distinct metabolic functions: primarily, it is a critical enzyme in cell wall biosynthesis, but also plays a role in gyrase inhibition. The ability of glutamate racemase and other proteins to serve two distinct functions is known as "moonlighting".
Moonlighting background.
Before the discovery of moonlighting proteins, it was generally believed by scientists that an enzyme only had one function which led to the concept of "one gene, one enzyme". However, this concept no longer applies in science after the discovery that some proteins consist of both major and minor functions. This led to numerous studies attempting to relate the two functions to each other. The minor functions of these unique enzymes are called moonlighting functions, in which a protein can have a secondary functions not dependent upon the main function. These two functions of the moonlighting protein are found in a single polypeptide chain. Proteins that are multifunctional are not included due to gene fusion, families of homologous proteins, splice variants or promiscuous enzyme activities. The enzyme glutamate racemase (MurI) is an example of a moonlighting protein, functioning both in bacterial cell wall biosynthesis as well as in gyrase inhibition.
Structure.
The dimensions of MurI is approximately 35 Å × 40 Å × 45 Å and consists of two compact domains of α/β structure. With the active site in between the two domains, the N-terminal domain contains residues 1-97 and 207-264 while the C-terminal domain includes residues 98-206. This allows the enzyme to produce L-isomer from D-glutamate. Also, the N-domain is composed of five-stranded β-sheets compared to four-stranded β-sheets of C-domain. These structural specifications are not identical between MurI of different species; "S. pyogenes" and "B. subtilis" actually possess the most structurally similar MurI enzymes yet found. It is also not rare to find MurI as a dimer.
The active site, as it is evenly between the N-domain and C-domain, is also between the two cysteine residues. It is accessible to solvents, as several water molecules, such as W1, are found in the active site. In some species, the active site also incorporates sulfate ions to undergo hydrogen bonding on the amide backbone and the side chains.
Function.
Bacterial wall synthesis.
Glutamate racemase is a bacterial enzyme that is encoded by the "murI" gene. This enzyme is most commonly known as being responsible for the synthesis of bacterial cell walls. Through experimentation it was found that this enzyme is able to construct these cell walls by synthesizing D-glutamate from L-glutamate through racemization. D-glutamate is a monomer of the peptidoglycan layer in prokaryotic cell walls. Peptidoglycan is an essential structural component of the bacterial cell wall. The peptidoglycan layer is also responsible for the rigidity of the cell wall. This process, in which MurI helps catalyze the interconversion of glutamate enantiomers, like L-Glutamate, into the essential D-glutamate, is also cofactor independent. As such it can proceed without needing an additional source, which would bind to an allosteric site, altering the enzyme shape to assist in catalyzing the reaction. Murl involves a two-step process to catalyze the glutamate enantiomers to D-glutamate. The first step is a deprotonation of the substrate to form an anion. Subsequently, the substrate gets reprotonated. Once the glutamate is in the active site of the enzyme it undergoes a very large conformational change of its domains. This change helps superimpose the two catalytic cysteine residues, Cys73 and Cys184, located on either sides of the substrate at equal positions. Those domains mentioned earlier are symmetric and this symmetry suggests that this racemase activity of the protein may have evolved from gene duplication. Due to this main function of biosynthesis of bacterial cell walls MurI has been targeted as an antibacterial in drug discovery.
Gyrase inhibition.
Along with its main function of cell wall biosynthesis, the moonlighting protein glutamate racemase also functions independently as a gyrase inhibitor. Present in certain forms of bacteria, MurI reduces the activity of DNA gyrase by preventing gyrase from binding to DNA. When gyrase binds to DNA, the enzyme decreases the tension in the DNA strands as they are unwound and causes the strands to become supercoiled. This is a critical step in DNA replication in these cells which results in the reproduction of bacterial cells. The presence of glutamate racemase in the process inhibits gyrase from effectively binding to DNA by deforming the shape of the enzyme's active site. It essentially disallows gyrase from catalyzing the reaction that coils unwinding DNA strands.
This function of MurI was discovered experimentally. DNA gyrase was incubated with the MurI enzyme and then added to a sample of DNA; the results of this experiment showed inhibition of supercoiling activity when MurI was present. The cell wall biosynthesis function of MurI is not directly related to its moonlighting function. MurI's ability to inhibit gyrase binding can proceed independently of its main function. This means that DNA gyrase, in turn, will not have any effect on MurI's racemization, which was confirmed in a study of the racemization with and without the presence of DNA gyrase. In an experimental analysis, it was determined that MurI employs the use of two different enzymatic active sites for its two functions. This was shown by the inclusion of the racemase substrate L-glutamate in an assay with the separated gyrase inhibition site. The gyrase inhibition occurs in both supercoiling and relaxing activities of the DNA gyrase, and the study concluded that the inhibition activity was able to proceed, unchanged, in the presence of the racemase substrate. This dictates that the two functions can be carried out independently of each other, on non-overlapping sites, making MurI a true moonlighting protein. Mutant forms of MurI that are unable to exhibit their racemase function, no matter how compromised their racemase abilities were, were still proven through a study to be able to perform the DNA gyrase inhibition, with comparable results to a non-mutated form of MurI.
Relationship between main and moonlighting functions.
Glutamate racemase (MurI) provides multiple functions for bacterial cells. MurI is an enzyme which is primarily known for its role in synthesizing bacterial cell walls. While performing the function of cell wall synthesis, MurI also acts as a gyrase inhibitor, preventing gyrase from binding to DNA. The two processes have been shown two be unrelated. In order to ascertain the effects of gyrase inhibition on cell wall synthesis, the efficiency of the conversion of D-glutamate to L-glutamate was measured while varying the concentration of DNA gyrase. Conversely, the effects of cell wall production on gyrase inhibition were discovered by varying the concentration of the racemization substrate. The results of these experiments conclude that there is no significant effect of racemization on gyrase inhibition or vice versa. The two functions of MurI act independently of each other reaffirming the fact that MurI is a moonlighting protein.
Relationship to active site.
Glutamate racemase is known to use its active site to undergo racemization and participate in the cell wall biosynthesis pathway of bacteria. Based on homology to other racemases and epimerases, glutamate racemase is thought to employ two active site cysteine residues as acid/base catalysts. Surprisingly however, substituting either of the two residues with serine did not appreciable change the rate of the reaction significantly; the kcat value remained within .3% to 3% compared to the wild-type enzyme. From previous studies, it is most likely that the active site of MurI that performs racemization is not the same active site that undergoes gyrase inhibition. In order to ascertain the effects of gyrase inhibition on cell wall synthesis, the efficiency of the conversion of D-glutamate to L-glutamate was measured while varying the concentration of DNA gyrase. Conversely, the effects of cell wall production on gyrase inhibition were discovered by varying the concentration of the racemization substrate. It has been shown that the two functions are neutral to each other. In other words, racemization substrates are neutral to gyrase inhibition, and DNA gyrase has no effect on racemization. This explains how glutamate racemase in certain bacteria, such as Glr from "B. subtilis", do not inhibit gyrase; if one active site is involved with both functions, this independence would not be possible. Consequently, a different site of MurI, distant from its active site, is involved in interacting with gyrase.
Enzyme regulation.
This protein may use the morpheein model of allosteric regulation.
Application.
Glutamate racemase has emerged as a potential antibacterial target since the product of this enzyme, D-glutamate, is an essential component of bacterial walls. Inhibiting the enzyme will prevent bacterial wall formation and ultimately result in lysis of the bacteria cell by osmotic pressure. Furthermore, glutamate racemase is not expressed nor is the product of this enzyme, D-glutamate is normally found in mammals, hence inhibiting this enzyme should not result in toxicity to the mammalian host organism. Possible inhibitors to MurI includes aziridino-glutamate that would alkylate the catalytic cysteines; N-hydroxy glutamate that by mimicking Wat2 (the bound water molecule that interacts with glutamate amino group) would prevent binding of the substrate; or 4-substituted D-glutamic acid analogs bearing aryl-, heteroaryl-, cinnamyl-, or biaryl-methyl substituents that would also prevent binding of substrate.
References.
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Further reading.
<templatestyles src="Refbegin/styles.css" />
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[
{
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"text": "\\rightleftharpoons"
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] |
https://en.wikipedia.org/wiki?curid=14215535
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14215558
|
(hydroxyamino)benzene mutase
|
Class of enzymes
In enzymology, a (hydroxyamino)benzene mutase (EC 5.4.4.1) is an enzyme that catalyzes the chemical reaction
(hydroxyamino)benzene formula_0 2-aminophenol
Hence, this enzyme has one substrate, (hydroxyamino)benzene, and one product, 2-aminophenol.
This enzyme belongs to the family of isomerases, specifically those intramolecular transferases transferring hydroxy groups. The systematic name of this enzyme class is (hydroxyamino)benzene hydroxymutase. Other names in common use include HAB mutase, hydroxylaminobenzene hydroxymutase, and hydroxylaminobenzene mutase. This enzyme participates in naphthalene and anthracene degradation.
References.
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[
{
"math_id": 0,
"text": "\\rightleftharpoons"
}
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https://en.wikipedia.org/wiki?curid=14215558
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14215583
|
Hydroxypyruvate isomerase
|
In enzymology, a hydroxypyruvate isomerase (EC 5.3.1.22) is an enzyme that catalyzes the chemical reaction
hydroxypyruvate formula_0 2-hydroxy-3-oxopropanoate
Hence, this enzyme has one substrate, hydroxypyruvate, and one product, 2-hydroxy-3-oxopropanoate.
This enzyme belongs to the family of isomerases, specifically those intramolecular oxidoreductases interconverting aldoses and ketoses. The systematic name of this enzyme class is hydroxypyruvate aldose-ketose-isomerase. This enzyme participates in glyoxylate and dicarboxylate metabolism.
References.
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[
{
"math_id": 0,
"text": "\\rightleftharpoons"
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https://en.wikipedia.org/wiki?curid=14215583
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14215610
|
Inositol-3-phosphate synthase
|
In enzymology, an inositol-3-phosphate synthase (EC 5.5.1.4) is an enzyme that catalyzes the chemical reaction
D-glucose 6-phosphate formula_0 1D-myo-inositol 3-phosphate
Hence, this enzyme has one substrate, D-glucose 6-phosphate, and one product, 1D-myo-inositol 3-phosphate.
This enzyme belongs to the family of isomerases, specifically the class of intramolecular lyases. The systematic name of this enzyme class is 1D-myo-inositol-3-phosphate lyase (isomerizing). Other names in common use include myo-inositol-1-phosphate synthase, D-glucose 6-phosphate cycloaldolase, inositol 1-phosphate synthatase, glucose 6-phosphate cyclase, inositol 1-phosphate synthetase, glucose-6-phosphate inositol monophosphate cycloaldolase, glucocycloaldolase, and 1L-myo-inositol-1-phosphate lyase (isomerizing).
This enzyme participates in streptomycin biosynthesis and inositol phosphate metabolism. It employs one cofactor, NAD+. The reaction this enzyme catalyses represents the first committed step in the production of all inositol-containing compounds, including phospholipids, either directly or by salvage. The enzyme exists in a cytoplasmic form in a wide range of plants, animals, and fungi. It has also been detected in several bacteria and a chloroplast form is observed in alga and higher plants. Inositol phosphates play an important role in signal transduction.
In "Saccharomyces cerevisiae" (Baker's yeast), the transcriptional regulation of the INO1 gene encoding inositol-3-phosphate synthase has been studied in detail and its expression is sensitive to the availability of phospholipid precursors as well as growth phase. The regulation of the structural gene encoding 1L-myo-inositol-1-phosphate synthase has also been analyzed at the transcriptional level in the aquatic angiosperm, "Spirodela polyrrhiza" (Giant duckweed) and the halophyte, "Mesembryanthemum crystallinum" (Common ice plant).
In prokaryotes, myo-D-inositol phosphate synthase was discovered by Bachhawat and Mande in 1999 (reported in Journal of Molecular Biology). The existence of inositol in prokaryotes is not extensive, but the discovery of this enzyme first in Mycobacterium tuberculosis, nucleated activity towards finding its inhibitors.
Structural studies.
As of late 2007, 12 structures have been solved for this class of enzymes, with PDB accession codes 1GR0, 1JKF, 1JKI, 1LA2, 1P1F, 1P1H, 1P1I, 1P1J, 1P1K, 1RM0, 1VJP, and 1VKO.
References.
<templatestyles src="Reflist/styles.css" />
Further reading.
Bachhawat N and Mande SC (1999) J. Mol. Biol. Identification of the INO1 gene of Mycobacterium tuberculosis H37Rv reveals a novel class of inositol-1-phosphate synthase enzyme. 291, 531–536.
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[
{
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"text": "\\rightleftharpoons"
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https://en.wikipedia.org/wiki?curid=14215610
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14215636
|
Isobutyryl-CoA mutase
|
In enzymology, an isobutyryl-CoA mutase (EC 5.4.99.13) is an enzyme that catalyzes the chemical reaction
2-methylpropanoyl-CoA formula_0 butanoyl-CoA
Hence, this enzyme has one substrate, 2-methylpropanoyl-CoA, and one product, butanoyl-CoA.
This enzyme belongs to the family of isomerases, specifically those intramolecular transferases transferring other groups. The systematic name of this enzyme class is 2-methylpropanoyl-CoA CoA-carbonylmutase. Other names in common use include isobutyryl coenzyme A mutase, and butyryl-CoA:isobutyryl-CoA mutase. It uses adenosylcobalamin as a cofactor, which is bound at the enzyme's vitamin B12-binding domain. The mechanism of action of the enzyme is to generate a 5′-deoxyadenosyl radical by homolytic cleavage of the cobalt-carbon bond of the cofactor. This radical abstracts a hydrogen atom from the substrate to initiate the rearrangement reaction.
References.
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<templatestyles src="Refbegin/styles.css" />
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[
{
"math_id": 0,
"text": "\\rightleftharpoons"
}
] |
https://en.wikipedia.org/wiki?curid=14215636
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14215703
|
Isomaltulose synthase
|
In enzymology, an isomaltulose synthase (EC 5.4.99.11) is an enzyme that catalyzes the chemical reaction
sucrose formula_0 6-O-Alpha-D-Glucopyranosyl-D-Fructofuranose
Hence, this enzyme has one substrate, sucrose (table sugar), and one product, 6-O-Alpha-D-Glucopyranosyl-D-Fructofuranose (also known as isomaltulose or Palatinose). It converts the α-1,2 glycosidic linkage between glucose and fructose in sucrose into the α-1,6 glycosidic linkage between glucose and fructose in isomaltulose.
This enzyme belongs to the family of isomerases. The systematic name of this enzyme class is sucrose glucosylmutase. Other names in common use include sucrose isomerase, sucrose alpha-glucosyltransferase, and trehalulose synthase.
The isomaltulose synthase of the bacterium "Protaminobacter rubrum" is commonly used in the industrial production of isomaltulose.
Structural studies.
As of late 2007, 7 structures have been solved for this class of enzymes, with PDB accession codes 1M53, 1ZJA, 1ZJB, 2PWD, 2PWE, 2PWF, and 2PWG.
References.
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[
{
"math_id": 0,
"text": "\\rightleftharpoons"
}
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https://en.wikipedia.org/wiki?curid=14215703
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14215729
|
Isopenicillin N epimerase
|
In enzymology, an isopenicillin N epimerase (EC 5.1.1.17) is an enzyme that catalyzes the chemical reaction
isopenicillin N formula_0 penicillin N
Hence, this enzyme has one substrate, isopenicillin N, and one product, penicillin N.
This enzyme belongs to the family of isomerases, specifically those racemases and epimerases acting on amino acids and derivatives. The systematic name of this enzyme class is penicillin N 5-amino-5-carboxypentanoyl-epimerase. This enzyme participates in penicillin and cephalosporin biosynthesis.
References.
<templatestyles src="Reflist/styles.css" />
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[
{
"math_id": 0,
"text": "\\rightleftharpoons"
}
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https://en.wikipedia.org/wiki?curid=14215729
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14215777
|
Isopiperitenone Delta-isomerase
|
Class of enzymes
In enzymology, an isopiperitenone Delta-isomerase (EC 5.3.3.11) is an enzyme that catalyzes the chemical reaction
isopiperitenone formula_0 piperitenone
Hence, this enzyme has one substrate, isopiperitenone, and one product, piperitenone.
This enzyme belongs to the family of isomerases, specifically those intramolecular oxidoreductases transposing C=C bonds. The systematic name of this enzyme class is isopiperitenone Delta8-Delta4-isomerase.
References.
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[
{
"math_id": 0,
"text": "\\rightleftharpoons"
}
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https://en.wikipedia.org/wiki?curid=14215777
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14215815
|
L-arabinose isomerase
|
In enzymology, a L-arabinose isomerase (EC 5.3.1.4) is an enzyme that catalyzes the chemical reaction
L-arabinose formula_0 L-ribulose
Hence, this enzyme has one substrate, L-arabinose, and one product, L-ribulose.
This enzyme belongs to the family of isomerases, specifically those intramolecular oxidoreductases interconverting aldoses and ketoses. The systematic name of this enzyme class is L-arabinose aldose-ketose-isomerase. This enzyme participates in pentose and glucuronate interconversions.
This enzyme catalyses the conversion of L-arabinose to L-ribulose as the first step in the pathway of L-arabinose utilization as a carbon source.
Structural studies.
As of late 2007, two structures have been solved for this class of enzymes, with PDB accession codes 2AJT and 2HXG.
References.
<templatestyles src="Reflist/styles.css" />
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[
{
"math_id": 0,
"text": "\\rightleftharpoons"
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https://en.wikipedia.org/wiki?curid=14215815
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14215853
|
L-dopachrome isomerase
|
In enzymology, a L-dopachrome isomerase (EC 5.3.3.12) is an enzyme that catalyzes the chemical reaction
L-dopachrome formula_0 5,6-dihydroxyindole-2-carboxylate
Hence, this enzyme has one substrate, L-dopachrome, and one product, 5,6-dihydroxyindole-2-carboxylate.
This enzyme belongs to the family of isomerases, specifically those intramolecular oxidoreductases transposing C=C bonds. The systematic name of this enzyme class is L-dopachrome keto-enol isomerase. Other names in common use include dopachrome tautomerase, tyrosinase-related protein 2, TRP-1, TRP2, TRP-2, tyrosinase-related protein-2, dopachrome Delta7,Delta2-isomerase, dopachrome Delta-isomerase, dopachrome conversion factor, dopachrome isomerase, dopachrome oxidoreductase, dopachrome-rearranging enzyme, DCF, DCT, dopachrome keto-enol isomerase, and L-dopachrome-methyl ester tautomerase. This enzyme participates in tyrosine metabolism and melanogenesis.
References.
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[
{
"math_id": 0,
"text": "\\rightleftharpoons"
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https://en.wikipedia.org/wiki?curid=14215853
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14215880
|
Leucine 2,3-aminomutase
|
In enzymology, a leucine 2,3-aminomutase (EC 5.4.3.7) is an enzyme that catalyzes the chemical reaction
(2S)-alpha-leucine formula_0 (3R)-beta-leucine
Hence, this enzyme is responsible for the conversion of L-leucine to β-leucine.
This enzyme belongs to the family of isomerases, specifically those intramolecular transferases transferring amino groups. The systematic name of this enzyme class is (2S)-alpha-leucine 2,3-aminomutase. This enzyme participates in valine, leucine and isoleucine degradation. It employs one cofactor, cobamide.
References.
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[
{
"math_id": 0,
"text": "\\rightleftharpoons"
}
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https://en.wikipedia.org/wiki?curid=14215880
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14215906
|
L-fucose isomerase
|
In enzymology, a L-fucose isomerase (EC 5.3.1.25) is an enzyme that catalyzes the chemical reaction
L-fucose formula_0 L-fuculose
Hence, this enzyme has one substrate, L-fucose, and one product, L-fuculose.
This enzyme belongs to the family of isomerases, specifically those intramolecular oxidoreductases interconverting aldoses and ketoses. The systematic name of this enzyme class is L-fucose aldose-ketose-isomerase. This enzyme participates in fructose and mannose metabolism.
The enzyme is a hexamer, forming the largest structurally known ketol isomerase, and has no sequence or structural similarity with other ketol isomerases. The structure was determined by X-ray crystallography at 2.5 Angstrom resolution. Each subunit of the hexameric enzyme is wedge-shaped and composed of three domains. Both domains 1 and 2 contain central parallel beta- sheets with surrounding alpha helices. The active centre is shared between pairs of subunits related along the molecular three-fold axis, with domains 2 and 3 from one subunit providing most of the substrate-contacting residues.
References.
<templatestyles src="Reflist/styles.css" />
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[
{
"math_id": 0,
"text": "\\rightleftharpoons"
}
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https://en.wikipedia.org/wiki?curid=14215906
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14215940
|
Linoleate isomerase
|
Enzyme
In enzymology, a linoleate isomerase (EC 5.2.1.5) is an enzyme that catalyzes the chemical reaction
9-cis,12-cis-octadecadienoate formula_0 9-cis,11-trans-octadecadienoate
Hence, this enzyme has one substrate, 9-cis,12-cis-octadecadienoate, and one product, 9-cis,11-trans-octadecadienoate.
This enzyme belongs to the family of isomerases, specifically cis-trans isomerases. The systematic name of this enzyme class is linoleate Delta12-cis-Delta11-trans-isomerase. This enzyme is also called linoleic acid isomerase. This enzyme participates in linoleic acid metabolism.
References.
<templatestyles src="Reflist/styles.css" />
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[
{
"math_id": 0,
"text": "\\rightleftharpoons"
}
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https://en.wikipedia.org/wiki?curid=14215940
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14215968
|
L-rhamnose isomerase
|
In enzymology, a L-rhamnose isomerase (EC 5.3.1.14) is an enzyme that catalyzes the chemical reaction
L-rhamnose formula_0 L-rhamnulose
Hence, this enzyme has one substrate, L-rhamnose, and one product, L-rhamnulose.
This enzyme belongs to the family of isomerases, specifically those intramolecular oxidoreductases interconverting aldoses and ketoses. The systematic name of this enzyme class is L-rhamnose aldose-ketose-isomerase. Other names in common use include rhamnose isomerase, and L-rhamnose ketol-isomerase. This enzyme participates in fructose and mannose metabolism.
Structural studies.
As of late 2007, 6 structures have been solved for this class of enzymes, with PDB accession codes 1D8W, 1DE5, 1DE6, 2HCV, 2I56, and 2I57.
References.
<templatestyles src="Reflist/styles.css" />
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[
{
"math_id": 0,
"text": "\\rightleftharpoons"
}
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https://en.wikipedia.org/wiki?curid=14215968
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