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14266568
|
Triglucosylalkylacylglycerol sulfotransferase
|
Class of enzymes
In enzymology, a triglucosylalkylacylglycerol sulfotransferase (EC 2.8.2.19) is an enzyme that catalyzes the chemical reaction
3'-phosphoadenylyl sulfate + α-D-glucosyl-(1→6)-alpha-D-glucosyl-(1→6)-α-D-glucosyl-(1→3)-1-O-alkyl-2-O-acylglycerol formula_0 adenosine 3',5'-bisphosphate + 6-sulfo-α-D-glucosyl-(1→6)-α-D-glucosyl-(1→6)-α-D-glucosyl-(1→3)-1-O-alkyl-2-O-acylglycerol
This enzyme belongs to the family of transferases, specifically the sulfotransferases, which transfer sulfur-containing groups. The systematic name of this enzyme class is 3'-phosphoadenylyl-sulfate:triglucosyl-1-O-alkyl-2-O-acylglycerol 6-sulfotransferase. This enzyme is also called triglucosylmonoalkylmonoacyl sulfotransferase.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "\\rightleftharpoons"
}
] |
https://en.wikipedia.org/wiki?curid=14266568
|
14266580
|
TRNA sulfurtransferase
|
Class of enzymes
In enzymology, a tRNA sulfurtransferase (EC 2.8.1.4) is an enzyme that catalyzes the chemical reaction
L-cysteine + 'activated' tRNA formula_0 L-serine + tRNA containing a thionucleotide
Thus, the two substrates of this enzyme are L-cysteine and 'activated' tRNA, whereas its two products are L-serine and tRNA containing a thionucleotide.
This enzyme belongs to the family of transferases, specifically the sulfurtransferases, which transfer sulfur-containing groups. The systematic name of this enzyme class is L-cysteine:tRNA sulfurtransferase. Other names in common use include transfer ribonucleate sulfurtransferase, RNA sulfurtransferase, ribonucleate sulfurtransferase, transfer RNA sulfurtransferase, and transfer RNA thiolase.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "\\rightleftharpoons"
}
] |
https://en.wikipedia.org/wiki?curid=14266580
|
14266597
|
Tyrosine-ester sulfotransferase
|
Class of enzymes
In enzymology, a tyrosine-ester sulfotransferase (EC 2.8.2.9) is an enzyme that catalyzes the chemical reaction
3'-phosphoadenylyl sulfate + L-tyrosine methyl ester formula_0 adenosine 3',5'-bisphosphate + L-tyrosine methyl ester 4-sulfate
Thus, the two substrates of this enzyme are 3'-phosphoadenylyl sulfate and L-tyrosine methyl ester, whereas its two products are adenosine 3',5'-bisphosphate and L-tyrosine methyl ester 4-sulfate.
This enzyme belongs to the family of transferases, specifically the sulfotransferases, which transfer sulfur-containing groups. The systematic name of this enzyme class is 3'-phosphoadenylyl-sulfate:L-tyrosine-methyl-ester sulfotransferase. Other names in common use include aryl sulfotransferase IV, and L-tyrosine methyl ester sulfotransferase.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "\\rightleftharpoons"
}
] |
https://en.wikipedia.org/wiki?curid=14266597
|
14266614
|
UDP-N-acetylgalactosamine-4-sulfate sulfotransferase
|
Class of enzymes
In enzymology, an UDP-N-acetylgalactosamine-4-sulfate sulfotransferase (EC 2.8.2.7) is an enzyme that catalyzes the chemical reaction
3'-phosphoadenylyl sulfate + UDP-N-acetyl-D-galactosamine 4-sulfate formula_0 adenosine 3',5'-bisphosphate + UDP-N-acetyl-D-galactosamine 4,6-bissulfate
Thus, the two substrates of this enzyme are 3'-phosphoadenylyl sulfate and UDP-N-acetyl-D-galactosamine 4-sulfate, whereas its two products are adenosine 3',5'-bisphosphate and UDP-N-acetyl-D-galactosamine 4,6-bissulfate.
This enzyme belongs to the family of transferases, specifically the sulfotransferases, which transfer sulfur-containing groups. The systematic name of this enzyme class is 3'-phosphoadenylyl-sulfate:UDP-N-acetyl-D-galactosamine-4-sulfate 6-sulfotransferase. Other names in common use include uridine diphosphoacetylgalactosamine 4-sulfate sulfotransferase, uridine diphospho-N-acetylgalactosamine 4-sulfate sulfotransferase, and uridine diphosphoacetylgalactosamine 4-sulfate sulfotransferase.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "\\rightleftharpoons"
}
] |
https://en.wikipedia.org/wiki?curid=14266614
|
1426858
|
William Brouncker, 2nd Viscount Brouncker
|
Anglo-Irish peer and mathematician (1620–1684)
William Brouncker, 2nd Viscount Brouncker FRS (c. 1620 – 5 April 1684) was an Anglo-Irish peer and mathematician who served as the president of the Royal Society from 1662 to 1677. Best known for introducing Brouncker's formula, he also worked as a civil servant, serving as a commissioner in the Royal Navy. Brouncker was a friend and colleague of Samuel Pepys, and features prominently in the Pepys' diary.
Biography.
Brouncker was born c. 1620 in Castlelyons, County Cork, the elder son of William Brouncker (1585–1649), 1st Viscount Brouncker and Winifred, daughter of Sir William Leigh of Newnham. His family came originally from Melksham in Wiltshire. His grandfather Sir Henry Brouncker (died 1607) had been Lord President of Munster 1603–1607, and settled his family in Ireland. His father was created a viscount in the Peerage of Ireland in 1645 for his services to the Crown. Although the first viscount had fought for the Crown in the Anglo-Scots war of 1639, malicious gossip said that he paid the then enormous sum of £1200 for the title and was almost ruined as a result. He died only a few months afterwards.
William obtained a DM at the University of Oxford in 1647. Until 1660 he played no part in public life: being a staunch Royalist, he felt it best to live quietly and devote himself to his mathematical studies. He was one of the founders and the first president of the Royal Society. In 1662, he became chancellor to Queen Catherine, then head of the Saint Catherine's Hospital. He was appointed one of the commissioners of the Royal Navy in 1664, and his career thereafter can be traced in the "Diary of Samuel Pepys"; despite their frequent disagreements, Samuel Pepys on the whole respected Brouncker more than most of his other colleagues, writing in 1668 that "in truth he is the best of them".
Although his attendance at the Royal Society had become infrequent, and he had quarrelled with some of his fellow members, he was nonetheless greatly displeased to be deprived of the presidency in 1677. He was commissioner for executing the office of Lord High Admiral of England from 1679.
Abigail Williams.
Brouncker never married, but lived for many years with the actress Abigail Williams (much to Pepys' disgust) and left most of his property to her. She was the daughter of Sir Henry Clere (died 1622), first and last of the Clere Baronets, and the estranged wife of John Williams, otherwise Cromwell, second son of Sir Oliver Cromwell, and first cousin to the renowned Oliver Cromwell. She and John had a son and a daughter. The fire of 1673 which destroyed the Royal Navy Office started in her private closet: this is unlikely to have improved her relations with Samuel Pepys, whose private apartments were also destroyed in the blaze.
On Brouncker's death in 1684, his title passed to his brother Henry, one of the most detested men of the era. William left him almost nothing in his will "for reasons I think not fit to mention".
Mathematical works.
His mathematical work concerned in particular the calculations of the lengths of the parabola and cycloid, and the quadrature of the hyperbola, which requires approximation of the natural logarithm function by infinite series. He was the first European to solve what is now known as Pell's equation. He was the first in England to take interest in generalized continued fractions and, following the work of John Wallis, he provided development in the generalized continued fraction of pi.
Brouncker's formula.
This formula provides a development of π/4 in a generalized continued fraction:
formula_0
The convergents are related to the Leibniz formula for pi: for instance
formula_1
and
formula_2
Because of its slow convergence, Brouncker's formula is not useful for practical computations of π.
Brouncker's formula can also be expressed as
formula_3
See Euler's continued fraction formula.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "\n\\frac \\pi 4 = \\cfrac{1}{1+\\cfrac{1^2}{2+\\cfrac{3^2}{2+\\cfrac{5^2}{2+\\cfrac{7^2}{2+\\cfrac{9^2}{2+\\ddots}}}}}}\n"
},
{
"math_id": 1,
"text": "\n\\frac{1}{1+\\frac{1^2}{2}} = \\frac{2}{3} = 1 - \\frac{1}{3}\n"
},
{
"math_id": 2,
"text": "\n\\frac{1}{1+\\frac{1^2}{2+\\frac{3^2}{2}}} = \\frac{13}{15} = 1 - \\frac{1}{3} + \\frac{1}{5}.\n"
},
{
"math_id": 3,
"text": "\n\\frac 4 \\pi = 1+\\cfrac{1^2}{2+\\cfrac{3^2}{2+\\cfrac{5^2}{2+\\cfrac{7^2}{2+\\cfrac{9^2}{2+\\ddots}}}}}\n"
}
] |
https://en.wikipedia.org/wiki?curid=1426858
|
14269777
|
Helicity basis
|
In the Standard Model, using quantum field theory it is conventional to use the helicity basis to simplify calculations (of cross sections, for example). In this basis, the spin is quantized along the axis in the direction of motion of the particle.
Spinors.
The two-component helicity eigenstates formula_0 satisfy
formula_1
where
formula_2 are the Pauli matrices,
formula_3 is the direction of the fermion momentum,
formula_4 depending on whether spin is pointing in the same direction as formula_3 or opposite.
To say more about the state, formula_5 we will use the generic form of fermion four-momentum:
formula_6
Then one can say the two helicity eigenstates are
formula_7
and
formula_8
These can be simplified by defining the z-axis such that the momentum direction is either parallel or anti-parallel, or rather:
formula_9.
In this situation the helicity eigenstates are for when the particle momentum is formula_10
formula_11 and formula_12
then for when momentum is formula_13
formula_14 and formula_15
Fermion (spin 1/2) wavefunction.
A fermion 4-component wave function, formula_16 may be decomposed into states with definite four-momentum:
formula_17
where
formula_18 and formula_19 are the creation and annihilation operators, and
formula_20 and formula_21 are the momentum-space Dirac spinors for a fermion and anti-fermion respectively.
Put it more explicitly, the Dirac spinors in the helicity basis for a fermion is
formula_22
and for an anti-fermion,
formula_23
Dirac matrices.
To use these helicity states, one can use the Weyl (chiral) representation for the Dirac matrices.
Spin-1 wavefunctions.
The plane wave expansion is
formula_24.
For a vector boson with mass "m" and a four-momentum formula_25, the polarization vectors quantized with respect to its momentum direction can be defined as
formula_26
where
formula_27 is transverse momentum, and
formula_28 is the energy of the boson.
|
[
{
"math_id": 0,
"text": "\\xi_\\lambda"
},
{
"math_id": 1,
"text": "\\sigma \\cdot \\hat{p} \\xi_\\lambda\\left(\\hat{p}\\right) = \\lambda \\xi_\\lambda\\left(\\hat{p}\\right) \\,"
},
{
"math_id": 2,
"text": "\\sigma \\,"
},
{
"math_id": 3,
"text": "\\hat{p} \\,"
},
{
"math_id": 4,
"text": "\\lambda = \\pm 1 \\,"
},
{
"math_id": 5,
"text": "\\xi_\\lambda \\,"
},
{
"math_id": 6,
"text": "p^\\mu = \\left(E, \\left|\\vec{p}\\right| \\sin{\\theta} \\cos{\\phi}, \\left|\\vec{p}\\right| \\sin{\\theta} \\sin{\\phi}, \\left|\\vec{p}\\right| \\cos{\\theta} \\right) \\,"
},
{
"math_id": 7,
"text": "\\xi_{+1}(\\vec{p}) =\n \\frac{1}{\\sqrt{2 \\left|\\vec{p}\\right|\\left(\\left|\\vec{p}\\right| + p_z\\right)}} \n \\begin{pmatrix}\n \\left|\\vec{p}\\right| + p_z\\\\\n p_x + i p_y\n \\end{pmatrix} = \\begin{pmatrix}\n \\cos{\\frac{\\theta}{2}} \\\\\n e^{i\\phi}\\sin{\\frac{\\theta}{2}}\n \\end{pmatrix}\\,\n"
},
{
"math_id": 8,
"text": "\n \\xi_{-1}(\\vec{p}) =\n \\frac{1}{\\sqrt{2 |\\vec{p}|(|\\vec{p}| + p_z)}} \n \\begin{pmatrix}\n -p_x + i p_y \\\\\n \\left|\\vec{p}\\right| + p_z\n \\end{pmatrix} = \n \\begin{pmatrix}\n -e^{-i\\phi}\\sin{\\frac{\\theta}{2}} \\\\\n \\cos{\\frac{\\theta}{2}}\n \\end{pmatrix}\\,\n"
},
{
"math_id": 9,
"text": "\\hat{z} = \\pm \\hat{p} \\,"
},
{
"math_id": 10,
"text": " \\hat{p} = + \\hat{z} \\,"
},
{
"math_id": 11,
"text": "\\xi_{+1}(\\hat{z}) = \\begin{pmatrix}\n 1 \\\\\n 0\n \\end{pmatrix} \\,"
},
{
"math_id": 12,
"text": "\\xi_{-1}(\\hat{z}) = \\begin{pmatrix}\n 0 \\\\\n 1\n \\end{pmatrix} \\,\n"
},
{
"math_id": 13,
"text": " \\hat{p} = - \\hat{z} \\,"
},
{
"math_id": 14,
"text": "\\xi_{+1}(-\\hat{z}) = \\begin{pmatrix}\n 0 \\\\\n 1\n \\end{pmatrix} \\,"
},
{
"math_id": 15,
"text": "\\xi_{-1}(-\\hat{z}) = \\begin{pmatrix}\n -1 \\\\\n 0\n \\end{pmatrix} \\,\n"
},
{
"math_id": 16,
"text": "\\psi\\,"
},
{
"math_id": 17,
"text": "\\psi(x) = \\int{\\frac{d^3p}{(2\\pi)^3 \\sqrt{2E} } \\sum_{\\lambda \\pm 1}{\\left(\\hat{a}_p^\\lambda u_\\lambda(p) e^{-i p \\cdot x} + \\hat{b}_p^\\lambda v_\\lambda(p) e^{i p \\cdot x} \\right)} } \\,"
},
{
"math_id": 18,
"text": "\\hat{a}_p^\\lambda \\,"
},
{
"math_id": 19,
"text": "\\hat{b}_p^\\lambda \\,"
},
{
"math_id": 20,
"text": " u_\\lambda(p) \\,"
},
{
"math_id": 21,
"text": "v_\\lambda(p) \\,"
},
{
"math_id": 22,
"text": "u_\\lambda(p) = \\begin{pmatrix}\n u_{-1} \\\\\n u_{+1}\n \\end{pmatrix} = \\begin{pmatrix}\n \\sqrt{E - \\lambda \\left|\\vec{p}\\right|} \\chi_\\lambda(\\hat{p}) \\\\\n \\sqrt{E + \\lambda \\left|\\vec{p}\\right|} \\chi_\\lambda(\\hat{p})\n \\end{pmatrix} \\,"
},
{
"math_id": 23,
"text": "v_\\lambda(p) = \\begin{pmatrix}\n v_{+1} \\\\\n v_{-1}\n \\end{pmatrix} = \\begin{pmatrix}\n -\\lambda \\sqrt{E + \\lambda \\left|\\vec{p}\\right|} \\chi_{-\\lambda}(\\hat{p}) \\\\\n \\lambda \\sqrt{E - \\lambda \\left|\\vec{p}\\right|} \\chi_{-\\lambda}(\\hat{p})\n \\end{pmatrix}\n"
},
{
"math_id": 24,
"text": "\\psi(x) =\n \\int{\\frac{d^3p}{(2\\pi)^3 \\sqrt{2E}} \\sum_{\\lambda = 0}^3 \\left(\n \\hat{a}_{p,\\lambda} \\epsilon_\\lambda(p) e^{-i p \\cdot x} +\n \\hat{a}_{p,\\lambda}^\\dagger \\epsilon^*_\\lambda(p) e^{i p \\cdot x}\n \\right)} \\,\n"
},
{
"math_id": 25,
"text": "q^\\mu = (E, q_x, q_y, q_z)"
},
{
"math_id": 26,
"text": "\\begin{align}\n \\epsilon^\\mu(q, x) &= \\frac{1}{\\left|\\vec{q}\\right| q_\\text{T}} \\left(0,q_x q_z, q_y q_z, -q_\\text{T}^2 \\right) \\\\\n \\epsilon^\\mu(q ,y) &= \\frac{1}{q_\\text{T}} \\left( 0, -q_y, q_x, 0 \\right) \\\\\n \\epsilon^\\mu(q, z) &= \\frac{E}{m\\left|\\vec{q}\\right|} \\left(\\frac{\\left|\\vec{q}\\right|^2}{E}, q_x, q_y, q_z \\right)\n\\end{align}"
},
{
"math_id": 27,
"text": "q_\\text{T} = \\sqrt{q_x^2 + q_y^2} \\,"
},
{
"math_id": 28,
"text": "E = \\sqrt{|\\vec{q}|^2 + m^2} \\,"
}
] |
https://en.wikipedia.org/wiki?curid=14269777
|
14271694
|
Exceptional divisor
|
In mathematics, specifically algebraic geometry, an exceptional divisor for a regular map
formula_0
of varieties is a kind of 'large' subvariety of formula_1 which is 'crushed' by formula_2, in a certain definite sense. More strictly, "f" has an associated exceptional locus which describes how it identifies nearby points in codimension one, and the exceptional divisor is an appropriate algebraic construction whose support is the exceptional locus. The same ideas can be found in the theory of holomorphic mappings of complex manifolds.
More precisely, suppose that
formula_0
is a regular map of varieties which is birational (that is, it is an isomorphism between open subsets of formula_1 and formula_3). A codimension-1 subvariety formula_4 is said to be "exceptional" if formula_5 has codimension at least 2 as a subvariety of formula_3. One may then define the "exceptional divisor" of formula_2 to be
formula_6
where the sum is over all exceptional subvarieties of formula_2, and is an element of the group of Weil divisors on formula_1.
Consideration of exceptional divisors is crucial in birational geometry: an elementary result (see for instance Shafarevich, II.4.4) shows (under suitable assumptions) that any birational regular map that is not an isomorphism has an exceptional divisor. A particularly important example is the blowup
formula_7
of a subvariety
formula_8:
in this case the exceptional divisor is exactly the preimage of formula_9.
|
[
{
"math_id": 0,
"text": "f: X \\rightarrow Y"
},
{
"math_id": 1,
"text": "X"
},
{
"math_id": 2,
"text": "f"
},
{
"math_id": 3,
"text": "Y"
},
{
"math_id": 4,
"text": "Z \\subset X"
},
{
"math_id": 5,
"text": "f(Z)"
},
{
"math_id": 6,
"text": "\\sum_i Z_i \\in Div(X),"
},
{
"math_id": 7,
"text": "\\sigma: \\tilde{X} \\rightarrow X"
},
{
"math_id": 8,
"text": "W \\subset X"
},
{
"math_id": 9,
"text": "W"
}
] |
https://en.wikipedia.org/wiki?curid=14271694
|
14272194
|
Saint-Venant's theorem
|
In solid mechanics, it is common to analyze the properties of beams with constant cross section. Saint-Venant's theorem states that the simply connected cross section with maximal torsional rigidity is a circle. It is named after the French mathematician Adhémar Jean Claude Barré de Saint-Venant.
Given a simply connected domain "D" in the plane with area "A", formula_0 the radius and formula_1 the area of its greatest inscribed circle, the torsional rigidity "P"
of "D" is defined by
formula_2
Here the supremum is taken over all the continuously differentiable functions vanishing on the boundary of "D". The existence of this supremum is a consequence of Poincaré inequality.
Saint-Venant conjectured in 1856 that
of all domains "D" of equal area "A" the circular one has the greatest torsional rigidity, that is
formula_3
A rigorous proof of this inequality was not given until 1948 by Pólya. Another proof was given by Davenport and reported in. A more general proof and an estimate
formula_4
is given by Makai.
Notes.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "\\rho"
},
{
"math_id": 1,
"text": "\\sigma"
},
{
"math_id": 2,
"text": " P= 4\\sup_f \\frac{\\left( \\iint\\limits_D f\\, dx\\, dy\\right)^2}{\\iint\\limits_D {f_x}^2+{f_y}^2\\, dx\\, dy}."
},
{
"math_id": 3,
"text": " P \\le P_{\\text{circle}} \\le \\frac{A^2}{2 \\pi}."
},
{
"math_id": 4,
"text": "P< 4 \\rho^2 A"
}
] |
https://en.wikipedia.org/wiki?curid=14272194
|
1427251
|
Alkalinity
|
Capacity of water to resist changes in pH that would make the water more acidic
Alkalinity (from ) is the capacity of water to resist acidification. It should not be confused with basicity, which is an absolute measurement on the pH scale. Alkalinity is the strength of a buffer solution composed of weak acids and their conjugate bases. It is measured by titrating the solution with an acid such as HCl until its pH changes abruptly, or it reaches a known endpoint where that happens. Alkalinity is expressed in units of concentration, such as meq/L (milliequivalents per liter), μeq/kg (microequivalents per kilogram), or mg/L CaCO3 (milligrams per liter of calcium carbonate). Each of these measurements corresponds to an amount of acid added as a titrant.
In freshwater, particularly those on non-limestone terrains, alkalinities are low and involve a lot of ions. In the ocean, on the other hand, alkalinity is completely dominated by carbonate and bicarbonate plus a small contribution from borate.
Although alkalinity is primarily a term used by limnologists and oceanographers, it is also used by hydrologists to describe temporary hardness. Moreover, measuring alkalinity is important in determining a stream's ability to neutralize acidic pollution from rainfall or wastewater. It is one of the best measures of the sensitivity of the stream to acid inputs. There can be long-term changes in the alkalinity of streams and rivers in response to human disturbances such as acid rain generated by SO"x" and NO"x" emissions.<ref name="10.1021/es401046s"></ref>
History.
In 1884, Professor Wilhelm (William) Dittmar of Anderson College, now the University of Strathclyde, analysed 77 pristine seawater samples from around the world brought back by the Challenger expedition. He found that in seawater the major ions were in a fixed ratio, confirming the hypothesis of Johan Georg Forchhammer, that is now known as the Principle of Constant Proportions. However, there was one exception. Dittmar found that the concentration of calcium was slightly greater in the deep ocean, and named this increase alkalinity.
Also in 1884, Svante Arrhenius submitted his PhD theses in which he advocated the existence of ions in solution, and defined acids as hydronium ion donors and bases as hydroxide ion donors. For that work, he received the Nobel Prize in Chemistry in 1903. See also Svante Arrhenius#Ionic disassociation.
Simplified summary.
Alkalinity roughly refers to the molar amount of bases in a solution that can be converted to uncharged species by a strong acid. For example, 1 mole of HCO3− in solution represents 1 molar equivalent, while 1 mole of CO32− is 2 molar equivalents because twice as many H+ ions would be necessary to balance the charge. The total charge of a solution always equals zero. This leads to a parallel definition of alkalinity that is based upon the charge balance of ions in a solution.
formula_0
Certain ions, including Na+, K+, Ca2+, Mg2+, Cl−, SO42−, and NO3− are "conservative" such that they are unaffected by changes in temperature, pressure or pH. Others such as HCO3− are affected by changes in pH, temperature, and pressure. By isolating the conservative ions on one side of this charge balance equation, the nonconservative ions which accept or donate protons and thus define alkalinity are clustered on the other side of the equation.
formula_1
This combined charge balance and proton balance is called total alkalinity. Total alkalinity is not (much) affected by temperature, pressure, or pH, and is thus itself a conservative measurement, which increases its usefulness in aquatic systems. All anions except HCO3− and CO32− have low concentrations in Earth's surface water (streams, rivers, and lakes). Thus carbonate alkalinity, which is equal to is also approximately equal to the total alkalinity in surface water.
Detailed description.
Alkalinity measures the ability of a solution to neutralize acids to the equivalence point of carbonate or bicarbonate, defined as pH 4.5 for many oceanographic/limnological studies. The alkalinity is equal to the stoichiometric sum of the bases in solution. In most Earth surface waters carbonate alkalinity tends to make up most of the total alkalinity due to the common occurrence and dissolution of carbonate rocks and other geological weathering processes that produce carbonate anions. Other common natural components that can contribute to alkalinity include borate, hydroxide, phosphate, silicate, dissolved ammonia, and the conjugate bases of organic acids (e.g., acetate). Solutions produced in a laboratory may contain a virtually limitless number of species that contribute to alkalinity. Alkalinity is frequently given as molar equivalents per liter of solution or per kilogram of solvent. In commercial (e.g. the swimming pool industry) and regulatory contexts, alkalinity might also be given in parts per million of equivalent calcium carbonate (ppm CaCO3). Alkalinity is sometimes incorrectly used interchangeably with basicity. For example, the addition of CO2 lowers the pH of a solution, thus reducing basicity while alkalinity remains unchanged (see example below).
A variety of titrants, endpoints, and indicators are specified for various alkalinity measurement methods. Hydrochloric and sulfuric acids are common acid titrants, while phenolpthalein, methyl red, and bromocresol green are common indicators.
Theoretical treatment.
In typical groundwater or seawater, the measured total alkalinity is set equal to:
AT = [HCO3-]T + 2[CO32-]T + [B(OH)4-]T + [OH−]T + 2[PO43-]T + [HPO42-]T + [SiO(OH)3-]T − [H+]sws − [HSO4-]
Alkalinity can be measured by titrating a sample with a strong acid until all the buffering capacity of the aforementioned ions above the pH of bicarbonate or carbonate is consumed. This point is functionally set to pH 4.5. At this point, all the bases of interest have been protonated to the zero level species, hence they no longer cause alkalinity. In the carbonate system the bicarbonate ions [HCO3-] and the carbonate ions [CO32-] have become converted to carbonic acid [H2CO3] at this pH. This pH is also called the CO2 equivalence point where the major component in water is dissolved CO2 which is converted to H2CO3 in an aqueous solution. There are no strong acids or bases at this point. Therefore, the alkalinity is modeled and quantified with respect to the CO2 equivalence point. Because the alkalinity is measured with respect to the CO2 equivalence point, the dissolution of CO2, although it adds acid and dissolved inorganic carbon, does not change the alkalinity. In natural conditions, the dissolution of basic rocks and addition of ammonia [NH3] or organic amines leads to the addition of base to natural waters at the CO2 equivalence point. The dissolved base in water increases the pH and titrates an equivalent amount of CO2 to bicarbonate ion and carbonate ion. At equilibrium, the water contains a certain amount of alkalinity contributed by the concentration of weak acid anions. Conversely, the addition of acid converts weak acid anions to CO2 and continuous addition of strong acids can cause the alkalinity to become less than zero. For example, the following reactions take place during the addition of acid to a typical seawater solution:
B(OH)4− + H+ → B(OH)3 + H2O
OH− + H+ → H2O
PO43− + 2 H+ → H2PO4−
HPO42− + H+ → H2PO4−
[SiO(OH)3-] + H+ → [Si(OH)4]
It can be seen from the above protonation reactions that most bases consume one proton (H+) to become a neutral species, thus increasing alkalinity by one per equivalent. CO32- however, will consume two protons before becoming a zero-level species (CO2), thus it increases alkalinity by two per mole of CO32-. [H+] and [HSO4-] decrease alkalinity, as they act as sources of protons. They are often represented collectively as [H+]T.
Alkalinity is typically reported as mg/L "as" CaCO3. (The conjunction "as" is appropriate in this case because the alkalinity results from a mixture of ions but is reported "as if" all of this is due to CaCO3.) This can be converted into milliequivalents per Liter (meq/L) by dividing by 50 (the approximate MW of CaCO3 divided by 2).
Carbon dioxide interactions.
Addition of CO2.
Addition (or removal) of CO2 to a solution does not change its alkalinity, since the net reaction produces the same number of equivalents of positively contributing species (H+) as negative contributing species (HCO3− and/or CO32−). Adding CO2 to the solution lowers its pH, but does not affect alkalinity.
At all pH values:
CO2 + H2O ⇌ HCO3− + H+
Only at high (basic) pH values:
HCO3− + H+ ⇌ CO32− + 2 H+
Dissolution of carbonate rock.
Addition of CO2 to a solution in contact with a solid can (over time) affect the alkalinity, especially for carbonate minerals in contact with groundwater or seawater. The dissolution (or precipitation) of carbonate rock has a strong influence on the alkalinity. This is because carbonate rock is composed of CaCO3 and its dissociation will add Ca2+ and CO32− into solution. Ca2+ will not influence alkalinity, but CO32− will increase alkalinity by 2 units. Increased dissolution of carbonate rock by acidification from acid rain and mining has contributed to increased alkalinity concentrations in some major rivers throughout the eastern U.S. The following reaction shows how acid rain, containing sulfuric acid, can have the effect of increasing river alkalinity by increasing the amount of bicarbonate ion:
2 CaCO3 + H2SO4 → 2 Ca2+ + 2 HCO3− + SO42−
Another way of writing this is:
CaCO3 + H+ ⇌ Ca2+ + HCO3−
The lower the pH, the higher the concentration of bicarbonate will be. This shows how a lower pH can lead to higher alkalinity if the amount of bicarbonate produced is greater than the amount of H+ remaining after the reaction. This is the case since the amount of acid in the rainwater is low. If this alkaline groundwater later comes into contact with the atmosphere, it can lose CO2, precipitate carbonate, and thereby become less alkaline again. When carbonate minerals, water, and the atmosphere are all in equilibrium, the reversible reaction
CaCO3 + 2 H+ ⇌ Ca2+ + CO2 + H2O
shows that pH will be related to calcium ion concentration, with lower pH going with higher calcium ion concentration. In this case, the higher the pH, the more bicarbonate and carbonate ion there will be, in contrast to the paradoxical situation described above, where one does not have equilibrium with the atmosphere.
Changes to oceanic alkalinity.
In the ocean, alkalinity is completely dominated by carbonate and bicarbonate plus a small contribution from borate.
Thus the chemical equation for alkalinity in seawater is:
AT = [HCO3-] + 2[CO32-] + [B(OH)4-]
There are many methods of alkalinity generation in the ocean. Perhaps the most well known is the dissolution of calcium carbonate to form Ca2+ and CO32− (carbonate). The carbonate ion has the potential to absorb two hydrogen ions. Therefore, it causes a net increase in ocean alkalinity. Calcium carbonate dissolution occurs in regions of the ocean which are undersaturated with respect to calcium carbonate.
The increasing carbon dioxide level in the atmosphere, due to carbon dioxide emissions, results in increasing absorption of CO2 from the atmosphere into the oceans. This does not affect the ocean's alkalinity but it does result in a reduction in pH value (called ocean acidification). Ocean alkalinity enhancement has been proposed as one option to add alkalinity to the ocean and therefore buffer against pH changes.
Biological processes have a much greater impact on oceanic alkalinity on short (minutes to centuries) timescales. Aerobic respiration of organic matter can decrease alkalinity by releasing protons. Denitrification and sulfate reduction occur in oxygen-limited environments. Both of these processes consume hydrogen ions (thus increasing alkalinity) and release gases (N2 or H2S), which eventually escape into the atmosphere. Nitrification and sulfide oxidation both decrease alkalinity by releasing protons as a byproduct of oxidation reactions.
Global temporal and spatial variability.
The ocean's alkalinity varies over time, most significantly over geologic timescales (millennia). Changes in the balance between terrestrial weathering and sedimentation of carbonate minerals (for example, as a function of ocean acidification) are the primary long-term drivers of alkalinity in the ocean. Over human timescales, mean ocean alkalinity is relatively stable. Seasonal and annual variability of mean ocean alkalinity is very low.
Alkalinity varies by location depending on evaporation/precipitation, advection of water, biological processes, and geochemical processes.
River dominated mixing also occurs close to the shore; it is strongest close to the mouth of a large river. Here, the rivers can act as either a source or a sink of alkalinity. AT follows the outflow of the river and has a linear relationship with salinity.
Oceanic alkalinity also follows general trends based on latitude and depth. It has been shown that AT is often inversely proportional to sea surface temperature (SST). Therefore, it generally increases with high latitudes and depths. As a result, upwelling areas (where water from the deep ocean is pushed to the surface) also have higher alkalinity values.
There are many programs to measure, record, and study oceanic alkalinity, together with many of the other characteristics of seawater, like temperature and salinity. These include: GEOSECS (Geochemical Ocean Sections Study), TTO/NAS (Transient Tracers in the Ocean/North Atlantic Study), JGOFS (Joint Global Ocean Flux Study), WOCE (World Ocean Circulation Experiment), CARINA (Carbon dioxide in the Atlantic Ocean).
References.
<templatestyles src="Reflist/styles.css" />
External links.
Carbonate system calculators.
The following packages calculate the state of the carbonate system in seawater (including pH):
|
[
{
"math_id": 0,
"text": "\\sum(\\text{cations})=\\sum(\\text{anions})"
},
{
"math_id": 1,
"text": "\\begin{align}\n&\\sum(\\text{conservative cations})-\\sum(\\text{conservative anions}) = \\\\\n&\\quad [\\mathrm{HCO_3^-}]+2[\\mathrm{CO_3^{2-}}]+[\\mathrm{B(OH)_4^-}]+\n[\\mathrm{OH^-}]+[\\mathrm{HPO_4^{2-}}]+2[\\mathrm{PO_4^{3-}}]+[\\mathrm{H_3SiO_4^-}]+[\\mathrm{NH_3}]+[\\mathrm{HS^-}]-[\\mathrm{H^+}]-[\\mathrm{HSO_4^-}]-[\\mathrm{HF}]-[\\mathrm{H_3PO_4}]-[\\mathrm{HNO_2}]\n\\end{align}"
}
] |
https://en.wikipedia.org/wiki?curid=1427251
|
14273099
|
Stribeck curve
|
Friction depending on speed when the surfaces are sliding lubricated.
The Stribeck curve is a fundamental concept in the field of tribology. It shows that friction in fluid-lubricated contacts is a non-linear function of the contact load, the lubricant viscosity and the lubricant entrainment speed. The discovery and underlying research is usually attributed to Richard Stribeck and Mayo D. Hersey, who studied friction in journal bearings for railway wagon applications during the first half of the 20th century; however, other researchers have arrived at similar conclusions before. The mechanisms along the Stribeck curve have been in parts also understood today on the atomistic level.
Concept.
For a contact of two fluid-lubricated surfaces, the Stribeck curve shows the relationship between the so-called "Hersey number", a dimensionless lubrication parameter, and the friction coefficient. The Hersey number is defined as:
formula_0
where "η" is the dynamic viscosity of the fluid, "N" is the entrainment speed of the fluid and "P" is the normal load per length of the tribological contact.
Hersey's original formula uses the rotational speed (revolutions per unit time) for "N" and the load per projected area (i.e. the product of a journal bearing's length and diameter) for "P".
Alternatively, the Hersey number is the dimensionless number obtained from the velocity (m/s) times the dynamic viscosity (Pa∙s = N∙s/m2), divided by the load per unit length of bearing (N/m).
Thus, for a given viscosity and load, the Stribeck curve shows how friction changes with increasing velocity. Based on the typical progression of the Stribeck curve (see right), three lubrication regimes can be identified.
History.
Richard Stribeck's research was performed in Berlin at the Royal Prussian Technical Testing Institute (MPA, now BAM), and his results were presented on 5 December 1901 during a public session of the railway society and published on 6 September 1902. Similar work was previously performed around 1885 by Adolf Martens at the same institute, and also in the mid-1870s by Robert Henry Thurston at the Stevens Institute of Technology in the U.S. The reason why the form of the friction curve for liquid lubricated surfaces was later attributed to Stribeck – although both Thurston and Martens achieved their results considerably earlier – may be because Stribeck published his findings in the most important technical journal in Germany at that time, "Zeitschrift des Vereins Deutscher Ingenieure" (VDI, Journal of German Mechanical Engineers). Martens published his results in the official journal of the Royal Prussian Technical Testing Institute, which has now become BAM. The VDI journal provided wide access to Stribeck's data and later colleagues rationalized the results into the three classical friction regimes. Thurston did not have the experimental means to record a continuous graph of the coefficient of friction but only measured it at discrete points. This may be the reason why the minimum in the coefficient of friction for a liquid-lubricated journal bearing was not discovered by him, but was demonstrated by the graphs of Martens and Stribeck.
The graphs plotted by Martens show the coefficient of friction either as a function of pressure, speed or temperature (i.e. viscosity), but not of their combination to the Hersey number. Schmidt attempts to do this using Marten's data. The curves' characteristic minima seem to correspond to very low Hersey numbers in the range 0.00005-0.00015.
Calculation of Stribeck Curve.
In general, there are two approaches for the calculation of Stribeck curve in all lubrication regimes. In the first approach, the governing flow and surface deformation equations (the system of the Elastohydrodynamic Lubrication equations) are solved numerically. Although the numerical solutions can be relatively accurate, this approach is computationally expensive and requires substantial computational resources. The second approach relies on the load-sharing concept that can be used to solve the problem approximately but at a significantly less computational cost.
In the second approach, the general problem is split up into two sub-problems: 1) lubrication problem assuming smooth surfaces and 2) a “dry” rough contact problem. The two sub-problems are coupled through the load carried by the lubricant and by the “dry” contact. In its simplest approximation, the lubrication sub-problem can be represented via a central film thickness fit to calculate the film thickness and the Greenwood-Williamson model for the “dry” contact sub-problem. This approach can give a reasonable qualitative prediction of the friction evolution; however, it is likely to overestimate friction due to the simplification assumptions used in central film thickness calculations and Greenwood-Williamson model.
An online calculator is available on www.tribonet.org that allows calculating Stribeck curve for line and point contacts. These tools are based on the load-sharing concept.
Also molecular simulation based on classical force fields can be used for predicting the Stribeck curve. Thereby, underlying molecular mechanisms can be elucidated.
|
[
{
"math_id": 0,
"text": "\n\\begin{align}\n \\text{Hersey number} = \\frac{\\eta \\cdot N}{P} ,\n\\end{align}\n"
}
] |
https://en.wikipedia.org/wiki?curid=14273099
|
1427502
|
Tournament (graph theory)
|
Directed graph where each vertex pair has one arc
In graph theory, a tournament is a directed graph with exactly one edge between each two vertices, in one of the two possible directions. Equivalently, a tournament is an orientation of an undirected complete graph. (However, as directed graphs, tournaments are not complete: complete directed graphs have two edges, in both directions, between each two vertices.) The name "tournament" comes from interpreting the graph as the outcome of a round-robin tournament, a game where each player is paired against every other exactly once. In a tournament, the vertices represent the players, and the edges between players point from the winner to the loser.
Many of the important properties of tournaments were investigated by H. G. Landau in 1953 to model dominance relations in flocks of chickens. Tournaments are also heavily studied in voting theory, where they can represent partial information about voter preferences among multiple candidates, and are central to the definition of Condorcet methods.
If every player beats the same number of other players (indegree − outdegree = 0) the tournament is called "regular".
Paths and cycles.
Any tournament on a finite number formula_0 of vertices contains a Hamiltonian path, i.e., directed path on all formula_0 vertices (Rédei 1934).
This is easily shown by induction on formula_0: suppose that the statement holds for formula_0, and consider any tournament formula_1 on formula_2 vertices. Choose a vertex formula_3 of formula_1 and consider a directed path formula_4 in formula_5. There is some formula_6 such that formula_7. (One possibility is to let formula_6 be maximal such that for every formula_8. Alternatively, let formula_9 be minimal such that formula_10.)
formula_11
is a directed path as desired. This argument also gives an algorithm for finding the Hamiltonian path. More efficient algorithms, that require examining only formula_12 of the edges, are known. The Hamiltonian paths are in one-to-one correspondence with the minimal feedback arc sets of the tournament. Rédei's theorem is the special case for complete graphs of the Gallai–Hasse–Roy–Vitaver theorem, relating the lengths of paths in orientations of graphs to the chromatic number of these graphs.
Another basic result on tournaments is that every strongly connected tournament has a Hamiltonian cycle. More strongly, every strongly connected tournament is vertex pancyclic: for each vertex formula_13, and each formula_14 in the range from three to the number of vertices in the tournament, there is a cycle of length formula_14 containing formula_13. A tournament formula_1is formula_14-strongly connected if for every set formula_15 of formula_16 vertices of formula_1, formula_17 is strongly connected.
If the tournament is 4‑strongly connected, then each pair of vertices can be connected with a Hamiltonian path. For every set formula_18 of at most formula_16 arcs of a formula_14-strongly connected tournament formula_1, we have that formula_19 has a Hamiltonian cycle. This result was extended by .
Transitivity.
A tournament in which formula_20 and formula_21 formula_22 formula_23 is called transitive. In other words, in a transitive tournament, the vertices may be (strictly) totally ordered by the edge relation, and the edge relation is the same as reachability.
Equivalent conditions.
The following statements are equivalent for a tournament formula_1 on formula_0 vertices:
Ramsey theory.
Transitive tournaments play a role in Ramsey theory analogous to that of cliques in undirected graphs. In particular, every tournament on formula_0 vertices contains a transitive subtournament on formula_25 vertices. The proof is simple: choose any one vertex formula_13 to be part of this subtournament, and form the rest of the subtournament recursively on either the set of incoming neighbors of formula_13 or the set of outgoing neighbors of formula_13, whichever is larger. For instance, every tournament on seven vertices contains a three-vertex transitive subtournament; the Paley tournament on seven vertices shows that this is the most that can be guaranteed. However, showed that this bound is not tight for some larger values of formula_0.
proved that there are tournaments on formula_0 vertices without a transitive subtournament of size formula_26 Their proof uses a counting argument: the number of ways that a formula_14-element transitive tournament can occur as a subtournament of a larger tournament on formula_0 labeled vertices is
formula_27
and when formula_14 is larger than formula_26, this number is too small to allow for an occurrence of a transitive tournament within each of the formula_28 different tournaments on the same set of formula_0 labeled vertices.
Paradoxical tournaments.
A player who wins all games would naturally be the tournament's winner. However, as the existence of non-transitive tournaments shows, there may not be such a player. A tournament for which every player loses at least one game is called a 1-paradoxical tournament. More generally, a tournament formula_29 is called formula_14-paradoxical if for every formula_14-element subset formula_30 of formula_31 there is a vertex formula_3 in formula_32 such that formula_33 for all formula_34. By means of the probabilistic method, Paul Erdős showed that for any fixed value of formula_14, if formula_35, then almost every tournament on formula_31 is formula_14-paradoxical. On the other hand, an easy argument shows that any formula_14-paradoxical tournament must have at least formula_36 players, which was improved to formula_37 by Esther and George Szekeres in 1965. There is an explicit construction of formula_14-paradoxical tournaments with formula_38 players by Graham and Spencer (1971) namely the Paley tournament.
Condensation.
The condensation of any tournament is itself a transitive tournament. Thus, even for tournaments that are not transitive, the strongly connected components of the tournament may be totally ordered.
Score sequences and score sets.
The score sequence of a tournament is the nondecreasing sequence of outdegrees of the vertices of a tournament. The score set of a tournament is the set of integers that are the outdegrees of vertices in that tournament.
Landau's Theorem (1953) A nondecreasing sequence of integers formula_39 is a score sequence if and only if:
Let formula_43 be the number of different score sequences of size formula_0. The sequence formula_43 (sequence in the OEIS) starts as:
1, 1, 1, 2, 4, 9, 22, 59, 167, 490, 1486, 4639, 14805, 48107, ...
Winston and Kleitman proved that for sufficiently large "n":
formula_44
where formula_45
Takács later showed, using some reasonable but unproven assumptions, that
formula_46
where formula_47
Together these provide evidence that:
formula_48
Here formula_49 signifies an asymptotically tight bound.
Yao showed that every nonempty set of nonnegative integers is the score set for some tournament.
Majority relations.
In social choice theory, tournaments naturally arise as majority relations of preference profiles. Let formula_50 be a finite set of alternatives, and consider a list formula_51 of linear orders over formula_50. We interpret each order formula_52 as the preference ranking of a voter formula_9. The (strict) majority relation formula_53 of formula_54 over formula_50 is then defined so that formula_55 if and only if a majority of the voters prefer formula_56 to formula_57, that is formula_58. If the number formula_0 of voters is odd, then the majority relation forms the dominance relation of a tournament on vertex set formula_50.
By a lemma of McGarvey, every tournament on formula_59 vertices can be obtained as the majority relation of at most formula_60 voters. Results by Stearns and Erdős & Moser later established that formula_61 voters are needed to induce every tournament on formula_59 vertices.
Laslier (1997) studies in what sense a set of vertices can be called the set of "winners" of a tournament. This revealed to be useful in Political Science to study, in formal models of political economy, what can be the outcome of a democratic process.
Notes.
<templatestyles src="Reflist/styles.css" />
References.
"This article incorporates material from tournament on PlanetMath, which is licensed under the ."
|
[
{
"math_id": 0,
"text": "n"
},
{
"math_id": 1,
"text": "T"
},
{
"math_id": 2,
"text": "n+1"
},
{
"math_id": 3,
"text": "v_0"
},
{
"math_id": 4,
"text": "v_1,v_2,\\ldots,v_n"
},
{
"math_id": 5,
"text": "T\\smallsetminus \\{v_0\\}"
},
{
"math_id": 6,
"text": "i \\in \\{0,\\ldots,n\\}"
},
{
"math_id": 7,
"text": "(i=0 \\vee v_i \\rightarrow v_0) \\wedge (v_0 \\rightarrow v_{i+1} \\vee i=n)"
},
{
"math_id": 8,
"text": "j \\leq i, v_j \\rightarrow v_0"
},
{
"math_id": 9,
"text": "i"
},
{
"math_id": 10,
"text": "\\forall j > i, v_0 \\rightarrow v_j"
},
{
"math_id": 11,
"text": "v_1,\\ldots,v_i,v_0,v_{i+1},\\ldots,v_n"
},
{
"math_id": 12,
"text": "O(n \\log n)"
},
{
"math_id": 13,
"text": "v"
},
{
"math_id": 14,
"text": "k"
},
{
"math_id": 15,
"text": "U"
},
{
"math_id": 16,
"text": "k-1"
},
{
"math_id": 17,
"text": "T-U"
},
{
"math_id": 18,
"text": "B"
},
{
"math_id": 19,
"text": "T-B"
},
{
"math_id": 20,
"text": "((a \\rightarrow b)"
},
{
"math_id": 21,
"text": "(b \\rightarrow c))"
},
{
"math_id": 22,
"text": "\\Rightarrow"
},
{
"math_id": 23,
"text": "(a \\rightarrow c)"
},
{
"math_id": 24,
"text": "\\{0,1,2,\\ldots,n-1\\}"
},
{
"math_id": 25,
"text": "1+\\lfloor\\log_2 n\\rfloor"
},
{
"math_id": 26,
"text": "2+2\\lfloor\\log_2 n\\rfloor"
},
{
"math_id": 27,
"text": "\\binom{n}{k}k!2^{\\binom{n}{2}-\\binom{k}{2}},"
},
{
"math_id": 28,
"text": "2^{\\binom{n}{2}}"
},
{
"math_id": 29,
"text": "T=(V,E)"
},
{
"math_id": 30,
"text": "S"
},
{
"math_id": 31,
"text": "V"
},
{
"math_id": 32,
"text": "V\\setminus S"
},
{
"math_id": 33,
"text": "v_0 \\rightarrow v"
},
{
"math_id": 34,
"text": "v \\in S"
},
{
"math_id": 35,
"text": "|V| \\geq k^22^k\\ln(2+o(1))"
},
{
"math_id": 36,
"text": "2^{k+1}-1"
},
{
"math_id": 37,
"text": "(k+2)2^{k-1}-1"
},
{
"math_id": 38,
"text": "k^24^{k-1}(1+o(1))"
},
{
"math_id": 39,
"text": "(s_1, s_2, \\ldots, s_n)"
},
{
"math_id": 40,
"text": "0 \\le s_1 \\le s_2 \\le \\cdots \\le s_n"
},
{
"math_id": 41,
"text": "s_1 + s_2 + \\cdots + s_i \\ge {i \\choose 2}, \\text{ for }i = 1, 2, \\ldots, n - 1"
},
{
"math_id": 42,
"text": "s_1 + s_2 + \\cdots + s_n = {n \\choose 2}."
},
{
"math_id": 43,
"text": "s(n)"
},
{
"math_id": 44,
"text": "s(n) > c_1 4^n n^{-5/2},"
},
{
"math_id": 45,
"text": "c_1 = 0.049."
},
{
"math_id": 46,
"text": "s(n) < c_2 4^n n^{-5/2},"
},
{
"math_id": 47,
"text": "c_2 < 4.858."
},
{
"math_id": 48,
"text": "s(n) \\in \\Theta (4^n n^{-5/2})."
},
{
"math_id": 49,
"text": "\\Theta"
},
{
"math_id": 50,
"text": "A"
},
{
"math_id": 51,
"text": "P = (\\succ_1, \\dots, \\succ_n)"
},
{
"math_id": 52,
"text": "\\succ_i"
},
{
"math_id": 53,
"text": "\\succ_{\\text{maj}}"
},
{
"math_id": 54,
"text": "P"
},
{
"math_id": 55,
"text": "a \\succ_{\\text{maj}} b"
},
{
"math_id": 56,
"text": "a"
},
{
"math_id": 57,
"text": "b"
},
{
"math_id": 58,
"text": "|\\{ i \\in [n] : a \\succ_i b \\}| > |\\{ i \\in [n] : b \\succ_i a \\}|"
},
{
"math_id": 59,
"text": "m"
},
{
"math_id": 60,
"text": "m(m-1)"
},
{
"math_id": 61,
"text": "\\Theta(m/\\log m)"
}
] |
https://en.wikipedia.org/wiki?curid=1427502
|
1427527
|
Electrical contact
|
Electrical circuit component
An electrical contact is an electrical circuit component found in electrical switches, relays, connectors and circuit breakers. Each contact is a piece of electrically conductive material, typically metal. When a pair of contacts touch, they can pass an electrical current with a certain contact resistance, dependent on surface structure, surface chemistry and contact time; when the pair is separated by an insulating gap, then the pair does not pass a current. When the contacts touch, the switch is "closed"; when the contacts are separated, the switch is "open". The gap must be an insulating medium, such as air, vacuum, oil, SF6. Contacts may be operated by humans in push-buttons and switches, by mechanical pressure in sensors or machine cams, and electromechanically in relays. The surfaces where contacts touch are usually composed of metals such as silver or gold alloys that have high electrical conductivity, wear resistance, oxidation resistance and other properties.
Materials.
Contacts can be produced from a wide variety of materials. Typical materials include:
Electrical ratings.
Contacts are rated for the current carrying capacity while closed, breaking capacity when opening (due to arcing) and voltage rating. Opening voltage rating may be an AC voltage rating, DC voltage rating or both.
Arc suppression.
When relay contacts open to interrupt a high current with an inductive load, a voltage spike will result, striking an arc across the contacts. If the voltage is high enough, an arc may be struck even without an inductive load. Regardless of how the arc forms, it will persist until the current through the arc falls to the point too low to sustain it. Arcing damages the electrical contacts, and a sustained arc may prevent the open contacts from removing power from the system being controlled.
In AC systems, where the current passes through zero twice for each cycle, all but the most energetic arcs are extinguished at the zero crossing. The problem is more severe with DC where such zero crossings do not occur. This is why contacts rated for one voltage for switching AC frequently have a lower voltage rating for DC.
Electrical contact theory.
Ragnar Holm contributed greatly to electrical contact theory and application.
Macroscopically smooth and clean surfaces are microscopically rough and, in air, contaminated with oxides, adsorbed water vapor, and atmospheric contaminants. When two metal electrical contacts touch, the actual metal-to-metal contact area is small compared to the total contact-to-contact area physically touching. In electrical contact theory, the relatively small area where electrical current flows between two contacts is called the a-spot where "a" stands for asperity. If the small a-spot is treated as a circular area and the resistivity of the metal is homogeneous, then the current and voltage in the metal conductor has spherical symmetry and a simple calculation can relate the size of the a-spot to the resistance of the electrical contact interface. If there is metal-to-metal contact between electrical contacts, then the electrical contact resistance, or ECR (as opposed to the bulk resistance of the contact metal) is mostly due to constriction of the current through a very small area, the a-spot. For contact spots of radii smaller than the mean free path of electrons formula_0, ballistic conduction of electrons occurs, resulting in a phenomenon known also as "Sharvin resistance". Contact force or pressure increases the size of the a-spot which decreases the constriction resistance and the electrical contact resistance. When the size of contacting asperities becomes larger than the mean free path of electrons, Holm-type contacts become the dominant transport mechanism, resulting in a relatively low contact resistance.
Relay contacts.
The National Association of Relay Manufacturers and its successor, the Relay and Switch Industry Association define 23 distinct forms of electrical contact found in relays and switches.
A "normally closed" ("<templatestyles src="Template:Visible anchor/styles.css" />NC") contact pair is closed (in a conductive state) when it, or the device operating it, is in a deenergized state or relaxed state.
A "normally open" ("<templatestyles src="Template:Visible anchor/styles.css" />NO") contact pair is open (in a non-conductive state) when it, or the device operating it, is in a deenergized state or relaxed state.
Contact form.
The National Association of Relay Manufacturers and its successor, the Relay and Switch Industry Association define 23 distinct electrical contact forms found in relays and switches. The following contact forms are particularly common:
Form A contacts.
"Form A" contacts ("make contacts") are "normally open" contacts. The contacts are open when the energizing force (magnet or relay solenoid) is "not" present. When the energizing force is present, the contact will close. An alternate notation for "Form A" is SPST-NO.
Form B contacts.
"Form B" contacts ("break contacts") are normally closed contacts. Its operation is logically inverted from Form A. An alternate notation for "Form B" is SPST-NC.
Form C contacts.
"Form C" contacts ("change over" or "transfer" contacts) are composed of a normally closed contact pair and a normally open contact pair that are operated by the same device; there is a common electrical connection between a contact of each pair that results in only three connection terminals. These terminals are usually labelled as "normally open", "common", and "normally closed" ("NO-C-NC"). An alternate notation for "Form C" is SPDT.
These contacts are quite frequently found in electrical switches and relays as the common contact element provides a mechanically economical method of providing a higher contact count.
Form D contacts.
"Form D" contacts ("continuity transfer" contacts) differ
from "Form C" in only one regard, the make-break order during transition. Where "Form C" guarantees that, briefly, both connections are open, "Form D" guarantees that, briefly, all three terminals will be connected. This is a relatively uncommon configuration.
Form E contacts.
"Form E" is a combination of form D and B.
Form K contacts.
"Form K" contacts (center-off) differ from "Form C" in that there is a center-off or normally-open position where neither connection is made. SPDT toggle switches with a center off position are common, but relays with this configuration are relatively rare.
Form X contacts.
"Form X" or double-make contacts are equivalent to two "Form A" contacts in series, mechanically linked and operated by a single actuator, and can also be described as SPST-NO contacts. These are commonly found in contactors and in toggle switches designed to handle high power inductive loads.
Form Y contacts.
"Form Y" or double-break contacts are equivalent to two "Form B" contacts in series, mechanically linked and operated by a single actuator, and can also be described as SPST-NC contacts.
Form Z contacts.
"Form Z" or double-make double-break contacts are comparable to "Form C" contacts, but they almost always have four external connections, two for the normally open path and two for the normally closed path. As with forms "X" and "Y", both current paths involve two contacts in series, mechanically linked and operated by a single actuator. Again, this is also described as an SPDT contact.
Make-break order.
Where a switch contains both normally open (NO) and normally closed (NC) contacts, the order in which they make and break may be significant. In most cases, the rule is "break-before-make" or "B-B-M"; that is, the NO and NC contacts are never simultaneously closed during the transition between states. This is not always the case, "Form C" contacts follow this rule, while the otherwise equivalent "Form D" contacts follow the opposite rule, make before break. The less common configuration, when the NO and NC contacts are simultaneously closed during the transition, is "make-before-break" or "M-B-B".
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "\\lambda"
}
] |
https://en.wikipedia.org/wiki?curid=1427527
|
14275961
|
Great ellipse
|
Ellipse on a spheroid centered on its origin
A great ellipse is an ellipse passing through two points on a spheroid and having the same center as that of the spheroid. Equivalently, it is an ellipse on the surface of a spheroid and centered on the origin, or the curve formed by intersecting the spheroid by a plane through its center.
For points that are separated by less than about a quarter of the circumference of the earth, about formula_0, the length of the great ellipse connecting the points is close (within one part in 500,000) to the geodesic distance.
The great ellipse therefore is sometimes proposed as a suitable route for marine navigation.
The great ellipse is special case of an earth section path.
Introduction.
Assume that the spheroid, an ellipsoid of revolution, has an equatorial radius formula_1 and polar semi-axis formula_2. Define the flattening formula_3, the eccentricity formula_4, and the second eccentricity formula_5. Consider two points: formula_6 at (geographic) latitude formula_7 and longitude formula_8 and formula_9 at latitude formula_10 and longitude formula_11. The connecting great ellipse (from formula_6 to formula_9) has length formula_12 and has azimuths formula_13 and formula_14 at the two endpoints.
There are various ways to map an ellipsoid into a sphere of radius formula_1 in such a way as to map the great ellipse into a great circle, allowing the methods of great-circle navigation to be used:
The last method gives an easy way to generate a succession of way-points on the great ellipse connecting two known points formula_6 and formula_9. Solve for the great circle between formula_19 and formula_20 and find the way-points on the great circle. These map into way-points on the corresponding great ellipse.
Mapping the great ellipse to a great circle.
If distances and headings are needed, it is simplest to use the first of the mappings. In detail, the mapping is as follows (this description is taken from ):
Solving the inverse problem.
The "inverse problem" is the determination of formula_12, formula_13, and formula_14, given the positions of formula_6 and formula_9. This is solved by computing formula_31 and formula_32 and solving for the great-circle between formula_33 and formula_34.
The spherical azimuths are relabeled as formula_24 (from formula_23). Thus formula_28, formula_35, and formula_36 and the spherical azimuths at the equator and at formula_6 and formula_9. The azimuths of the endpoints of great ellipse, formula_13 and formula_14, are computed from formula_35 and formula_36.
The semi-axes of the great ellipse can be found using the value of formula_28.
Also determined as part of the solution of the great circle problem are the arc lengths, formula_37 and formula_38, measured from the equator crossing to formula_6 and formula_9. The distance formula_12 is found by computing the length of a portion of perimeter of the ellipse using the formula giving the meridian arc in terms the parametric latitude. In applying this formula, use the semi-axes for the great ellipse (instead of for the meridian) and substitute formula_37 and formula_38 for formula_16.
The solution of the "direct problem", determining the position of formula_9 given formula_6, formula_13, and formula_12, can be similarly be found (this requires, in addition, the inverse meridian distance formula). This also enables way-points (e.g., a series of equally spaced intermediate points) to be found in the solution of the inverse problem.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "10\\,000\\,\\mathrm{km}"
},
{
"math_id": 1,
"text": "a"
},
{
"math_id": 2,
"text": "b"
},
{
"math_id": 3,
"text": "f=(a-b)/a"
},
{
"math_id": 4,
"text": "e=\\sqrt{f(2-f)}"
},
{
"math_id": 5,
"text": "e'=e/(1-f)"
},
{
"math_id": 6,
"text": "A"
},
{
"math_id": 7,
"text": "\\phi_1"
},
{
"math_id": 8,
"text": "\\lambda_1"
},
{
"math_id": 9,
"text": "B"
},
{
"math_id": 10,
"text": "\\phi_2"
},
{
"math_id": 11,
"text": "\\lambda_2"
},
{
"math_id": 12,
"text": "s_{12}"
},
{
"math_id": 13,
"text": "\\alpha_1"
},
{
"math_id": 14,
"text": "\\alpha_2"
},
{
"math_id": 15,
"text": "\\phi"
},
{
"math_id": 16,
"text": "\\beta"
},
{
"math_id": 17,
"text": "\\theta"
},
{
"math_id": 18,
"text": "a^2/b"
},
{
"math_id": 19,
"text": "(\\phi_1,\\lambda_1)"
},
{
"math_id": 20,
"text": "(\\phi_2,\\lambda_2)"
},
{
"math_id": 21,
"text": "a\\tan\\beta = b\\tan\\phi."
},
{
"math_id": 22,
"text": "\\lambda"
},
{
"math_id": 23,
"text": "\\alpha"
},
{
"math_id": 24,
"text": "\\gamma"
},
{
"math_id": 25,
"text": "\n\\begin{align}\n\\tan\\alpha &= \\frac{\\tan\\gamma}{\\sqrt{1-e^2\\cos^2\\beta}}, \\\\\n\\tan\\gamma &= \\frac{\\tan\\alpha}{\\sqrt{1+e'^2\\cos^2\\phi}},\n\\end{align}\n"
},
{
"math_id": 26,
"text": "\\sigma"
},
{
"math_id": 27,
"text": "a \\sqrt{1 - e^2\\cos^2\\gamma_0}"
},
{
"math_id": 28,
"text": "\\gamma_0"
},
{
"math_id": 29,
"text": "\\omega"
},
{
"math_id": 30,
"text": "b \\sqrt{1 + e'^2\\cos^2\\alpha_0}"
},
{
"math_id": 31,
"text": "\\beta_1"
},
{
"math_id": 32,
"text": "\\beta_2"
},
{
"math_id": 33,
"text": "(\\beta_1,\\lambda_1)"
},
{
"math_id": 34,
"text": "(\\beta_2,\\lambda_2)"
},
{
"math_id": 35,
"text": "\\gamma_1"
},
{
"math_id": 36,
"text": "\\gamma_2"
},
{
"math_id": 37,
"text": "\\sigma_{01}"
},
{
"math_id": 38,
"text": "\\sigma_{02}"
}
] |
https://en.wikipedia.org/wiki?curid=14275961
|
14276188
|
2,6-dioxo-6-phenylhexa-3-enoate hydrolase
|
Class of enzymes
In enzymology, a 2,6-dioxo-6-phenylhexa-3-enoate hydrolase (EC 3.7.1.8) is an enzyme that catalyzes the chemical reaction
2,6-dioxo-6-phenylhexa-3-enoate + H2O formula_0 benzoate + 2-oxopent-4-enoate
Thus, the two substrates of this enzyme are 2,6-dioxo-6-phenylhexa-3-enoate and H2O, whereas its two products are benzoate and 2-oxopent-4-enoate.
This enzyme belongs to the family of hydrolases, specifically those acting on carbon-carbon bonds in ketonic substances. The systematic name of this enzyme class is 2,6-dioxo-6-phenylhexa-3-enoate benzoylhydrolase. This enzyme is also called HOHPDA hydrolase. This enzyme participates in biphenyl degradation and fluorene degradation.
Structural studies.
As of late 2007, 3 structures have been solved for this class of enzymes, with PDB accession codes 1C4X, 1J1I, and 2OG1.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "\\rightleftharpoons"
}
] |
https://en.wikipedia.org/wiki?curid=14276188
|
14276201
|
2'-hydroxybiphenyl-2-sulfinate desulfinase
|
Class of enzymes
In enzymology, a 2'-hydroxybiphenyl-2-sulfinate desulfinase (EC 3.13.1.3) is an enzyme that catalyzes the chemical reaction
2'-hydroxybiphenyl-2-sulfinate + H2O formula_0 2-hydroxybiphenyl + sulfite
Thus, the two substrates of this enzyme are 2'-hydroxybiphenyl-2-sulfinate and H2O, whereas its two products are 2-hydroxybiphenyl and sulfite.
This enzyme belongs to the family of hydrolases, specifically those acting on carbon-sulfur bonds. The systematic name of this enzyme class is 2'-hydroxybiphenyl-2-sulfinate sulfohydrolase. Other names in common use include gene dszB-encoded hydrolase, 2-(2-hydroxyphenyl) benzenesulfinate:H2O hydrolase, DszB, HBPSi desulfinase, 2-(2-hydroxyphenyl) benzenesulfinate sulfohydrolase, HPBS desulfinase, 2-(2-hydroxyphenyl)benzenesulfinate hydrolase, 2-(2'-hydroxyphenyl)benzenesulfinate desulfinase, and 2-(2-hydroxyphenyl)benzenesulfinate desulfinase.
Structural studies.
As of late 2007, 3 structures have been solved for this class of enzymes, with PDB accession codes 2DE2, 2DE3, and 2DE4.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "\\rightleftharpoons"
}
] |
https://en.wikipedia.org/wiki?curid=14276201
|
14276211
|
2-hydroxymuconate-semialdehyde hydrolase
|
Class of enzymes
In enzymology, a 2-hydroxymuconate-semialdehyde hydrolase (EC 3.7.1.9) is an enzyme that catalyzes the chemical reaction
2-hydroxymuconate semialdehyde + H2O formula_0 formate + 2-oxopent-4-enoate
Thus, the two substrates of this enzyme are 2-hydroxymuconate semialdehyde and H2O, whereas its two products are formate and 2-oxopent-4-enoate.
This enzyme belongs to the family of hydrolases, specifically those acting on carbon-carbon bonds in ketonic substances. The systematic name of this enzyme class is 2-hydroxymuconate-semialdehyde formylhydrolase. Other names in common use include 2-hydroxy-6-oxohepta-2,4-dienoate hydrolase, 2-hydroxymuconic semialdehyde hydrolase, HMSH, and HOD hydrolase. This enzyme participates in 5 metabolic pathways: benzoate degradation via hydroxylation, toluene and xylene degradation, 1,4-dichlorobenzene degradation, carbazole degradation, and styrene degradation.
Structural studies.
As of late 2007, 10 structures have been solved for this class of enzymes, with PDB accession codes 1IUN, 1IUO, 1IUP, 1UK6, 1UK7, 1UK8, 1UK9, 1UKA, 1UKB, and 2D0D.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "\\rightleftharpoons"
}
] |
https://en.wikipedia.org/wiki?curid=14276211
|
14276224
|
4-chlorobenzoate dehalogenase
|
Class of enzymes
In enzymology, a 4-chlorobenzoate dehalogenase (EC 3.8.1.6) is an enzyme that catalyzes the chemical reaction
4-chlorobenzoate + H2O formula_0 4-hydroxybenzoate + chloride
Thus, the two substrates of this enzyme are 4-chlorobenzoate and H2O, whereas its two products are 4-hydroxybenzoate and chloride.
This enzyme belongs to the family of hydrolases, specifically those acting on halide bonds in carbon-halide compounds. The systematic name of this enzyme class is 4-chlorobenzoate chlorohydrolase. This enzyme is also called halobenzoate dehalogenase.
Structural studies.
As of late 2007, 3 structures have been solved for this class of enzymes, with PDB accession codes 1BVQ, 1JXZ, and 1NZY.
References.
<templatestyles src="Reflist/styles.css" />
<templatestyles src="Refbegin/styles.css" />
|
[
{
"math_id": 0,
"text": "\\rightleftharpoons"
}
] |
https://en.wikipedia.org/wiki?curid=14276224
|
14276245
|
4-chlorobenzoyl-CoA dehalogenase
|
Class of enzymes
In enzymology, a 4-chlorobenzoyl-CoA dehalogenase (EC 3.8.1.7) is an enzyme that catalyzes the chemical reaction
4-chlorobenzoyl-CoA + H2O formula_0 4-hydroxybenzoyl CoA + chloride
Thus, the two substrates of this enzyme are 4-chlorobenzoyl-CoA and H2O, whereas its two products are 4-hydroxybenzoyl CoA and chloride.
This enzyme belongs to the family of hydrolases, specifically those acting on halide bonds in carbon-halide compounds. The systematic name of this enzyme class is 4-chlorobenzoyl CoA chlorohydrolase. This enzyme participates in 2,4-dichlorobenzoate degradation.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "\\rightleftharpoons"
}
] |
https://en.wikipedia.org/wiki?curid=14276245
|
14276255
|
Acetylpyruvate hydrolase
|
Class of enzymes
In enzymology, an acetylpyruvate hydrolase (EC 3.7.1.6) is an enzyme that catalyzes the chemical reaction
acetylpyruvate + H2O formula_0 acetate + pyruvate
Thus, the two substrates of this enzyme are acetylpyruvate and H2O, whereas its two products are acetate and pyruvate.
This enzyme belongs to the family of hydrolases, specifically those acting on carbon-carbon bonds in ketonic substances. The systematic name of this enzyme class is 2,4-dioxopentanoate acetylhydrolase.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "\\rightleftharpoons"
}
] |
https://en.wikipedia.org/wiki?curid=14276255
|
14276270
|
Acylpyruvate hydrolase
|
Class of enzymes
In enzymology, an acylpyruvate hydrolase (EC 3.7.1.5) is an enzyme that catalyzes the chemical reaction
a 3-acylpyruvate + H2O formula_0 a carboxylate + pyruvate
Thus, the two substrates of this enzyme are 3-acylpyruvate and water, whereas its two products are carboxylate and pyruvate.
This enzyme belongs to the family of hydrolases, specifically those acting on carbon-carbon bonds in ketonic substances. The systematic name of this enzyme class is 3-acylpyruvate acylhydrolase. This enzyme participates in tyrosine metabolism.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "\\rightleftharpoons"
}
] |
https://en.wikipedia.org/wiki?curid=14276270
|
14276280
|
Alkylhalidase
|
In enzymology, an alkylhalidase (EC 3.8.1.1) is an enzyme that catalyzes the chemical reaction
bromochloromethane + H2O formula_0 formaldehyde + bromide + chloride
Thus, the two substrates of this enzyme are bromochloromethane and H2O, whereas its 3 products are formaldehyde, bromide, and chloride.
This enzyme belongs to the family of hydrolases, specifically those acting on halide bonds in carbon-halide compounds. The systematic name of this enzyme class is alkyl-halide halidohydrolase. Other names in common use include halogenase, haloalkane halidohydrolase, and haloalkane dehalogenase.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "\\rightleftharpoons"
}
] |
https://en.wikipedia.org/wiki?curid=14276280
|
14276291
|
Beta-diketone hydrolase
|
In enzymology, a beta-diketone hydrolase (EC 3.7.1.7) is an enzyme that catalyzes the chemical reaction
nonane-4,6-dione + H2O formula_0 pentan-2-one + butanoate
Thus, the two substrates of this enzyme are nonane-4,6-dione and H2O, whereas its two products are 2-pentanone and butanoate.
This enzyme belongs to the family of hydrolases, specifically those acting on carbon-carbon bonds in ketonic substances. The systematic name of this enzyme class is nonane-4,6-dione acylhydrolase. This enzyme is also called oxidized PVA hydrolase.
Structural studies.
As of late 2007, two structures have been solved for this class of enzymes, with PDB accession codes 2J5G and 2J5S.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "\\rightleftharpoons"
}
] |
https://en.wikipedia.org/wiki?curid=14276291
|
14276300
|
Cyclamate sulfohydrolase
|
In enzymology, a cyclamate sulfohydrolase (EC 3.10.1.2) is an enzyme that catalyzes the chemical reaction
cyclohexylsulfamate + H2O formula_0 cyclohexylamine + sulfate
Thus, the two substrates of this enzyme are cyclohexylsulfamate and H2O, whereas its two products are cyclohexylamine and sulfate.
This enzyme belongs to the family of hydrolases, specifically those acting on sulfur-nitrogen bonds. The systematic name of this enzyme class is cyclohexylsulfamate sulfohydrolase. Other names in common use include cyclamate sulfamatase, cyclamate sulfamidase, and cyclohexylsulfamate sulfamidase. This enzyme participates in caprolactam degradation.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "\\rightleftharpoons"
}
] |
https://en.wikipedia.org/wiki?curid=14276300
|
14276311
|
Cyclohexane-1,3-dione hydrolase
|
In enzymology, a cyclohexane-1,3-dione hydrolase (EC 3.7.1.10) is an enzyme that catalyzes the chemical reaction
cyclohexane-1,3-dione + H2O formula_0 5-oxohexanoate
Thus, the two substrates of this enzyme are cyclohexane-1,3-dione and H2O, whereas its product is 5-oxohexanoate.
This enzyme belongs to the family of hydrolases, specifically those acting on carbon-carbon bonds in ketonic substances. The systematic name of this enzyme class is cyclohexane-1,3-dione acylhydrolase (decyclizing). This enzyme is also called 1,3-cyclohexanedione hydrolase.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "\\rightleftharpoons"
}
] |
https://en.wikipedia.org/wiki?curid=14276311
|
14276324
|
Haloacetate dehalogenase
|
Class of enzymes
In enzymology, a haloacetate dehalogenase (EC 3.8.1.3) is an enzyme that catalyzes the chemical reaction
haloacetate + H2O formula_0 glycolate + halide
Thus, the two substrates of this enzyme are haloacetate and H2O, whereas its two products are glycolate and halide. For examples, in the case of fluoroacetate in will produce glycolate and fluoride.
This enzyme belongs to the family of hydrolases, one of the largest known enzyme families comprising approximately 1% of the genes in the human genome, exists as a homodimer, and acts specifically on halide bonds in carbon-halide compounds. The systematic name of this enzyme class is haloacetate halidohydrolase. This enzyme is also called monohaloacetate dehalogenase and fluoroacetate dehalogenase. This enzyme participates in gamma-hexachlorocyclohexane degradation and 1,2-dichloroethane degradation.
Reactions.
Haloacetate dehalogenase is unique because it catalyzes the cleavage of the remarkably stable carbon–fluorine bond of a fluorinated aliphatic compound. In the reaction of L-2-haloacid dehalogenase and fluoroacetate dehalogenase, the carboxylate group performs a nucleophilic attack on the alpha-carbon atom, moving the halogen atom. This action is common to haloalkane dehalogenase and 4-chlorobenzoyl-CoA dehalogenase. DL-2-Haloacid dehalogenase is unique in that a water molecule directly attacks the substrate, displacing the halogen atom.
Significance.
As fluoroacetate is poisonous and present in plants endemic to Australia, Africa, and Central America, livestock are often killed by fluoroacetate poisoning. Fluoroacetate is lethal to sheep and cattle at doses of 0.25 to 0.5 mg/kg of body weight, and is a problem in the livestock industry. A fluoroacetate dehalogenase gene from the soil bacterium "Moraxella" species strain B was transferred into the rumen bacterium "Butyrivibrio fibrisolvens" and expressed in vitro at sufficiently high levels to detoxify fluoroacetate in the surrounding medium. Scientists and farmers want to determine a way to get "B. fibrisolvens" into either the animals or plants.
Structural studies.
As of late 2007, only one structure has been solved for this class of enzymes, with the PDB accession code 1Y37.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "\\rightleftharpoons"
}
] |
https://en.wikipedia.org/wiki?curid=14276324
|
14276338
|
Haloalkane dehalogenase
|
In enzymology, a haloalkane dehalogenase (EC 3.8.1.5) is an enzyme that catalyzes the chemical reaction
1-haloalkane + H2O formula_0 a primary alcohol + halide
Thus, the two substrates of this enzyme are 1-haloalkane and H2O, whereas its two products are primary alcohol and halide.
This enzyme belongs to the family of hydrolases, specifically those acting on halide bonds in carbon-halide compounds. The systematic name of this enzyme class is 1-haloalkane halidohydrolase. Other names in common use include 1-chlorohexane halidohydrolase, and 1-haloalkane dehalogenase. Haloalkane dehalogenases are found in certain bacteria and belong the alpha-beta hydrolase fold superfamily of enzymes. They participate in several metabolic pathways: 1,2-dichloroethane degradation, 1-chloro-n-butane degradation, hexachlorocyclohexane degradation, 1,2-dibromoethane degradation, 2-chloroethyl-vinylether degradation, and 1,3-dichloropropene degradation.
Enzyme Structure and Structural studies.
Structurally, haloalkane dehalogenases belong to the alpha/beta-hydrolase superfamily. Their active site is buried in a predominantly hydrophobic cavity at the interface of the alpha/beta-hydrolase core domain and the helical cap domain, and is connected to the bulk solvent by access tunnels. The active-site residues that are essential for catalysis are referred to as the catalytic pentad, and comprise a nucleophilic aspartate residue, a basic histidine residue, an aspartic or glutamic acid moiety that serves as a general acid and either two tryptophan residues or a tryptophan-asparagine pair that serve to stabilize the leaving halide ion. The haloalkane dehalogenase family currently includes 14 distinct enzymes with experimentally confirmed dehalogenation activity. An analysis of the sequences and structures of haloalkane dehalogenase and their homologues divided the family into three subfamilies, which differ mainly in the composition of their catalytic pentad and cap domain.
As of late 2007, 25 structures have been solved for this class of enzymes, with PDB accession codes 1B6G, 1BE0, 1BEE, 1BEZ, 1BN6, 1BN7, 1CIJ, 1CQW, 1CV2, 1D07, 1EDB, 1EDD, 1EDE, 1HDE, 1K5P, 1K63, 1K6E, 1MJ5, 2DHC, 2DHD, 2DHE, 2EDA, 2EDC, 2PKY, and 2YXP.
Enzyme mechanism.
The main reaction is an SN2 displacement of the halogen for a hydroxyl group derived from water. To begin, aspartate 124 is perfectly aligned with the substrate. It will drive off the halogen and form an ester functionality carbon-oxygen bond. Following this displacement is a hydrolysis reaction by utilizing the imidazole ring of histidine 289 as the general base. This will deprotonate water, form a tetrahedral intermediate at the original ester, and create an imidazolium cation at histidine. The final step is beta-elimination. With a newly formed imidazolium cation ready to be an acid, aspartate 124 reverts to its original acidic state and breaks the ester linkage, as well as deprotonating histidine 289. The alcohol is eliminated and the halogen is now a free anion.
Also taking place in a facilitating role are tryptophan groups in the periphery of the active site. These residues provide hydrogen bond donor groups to the chloride as it begins to undergo the SN2 reaction and become an anion. A second tryptophan also provides rigidity through a stable peptide bond to aspartate 124. It holds the beta-carbon oxygen in place so that it’s in prime position to make the ester linkage.
Industrial functionality.
A number of halogenated compounds are environmentally toxic industrial by-products, and it has been suggested that haloalkane dehalogenases may be useful catalysts for their biodegradation, with potential applications in bioremediation. In biocatalysis, there is a standing interest in these enzymes, particularly for the production of optically pure alcohols. Therefore, the identification of dehalogenating enzymes with appropriate selectivity patterns is very important in terms of their industrial utility.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "\\rightleftharpoons"
}
] |
https://en.wikipedia.org/wiki?curid=14276338
|
1427634
|
Rate-determining step
|
Slowest step of a chemical reaction
In chemical kinetics, the overall rate of a reaction is often approximately determined by the slowest step, known as the rate-determining step (RDS or RD-step or r/d step) or rate-limiting step. For a given reaction mechanism, the prediction of the corresponding rate equation (for comparison with the experimental rate law) is often simplified by using this approximation of the rate-determining step.
In principle, the time evolution of the reactant and product concentrations can be determined from the set of simultaneous rate equations for the individual steps of the mechanism, one for each step. However, the analytical solution of these differential equations is not always easy, and in some cases numerical integration may even be required. The hypothesis of a single rate-determining step can greatly simplify the mathematics. In the simplest case the initial step is the slowest, and the overall rate is just the rate of the first step.
Also, the rate equations for mechanisms with a single rate-determining step are usually in a simple mathematical form, whose relation to the mechanism and choice of rate-determining step is clear. The correct rate-determining step can be identified by predicting the rate law for each possible choice and comparing the different predictions with the experimental law, as for the example of and CO below.
The concept of the rate-determining step is very important to the optimization and understanding of many chemical processes such as catalysis and combustion.
Example reaction: + CO.
As an example, consider the gas-phase reaction + CO → NO + CO2. If this reaction occurred in a single step, its reaction rate ("r") would be proportional to the rate of collisions between and CO molecules: "r" = "k"[][CO], where "k" is the reaction rate constant, and square brackets indicate a molar concentration. Another typical example is the Zel'dovich mechanism.
First step rate-determining.
In fact, however, the observed reaction rate is second-order in and zero-order in CO, with rate equation "r" = "k"[]2. This suggests that the rate is determined by a step in which two molecules react, with the CO molecule entering at another, faster, step. A possible mechanism in two elementary steps that explains the rate equation is:
In this mechanism the reactive intermediate species is formed in the first step with rate "r"1 and reacts with CO in the second step with rate "r"2. However, can also react with NO if the first step occurs in the "reverse direction" (NO + → 2 ) with rate "r"−1, where the minus sign indicates the rate of a reverse reaction.
The concentration of a reactive intermediate such as [] remains low and almost constant. It may therefore be estimated by the steady-state approximation, which specifies that the rate at which it is formed equals the (total) rate at which it is consumed. In this example is formed in one step and reacts in two, so that
formula_0
The statement that the first step is the slow step actually means that the first step "in the reverse direction" is slower than the second step in the forward direction, so that almost all is consumed by reaction with CO and not with NO. That is, "r"−1 ≪ "r"2, so that "r"1 − "r"2 ≈ 0. But the overall rate of reaction is the rate of formation of final product (here CO2), so that "r" = "r"2 ≈ "r"1. That is, the overall rate is determined by the rate of the first step, and (almost) all molecules that react at the first step continue to the fast second step.
Pre-equilibrium: if the second step were rate-determining.
The other possible case would be that the second step is slow and rate-determining, meaning that it is slower than the first step in the reverse direction: "r"2 ≪ "r"−1. In this hypothesis, "r"1 − r−1 ≈ 0, so that the first step is (almost) at equilibrium. The overall rate is determined by the second step: "r" = "r"2 ≪ "r"1, as very few molecules that react at the first step continue to the second step, which is much slower. Such a situation in which an intermediate (here ) forms an equilibrium with reactants "prior" to the rate-determining step is described as a "pre-equilibrium" For the reaction of and CO, this hypothesis can be rejected, since it implies a rate equation that disagrees with experiment.
If the first step were at equilibrium, then its equilibrium constant expression permits calculation of the concentration of the intermediate in terms of more stable (and more easily measured) reactant and product species:
formula_1
formula_2
The overall reaction rate would then be
formula_3
which disagrees with the experimental rate law given above, and so disproves the hypothesis that the second step is rate-determining for this reaction. However, some other reactions are believed to involve rapid pre-equilibria prior to the rate-determining step, as shown below.
Nucleophilic substitution.
Another example is the unimolecular nucleophilic substitution (SN1) reaction in organic chemistry, where it is the first, rate-determining step that is unimolecular. A specific case is the basic hydrolysis of tert-butyl bromide (t-C4H9Br) by aqueous sodium hydroxide. The mechanism has two steps (where R denotes the tert-butyl radical t-C4H9):
This reaction is found to be first-order with "r" = "k"[R−Br], which indicates that the first step is slow and determines the rate. The second step with OH− is much faster, so the overall rate is independent of the concentration of OH−.
In contrast, the alkaline hydrolysis of methyl bromide (CH3Br) is a bimolecular nucleophilic substitution (SN2) reaction in a single bimolecular step. Its rate law is second-order: "r" = "k"[R−Br][OH-].
Composition of the transition state.
A useful rule in the determination of mechanism is that the concentration factors in the rate law indicate the composition and charge of the activated complex or transition state. For the –CO reaction above, the rate depends on []2, so that the activated complex has composition N2O4, with 2 entering the reaction before the transition state, and CO reacting after the transition state.
A multistep example is the reaction between oxalic acid and chlorine in aqueous solution: H2C2O4 + Cl2 → 2 CO2 + 2 H+ + 2 Cl-.
The observed rate law is
formula_4
which implies an activated complex in which the reactants lose 2H+ + Cl- before the rate-determining step. The formula of the activated complex is Cl2 + H2C2O4 − 2 H+ − Cl- + x , or C2O4Cl(H2O)– (an unknown number of water molecules are added because the possible dependence of the reaction rate on was not studied, since the data were obtained in water solvent at a large and essentially unvarying concentration).
One possible mechanism in which the preliminary steps are assumed to be rapid pre-equilibria occurring prior to the transition state is
Cl2 + ⇌ HOCl + Cl- + H+
H2C2O4 ⇌ H+ + HC2O4-
HOCl + HC2O4- → + Cl- + 2 CO2
Reaction coordinate diagram.
In a multistep reaction, the rate-determining step does not necessarily correspond to the highest Gibbs energy on the reaction coordinate diagram. If there is a reaction intermediate whose energy is lower than the initial reactants, then the activation energy needed to pass through any subsequent transition state depends on the Gibbs energy of that state relative to the lower-energy intermediate. The rate-determining step is then the step with the largest Gibbs energy difference relative either to the starting material or to any previous intermediate on the diagram.
Also, for reaction steps that are not first-order, concentration terms must be considered in choosing the rate-determining step.
Chain reactions.
Not all reactions have a single rate-determining step. In particular, the rate of a chain reaction is usually not controlled by any single step.
Diffusion control.
In the previous examples the rate determining step was one of the sequential chemical reactions leading to a product. The rate-determining step can also be the transport of reactants to where they can interact and form the product. This case is referred to as diffusion control and, in general, occurs when the formation of product from the activated complex is very rapid and thus the provision of the supply of reactants is rate-determining.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "\\frac{d\\ce{[NO3]}}{dt} = r_1 - r_2 - r_{-1} \\approx 0."
},
{
"math_id": 1,
"text": "K_1 = \\frac{\\ce{[NO][NO3]}}{\\ce{[NO2]^2}},"
},
{
"math_id": 2,
"text": "[\\ce{NO3}] = K_1 \\frac{\\ce{[NO2]^2}}{\\ce{[NO]}}."
},
{
"math_id": 3,
"text": "r = r_2 = k_2 \\ce{[NO3][CO]} = k_2 K_1 \\frac{\\ce{[NO2]^2 [CO]}}{\\ce{[NO]}},"
},
{
"math_id": 4,
"text": "v = k \\frac{\\ce{[Cl2][H2C2O4]}}{[\\ce{H+}]^2[\\ce{Cl^-}]},"
}
] |
https://en.wikipedia.org/wiki?curid=1427634
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14276351
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N-sulfoglucosamine sulfohydrolase
|
Class of enzymes
In enzymology, a N-sulfoglucosamine sulfohydrolase (EC 3.10.1.1), otherwise known as SGSH, is an enzyme that catalyzes the chemical reaction
N-sulfo-D-glucosamine + H2O formula_0 D-glucosamine + sulfate
Thus, the two substrates of this enzyme are N-sulfo-D-glucosamine and H2O, whereas its two products are D-glucosamine and sulfate.
This enzyme belongs to the family of hydrolases, specifically those acting on sulfur-nitrogen bonds. The systematic name of this enzyme class is N-sulfo-D-glucosamine sulfohydrolase. Other names in common use include sulfoglucosamine sulfamidase, heparin sulfamidase, 2-desoxy-D-glucoside-2-sulphamate sulphohydrolase (sulphamate, and sulphohydrolase). This enzyme participates in glycosaminoglycan degradation and glycan structures - degradation.
Structure.
N-glusoamine sulfohydrolase is a homodimer of two identical monomeric subunits that associate non-covalently. The crystal structure, solved using molecular replacement, consists of two domains: a large N-terminal domain (domain 1) and a smaller C-terminal domain (domain 2) that are both centered about a β-sheet. The core of Domain 1 is formed by eight β-strands surrounded by nine ⍺-helices while the core of Domain 2 is formed by a four-stranded antiparallel β-sheet surrounded by 4 ⍺-helices, accompanied by a C-terminal extension of a small two-stranded β-sheet. There are two disulfide bonds that serve to stabilize a large loop segment in Domain 1 and connect the C-terminal extension to a nearby loop in Domain 2.
The active site is located through a short tunnel at the bottom of the surface cleft of the enzyme close to the end of the first β-strand in Domain 1 of the dimer subunits. A calcium ion, Ca2+, is coordinated in an octahedral arrangement by oxygen atoms of nearby side chains of residues Asp31, Asp32, Asp273, Asn274, and phosphorylated formylglycine, PFGly70. The PFGly70, which is post-translationally converted from a cysteine and is key in the catalytic activity of the enzyme, is stabilized by the side chains of residues Arg74, Lys123, His125, His181, Asp273, and Arg282 and the coordinate calcium ion.
Mechanism.
One of the proposed mechanisms for the N-sulfoglucosamine sulfohydrolase, depicted in the figure above, was inspired by a mechanism determined for a sulfatase enzymes, which similarly cleave sulfate ester groups from their substrates. The N-sulfoglucosamine sulfohydrolase mechanism involves four key residues found in the enzyme active site: formylglycine (FGly70), two histidines (His125 and His181), and aspartic acid (Asp273). The formylglycine residue (FGly70) is first hydrated to form a geminal diol whose hydroxyl group subsequently coordinates with a Calcium(II) ion in the active site. A neighboring aspartic acid residue (Asp273) then acts as a base to deprotonate the coordinated hydroxyl group in the geminal diol, which acts as a nucleophile to the sulfate group on the substrate and leads to the cleavage of the substrate nitrogen-sulfur bond. A proximal histidine residue (His181) is proposed to donate a proton as a replacement for the sulfate group that was involved in the nitrogen-sulfur bond. At this point, the formylglycine (FGly70) has the sulfate group attached adjacent to a hydroxyl group, which is believed to be deprotonated by a second histidine group (His125) so that the negatively-charged oxygen atom can participate in the removal of the sulfate group from the residue. This final step liberates the enzyme from the sulfate group.
This mechanism, however, is still being studied for N-sulfoglucosamine sulfohydrolase, meaning other mechanisms are still plausible for the reaction. For instance, a second mechanism proposal involves the aldehyde form of the formylglycine residue (FGly70) serving as an electrophile as it is attacked by one of the oxygen atoms on the sulfate group to transfer the sulfate from the substrate to the enzyme. The order of attack for the sulfate transfer step in this proposed mechanism is therefore inverted to that of the proposed mechanism above.
Overall, the N-sulfoglucosamine sulfohydrolase mechanisms proposed involve the cleavage of the substrate nitrogen-sulfur bond - transferring the sulfate group from the substrate to the enzyme - and freeing the enzyme from its bond to the transferred sulfate group.
Function.
As previously noted, N-sulfoglucosamine sulfohydrolase plays a crucial role in the degradation of glycosaminoglycans (GAGs), including heparin and heparan sulfate. Since these GAGs, particularly haparan sulfates, are integral to several biochemical processes, such as signaling pathways, any disruption in N-sulfoglucosamine sulfohydrolase activity could lead to serious diseases as noted below.
Disease relevance.
Sanfillipo Syndrome or Mucopolysaccharidosis III, MPS III, is a lysosomal storage disease resulting from a deficiency in one of five lysosomal enzymes: N-glusoamine sulfohydrolase (Type A), a-N-acetylglucoaminidase (Type B), acetyl CoA a-glusoaminide acetyltransferase (Type C), a-N-acetylglusoamine 6-sulfatase (Type D), and N-glusoamine 3-O-sulfatase (Type E) caused by a dysfunction of one of the genes encoding the enzyme. These enzymes are responsible for the degradation of heparin sulfate which, in MPS III, accumulates in the lysosomes and outside of the cell, as the primary storage material. In Mucopolysaccharidosis type IIIA, where there are genetic changes in the SGSH gene, there are initial signs of neurodegeneration, developmental delays, and behavioral problems with a wide phenotypic variability. This is the most common form of Mucopolysaccharidosis III with a prevalence of 1 in every 100,000 individuals.
References.
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Further reading.
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[
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https://en.wikipedia.org/wiki?curid=14276351
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14276361
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Oxaloacetase
|
Enzyme
In enzymology, an oxaloacetase (EC 3.7.1.1) is an enzyme that catalyzes the chemical reaction:
oxaloacetate + H2O formula_0 oxalate + acetate
Thus, the two substrates of this enzyme are oxaloacetate and H2O, whereas its two products are oxalate and acetate.
This enzyme belongs to the family of hydrolases, specifically those acting on carbon-carbon bonds in ketonic substances. The systematic name of this enzyme class is oxaloacetate acetylhydrolase. This enzyme is also called oxalacetic hydrolase.
References.
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{
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14276379
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Phloretin hydrolase
|
In enzymology, a phloretin hydrolase (EC 3.7.1.4) is an enzyme that catalyzes the chemical reaction
phloretin + H2O formula_0 phloretate + phloroglucinol
Thus, the two substrates of this enzyme are phloretin and H2O, whereas its two products are phloretate and phloroglucinol.
This enzyme belongs to the family of hydrolases, specifically those acting on carbon–carbon bonds in ketonic substances. The systematic name of this enzyme class is 2',4,4',6'-tetrahydroxydehydrochalcone 1,3,5-trihydroxybenzenehydrolase. This enzyme is also called lactase-phlorizin hydrolase.
References.
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14276393
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Phosphoamidase
|
In enzymology, a phosphoamidase (EC 3.9.1.1) is an enzyme that catalyzes the chemical reaction
N-phosphocreatine + H2O formula_0 creatine + phosphate
Thus, the two substrates of this enzyme are N-phosphocreatine and H2O, whereas its two products are creatine and phosphate.
This enzyme belongs to the family of hydrolases, specifically those acting on phosphorus-nitrogen bonds. The systematic name of this enzyme class is phosphamide hydrolase. This enzyme is also called creatine phosphatase.
References.
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{
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14276415
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Phosphonoacetaldehyde hydrolase
|
In enzymology, a phosphonoacetaldehyde hydrolase (EC 3.11.1.1) is an enzyme that catalyzes the chemical reaction
phosphonoacetaldehyde + H2O formula_0 acetaldehyde + phosphate
Thus, the two substrates of this enzyme are phosphonoacetaldehyde and H2O, whereas its two products are acetaldehyde and phosphate.
This enzyme belongs to the family of hydrolases, specifically those acting on carbon-phosphorus bonds. The systematic name of this enzyme class is 2-oxoethylphosphonate phosphonohydrolase. Other names in common use include phosphonatase, and 2-phosphonoacetylaldehyde phosphonohydrolase. This enzyme participates in aminophosphonate metabolism.
Structural studies.
As of late 2007, two structures have been solved for this class of enzymes, with PDB accession codes 1SWV and 1SWW.
References.
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{
"math_id": 0,
"text": "\\rightleftharpoons"
}
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https://en.wikipedia.org/wiki?curid=14276415
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14276433
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Phosphonoacetate hydrolase
|
In enzymology, a phosphonoacetate hydrolase (EC 3.11.1.2) is an enzyme that catalyzes the chemical reaction
phosphonoacetate + H2O formula_0 acetate + phosphate
Thus, the two substrates of this enzyme are phosphonoacetate and H2O, whereas its two products are acetate and phosphate.
This enzyme belongs to the family of hydrolases, specifically those acting on carbon-phosphorus bonds. The systematic name of this enzyme class is phosphonoacetate phosphonohydrolase. This enzyme participates in aminophosphonate metabolism. It employs one cofactor, zinc.
Structural studies.
The structure of this enzyme, with the PDB accession code 1EI6, shows it adopts the alkaline phosphatase fold.
References.
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{
"math_id": 0,
"text": "\\rightleftharpoons"
}
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https://en.wikipedia.org/wiki?curid=14276433
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14276446
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Phosphonopyruvate hydrolase
|
In enzymology, a phosphonopyruvate hydrolase (EC 3.11.1.3) is an enzyme that catalyzes the chemical reaction
3-phosphonopyruvate + H2O formula_0 pyruvate + phosphate
Thus, the two substrates of this enzyme are 3-phosphonopyruvate and H2O, whereas its two products are pyruvate and phosphate.
This enzyme belongs to the family of hydrolases, specifically those acting on carbon-phosphorus bonds. The systematic name of this enzyme class is . This enzyme is also called PPH.
References.
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{
"math_id": 0,
"text": "\\rightleftharpoons"
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https://en.wikipedia.org/wiki?curid=14276446
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14276463
|
(R)-2-haloacid dehalogenase
|
Class of enzymes
In enzymology, a (R)-2-haloacid dehalogenase "(EC 3.8.1.9)", "DL-2-haloacid halidohydrolase (inversion of configuration)", "DL-DEXi", "(R,S)-2-haloacid dehalogenase (configuration-inverting)") is an enzyme that catalyzes the chemical reaction
(R)-2-haloacid + H2O formula_0 (S)-2-hydroxyacid + halide
Thus, the two substrates of this enzyme are (R)-2-haloacid and H2O, whereas its two products are (S)-2-hydroxyacid and halide.
This enzyme belongs to the family of hydrolases, specifically those acting on halide bonds in carbon-halide compounds. The systematic name of this enzyme class is (R)-2-haloacid halidohydrolase. Other names in common use include 2-haloalkanoic acid dehalogenase[ambiguous], 2-haloalkanoid acid halidohydrolase[ambiguous], D-2-haloacid dehalogenase, and D-DEX.
References.
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14276478
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(S)-2-haloacid dehalogenase
|
Class of enzymes
In enzymology, a (S)-2-haloacid dehalogenase (EC 3.8.1.2) is an enzyme that catalyzes the chemical reaction
(S)-2-haloacid + H2O formula_0 (R)-2-hydroxyacid + halide
Thus, the two substrates of this enzyme are (S)-2-haloacid and H2O, whereas its two products are (R)-2-hydroxyacid and halide.
This enzyme belongs to the family of hydrolases, specifically those acting on halide bonds in carbon-halide compounds. The systematic name of this enzyme class is (S)-2-haloacid halidohydrolase. Other names in common use include 2-haloacid dehalogenase[ambiguous], 2-haloacid halidohydrolase [ambiguous][ambiguous], 2-haloalkanoic acid dehalogenase, 2-haloalkanoid acid halidohydrolase, 2-halocarboxylic acid dehalogenase II, DL-2-haloacid dehalogenase[ambiguous], L-2-haloacid dehalogenase, and L-DEX. This enzyme participates in gamma-hexachlorocyclohexane degradation and 1,2-dichloroethane degradation.
Structural studies.
As of late 2007, 10 structures have been solved for this class of enzymes, with PDB accession codes 1AQ6, 1JUD, 1QH9, 1QQ5, 1QQ6, 1QQ7, 1ZRM, 1ZRN, 2NO4, and 2NO5.
References.
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"text": "\\rightleftharpoons"
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https://en.wikipedia.org/wiki?curid=14276478
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14276496
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Trithionate hydrolase
|
Class of enzymes
In enzymology, a trithionate hydrolase (EC 3.12.1.1) is an enzyme that catalyzes the chemical reaction
trithionate + H2O formula_0 thiosulfate + sulfate + 2 H+
Thus, the two substrates of this enzyme are trithionate and H2O, whereas its 3 products are thiosulfate, sulfate, and H+.
This enzyme belongs to the family of hydrolases, specifically those acting on sulfur-sulfur bonds. The systematic name of this enzyme class is trithionate thiosulfohydrolase. This enzyme participates in sulfur metabolism.
References.
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"text": "\\rightleftharpoons"
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https://en.wikipedia.org/wiki?curid=14276496
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14276508
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UDP-sulfoquinovose synthase
|
Class of enzymes
UDP-sulfoquinovose synthase (EC 3.13.1.1) is an enzyme that catalyzes the chemical reaction
UDP-glucose + sulfite formula_0 UDP-6-sulfoquinovose + H2O
Thus, the two substrates of this enzyme are UDP-glucose and sulfite, whereas its two products are UDP-6-sulfoquinovose and H2O.
In a subsequent reaction catalyzed by sulfoquinovosyl diacylglycerol synthase, the sulfoquinovose portion of UDP-sulfoquinovose is combined with diacyglycerol to produce the sulfolipid sulfoquinovosyl diacylglycerol (SQDG).
This enzyme belongs to the family of hydrolases, specifically those acting on carbon-sulfur bonds. The systematic name of this enzyme class is UDP-6-sulfo-6-deoxyglucose sulfohydrolase. Other names in common use include sulfite:UDP-glucose sulfotransferase, and UDP-sulfoquinovose synthase. This enzyme participates in nucleotide sugars metabolism and glycerolipid metabolism.
The 3-dimensional structure of the enzyme is known from Protein Data Bank entries 1qrr (Mulichak et al., 1999), 1i24, 1i2b and 1i2c.
References.
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1427763
|
Collision theory
|
Chemistry principle
"Collision theory" is a principle of chemistry used to predict the rates of chemical reactions. It states that when suitable particles of the reactant hit each other with the correct orientation, only a certain amount of collisions result in a perceptible or notable change; these successful changes are called successful collisions. The successful collisions must have enough energy, also known as activation energy, at the moment of impact to break the pre-existing bonds and form all new bonds. This results in the products of the reaction. The activation energy is often predicted using the Transition state theory. Increasing the concentration of the reactant brings about more collisions and hence more successful collisions. Increasing the temperature increases the average kinetic energy of the molecules in a solution, increasing the number of collisions that have enough energy. Collision theory was proposed independently by Max Trautz in 1916 and William Lewis in 1918.
When a catalyst is involved in the collision between the reactant molecules, less energy is required for the chemical change to take place, and hence more collisions have sufficient energy for the reaction to occur. The reaction rate therefore increases.
Collision theory is closely related to chemical kinetics.
Collision theory was initially developed for the gas reaction system with no dilution. But most reactions involve solutions, for example, gas reactions in a carrying inert gas, and almost all reactions in solutions. The collision frequency of the solute molecules in these solutions is now controlled by diffusion or Brownian motion of individual molecules. The flux of the diffusive molecules follows Fick's laws of diffusion. For particles in a solution, an example model to calculate the collision frequency and associated coagulation rate is the Smoluchowski coagulation equation proposed by Marian Smoluchowski in a seminal 1916 publication. In this model, Fick's flux at the infinite time limit is used to mimic the particle speed of the collision theory. Jixin Chen proposed a finite-time solution to the diffusion flux in 2022 which significantly changes the estimated collision frequency of two particles in a solution.
Rate equations.
The rate for a bimolecular gas-phase reaction, A + B → product, predicted by collision theory is
formula_0
where:
The unit of "r"("T") can be converted to mol⋅L−1⋅s−1, after divided by (1000×"N"A), where "N"A is the Avogadro constant.
For a reaction between A and B, the collision frequency calculated with the hard-sphere model with the unit number of collisions per m3 per second is:
formula_2
where:
If all the units that are related to dimension are converted to dm, i.e. mol⋅dm−3 for [A] and [B], dm2 for "σ"AB, dm2⋅kg⋅s−2⋅K−1 for the Boltzmann constant, then
formula_5
unit mol⋅dm−3⋅s−1.
Quantitative insights.
Derivation.
Consider the bimolecular elementary reaction:
A + B → C
In collision theory it is considered that two particles A and B will collide if their nuclei get closer than a certain distance. The area around a molecule A in which it can collide with an approaching B molecule is called the cross section (σAB) of the reaction and is, in simplified terms, the area corresponding to a circle whose radius (formula_6) is the sum of the radii of both reacting molecules, which are supposed to be spherical.
A moving molecule will therefore sweep a volume formula_7 per second as it moves, where formula_8 is the average velocity of the particle. (This solely represents the classical notion of a collision of solid balls. As molecules are quantum-mechanical many-particle systems of electrons and nuclei based upon the Coulomb and exchange interactions, generally they neither obey rotational symmetry nor do they have a box potential. Therefore, more generally the cross section is defined as the reaction probability of a ray of A particles per areal density of B targets, which makes the definition independent from the nature of the interaction between A and B. Consequently, the radius formula_6 is related to the length scale of their interaction potential.)
From kinetic theory it is known that a molecule of A has an average velocity (different from root mean square velocity) of formula_9, where formula_10 is the Boltzmann constant, and formula_11 is the mass of the molecule.
The solution of the two-body problem states that two different moving bodies can be treated as one body which has the reduced mass of both and moves with the velocity of the center of mass, so, in this system formula_12 must be used instead of formula_11.
Thus, for a given molecule A, it travels formula_13 before hitting a molecule B if all B is fixed with no movement, where formula_14 is the average traveling distance. Since B also moves, the relative velocity can be calculated using the reduced mass of A and B.
Therefore, the total collision frequency, of all A molecules, with all B molecules, is
formula_15
From Maxwell–Boltzmann distribution it can be deduced that the fraction of collisions with more energy than the activation energy is formula_16. Therefore, the rate of a bimolecular reaction for ideal gases will be
formula_17 in unit number of molecular reactions s−1⋅m−3,
where:
The product "zρ" is equivalent to the preexponential factor of the Arrhenius equation.
Validity of the theory and steric factor.
Once a theory is formulated, its validity must be tested, that is, compare its predictions with the results of the experiments.
When the expression form of the rate constant is compared with the rate equation for an elementary bimolecular reaction, formula_18, it is noticed that
formula_19
unit M−1⋅s−1 (= dm3⋅mol−1⋅s−1), with all dimension unit dm including "k"B.
This expression is similar to the Arrhenius equation and gives the first theoretical explanation for the Arrhenius equation on a molecular basis. The weak temperature dependence of the preexponential factor is so small compared to the exponential factor that it cannot be measured experimentally, that is, "it is not feasible to establish, on the basis of temperature studies of the rate constant, whether the predicted "T" dependence of the preexponential factor is observed experimentally".
Steric factor.
If the values of the predicted rate constants are compared with the values of known rate constants, it is noticed that collision theory fails to estimate the constants correctly, and the more complex the molecules are, the more it fails. The reason for this is that particles have been supposed to be spherical and able to react in all directions, which is not true, as the orientation of the collisions is not always proper for the reaction. For example, in the hydrogenation reaction of ethylene the H2 molecule must approach the bonding zone between the atoms, and only a few of all the possible collisions fulfill this requirement.
To alleviate this problem, a new concept must be introduced: the steric factor "ρ". It is defined as the ratio between the experimental value and the predicted one (or the ratio between the frequency factor and the collision frequency):
formula_20
and it is most often less than unity.
Usually, the more complex the reactant molecules, the lower the steric factor. Nevertheless, some reactions exhibit steric factors greater than unity: the harpoon reactions, which involve atoms that exchange electrons, producing ions. The deviation from unity can have different causes: the molecules are not spherical, so different geometries are possible; not all the kinetic energy is delivered into the right spot; the presence of a solvent (when applied to solutions), etc.
Collision theory can be applied to reactions in solution; in that case, the "solvent cage" has an effect on the reactant molecules, and several collisions can take place in a single encounter, which leads to predicted preexponential factors being too large. "ρ" values greater than unity can be attributed to favorable entropic contributions.
Alternative collision models for diluted solutions.
Collision in diluted gas or liquid solution is regulated by diffusion instead of direct collisions, which can be calculated from Fick's laws of diffusion. Theoretical models to calculate the collision frequency in solutions have been proposed by Marian Smoluchowski in a seminal 1916 publication at the infinite time limit, and Jixin Chen in 2022 at a finite-time approximation. A scheme of comparing the rate equations in pure gas and solution is shown in the right figure.
For a diluted solution in the gas or the liquid phase, the collision equation developed for neat gas is not suitable when diffusion takes control of the collision frequency, i.e., the direct collision between the two molecules no longer dominates. For any given molecule A, it has to collide with a lot of solvent molecules, let's say molecule C, before finding the B molecule to react with. Thus the probability of collision should be calculated using the Brownian motion model, which can be approximated to a diffusive flux using various boundary conditions that yield different equations in the Smoluchowski model and the JChen Model.
For the diffusive collision, at the infinite time limit when the molecular flux can be calculated from the Fick's laws of diffusion, in 1916 Smoluchowski derived a collision frequency between molecule A and B in a diluted solution:
formula_21
where:
or
formula_28
where:
There have been a lot of extensions and modifications to the Smoluchowski model since it was proposed in 1916.
In 2022, Chen rationales that because the diffusive flux is evolving over time and the distance between the molecules has a finite value at a given concentration, there should be a critical time to cut off the evolution of the flux that will give a value much larger than the infinite solution Smoluchowski has proposed. So he proposes to use the average time for two molecules to switch places in the solution as the critical cut-off time, i.e., first neighbor visiting time. Although an alternative time could be the mean free path time or the average first passenger time, it overestimates the concentration gradient between the original location of the first passenger to the target. This hypothesis yields a fractal reaction kinetic rate equation of diffusive collision in a diluted solution:
formula_33
where:
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "r(T) = kn_\\text{A}n_\\text{B}= Z \\rho \\exp \\left( \\frac{-E_\\text{a}}{RT} \\right)"
},
{
"math_id": 1,
"text": "\\rho"
},
{
"math_id": 2,
"text": " Z = n_\\text{A} n_\\text{B} \\sigma_\\text{AB} \\sqrt\\frac{8 k_\\text{B} T}{\\pi \\mu_\\text{AB}} = 10^6N_A^2\\text{[A][B]} \\sigma_\\text{AB} \\sqrt\\frac{8 k_\\text{B} T}{\\pi \\mu_\\text{AB}}"
},
{
"math_id": 3,
"text": " \\sigma_\\text{AB} = \\pi(r_\\text{A}+r_\\text{B})^2 "
},
{
"math_id": 4,
"text": " \\mu_\\text{AB} = \\frac{{m_\\text{A}}{m_\\text{B}}}{{m_\\text{A}} + {m_\\text{B}}} "
},
{
"math_id": 5,
"text": " Z = N_\\text{A} \\sigma_\\text{AB} \\sqrt\\frac{8 k_\\text{B} T}{\\pi \\mu_\\text{AB}}[\\text{A}][\\text{B}] = k [A][B]"
},
{
"math_id": 6,
"text": "r_{AB}"
},
{
"math_id": 7,
"text": "\\pi r^{2}_{AB} c_A"
},
{
"math_id": 8,
"text": "c_A"
},
{
"math_id": 9,
"text": "c_A = \\sqrt \\frac{8 k_\\text{B} T}{\\pi m_A}"
},
{
"math_id": 10,
"text": "k_\\text{B}"
},
{
"math_id": 11,
"text": "m_A"
},
{
"math_id": 12,
"text": "\\mu_{AB}"
},
{
"math_id": 13,
"text": "t=l/c_A=1/(n_B\\sigma_{AB}c_A)"
},
{
"math_id": 14,
"text": "l"
},
{
"math_id": 15,
"text": " Z = n_\\text{A} n_\\text{B} \\sigma_{AB} \\sqrt\\frac{8 k_\\text{B} T}{\\pi \\mu_{AB}} = 10^6N_A^2[A][B] \\sigma_{AB} \\sqrt\\frac{8 k_\\text{B} T}{\\pi \\mu_{AB}} = z[A][B],"
},
{
"math_id": 16,
"text": "e^{\\frac{-E_\\text{a}}{RT}}"
},
{
"math_id": 17,
"text": "r = z \\rho [A][B] \\exp\\left( \\frac{-E_\\text{a}}{RT} \\right),"
},
{
"math_id": 18,
"text": "r = k(T) [A][B]"
},
{
"math_id": 19,
"text": "k(T) = N_A \\sigma_{AB}\\rho \\sqrt \\frac{8 k_\\text{B} T}{\\pi \\mu_{AB}} \\exp \\left( \\frac{-E_\\text{a}}{RT} \\right)"
},
{
"math_id": 20,
"text": "\\rho = \\frac{A_\\text{observed}}{Z_\\text{calculated}},"
},
{
"math_id": 21,
"text": "Z_{AB} = 4 \\pi R D_r C_A C_B "
},
{
"math_id": 22,
"text": "Z_{AB}"
},
{
"math_id": 23,
"text": "R"
},
{
"math_id": 24,
"text": "D_r"
},
{
"math_id": 25,
"text": "D_r = D_A + D_B"
},
{
"math_id": 26,
"text": "C_A"
},
{
"math_id": 27,
"text": "C_B"
},
{
"math_id": 28,
"text": "Z_{AB} = 1000 N_A * 4 \\pi R D_r [A] [B] = k [A] [B] "
},
{
"math_id": 29,
"text": "N_\\text{A}"
},
{
"math_id": 30,
"text": "[A]"
},
{
"math_id": 31,
"text": "[B]"
},
{
"math_id": 32,
"text": "k"
},
{
"math_id": 33,
"text": "Z_{AB} = (1000 N_A)^{4/3} * 8 \\pi^{-1} A \\beta D_r ([A] + [B])^{1/3}[A] [B] = k ([A] + [B])^{1/3}[A] [B] "
},
{
"math_id": 34,
"text": "A"
},
{
"math_id": 35,
"text": "\\beta"
},
{
"math_id": 36,
"text": "\\beta A"
}
] |
https://en.wikipedia.org/wiki?curid=1427763
|
1427967
|
Four-wave mixing
|
Four-wave mixing (FWM) is an intermodulation phenomenon in nonlinear optics, whereby interactions between two or three wavelengths produce two or one new wavelengths. It is similar to the third-order intercept point in electrical systems. Four-wave mixing can be compared to the intermodulation distortion in standard electrical systems. It is a parametric nonlinear process, in that the energy of the incoming photons is conserved. FWM is a phase-sensitive process, in that the efficiency of the process is strongly affected by phase matching conditions.
Mechanism.
When three frequencies (f1, f2, and f3) interact in a nonlinear medium, they give rise to a fourth frequency (f4) which is formed by the scattering of the incident photons, producing the fourth photon.
Given inputs "f1, f2," and "f3", the nonlinear system will produce
formula_0
From calculations with the three input signals, it is found that 12 interfering frequencies are produced, three of which lie on one of the original incoming frequencies. Note that these three frequencies which lie at the original incoming frequencies are typically attributed to self-phase modulation and cross-phase modulation, and are naturally phase-matched unlike FWM.
Sum- and difference-frequency generation.
Two common forms of four-wave mixing are dubbed sum-frequency generation and difference-frequency generation. In sum-frequency generation three fields are input and the output is a new high frequency field at the sum of the three input frequencies. In difference-frequency generation, the typical output is the sum of two minus the third.
A condition for efficient generation of FWM is phase matching: the associated k-vectors of the four components must add to zero when they are plane waves. This becomes significant since sum- and difference-frequency generation are often enhanced when resonance in the mixing media is exploited. In many configurations the sum of the first two photons will be tuned close to a resonant state. However, close to resonances the index of refraction changes rapidly and makes addition four co-linear k-vectors fail to add exactly to zero—thus long mixing path lengths are not always possible as the four component lose phase lock. Consequently, beams are often focused both for intensity but also to shorten the mixing zone.
In gaseous media an often overlooked complication is that light beams are rarely plane waves but are often focused for extra intensity, this can add an addition pi-phase shift to each k-vector in the phase matching condition. It is often very hard to satisfy this in the sum-frequency configuration but it is more easily satisfied in the difference-frequency configuration (where the pi phase shifts cancel out). As a result, difference-frequency is usually more broadly tunable and easier to set up than sum-frequency generation, making it preferable as a light source even though it's less quantum efficient than sum-frequency generation.
The special case of sum-frequency generation where all the input photons have the same frequency (and wavelength) is Third-Harmonic Generation (THG).
Degenerate four-wave mixing.
Four-wave mixing is also present if only two components interact. In this case the term
formula_1
couples three components, thus generating so-called degenerate four-wave mixing, showing identical properties to the case of three interacting waves.
Adverse effects of FWM in fiber-optic communications.
FWM is a fiber-optic characteristic that affects wavelength-division multiplexing (WDM) systems, where multiple optical wavelengths are spaced at equal intervals or channel spacing. The effects of FWM are pronounced with decreased channel spacing of wavelengths (such as in dense WDM systems) and at high signal power levels. High chromatic dispersion "decreases" FWM effects, as the signals lose coherence, or in other words, the phase mismatch between the signals increases. The interference FWM caused in WDM systems is known as interchannel crosstalk. FWM can be mitigated by using uneven channel spacing or fiber that increases dispersion. For the special case where the three frequencies are close to degenerate, then optical separation of the difference frequency can be technically challenging.
formula_2
Applications.
FWM finds applications in optical phase conjugation, parametric amplification, supercontinuum generation, Vacuum Ultraviolet light generation and in microresonator based frequency comb generation. Parametric amplifiers and oscillators based on four-wave mixing use the third order nonlinearity, as opposed to most typical parametric oscillators which use the second-order nonlinearity. Apart from these classical applications, four-wave mixing has shown promise in the quantum optical regime for generating single photons, correlated photon pairs, squeezed light and entangled photons.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": " \\pm f_{1} \\pm f_{2} \\pm f_{3}"
},
{
"math_id": 1,
"text": " f_{0} = f_{1} + f_{1} - f_{2} "
},
{
"math_id": 2,
"text": " f_{ijk} = f_{i} + f_{j} - f_{k}, \\mathrm{where}\\, i, j \\neq k "
}
] |
https://en.wikipedia.org/wiki?curid=1427967
|
1428088
|
Pseudorange
|
Pseudo distance between a satellite and a navigation satellite receiver
The pseudorange (from pseudo- and range) is the "pseudo" distance between a satellite and a navigation satellite receiver (see GNSS positioning calculation), for instance Global Positioning System (GPS) receivers.
To determine its position, a satellite navigation receiver will determine the ranges to (at least) four satellites as well as their positions at time of transmitting. Knowing the satellites' orbital parameters, these positions can be calculated for any point in time. The pseudoranges of each satellite are obtained by multiplying the speed of light by the time the signal has taken from the satellite to the receiver. As there are accuracy errors in the time measured, the term "pseudo"-ranges is used rather than ranges for such distances.
Pseudorange and time error estimation.
Typically a quartz oscillator is used in the receiver to do the timing. The accuracy of quartz clocks in general is worse (i.e. more) than one part in a million; thus, if the clock hasn't been corrected for a week, the deviation may be so great as to result in a reported location not on the Earth, but outside the Moon's orbit. Even if the clock is corrected, a second later the clock may no longer be usable for positional calculation, because after a second the error will be hundreds of meters for a typical quartz clock. But in a GPS receiver the clock's time is used to measure the ranges to different satellites at almost the same time, meaning all the measured ranges have the same error. Ranges with the same error are called pseudoranges. By finding the pseudo-range of an additional fourth satellite for precise position calculation, the time error can also be estimated. Therefore, by having the pseudoranges and the locations of four satellites, the actual receiver's position along the "x", "y", "z" axes and the time error formula_0 can be computed accurately.
The reason we speak of "pseudo"-ranges rather than ranges, is precisely this "contamination" with unknown receiver clock offset. GPS positioning is sometimes referred to as trilateration, but would be more accurately referred to as "pseudo-trilateration".
Following the laws of error propagation, neither the receiver position nor the clock offset are computed exactly, but rather "estimated" through a least squares adjustment procedure known from geodesy.
To describe this imprecision, so-called GDOP quantities have been defined: geometric dilution of precision (x,y,z,t).
Pseudorange calculations therefore use the signals of four satellites to compute the receiver's location and the clock error. A clock with an accuracy of one in a million will introduce an error of one millionth of a second each second. This error multiplied by the speed of light gives an error of 300 meters. For a typical satellite constellation this error will increase by about formula_1 (less if satellites are close together, more if satellites are all near the horizon). If positional calculation was done using this clock and only using three satellites, just standing still the GPS would indicate that you are traveling at a speed in excess of 300 meters per second, (over 1000 km/hour or 600 miles an hour). With only signals from three satellites the GPS receiver would not be able to determine whether the 300m/s was due to clock error or actual movement of the GPS receiver.
If the satellites being used are scattered throughout the sky, then the value of geometric dilution of precision (GDOP) is low while if satellites are clustered near each other from the receiver's vantage point the GDOP values are higher. The lower the value of GDOP then the better the ratio of position error to range error computing will be, so GDOP plays an important role in calculating the receiver's position on the surface of the earth using pseudoranges. The larger the number of satellites, the better the value of GDOP will be.
|
[
{
"math_id": 0,
"text": "\\Delta t"
},
{
"math_id": 1,
"text": "\\textstyle{\\sqrt 2}"
}
] |
https://en.wikipedia.org/wiki?curid=1428088
|
1428123
|
BF model
|
Topological field
The BF model or BF theory is a topological field, which when quantized, becomes a topological quantum field theory. BF stands for background field B and F, as can be seen below, are also the variables appearing in the Lagrangian of the theory, which is helpful as a mnemonic device.
We have a 4-dimensional differentiable manifold M, a gauge group G, which has as "dynamical" fields a 2-form B taking values in the adjoint representation of G, and a connection form A for G.
The action is given by
formula_0
where K is an invariant nondegenerate bilinear form over formula_1 (if G is semisimple, the Killing form will do) and F is the curvature form
formula_2
This action is diffeomorphically invariant and gauge invariant. Its Euler–Lagrange equations are
formula_3 (no curvature)
and
formula_4 (the covariant exterior derivative of B is zero).
In fact, it is always possible to gauge away any local degrees of freedom, which is why it is called a topological field theory.
However, if M is topologically nontrivial, A and B can have nontrivial solutions globally.
In fact, BF theory can be used to formulate discrete gauge theory. One can add additional twist terms allowed by group cohomology theory such as Dijkgraaf–Witten topological gauge theory. There are many kinds of modified BF theories as topological field theories, which give rise to link invariants in 3 dimensions, 4 dimensions, and other general dimensions.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "S=\\int_M K[\\mathbf{B}\\wedge \\mathbf{F}]"
},
{
"math_id": 1,
"text": "\\mathfrak{g}"
},
{
"math_id": 2,
"text": "\\mathbf{F}\\equiv d\\mathbf{A}+\\mathbf{A}\\wedge \\mathbf{A}"
},
{
"math_id": 3,
"text": "\\mathbf{F}=0"
},
{
"math_id": 4,
"text": "d_\\mathbf{A}\\mathbf{B}=0"
}
] |
https://en.wikipedia.org/wiki?curid=1428123
|
14281313
|
17-alpha-hydroxyprogesterone aldolase
|
Class of enzymes
In enzymology, a 17α-hydroxyprogesterone aldolase (EC 4.1.2.30) is an enzyme that catalyzes the chemical reaction
17α-hydroxyprogesterone formula_0 androst-4-en-3,17-dione + acetaldehyde
Hence, this enzyme has one substrate, 17α-hydroxyprogesterone, and two products, androst-4-en-3,17-dione and acetaldehyde.
This enzyme belongs to the family of lyases, specifically the aldehyde-lyases, which cleave carbon-carbon bonds. The systematic name of this enzyme class is 17α-hydroxyprogesterone acetaldehyde-lyase (4-androstene-3,17-dione-forming). Other names in common use include C-17/C-20-lyase, and 17α-hydroxyprogesterone acetaldehyde-lyase. This enzyme participates in androgen and estrogen metabolism.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "\\rightleftharpoons"
}
] |
https://en.wikipedia.org/wiki?curid=14281313
|
14281321
|
(1-hydroxycyclohexan-1-yl)acetyl-CoA lyase
|
Class of enzymes
The enzyme (1-hydroxycyclohexan-1-yl)acetyl-CoA lyase (EC 4.1.3.35) catalyzes the chemical reaction
(1-hydroxycyclohexan-1-yl)acetyl-CoA formula_0 acetyl-CoA + cyclohexanone
Hence, this enzyme has one substrate, (1-hydroxycyclohexan-1-yl)acetyl-CoA, and two products, acetyl-CoA and cyclohexanone.
This enzyme belongs to the family of lyases, specifically the oxo-acid-lyases, which cleave carbon-carbon bonds. The systematic name of this enzyme class is (1-hydroxycyclohexan-1-yl)acetyl-CoA cyclohexanone-lyase (acetyl-CoA-forming). This enzyme is also called (1-hydroxycyclohexan-1-yl)acetyl-CoA cyclohexanone-lyase.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "\\rightleftharpoons"
}
] |
https://en.wikipedia.org/wiki?curid=14281321
|
14281330
|
2,2-dialkylglycine decarboxylase (pyruvate)
|
Class of enzymes
The enzyme 2,2-dialkylglycine decarboxylase (pyruvate) (EC 4.1.1.64) catalyzes the chemical reaction
2,2-dialkylglycine + pyruvate formula_0 dialkyl ketone + CO2 + L-alanine
This enzyme belongs to the family of lyases, specifically the carboxy-lyases, which cleave carbon-carbon bonds. The systematic name of this enzyme class is 2,2-dialkylglycine carboxy-lyase (amino-transferring L-alanine-forming). Other names in common use include dialkyl amino acid (pyruvate) decarboxylase, alpha-dialkyl amino acid transaminase, 2,2-dialkyl-2-amino acid-pyruvate aminotransferase, L-alanine-alpha-ketobutyrate aminotransferase, dialkylamino-acid decarboxylase (pyruvate), and 2,2-dialkylglycine carboxy-lyase (amino-transferring). It employs one cofactor, pyridoxal phosphate.
Structural studies.
As of late 2007, 16 structures have been solved for this class of enzymes, with PDB accession codes 1D7R, 1D7S, 1D7U, 1D7V, 1DGD, 1DGE, 1DKA, 1M0N, 1M0O, 1M0P, 1M0Q, 1Z3Z, 1ZC9, 1ZOB, 1ZOD, and 2DKB.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "\\rightleftharpoons"
}
] |
https://en.wikipedia.org/wiki?curid=14281330
|
14281347
|
2,3-dimethylmalate lyase
|
Class of enzymes
The enzyme 2,3-dimethylmalate lyase (EC 4.1.3.32) catalyzes the chemical reaction
(2"R",3"S")-2,3-dimethylmalate formula_0 propanoate + pyruvate
This enzyme belongs to the family of lyases, specifically the oxo-acid-lyases, which cleave carbon-carbon bonds. The systematic name of this enzyme class is (2"R",3"S")-2,3-dimethylmalate pyruvate-lyase (propanoate-forming). Other names in common use include 2,3-dimethylmalate pyruvate-lyase, and (2"R",3"S")-2,3-dimethylmalate pyruvate-lyase. This enzyme participates in c5-branched dibasic acid metabolism.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "\\rightleftharpoons"
}
] |
https://en.wikipedia.org/wiki?curid=14281347
|
14281357
|
2-dehydro-3-deoxy-6-phosphogalactonate aldolase
|
Class of enzymes
The enzyme 2-dehydro-3-deoxy-6-phosphogalactonate aldolase (EC 4.1.2.21) catalyzes the chemical reaction
2-dehydro-3-deoxy--galactonate 6-phosphate formula_0 pyruvate + -glyceraldehyde 3-phosphate
This enzyme belongs to the family of lyases, specifically the aldehyde-lyases, which cleave carbon-carbon bonds. The systematic name of this enzyme class is 2-dehydro-3-deoxy-D-galactonate-6-phosphate D-glyceraldehyde-3-phosphate-lyase (pyruvate-forming). Other names in common use include 6-phospho-2-keto-3-deoxygalactonate aldolase, phospho-2-keto-3-deoxygalactonate aldolase, 2-keto-3-deoxy-6-phosphogalactonic aldolase, phospho-2-keto-3-deoxygalactonic aldolase, 2-keto-3-deoxy-6-phosphogalactonic acid aldolase, (KDPGal)aldolase, 2-dehydro-3-deoxy-D-galactonate-6-phosphate, and D-glyceraldehyde-3-phosphate-lyase. This enzyme participates in galactose metabolism.
Structural studies.
As of late 2007, two structures have been solved for this class of enzymes, with PDB accession codes 2V81 and 2V82.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "\\rightleftharpoons"
}
] |
https://en.wikipedia.org/wiki?curid=14281357
|
14281365
|
2-dehydro-3-deoxy-D-pentonate aldolase
|
Class of enzymes
The enzyme 2-dehydro-3-deoxy--pentonate aldolase (EC 4.1.2.28) catalyzes the chemical reaction
2-dehydro-3-deoxy-pentonate formula_0 pyruvate + glycolaldehyde
This enzyme belongs to the family of lyases, specifically the aldehyde-lyases, which cleave carbon-carbon bonds. The systematic name of this enzyme class is 2-dehydro-3-deoxy-D-pentonate glycolaldehyde-lyase (pyruvate-forming). Other names in common use include 2-keto-3-deoxy-D-pentonate aldolase, 3-deoxy-D-pentulosonic acid aldolase, and 2-dehydro-3-deoxy-D-pentonate glycolaldehyde-lyase. This enzyme participates in pentose and glucuronate interconversions.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "\\rightleftharpoons"
}
] |
https://en.wikipedia.org/wiki?curid=14281365
|
14281383
|
2-dehydro-3-deoxyglucarate aldolase
|
Class of enzymes
The enzyme 2-dehydro-3-deoxyglucarate aldolase (EC 4.1.2.20) catalyzes the chemical reaction
2-dehydro-3-deoxy--glucarate formula_0 pyruvate + tartronate semialdehyde
This enzyme belongs to the family of lyases, specifically the aldehyde-lyases, which cleave carbon-carbon bonds. The systematic name of this enzyme class is 2-dehydro-3-deoxy-D-glucarate tartronate-semialdehyde-lyase (pyruvate-forming). Other names in common use include 2-keto-3-deoxyglucarate aldolase, alpha-keto-beta-deoxy-D-glucarate aldolase, and 2-dehydro-3-deoxy-D-glucarate tartronate-semialdehyde-lyase. This enzyme participates in ascorbate and aldarate metabolism.
Structural studies.
As of late 2007, 6 structures have been solved for this class of enzymes, with PDB accession codes 1DXE, 1DXF, 1W37, 1W3I, 1W3N, and 1W3T.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "\\rightleftharpoons"
}
] |
https://en.wikipedia.org/wiki?curid=14281383
|
14281397
|
2-dehydro-3-deoxy-L-pentonate aldolase
|
Class of enzymes
The enzyme 2-dehydro-3-deoxy-L-pentonate aldolase (EC 4.1.2.18) catalyzes the chemical reaction
2-dehydro-3-deoxy--pentonate formula_0 pyruvate + glycolaldehyde
This enzyme belongs to the family of lyases, specifically the aldehyde-lyases, which cleave carbon-carbon bonds. The systematic name of this enzyme class is 2-dehydro-3-deoxy-L-pentonate glycolaldehyde-lyase (pyruvate-forming). Other names in common use include 2-keto-3-deoxy-L-pentonate aldolase, 2-keto-3-deoxy-L-arabonate aldolase, 2-keto-3-deoxy-D-xylonate aldolase, 3-deoxy-D-pentulosonic acid aldolase, and 2-dehydro-3-deoxy-L-pentonate glycolaldehyde-lyase. This enzyme participates in fructose and mannose metabolism.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "\\rightleftharpoons"
}
] |
https://en.wikipedia.org/wiki?curid=14281397
|
14281409
|
2-Dehydro-3-deoxy-phosphogluconate aldolase
|
Class of enzymes
The enzyme 2-dehydro-3-deoxy-phosphogluconate aldolase (EC 4.1.2.14), commonly known as KDPG aldolase, catalyzes the chemical reaction
2-dehydro-3-deoxy--gluconate 6-phosphate formula_0 pyruvate + -glyceraldehyde 3-phosphate
This enzyme belongs to the family of lyases, specifically the aldehyde-lyases, which cleave carbon-carbon bonds. It is used in the Entner–Doudoroff pathway in prokaryotes, feeding into glycolysis. 2-dehydro-3-deoxy-phosphogluconate aldolase is one of the two enzymes distinguishing this pathway from the more commonly known Embden–Meyerhof–Parnas pathway. This enzyme also participates in following 3 metabolic pathways: pentose phosphate pathway, pentose and glucuronate interconversions, and arginine and proline metabolism.
In addition to the cleavage of 2-dehydro-3-deoxy-D-gluconate 6-phosphate, it is also found to naturally catalyze Schiff base formation between a lysine E-amino acid group and carbonyl compounds, decarboxylation of oxaloacetate, and exchange of solvent protons with the methyl hydrogen atoms of pyruvate.
Nomenclature.
The systematic name of this enzyme class is 2-dehydro-3-deoxy-D-gluconate-6-phosphate D-glyceraldehyde-3-phosphate-lyase (pyruvate-forming). Other names in common use include:
Enzyme structure.
KDPG Aldolase was recently determined to be a trimer through crystallographic three-fold symmetry, with 225 residues. The enzyme was determined to have a molecular weight of 23,942. The trimer is stabilized primarily through hydrophobic interactions. The molecule has tertiary folding similar to triosephosphate isomerase and the A-domain of pyruvate kinase, forming an eight strand α/β-barrel structure. The α/β-barrel structure is capped on one side by the N-terminal helix. The other side, the carboxylic side, contains the active site. Each subunit contains a phosphate-ion bound in position of the aldolase binding site. It has been found that there are four cysteinyl groups present in each subunit, with two readily accessible and two buried in the subunit.
The active site contains the zwitterionic pair Glu-45/Lys-133. The Lysine, which is involved in the formation of the Schiff base is coordinated by a phosphate ion and two solvent water molecules. The first water molecule serves as a shuttle between the Glutamate and the substrate, staying bound to the enzyme throughout the catalytic cycle. The second water molecule is a product of the dehydration of the carbinolamine that leads to the formation of the Schiff base. It also functions as the nucleophile during hydrolysis of the enzyme-product Schiff base, leading to the release of pyruvate.
As of late 2007, 13 structures have been solved for this class of enzymes, with PDB accession codes 1EUA, 1EUN, 1FQ0, 1FWR, 1KGA, 1MXS, 1WA3, 1WAU, 1WBH, 2C0A, 2NUW, 2NUX, and 2NUY.
Enzyme mechanism.
One of the reactions KDPG Aldolase catalyzes, as in the Entner–Doudoroff pathway, is the reversible cleavage of 2-keto-3-deoxy-6-phosphogluconate (KDPG) into pyruvate and D-glyceraldehyde-3-phosphate. This occurs through a stereospecific retro-aldol cleavage. A proton transfer between the zwitterionic pair Glu-45/Lys-133 in the active site activates Lysine to serve as the nucleophile in the first step and Glutamate to aid in the base catalysis involved in the carbon-carbon cleavage. Lysine Residue 133 serves as the nucleophile and attacks the carbonyl group of 2-Keto-3-deoxy-6-phosphogluconate to form a protonated carbinolamine intermediate, also known as a Schiff base intermediate. The intermediate is stabilized by hydrogen bonding with residues in the active site. A three carbon residue, glyceraldehyde 3-phosphate, is cleaved off through base catalysis with a water molecule and residue Glu-45. The pyruvate is generated through the nucleophilic attack of water on the Schiff-base to reform a ketone. Aromatic interaction with Phe-135 ensures the stereospecific addition involved in the reverse process.
KDPG aldolase has also been shown to catalyze the exchange of hydrogen atoms of the methyl groups of pyruvate with protons of the solvent.
Evolutionary significance.
History.
Arguments have been made for both the convergent and divergent evolution of α/β-barrel structured enzymes such as KDPG Aldolase, triosephosphate isomerase, and the A-domain of pyruvate kinase.
Convergent evolution can lead to geometrically similar active sites while each enzyme has a distinct backbone conformation. Convergence to a common backbone structure, as is the case here however, has not been observed, although it is argues that it might be possible for a symmetrically repetitive structure as the one observed here. The similarity in the folding of the three enzymes and the exceptional symmetry commonly suggests divergent evolution from a common ancestor. The functional similarity of the enzymes remains the strongest argument for divergent evolution. All three enzymes activate a C–H bond adjacent to a carbonyl group. The active sites are located at the carboxylic ends of the β strands. Such congruence is in favor of divergent evolution.
Should the divergent evolution hypothesis prevail, this would suggest the existence of a class of enzymes with unrelated amino acid sequences yet analogous symmetrical structure and folding.
Directed Evolution
KDPG aldolase has limited utility due to its high specificity for its natural substrates in the cleavage of KDPG and the reverse addition of D-glyceraldehyde-3-phosphate and pyruvate. In vitro evolution has allowed KDPG aldolase to be converted into a more efficient aldolase with altered substrate specificity and stereoselectivity thereby improving its utility in asymmetric synthesis. Rather than modifying the recognition site, the substrate is modified by moving the active site lysine from one β strand to a neighboring one. The evolved aldolase is capable of accepting both D- and L-glyceraldehyde in their non-phosphorylated form.
References.
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[
{
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https://en.wikipedia.org/wiki?curid=14281409
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14281424
|
2-dehydropantoate aldolase
|
Class of enzymes
The enzyme 2-dehydropantoate aldolase (EC 4.1.2.12) catalyzes the chemical reaction:
2-dehydropantoate formula_0 3-methyl-2-oxobutanoate + formaldehyde
This enzyme belongs to the family of lyases, specifically the aldehyde-lyases, which cleave carbon-carbon bonds. The systematic name of this enzyme class is 2-dehydropantoate formaldehyde-lyase (3-methyl-2-oxobutanoate-forming). Other names in common use include ketopantoaldolase, and 2-dehydropantoate formaldehyde-lyase.
References.
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[
{
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"text": "\\rightleftharpoons"
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https://en.wikipedia.org/wiki?curid=14281424
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14281439
|
2-oxoglutarate decarboxylase
|
Class of enzymes
The enzyme 2-oxoglutarate decarboxylase (EC 4.1.1.71) catalyzes the chemical reaction: 2-oxoglutarate formula_0 succinate semialdehyde + CO2
This enzyme belongs to the family of lyases, specifically the carboxy-lyases, which cleave carbon-carbon bonds. The systematic name of this enzyme class is 2-oxoglutarate carboxy-lyase (succinate-semialdehyde-forming). Other names in common use include oxoglutarate decarboxylase, alpha-ketoglutarate decarboxylase, alpha-ketoglutaric decarboxylase, oxoglutarate decarboxylase, pre-2-oxoglutarate decarboxylase, and 2-oxoglutarate carboxy-lyase. It employs one cofactor, thiamin diphosphate.
References.
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[
{
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https://en.wikipedia.org/wiki?curid=14281439
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14281454
|
3,4-dihydroxyphthalate decarboxylase
|
Class of enzymes
THe enzyme 3,4-dihydroxyphthalate decarboxylase (EC 4.1.1.69) catalyzes the chemical reaction
3,4-dihydroxyphthalate formula_0 3,4-dihydroxybenzoate + CO2
This enzyme belongs to the family of lyases, specifically the carboxy-lyases, which cleave carbon-carbon bonds. The systematic name of this enzyme class is 3,4-dihydroxyphthalate carboxy-lyase (3,4-dihydroxybenzoate-forming). This enzyme is also called 3,4-dihydroxyphthalate carboxy-lyase.
References.
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[
{
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"text": "\\rightleftharpoons"
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https://en.wikipedia.org/wiki?curid=14281454
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14281466
|
3-dehydro-L-gulonate-6-phosphate decarboxylase
|
Class of enzymes
The enzyme 3-dehydro-L-gulonate-6-phosphate decarboxylase (EC 4.1.1.85) catalyzes the chemical reaction
3-dehydro--gulonate 6-phosphate + H+ formula_0 -xylulose 5-phosphate + CO2
This enzyme belongs to the family of lyases, specifically the carboxy-lyases, which cleave carbon-carbon bonds. The systematic name of this enzyme class is 3-dehydro-L-gulonate-6-phosphate carboxy-lyase (L-xylulose-5-phosphate-forming). Other names in common use include 3-keto-L-gulonate 6-phosphate decarboxylase, UlaD, SgaH, SgbH, KGPDC, and 3-dehydro-L-gulonate-6-phosphate carboxy-lyase. This enzyme participates in pentose and glucuronate interconversions and ascorbate and aldarate metabolism.
References.
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[
{
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"text": "\\rightleftharpoons"
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https://en.wikipedia.org/wiki?curid=14281466
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14281476
|
3-deoxy-D-manno-octulosonate aldolase
|
Class of enzymes
The enzyme 3-deoxy--"manno"-octulosonate aldolase (EC 4.1.2.23) catalyzes the chemical reaction
3-deoxy--"manno"-octulosonate formula_0 pyruvate + -arabinose
This enzyme belongs to the family of lyases, specifically the aldehyde-lyases, which cleave carbon-carbon bonds. The systematic name of this enzyme class is 3-deoxy-D-manno-octulosonate D-arabinose-lyase (pyruvate-forming). Other names in common use include 2-keto-3-deoxyoctonate aldolase, KDOaldolase, 3-deoxyoctulosonic aldolase, 2-keto-3-deoxyoctonic aldolase, 3-deoxy-D-manno-octulosonic aldolase, and 3-deoxy-D-manno-octulosonate D-arabinose-lyase.
References.
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[
{
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https://en.wikipedia.org/wiki?curid=14281476
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14281482
|
3-hydroxy-2-methylpyridine-4,5-dicarboxylate 4-decarboxylase
|
Class of enzymes
The enzyme 3-hydroxy-2-methylpyridine-4,5-dicarboxylate 4-decarboxylase (EC 4.1.1.51) catalyzes the chemical reaction
3-hydroxy-2-methylpyridine-4,5-dicarboxylate formula_0 3-hydroxy-2-methylpyridine-5-carboxylate + CO2
This enzyme belongs to the family of lyases, specifically the carboxy-lyases, which cleave carbon-carbon bonds. The systematic name of this enzyme class is 3-hydroxy-2-methylpyridine-4,5-dicarboxylate 4-carboxy-lyase (3-hydroxy-2-methylpyridine-5-carboxylate-forming). This enzyme is also called 3-hydroxy-2-methylpyridine-4,5-dicarboxylate 4-carboxy-lyase. This enzyme participates in vitamin B6 metabolism.
References.
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[
{
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https://en.wikipedia.org/wiki?curid=14281482
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14281491
|
3-hydroxy-3-isohexenylglutaryl-CoA lyase
|
Class of enzymes
The enzyme 3-hydroxy-3-isohexenylglutaryl-CoA lyase (EC 4.1.3.26) catalyzes the chemical reaction
3-hydroxy-3-(4-methylpent-3-en-1-yl)glutaryl-CoA formula_0 7-methyl-3-oxooct-6-enoyl-CoA + acetate
This enzyme belongs to the family of lyases, specifically the oxo-acid-lyases, which cleave carbon-carbon bonds. The systematic name of this enzyme class is 3-hydroxy-3-(4-methylpent-3-en-1-yl)glutaryl-CoA acetate-lyase (7-methyl-3-oxooct-6-enoyl-CoA-forming). Other names in common use include beta-hydroxy-beta-isohexenylglutaryl CoA-lyase, hydroxyisohexenylglutaryl-CoA:acetatelyase, 3-hydroxy-3-isohexenylglutaryl coenzyme A lyase, 3-hydroxy-3-isohexenylglutaryl-CoA isopentenylacetoacetyl-CoA-lyase, and 3-hydroxy-3-(4-methylpent-3-en-1-yl)glutaryl-CoA acetate-lyase.
References.
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[
{
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https://en.wikipedia.org/wiki?curid=14281491
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14281506
|
3-hydroxyaspartate aldolase
|
Class of enzymes
The enzyme -"erythro"-3-hydroxyaspartate aldolase (EC 4.1.3.14) catalyzes the chemical reaction
-"erythro"-3-hydroxy-aspartate formula_0 glycine + glyoxylate
This enzyme belongs to the family of lyases, specifically the oxo-acid-lyases, which cleave carbon-carbon bonds. The systematic name of this enzyme class is -"erythro"-3-hydroxy-aspartate glyoxylate-lyase (glycine-forming). Other names in common use include erythro-beta-hydroxyaspartate aldolase, erythro-beta-hydroxyaspartate glycine-lyase, and erythro-3-hydroxy-Ls-aspartate glyoxylate-lyase.
References.
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[
{
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https://en.wikipedia.org/wiki?curid=14281506
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14281518
|
3-oxolaurate decarboxylase
|
Class of enzymes
The enzyme 3-oxolaurate decarboxylase (EC 4.1.1.56) catalyzes the chemical reaction
3-oxododecanoate formula_0 2-undecanone + CO2
This enzyme belongs to the family of lyases, specifically the carboxy-lyases, which cleave carbon-carbon bonds. The systematic name of this enzyme class is 3-oxododecanoate carboxy-lyase (2-undecanone-forming). Other names in common use include beta-ketolaurate decarboxylase, beta-ketoacyl decarboxylase, and 3-oxododecanoate carboxy-lyase.
References.
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{
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https://en.wikipedia.org/wiki?curid=14281518
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14281537
|
4-(2-carboxyphenyl)-2-oxobut-3-enoate aldolase
|
Class of enzymes
The enzyme 4-(2-carboxyphenyl)-2-oxobut-3-enoate aldolase (EC 4.1.2.34) catalyzes the chemical reaction
(3"Z")-4-(2-carboxyphenyl)-2-oxobut-3-enoate + H2O formula_0 2-formylbenzoate + pyruvate
This enzyme participates in naphthalene and anthracene degradation.
Nomenclature.
This enzyme belongs to the family of lyases, specifically the aldehyde-lyases, which cleave carbon-carbon bonds. The systematic name of this enzyme class is (3Z)-4-(2-carboxyphenyl)-2-oxobut-3-enoate 2-formylbenzoate-lyase (pyruvate-forming). Other names in common use include 2'-carboxybenzalpyruvate aldolase, (3E)-4-(2-carboxyphenyl)-2-oxobut-3-enoate, 2-carboxybenzaldehyde-lyase, and (3Z)-4-(2-carboxyphenyl)-2-oxobut-3-enoate 2-formylbenzoate-lyase.
References.
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[
{
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https://en.wikipedia.org/wiki?curid=14281537
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14281551
|
4,5-dihydroxyphthalate decarboxylase
|
Class of enzymes
The enzyme 4,5-dihydroxyphthalate decarboxylase (EC 4.1.1.55) catalyzes the chemical reaction
4,5-dihydroxyphthalate formula_0 3,4-dihydroxybenzoate + CO2
This enzyme belongs to the family of lyases, specifically the carboxy-lyases, which cleave carbon-carbon bonds. The systematic name of this enzyme class is 4,5-dihydroxyphthalate carboxy-lyase (3,4-dihydroxybenzoate-forming). This enzyme is also called 4,5-dihydroxyphthalate carboxy-lyase. This enzyme participates in 2,4-dichlorobenzoate degradation.
References.
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[
{
"math_id": 0,
"text": "\\rightleftharpoons"
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https://en.wikipedia.org/wiki?curid=14281551
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14281564
|
4-carboxymuconolactone decarboxylase
|
Class of enzymes
The enzyme 4-carboxymuconolactone decarboxylase (EC 4.1.1.44) catalyzes the chemical reaction
2-carboxy-2,5-dihydro-5-oxofuran-2-acetate formula_0 4,5-dihydro-5-oxofuran-2-acetate + CO2
This enzyme belongs to the family of lyases, specifically the carboxy-lyases, which cleave carbon-carbon bonds. The systematic name of this enzyme class is 2-carboxy-2,5-dihydro-5-oxofuran-2-acetate carboxy-lyase (4,5-dihydro-5-oxofuran-2-acetate-forming). Other names in common use include gamma-4-carboxymuconolactone decarboxylase, and 4-carboxymuconolactone carboxy-lyase. 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 2AF7.
References.
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{
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https://en.wikipedia.org/wiki?curid=14281564
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14281576
|
4-hydroxy-2-oxoglutarate aldolase
|
Class of enzymes
The enzyme 4-hydroxy-2-oxoglutarate aldolase (EC 4.1.3.16) catalyzes the chemical reaction
4-hydroxy-2-oxoglutarate formula_0 pyruvate + glyoxylate
This enzyme belongs to the family of lyases, specifically the oxo-acid-lyases, which cleave carbon-carbon bonds. The systematic name of this enzyme class is 4-hydroxy-2-oxoglutarate glyoxylate-lyase (pyruvate-forming). Other names in common use include 2-oxo-4-hydroxyglutarate aldolase, hydroxyketoglutaric aldolase, 4-hydroxy-2-ketoglutaric aldolase, 2-keto-4-hydroxyglutaric aldolase, 4-hydroxy-2-ketoglutarate aldolase, 2-keto-4-hydroxyglutarate aldolase, 2-oxo-4-hydroxyglutaric aldolase, DL-4-hydroxy-2-ketoglutarate aldolase, hydroxyketoglutarate aldolase, 2-keto-4-hydroxybutyrate aldolase, and 4-hydroxy-2-oxoglutarate glyoxylate-lyase. This enzyme participates in arginine and proline metabolism and glyoxylate and dicarboxylate metabolism.
Structural studies.
As of late 2007, two structures have been solved for this class of enzymes, with PDB accession codes 1WAU and 2C0A.
References.
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[
{
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https://en.wikipedia.org/wiki?curid=14281576
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14281586
|
4-hydroxy-2-oxovalerate aldolase
|
InterPro Family
The enzyme 4-hydroxy-2-oxovalerate aldolase (EC 4.1.3.39) catalyzes the chemical reaction
4-hydroxy-2-oxopentanoate formula_0 acetaldehyde + pyruvate
Baker "et al." showed that BphI, a member of this family from Burkholderia xenovorans LB400 was able to utilize 4-hydroxy- 2-oxohexanoate (HOHA) with equal catalytic efficiency as 4-hydroxy-2-oxopentanoate, producing propionaldehyde + pyruvate. Furthermore, the enzyme was also able to catalyze the cleavage of 4-hydroxy-2-oxoheptanoate (HOHEP), forming butyraldehyde + pyruvate. Baker "et al." we also able to show that acetaldehyde and propionaldehyde are not released into the bulk solvent, but are channeled to an associated acetaldehyde dehydrogenase known as BphJ. This is the first demonstration of substrate channeling in this family of enzymes.
This enzyme belongs to the family of lyases, specifically the oxo-acid-lyases, which cleave carbon-carbon bonds. The systematic name of this enzyme class is 4-hydroxy-2-oxopentanoate pyruvate-lyase (acetaldehyde-forming). Other names in common use include 4-hydroxy-2-ketovalerate aldolase, HOA, DmpG, BphI, 4-hydroxy-2-oxovalerate pyruvate-lyase, and 4-hydroxy-2-oxopentanoate pyruvate-lyase. This enzyme participates in 8 metabolic pathways: phenylalanine metabolism, benzoate degradation via hydroxylation, biphenyl degradation, toluene and xylene degradation, 1,4-dichlorobenzene degradation, fluorene degradation, carbazole degradation, and styrene degradation.
References.
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Further reading.
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https://en.wikipedia.org/wiki?curid=14281586
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14281598
|
4-hydroxy-4-methyl-2-oxoglutarate aldolase
|
Class of enzymes
The enzyme 4-hydroxy-4-methyl-2-oxoglutarate aldolase (EC 4.1.3.17) catalyzes the chemical reaction
4-hydroxy-4-methyl-2-oxoglutarate formula_0 2 pyruvate
This enzyme belongs to the family of lyases, specifically the oxo-acid-lyases, which cleave carbon-carbon bonds. The systematic name of this enzyme class is 4-hydroxy-4-methyl-2-oxoglutarate pyruvate-lyase (pyruvate-forming). Other names in common use include pyruvate aldolase, gamma-methyl-gamma-hydroxy-alpha-ketoglutaric aldolase, 4-hydroxy-4-methyl-2-ketoglutarate aldolase, and 4-hydroxy-4-methyl-2-oxoglutarate pyruvate-lyase. This enzyme participates in benzoate degradation via hydroxylation and c5-branched dibasic acid metabolism.
References.
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{
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https://en.wikipedia.org/wiki?curid=14281598
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14281612
|
4-hydroxybenzoate decarboxylase
|
Class of enzymes
The enzyme 4-hydroxybenzoate decarboxylase (EC 4.1.1.61) catalyzes the chemical reaction
4-hydroxybenzoate formula_0 phenol + CO2
This enzyme belongs to the family of lyases, specifically the carboxy-lyases, which cleave carbon-carbon bonds. The systematic name of this enzyme class is 4-hydroxybenzoate carboxy-lyase (phenol-forming). Other names in common use include p-hydroxybenzoate decarboxylase, and 4-hydroxybenzoate carboxy-lyase. This enzyme participates in benzoate degradation via CoA ligation.
References.
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[
{
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https://en.wikipedia.org/wiki?curid=14281612
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14281622
|
4-hydroxyphenylacetate decarboxylase
|
Class of enzymes
The enzyme 4-hydroxyphenylacetate decarboxylase (EC 4.1.1.83) catalyzes the chemical reaction
(4-hydroxyphenyl)acetate + H+ formula_0 4-methylphenol + CO2
This enzyme belongs to the family of lyases, specifically the carboxy-lyases, which cleave carbon-carbon bonds. The systematic name of this enzyme class is 4-(hydroxyphenyl)acetate carboxy-lyase (4-methylphenol-forming). Other names in common use include p-hydroxyphenylacetate decarboxylase, p-Hpd, 4-Hpd, and 4-hydroxyphenylacetate carboxy-lyase.
References.
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[
{
"math_id": 0,
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https://en.wikipedia.org/wiki?curid=14281622
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14281649
|
4-hydroxyphenylpyruvate decarboxylase
|
Class of enzymes
The enzyme 4-hydroxyphenylpyruvate decarboxylase (EC 4.1.1.80) catalyzes the chemical reaction
4-hydroxyphenylpyruvate formula_0 4-hydroxyphenylacetaldehyde + CO2
This enzyme belongs to the family of lyases, specifically the carboxy-lyases, which cleave carbon-carbon bonds. The systematic name of this enzyme class is 4-hydroxyphenylpyruvate carboxy-lyase (4-hydroxyphenylacetaldehyde-forming). This enzyme is also called 4-hydroxyphenylpyruvate carboxy-lyase.
References.
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[
{
"math_id": 0,
"text": "\\rightleftharpoons"
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https://en.wikipedia.org/wiki?curid=14281649
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14281661
|
4-oxalocrotonate decarboxylase
|
Class of enzymes
The enzyme 4-oxalocrotonate decarboxylase (EC 4.1.1.77) catalyzes the chemical reaction
4-oxalocrotonate formula_0 2-oxopent-4-enoate + CO2
This enzyme belongs to the family of lyases, specifically the carboxy-lyases, which cleave carbon-carbon bonds. The systematic name of this enzyme class is 4-oxalocrotonate carboxy-lyase (2-oxopent-4-enoate-forming). This enzyme is also called 4-oxalocrotonate carboxy-lyase. This enzyme participates in 3 metabolic pathways: benzoate degradation via hydroxylation, toluene and xylene degradation, and fluorene degradation.
References.
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[
{
"math_id": 0,
"text": "\\rightleftharpoons"
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https://en.wikipedia.org/wiki?curid=14281661
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14281678
|
5-dehydro-2-deoxyphosphogluconate aldolase
|
Class of enzymes
The enzyme 5-dehydro-2-deoxyphosphogluconate aldolase (EC 4.1.2.29) catalyzes the chemical reaction
5-dehydro-2-deoxy--gluconate 6-phosphate formula_0 glycerone phosphate + malonate semialdehyde
This enzyme belongs to the family of lyases, specifically the aldehyde-lyases, which cleave carbon-carbon bonds. The systematic name of this enzyme class is 5-dehydro-2-deoxy-D-gluconate-6-phosphate malonate-semialdehyde-lyase (glycerone-phosphate-forming). Other names in common use include phospho-5-keto-2-deoxygluconate aldolase, 5-dehydro-2-deoxy-D-gluconate-6-phosphate, and malonate-semialdehyde-lyase.
References.
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[
{
"math_id": 0,
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https://en.wikipedia.org/wiki?curid=14281678
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14281687
|
5-guanidino-2-oxopentanoate decarboxylase
|
Class of enzymes
The enzyme 5-guanidino-2-oxopentanoate decarboxylase (EC 4.1.1.75) catalyzes the chemical reaction
5-guanidino-2-oxo-pentanoate formula_0 4-guanidinobutanal + CO2
This enzyme belongs to the family of lyases, specifically the carboxy-lyases, which cleave carbon-carbon bonds. The systematic name of this enzyme class is 5-guanidino-2-oxo-pentanoate carboxy-lyase (4-guanidinobutanal-forming). Other names in common use include alpha-ketoarginine decarboxylase, and 2-oxo-5-guanidinopentanoate carboxy-lyase. It has 2 cofactors: thiamin diphosphate, and Divalent cation.
References.
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[
{
"math_id": 0,
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https://en.wikipedia.org/wiki?curid=14281687
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14281695
|
5-oxopent-3-ene-1,2,5-tricarboxylate decarboxylase
|
Enzyme
The enzyme 5-oxopent-3-ene-1,2,5-tricarboxylate decarboxylase (EC 4.1.1.68) catalyzes the chemical reaction
5-oxopent-3-ene-1,2,5-tricarboxylate formula_0 2-oxohept-3-enedioate + CO2
This enzyme belongs to the family of lyases, specifically the carboxy-lyases, which cleave carbon-carbon bonds. The systematic name of this enzyme class is 5-oxopent-3-ene-1,2,5-tricarboxylate carboxy-lyase (2-oxohept-3-enedioate-forming). Other names in common use include 5-carboxymethyl-2-oxo-hex-3-ene-1,6-dioate decarboxylase, and 5-oxopent-3-ene-1,2,5-tricarboxylate carboxy-lyase. This enzyme participates in tyrosine metabolism.
Structural studies.
As of late 2007, only one structure has been solved for this class of enzymes, with the PDB accession code 1I7O.
References.
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[
{
"math_id": 0,
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https://en.wikipedia.org/wiki?curid=14281695
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14281705
|
6-methylsalicylate decarboxylase
|
Class of enzymes
THe enzyme 6-methylsalicylate decarboxylase (EC 4.1.1.52) catalyzes the chemical reaction
6-methylsalicylate formula_0 3-cresol + CO2
This enzyme belongs to the family of lyases, specifically the carboxy-lyases, which cleave carbon-carbon bonds. The systematic name of this enzyme class is 6-methylsalicylate carboxy-lyase (3-cresol-forming). Other names in common use include 6-methylsalicylic acid (2,6-cresotic acid) decarboxylase, 6-MSA decarboxylase, and 6-methylsalicylate carboxy-lyase.
References.
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[
{
"math_id": 0,
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https://en.wikipedia.org/wiki?curid=14281705
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14281724
|
Acetolactate decarboxylase
|
InterPro Family
The enzyme acetolactate decarboxylase (EC 4.1.1.5) catalyzes the chemical reaction
(S)-2-hydroxy-2-methyl-3-oxobutanoate formula_0 (R)-2-acetoin + CO2
Hence, this enzyme has one substrate, (S)-2-hydroxy-2-methyl-3-oxobutanoate, and two products, (R)-2-acetoin and CO2.
This enzyme belongs to the family of lyases, specifically the carboxy-lyases, which cleave carbon-carbon bonds. The systematic name of this enzyme class is (S)-2-hydroxy-2-methyl-3-oxobutanoate carboxy-lyase [(R)-2-acetoin-forming]. Other names in common use include alpha-acetolactate decarboxylase, and (S)-2-hydroxy-2-methyl-3-oxobutanoate carboxy-lyase. This enzyme participates in butanoate metabolism and c5-branched dibasic acid metabolism.
Structural studies.
As of late 2007, only one structure has been solved for this class of enzymes, with the PDB accession code 1XV2.
References.
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[
{
"math_id": 0,
"text": "\\rightleftharpoons"
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https://en.wikipedia.org/wiki?curid=14281724
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14281731
|
Acetylenedicarboxylate decarboxylase
|
Class of enzymes
The enzyme acetylenedicarboxylate decarboxylase (EC 4.1.1.78) catalyzes the chemical reaction
acetylenedicarboxylate + H2O formula_0 pyruvate + CO2
This enzyme belongs to the family of lyases, specifically the carboxy-lyases, which cleave carbon-carbon bonds. The systematic name of this enzyme class is acetylenedicarboxylate carboxy-lyase (pyruvate-forming). Other names in common use include acetylenedicarboxylate hydratase, acetylenedicarboxylate hydrase, and acetylenedicarboxylate carboxy-lyase. This enzyme participates in pyruvate metabolism.
References.
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[
{
"math_id": 0,
"text": "\\rightleftharpoons"
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https://en.wikipedia.org/wiki?curid=14281731
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14281745
|
Aconitate decarboxylase
|
Class of enzymes
The enzyme aconitate decarboxylase (EC 4.1.1.6) (i.e., ACOD1, also termed cis-aconitate decarboxylase, immune-responsive gene 1, immune response gene 1, and IRK1) is a protein enzyme that in humans is encoded by the "decarboxylase 1 aconitate decarboxylase 1" gene located at position 22.3 on the long arm (i.e., p-arm) of chromosome 13. ACOD1 catalyzes the following reversible (i.e., runs in both directions, as indicated by formula_0) decarboxylation chemical reaction:
cis-aconitate formula_0 itaconate + CO2
Hence, ACOD1 converts cis-aconitate into two products, itaconate and CO2 or itaconate and CO2 into one product, aconitate.
ACOD1 belongs to the family of lyases, specifically the carboxy-lyases, which cleave carbon-carbon bonds. The systematic name of this enzyme class is cis-aconitate carboxy-lyase (itaconate-forming). Other names once in common use for this enzyme class include CAD and cis-aconitate carboxy-lyase. ACOD1 participates in c5-branched dibasic acid metabolism.
"Ustilago maydis" (a species of Ustilago fungi) converts "cis"-aconitate to its thermodynamically favored product, trans-aconitate, by the enzyme aconitate delta-isomerase (i.e., Adi1). The trans-aconitate product is decarboxylated to itaconate by trans-aconitate decarboxylase (i.e., Tad1). This Adi followed by Tad 1 enzymatic metabolic pathway is:
cis-aconitate formula_0 trans-itaconate → itaconate + CO2
Trans-aconitate decarboxylase does not metabolize cis-aconitate to itaconate. (The genes for aconitate delta-isomerase and trans-aconitate decarboxylase have been reported in several types of fungi hut not in other organisms, including humand, and are classified as provisional, i.e., accepted provisional to further studies.)
References.
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https://en.wikipedia.org/wiki?curid=14281745
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14281753
|
Aminobenzoate decarboxylase
|
The enzyme aminobenzoate decarboxylase (EC 4.1.1.24) catalyzes the chemical reaction
4(or 2)-aminobenzoate formula_0 aniline + CO2
Thus, the two substrates of this enzyme are 4-aminobenzoate and 2-aminobenzoate, whereas its two products are aniline and CO2.
This enzyme belongs to the family of lyases, specifically the carboxy-lyases, which cleave carbon-carbon bonds. The systematic name of this enzyme class is aminobenzoate carboxy-lyase (aniline-forming). It employs one cofactor, pyridoxal phosphate.
References.
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[
{
"math_id": 0,
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https://en.wikipedia.org/wiki?curid=14281753
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14281762
|
Aminocarboxymuconate-semialdehyde decarboxylase
|
The enzyme aminocarboxymuconate-semialdehyde decarboxylase (EC 4.1.1.45) catalyzes the chemical reaction
2-amino-3-(3-oxoprop-1-en-1-yl)but-2-enedioate formula_0 2-aminomuconate semialdehyde + CO2
This enzyme belongs to the family of lyases, specifically the carboxy-lyases, which cleave carbon-carbon bonds. This enzyme participates in tryptophan metabolism. It has been identified as a marker in nonverbal autism.
Nomenclature.
The systematic name of this enzyme class is 2-amino-3-(3-oxoprop-1-en-1-yl)but-2-enedioate carboxy-lyase (2-aminomuconate-semialdehyde-forming). Other names in common use include picolinic acid carboxylase, picolinic acid decarboxylase, alpha-amino-beta-carboxymuconate-epsilon-semialdehade decarboxylase, alpha-amino-beta-carboxymuconate-epsilon-semialdehyde, beta-decarboxylase, 2-amino-3-(3-oxoprop-2-enyl)but-2-enedioate carboxy-lyase, and 2-amino-3-(3-oxoprop-1-en-1-yl)but-2-enedioate carboxy-lyase.
References.
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Further reading.
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https://en.wikipedia.org/wiki?curid=14281762
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14281774
|
Aminodeoxychorismate lyase
|
4-amino-4-deoxychorismate lyase (EC 4.1.3.38) is an enzyme that participates in folate biosynthesis by catalyzing the production of PABA by the following reaction
4-amino-4-deoxychorismate formula_0 4-aminobenzoate + pyruvate
This enzyme belongs to the family of lyases, specifically the oxo-acid-lyases, which cleave carbon-carbon bonds. This enzyme, encoded by the "pabC" gene in bacteria and plants, is also known as PabC or ADC lyase. The fungal enzyme has been designated ABZ2.
All known examples of 4-amino-4-deoxychorismate lyase bind PLP (pyridoxal-5'-phosphate), a cofactor employed during catalysis.
References.
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{
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https://en.wikipedia.org/wiki?curid=14281774
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14281786
|
Anthranilate synthase
|
The enzyme anthranilate synthase (EC 4.1.3.27) catalyzes the chemical reaction
chorismate + -glutamine formula_0 anthranilate + pyruvate + -glutamate
Function.
Anthranilate synthase creates anthranilate, an important intermediate in the biosynthesis of indole, and by extension, the amino acid tryptophan. Tryptophan can then be metabolized further into serotonin, melatonin, or various auxins.
Reaction.
Anthranilate synthase catalyzes the change from chorismate to anthranilate. As its other substrate, it can use either glutamine or ammonia. During the reaction, both a hydroxyl group and an enolpyruvyl group are removed from the aromatic ring. The enolpyruvyl group gains a proton to form pyruvate. It has been shown that the proton comes from the surrounding water and not from an intramolecular shift of a hydrogen atom on the substrates. The amino group of glutamine (or ammonia itself) attacks chorismate in position 2, which leads to the elimination of enolpyruvyl group from position 3. In the process, an aromatic ring is re-formed.
Structure and assembly.
The complex is made up of α and β subunits. Gel filtration experiments reveal that the complex occurs as an α2β2 tetramer under native conditions, and as an αβ dimer under high salt concentrations. The αβ dimers interact through the α subunits to form the complex.
The subunits of anthranilate synthase are encoded by the trpE and trpD genes in "E. coli", both of which appear in the trp operon.
In Enterobacteriaceae, this enzyme exists in the form of an aggregate with anthranilate phosphoribosyltransferase. If these two enzymes are not clustered, the complex is unable to use glutamine as a substrate and can only use ammonia.
Nomenclature.
This enzyme belongs to the family of lyases, to be specific the oxo-acid-lyases, which cleave carbon-carbon bonds. The systematic name of this enzyme class is chorismate pyruvate-lyase (amino-accepting; anthranilate-forming). Other names in common use include anthranilate synthetase, chorismate lyase, and chorismate pyruvate-lyase (amino-accepting). This enzyme participates in phenylalanine, tyrosine and tryptophan biosynthesis and two-component system - general.
Structural studies.
As of late 2007, five structures have been solved for this class of enzymes, with PDB accession codes 1I1Q, 1I7Q, 1I7S, 1QDL, and 2I6Y.
References.
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{
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https://en.wikipedia.org/wiki?curid=14281786
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14281812
|
Arylmalonate decarboxylase
|
Class of enzymes
The enzyme arylmalonate decarboxylase (EC 4.1.1.76) catalyzes the chemical reaction
2-aryl-2-methylmalonate formula_0 2-arylpropanoate + CO2
This enzyme belongs to the family of lyases, specifically the carboxy-lyases, which cleave carbon-carbon bonds. The systematic name of this enzyme class is 2-aryl-2-methylmalonate carboxy-lyase (2-arylpropanoate-forming). Other names in common use include AMDASE, 2-aryl-2-methylmalonate carboxy-lyase, and 2-aryl-2-methylmalonate carboxy-lyase (2-arylpropionate-forming).
Structural studies.
As of late 2007, only one structure has been solved for this class of enzymes, with the PDB accession code 2DGD.
References.
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{
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https://en.wikipedia.org/wiki?curid=14281812
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14281825
|
Aspartate 1-decarboxylase
|
The enzyme aspartate 1-decarboxylase (EC 4.1.1.11) catalyzes the chemical reaction
L-aspartate formula_0 beta-alanine + CO2
Hence, this enzyme has one substrate, L-aspartate, and two products, beta-alanine and CO2.
This enzyme belongs to the family of lyases, specifically the carboxy-lyases, which cleave carbon-carbon bonds. The systematic name of this enzyme class is L-aspartate 1-carboxy-lyase (beta-alanine-forming). Other names in common use include aspartate alpha-decarboxylase, L-aspartate alpha-decarboxylase, aspartic alpha-decarboxylase, and L-aspartate 1-carboxy-lyase. This enzyme participates in alanine and aspartate metabolism and beta-alanine metabolism.
Structural studies.
As of late 2007[ [update]], 12 structures have been solved for this class of enzymes, with PDB accession codes 1AW8, 1PPY, 1PQE, 1PQF, 1PQH, 1PT0, 1PT1, 1PYQ, 1PYU, 1UHD, 1UHE, and 2C45.
References.
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[
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https://en.wikipedia.org/wiki?curid=14281825
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14281987
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Aspartate 4-decarboxylase
|
In enzymology, an aspartate 4-decarboxylase (EC 4.1.1.12) is an enzyme that catalyzes the chemical reaction
L-aspartate formula_0 L-alanine + CO2
Hence, this enzyme has one substrate, L-aspartate, and two products, L-alanine and CO2. This reaction is the basis of the industrial synthesis of L-alanine.
This enzyme belongs to the family of lyases, specifically the carboxy-lyases, which cleave carbon-carbon bonds. The systematic name of this enzyme class is L-aspartate 4-carboxy-lyase (L-alanine-forming). Other names in common use include desulfinase, aminomalonic decarboxylase, aspartate beta-decarboxylase, aspartate omega-decarboxylase, aspartic omega-decarboxylase, aspartic beta-decarboxylase, L-aspartate beta-decarboxylase, cysteine sulfinic desulfinase, L-cysteine sulfinate acid desulfinase, and L-aspartate 4-carboxy-lyase. This enzyme participates in alanine and aspartate metabolism and cysteine metabolism. It employs one cofactor, pyridoxal phosphate.
References.
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{
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https://en.wikipedia.org/wiki?curid=14281987
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14282002
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Benzoin aldolase
|
Enzyme
The enzyme benzoin aldolase (EC 4.1.2.38) catalyzes the chemical reaction:
2-hydroxy-1,2-diphenylethanone formula_0 2 benzaldehyde
This enzyme belongs to the family of lyases, specifically the aldehyde-lyases, which cleave carbon-carbon bonds. The systematic name of benzoin is indeed is 2-hydroxy-1,2-diphenylethanone benzaldehyde-lyase (benzaldehyde-forming)—the systematic name of benzoin is 2-hydroxy-1,2-diphenylethanone. Other names in common use include benzaldehyde lyase, and 2-hydroxy-1,2-diphenylethanone benzaldehyde-lyase. This enzyme employs one cofactor: thiamin diphosphate.
Structural studies.
As of late 2007, 3 structures have been solved for this class of enzymes, with PDB accession codes 2AG0, 2AG1, and 2UZ1.
References.
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https://en.wikipedia.org/wiki?curid=14282002
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14282032
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Benzoylformate decarboxylase
|
The enzyme benzoylformate decarboxylase (EC 4.1.1.7) catalyzes the following chemical reaction:
benzoylformate + H+ formula_0 benzaldehyde + CO2
Hence, this enzyme has one substrate, benzoylformate, and two products, benzaldehyde and CO2.
This enzyme belongs to the family of lyases, specifically the carboxy-lyases, which cleave carbon-carbon bonds. The systematic name of this enzyme class is benzoylformate carboxy-lyase (benzaldehyde-forming). Other names in common use include phenylglyoxylate decarboxylase, and benzoylformate carboxy-lyase. This enzyme participates in benzoate degradation via hydroxylation and toluene and xylene degradation. It employs one cofactor, thiamin diphosphate.
Structural studies.
As of late 2007, 8 structures have been solved for this class of enzymes, with PDB accession codes 1BFD, 1MCZ, 1PI3, 1PO7, 1Q6Z, 1YNO, 2FN3, and 2FWN.
References.
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https://en.wikipedia.org/wiki?curid=14282032
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14282049
|
Benzylsuccinate synthase
|
The enzyme benzylsuccinate synthase (EC 4.1.99.11) catalyzes the chemical reaction
toluene + fumarate formula_0 benzylsuccinate
This enzyme catalyses a radical-type addition of toluene and fumarate as substrates to generate ("R")-benzylsuccinate as product, the first step of anaerobic toluene degradation.
Benzylsuccinate synthase is a glycyl-radical enzyme found in many microorganisms such as "Thauera aromatica" and "Aromatoleum tuluolicum" responsible for degrading aromatic hydrocarbons, primarily toluene. It operates by catalyzing carbon-carbon bond formation between toluene and fumarate into benzylsuccinate. Benzylsuccinate is then converted to benzoyl CoA in a scheme resembling β-oxidation, and then reductively de-aromatized into metabolites. BSS comprises an α, β, and γ subunit, with the α-subunit containing the activated glycyl-radical, and the β- and γ-subunits being necessary for formation and stability of the cataytically active enzyme. In terms of its catalytic residues, mutagenesis experiments suggest that in addition to Cys493 and Gly829 being critical for catalysis in a "T. aromatica" T1 strain, mutation of a conserved Arg508 is also critical for benzylsuccinate synthase activity.
Nomenclature.
This enzyme belongs to the family of glycyl-radical enzymes and is listed under the enzyme commission category of lyases, specifically in the "catch-all" class of carbon-carbon lyases. The systematic name of this enzyme class is benzylsuccinate fumarate-lyase (toluene-forming). This enzyme is also called benzylsuccinate fumarate-lyase, although it does not catalyse t, andhe cleavage of benzylsuccinate. This enzyme participates in feeding toluene into the pathway of benzoate degradation via CoA ligation.
Mechanism.
As a result of its intrinsic function, its sequence is being used as a gene marker when studying anoxic toluene contamination sites for active degradation
The mechanism proposed by Heider and coworkers is as follows: activated benzylsuccinate synthase contains a low-reactivity glycine radical at Gly829. Upon binding of both substrates to the active site, the glycine radical generates the thiyl radical on Cys429, which is in close proximity. This thiyl radical abstracts a proton from the methyl group of toluene, which adds to the double bond of fumarate. The benzylsuccinyl radical intermediate then abstracts a proton from Cys429, returning it to the thiyl state which can then restore the glycyl radical resting state.
Environmental significance.
Aromatic hydrocarbons such as toluene, ethylbenzene, and phenol are persistent pollutants in ecological systems, particularly in groundwater. Moreover, they are difficult to degrade due to their inertness, aromaticity, and lack of easily oxidizable functionalities. Microorganisms such as "Magnetospirillum sp." have shown an ability to overcome these chemical obstacles by utilizing radical chemistry to functionalize these hydrocarbons for additional oxidation. However, the industrial utility of benzylsuccinate synthase appears to be limited to anaerobic conditions since the active form of the enzyme is easily and quickly degraded by molecular oxygen. This is present in many ambient environments, and interferes with the radical-glycyl chemistry performed at the active site. It is also believed that the inactive form of benzylsuccinate synthase is not compatible in oxic conditions due to a [4Fe-4S]-cluster that is oxygen sensitive and displays a low midpoint potential.
Salii and coworkers have shown that it is possible to expand the scope of substrates for benzylsuccinate synthase, expanding its applications for the biodegradation of aromatic hydrocarbons. This include "m-, p-", and "o-"cresols which display a hydroxyl group in addition to the methyl group characteristic of toluene. In their work, substrate expansion was accomplished by mutating Ile617 to Val, as Ile617 and Ile620 were two amino acid residues predicted to form a protective hydrophobic wall around the active site.
References.
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Further reading.
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https://en.wikipedia.org/wiki?curid=14282049
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14282061
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Branched-chain-2-oxoacid decarboxylase
|
The enzyme branched-chain-2-oxoacid decarboxylase (EC 4.1.1.72) catalyzes the chemical reaction
(3S)-3-methyl-2-oxopentanoate formula_0 2-methylbutanal + CO2
This enzyme belongs to the family of lyases, specifically the carboxy-lyases, which cleave carbon-carbon bonds. The systematic name of this enzyme class is (3S)-3-methyl-2-oxopentanoate carboxy-lyase (2-methylbutanal-forming). Other names in common use include branched-chain oxo acid decarboxylase, branched-chain alpha-keto acid decarboxylase, branched-chain keto acid decarboxylase, BCKA, and (3S)-3-methyl-2-oxopentanoate carboxy-lyase.
References.
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https://en.wikipedia.org/wiki?curid=14282061
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14282084
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Carnitine decarboxylase
|
Enzyme
The enzyme carnitine decarboxylase (EC 4.1.1.42) catalyzes the chemical reaction
carnitine formula_0 2-methylcholine + CO2
This enzyme belongs to the family of lyases, specifically the carboxy-lyases, which cleave carbon-carbon bonds. The systematic name of this enzyme class is carnitine carboxy-lyase (2-methylcholine-forming). This enzyme is also called carnitine carboxy-lyase. It employs one cofactor, ATP.
References.
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{
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https://en.wikipedia.org/wiki?curid=14282084
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14282104
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Chorismate lyase
|
The enzyme chorismate lyase (EC 4.1.3.40) catalyzes the first step in ubiquinone biosynthesis, the removal of pyruvate from chorismate, to yield 4-hydroxybenzoate in "Escherichia coli" and other Gram-negative bacteria. It belongs to the family of lyases, specifically the oxo-acid-lyases, which cleave carbon-carbon bonds. The systematic name of this enzyme class is chorismate pyruvate-lyase (4-hydroxybenzoate-forming). Other names in common use include CL, CPL, and UbiC.
This enzyme catalyses the chemical reaction:
chorismate formula_0 4-hydroxybenzoate + pyruvate
Its activity does not require metal cofactors.
Activity.
Enzymatic activity.
Inhibited by:
Pathway.
The pathway used is called the ubiquinone biosynthesis pathway, it catalyzes the first step in the biosynthesis of ubiquinone in "E. coli." Ubiquinone is a lipid-soluble electron-transporting coenzyme. They are essential electron carriers in prokaryotes and are essential in aerobic organisms to achieve ATP synthesis.
Nomenclature.
There are several different names for chorismate lyase. It is also called chorismate pyruvate lyase (4-hydroxybenzoate-forming) and it is also abbreviated several different ways: CPL, CL, and ubiC. It is sometimes referred to as ubiC, because that is the gene name. This enzyme belongs to the class lyases; more specifically the ox-acid-lyase or the carbon-carbon-lyases.
Taxonomic lineage:
Structure.
This enzyme is a monomer. Its secondary structure contains helixes, turns, and beta-strands. It has a mass of 18,777 daltons and its sequence is 165 amino acids long.
References.
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https://en.wikipedia.org/wiki?curid=14282104
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14282127
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Citramalate lyase
|
The enzyme citramalate lyase (EC 4.1.3.22) catalyzes the chemical reaction
(2"S")-2-hydroxy-2-methylbutanedioate formula_0 acetate + pyruvate
This enzyme belongs to the family of lyases, specifically the oxo-acid-lyases, which cleave carbon-carbon bonds. The systematic name of this enzyme class is (2"S")-2-hydroxy-2-methylbutanedioate pyruvate-lyase (acetate-forming). Other names in common use include citramalate pyruvate-lyase, citramalate synthase, citramalic-condensing enzyme, citramalate synthetase, citramalic synthase, (S)-citramalate lyase, (+)-citramalate pyruvate-lyase, citramalate pyruvate lyase, (3S)-citramalate pyruvate-lyase, and (2S)-2-hydroxy-2-methylbutanedioate pyruvate-lyase. This enzyme participates in c5-branched dibasic acid metabolism.
References.
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{
"math_id": 0,
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https://en.wikipedia.org/wiki?curid=14282127
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14282140
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Citramalyl-CoA lyase
|
The enzyme citramalyl-CoA lyase (EC 4.1.3.25) catalyzes the chemical reaction
(3"S")-citramalyl-CoA formula_0 acetyl-CoA + pyruvate
This enzyme belongs to the family of lyases, specifically the oxo-acid-lyases, which cleave carbon-carbon bonds. The systematic name of this enzyme class is (3"S")-citramalyl-CoA pyruvate-lyase (acetyl-CoA-forming). Other names in common use include citramalyl coenzyme A lyase, (+)-CMA-CoA lyase, and (3"S")-citramalyl-CoA pyruvate-lyase. This enzyme participates in pyruvate metabolism and c5-branched dibasic acid metabolism.
References.
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{
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https://en.wikipedia.org/wiki?curid=14282140
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14283919
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Citrate (pro-3S)-lyase
|
The enzyme citrate ("pro"-3"S")-lyase (EC 4.1.3.6) catalyzes the chemical reaction
citrate formula_0 acetate + oxaloacetate
This enzyme belongs to the family of lyases, specifically the oxo-acid-lyases, which cleave carbon-carbon bonds. The systematic name of this enzyme class is citrate oxaloacetate-lyase (forming acetate from the pro-S carboxymethyl group of citrate). Other names in common use include citrase, citratase, citritase, citridesmolase, citrate aldolase, citric aldolase, citrate lyase, citrate oxaloacetate-lyase, and citrate oxaloacetate-lyase [(pro-3S)-CH2COO-->acetate]. This enzyme participates in citrate cycle and two-component system - general.
Structural studies.
As of late 2007, 4 structures have been solved for this class of enzymes, with PDB accession codes 1U5H, 1U5V, 1Z6K, and 2HJ0.
References.
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[
{
"math_id": 0,
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https://en.wikipedia.org/wiki?curid=14283919
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14283936
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Citryl-CoA lyase
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Enzyme
The enzyme citryl-CoA lyase (EC 4.1.3.34) catalyzes the chemical reaction
(3"S")-citryl-CoA formula_0 acetyl-CoA + oxaloacetate
This enzyme belongs to the family of lyases, specifically the oxo-acid-lyases, which cleave carbon–carbon bonds. The systematic name of this enzyme class is (3"S")-citryl-CoA oxaloacetate-lyase (acetyl-CoA-forming). This enzyme is also called (3"S")-citryl-CoA oxaloacetate-lyase. This enzyme participates in citrate cycle (tca cycle).
References.
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https://en.wikipedia.org/wiki?curid=14283936
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14283950
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D-dopachrome decarboxylase
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The enzyme -dopachrome decarboxylase (EC 4.1.1.84) catalyzes the chemical reaction
-dopachrome formula_0 5,6-dihydroxyindole + CO2
This enzyme belongs to the family of lyases, specifically the carboxy-lyases, which cleave carbon-carbon bonds. The systematic name of this enzyme class is D-dopachrome carboxy-lyase (5,6-dihydroxyindole-forming). Other names in common use include phenylpyruvate tautomerase II, D-tautomerase, D-dopachrome tautomerase, and D-dopachrome carboxy-lyase.
References.
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https://en.wikipedia.org/wiki?curid=14283950
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14283966
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Dehydro-L-gulonate decarboxylase
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The enzyme dehydro--gulonate decarboxylase (EC 4.1.1.34) catalyzes the chemical reaction
3-dehydro-L-gulonate formula_0 L-xylulose + CO2
This enzyme belongs to the family of lyases, specifically the carboxy-lyases, which cleave carbon-carbon bonds. The systematic name of this enzyme class is 3-dehydro-L-gulonate carboxy-lyase (L-xylulose-forming). This enzyme participates in pentose and glucuronate interconversions.
References.
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https://en.wikipedia.org/wiki?curid=14283966
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14284002
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Deoxyribose-phosphate aldolase
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InterPro Family
The enzyme deoxyribose-phosphate aldolase (EC 4.1.2.4) catalyzes the reversible chemical reaction
2-deoxy-D-ribose 5-phosphate formula_0 D-glyceraldehyde 3-phosphate + acetaldehyde
This enzyme belongs to the family of lyases, specifically the aldehyde-lyases, which cleave carbon-carbon bonds. The systematic name of this enzyme class is 2-deoxy-D-ribose-5-phosphate acetaldehyde-lyase (D-glyceraldehyde-3-phosphate-forming). Other names in common use include phosphodeoxyriboaldolase, deoxyriboaldolase, deoxyribose-5-phosphate aldolase, 2-deoxyribose-5-phosphate aldolase, and 2-deoxy-D-ribose-5-phosphate acetaldehyde-lyase.
Enzyme Mechanism.
Amongst aldolases, DERA is one of the 2 only aldolases able to use two aldehydes as substrate (the other one being FSA). Crystallography shows that the enzyme is a Class I aldolase, so the mechanism proceeds via the formation of a Schiff base with Lys167 at the active site. A nearby residue, Lys201, is critical to reaction by increasing the acidity of protonated Lys167, so Schiff base formation can occur more readily.
As equilibrium of the reaction as written lies on the side of reactant, DERA can also used to catalyze the backward aldol reaction. The enzyme has been found to exhibit some promiscuity by accepting various carbonyl compounds as substrates: acetaldehyde can be replaced with other small aldehydes or acetone; and a variety of aldehydes can be used in place of D-glyceraldehyde 3-phosphate. However, due to the spatial arrangement of stabilizing interactions of the electrophilic aldehyde at the active site, the aldol reaction is stereospecific and gives the "(S)"-configuration at the reactive carbon. Molecular modeling of the active site showed a hydrophilic pocket formed by Thr170 and Lys172 to stabilize C2-hydroxy group of D-glyceraldehyde 3-phosphate, while the C2-hydrogen atom is stabilized in a hydrophobic pocket. When a racemic mixture of glyceraldehyde 3-phosphate is used as the substrate, only the D-isomer reacted.
Enzyme Structure.
The DERA monomer contains a TIM α/β barrel fold, consistent with other Class I aldolases. The structure of DERAs across many organisms: DERAs from "Escherichia coli" and "Aeropyrum pernix" shares 37.7% sequence identity with DERA from "Thermus thermophilus" HB8. The reaction mechanism is also conserved between DERAs.
In solution, DERAs are found in homodimers or homotetramers. The oligomeric nature of the enzyme does not contribute to enzymatic activity, but serves to increase thermal stability through hydrophobic interactions and hydrogen bonding between interfacial residues.
As of late 2007, 10 structures have been solved for this class of enzymes, with PDB accession codes 1JCJ, 1JCL, 1KTN, 1MZH, 1N7K, 1O0Y, 1P1X, 1UB3, 1VCV, and 2A4A.
Biological Function.
DERA is part of the inducible "deo" operon in bacteria which allows for the conversion of exogenous deoxyribonucleosides for energy generation. The products of DERA, glyceraldehyde-3-phosphate and acetaldehyde (subsequently converted to acetyl CoA) can enter the glycolysis and Kreb’s cycle pathways respectively.
In humans, DERA is mainly expressed in lungs, liver and colon and is necessary for the cellular stress response. After induction of oxidative stress or mitochondrial stress, DERA colocalizes with stress granules and associates with YBX1, a known stress granule protein. Cells with high DERA expression were able to utilize exogenous deoxyinosine to produce ATP when starved of glucose and incubated with mitochondrial uncoupler FCCP.
Industrial Relevance.
DERA is being used in chemical syntheses as a tool for green, enantioselective aldol reactions. Formation of the deoxyribose skeleton from small molecules can facilitate the synthesis of nucleoside reverse transcriptase inhibitors. For example, DERA was used in a mixture of five enzymes in the biocatalytic synthesis of islatravir.
DERA has also been used to perform tandem aldol reactions with three aldehyde substrates, with reaction equilibrium driven by the formation of the six-membered cyclic hemiacetal. This intermediate has been used in the synthesis of statin drugs, such as atorvastatin, rosuvastatin and mevastatin.
Natural DERAs show low tolerance to high concentrations of acetaldehyde due to the formation of the highly reactive crotonaldehyde intermediate that irreversibly inactivates the enzyme. This features hampers the industrial applications of DERA as the concentration of acetaldehyde used will be limited. To overcome this, directed evolution has been used to improve the acetaldehyde tolerance of DERA to up to 400mM.
References.
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14284014
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Diaminobutyrate decarboxylase
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The enzyme diaminobutyrate decarboxylase (EC 4.1.1.86) catalyzes the chemical reaction
-2,4-diaminobutanoate formula_0 propane-1,3-diamine + CO2
This enzyme belongs to the family of lyases, specifically the carboxy-lyases, which cleave carbon-carbon bonds. The systematic name of this enzyme class is L-2,4-diaminobutanoate carboxy-lyase (propane-1,3-diamine-forming). Other names in common use include DABA DC, L-2,4-diaminobutyrate decarboxylase, and L-2,4-diaminobutanoate carboxy-lyase.
References.
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https://en.wikipedia.org/wiki?curid=14284014
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14284045
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Dihydroneopterin aldolase
|
The enzyme dihydroneopterin aldolase (EC 4.1.2.25) catalyzes the chemical reaction
2-amino-4-hydroxy-6-(-"erythro"-1,2,3-trihydroxypropyl)-7,8- dihydropteridine formula_0 2-amino-4-hydroxy-6-hydroxymethyl-7,8-dihydropteridine + glycolaldehyde
This enzyme belongs to the family of lyases, specifically the aldehyde-lyases, which cleave carbon-carbon bonds. The systematic name of this enzyme class is 2-amino-4-hydroxy-6-(D-erythro-1,2,3-trihydroxypropyl)-7,8-dihydropt eridine glycolaldehyde-lyase (2-amino-4-hydroxy-6-hydroxymethyl-7,8-dihydropteridine-forming). Other names in common use include 2-amino-4-hydroxy-6-(D-erythro-1,2,3-trihydroxypropyl)-7,8-, and dihydropteridine glycolaldehyde-lyase. This enzyme participates in folate biosynthesis.
Structural studies.
As of late 2007, 13 structures have been solved for this class of enzymes, with PDB accession codes 1NBU, 1RRI, 1RRW, 1RRY, 1RS2, 1RS4, 1RSD, 1RSI, 1U68, 1Z9W, 2CG8, 2NM2, and 2NM3.
References.
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https://en.wikipedia.org/wiki?curid=14284045
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14284065
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Dihydroxyfumarate decarboxylase
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The enzyme dihydroxyfumarate decarboxylase (EC 4.1.1.54) catalyzes the chemical reaction
dihydroxyfumarate formula_0 tartronate semialdehyde + CO2
This enzyme belongs to the family of lyases, specifically the carboxy-lyases, which cleave carbon-carbon bonds. The systematic name of this enzyme class is dihydroxyfumarate carboxy-lyase (tartronate-semialdehyde-forming). This enzyme is also called dihydroxyfumarate carboxy-lyase. This enzyme participates in glyoxylate and dicarboxylate metabolism.
References.
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https://en.wikipedia.org/wiki?curid=14284065
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