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14105833 | Francis Birch (geophysicist) | American geophysicist (1903–1992)
Albert Francis Birch (August 22, 1903 – January 30, 1992) was an American geophysicist. He is considered one of the founders of solid Earth geophysics. He is also known for his part in the atomic bombing of Hiroshima and Nagasaki.
During World War II, Birch participated in the Manhattan Project, working on the design and development of the gun-type nuclear weapon known as Little Boy. He oversaw its manufacture, and went to Tinian to supervise its assembly and loading into "Enola Gay", the Boeing B-29 Superfortress tasked with dropping the bomb.
A graduate of Harvard University, Birch began working on geophysics as a research assistant. He subsequently spent his entire career at Harvard working in the field, becoming an Associate Professor of Geology in 1943, a professor in 1946, and Sturgis Hooper Professor of Geology in 1949, and professor emeritus in 1974.
Birch published over 100 papers. He developed what is now known as the Birch-Murnaghan equation of state in 1947. In 1952 he demonstrated that Earth's mantle is chiefly composed of silicate minerals, with an inner and outer core of molten iron. In two 1961 papers on compressional wave velocities, he established what is now called Birch's law.
Early life.
Birch was born in Washington, D.C., on August 22, 1903, the son of George Albert Birch, who was involved in banking and real estate, and Mary Hemmick Birch, a church choir singer and soloist at St. Matthew's Cathedral in Washington, D.C. He had three younger brothers: David, who became a banker; John, who became a diplomat; and Robert, who became a songwriter. He was educated at Washington, D.C., schools, and Western High School, where he joined the High School Cadets in 1916.
In 1920 Birch entered Harvard University on a scholarship. While there he served in Harvard's Reserve Officers' Training Corps Field Artillery Battalion. He graduated magna cum laude in 1924, and received his Bachelor of Science (S.B.) degree in electrical engineering.
Birch went to work in the Engineering Department of the New York Telephone Company. He applied for and received an American Field Service Fellowship in 1926, which he used to travel to Strasbourg, and study at the University of Strasbourg's Institut de Physique under the tutelage of Pierre Weiss. There, he wrote or co-wrote four papers, in French, on topics such as the paramagnetic properties of potassium cyanide, and the magnetic moment of Cu++ ions.
On returning to the United States in 1928, Birch went back to Harvard to pursue physics. He was awarded his Master of Arts (A.M.) degree in 1929, and then commenced work on his 1932 Doctor of Philosophy (Ph.D.) degree under the supervision of Percy Bridgman, who would receive the Nobel Prize for Physics in 1946. For his thesis, Birch measured the vapor-liquid critical point of mercury. He determined this as 1460±20 °C and 1640±50 kg/cm2, results he published in 1932 in the Physical Review.
Around this time, there was an increased interest in geophysics at Harvard University, and Reginald Aldworth Daly established a Committee for Experimental Geology and Geophysics that included Bridgman, astronomer Harlow Shapley, geologists Louis Caryl Graton and D. H. McLaughlin and chemist G. P. Baxter. William Zisman, another one of Bridgman's Ph.D. students, was hired as the committee's research associate, but, having little interest in the study of rocks, he resigned in 1932. The position was then offered to Birch, who had little interest or experience in geology either, but with the advent of the Great Depression, jobs were hard to find, and he accepted.
On July 15, 1933, Birch married Barbara Channing, a Bryn Mawr College alumna, and a collateral descendant of the theologian William Ellery Channing. They had three children: Anne Campaspe, Francis (Frank) Sylvanus and Mary Narcissa. Frank later became a professor of geophysics at the University of New Hampshire.
World War II.
In 1942, during World War II, Birch took a leave of absence from Harvard, in order to work at the Massachusetts Institute of Technology Radiation Laboratory, which was developing radar. He worked on the proximity fuze, a radar-triggered fuze that would explode a shell in the proximity of a target. The following year he accepted a commission in the United States Navy as a lieutenant commander, and was posted to the Bureau of Ships in Washington, D.C.
Later that year he was assigned to the Manhattan Project, and moved with his family to Los Alamos, New Mexico. There he joined the Los Alamos Laboratory's Ordnance (O) Division, which was under the command of another Naval officer, Captain William S. Parsons. Initially the goal of the O Division was to design a gun-type nuclear weapon known as Thin Man. This proved to be impractical due to contamination of the reactor-bred plutonium with plutonium-240, and in February 1944, the Division switched its attention to the development of the Little Boy, a smaller device using uranium-235. Birch used unenriched uranium to create scale models and later full-scale mock-ups of the device.
Birch supervised the manufacture of the Little Boy, and went to Tinian to supervise its assembly and loading it onto "Enola Gay", the Boeing B-29 Superfortress tasked with dropping the bomb. He devised the 'double plug' system that allowed for actually arming the bomb after "Enola Gay" took off so that if it crashed, there would not be a nuclear explosion. He was awarded the Legion of Merit. His citation read:<templatestyles src="Template:Blockquote/styles.css" />for exceptionally meritorious conduct in the performance of outstanding services to the Government of the United States in connection with the development of the greatest military weapon of all time, the atomic bomb. His initial assignment was the instrumentation of laboratory and field tests. He carried out this assignment in such outstanding fashion that he was placed in charge of the engineering and development of the first atomic bomb. He carried out this assignment with outstanding judgment and skill, and finally, went with the bomb to the advanced base where he insured, by his care and leadership, that the bomb was adequately prepared in every respect. Commander Birch's engineering ability, understanding of all principles involved, professional skill and devotion to duty throughout the development and delivery of the atomic bomb were outstanding and were in keeping with the highest traditions of the United States Naval Service.
Birch was promoted to commander and released from the Navy in 1945.
Post-war.
Birch returned to Harvard after the war ended, having been promoted to Associate Professor of Geology in 1943 while he was away. He would remain at Harvard for the rest of his career, becoming a professor in 1946, and Sturgis Hooper Professor of Geology in 1949, and professor emeritus in 1974. Professor Birch published over 100 papers. He served as president of The Geological Society of America in 1964 and was awarded their Penrose Medal in 1969.
In 1947, he adapted the isothermal Murnaghan equation of state, which had been developed for infinitesimal strain, for Eulerian finite strain, developing what is now known as the Birch-Murnaghan equation of state.
Albert Francis Birch is known for his experimental work on the properties of Earth-forming minerals at high pressure and temperature, in 1952 he published a well-known paper in the Journal of Geophysical Research, where he demonstrated that the mantle is chiefly composed of silicate minerals, the upper and lower mantle are separated by a thin transition zone associated with silicate phase transitions, and the inner and outer core are alloys of crystalline and molten iron. His conclusions are still accepted as correct today. The most famous portion of the paper, however, is a humorous footnote he included in the introduction:
Unwary readers should take warning that ordinary language undergoes modification to a high-pressure form when applied to the interior of the Earth. A few examples of equivalents follow:
In 1961, Birch published two papers on compressional wave velocities establishing a linear relation of the compressional wave velocity Vp of rocks and minerals of a constant average atomic weight formula_0 with density formula_1 as:
formula_2.
This relationship became known as Birch's law. Birch was elected to the American Academy of Arts and Sciences in 1942, the National Academy of Sciences in 1950, the American Philosophical Society in 1955, and served as the president of the Geological Society of America in 1963 and 1964. He received numerous honors in his career, including the Geological Society of America's Arthur L. Day Medal on 1950 and Penrose Medal in 1969, the American Geophysical Union's William Bowie Medal in 1960, the National Medal of Science from President Lyndon Johnson in 1967, the Vetlesen Prize (shared with Sir Edward Bullard) in 1968, the Gold Medal of the Royal Astronomical Society in 1973, and the International Association for the Advancement of High Pressure Research's Bridgman Award in 1983. Since 1992, the American Geophysical Union's Tectonophysics section has sponsored a Francis Birch Lecture, given at its annual meeting by a noted researcher in this field.
Birch died of prostate cancer at his home in Cambridge, Massachusetts, on January 30, 1992. He was survived by wife Barbara, his three children and his three brothers. His papers are in the Harvard University Archives.
Notes.
<templatestyles src="Reflist/styles.css" /> | [
{
"math_id": 0,
"text": "\\bar{ M}"
},
{
"math_id": 1,
"text": "\\rho"
},
{
"math_id": 2,
"text": " V_p = a (\\bar{ M}) + b \\rho "
}
]
| https://en.wikipedia.org/wiki?curid=14105833 |
1410631 | Wiedemann–Franz law | Law of physics
In physics, the Wiedemann–Franz law states that the ratio of the electronic contribution of the thermal conductivity ("κ") to the electrical conductivity ("σ") of a metal is proportional to the temperature ("T").
formula_0
Theoretically, the proportionality constant "L", known as the Lorenz number, is equal to
formula_1
where "k"B is Boltzmann's constant and "e" is the elementary charge.
This empirical law is named after Gustav Wiedemann and Rudolph Franz, who in 1853 reported that "κ"/"σ" has approximately the same value for different metals at the same temperature. The proportionality of "κ"/"σ" with temperature was discovered by Ludvig Lorenz in 1872.
Derivation.
Qualitatively, this relationship is based upon the fact that the heat and electrical transport both involve the free electrons in the metal.
The mathematical expression of the law can be derived as following. Electrical conduction of metals is a well-known phenomenon and is attributed to the free conduction electrons, which can be measured as sketched in the figure. The current density "j" is observed to be proportional to the applied electric field and follows Ohm's law where the prefactor is the specific electrical conductivity. Since the electric field and the current density are vectors Ohm's law is expressed here in bold face. The conductivity can in general be expressed as a tensor of the second rank (3×3 matrix). Here we restrict the discussion to isotropic, i.e. scalar conductivity. The specific resistivity is the inverse of the conductivity. Both parameters will be used in the following.
Drude model derivation.
Paul Drude (c. 1900) realized that the phenomenological description of conductivity can be formulated quite generally (electron-, ion-, heat- etc. conductivity). Although the phenomenological description is incorrect for conduction electrons, it can serve as a preliminary treatment.
The assumption is that the electrons move freely in the solid like in an ideal gas. The force applied to the electron by the electric field leads to an acceleration according to
formula_2
formula_3
This would lead, however, to a constant acceleration and, ultimately, to an infinite velocity. The further assumption therefore is that the electrons bump into obstacles (like defects or phonons) once in a while which limits their free flight. This establishes an average or drift velocity "V"d. The drift velocity is related to the average scattering time as becomes evident from the following relations.
formula_4
From kinetic theory of gases, formula_5, where formula_6 is the heat capacity per electron, formula_7 is the mean free path of the electrons, and
formula_8,
is the average speed of the particles in the gas.
From the Drude model,
formula_9.
Therefore,
formula_10,
which is the Wiedemann–Franz law with an erroneous proportionality constant formula_11.
In Drude's original paper he used formula_12 instead of formula_13, and also accidentally used a factor of 2. This meant his result is formula_14which is very close to experimental values. This is in fact due to 3 mistakes that conspired to make his result more accurate than warranted: the factor of 2 mistake; the specific heat per electron is in fact about 100 times less than formula_15; the mean squared velocity of an electron is in fact about 100 times larger.
Free electron model.
After taking into account the quantum effects, as in the free electron model, the heat capacity, mean free path and average speed of electrons are modified and the proportionality constant is then corrected to formula_16, which agrees with experimental values.
Temperature dependence.
The value "L"0 = 2.44×10−8 V2K−2 results from the fact that at low temperatures (formula_17 K) the heat and charge currents are carried by the same quasi-particles: electrons or holes. At finite temperatures two mechanisms produce a deviation of the ratio formula_18 from the theoretical Lorenz value "L"0: (i) other thermal carriers such as phonons or magnons, (ii) Inelastic scattering.
As the temperature tends to 0 K, inelastic scattering becomes weak and promotes large q scattering values (trajectory "a" in the figure). For each electron transported, a thermal excitation is also carried and the Lorenz number is reached "L" = "L"0. Note that in a perfect metal, inelastic scattering would be completely absent in the limit formula_17 K and the thermal conductivity would vanish formula_19.
At finite temperature small q scattering values are possible (trajectory b in the figure) and electrons can be transported without the transport of a thermal excitation "L"("T") < "L"0.
At higher temperatures, the contribution of phonons to thermal transport in a system becomes important. This can lead to "L"("T") > "L"0. Above the Debye temperature the phonon contribution to thermal transport is constant and the ratio "L"("T") is again found constant.
Limitations of the theory.
Experiments have shown that the value of "L", while roughly constant, is not exactly the same for all materials.
Kittel gives some values of "L" ranging from "L" = 2.23×10−8V2K−2 for copper at 0 °C to "L" = 3.2×10−8V2K−2 for tungsten at 100 °C. Rosenberg notes that the Wiedemann–Franz law is generally valid for high temperatures and for low (i.e., a few Kelvins) temperatures, but may not hold at intermediate temperatures.
In many high purity metals both the electrical and thermal conductivities rise as temperature is decreased. In certain materials (such as silver or aluminum) however, the value of "L" also may decrease with temperature. In the purest samples of silver and at very low temperatures, "L" can drop by as much as a factor of 10.
In degenerate semiconductors, the Lorenz number "L" has a strong dependency on certain system parameters: dimensionality, strength of interatomic interactions and Fermi level. This law is not valid or the value of the Lorenz
number can be reduced at least in the following cases: manipulating electronic density of states, varying doping density and layer thickness in superlattices and materials with correlated carriers. In thermoelectric materials there are also corrections due to boundary conditions, specifically open circuit vs. closed circuit.
Violations.
In 2011, N. Wakeham et al. found that the ratio of the thermal and electrical Hall conductivities in the metallic phase of quasi-one-dimensional lithium molybdenum purple bronze Li0.9Mo6O17 diverges with decreasing temperature, reaching a value five orders of magnitude larger than that found in conventional metals obeying the Wiedemann–Franz law. This due to spin-charge separation and it behaving as a Luttinger liquid.
A Berkeley-led study in 2016 by S. Lee "et al." also found a large violation of the Wiedemann–Franz law near the insulator-metal transition in VO2 nanobeams. In the metallic phase, the electronic contribution to thermal conductivity was much smaller than what would be expected from the Wiedemann–Franz law. The results can be explained in terms of independent propagation of charge and heat in a strongly correlated system.
Molecular systems.
In 2020, Galen Craven and Abraham Nitzan derived a Wiedemann-Franz law for molecular systems in which electronic conduction is dominated not by free electron motion as in metals, but instead by electron transfer between molecular sites. The molecular Wiedemann-Franz law is given by
formula_20
where
formula_21
is the Lorenz number for molecules and formula_22 is the reorganization energy for electron transfer.
References.
<templatestyles src="Reflist/styles.css" /> | [
{
"math_id": 0,
"text": "\\frac \\kappa \\sigma = LT"
},
{
"math_id": 1,
"text": "L = \\frac \\kappa {\\sigma T} = \\frac{\\pi^2} 3 \\left(\\frac{k_{\\rm B}} e \\right)^2 = 2.44\\times 10^{-8}\\;\\mathrm{V}^2\\mathrm{K}^{-2},"
},
{
"math_id": 2,
"text": " \\mathbf{F} = - e \\mathbf{E} = m \\frac{\\;d\\mathbf{v}}{dt}"
},
{
"math_id": 3,
"text": "\\;d\\mathbf{v}= - \\frac{e \\mathbf{E}} m dt"
},
{
"math_id": 4,
"text": " \\frac{d\\mathbf{v}}{dt}= - \\frac{e \\mathbf{E}} m - \\frac{1}{\\tau} \\mathbf{v} "
},
{
"math_id": 5,
"text": "\\kappa = \\frac{1}{3}c n\\,\\ell \\,\\langle v\\rangle "
},
{
"math_id": 6,
"text": "c = \\frac 32 k_{\\rm B} "
},
{
"math_id": 7,
"text": "\\ell"
},
{
"math_id": 8,
"text": " \\langle v\\rangle = \\sqrt{\\frac{8k_{\\rm B} T}{\\pi m}}=\\sqrt{\\frac{8}{3\\pi}}v_{\\rm rms}"
},
{
"math_id": 9,
"text": "\\sigma = \\frac{ne^2\\tau}{m} = \\frac{ne^2\\ell}{m\\langle v\\rangle}"
},
{
"math_id": 10,
"text": "\\frac \\kappa \\sigma = \\frac{c m \\, \\langle {v} \\rangle^2}{3e^2} = \\frac{4}{\\pi} \\frac{k_{\\rm B}^2T}{e^2} = 0.94\\times 10^{-8}\\;\\mathrm{V}^2\\mathrm{K}^{-2} "
},
{
"math_id": 11,
"text": "\\frac{4}{\\pi}\\approx 1.27"
},
{
"math_id": 12,
"text": "\\langle v^2\\rangle"
},
{
"math_id": 13,
"text": "\\langle v\\rangle^2"
},
{
"math_id": 14,
"text": "L = 3 \\left(\\frac{k_{\\rm B}} e \\right)^2 = 2.22\\times 10^{-8}\\;\\mathrm{V}^2\\mathrm{K}^{-2},"
},
{
"math_id": 15,
"text": "\\frac 32 k_{\\rm B}"
},
{
"math_id": 16,
"text": "\\frac{\\pi^2} 3\\approx3.29"
},
{
"math_id": 17,
"text": "T\\rightarrow 0"
},
{
"math_id": 18,
"text": "L = \\kappa/(\\sigma T)"
},
{
"math_id": 19,
"text": "\\kappa\\rightarrow 0; L\\rightarrow 0"
},
{
"math_id": 20,
"text": "\\frac{\\kappa}{\\sigma} = L_\\text{M}\\frac{\\lambda}{k_{\\rm B}}"
},
{
"math_id": 21,
"text": "L_\\text{M} = \\frac{1} 2 \\left(\\frac{k_{\\rm B}} e \\right)^2"
},
{
"math_id": 22,
"text": "\\lambda"
}
]
| https://en.wikipedia.org/wiki?curid=1410631 |
1410760 | Molar heat capacity | Intensive quantity, heat capacity per amount of substance
The molar heat capacity of a chemical substance is the amount of energy that must be added, in the form of heat, to one mole of the substance in order to cause an increase of one unit in its temperature. Alternatively, it is the heat capacity of a sample of the substance divided by the amount of substance of the sample; or also the specific heat capacity of the substance times its molar mass. The SI unit of molar heat capacity is joule per kelvin per mole, J⋅K−1⋅mol−1.
Like the specific heat, the measured molar heat capacity of a substance, especially a gas, may be significantly higher when the sample is allowed to expand as it is heated (at constant pressure, or isobaric) than when it is heated in a closed vessel that prevents expansion (at constant volume, or isochoric). The ratio between the two, however, is the same heat capacity ratio obtained from the corresponding specific heat capacities.
This property is most relevant in chemistry, when amounts of substances are often specified in moles rather than by mass or volume. The molar heat capacity generally increases with the molar mass, often varies with temperature and pressure, and is different for each state of matter. For example, at atmospheric pressure, the (isobaric) molar heat capacity of water just above the melting point is about 76 J⋅K−1⋅mol−1, but that of ice just below that point is about 37.84 J⋅K−1⋅mol−1. While the substance is undergoing a phase transition, such as melting or boiling, its molar heat capacity is technically infinite, because the heat goes into changing its state rather than raising its temperature. The concept is not appropriate for substances whose precise composition is not known, or whose molar mass is not well defined, such as polymers and oligomers of indeterminate molecular size.
A closely related property of a substance is the heat capacity per mole of atoms, or atom-molar heat capacity, in which the heat capacity of the sample is divided by the number of moles of atoms instead of moles of molecules. So, for example, the atom-molar heat capacity of water is 1/3 of its molar heat capacity, namely 25.3 J⋅K−1⋅mol−1.
In informal chemistry contexts, the molar heat capacity may be called just "heat capacity" or "specific heat". However, international standards now recommend that "specific heat capacity" always refer to capacity per unit of mass, to avoid possible confusion. Therefore, the word "molar", not "specific", should always be used for this quantity.
Definition.
The molar heat capacity of a substance, which may be denoted by "c"m, is the heat capacity "C" of a sample of the substance, divided by the amount (moles) "n" of the substance in the sample:
"c"mformula_0
where "Q" is the amount of heat needed to raise the temperature of the sample by "ΔT". Obviously, this parameter cannot be computed when "n" is not known or defined.
Like the heat capacity of an object, the molar heat capacity of a substance may vary, sometimes substantially, depending on the starting temperature "T" of the sample and the pressure "P" applied to it. Therefore, it should be considered a function "c"m("P","T") of those two variables.
These parameters are usually specified when giving the molar heat capacity of a substance. For example, "H2O: 75.338 J⋅K−1⋅mol−1 (25 °C, 101.325 kPa)" When not specified, published values of the molar heat capacity "c"m generally are valid for some standard conditions for temperature and pressure.
However, the dependency of "c"m("P","T") on starting temperature and pressure can often be ignored in practical contexts, e.g. when working in narrow ranges of those variables. In those contexts one can usually omit the qualifier ("P","T"), and approximate the molar heat capacity by a constant "c"m suitable for those ranges.
Since the molar heat capacity of a substance is the specific heat "c" times the molar mass of the substance "M"/"N" its numerical value is generally smaller than that of the specific heat. Paraffin wax, for example, has a specific heat of about but a molar heat capacity of about .
The molar heat capacity is an "intensive" property of a substance, an intrinsic characteristic that does not depend on the size or shape of the amount in consideration. (The qualifier "specific" in front of an extensive property often indicates an intensive property derived from it.)
Variations.
The injection of heat energy into a substance, besides raising its temperature, usually causes an increase in its volume or its pressure, depending on how the sample is confined. The choice made about the latter affects the measured molar heat capacity, even for the same starting pressure "P" and starting temperature "T". Two particular choices are widely used:
The value of "c""V",m is always less than the value of "c""P",m. This difference is particularly notable in gases where values under constant pressure are typically 30% to 66.7% greater than those at constant volume.
All methods for the measurement of specific heat apply to molar heat capacity as well.
Units.
The SI unit of molar heat capacity heat is joule per kelvin per mole (J/(K⋅mol), J/(K mol), J K−1 mol−1, etc.). Since an increment of temperature of one degree Celsius is the same as an increment of one kelvin, that is the same as joule per degree Celsius per mole (J/(°C⋅mol)).
In chemistry, heat amounts are still often measured in calories. Confusingly, two units with that name, denoted "cal" or "Cal", have been commonly used to measure amounts of heat:
When heat is measured in these units, the unit of specific heat is usually
1 cal/(°C⋅mol) ("small calorie") = 4.184 J⋅K−1⋅mol−1
1 kcal/(°C⋅mol) ("large calorie") = 4184 J⋅K−1⋅mol−1.
The molar heat capacity of a substance has the same dimension as the heat capacity of an object; namely, L2⋅M⋅T−2⋅Θ−1, or M(L/T)2/Θ. (Indeed, it is the heat capacity of the object that consists of an Avogadro number of molecules of the substance.) Therefore, the SI unit J⋅K−1⋅mol−1 is equivalent to kilogram metre squared per second squared per kelvin (kg⋅m2⋅K−1⋅s−2).
Physical basis.
Monatomic gases.
The temperature of a sample of a substance reflects the average kinetic energy of its constituent particles (atoms or molecules) relative to its center of mass. Quantum mechanics predicts that, at room temperature and ordinary pressures, an isolated atom in a gas cannot store any significant amount of energy except in the form of kinetic energy. Therefore, when a certain number "N" of atoms of a monatomic gas receives an input "Q" of heat energy, in a container of fixed volume, the kinetic energy of each atom will increase by "Q"/"N", independently of the atom's mass. This assumption is the foundation of the theory of ideal gases.
In other words, that theory predicts that the molar heat capacity "at constant volume" "c""V",m of all monatomic gases will be the same; specifically,
"c""V",m = "R"
where "R" is the ideal gas constant, about 8.31446 J⋅K−1⋅mol−1 (which is the product of the Boltzmann constant "k"B and the Avogadro constant). And, indeed, the experimental values of "c""V",m for the noble gases helium, neon, argon, krypton, and xenon (at 1 atm and 25 °C) are all 12.5 J⋅K−1⋅mol−1, which is "R"; even though their atomic weights range from 4 to 131.
The same theory predicts that the molar heat capacity of a monatomic gas "at constant pressure" will be
"c""P",m = "c""V",m + "R" = "R"
This prediction matches the experimental values, which, for helium through xenon, are 20.78, 20.79, 20.85, 20.95, and 21.01 J⋅K−1⋅mol−1, respectively; very close to the theoretical "R" = 20.78 J⋅K−1⋅mol−1.
Therefore, the specific heat (per unit of mass, not per mole) of a monatomic gas will be inversely proportional to its (adimensional) relative atomic mass "A". That is, approximately,
"c""V" = (12470 J⋅K−1⋅kg−1)/"A" "c""P" = (20786 J⋅K−1⋅kg−1)/"A"
Polyatomic gases.
Degrees of freedom.
A polyatomic molecule (consisting of two or more atoms bound together) can store heat energy in other forms besides its kinetic energy. These forms include rotation of the molecule, and vibration of the atoms relative to its center of mass.
These extra degrees of freedom contribute to the molar heat capacity of the substance. Namely, when heat energy is injected into a gas with polyatomic molecules, only part of it will go into increasing their kinetic energy, and hence the temperature; the rest will go to into those other degrees of freedom. Thus, in order to achieve the same increase in temperature, more heat energy will have to be provided to a mol of that substance than to a mol of a monatomic gas. Substances with high atomic count per molecule, like octane, can therefore have a very large heat capacity per mole, and yet a relatively small specific heat (per unit mass).
If the molecule could be entirely described using classical mechanics, then the theorem of equipartition of energy could be used to predict that each degree of freedom would have an average energy in the amount of "kT", where "k" is the Boltzmann constant, and "T" is the temperature. If the number of degrees of freedom of the molecule is "f", then each molecule would be holding, on average, a total energy equal to "fkT". Then the molar heat capacity (at constant volume) would be
"c""V",m = "fR"
where "R" is the ideal gas constant. According to Mayer's relation, the molar heat capacity at constant pressure would be
"c""P",m = "c""V",m + "R" = "fR" + "R" = (f + 2)"R"
Thus, each additional degree of freedom will contribute "R" to the molar heat capacity of the gas (both "c""V",m and "c""P",m).
In particular, each molecule of a monatomic gas has only "f" = 3 degrees of freedom, namely the components of its velocity vector; therefore "c""V",m = "R" and "c""P",m = "R".
Rotational modes of a diatomic molecule.
For example, the molar heat capacity of nitrogen N2 at constant volume is 20.6 J⋅K−1⋅mol−1 (at 15 °C, 1 atm), which is 2.49 "R". From the theoretical equation "c""V",m = "fR", one concludes that each molecule has "f" = 5 degrees of freedom. These turn out to be three degrees of the molecule's velocity vector, plus two degrees from its rotation about an axis through the center of mass and perpendicular to the line of the two atoms. The degrees of freedom due to translations and rotations are called the rigid degrees of freedom, since they do not involve any deformation of the molecule.
Because of those two extra degrees of freedom, the molar heat capacity "c""V",m of N2 (20.6 J⋅K−1⋅mol−1) is greater than that of an hypothetical monatomic gas (12.5 J⋅K−1⋅mol−1) by a factor of .
Frozen and active degrees of freedom.
According to classical mechanics, a diatomic molecule like nitrogen should have more degrees of internal freedom, corresponding to vibration of the two atoms that stretch and compress the bond between them.
For thermodynamic purposes, each direction in which an atom can independently vibrate relative to the rest of the molecule introduces two degrees of freedom: one associated with the potential energy from distorting the bonds, and one for the kinetic energy of the atom's motion. In a diatomic molecule like N2, there is only one direction for the vibration, and the motions of the two atoms must be opposite but equal; so there are only two degrees of vibrational freedom. That would bring "f" up to 7, and "c""V",m to 3.5 "R".
The reason why these vibrations are not absorbing their expected fraction of heat energy input is provided by quantum mechanics. According to that theory, the energy stored in each degree of freedom must increase or decrease only in certain amounts (quanta). Therefore, if the temperature "T" of the system is not high enough, the average energy that would be available for some of the theoretical degrees of freedom ("kT"/"f") may be less than the corresponding minimum quantum. If the temperature is low enough, that may be the case for practically all molecules. One then says that those degrees of freedom are "frozen". The molar heat capacity of the gas will then be determined only by the "active" degrees of freedom — that, for most molecules, can receive enough energy to overcome that quantum threshold.
For each degree of freedom, there is an approximate critical temperature at which it "thaws" ("unfreezes") and becomes active, thus being able to hold heat energy. For the three translational degrees of freedom of molecules in a gas, this critical temperature is extremely small, so they can be assumed to be always active. For the rotational degrees of freedom, the thawing temperature is usually a few tens of kelvins (although with a very light molecule such as hydrogen the rotational energy levels will be spaced so widely that rotational heat capacity may not completely "unfreeze" until considerably higher temperatures are reached). Vibration modes of diatomic molecules generally start to activate only well above room temperature.
In the case of nitrogen, the rotational degrees of freedom are fully active already at −173 °C (100 K, just 23 K above the boiling point). On the other hand, the vibration modes only start to become active around 350 K (77 °C) Accordingly, the molar heat capacity "c""P",m is nearly constant at 29.1 J⋅K−1⋅mol−1 from 100 K to about 300 °C. At about that temperature, it starts to increase rapidly, then it slows down again. It is 35.5 J⋅K−1⋅mol−1 at 1500 °C, 36.9 at 2500 °C, and 37.5 at 3500 °C. The last value corresponds almost exactly to the predicted value for "f" = 7.
The following is a table of some constant-pressure molar heat capacities "c""P",m of various diatomic gases at standard temperature (25 °C = 298 K), at 500 °C, and at 5000 °C, and the apparent number of degrees of freedom "f"* estimated by the formula "f"* = 2"c""P",m/"R" − 2:
The quantum harmonic oscillator approximation implies that the spacing of energy levels of vibrational modes are inversely proportional to the square root of the reduced mass of the atoms composing the diatomic molecule. This fact explains why the vibrational modes of heavier molecules like Br2 are active at lower temperatures. The molar heat capacity of Br2 at room temperature is consistent with "f" = 7 degrees of freedom, the maximum for a diatomic molecule. At high enough temperatures, all diatomic gases approach this value.
Rotational modes of single atoms.
Quantum mechanics also explains why the specific heat of monatomic gases is well predicted by the ideal gas theory with the assumption that each molecule is a point mass that has only the "f" = 3 translational degrees of freedom.
According to classical mechanics, since atoms have non-zero size, they should also have three rotational degrees of freedom, or "f" = 6 in total. Likewise, the diatomic nitrogen molecule should have an additional rotation mode, namely about the line of the two atoms; and thus have "f" = 6 too. In the classical view, each of these modes should store an equal share of the heat energy.
However, according to quantum mechanics, the energy difference between the allowed (quantized) rotation states is inversely proportional to the moment of inertia about the corresponding axis of rotation. Because the moment of inertia of a single atom is exceedingly small, the activation temperature for its rotational modes is extremely high. The same applies to the moment of inertia of a diatomic molecule (or a linear polyatomic one) about the internuclear axis, which is why that mode of rotation is not active in general.
On the other hand, electrons and nuclei can exist in excited states and, in a few exceptional cases, they may be active even at room temperature, or even at cryogenic temperatures.
Polyatomic gases.
The set of all possible ways to infinitesimally displace the "n" atoms of a polyatomic gas molecule is a linear space of dimension 3"n", because each atom can be independently displaced in each of three orthogonal axis directions. However, some three of these dimensions are just translation of the molecule by an infinitesimal displacement vector, and others are just rigid rotations of it by an infinitesimal angle about some axis. Still others may correspond to relative rotation of two parts of the molecule about a single bond that connects them.
The independent "deformation modes"—linearly independent ways to actually deform the molecule, that strain its bonds—are only the remaining dimensions of this space. As in the case diatomic molecules, each of these deformation modes counts as two vibrational degrees of freedom for energy storage purposes: one for the potential energy stored in the strained bonds, and one for the extra kinetic energy of the atoms as they vibrate about the rest configuration of the molecule.
In particular, if the molecule is linear (with all atoms on a straight line), it has only two non-trivial rotation modes, since rotation about its own axis does not displace any atom. Therefore, it has 3"n" − 5 actual deformation modes. The number of energy-storing degrees of freedom is then "f" = 3 + 2 + 2(3"n" − 5) = 6"n" − 5.
For example, the linear nitrous oxide molecule (with "n" = 3) has 3"n" − 5 = 4 independent infinitesimal deformation modes. Two of them can be described as stretching one of the bonds while the other retains its normal length. The other two can be identified which the molecule bends at the central atom, in the two directions that are orthogonal to its axis. In each mode, one should assume that the atoms get displaced so that the center of mass remains stationary and there is no rotation. The molecule then has "f" = 6"n" − 5 = 13 total energy-storing degrees of freedom (3 translational, 2 rotational, 8 vibrational). At high enough temperature, its molar heat capacity then should be "c""P",m = 7.5 "R" = 62.63 J⋅K−1⋅mol−1. For cyanogen and acetylene ("n" = 4) the same analysis yields "f" = 19 and predicts "c""P",m = 10.5 "R" = 87.3 J⋅K−1⋅mol−1.
A molecule with "n" atoms that is rigid and not linear has 3 translation modes and 3 non-trivial rotation modes, hence only 3"n" − 6 deformation modes. It therefore has "f" = 3 + 3 + 2(3"n" − 6) = 6"n" − 6 energy-absorbing degrees of freedom (one less than a linear molecule with the same atom count). Water ("n" = 3) is bent in its non-strained state, therefore it is predicted to have "f" = 12 degrees of freedom. Methane ("n" = 5) is tridimensional, and the formula predicts "f" = 24.
Ethane ("n" = 8) has 4 degrees of rotational freedom: two about axes that are perpendicular to the central bond, and two more because each methyl group can rotate independently about that bond, with negligible resistance. Therefore, the number of independent deformation modes is 3"n" − 7, which gives "f" = 3 + 4 + 2(3"n" − 7) = 6n − 7 = 41.
The following table shows the experimental molar heat capacities at constant pressure "c""P",m of the above polyatomic gases at standard temperature (25 °C = 298 K), at 500 °C, and at 5000 °C, and the apparent number of degrees of freedom "f"* estimated by the formula "f"* = 2"c""P",m/"R" − 2:
(*) At 3000C
Specific heat of solids.
In most solids (but not all), the molecules have a fixed mean position and orientation, and therefore the only degrees of freedom available are the vibrations of the atoms. Thus the specific heat is proportional to the number of atoms (not molecules) per unit of mass, which is the Dulong–Petit law. Other contributions may come from magnetic degrees of freedom in solids, but these rarely make substantial contributions. and electronic
Since each atom of the solid contributes one independent vibration mode, the number of degrees of freedom in "n" atoms is 6"n". Therefore, the heat capacity of a sample of a solid substance is expected to be 3"RN"a, or (24.94 J/K)"N"a, where "N"a is the number of moles of "atoms" in the sample, not molecules. Said another way, the "atom-molar heat capacity" of a solid substance is expected to be 3"R" = 24.94 J⋅K−1⋅mol−1, where "amol" denotes an amount of the solid that contains the Avogadro number of atoms.
It follows that, in molecular solids, the heat capacity per mole "of molecules" will usually be close to 3"nR", where "n" is the number of atoms per molecule.
Thus "n" atoms of a solid should in principle store twice as much energy as "n" atoms of a monatomic gas. One way to look at this result is to observe that the monatomic gas can only store energy as kinetic energy of the atoms, whereas the solid can store it also as potential energy of the bonds strained by the vibrations. The atom-molar heat capacity of a polyatomic gas approaches that of a solid as the number "n" of atoms per molecule increases.
As in the case f gases, some of the vibration modes will be "frozen out" at low temperatures, especially in solids with light and tightly bound atoms, causing the atom-molar heat capacity to be less than this theoretical limit. Indeed, the atom-molar (or specific) heat capacity of a solid substance tends toward zero, as the temperature approaches absolute zero.
Dulong–Petit law.
As predicted by the above analysis, the heat capacity "per mole of atoms", rather than per mole of molecules, is found to be remarkably constant for all solid substances at high temperatures. This relationship was noticed empirically in 1819, and is called the Dulong–Petit law, after its two discoverers. This discovery was an important argument in support of the atomic theory of matter.
Indeed, for solid metallic chemical elements at room temperature, atom-molar heat capacities range from about 2.8 "R" to 3.4 "R". Large exceptions at the lower end involve solids composed of relatively low-mass, tightly bonded atoms, such as beryllium (2.0 "R", only of 66% of the theoretical value), and diamond (0.735 "R", only 24%). Those conditions imply larger quantum vibrational energy spacing, thus many vibrational modes are "frozen out" at room temperature. Water ice close to the melting point, too, has an anomalously low heat capacity per atom (1.5 "R", only 50% of the theoretical value).
At the higher end of possible heat capacities, heat capacity may exceed "R" by modest amounts, due to contributions from anharmonic vibrations in solids, and sometimes a modest contribution from conduction electrons in metals. These are not degrees of freedom treated in the Einstein or Debye theories.
Specific heat of solid elements.
Since the bulk density of a solid chemical element is strongly related to its molar mass, there exists a noticeable inverse correlation between a solid's density and its specific heat capacity on a per-mass basis.
This is due to a very approximate tendency of atoms of most elements to be about the same size, despite much wider variations in density and atomic weight. These two factors (constancy of atomic volume and constancy of mole-specific heat capacity) result in a good correlation between the "volume" of any given solid chemical element and its total heat capacity.
Another way of stating this, is that the volume-specific heat capacity (volumetric heat capacity) of solid elements is roughly a constant. The molar volume of solid elements is very roughly constant, and (even more reliably) so also is the molar heat capacity for most solid substances. These two factors determine the volumetric heat capacity, which as a bulk property may be striking in consistency. For example, the element uranium is a metal that has a density almost 36 times that of the metal lithium, but uranium's specific heat capacity on a volumetric basis (i.e. per given volume of metal) is only 18% larger than lithium's.
However, the average atomic volume in solid elements is not quite constant, so there are deviations from this principle. For instance, arsenic, which is only 14.5% less dense than antimony, has nearly 59% more specific heat capacity on a mass basis. In other words; even though an ingot of arsenic is only about 17% larger than an antimony one of the same mass, it absorbs about 59% more heat for a given temperature rise. The heat capacity ratios of the two substances closely follows the ratios of their molar volumes (the ratios of numbers of atoms in the same volume of each substance); the departure from the correlation to simple volumes, in this case, is due to lighter arsenic atoms being significantly more closely packed than antimony atoms, instead of similar size. In other words, similar-sized atoms would cause a mole of arsenic to be 63% larger than a mole of antimony, with a correspondingly lower density, allowing its volume to more closely mirror its heat capacity behavior.
Effect of impurities.
Sometimes small impurity concentrations can greatly affect the specific heat, for example in semiconducting ferromagnetic alloys.
Specific heat of liquids.
A general theory of the heat capacity of liquids has not yet been achieved, and is still an active area of research. It was long thought that phonon theory is not able to explain the heat capacity of liquids, since liquids only sustain longitudinal, but not transverse phonons, which in solids are responsible for 2/3 of the heat capacity. However, Brillouin scattering experiments with neutrons and with X-rays, confirming an intuition of Yakov Frenkel, have shown that transverse phonons do exist in liquids, albeit restricted to frequencies above a threshold called the Frenkel frequency. Since most energy is contained in these high-frequency modes, a simple modification of the Debye model is sufficient to yield a good approximation to experimental heat capacities of simple liquids.
Because of high crystal binding energies, the effects of vibrational mode freezing are observed in solids more often than liquids: for example the heat capacity of liquid water is twice that of ice at near the same temperature, and is again close to the 3"R" per mole of atoms of the Dulong–Petit theoretical maximum.
Amorphous materials can be considered a type of liquid at temperatures above the glass transition temperature. Below the glass transition temperature amorphous materials are in the solid (glassy) state form. The specific heat has characteristic discontinuities at the glass transition temperature which are caused by the absence in the glassy state of percolating clusters made of broken bonds (configurons) that are present only in the liquid phase. Above the glass transition temperature percolating clusters formed by broken bonds enable a more floppy structure and hence a larger degree of freedom for atomic motion which results in a higher heat capacity of liquids. Below the glass transition temperature there are no extended clusters of broken bonds and the heat capacity is smaller because the solid-state (glassy) structure of amorphous material is more rigid.
The discontinuities in the heat capacity are typically used to detect the glass transition temperature where a supercooled liquid transforms to a glass.
Effect of hydrogen bonds.
Hydrogen-containing polar molecules like ethanol, ammonia, and water have powerful, intermolecular hydrogen bonds when in their liquid phase. These bonds provide another place where heat may be stored as potential energy of vibration, even at comparatively low temperatures. Hydrogen bonds account for the fact that liquid water stores nearly the theoretical limit of 3"R" per mole of atoms, even at relatively low temperatures (i.e. near the freezing point of water).
See also.
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References.
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"text": "{} \\;=\\; \\frac{C}{n} \\;=\\; \\frac{1}{n} \\lim_{\\Delta T \\rightarrow 0}\\frac{Q}{\\Delta T}"
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| https://en.wikipedia.org/wiki?curid=1410760 |
14107961 | Antony Valentini | British theoretical physicist (born 1965)
Antony Valentini is a British-Italian theoretical physicist known for his work on the foundations of quantum physics.
Education and career.
Valentini obtained an undergraduate degree from Cambridge University, then earned his Ph.D. in 1992 with Dennis Sciama at the International School for Advanced Studies (ISAS-SISSA) in Trieste, Italy. In 1999, after seven years in Italy, he took up a post-doc grant to work at the Imperial College with Lee Smolin and Christopher Isham.
He worked at the Perimeter Institute for Theoretical Physics. In February 2011, he became professor of physics and astronomy at Clemson University; as of 2022, he was listed as adjunct faculty.
Together with Mike Towler, Royal Society research fellow of the University of Cambridge's Cavendish Laboratory, he organized a conference on the de Broglie-Bohm theory the "Apuan Alps Centre for Physics" in August 2010, hosted by the "Towler Institute" located in Vallico di Sotto in Tuscany, Italy, which is loosely associated with the Theory of Condensed Matter group of the Cavendish Laboratory. Among the questions announced for discussion, the organizers included "Why should young people be interested in these ideas, when showing interest in quantum foundations still might harm their careers?"
Work.
Valentini has been working on an extension of the causal interpretation of quantum theory. This interpretation had been proposed in conceptual terms in 1927 by Louis de Broglie, was independently re-discovered by David Bohm who brought it to a complete and systematic form in 1952, and was expanded on by Bohm and Hiley. Emphasizing de Broglie's contribution, Valentini has consistently referred to the causal interpretation of quantum mechanics underlying his work as the "de Broglie–Bohm theory".
Quantum equilibrium, locality and uncertainty.
In 1991, Valentini provided indications for deriving the quantum equilibrium hypothesis which states that formula_0 in the frame work of the pilot wave theory. Valentini showed that the relaxation formula_1 → formula_2 may be accounted for by a "H-theorem" constructed in analogy to the Boltzmann H-theorem of statistical mechanics. Valentini showed that his expansion of the De Broglie–Bohm theory would allow "signal nonlocality" for non-equilibrium cases in which formula_1 ≠formula_2. According to Valentini, the universe is fundamentally nonlocal, and quantum theory merely describes a special equilibrium state in which nonlocality is hidden in statistical noise. He furthermore showed that an ensemble of particles with "known" wave function and "known" nonequilibrium distribution could be used to perform, on another system, measurements that violate the uncertainty principle.
In 1992, Valentini extended pilot wave theory to spin-formula_3 fields and to gravitation.
Background and implications.
Valentini has been described as an "ardent admirer of de Broglie". He noted that "de Broglie (rather like Maxwell) emphasized an underlying 'mechanical' picture: particles were assumed to be singularities of physical waves in space". He emphasized that de Broglie, with the assistance of Erwin Schrödinger, had constructed pilot wave theory, but later abandoned it in favor of quantum formalism.
Valentini's derivation of the quantum equilibrium hypothesis was criticized by Detlef Dürr and co-workers in 1992, and the derivation of the quantum equilibrium hypothesis has remained a topic of active investigation.
"Signal nonlocality", which is forbidden in orthodox quantum theory, would allow nonlocal quantum entanglement to be used as a stand-alone communication channel without the need of a classical light-speed limited retarded signal to unlock the entangled message from the sender to the receiver. This would be a major revolution in physics and would possibly make the cosmic landscape string theory Popper falsifiable.
References.
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"text": "\\rho(x,y,z,t)=|\\psi(x,y,z,t)|^2"
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"text": "\\rho(x,y,z,t)"
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"text": "|\\psi(x,y,z,t)|^2"
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| https://en.wikipedia.org/wiki?curid=14107961 |
14108507 | U-quadratic distribution | In probability theory and statistics, the U-quadratic distribution is a continuous probability distribution defined by a unique convex quadratic function with lower limit "a" and upper limit "b".
formula_0
Parameter relations.
This distribution has effectively only two parameters "a", "b", as the other two are explicit functions of the support defined by the former two parameters:
formula_1
(gravitational balance center, offset), and
formula_2
(vertical scale).
Related distributions.
One can introduce a vertically inverted (formula_3)-quadratic distribution in analogous fashion. That inverted distribution is also closely related to the Epanechnikov distribution.
Applications.
This distribution is a useful model for symmetric bimodal processes. Other continuous distributions allow more flexibility, in terms of relaxing the symmetry and the quadratic shape of the density function, which are enforced in the U-quadratic distribution – e.g., beta distribution and gamma distribution.
formula_4
formula_5 | [
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"text": "f(x|a,b,\\alpha, \\beta)=\\alpha \\left ( x - \\beta \\right )^2, \\quad\\text{for } x \\in [a , b]."
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"text": "\\beta = {b+a \\over 2}"
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"text": "\\alpha = {12 \\over \\left ( b-a \\right )^3}"
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"text": "\\cap"
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"text": "M_X(t) = {-3\\left(e^{at}(4+(a^2+2a(-2+b)+b^2)t)- e^{bt} (4 + (-4b + (a+b)^2)t)\\right) \\over (a-b)^3 t^2 }"
},
{
"math_id": 5,
"text": "\\phi_X(t) = {3i\\left(e^{iate^{ibt}} (4i - (-4b + (a+b)^2)t)\\right) \\over (a-b)^3 t^2 }"
}
]
| https://en.wikipedia.org/wiki?curid=14108507 |
14110 | Holomorphic function | Complex-differentiable (mathematical) function
In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space &NoBreak;&NoBreak;. The existence of a complex derivative in a neighbourhood is a very strong condition: It implies that a holomorphic function is infinitely differentiable and locally equal to its own Taylor series (is "analytic"). Holomorphic functions are the central objects of study in complex analysis.
Though the term "analytic function" is often used interchangeably with "holomorphic function", the word "analytic" is defined in a broader sense to denote any function (real, complex, or of more general type) that can be written as a convergent power series in a neighbourhood of each point in its domain. That all holomorphic functions are complex analytic functions, and vice versa, is a major theorem in complex analysis.
Holomorphic functions are also sometimes referred to as "regular functions". A holomorphic function whose domain is the whole complex plane is called an entire function. The phrase "holomorphic at a point &NoBreak;&NoBreak;" means not just differentiable at &NoBreak;&NoBreak;, but differentiable everywhere within some close neighbourhood of &NoBreak;&NoBreak; in the complex plane.
Definition.
Given a complex-valued function &NoBreak;&NoBreak; of a single complex variable, the derivative of &NoBreak;&NoBreak; at a point &NoBreak;&NoBreak; in its domain is defined as the limit
formula_0
This is the same definition as for the derivative of a real function, except that all quantities are complex. In particular, the limit is taken as the complex number &NoBreak;&NoBreak; tends to &NoBreak;&NoBreak;, and this means that the same value is obtained for any sequence of complex values for &NoBreak;&NoBreak; that tends to &NoBreak;&NoBreak;. If the limit exists, &NoBreak;&NoBreak; is said to be complex differentiable at &NoBreak;&NoBreak;. This concept of complex differentiability shares several properties with real differentiability: It is linear and obeys the product rule, quotient rule, and chain rule.
A function is holomorphic on an open set &NoBreak;&NoBreak; if it is "complex differentiable" at "every" point of &NoBreak;&NoBreak;. A function &NoBreak;&NoBreak; is "holomorphic" at a point &NoBreak;&NoBreak; if it is holomorphic on some neighbourhood of &NoBreak;&NoBreak;.
A function is "holomorphic" on some non-open set &NoBreak;&NoBreak; if it is holomorphic at every point of &NoBreak;&NoBreak;.
A function may be complex differentiable at a point but not holomorphic at this point. For example, the function formula_1 "is" complex differentiable at &NoBreak;&NoBreak;, but "is not" complex differentiable anywhere else, esp. including in no place close to &NoBreak;&NoBreak; (see the Cauchy–Riemann equations, below). So, it is "not" holomorphic at &NoBreak;&NoBreak;.
The relationship between real differentiability and complex differentiability is the following: If a complex function &NoBreak;&NoBreak; is holomorphic, then &NoBreak;&NoBreak; and &NoBreak;&NoBreak; have first partial derivatives with respect to &NoBreak;&NoBreak; and &NoBreak;&NoBreak;, and satisfy the Cauchy–Riemann equations:
formula_2
or, equivalently, the Wirtinger derivative of &NoBreak;&NoBreak; with respect to &NoBreak;&NoBreak;, the complex conjugate of &NoBreak;&NoBreak;, is zero:
formula_3
which is to say that, roughly, &NoBreak;&NoBreak; is functionally independent from &NoBreak;&NoBreak;, the complex conjugate of &NoBreak;&NoBreak;.
If continuity is not given, the converse is not necessarily true. A simple converse is that if &NoBreak;&NoBreak; and &NoBreak;&NoBreak; have "continuous" first partial derivatives and satisfy the Cauchy–Riemann equations, then &NoBreak;&NoBreak; is holomorphic. A more satisfying converse, which is much harder to prove, is the Looman–Menchoff theorem: if &NoBreak;&NoBreak; is continuous, &NoBreak;&NoBreak; and &NoBreak;&NoBreak; have first partial derivatives (but not necessarily continuous), and they satisfy the Cauchy–Riemann equations, then &NoBreak;&NoBreak; is holomorphic.
Terminology.
The term "holomorphic" was introduced in 1875 by Charles Briot and Jean-Claude Bouquet, two of Cauchy's students, and derives from the Greek ὅλος ("hólos") meaning "whole", and μορφή ("morphḗ") meaning "form" or "appearance" or "type", in contrast to the term "meromorphic" derived from μέρος ("méros") meaning "part". A holomorphic function resembles an entire function ("whole") in a domain of the complex plane while a meromorphic function (defined to mean holomorphic except at certain isolated poles), resembles a rational fraction ("part") of entire functions in a domain of the complex plane.
Cauchy had instead used the term "synectic".
Today, the term "holomorphic function" is sometimes preferred to "analytic function". An important result in complex analysis is that every holomorphic function is complex analytic, a fact that does not follow obviously from the definitions. The term "analytic" is however also in wide use.
Properties.
Because complex differentiation is linear and obeys the product, quotient, and chain rules, the sums, products and compositions of holomorphic functions are holomorphic, and the quotient of two holomorphic functions is holomorphic wherever the denominator is not zero. That is, if functions &NoBreak;&NoBreak; and &NoBreak;&NoBreak; are holomorphic in a domain &NoBreak;&NoBreak;, then so are &NoBreak;&NoBreak;, &NoBreak;&NoBreak;, &NoBreak;&NoBreak;, and &NoBreak;&NoBreak;. Furthermore, &NoBreak;&NoBreak; is holomorphic if &NoBreak;&NoBreak; has no zeros in &NoBreak;&NoBreak;; otherwise it is meromorphic.
If one identifies &NoBreak;&NoBreak; with the real plane &NoBreak;&NoBreak;, then the holomorphic functions coincide with those functions of two real variables with continuous first derivatives which solve the Cauchy–Riemann equations, a set of two partial differential equations.
Every holomorphic function can be separated into its real and imaginary parts &NoBreak;&NoBreak;, and each of these is a harmonic function on &NoBreak;&NoBreak; (each satisfies Laplace's equation &NoBreak;&NoBreak;), with &NoBreak;&NoBreak; the harmonic conjugate of &NoBreak;&NoBreak;.
Conversely, every harmonic function &NoBreak;&NoBreak; on a simply connected domain &NoBreak;&NoBreak; is the real part of a holomorphic function: If &NoBreak;&NoBreak; is the harmonic conjugate of &NoBreak;&NoBreak;, unique up to a constant, then &NoBreak;&NoBreak; is holomorphic.
Cauchy's integral theorem implies that the contour integral of every holomorphic function along a loop vanishes:
formula_4
Here &NoBreak;&NoBreak; is a rectifiable path in a simply connected complex domain &NoBreak;&NoBreak; whose start point is equal to its end point, and &NoBreak;&NoBreak; is a holomorphic function.
Cauchy's integral formula states that every function holomorphic inside a disk is completely determined by its values on the disk's boundary. Furthermore: Suppose &NoBreak;&NoBreak; is a complex domain, &NoBreak;&NoBreak; is a holomorphic function and the closed disk &NoBreak;&NoBreak; is completely contained in &NoBreak;&NoBreak;. Let &NoBreak;&NoBreak; be the circle forming the boundary of &NoBreak;&NoBreak;. Then for every &NoBreak;&NoBreak; in the interior of &NoBreak;&NoBreak;:
formula_5
where the contour integral is taken counter-clockwise.
The derivative &NoBreak;&NoBreak; can be written as a contour integral using Cauchy's differentiation formula:
formula_6
for any simple loop positively winding once around &NoBreak;&NoBreak;, and
formula_7
for infinitesimal positive loops &NoBreak;&NoBreak; around &NoBreak;&NoBreak;.
In regions where the first derivative is not zero, holomorphic functions are conformal: they preserve angles and the shape (but not size) of small figures.
Every holomorphic function is analytic. That is, a holomorphic function &NoBreak;&NoBreak; has derivatives of every order at each point &NoBreak;&NoBreak; in its domain, and it coincides with its own Taylor series at &NoBreak;&NoBreak; in a neighbourhood of &NoBreak;&NoBreak;. In fact, &NoBreak;&NoBreak; coincides with its Taylor series at &NoBreak;&NoBreak; in any disk centred at that point and lying within the domain of the function.
From an algebraic point of view, the set of holomorphic functions on an open set is a commutative ring and a complex vector space. Additionally, the set of holomorphic functions in an open set &NoBreak;&NoBreak; is an integral domain if and only if the open set &NoBreak;&NoBreak; is connected. In fact, it is a locally convex topological vector space, with the seminorms being the suprema on compact subsets.
From a geometric perspective, a function &NoBreak;&NoBreak; is holomorphic at &NoBreak;&NoBreak; if and only if its exterior derivative &NoBreak;&NoBreak; in a neighbourhood &NoBreak;&NoBreak; of &NoBreak;&NoBreak; is equal to &NoBreak;&NoBreak; for some continuous function &NoBreak;&NoBreak;. It follows from
formula_8
that &NoBreak;&NoBreak; is also proportional to &NoBreak;&NoBreak;, implying that the derivative &NoBreak;&NoBreak; is itself holomorphic and thus that &NoBreak;&NoBreak; is infinitely differentiable. Similarly, &NoBreak;&NoBreak; implies that any function &NoBreak;&NoBreak; that is holomorphic on the simply connected region &NoBreak;&NoBreak; is also integrable on &NoBreak;&NoBreak;.
(For a path &NoBreak;&NoBreak; from &NoBreak;&NoBreak; to &NoBreak;&NoBreak; lying entirely in &NoBreak;&NoBreak;, define &NoBreak;&NoBreak;; in light of the Jordan curve theorem and the generalized Stokes' theorem, &NoBreak;&NoBreak; is independent of the particular choice of path &NoBreak;&NoBreak;, and thus &NoBreak;&NoBreak; is a well-defined function on &NoBreak;&NoBreak; having &NoBreak;&NoBreak; or &NoBreak;}&NoBreak;.
Examples.
All polynomial functions in &NoBreak;&NoBreak; with complex coefficients are entire functions (holomorphic in the whole complex plane &NoBreak;&NoBreak;), and so are the exponential function &NoBreak;&NoBreak; and the trigonometric functions &NoBreak;&NoBreak; and &NoBreak;&NoBreak; (cf. Euler's formula). The principal branch of the complex logarithm function &NoBreak;&NoBreak; is holomorphic on the domain &NoBreak;}&NoBreak;. The square root function can be defined as &NoBreak;&NoBreak; and is therefore holomorphic wherever the logarithm &NoBreak;&NoBreak; is. The reciprocal function &NoBreak;}&NoBreak; is holomorphic on &NoBreak;}&NoBreak;. (The reciprocal function, and any other rational function, is meromorphic on &NoBreak;&NoBreak;.)
As a consequence of the Cauchy–Riemann equations, any real-valued holomorphic function must be constant. Therefore, the absolute value formula_9, the argument &NoBreak;&NoBreak;, the real part &NoBreak;&NoBreak; and the imaginary part &NoBreak;&NoBreak; are not holomorphic. Another typical example of a continuous function which is not holomorphic is the complex conjugate &NoBreak;&NoBreak; (The complex conjugate is antiholomorphic.)
Several variables.
The definition of a holomorphic function generalizes to several complex variables in a straightforward way. A function &NoBreak;&NoBreak; in &NoBreak;&NoBreak; complex variables is analytic at a point &NoBreak;&NoBreak; if there exists a neighbourhood of &NoBreak;&NoBreak; in which &NoBreak;&NoBreak; is equal to a convergent power series in &NoBreak;&NoBreak; complex variables;
the function &NoBreak;&NoBreak; is holomorphic in an open subset &NoBreak;&NoBreak; of &NoBreak;&NoBreak; if it is analytic at each point in &NoBreak;&NoBreak;. Osgood's lemma shows (using the multivariate Cauchy integral formula) that, for a continuous function &NoBreak;&NoBreak;, this is equivalent to &NoBreak;&NoBreak; being holomorphic in each variable separately (meaning that if any &NoBreak;&NoBreak; coordinates are fixed, then the restriction of &NoBreak;&NoBreak; is a holomorphic function of the remaining coordinate). The much deeper Hartogs' theorem proves that the continuity assumption is unnecessary: &NoBreak;&NoBreak; is holomorphic if and only if it is holomorphic in each variable separately.
More generally, a function of several complex variables that is square integrable over every compact subset of its domain is analytic if and only if it satisfies the Cauchy–Riemann equations in the sense of distributions.
Functions of several complex variables are in some basic ways more complicated than functions of a single complex variable. For example, the region of convergence of a power series is not necessarily an open ball; these regions are logarithmically-convex Reinhardt domains, the simplest example of which is a polydisk. However, they also come with some fundamental restrictions. Unlike functions of a single complex variable, the possible domains on which there are holomorphic functions that cannot be extended to larger domains are highly limited. Such a set is called a domain of holomorphy.
A complex differential &NoBreak;&NoBreak;-form &NoBreak;&NoBreak; is holomorphic if and only if its antiholomorphic Dolbeault derivative is zero: &NoBreak;&NoBreak;.
Extension to functional analysis.
The concept of a holomorphic function can be extended to the infinite-dimensional spaces of functional analysis. For instance, the Fréchet or Gateaux derivative can be used to define a notion of a holomorphic function on a Banach space over the field of complex numbers.
See also.
<templatestyles src="Div col/styles.css"/>
Footnotes.
<templatestyles src="Reflist/styles.css" />
References.
<templatestyles src="Reflist/styles.css" /> | [
{
"math_id": 0,
"text": "f'(z_0) = \\lim_{z \\to z_0} \\frac{f(z) - f(z_0)}{ z - z_0 }."
},
{
"math_id": 1,
"text": "\\textstyle f(z) = |z|\\vphantom{l}^2 = z\\bar{z}"
},
{
"math_id": 2,
"text": "\\frac{\\partial u}{\\partial x} = \\frac{\\partial v}{\\partial y} \\qquad \\mbox{and} \\qquad \\frac{\\partial u}{\\partial y} = -\\frac{\\partial v}{\\partial x}\\,"
},
{
"math_id": 3,
"text": "\\frac{\\partial f}{\\partial\\bar{z}} = 0,"
},
{
"math_id": 4,
"text": "\\oint_\\gamma f(z)\\,\\mathrm{d}z = 0."
},
{
"math_id": 5,
"text": "f(a) = \\frac{ 1 }{2\\pi i} \\oint_\\gamma \\frac{f(z)}{z-a}\\,\\mathrm{d}z"
},
{
"math_id": 6,
"text": " f'\\!(a) = \\frac{ 1 }{2\\pi i} \\oint_\\gamma \\frac{f(z)}{(z-a)^2}\\,\\mathrm{d}z,"
},
{
"math_id": 7,
"text": " f'\\!(a) = \\lim\\limits_{\\gamma\\to a} \\frac{ i }{2\\mathcal{A}(\\gamma)} \\oint_{\\gamma}f(z)\\,\\mathrm{d}\\bar{z},"
},
{
"math_id": 8,
"text": "0 = \\mathrm{d}^2 f = \\mathrm{d}(f'\\,\\mathrm{d}z) = \\mathrm{d}f' \\wedge \\mathrm{d}z"
},
{
"math_id": 9,
"text": "|z|"
}
]
| https://en.wikipedia.org/wiki?curid=14110 |
1411087 | Weak derivative | Generalisation of the derivative of a function
In mathematics, a weak derivative is a generalization of the concept of the derivative of a function ("strong derivative") for functions not assumed differentiable, but only integrable, i.e., to lie in the L"p" space formula_0.
The method of integration by parts holds that for differentiable functions formula_1 and formula_2 we have
formula_3
A function "u"' being the weak derivative of "u" is essentially defined by the requirement that this equation must hold for all infinitely differentiable functions formula_2 vanishing at the boundary points (formula_4).
Definition.
Let formula_1 be a function in the Lebesgue space formula_0. We say that formula_5 in formula_0 is a weak derivative of formula_1 if
formula_6
for "all" infinitely differentiable functions formula_7 with formula_4.
Generalizing to formula_8 dimensions, if formula_1 and formula_5 are in the space formula_9 of locally integrable functions for some open set formula_10, and if formula_11 is a multi-index, we say that formula_5 is the formula_12-weak derivative of formula_1 if
formula_13
for all formula_14, that is, for all infinitely differentiable functions formula_2 with compact support in formula_15. Here formula_16 is defined as
formula_17
If formula_1 has a weak derivative, it is often written formula_18 since weak derivatives are unique (at least, up to a set of measure zero, see below).
Properties.
If two functions are weak derivatives of the same function, they are equal except on a set with Lebesgue measure zero, i.e., they are equal almost everywhere. If we consider equivalence classes of functions such that two functions are equivalent if they are equal almost everywhere, then the weak derivative is unique.
Also, if "u" is differentiable in the conventional sense then its weak derivative is identical (in the sense given above) to its conventional (strong) derivative. Thus the weak derivative is a generalization of the strong one. Furthermore, the classical rules for derivatives of sums and products of functions also hold for the weak derivative.
Extensions.
This concept gives rise to the definition of weak solutions in Sobolev spaces, which are useful for problems of differential equations and in functional analysis. | [
{
"math_id": 0,
"text": "L^1([a,b])"
},
{
"math_id": 1,
"text": "u"
},
{
"math_id": 2,
"text": "\\varphi"
},
{
"math_id": 3,
"text": "\\begin{align}\n \\int_a^b u(x) \\varphi'(x) \\, dx \n & = \\Big[u(x) \\varphi(x)\\Big]_a^b - \\int_a^b u'(x) \\varphi(x) \\, dx. \\\\[6pt]\n \\end{align}"
},
{
"math_id": 4,
"text": "\\varphi(a)=\\varphi(b)=0"
},
{
"math_id": 5,
"text": "v"
},
{
"math_id": 6,
"text": "\\int_a^b u(t)\\varphi'(t) \\, dt=-\\int_a^b v(t)\\varphi(t) \\, dt"
},
{
"math_id": 7,
"text": " \\varphi "
},
{
"math_id": 8,
"text": "n"
},
{
"math_id": 9,
"text": "L_\\text{loc}^1(U)"
},
{
"math_id": 10,
"text": "U \\subset \\mathbb{R}^n"
},
{
"math_id": 11,
"text": "\\alpha"
},
{
"math_id": 12,
"text": "\\alpha^\\text{th}"
},
{
"math_id": 13,
"text": "\\int_U u D^\\alpha \\varphi=(-1)^{|\\alpha|} \\int_U v\\varphi,"
},
{
"math_id": 14,
"text": "\\varphi \\in C^\\infty_c (U)"
},
{
"math_id": 15,
"text": "U"
},
{
"math_id": 16,
"text": " D^{\\alpha}\\varphi"
},
{
"math_id": 17,
"text": " D^{\\alpha}\\varphi = \\frac{\\partial^{| \\alpha |} \\varphi }{\\partial x_1^{\\alpha_1} \\cdots \\partial x_n^{\\alpha_n}}."
},
{
"math_id": 18,
"text": "D^{\\alpha}u"
},
{
"math_id": 19,
"text": "u : \\mathbb{R} \\rightarrow \\mathbb{R}_+, u(t) = |t|"
},
{
"math_id": 20,
"text": "t = 0"
},
{
"math_id": 21,
"text": "v: \\mathbb{R} \\rightarrow \\mathbb{R}"
},
{
"math_id": 22,
"text": "\nv(t) = \\begin{cases} 1 & \\text{if } t > 0; \\\\[6pt] 0 & \\text{if } t = 0; \\\\[6pt] -1 & \\text{if } t < 0. \\end{cases}"
},
{
"math_id": 23,
"text": " 1_{\\mathbb{Q}} "
},
{
"math_id": 24,
"text": " \\int 1_{\\mathbb{Q}}(t) \\varphi(t) \\, dt = 0."
},
{
"math_id": 25,
"text": " v(t)=0 "
}
]
| https://en.wikipedia.org/wiki?curid=1411087 |
1411100 | Introduction to general relativity | Theory of gravity by Albert Einstein
General relativity is a theory of gravitation developed by Albert Einstein between 1907 and 1915. The theory of general relativity says that the observed gravitational effect between masses results from their warping of spacetime.
By the beginning of the 20th century, Newton's law of universal gravitation had been accepted for more than two hundred years as a valid description of the gravitational force between masses. In Newton's model, gravity is the result of an attractive force between massive objects. Although even Newton was troubled by the unknown nature of that force, the basic framework was extremely successful at describing motion.
Experiments and observations show that Einstein's description of gravitation accounts for several effects that are unexplained by Newton's law, such as minute anomalies in the orbits of Mercury and other planets. General relativity also predicts novel effects of gravity, such as gravitational waves, gravitational lensing and an effect of gravity on time known as gravitational time dilation. Many of these predictions have been confirmed by experiment or observation, most recently gravitational waves.
General relativity has developed into an essential tool in modern astrophysics. It provides the foundation for the current understanding of black holes, regions of space where the gravitational effect is strong enough that even light cannot escape. Their strong gravity is thought to be responsible for the intense radiation emitted by certain types of astronomical objects (such as active galactic nuclei or microquasars). General relativity is also part of the framework of the standard Big Bang model of cosmology.
Although general relativity is not the only relativistic theory of gravity, it is the simplest one that is consistent with the experimental data. Nevertheless, a number of open questions remain, the most fundamental of which is how general relativity can be reconciled with the laws of quantum physics to produce a complete and self-consistent theory of quantum gravity.
From special to general relativity.
In September 1905, Albert Einstein published his theory of special relativity, which reconciles Newton's laws of motion with electrodynamics (the interaction between objects with electric charge). Special relativity introduced a new framework for all of physics by proposing new concepts of space and time. Some then-accepted physical theories were inconsistent with that framework; a key example was Newton's theory of gravity, which describes the mutual attraction experienced by bodies due to their mass.
Several physicists, including Einstein, searched for a theory that would reconcile Newton's law of gravity and special relativity. Only Einstein's theory proved to be consistent with experiments and observations. To understand the theory's basic ideas, it is instructive to follow Einstein's thinking between 1907 and 1915, from his simple thought experiment involving an observer in free fall to his fully geometric theory of gravity.
Equivalence principle.
A person in a free-falling elevator experiences weightlessness; objects either float motionless or drift at constant speed. Since everything in the elevator is falling together, no gravitational effect can be observed. In this way, the experiences of an observer in free fall are indistinguishable from those of an observer in deep space, far from any significant source of gravity. Such observers are the privileged ("inertial") observers Einstein described in his theory of special relativity: observers for whom light travels along straight lines at constant speed.
Einstein hypothesized that the similar experiences of weightless observers and inertial observers in special relativity represented a fundamental property of gravity, and he made this the cornerstone of his theory of general relativity, formalized in his equivalence principle. Roughly speaking, the principle states that a person in a free-falling elevator cannot tell that they are in free fall. Every experiment in such a free-falling environment has the same results as it would for an observer at rest or moving uniformly in deep space, far from all sources of gravity.
Gravity and acceleration.
Most effects of gravity vanish in free fall, but effects that seem the same as those of gravity can be "produced" by an accelerated frame of reference. An observer in a closed room cannot tell which of the following two scenarios is true:
Conversely, any effect observed in an accelerated reference frame should also be observed in a gravitational field of corresponding strength. This principle allowed Einstein to predict several novel effects of gravity in 1907, as explained in the next section.
An observer in an accelerated reference frame must introduce what physicists call fictitious forces to account for the acceleration experienced by the observer and objects around them. In the example of the driver being pressed into their seat, the force felt by the driver is one example; another is the force one can feel while pulling the arms up and out if attempting to spin around like a top. Einstein's master insight was that the constant, familiar pull of the Earth's gravitational field is fundamentally the same as these fictitious forces. The apparent magnitude of the fictitious forces always appears to be proportional to the mass of any object on which they act – for instance, the driver's seat exerts just enough force to accelerate the driver at the same rate as the car. By analogy, Einstein proposed that an object in a gravitational field should feel a gravitational force proportional to its mass, as embodied in Newton's law of gravitation.
Physical consequences.
In 1907, Einstein was still eight years away from completing the general theory of relativity. Nonetheless, he was able to make a number of novel, testable predictions that were based on his starting point for developing his new theory: the equivalence principle.
The first new effect is the gravitational frequency shift of light. Consider two observers aboard an accelerating rocket-ship. Aboard such a ship, there is a natural concept of "up" and "down": the direction in which the ship accelerates is "up", and free-floating objects accelerate in the opposite direction, falling "downward". Assume that one of the observers is "higher up" than the other. When the lower observer sends a light signal to the higher observer, the acceleration of the ship causes the light to be red-shifted, as may be calculated from special relativity; the second observer will measure a lower frequency for the light than the first sent out. Conversely, light sent from the higher observer to the lower is blue-shifted, that is, shifted towards higher frequencies. Einstein argued that such frequency shifts must also be observed in a gravitational field. This is illustrated in the figure at left, which shows a light wave that is gradually red-shifted as it works its way upwards against the gravitational acceleration. This effect has been confirmed experimentally, as described below.
This gravitational frequency shift corresponds to a gravitational time dilation: Since the "higher" observer measures the same light wave to have a lower frequency than the "lower" observer, time must be passing faster for the higher observer. Thus, time runs more slowly for observers the lower they are in a gravitational field.
It is important to stress that, for each observer, there are no observable changes of the flow of time for events or processes that are at rest in his or her reference frame. Five-minute-eggs as timed by each observer's clock have the same consistency; as one year passes on each clock, each observer ages by that amount; each clock, in short, is in perfect agreement with all processes happening in its immediate vicinity. It is only when the clocks are compared between separate observers that one can notice that time runs more slowly for the lower observer than for the higher. This effect is minute, but it too has been confirmed experimentally in multiple experiments, as described below.
In a similar way, Einstein predicted the gravitational deflection of light: in a gravitational field, light is deflected downward, to the center of the gravitational field. Quantitatively, his results were off by a factor of two; the correct derivation requires a more complete formulation of the theory of general relativity, not just the equivalence principle.
Tidal effects.
The equivalence between gravitational and inertial effects does not constitute a complete theory of gravity. When it comes to explaining gravity near our own location on the Earth's surface, noting that our reference frame is not in free fall, so that fictitious forces are to be expected, provides a suitable explanation. But a freely falling reference frame on one side of the Earth cannot explain why the people on the opposite side of the Earth experience a gravitational pull in the opposite direction.
A more basic manifestation of the same effect involves two bodies that are falling side by side towards the Earth, with a similar position and velocity. In a reference frame that is in free fall alongside these bodies, they appear to hover weightlessly – but not exactly so. These bodies are not falling in precisely the same direction, but towards a single point in space: namely, the Earth's center of gravity. Consequently, there is a component of each body's motion towards the other (see the figure). In a small environment such as a freely falling lift, this relative acceleration is minuscule, while for skydivers on opposite sides of the Earth, the effect is large. Such differences in force are also responsible for the tides in the Earth's oceans, so the term "tidal effect" is used for this phenomenon.
The equivalence between inertia and gravity cannot explain tidal effects – it cannot explain variations in the gravitational field. For that, a theory is needed which describes the way that matter (such as the large mass of the Earth) affects the inertial environment around it.
From acceleration to geometry.
While Einstein was exploring the equivalence of gravity and acceleration as well as the role of tidal forces, he discovered several analogies with the geometry of surfaces. An example is the transition from an inertial reference frame (in which free particles coast along straight paths at constant speeds) to a rotating reference frame (in which fictitious forces have to be introduced in order to explain particle motion): this is analogous to the transition from a Cartesian coordinate system (in which the coordinate lines are straight lines) to a curved coordinate system (where coordinate lines need not be straight).
A deeper analogy relates tidal forces with a property of surfaces called "curvature". For gravitational fields, the absence or presence of tidal forces determines whether or not the influence of gravity can be eliminated by choosing a freely falling reference frame. Similarly, the absence or presence of curvature determines whether or not a surface is equivalent to a plane. In the summer of 1912, inspired by these analogies, Einstein searched for a geometric formulation of gravity.
The elementary objects of geometry – points, lines, triangles – are traditionally defined in three-dimensional space or on two-dimensional surfaces. In 1907, Hermann Minkowski, Einstein's former mathematics professor at the Swiss Federal Polytechnic, introduced Minkowski space, a geometric formulation of Einstein's special theory of relativity where the geometry included not only space but also time. The basic entity of this new geometry is four-dimensional spacetime. The orbits of moving bodies are curves in spacetime; the orbits of bodies moving at constant speed without changing direction correspond to straight lines.
The geometry of general curved surfaces was developed in the early 19th century by Carl Friedrich Gauss. This geometry had in turn been generalized to higher-dimensional spaces in Riemannian geometry introduced by Bernhard Riemann in the 1850s. With the help of Riemannian geometry, Einstein formulated a geometric description of gravity in which Minkowski's spacetime is replaced by distorted, curved spacetime, just as curved surfaces are a generalization of ordinary plane surfaces. Embedding Diagrams are used to illustrate curved spacetime in educational contexts.
After he had realized the validity of this geometric analogy, it took Einstein a further three years to find the missing cornerstone of his theory: the equations describing how matter influences spacetime's curvature. Having formulated what are now known as Einstein's equations (or, more precisely, his field equations of gravity), he presented his new theory of gravity at several sessions of the Prussian Academy of Sciences in late 1915, culminating in his final presentation on November 25, 1915.
Geometry and gravitation.
Paraphrasing John Wheeler, Einstein's geometric theory of gravity can be summarized as: "spacetime tells matter how to move; matter tells spacetime how to curve". What this means is addressed in the following three sections, which explore the motion of so-called test particles, examine which properties of matter serve as a source for gravity, and, finally, introduce Einstein's equations, which relate these matter properties to the curvature of spacetime.
Probing the gravitational field.
In order to map a body's gravitational influence, it is useful to think about what physicists call probe or test particles: particles that are influenced by gravity, but are so small and light that we can neglect their own gravitational effect. In the absence of gravity and other external forces, a test particle moves along a straight line at a constant speed. In the language of spacetime, this is equivalent to saying that such test particles move along straight world lines in spacetime. In the presence of gravity, spacetime is non-Euclidean, or curved, and in curved spacetime straight world lines may not exist. Instead, test particles move along lines called geodesics, which are "as straight as possible", that is, they follow the shortest path between starting and ending points, taking the curvature into consideration.
A simple analogy is the following: In geodesy, the science of measuring Earth's size and shape, a geodesic is the shortest route between two points on the Earth's surface. Approximately, such a route is a segment of a great circle, such as a line of longitude or the equator. These paths are certainly not straight, simply because they must follow the curvature of the Earth's surface. But they are as straight as is possible subject to this constraint.
The properties of geodesics differ from those of straight lines. For example, on a plane, parallel lines never meet, but this is not so for geodesics on the surface of the Earth: for example, lines of longitude are parallel at the equator, but intersect at the poles. Analogously, the world lines of test particles in free fall are spacetime geodesics, the straightest possible lines in spacetime. But still there are crucial differences between them and the truly straight lines that can be traced out in the gravity-free spacetime of special relativity. In special relativity, parallel geodesics remain parallel. In a gravitational field with tidal effects, this will not, in general, be the case. If, for example, two bodies are initially at rest relative to each other, but are then dropped in the Earth's gravitational field, they will move towards each other as they fall towards the Earth's center.
Compared with planets and other astronomical bodies, the objects of everyday life (people, cars, houses, even mountains) have little mass. Where such objects are concerned, the laws governing the behavior of test particles are sufficient to describe what happens. Notably, in order to deflect a test particle from its geodesic path, an external force must be applied. A chair someone is sitting on applies an external upwards force preventing the person from falling freely towards the center of the Earth and thus following a geodesic, which they would otherwise be doing without the chair there, or any other matter in between them and the center point of the Earth. In this way, general relativity explains the daily experience of gravity on the surface of the Earth "not" as the downwards pull of a gravitational force, but as the upwards push of external forces. These forces deflect all bodies resting on the Earth's surface from the geodesics they would otherwise follow. For objects massive enough that their own gravitational influence cannot be neglected, the laws of motion are somewhat more complicated than for test particles, although it remains true that spacetime tells matter how to move.
Sources of gravity.
In Newton's description of gravity, the gravitational force is caused by matter. More precisely, it is caused by a specific property of material objects: their mass. In Einstein's theory and related theories of gravitation, curvature at every point in spacetime is also caused by whatever matter is present. Here, too, mass is a key property in determining the gravitational influence of matter. But in a relativistic theory of gravity, mass cannot be the only source of gravity. Relativity links mass with energy, and energy with momentum.
The equivalence between mass and energy, as expressed by the formula "E" = "mc", is the most famous consequence of special relativity. In relativity, mass and energy are two different ways of describing one physical quantity. If a physical system has energy, it also has the corresponding mass, and vice versa. In particular, all properties of a body that are associated with energy, such as its temperature or the binding energy of systems such as nuclei or molecules, contribute to that body's mass, and hence act as sources of gravity.
In special relativity, energy is closely connected to momentum. In special relativity, just as space and time are different aspects of a more comprehensive entity called spacetime, energy and momentum are merely different aspects of a unified, four-dimensional quantity that physicists call four-momentum. In consequence, if energy is a source of gravity, momentum must be a source as well. The same is true for quantities that are directly related to energy and momentum, namely internal pressure and tension. Taken together, in general relativity it is mass, energy, momentum, pressure and tension that serve as sources of gravity: they are how matter tells spacetime how to curve. In the theory's mathematical formulation, all these quantities are but aspects of a more general physical quantity called the energy–momentum tensor.
Einstein's equations.
Einstein's equations are the centerpiece of general relativity. They provide a precise formulation of the relationship between spacetime geometry and the properties of matter, using the language of mathematics. More concretely, they are formulated using the concepts of Riemannian geometry, in which the geometric properties of a space (or a spacetime) are described by a quantity called a metric. The metric encodes the information needed to compute the fundamental geometric notions of distance and angle in a curved space (or spacetime).
A spherical surface like that of the Earth provides a simple example. The location of any point on the surface can be described by two coordinates: the geographic latitude and longitude. Unlike the Cartesian coordinates of the plane, coordinate differences are not the same as distances on the surface, as shown in the diagram on the right: for someone at the equator, moving 30 degrees of longitude westward (magenta line) corresponds to a distance of roughly , while for someone at a latitude of 55 degrees, moving 30 degrees of longitude westward (blue line) covers a distance of merely . Coordinates therefore do not provide enough information to describe the geometry of a spherical surface, or indeed the geometry of any more complicated space or spacetime. That information is precisely what is encoded in the metric, which is a function defined at each point of the surface (or space, or spacetime) and relates coordinate differences to differences in distance. All other quantities that are of interest in geometry, such as the length of any given curve, or the angle at which two curves meet, can be computed from this metric function.
The metric function and its rate of change from point to point can be used to define a geometrical quantity called the Riemann curvature tensor, which describes exactly how the Riemannian manifold, the spacetime in the theory of relativity, is curved at each point. As has already been mentioned, the matter content of the spacetime defines another quantity, the energy–momentum tensor T, and the principle that "spacetime tells matter how to move, and matter tells spacetime how to curve" means that these quantities must be related to each other. Einstein formulated this relation by using the Riemann curvature tensor and the metric to define another geometrical quantity G, now called the Einstein tensor, which describes some aspects of the way spacetime is curved. "Einstein's equation" then states that
formula_0
i.e., up to a constant multiple, the quantity G (which measures curvature) is equated with the quantity T (which measures matter content). Here, "G" is the gravitational constant of Newtonian gravity, and "c" is the speed of light from special relativity.
This equation is often referred to in the plural as "Einstein's equations", since the quantities G and T are each determined by several functions of the coordinates of spacetime, and the equations equate each of these component functions. A solution of these equations describes a particular geometry of spacetime; for example, the Schwarzschild solution describes the geometry around a spherical, non-rotating mass such as a star or a black hole, whereas the Kerr solution describes a rotating black hole. Still other solutions can describe a gravitational wave or, in the case of the Friedmann–Lemaître–Robertson–Walker solution, an expanding universe. The simplest solution is the uncurved Minkowski spacetime, the spacetime described by special relativity.
Experiments.
No scientific theory is self-evidently true; each is a model that must be checked by experiment. Newton's law of gravity was accepted because it accounted for the motion of planets and moons in the Solar System with considerable accuracy. As the precision of experimental measurements gradually improved, some discrepancies with Newton's predictions were observed, and these were accounted for in the general theory of relativity. Similarly, the predictions of general relativity must also be checked with experiment, and Einstein himself devised three tests now known as the classical tests of the theory:
Of these tests, only the perihelion advance of Mercury was known prior to Einstein's final publication of general relativity in 1916. The subsequent experimental confirmation of his other predictions, especially the first measurements of the deflection of light by the sun in 1919, catapulted Einstein to international stardom. These three experiments justified adopting general relativity over Newton's theory and, incidentally, over a number of alternatives to general relativity that had been proposed.
Further tests of general relativity include precision measurements of the Shapiro effect or gravitational time delay for light, measured in 2002 by the Cassini space probe. One set of tests focuses on effects predicted by general relativity for the behavior of gyroscopes travelling through space. One of these effects, geodetic precession, has been tested with the Lunar Laser Ranging Experiment (high-precision measurements of the orbit of the Moon). Another, which is related to rotating masses, is called frame-dragging. The geodetic and frame-dragging effects were both tested by the Gravity Probe B satellite experiment launched in 2004, with results confirming relativity to within 0.5% and 15%, respectively, as of December 2008.
By cosmic standards, gravity throughout the solar system is weak. Since the differences between the predictions of Einstein's and Newton's theories are most pronounced when gravity is strong, physicists have long been interested in testing various relativistic effects in a setting with comparatively strong gravitational fields. This has become possible thanks to precision observations of binary pulsars. In such a star system, two highly compact neutron stars orbit each other. At least one of them is a pulsar – an astronomical object that emits a tight beam of radiowaves. These beams strike the Earth at very regular intervals, similarly to the way that the rotating beam of a lighthouse means that an observer sees the lighthouse blink, and can be observed as a highly regular series of pulses. General relativity predicts specific deviations from the regularity of these radio pulses. For instance, at times when the radio waves pass close to the other neutron star, they should be deflected by the star's gravitational field. The observed pulse patterns are impressively close to those predicted by general relativity.
One particular set of observations is related to eminently useful practical applications, namely to satellite navigation systems such as the Global Positioning System that are used for both precise positioning and timekeeping. Such systems rely on two sets of atomic clocks: clocks aboard satellites orbiting the Earth, and reference clocks stationed on the Earth's surface. General relativity predicts that these two sets of clocks should tick at slightly different rates, due to their different motions (an effect already predicted by special relativity) and their different positions within the Earth's gravitational field. In order to ensure the system's accuracy, either the satellite clocks are slowed down by a relativistic factor, or that same factor is made part of the evaluation algorithm. In turn, tests of the system's accuracy (especially the very thorough measurements that are part of the definition of universal coordinated time) are testament to the validity of the relativistic predictions.
A number of other tests have probed the validity of various versions of the equivalence principle; strictly speaking, all measurements of gravitational time dilation are tests of the weak version of that principle, not of general relativity itself. So far, general relativity has passed all observational tests.
Astrophysical applications.
Models based on general relativity play an important role in astrophysics; the success of these models is further testament to the theory's validity.
Gravitational lensing.
Since light is deflected in a gravitational field, it is possible for the light of a distant object to reach an observer along two or more paths. For instance, light of a very distant object such as a quasar can pass along one side of a massive galaxy and be deflected slightly so as to reach an observer on Earth, while light passing along the opposite side of that same galaxy is deflected as well, reaching the same observer from a slightly different direction. As a result, that particular observer will see one astronomical object in two different places in the night sky. This kind of focussing is well known when it comes to optical lenses, and hence the corresponding gravitational effect is called gravitational lensing.
Observational astronomy uses lensing effects as an important tool to infer properties of the lensing object. Even in cases where that object is not directly visible, the shape of a lensed image provides information about the mass distribution responsible for the light deflection. In particular, gravitational lensing provides one way to measure the distribution of dark matter, which does not give off light and can be observed only by its gravitational effects. One particularly interesting application are large-scale observations, where the lensing masses are spread out over a significant fraction of the observable universe, and can be used to obtain information about the large-scale properties and evolution of our cosmos.
Gravitational waves.
Gravitational waves, a direct consequence of Einstein's theory, are distortions of geometry that propagate at the speed of light, and can be thought of as ripples in spacetime. They should not be confused with the gravity waves of fluid dynamics, which are a different concept.
In February 2016, the Advanced LIGO team announced that they had directly observed gravitational waves from a black hole merger.
Indirectly, the effect of gravitational waves had been detected in observations of specific binary stars. Such pairs of stars orbit each other and, as they do so, gradually lose energy by emitting gravitational waves. For ordinary stars like the Sun, this energy loss would be too small to be detectable, but this energy loss was observed in 1974 in a binary pulsar called PSR1913+16. In such a system, one of the orbiting stars is a pulsar. This has two consequences: a pulsar is an extremely dense object known as a neutron star, for which gravitational wave emission is much stronger than for ordinary stars. Also, a pulsar emits a narrow beam of electromagnetic radiation from its magnetic poles. As the pulsar rotates, its beam sweeps over the Earth, where it is seen as a regular series of radio pulses, just as a ship at sea observes regular flashes of light from the rotating light in a lighthouse. This regular pattern of radio pulses functions as a highly accurate "clock". It can be used to time the double star's orbital period, and it reacts sensitively to distortions of spacetime in its immediate neighborhood.
The discoverers of PSR1913+16, Russell Hulse and Joseph Taylor, were awarded the Nobel Prize in Physics in 1993. Since then, several other binary pulsars have been found. The most useful are those in which both stars are pulsars, since they provide accurate tests of general relativity.
Currently, a number of land-based gravitational wave detectors are in operation, and a mission to launch a space-based detector, LISA, is currently under development, with a precursor mission (LISA Pathfinder) which was launched in 2015. Gravitational wave observations can be used to obtain information about compact objects such as neutron stars and black holes, and also to probe the state of the early universe fractions of a second after the Big Bang.
Black holes.
When mass is concentrated into a sufficiently compact region of space, general relativity predicts the formation of a black hole – a region of space with a gravitational effect so strong that not even light can escape. Certain types of black holes are thought to be the final state in the evolution of massive stars. On the other hand, supermassive black holes with the mass of millions or billions of Suns are assumed to reside in the cores of most galaxies, and they play a key role in current models of how galaxies have formed over the past billions of years.
Matter falling onto a compact object is one of the most efficient mechanisms for releasing energy in the form of radiation, and matter falling onto black holes is thought to be responsible for some of the brightest astronomical phenomena imaginable. Notable examples of great interest to astronomers are quasars and other types of active galactic nuclei. Under the right conditions, falling matter accumulating around a black hole can lead to the formation of jets, in which focused beams of matter are flung away into space at speeds near that of light.
There are several properties that make black holes the most promising sources of gravitational waves. One reason is that black holes are the most compact objects that can orbit each other as part of a binary system; as a result, the gravitational waves emitted by such a system are especially strong. Another reason follows from what are called black-hole uniqueness theorems: over time, black holes retain only a minimal set of distinguishing features (these theorems have become known as "no-hair" theorems), regardless of the starting geometric shape. For instance, in the long term, the collapse of a hypothetical matter cube will not result in a cube-shaped black hole. Instead, the resulting black hole will be indistinguishable from a black hole formed by the collapse of a spherical mass. In its transition to a spherical shape, the black hole formed by the collapse of a more complicated shape will emit gravitational waves.
Cosmology.
One of the most important aspects of general relativity is that it can be applied to the universe as a whole. A key point is that, on large scales, our universe appears to be constructed along very simple lines: all current observations suggest that, on average, the structure of the cosmos should be approximately the same, regardless of an observer's location or direction of observation: the universe is approximately homogeneous and isotropic. Such comparatively simple universes can be described by simple solutions of Einstein's equations. The current cosmological models of the universe are obtained by combining these simple solutions to general relativity with theories describing the properties of the universe's matter content, namely thermodynamics, nuclear- and particle physics. According to these models, our present universe emerged from an extremely dense high-temperature state – the Big Bang – roughly 14 billion years ago and has been expanding ever since.
Einstein's equations can be generalized by adding a term called the cosmological constant. When this term is present, empty space itself acts as a source of attractive (or, less commonly, repulsive) gravity. Einstein originally introduced this term in his pioneering 1917 paper on cosmology, with a very specific motivation: contemporary cosmological thought held the universe to be static, and the additional term was required for constructing static model universes within the framework of general relativity. When it became apparent that the universe is not static, but expanding, Einstein was quick to discard this additional term. Since the end of the 1990s, however, astronomical evidence indicating an accelerating expansion consistent with a cosmological constant – or, equivalently, with a particular and ubiquitous kind of dark energy – has steadily been accumulating.
Modern research.
General relativity is very successful in providing a framework for accurate models which describe an impressive array of physical phenomena. On the other hand, there are many interesting open questions, and in particular, the theory as a whole is almost certainly incomplete.
In contrast to all other modern theories of fundamental interactions, general relativity is a classical theory: it does not include the effects of quantum physics. The quest for a quantum version of general relativity addresses one of the most fundamental open questions in physics. While there are promising candidates for such a theory of quantum gravity, notably string theory and loop quantum gravity, there is at present no consistent and complete theory. It has long been hoped that a theory of quantum gravity would also eliminate another problematic feature of general relativity: the presence of spacetime singularities. These singularities are boundaries ("sharp edges") of spacetime at which geometry becomes ill-defined, with the consequence that general relativity itself loses its predictive power. Furthermore, there are so-called singularity theorems which predict that such singularities "must" exist within the universe if the laws of general relativity were to hold without any quantum modifications. The best-known examples are the singularities associated with the model universes that describe black holes and the beginning of the universe.
Other attempts to modify general relativity have been made in the context of cosmology. In the modern cosmological models, most energy in the universe is in forms that have never been detected directly, namely dark energy and dark matter. There have been several controversial proposals to remove the need for these enigmatic forms of matter and energy, by modifying the laws governing gravity and the dynamics of cosmic expansion, for example modified Newtonian dynamics.
Beyond the challenges of quantum effects and cosmology, research on general relativity is rich with possibilities for further exploration: mathematical relativists explore the nature of singularities and the fundamental properties of Einstein's equations, and ever more comprehensive computer simulations of specific spacetimes (such as those describing merging black holes) are run.
More than one hundred years after the theory was first published, research is more active than ever.
See also.
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References.
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Bibliography.
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External links.
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| https://en.wikipedia.org/wiki?curid=1411100 |
1411188 | Activated sludge | Wastewater treatment process using aeration and a biological floc
The activated sludge process is a type of biological wastewater treatment process for treating sewage or industrial wastewaters using aeration and a biological floc composed of bacteria and protozoa. It is one of several biological wastewater treatment alternatives in secondary treatment, which deals with the removal of biodegradable organic matter and suspended solids. It uses air (or oxygen) and microorganisms to biologically oxidize organic pollutants, producing a waste sludge (or floc) containing the oxidized material.
The activated sludge process for removing carbonaceous pollution begins with an aeration tank where air (or oxygen) is injected into the waste water. This is followed by a settling tank to allow the biological flocs (the sludge blanket) to settle, thus separating the biological sludge from the clear treated water. Part of the waste sludge is recycled to the aeration tank and the remaining waste sludge is removed for further treatment and ultimate disposal.
Plant types include package plants, oxidation ditch, deep shaft/vertical treatment, surface-aerated basins, and sequencing batch reactors (SBRs). Aeration methods include diffused aeration, surface aerators (cones) or, rarely, pure oxygen aeration.
Sludge bulking can occur which makes activated sludge difficult to settle and frequently has an adverse impact on final effluent quality. Treating sludge bulking and managing the plant to avoid a recurrence requires skilled management and may require full-time staffing of a works to allow immediate intervention. A new development of the activated sludge process is the Nereda process which produces a granular sludge that settles very well.
Purpose.
The activated sludge process is a biological process used to oxidise carbonaceous biological matter, oxidising nitrogenous matter (mainly ammonium and nitrogen) in biological matter, and removing nutrients (nitrogen and phosphorus).
Process description.
The process takes advantage of aerobic micro-organisms that can digest organic matter in sewage, and clump together by flocculation entrapping fine particulate matter as they do so. It thereby produces a liquid that is relatively free from suspended solids and organic material, and flocculated particles that will readily settle out and can be removed.
The general arrangement of an activated sludge process for removing carbonaceous pollution includes the following items:
Treatment of nitrogenous or phosphorous matter comprises the addition of an anoxic compartment inside the aeration tank in order to perform the nitrification-denitrification process more efficiently. First, ammonia is oxidized to nitrite, which is then converted into nitrate in aerobic conditions (aeration compartment). Facultative bacteria then reduce the nitrate to nitrogen gas in anoxic conditions (anoxic compartment). Moreover, the organisms used for the phosphorus uptake (Polyphosphate Accumulating Organisms) are more efficient under anoxic conditions. These microorganisms accumulate large amounts of phosphates in their cells and are settled in the secondary clarifier. The settled sludge is either disposed of as waste activated sludge or reused in the aeration tank as return activated sludge. Some sludge must always be returned to the aeration tanks to maintain an adequate population of organisms.
The yield of PAOs (Polyphosphate Accumulating Organisms) is reduced between 70 and 80% under aerobic conditions. Even though the phosphorus can be removed upstream of the aeration tank by chemical precipitation (adding metal ions such as: calcium, aluminum or iron), the biological phosphorus removal is more economic due to the saving of chemicals.
Bioreactor and final clarifier.
The process involves air or oxygen being introduced into a mixture of screened, and primary treated sewage or industrial wastewater (wastewater) combined with organisms to develop a biological floc which reduces the organic content of the sewage. This material, which in healthy sludge is a brown floc, is largely composed of Saprotrophic bacteria but also has an important protozoan flora component mainly composed of amoebae, Spirotrichs, Peritrichs including Vorticellids and a range of other filter-feeding species. Other important constituents include motile and sedentary Rotifers. In poorly managed activated sludge, a range of mucilaginous filamentous bacteria can develop - including "Sphaerotilus natans", "Gordonia", and other microorganisms - which produces a sludge that is difficult to settle and can result in the sludge blanket decanting over the weirs in the settlement tank to severely contaminate the final effluent quality. This material is often described as sewage fungus but true fungal communities are relatively uncommon.
The combination of wastewater and biological mass is commonly known as "mixed liquor". In all activated sludge plants, once the wastewater has received sufficient treatment, excess mixed liquor is discharged into settling tanks and the treated supernatant is run off to undergo further treatment before discharge. Part of the settled material, the sludge, is returned to the head of the aeration system to re-seed the new wastewater entering the tank. This fraction of the floc is called "return activated sludge" (R.A.S.).
The space required for a sewage treatment plant can be reduced by using a membrane bioreactor to remove some wastewater from the mixed liquor prior to treatment. This results in a more concentrated waste product that can then be treated using the activated sludge process.
Many sewage treatment plants use axial flow pumps to transfer nitrified mixed liquor from the aeration zone to the anoxic zone for denitrification. These pumps are often referred to as internal mixed liquor recycle pumps (IMLR pumps). The raw sewage, the RAS, and the nitrified mixed liquor are mixed by submersible mixers in the anoxic zones in order to achieve denitrification.
Sludge production.
Activated sludge is also the name given to the active biological material produced by activated sludge plants. Excess sludge is called "surplus activated sludge" or "waste activated sludge" and is removed from the treatment process to keep "food to biomass" (F/M) ratio in balance (where biomass refers to the activated sludge). This sewage sludge is usually mixed with primary sludge from the primary clarifiers and undergoes further sludge treatment for example by anaerobic digestion, followed by thickening, dewatering, composting and land application.
The amount of sewage sludge produced from the activated sludge process is directly proportional to the amount of wastewater treated. The total sludge production consists of the sum of primary sludge from the primary sedimentation tanks as well as waste activated sludge from the bioreactors. The activated sludge process produces about of waste activated sludge (that is grams of dry solids produced per cubic metre of wastewater treated). is regarded as being typical. In addition, about of primary sludge is produced in the primary sedimentation tanks which most - but not all - of the activated sludge process configurations use.
Process control.
The general process control method is to monitor sludge blanket level, SVI (Sludge Volume Index), MCRT (Mean Cell Residence Time), F/M (Food to Microorganism), as well as the biota of the activated sludge and the major nutrients DO (Dissolved oxygen), nitrogen, phosphate, BOD (Biochemical oxygen demand), and COD (Chemical oxygen demand). In the reactor/aerator and clarifier system, the sludge blanket is measured from the bottom of the clarifier to the level of settled solids in the clarifier's water column; this, in large plants, can be done up to three times a day.
The SVI is the volume of settled sludge occupied by a given mass of dry sludge solids. It is calculated by dividing the volume of settled sludge in a mixed liquor sample, measured in milliliters per liter of sample (after 30 minutes of settling), by the MLSS (Mixed Liquor Suspended Solids), measured in grams per liter. The MCRT is the total mass (in kilograms or pounds) of mixed liquor suspended solids in the aerator and clarifier divided by the mass flow rate (in kilograms/pounds per day) of mixed liquor suspended solids leaving as WAS and final effluent. The F/M is the ratio of food fed to the microorganisms each day to the mass of microorganisms held under aeration. Specifically, it is the amount of BOD fed to the aerator (in kilograms/pounds per day) divided by the amount (in kilograms or pounds) of Mixed Liquor Volatile Suspended Solids (MLVSS) under aeration. Note: Some references use MLSS (Mixed Liquor Suspended Solids) for expedience, but MLVSS is considered more accurate for the measure of microorganisms. Again, due to expedience, COD is generally used, in lieu of BOD, as BOD takes five days for results.
To ensure good bacterial settlement and to avoid sedimentation problems caused by filamentous bacteria, plants using atmospheric air as an oxygen source should maintain a dissolved oxygen (DO) level of about 2 mg/L in the aeration tank. In pure oxygen systems, DO levels are usually in the range of 4 to 10 mg/L. Operators should monitor the tank for low DO bacteria, such as S. natans, type 1701 and H. hydrossis, which indicate low DO conditions by elevated effluent turbidity and dark activated sludge with foul odours. Many plants have on-line monitoring equipment that continuously measures and records DO levels at specific points within the aeration tank. These on-line analysers send data to the SCADA system and allow automatic control of the aeration system to maintain a predetermined DO level. Whether generated automatically or taken manually, regular monitoring is necessary to favour organisms that settle well rather than filaments. However, operating the aeration system involves finding a balance between sufficient oxygen for proper treatment and the energy cost, which represents approximately 90% of the total treatment cost.
Based on these control methods, the amount of settled solids in the mixed liquor can be varied by wasting activated sludge (WAS) or returning activated sludge (RAS).
The returning activated sludge is designed to recycle a portion of the activated sludge from the secondary clarifier back t the aeration tank. It usually includes a pump that draws the portion back.
The RAS line is designed considering the potential for clogging, settling, and other relatable issues that manage to impact the flow of the activated sludge back to the aeration tank. This line must handle the required flow of the plant and has to be designed to minimize the risk of solids settling or accumulating.
Nitrification and Denitrification.
Ammonium can have toxic effort on aquatic organism. Nitrification also takes places in bodies of water, which leads to oxygen depletion. Furthermore, nitrate and ammonium are eutophying (fertilizing) nutrients that can impair water bodies. For these reasons, nitrification and, in many cases, nitrogen removal is necessary.
Two special steps are required for nitrogen removal:
a) Nitrification: Oxidation of ammonium nitrogen and organically bound nitrogen to nitrate. Nitrification is very sensitive to inhibitors and can lead to a pH value in poorly buffered water.
Nitrification takes places in following steps:
this results in:
formula_2
Nitrification is associated with the production of acid (H+). This puts a strain on the buffering capacity of the water or a pH value shift may occur, which impairs the process.
b) Denitrification: Reduction of nitrate nitrogen to molecular nitrogen, which escapes from the wastewater into the atmosphere. This step can be carried out by microorganism commonly found in sewage treatment plants. However, these only use the nitrate as an electron acceptor if no dissolved oxygen is present.
formula_3
In order for denitrification to take place in the activated sludge process, an electron source, a reductant, must therefore also be present that can reduce sufficient nitrate to N2. If there is too little substrate in the raw wastewater, this can be added artificially. In addition, denitrification corrects the change in H+ concentration (pH value shift) that occurs during nitrification. This is particularly important for poorly buffered water.
Nitrification and denitrification are in considerable contradiction with regard to the required environmental conditions. Nitrification requires oxygen and CO2. Denitrification only takes place in the absence of dissolved oxygen and with a sufficient supply of oxidizable substances.
Plant types.
There are a variety of types of activated sludge plants. These include:
Package plants.
There are a wide range of types of package plants, often serving small communities or industrial plants that may use hybrid treatment processes often involving the use of aerobic sludge to treat the incoming sewage. In such plants the primary settlement stage of treatment may be omitted. In these plants, a biotic floc is created which provides the required substrate. Package plants are designed and fabricated by specialty engineering firms in dimensions that allow for their transportation to the job site in public highways, typically width and height of . Length varies with capacity with larger plants being fabricated in pieces and welded on site. Steel is preferred over synthetic materials (e.g., plastic) for its durability. Package plants are commonly variants of extended aeration, to promote the "fit and forget" approach required for small communities without dedicated operational staff. There are various standards to assist with their design.
To use less space, treat difficult waste, and intermittent flows, a number of designs of hybrid treatment plants have been produced. Such plants often combine at least two stages of the three main treatment stages into one combined stage. In the UK, where a large number of wastewater treatment plants serve small populations, package plants are a viable alternative to building a large structure for each process stage. In the US, package plants are typically used in rural areas, highway rest stops and trailer parks.
Package plants may be referred to as "high charged" or "low charged". This refers to the way the biological load is processed. In high charged systems, the biological stage is presented with a high organic load and the combined floc and organic material is then oxygenated for a few hours before being charged again with a new load. In the low charged system the biological stage contains a low organic load and is combined with flocculate for longer times.
Oxidation ditch.
In some areas, where more land is available, sewage is treated in large round or oval ditches with one or more horizontal aerators typically called brush or disc aerators which drive the mixed liquor around the ditch and provide aeration. These are oxidation ditches, often referred to by manufacturer's trade names such as Pasveer, Orbal, or Carrousel. They have the advantage that they are relatively easy to maintain and are resilient to shock loads that often occur in smaller communities (i.e. at breakfast time and in the evening).
Oxidation ditches are installed commonly as 'fit & forget' technology, with typical design parameters of a hydraulic retention time of 24 – 48 hours, and a sludge age of 12 – 20 days. This compares with nitrifying activated sludge plants having a retention time of 8 hours, and a sludge age of 8 – 12 days.
Deep shaft / Vertical treatment.
Where land is in short supply sewage may be treated by injection of oxygen into a pressured return sludge stream which is injected into the base of a deep columnar tank buried in the ground. Such shafts may be up to deep and are filled with sewage liquor. As the sewage rises the oxygen forced into solution by the pressure at the base of the shaft breaks out as molecular oxygen providing a highly efficient source of oxygen for the activated sludge biota. The rising oxygen and injected return sludge provide the physical mechanism for mixing of the sewage and sludge. Mixed sludge and sewage is decanted at the surface and separated into supernatant and sludge components. The efficiency of deep shaft treatment can be high.
Surface aerators are commonly quoted as having an aeration efficiency of 0.5–1.5 kg O2/kWh (1.1–3.3 lb O2/kWh), diffused aeration as 1.5–2.5 kg O2/kWh (3.3–5.5 lb O2/kWh). Deep Shaft claims 5–8 kg O2/kWh (11–18 lb O2/kWh).
However, the costs of construction are high. Deep Shaft has seen the greatest uptake in Japan, because of the land area issues. Deep Shaft was developed by ICI, as a spin-off from their Pruteen process. In the UK it is found at three sites: Tilbury, Anglian water, treating a wastewater with a high industrial contribution; Southport, United Utilities, because of land space issues; and Billingham, ICI, again treating industrial effluent, and built (after the Tilbury shafts) by ICI to help the agent sell more.
DeepShaft is a patented, licensed, process. The licensee has changed several times and currently (2015) Noram Engineering sells it.
Surface-aerated basins.
Most biological oxidation processes for treating industrial wastewaters have in common the use of oxygen (or air) and microbial action. Surface-aerated basins achieve 80 to 90% removal of BOD with retention times of 1 to 10 days. The basins may range in depth from and utilize motor-driven aerators floating on the surface of the wastewater.
In an aerated basin system, the aerators provide two functions: they transfer air into the basins required by the biological oxidation reactions, and they provide the mixing required for dispersing the air and for contacting the reactants (that is, oxygen, wastewater and microbes). Typically, the floating surface aerators are rated to deliver the amount of air equivalent to 1.8 to 2.7 kilograms O2/kWh (4.0 to 6.0 lb O2/kWh). However, they do not provide as good mixing as is normally achieved in activated sludge systems and therefore aerated basins do not achieve the same performance level as activated sludge units.
Biological oxidation processes are sensitive to temperature and, between , the rate of biological reactions increase with temperature. Most surface aerated vessels operate at between .
Sequencing batch reactors (SBRs).
Sequencing batch reactors (SBRs) treat wastewater in batches within the same vessel. This means that the bioreactor and final clarifier are not separated in space but in a timed sequence. The installation consists of at least two identically equipped tanks with a common inlet, which can be alternated between them. While one tank is in settle/decant mode the other is aerating and filling.
Aeration methods.
Diffused aeration.
Sewage liquor is run into deep tanks with diffuser grid aeration systems that are attached to the floor. These are like the diffused airstone used in tropical fish tanks but on a much larger scale. Air is pumped through the blocks and the curtain of bubbles formed both oxygenates the liquor and also provides the necessary mixing action. Where capacity is limited or the sewage is unusually strong or difficult to treat, oxygen may be used instead of air. Typically, the air is generated by some type of air blower.
Surface aerators (cones).
Vertically mounted tubes of up to diameter extending from just above the base of a deep concrete tank to just below the surface of the sewage liquor. A typical shaft might be high. At the surface end, the tube is formed into a cone with helical vanes attached to the inner surface. When the tube is rotated, the vanes spin liquor up and out of the cones drawing new sewage liquor from the base of the tank. In many works, each cone is located in a separate cell that can be isolated from the remaining cells if required for maintenance. Some works may have two cones to a cell and some large works may have 4 cones per cell.
Pure oxygen aeration.
Pure oxygen activated sludge aeration systems are sealed-tank reactor vessels with surface aerator type impellers mounted within the tanks at the oxygen carbon liquor surface interface. The amount of oxygen entrainment, or DO (Dissolved Oxygen), can be controlled by a weir adjusted level control, and a vent gas oxygen controlled oxygen feed valve. Oxygen is generated on site by cryogenic distillation of air, pressure swing adsorption, or other methods. These systems are used where wastewater plant space is at a premium and high sewage throughput is required as high energy costs are involved in purifying oxygen.
Recent developments.
A new development of the activated sludge process is the Nereda process which produces a granular sludge that settles very well (the sludge volume index is reduced from ). A new process reactor system is created to take advantage of this quick settling sludge and is integrated into the aeration tank instead of having a separate unit outside. About 30 Nereda wastewater treatment plants worldwide are operational, under construction or under design, varying in size from 5,000 up to 858,000 person equivalent.
Issues.
Process upsets.
Sludge bulking can occur which makes activated sludge difficult to settle and frequently has an adverse impact on final effluent quality. Treating sludge bulking and managing the plant to avoid a recurrence requires skilled management and may require full-time staffing of a works to allow immediate intervention.
The discharge of toxic industrial pollution to treatment plants designed primarily to treat domestic sewage can create process upsets.
Costs and technology choice.
The activated sludge process is an example for a more high-tech, energy intensive or "mechanized" process that is relatively expensive compared to some other wastewater treatment systems. It can provide a very high level of treatment.
Activated sludge plants are wholly dependent on an electrical supply to power the aerators to transfer settled solids back to the aeration tank inlet, and in many cases to pump waste sludge and final effluent. In some works untreated sewage is lifted by pumps to the head-works to provide sufficient fall through the works to enable a satisfactory discharge head for the final effluent. Alternative technologies such as trickling filter treatment requires much less power and can operate on gravity alone.
History.
The activated sludge process was discovered in 1913 in the United Kingdom by two engineers, Edward Ardern and W.T. Lockett, who were conducting research for the Manchester Corporation Rivers Department at Davyhulme Sewage Works. In 1912, Gilbert Fowler, a scientist at the University of Manchester, observed experiments being conducted at the Lawrence Experiment Station at Massachusetts involving the aeration of sewage in a bottle that had been coated with algae. Fowler's engineering colleagues, Ardern and Lockett, experimented on treating sewage in a draw-and-fill reactor, which produced a highly treated effluent. They aerated the waste-water continuously for about a month and were able to achieve a complete nitrification of the sample material. Believing that the sludge had been activated (in a similar manner to activated carbon) the process was named "activated sludge". Not until much later was it realized that what had actually occurred was a means to concentrate biological organisms, decoupling the liquid retention time (ideally, low, for a compact treatment system) from the solids retention time (ideally, fairly high, for an effluent low in BOD5 and ammonia.)
Their results were published in their seminal 1914 paper, and the first full-scale continuous-flow system was installed at Worcester two years later. In the aftermath of the First World War the new treatment method spread rapidly, especially to the US, Denmark, Germany and Canada. By the late 1930s, the activated sludge treatment became a well-known biological wastewater treatment process in those countries where sewer systems and sewage treatment plants were common.
References.
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"text": "\\mathrm {\\ NH_4^+ + 1,5 \\ O_2 \\longrightarrow \\ NO_2^- + 2 H^+ + H_2O + Energy}"
},
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"math_id": 1,
"text": "\\mathrm {\\ NO_2^- + 0,5 \\ O_2 \\longrightarrow \\ NO_3^- + Energy}\n\n"
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"text": "\\mathrm {\\ NH_4^+ + 2 \\ O_2 \\longrightarrow \\ NO_3^- + 2H^+ + H_2O + Energy}\n"
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"text": "\\mathrm {\\ 2 \\ NO_3^- + 2 \\ H^+ + 10 \\ H \\longrightarrow \\ N_2 + 6 \\ H_2O}\n"
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| https://en.wikipedia.org/wiki?curid=1411188 |
14114496 | Metric space aimed at its subspace | Universal property of metric spaces
In mathematics, a metric space aimed at its subspace is a categorical construction that has a direct geometric meaning. It is also a useful step toward the construction of the "metric envelope", or tight span, which are basic (injective) objects of the category of metric spaces.
Following , a notion of a metric space "Y" aimed at its subspace "X" is defined.
Informal introduction.
Informally, imagine terrain "Y", and its part "X", such that wherever in "Y" you place a sharpshooter, and an apple at another place in "Y", and then let the sharpshooter fire, the bullet will go through the apple and will always hit a point of "X", or at least it will fly arbitrarily close to points of "X" – then we say that "Y" is aimed at "X".
A priori, it may seem plausible that for a given "X" the superspaces "Y" that aim at "X" can be arbitrarily large or at least huge. We will see that this is not the case. Among the spaces which aim at a subspace isometric to "X", there is a unique (up to isometry) universal one, Aim("X"), which in a sense of canonical isometric embeddings contains any other space aimed at (an isometric image of) "X". And in the special case of an arbitrary compact metric space "X" every bounded subspace of an arbitrary metric space "Y" aimed at "X" is totally bounded (i.e. its metric completion is compact).
Definitions.
Let formula_0 be a metric space. Let formula_1 be a subset of formula_2, so that formula_3 (the set formula_1 with the metric from formula_2 restricted to formula_1) is a metric subspace of formula_4. Then
Definition. Space formula_2 aims at formula_1 if and only if, for all points formula_5 of formula_2, and for every real formula_6, there exists a point formula_7 of formula_1 such that
formula_8
Let formula_9 be the space of all real valued metric maps (non-contractive) of formula_1. Define
formula_10
Then
formula_11
for every formula_12 is a metric on formula_13. Furthermore, formula_14, where formula_15, is an isometric embedding of formula_1 into formula_16; this is essentially a generalisation of the Kuratowski-Wojdysławski embedding of bounded metric spaces formula_1 into formula_17, where we here consider arbitrary metric spaces (bounded or unbounded). It is clear that the space formula_16 is aimed at formula_18.
Properties.
Let formula_19 be an isometric embedding. Then there exists a natural metric map formula_20 such that formula_21:
formula_22
for every formula_23 and formula_24.
Theorem The space "Y" above is aimed at subspace "X" if and only if the natural mapping formula_20 is an isometric embedding.
Thus it follows that every space aimed at "X" can be isometrically mapped into Aim(X), with some additional (essential) categorical requirements satisfied.
The space Aim(X) is injective (hyperconvex in the sense of Aronszajn-Panitchpakdi) – given a metric space "M," which contains Aim(X) as a metric subspace, there is a canonical (and explicit) metric retraction of "M" onto Aim(X) . | [
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{
"math_id": 11,
"text": "d(f,g) := \\sup_{x\\in X} |f(x)-g(x)| < \\infty"
},
{
"math_id": 12,
"text": "f, g\\in \\text{Aim}(X)"
},
{
"math_id": 13,
"text": "\\text{Aim}(X)"
},
{
"math_id": 14,
"text": "\\delta_X\\colon x\\mapsto d_x"
},
{
"math_id": 15,
"text": "d_x(p) := d(x,p)\\,"
},
{
"math_id": 16,
"text": "\\operatorname{Aim}(X)"
},
{
"math_id": 17,
"text": "C(X)"
},
{
"math_id": 18,
"text": "\\delta_X(X)"
},
{
"math_id": 19,
"text": "i\\colon X \\to Y"
},
{
"math_id": 20,
"text": "j\\colon Y \\to \\operatorname{Aim}(X)"
},
{
"math_id": 21,
"text": "j \\circ i = \\delta_X"
},
{
"math_id": 22,
"text": "(j(y))(x) := d(x,y)\\,"
},
{
"math_id": 23,
"text": "x\\in X\\,"
},
{
"math_id": 24,
"text": "y\\in Y\\,"
}
]
| https://en.wikipedia.org/wiki?curid=14114496 |
1411451 | BSO | BSO may refer to:
Other uses.
Topics referred to by the same term
<templatestyles src="Dmbox/styles.css" />
This page lists associated with the title . | [
{
"math_id": 0,
"text": "\\operatorname{BSO}(n)"
},
{
"math_id": 1,
"text": "\\operatorname{BSO}"
}
]
| https://en.wikipedia.org/wiki?curid=1411451 |
1411533 | Connected category | In category theory, a branch of mathematics, a connected category is a category in which, for every two objects "X" and "Y" there is a finite sequence of objects
formula_0
with morphisms
formula_1
or
formula_2
for each 0 ≤ "i" < "n" (both directions are allowed in the same sequence). Equivalently, a category "J" is connected if each functor from "J" to a discrete category is constant. In some cases it is convenient to not consider the empty category to be connected.
A stronger notion of connectivity would be to require at least one morphism "f" between any pair of objects "X" and "Y". Any category with this property is connected in the above sense.
A small category is connected if and only if its underlying graph is weakly connected, meaning that it is connected if one disregards the direction of the arrows.
Each category "J" can be written as a disjoint union (or coproduct) of a collection of connected categories, which are called the connected components of "J". Each connected component is a full subcategory of "J". | [
{
"math_id": 0,
"text": "X = X_0, X_1, \\ldots, X_{n-1}, X_n = Y"
},
{
"math_id": 1,
"text": "f_i : X_i \\to X_{i+1}"
},
{
"math_id": 2,
"text": "f_i : X_{i+1} \\to X_i"
}
]
| https://en.wikipedia.org/wiki?curid=1411533 |
14116393 | Wilson–Cowan model | Mathematical model of the interactions between neuronal populations
In computational neuroscience, the Wilson–Cowan model describes the dynamics of interactions between populations of very simple excitatory and inhibitory model neurons. It was developed by Hugh R. Wilson and Jack D. Cowan and extensions of the model have been widely used in modeling neuronal populations. The model is important historically because it uses phase plane methods and numerical solutions to describe the responses of neuronal populations to stimuli. Because the model neurons are simple, only elementary limit cycle behavior, i.e. neural oscillations, and stimulus-dependent evoked responses are predicted. The key findings include the existence of multiple stable states, and hysteresis, in the population response.
Mathematical description.
The Wilson–Cowan model considers a homogeneous population of interconnected neurons of excitatory and inhibitory subtypes. All cells receive the same number of excitatory and inhibitory afferents, that is, all cells receive the same average excitation, x(t). The target is to analyze the evolution in time of number of excitatory and inhibitory cells firing at time t, formula_0 and formula_1 respectively.
The equations that describes this evolution are the Wilson-Cowan model:
formula_2
formula_3
where:
If formula_13 denotes a cell's threshold potential and formula_14 is the distribution of thresholds in all cells, then the expected proportion of neurons receiving an excitation at or above threshold level per unit time is:
formula_15,
that is a function of sigmoid form if formula_16 is unimodal.
If, instead of all cells receiving same excitatory inputs and different threshold, we consider that all cells have same threshold but different number of afferent synapses per cell, being formula_17 the distribution of the number of afferent synapses, a variant of function formula_18 must be used:
formula_19
Derivation of the model.
If we denote by formula_20 the refractory period after a trigger, the proportion of cells in refractory period isformula_21 and the proportion of sensitive (able to trigger) cells is formula_22.
The average excitation level of an excitatory cell at time formula_23 is:
formula_24
Thus, the number of cells that triggers at some time formula_25 is the number of cells not in refractory interval, formula_22 AND that have reached the excitatory level, formula_26, obtaining in this way the product at right side of the first equation of the model (with the assumption of uncorrelated terms). Same rationale can be done for inhibitory cells, obtaining second equation.
Simplification of the model assuming time coarse graining.
When time coarse-grained modeling is assumed the model simplifies, being the new equations of the model:
formula_27
formula_28
where bar terms are the time coarse-grained versions of original ones.
Application to epilepsy.
The determination of three concepts is fundamental to an understanding of hypersynchronization of neurophysiological activity at the global (system) level:
A canonical analysis of these issues, developed in 2008 by Shusterman and Troy using the Wilson–Cowan model, predicts qualitative and quantitative features of epileptiform activity. In particular, it accurately predicts the propagation speed of epileptic seizures (which is approximately 4–7 times slower than normal brain wave activity) in a human subject with chronically implanted electroencephalographic electrodes.
Transition into hypersynchronization.
The transition from normal state of brain activity to epileptic seizures was not formulated theoretically until 2008, when a theoretical path from a baseline state to large-scale self-sustained oscillations, which spread out uniformly from the point of stimulus, has been mapped for the first time.
A realistic state of baseline physiological activity has been defined, using the following two-component definition:
(1) A time-independent component represented by subthreshold excitatory activity E and superthreshold inhibitory activity I.
(2) A time-varying component which may include singlepulse waves, multipulse waves, or periodic waves caused by spontaneous neuronal activity.
This baseline state represents activity of the brain in the state of relaxation, in which neurons receive some level of spontaneous, weak stimulation by small, naturally present concentrations of neurohormonal substances. In waking adults this state is commonly associated with alpha rhythm, whereas slower (theta and delta) rhythms are usually observed during deeper relaxation and sleep. To describe this general setting, a 3-variable formula_29 spatially dependent extension of the classical Wilson–Cowan model can be utilized. Under appropriate initial conditions, the excitatory component, u, dominates over the inhibitory component, I, and the three-variable system reduces to the two-variable Pinto-Ermentrout type model
formula_30
formula_31
The variable v governs the recovery of excitation u; formula_32 and formula_33 determine the rate of change of recovery. The connection function formula_34 is positive, continuous, symmetric, and has the typical form formula_35. In Ref. formula_36 The firing rate function, which is generally accepted to have a sharply increasing sigmoidal shape, is approximated by formula_37, where H denotes the Heaviside function; formula_38 is a short-time stimulus. This formula_39 system has been successfully used in a wide variety of neuroscience research studies. In particular, it predicted the existence of spiral waves, which can occur during seizures; this theoretical prediction was subsequently confirmed experimentally using optical imaging of slices from the rat cortex.
Rate of expansion.
The expansion of hypersynchronized regions exhibiting large-amplitude stable bulk oscillations occurs when the oscillations coexist with the stable rest state formula_40. To understand the mechanism responsible for the expansion, it is necessary to linearize the formula_41 system around formula_42 when formula_43 is held fixed. The linearized system exhibits subthreshold decaying oscillations whose frequency increases as formula_44 increases. At a critical value formula_45 where the oscillation frequency is high enough, bistability occurs in the formula_46 system: a stable, spatially independent, periodic solution (bulk oscillation) and a stable rest state coexist over a continuous range of parameters. When formula_47 where bulk oscillations occur, "The rate of expansion of the hypersynchronization region is determined by an interplay between two key features: (i) the speed c of waves that form and propagate outward from the edge of the region, and (ii) the concave shape of the graph of the activation variable u as it rises, during each bulk oscillation cycle, from the rest state u=0 to the activation threshold. Numerical experiments show that during the rise of
u towards threshold, as the rate of vertical increase slows down, over time interval formula_48 due to the concave component, the stable solitary wave emanating from the region causes the region to expand spatially at a Rate proportional to the wave speed. From this initial observation it is natural to expect that the proportionality constant should be the fraction of the time that the solution is concave during one cycle." Therefore, when formula_47, the rate of expansion of the region is estimated by
formula_49
where formula_50 is the length of subthreshold time interval, T is period of the periodic solution; c is the speed of waves emanating from the hypersynchronization region. A realistic value of c, derived by Wilson et al., is c=22.4 mm/s.
How to evaluate the ratio formula_51 To determine values for formula_52 it is necessary to analyze the underlying bulk oscillation which satisfies the spatially independent system
formula_53
formula_54
This system is derived using standard functions and parameter values formula_55, formula_56 and formula_57 Bulk oscillations occur when formula_58. When formula_59, Shusterman and Troy analyzed the bulk oscillations and found formula_60. This gives the range
formula_61
Since formula_60, Eq. (1) shows that the migration Rate is a fraction of the traveling wave speed, which is consistent with experimental and clinical observations regarding the slow spread of epileptic activity. This migration mechanism also provides a plausible explanation for spread and sustenance of epileptiform activity without a driving source that, despite a number of experimental studies, has never been observed.
Comparing theoretical and experimental migration rates.
The rate of migration of hypersynchronous activity that was experimentally recorded during seizures in a human subject, using chronically implanted subdural electrodes on the surface of the left temporal lobe, has been estimated as
formula_62,
which is consistent with the theoretically predicted range given above in (2). The ratio formula_63 in formula (1) shows that the leading edge of the region of synchronous seizure activity migrates approximately 4–7 times more slowly than normal brain wave activity, which is in agreement with the experimental data described above.
To summarize, mathematical modeling and theoretical analysis of large-scale electrophysiological activity provide tools for predicting the spread and migration of hypersynchronous brain activity, which can be useful for diagnostic evaluation and management of patients with epilepsy. It might be also useful for predicting migration and spread of electrical activity over large regions of the brain that occur during deep sleep (Delta wave), cognitive activity and in other functional settings.
References.
<templatestyles src="Reflist/styles.css" /> | [
{
"math_id": 0,
"text": "E(t)"
},
{
"math_id": 1,
"text": "I(t)"
},
{
"math_id": 2,
"text": "E(t+\\tau)=\\left[1-\\int_{t-r}^{t}E(t')dt'\\right] \\; S_e\\left( \\int_{-\\infty}^{t}\\alpha(t-t')[c_1E(t')-c_2I(t')+P(t')]dt'\\right )"
},
{
"math_id": 3,
"text": "I(t+\\tau)=\\left[ 1-\\int_{t-r}^{t}I(t')dt'\\right] \\; S_i \\left( \\int_{-\\infty}^{t}\\alpha(t-t')[c_3E(t')-c_4I(t')+Q(t')]dt'\\right)"
},
{
"math_id": 4,
"text": "S_e\\{\\}"
},
{
"math_id": 5,
"text": "S_i\\{\\}"
},
{
"math_id": 6,
"text": "\\alpha(t)"
},
{
"math_id": 7,
"text": "c_1"
},
{
"math_id": 8,
"text": "c_2"
},
{
"math_id": 9,
"text": "c_3"
},
{
"math_id": 10,
"text": "c_4"
},
{
"math_id": 11,
"text": "P(t)"
},
{
"math_id": 12,
"text": "Q(t)"
},
{
"math_id": 13,
"text": "\\theta"
},
{
"math_id": 14,
"text": "D(\\theta)"
},
{
"math_id": 15,
"text": "S(x)=\\int_{0}^{x}D(\\theta)d\\theta"
},
{
"math_id": 16,
"text": "D()"
},
{
"math_id": 17,
"text": "C(w)"
},
{
"math_id": 18,
"text": "S()"
},
{
"math_id": 19,
"text": "S(x)=\\int_{\\frac{\\theta}{x}}^{\\infty}C(w)dw"
},
{
"math_id": 20,
"text": "\\tau"
},
{
"math_id": 21,
"text": "\\int_{t-r}^{t}E(t')dt'"
},
{
"math_id": 22,
"text": "1-\\int_{t-r}^{t}E(t')dt'"
},
{
"math_id": 23,
"text": "t"
},
{
"math_id": 24,
"text": "x(t) = \\int_{-\\infty}^{t}\\alpha(t-t')[c_1 E(t')-c_2 I(t')+P(t')]dt'"
},
{
"math_id": 25,
"text": "E(t+\\tau)"
},
{
"math_id": 26,
"text": "S_e(x(t))"
},
{
"math_id": 27,
"text": "\\tau\\frac{d\\bar{E}}{dt}=-\\bar{E}+(1-r\\bar{E})S_e[kc_1\\bar{E}(t)-kc_2\\bar{I}(t)+kP(t)]"
},
{
"math_id": 28,
"text": "\\tau'\\frac{d\\bar{I}}{dt}=-\\bar{I}+(1-r'\\bar{I})S_i[k'c_3\\bar{E}(t)-k'c_4\\bar{I}(t)+k'Q(t)]"
},
{
"math_id": 29,
"text": " (u,I,v) "
},
{
"math_id": 30,
"text": "{\\partial u \\over \\partial t}=u-v+ \\int_{R^2}\\omega(x-x',y-y')f(u-\\theta)\\,dxdy + \\zeta(x,y,t),"
},
{
"math_id": 31,
"text": "{\\partial v \\over \\partial t}=\\epsilon (\\beta u-v)."
},
{
"math_id": 32,
"text": " \\epsilon>0 "
},
{
"math_id": 33,
"text": " \\beta>0 "
},
{
"math_id": 34,
"text": " \\omega(x,y) "
},
{
"math_id": 35,
"text": " \\omega=Ae^{-\\lambda\\sqrt {-(x^2+y^2)}}"
},
{
"math_id": 36,
"text": "(A,\\lambda)=(2.1,1)."
},
{
"math_id": 37,
"text": " f(u-\\theta)=H(u-\\theta) "
},
{
"math_id": 38,
"text": " \\zeta(x,y,t) "
},
{
"math_id": 39,
"text": " (u,v) "
},
{
"math_id": 40,
"text": "(u,v)=(0,0)"
},
{
"math_id": 41,
"text": " (u, v) "
},
{
"math_id": 42,
"text": "(0,0)"
},
{
"math_id": 43,
"text": "\\epsilon>0"
},
{
"math_id": 44,
"text": "\\beta"
},
{
"math_id": 45,
"text": "\\beta^{*}"
},
{
"math_id": 46,
"text": "(u,v)"
},
{
"math_id": 47,
"text": "\\beta\\ge\\beta^{*}"
},
{
"math_id": 48,
"text": "\\Delta t,"
},
{
"math_id": 49,
"text": " Rate =(\\Delta t/T)*c~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(1) "
},
{
"math_id": 50,
"text": "\\Delta t "
},
{
"math_id": 51,
"text": "\\Delta t/T?"
},
{
"math_id": 52,
"text": " \\Delta t/T "
},
{
"math_id": 53,
"text": "{{du} \\over {dt}}=u-v+H(u-\\theta),"
},
{
"math_id": 54,
"text": "{{dv} \\over {dt}}=\\epsilon (\\beta u-v)."
},
{
"math_id": 55,
"text": " \\omega=2.1e^{-\\lambda\\sqrt {-(x^2+y^2)}}"
},
{
"math_id": 56,
"text": " \\epsilon=0.1"
},
{
"math_id": 57,
"text": " \\theta=0.1 "
},
{
"math_id": 58,
"text": " \\beta \\ge \\beta^{*}=12.61"
},
{
"math_id": 59,
"text": "12.61 \\le \\beta \\le 17"
},
{
"math_id": 60,
"text": "0.136 \\le \\Delta t/T \\le 0.238"
},
{
"math_id": 61,
"text": " 3.046 mm/s \\le Rate \\le 5.331 mm/s~~~~~~~~~~~~(2) "
},
{
"math_id": 62,
"text": "Rate \\approx 4 mm/s"
},
{
"math_id": 63,
"text": "Rate/c"
}
]
| https://en.wikipedia.org/wiki?curid=14116393 |
1412 | Amine | Chemical compounds and groups containing nitrogen with a lone pair (:N)
In chemistry, amines (, UK also ) are compounds and functional groups that contain a basic nitrogen atom with a lone pair. Formally, amines are derivatives of ammonia (), wherein one or more hydrogen atoms have been replaced by a substituent such as an alkyl or aryl group (these may respectively be called alkylamines and arylamines; amines in which both types of substituent are attached to one nitrogen atom may be called alkylarylamines). Important amines include amino acids, biogenic amines, trimethylamine, and aniline. Inorganic derivatives of ammonia are also called amines, such as monochloramine ().
The substituent is called an amino group.
Compounds with a nitrogen atom attached to a carbonyl group, thus having the structure , are called amides and have different chemical properties from amines.
Classification of amines.
Amines can be classified according to the nature and number of substituents on nitrogen. Aliphatic amines contain only H and alkyl substituents. Aromatic amines have the nitrogen atom connected to an aromatic ring.
Amines, alkyl and aryl alike, are organized into three subcategories based on the number of carbon atoms adjacent to the nitrogen (how many hydrogen atoms of the ammonia molecule are replaced by hydrocarbon groups):
A fourth subcategory is determined by the connectivity of the substituents attached to the nitrogen:
It is also possible to have four organic substituents on the nitrogen. These species are not amines but are quaternary ammonium cations and have a charged nitrogen center. Quaternary ammonium salts exist with many kinds of anions.
Naming conventions.
Amines are named in several ways. Typically, the compound is given the prefix "amino-" or the suffix "-amine". The prefix ""N"-" shows substitution on the nitrogen atom. An organic compound with multiple amino groups is called a diamine, triamine, tetraamine and so forth.
Lower amines are named with the suffix "-amine".
Higher amines have the prefix "amino" as a functional group. IUPAC however does not recommend this convention, but prefers the alkanamine form, e.g. butan-2-amine.
Physical properties.
Hydrogen bonding significantly influences the properties of primary and secondary amines. For example, methyl and ethyl amines are gases under standard conditions, whereas the corresponding methyl and ethyl alcohols are liquids. Amines possess a characteristic ammonia smell, liquid amines have a distinctive "fishy" and foul smell.
The nitrogen atom features a lone electron pair that can bind H+ to form an ammonium ion R3NH+. The lone electron pair is represented in this article by two dots above or next to the N. The water solubility of simple amines is enhanced by hydrogen bonding involving these lone electron pairs. Typically salts of ammonium compounds exhibit the following order of solubility in water: primary ammonium (RNH3+) > secondary ammonium (R2NH2+) > tertiary ammonium (R3NH+). Small aliphatic amines display significant solubility in many solvents, whereas those with large substituents are lipophilic. Aromatic amines, such as aniline, have their lone pair electrons conjugated into the benzene ring, thus their tendency to engage in hydrogen bonding is diminished. Their boiling points are high and their solubility in water is low.
Spectroscopic identification.
Typically the presence of an amine functional group is deduced by a combination of techniques, including mass spectrometry as well as NMR and IR spectroscopies. 1H NMR signals for amines disappear upon treatment of the sample with D2O. In their infrared spectrum primary amines exhibit two N-H bands, whereas secondary amines exhibit only one. In their IR spectra, primary and secondary amines exhibit distinctive N-H stretching bands near 3300 cm-1. Somewhat less distinctive are the bands appearing below 1600 cm-1, which are weaker and overlap with C-C and C-H modes. For the case of propyl amine, the H-N-H scissor mode appears near 1600 cm-1, the C-N stretch near 1000 cm-1, and the R2N-H bend near 810 cm-1.
Structure.
Alkyl amines.
Alkyl amines characteristically feature tetrahedral nitrogen centers. C-N-C and C-N-H angles approach the idealized angle of 109°. C-N distances are slightly shorter than C-C distances. The energy barrier for the nitrogen inversion of the stereocenter is about 7 kcal/mol for a trialkylamine. The interconversion has been compared to the inversion of an open umbrella into a strong wind.
Amines of the type NHRR' and NRR′R″ are chiral: the nitrogen center bears four substituents counting the lone pair. Because of the low barrier to inversion, amines of the type NHRR' cannot be obtained in optical purity. For chiral tertiary amines, NRR′R″ can only be resolved when the R, R', and R″ groups are constrained in cyclic structures such as "N"-substituted aziridines (quaternary ammonium salts are resolvable).
Aromatic amines.
In aromatic amines ("anilines"), nitrogen is often nearly planar owing to conjugation of the lone pair with the aryl substituent. The C-N distance is correspondingly shorter. In aniline, the C-N distance is the same as the C-C distances.
Basicity.
Like ammonia, amines are bases. Compared to alkali metal hydroxides, amines are weaker.
The basicity of amines depends on:
Electronic effects.
Owing to inductive effects, the basicity of an amine might be expected to increase with the number of alkyl groups on the amine. Correlations are complicated owing to the effects of solvation which are opposite the trends for inductive effects. Solvation effects also dominate the basicity of aromatic amines (anilines). For anilines, the lone pair of electrons on nitrogen delocalizes into the ring, resulting in decreased basicity. Substituents on the aromatic ring, and their positions relative to the amino group, also affect basicity as seen in the table.
Solvation effects.
Solvation significantly affects the basicity of amines. N-H groups strongly interact with water, especially in ammonium ions. Consequently, the basicity of ammonia is enhanced by 1011 by solvation. The intrinsic basicity of amines, i.e. the situation where solvation is unimportant, has been evaluated in the gas phase. In the gas phase, amines exhibit the basicities predicted from the electron-releasing effects of the organic substituents. Thus tertiary amines are more basic than secondary amines, which are more basic than primary amines, and finally ammonia is least basic. The order of pKb's (basicities in water) does not follow this order. Similarly aniline is more basic than ammonia in the gas phase, but ten thousand times less so in aqueous solution.
In aprotic polar solvents such as DMSO, DMF, and acetonitrile the energy of solvation is not as high as in protic polar solvents like water and methanol. For this reason, the basicity of amines in these aprotic solvents is almost solely governed by the electronic effects.
Synthesis.
From alcohols.
Industrially significant alkyl amines are prepared from ammonia by alkylation with alcohols:
<chem>ROH + NH3 -> RNH2 + H2O</chem>
From alkyl and aryl halides.
Unlike the reaction of amines with alcohols the reaction of amines and ammonia with alkyl halides is used for synthesis in the laboratory:
<chem>RX + 2 R'NH2 -> RR'NH + [RR'NH2]X</chem>
In such reactions, which are more useful for alkyl iodides and bromides, the degree of alkylation is difficult to control such that one obtains mixtures of primary, secondary, and tertiary amines, as well as quaternary ammonium salts.
Selectivity can be improved via the Delépine reaction, although this is rarely employed on an industrial scale. Selectivity is also assured in the Gabriel synthesis, which involves organohalide reacting with potassium phthalimide.
Aryl halides are much less reactive toward amines and for that reason are more controllable. A popular way to prepare aryl amines is the Buchwald-Hartwig reaction.
From alkenes.
Disubstituted alkenes react with HCN in the presence of strong acids to give formamides, which can be decarbonylated. This method, the Ritter reaction, is used industrially to produce tertiary amines such as "tert"-octylamine.
Hydroamination of alkenes is also widely practiced. The reaction is catalyzed by zeolite-based solid acids.
Reductive routes.
Via the process of hydrogenation, unsaturated N-containing functional groups are reduced to amines using hydrogen in the presence of a nickel catalyst. Suitable groups include nitriles, azides, imines including oximes, amides, and nitro. In the case of nitriles, reactions are sensitive to acidic or alkaline conditions, which can cause hydrolysis of the group. is more commonly employed for the reduction of these same groups on the laboratory scale.
Many amines are produced from aldehydes and ketones via reductive amination, which can either proceed catalytically or stoichiometrically.
Aniline () and its derivatives are prepared by reduction of the nitroaromatics. In industry, hydrogen is the preferred reductant, whereas, in the laboratory, tin and iron are often employed.
Specialized methods.
Many methods exist for the preparation of amines, many of these methods being rather specialized.
Reactions.
Alkylation, acylation, and sulfonation, etc..
Aside from their basicity, the dominant reactivity of amines is their nucleophilicity. Most primary amines are good ligands for metal ions to give coordination complexes. Amines are alkylated by alkyl halides. Acyl chlorides and acid anhydrides react with primary and secondary amines to form amides (the "Schotten–Baumann reaction").
Similarly, with sulfonyl chlorides, one obtains sulfonamides. This transformation, known as the Hinsberg reaction, is a chemical test for the presence of amines.
Because amines are basic, they neutralize acids to form the corresponding ammonium salts . When formed from carboxylic acids and primary and secondary amines, these salts thermally dehydrate to form the corresponding amides.
formula_0
Amines undergo sulfamation upon treatment with sulfur trioxide or sources thereof:
<chem>R2NH + SO3 -> R2NSO3H</chem>
Diazotization.
Amines reacts with nitrous acid to give diazonium salts. The alkyl diazonium salts are of little importance because they are too unstable. The most important members are derivatives of aromatic amines such as aniline ("phenylamine") (A = aryl or naphthyl):
<chem>ANH2 + HNO2 + HX -> AN2+ + X- + 2 H2O</chem>
Anilines and naphthylamines form more stable diazonium salts, which can be isolated in the crystalline form. Diazonium salts undergo a variety of useful transformations involving replacement of the group with anions. For example, cuprous cyanide gives the corresponding nitriles:
<chem>AN2+ + Y- -> AY + N2</chem>
Aryldiazoniums couple with electron-rich aromatic compounds such as a phenol to form azo compounds. Such reactions are widely applied to the production of dyes.
Conversion to imines.
Imine formation is an important reaction. Primary amines react with ketones and aldehydes to form imines. In the case of formaldehyde (R'
H), these products typically exist as cyclic trimers.
<chem>RNH2 + R'_2C=O -> R'_2C=NR + H2O</chem>
Reduction of these imines gives secondary amines:
<chem>R'_2C=NR + H2 -> R'_2CH-NHR</chem>
Similarly, secondary amines react with ketones and aldehydes to form enamines:
<chem> R2NH + R'(R"CH2)C=O -> R"CH=C(NR2)R' + H2O</chem>
Overview.
An overview of the reactions of amines is given below:
Biological activity.
Amines are ubiquitous in biology. The breakdown of amino acids releases amines, famously in the case of decaying fish which smell of trimethylamine. Many neurotransmitters are amines, including epinephrine, norepinephrine, dopamine, serotonin, and histamine. Protonated amino groups (–NH3+) are the most common positively charged moieties in proteins, specifically in the amino acid lysine. The anionic polymer DNA is typically bound to various amine-rich proteins. Additionally, the terminal charged primary ammonium on lysine forms salt bridges with carboxylate groups of other amino acids in polypeptides, which is one of the primary influences on the three-dimensional structures of proteins.
Amine hormones.
Hormones derived from the modification of amino acids are referred to as amine hormones. Typically, the original structure of the amino acid is modified such that a –COOH, or carboxyl, group is removed, whereas the –NH3+, or amine, group remains. Amine hormones are synthesized from the amino acids tryptophan or tyrosine.
Application of amines.
Dyes.
Primary aromatic amines are used as a starting material for the manufacture of azo dyes. It reacts with nitrous acid to form diazonium salt, which can undergo coupling reaction to form an azo compound. As azo-compounds are highly coloured, they are widely used in dyeing industries, such as:
Drugs.
Most drugs and drug candidates contain amine functional groups:
Gas treatment.
Aqueous monoethanolamine (MEA), diglycolamine (DGA), diethanolamine (DEA), diisopropanolamine (DIPA) and methyldiethanolamine (MDEA) are widely used industrially for removing carbon dioxide (CO2) and hydrogen sulfide (H2S) from natural gas and refinery process streams. They may also be used to remove CO2 from combustion gases and flue gases and may have potential for abatement of greenhouse gases. Related processes are known as sweetening.
Epoxy resin curing agents.
Amines are often used as epoxy resin curing agents. These include dimethylethylamine, cyclohexylamine, and a variety of diamines such as 4,4-diaminodicyclohexylmethane. Multifunctional amines such as tetraethylenepentamine and triethylenetetramine are also widely used in this capacity. The reaction proceeds by the lone pair of electrons on the amine nitrogen attacking the outermost carbon on the oxirane ring of the epoxy resin. This relieves ring strain on the epoxide and is the driving force of the reaction. Molecules with tertiary amine functionality are often used to accelerate the epoxy-amine curing reaction and include substances such as 2,4,6-Tris(dimethylaminomethyl)phenol. It has been stated that this is the most widely used room temperature accelerator for two-component epoxy resin systems.
Safety.
Low molecular weight simple amines, such as ethylamine, are only weakly toxic with LD50 between 100 and 1000 mg/kg. They are skin irritants, especially as some are easily absorbed through the skin. Amines are a broad class of compounds, and more complex members of the class can be extremely bioactive, for example strychnine.
References.
<templatestyles src="Reflist/styles.css" /> | [
{
"math_id": 0,
"text": "\n{\n \\underbrace\\ce{H-\\!\\!\\overset{\\displaystyle R1 \\atop |}{\\underset{| \\atop \\displaystyle R2}N}\\!\\!\\!\\!:}_\\text{amine} +\n \\underbrace\\ce{R3-\\overset{\\displaystyle O \\atop \\|}C-OH}_\\text{carboxylic acid} ->\n}\\ \n\\underbrace\\ce{{H-\\overset{\\displaystyle R1 \\atop |}{\\underset{| \\atop \\displaystyle R2}{N+}}-H} + R3-COO^-}\n_{\\text{substituted-ammonium} \\atop \\text{carboxylate salt}}\n\\ce{->[\\text{heat}][\\text{dehydration}]}{\n \\underbrace\\ce{\\overset{\\displaystyle R1 \\atop |}{\\underset{| \\atop \\displaystyle R2}N}\\!\\!-\\overset{\\displaystyle O \\atop \\|}C-R3}_\\text{amide} +\n \\underbrace\\ce{H2O}_\\text{water}\n}"
}
]
| https://en.wikipedia.org/wiki?curid=1412 |
14121 | Hertz | SI unit for frequency
<templatestyles src="Template:Infobox/styles-images.css" />
The hertz (symbol: Hz) is the unit of frequency in the International System of Units (SI), often described as being equivalent to one event (or cycle) per second. The hertz is an SI derived unit whose formal expression in terms of SI base units is s−1, meaning that one hertz is one per second or the reciprocal of one second. It is used only in the case of periodic events. It is named after Heinrich Rudolf Hertz (1857–1894), the first person to provide conclusive proof of the existence of electromagnetic waves. For high frequencies, the unit is commonly expressed in multiples: kilohertz (kHz), megahertz (MHz), gigahertz (GHz), terahertz (THz).
Some of the unit's most common uses are in the description of periodic waveforms and musical tones, particularly those used in radio- and audio-related applications. It is also used to describe the clock speeds at which computers and other electronics are driven. The units are sometimes also used as a representation of the energy of a photon, via the Planck relation "E" = "hν", where "E" is the photon's energy, "ν" is its frequency, and "h" is the Planck constant.
Definition.
The hertz is defined as one per second for periodic events. The International Committee for Weights and Measures defined the second as "the duration of periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium-133 atom" and then adds: "It follows that the hyperfine splitting in the ground state of the caesium 133 atom is exactly , "ν"hfs Cs =." The dimension of the unit hertz is 1/time (T−1). Expressed in base SI units, the unit is the reciprocal second (1/s).
In English, "hertz" is also used as the plural form. As an SI unit, Hz can be prefixed; commonly used multiples are kHz (kilohertz, ), MHz (megahertz, ), GHz (gigahertz, ) and THz (terahertz, ). One hertz (i.e. one per second) simply means "one periodic event occurs per second" (where the event being counted may be a complete cycle); means "one hundred periodic events occur per second", and so on. The unit may be applied to any periodic event—for example, a clock might be said to tick at , or a human heart might be said to beat at .
The occurrence rate of aperiodic or stochastic events is expressed in "reciprocal second" or "inverse second" (1/s or s−1) in general or, in the specific case of radioactivity, in becquerels. Whereas (one per second) specifically refers to one cycle (or periodic event) per second, (also one per second) specifically refers to one radionuclide event per second on average.
Even though frequency, angular velocity, angular frequency and radioactivity all have the dimension T−1, of these only frequency is expressed using the unit hertz. Thus a disc rotating at 60 revolutions per minute (rpm) is said to have an angular velocity of 2π rad/s and a frequency of rotation of . The correspondence between a frequency "f" with the unit hertz and an angular velocity "ω" with the unit radians per second is
formula_0 and formula_1
The hertz is named after Heinrich Hertz. As with every SI unit named for a person, its symbol starts with an upper case letter (Hz), but when written in full, it follows the rules for capitalisation of a common noun; i.e., "hertz" becomes capitalised at the beginning of a sentence and in titles but is otherwise in lower case.
History.
The hertz is named after the German physicist Heinrich Hertz (1857–1894), who made important scientific contributions to the study of electromagnetism. The name was established by the International Electrotechnical Commission (IEC) in 1935. It was adopted by the General Conference on Weights and Measures (CGPM) ("Conférence générale des poids et mesures") in 1960, replacing the previous name for the unit, "cycles per second" (cps), along with its related multiples, primarily "kilocycles per second" (kc/s) and "megacycles per second" (Mc/s), and occasionally "kilomegacycles per second" (kMc/s). The term "cycles per second" was largely replaced by "hertz" by the 1970s.
In some usage, the "per second" was omitted, so that "megacycles" (Mc) was used as an abbreviation of "megacycles per second" (that is, megahertz (MHz)).
Applications.
Sound and vibration.
Sound is a traveling longitudinal wave, which is an oscillation of pressure. Humans perceive the frequency of a sound as its pitch. Each musical note corresponds to a particular frequency. An infant's ear is able to perceive frequencies ranging from to ; the average adult human can hear sounds between and . The range of ultrasound, infrasound and other physical vibrations such as molecular and atomic vibrations extends from a few femtohertz into the terahertz range and beyond.
Electromagnetic radiation.
Electromagnetic radiation is often described by its frequency—the number of oscillations of the perpendicular electric and magnetic fields per second—expressed in hertz.
Radio frequency radiation is usually measured in kilohertz (kHz), megahertz (MHz), or gigahertz (GHz). Light is electromagnetic radiation that is even higher in frequency, and has frequencies in the range of tens (infrared) to thousands (ultraviolet) of terahertz. Electromagnetic radiation with frequencies in the low terahertz range (intermediate between those of the highest normally usable radio frequencies and long-wave infrared light) is often called terahertz radiation. Even higher frequencies exist, such as that of gamma rays, which can be measured in exahertz (EHz). (For historical reasons, the frequencies of light and higher frequency electromagnetic radiation are more commonly specified in terms of their wavelengths or photon energies: for a more detailed treatment of this and the above frequency ranges, see "Electromagnetic spectrum".)
Computers.
In computers, most central processing units (CPU) are labeled in terms of their clock rate expressed in megahertz () or gigahertz (). This specification refers to the frequency of the CPU's master clock signal. This signal is nominally a square wave, which is an electrical voltage that switches between low and high logic levels at regular intervals. As the hertz has become the primary unit of measurement accepted by the general populace to determine the performance of a CPU, many experts have criticized this approach, which they claim is an easily manipulable benchmark. Some processors use multiple clock cycles to perform a single operation, while others can perform multiple operations in a single cycle. For personal computers, CPU clock speeds have ranged from approximately in the late 1970s (Atari, Commodore, Apple computers) to up to in IBM Power microprocessors.
Various computer buses, such as the front-side bus connecting the CPU and northbridge, also operate at various frequencies in the megahertz range.
SI multiples.
Higher frequencies than the International System of Units provides prefixes for are believed to occur naturally in the frequencies of the quantum-mechanical vibrations of massive particles, although these are not directly observable and must be inferred through other phenomena. By convention, these are typically not expressed in hertz, but in terms of the equivalent energy, which is proportional to the frequency by the factor of the Planck constant.
Unicode.
The CJK Compatibility block in Unicode contains characters for common SI units for frequency. These are intended for compatibility with East Asian character encodings, and not for use in new documents (which would be expected to use Latin letters, e.g. "MHz").
Notes.
<templatestyles src="Reflist/styles.css" />
References.
<templatestyles src="Reflist/styles.css" /> | [
{
"math_id": 0,
"text": "\\omega = 2\\pi f"
},
{
"math_id": 1,
"text": "f = \\frac{\\omega}{2\\pi} ."
}
]
| https://en.wikipedia.org/wiki?curid=14121 |
141270 | Syntonic comma | Musical interval
In music theory, the syntonic comma, also known as the chromatic diesis, the Didymean comma, the Ptolemaic comma, or the diatonic comma is a small comma type interval between two musical notes, equal to the frequency ratio 81:80 (= 1.0125) (around 21.51 cents). Two notes that differ by this interval would sound different from each other even to untrained ears, but would be close enough that they would be more likely interpreted as out-of-tune versions of the same note than as different notes. The comma is also referred to as a Didymean comma because it is the amount by which Didymus corrected the Pythagorean major third (81:64, around 407.82 cents) to a just major third (5:4, around 386.31 cents).
The word "comma" came via Latin from Greek κόμμα, from earlier *κοπ-μα = "a thing cut off".
Relationships.
The prime factors of the just interval 81/80 known as the syntonic comma can be separated out and reconstituted into various sequences of two or more intervals that arrive at the comma, such as 81/1 × 1/80 or (fully expanded and sorted by prime) 1/2 × 1/2 × 1/2 × 1/2 × 3/1 × 3/1 × 3/1 × 3/1 × 1/5. All sequences are mathematically valid, but some of the more musical sequences people use to remember and explain the comma's composition, occurrence, and usage are listed below:
On a piano keyboard (typically tuned with 12-tone equal temperament) a stack of four fifths (700 × 4 = 2800 cents) is exactly equal to two octaves (1200 × 2 = 2400 cents) plus a major third (400 cents). In other words, starting from a C, both combinations of intervals will end up at E. Using justly tuned octaves (2:1), fifths (3:2), and thirds (5:4), however, yields two slightly different notes. The ratio between their frequencies, as explained above, is a syntonic comma (81:80). Pythagorean tuning uses justly tuned fifths (3:2) as well, but uses the relatively complex ratio of 81:64 for major thirds. Quarter-comma meantone uses justly tuned major thirds (5:4), but flattens each of the fifths by a quarter of a syntonic comma, relative to their just size (3:2). Other systems use different compromises. This is one of the reasons why 12-tone equal temperament is currently the preferred system for tuning most musical instruments.
Mathematically, by Størmer's theorem, 81:80 is the closest superparticular ratio possible with regular numbers as numerator and denominator. A superparticular ratio is one whose numerator is 1 greater than its denominator, such as 5:4, and a regular number is one whose prime factors are limited to 2, 3, and 5. Thus, although smaller intervals can be described within 5-limit tunings, they cannot be described as superparticular ratios.
Syntonic comma in the history of music.
The syntonic comma has a crucial role in the history of music. It is the amount by which some of the notes produced in Pythagorean tuning were flattened or sharpened to produce just minor and major thirds. In Pythagorean tuning, the only highly consonant intervals were the perfect fifth and its inversion, the perfect fourth. The Pythagorean major third (81:64) and minor third (32:27) were dissonant, and this prevented musicians from using triads and chords, forcing them for centuries to write music with relatively simple texture.
The syntonic tempering dates to Didymus the Musician, whose tuning of the diatonic genus of the tetrachord replaced one 9:8 interval with a 10:9 interval (lesser tone), obtaining a just major third (5:4) and semitone (16:15). This was later revised by Ptolemy (swapping the two tones) in his "syntonic diatonic" scale (συντονόν διατονικός, "syntonón diatonikós", from συντονός + διάτονος). The term "syntonón" was based on Aristoxenus, and may be translated as "tense" (conventionally "intense"), referring to tightened strings (hence sharper), in contrast to μαλακόν ("malakón", from μαλακός), translated as "relaxed" (conventional "soft"), referring to looser strings (hence flatter or "softer").
This was rediscovered in the late Middle Ages, where musicians realized that by slightly tempering the pitch of some notes, the Pythagorean thirds could be made consonant. For instance, if the frequency of E is decreased by a syntonic comma (81:80), C-E (a major third), and E-G (a minor third) become just. Namely, C-E is narrowed to a justly intonated ratio of
formula_0
and at the same time E-G is widened to the just ratio of
formula_1
The drawback is that the fifths A-E and E-B, by flattening E, become almost as dissonant as the Pythagorean wolf fifth. But the fifth C-G stays consonant, since only E has been flattened (C-E × E-G = 5/4 × 6/5 = 3/2), and can be used together with C-E to produce a C-major triad (C-E-G). These experiments eventually brought to the creation of a new tuning system, known as quarter-comma meantone, in which the number of major thirds was maximized, and most minor thirds were tuned to a ratio which was very close to the just 6:5. This result was obtained by narrowing each fifth by a quarter of a syntonic comma, an amount which was considered negligible, and permitted the full development of music with complex texture, such as polyphonic music, or melody with instrumental accompaniment. Since then, other tuning systems were developed, and the syntonic comma was used as a reference value to temper the perfect fifths in an entire family of them. Namely, in the family belonging to the syntonic temperament continuum, including meantone temperaments.
Comma pump.
The syntonic comma arises in "comma pump" ("comma drift") sequences such as C G D A E C, when each interval from one note to the next is played with certain specific intervals in just intonation tuning. If we use the frequency ratio 3/2 for the perfect fifths (C-G and D-A), 3/4 for the descending perfect fourths (G-D and A-E), and 4/5 for the descending major third (E-C), then the sequence of intervals from one note to the next in that sequence goes 3/2, 3/4, 3/2, 3/4, 4/5. These multiply together to give
formula_2
which is the syntonic comma (musical intervals stacked in this way are multiplied together). The "drift" is created by the combination of Pythagorean and 5-limit intervals in just intonation, and would not occur in Pythagorean tuning due to the use only of the Pythagorean major third (64/81) which would thus return the last step of the sequence to the original pitch.
So in that sequence, the second C is sharper than the first C by a syntonic comma . That sequence, or any transposition of it, is known as the comma pump. If a line of music follows that sequence, and if each of the intervals between adjacent notes is justly tuned, then every time the sequence is followed, the pitch of the piece rises by a syntonic comma (about a fifth of a semitone).
Study of the comma pump dates back at least to the sixteenth century when the Italian scientist Giovanni Battista Benedetti composed a piece of music to illustrate syntonic comma drift.
Note that a descending perfect fourth (3/4) is the same as a descending octave (1/2) followed by an ascending perfect fifth (3/2). Namely, (3/4) = (1/2) × (3/2). Similarly, a descending major third (4/5) is the same as a descending octave (1/2) followed by an ascending minor sixth (8/5). Namely, (4/5) = (1/2) × (8/5). Therefore, the above-mentioned sequence is equivalent to:
formula_3
or, by grouping together similar intervals,
formula_4
This means that, if all intervals are justly tuned, a syntonic comma can be obtained with a stack of four perfect fifths plus one minor sixth, followed by three descending octaves (in other words, four P5 plus one m6 minus three P8).
Notation.
Moritz Hauptmann developed a method of notation used by Hermann von Helmholtz. Based on Pythagorean tuning, subscript numbers are then added to indicate the number of syntonic commas to lower a note by. Thus a Pythagorean scale is C D E F G A B, while a just scale is C D E1 F G A1 B1. Carl Eitz developed a similar system used by J. Murray Barbour. Superscript positive and negative numbers are added, indicating the number of syntonic commas to raise or lower from Pythagorean tuning. Thus a Pythagorean scale is C D E F G A B, while the 5-limit Ptolemaic scale is C D E−1 F G A−1 B−1.
In Helmholtz-Ellis notation, a syntonic comma is indicated with up and down arrows added to the traditional accidentals. Thus a Pythagorean scale is C D E F G A B, while the 5-limit Ptolemaic scale is C D E F G A B.
Composer Ben Johnston uses a "−" as an accidental to indicate a note is lowered by a syntonic comma, or a "+" to indicate a note is raised by a syntonic comma. Thus a Pythagorean scale is C D E+ F G A+ B+, while the 5-limit Ptolemaic scale is C D E F G A B.
References.
<templatestyles src="Reflist/styles.css" /> | [
{
"math_id": 0,
"text": " {81\\over64} \\cdot {80\\over81} = {{1\\cdot5}\\over{4\\cdot1}} = {5\\over4}"
},
{
"math_id": 1,
"text": " {32\\over27} \\cdot {81\\over80} = {{2\\cdot3}\\over{1\\cdot5}} = {6\\over5}"
},
{
"math_id": 2,
"text": " {3\\over2} \\cdot {3\\over4} \\cdot {3\\over2} \\cdot {3\\over4} \\cdot {4\\over5} = {81\\over80}"
},
{
"math_id": 3,
"text": " {3\\over2} \\cdot {1\\over2} \\cdot {3\\over2} \\cdot {3\\over2} \\cdot {1\\over2} \\cdot {3\\over2} \\cdot {1\\over2} \\cdot {8\\over5} = {81\\over80}"
},
{
"math_id": 4,
"text": " {3\\over2} \\cdot {3\\over2} \\cdot {3\\over2} \\cdot {3\\over2} \\cdot {8\\over5} \\cdot {1\\over2} \\cdot {1\\over2} \\cdot {1\\over2} = {81\\over80}"
}
]
| https://en.wikipedia.org/wiki?curid=141270 |
1412703 | 1/N expansion | Perturbative analysis of quantum field theories
In quantum field theory and statistical mechanics, the 1/"N" expansion (also known as the "large "N"" expansion) is a particular perturbative analysis of quantum field theories with an internal symmetry group such as SO(N) or SU(N). It consists in deriving an expansion for the properties of the theory in powers of formula_0, which is treated as a small parameter.
This technique is used in QCD (even though formula_1 is only 3 there) with the gauge group SU(3). Another application in particle physics is to the study of AdS/CFT dualities.
It is also extensively used in condensed matter physics where it can be used to provide a rigorous basis for mean-field theory.
Example.
Starting with a simple example — the O(N) φ4 — the scalar field φ takes on values in the real vector representation of O(N). Using the index notation for the N "flavors" with the Einstein summation convention and because O(N) is orthogonal, no distinction will be made between covariant and contravariant indices. The Lagrangian density is given by
formula_2
where formula_3 runs from 1 to N. Note that N has been absorbed into the coupling strength λ. This is crucial here.
Introducing an auxiliary field F;
formula_4
In the Feynman diagrams, the graph breaks up into disjoint cycles, each made up of φ edges of the same flavor and the cycles are connected by F edges (which have no propagator line as auxiliary fields do not propagate).
Each 4-point vertex contributes λ/N and hence, 1/N. Each flavor cycle contributes N because there are N such flavors to sum over. Note that not all momentum flow cycles are flavor cycles.
At least perturbatively, the dominant contribution to the 2k-point connected correlation function is of the order (1/N)k-1 and the other terms are higher powers of 1/N. Performing a 1/N expansion gets more and more accurate in the large N limit. The vacuum energy density is proportional to N, but can be ignored due to non-compliance with general relativity assumptions.
Due to this structure, a different graphical notation to denote the Feynman diagrams can be used. Each flavor cycle can be represented by a vertex. The flavor paths connecting two external vertices are represented by a single vertex. The two external vertices along the same flavor path are naturally paired and can be replaced by a single vertex and an edge (not an F edge) connecting it to the flavor path. The F edges are edges connecting two flavor cycles/paths to each other (or a flavor cycle/path to itself). The interactions along a flavor cycle/path have a definite cyclic order and represent a special kind of graph where the order of the edges incident to a vertex matters, but only up to a cyclic permutation, and since this is a theory of real scalars, also an order reversal (but if we have SU(N) instead of SU(2), order reversals aren't valid). Each F edge is assigned a momentum (the momentum transfer) and there is an internal momentum integral associated with each flavor cycle.
QCD.
QCD is an SU(3) gauge theory involving gluons and quarks. The left-handed quarks belong to a triplet representation, the right-handed to an antitriplet representation (after charge-conjugating them) and the gluons to a real adjoint representation. A quark edge is assigned a color and orientation and a gluon edge is assigned a color pair.
In the large N limit, we only consider the dominant term. See AdS/CFT. | [
{
"math_id": 0,
"text": "1/N"
},
{
"math_id": 1,
"text": "N"
},
{
"math_id": 2,
"text": "\\mathcal{L}={1\\over 2}\\partial^\\mu \\phi_a \\partial_\\mu \\phi_a-{m^2\\over 2}\\phi_a \\phi_a-{\\lambda\\over 8N}(\\phi_a \\phi_a)^2"
},
{
"math_id": 3,
"text": "a"
},
{
"math_id": 4,
"text": "\\mathcal{L}={1\\over 2}\\partial^\\mu \\phi_a \\partial_\\mu \\phi_a -{m^2\\over 2}\\phi_a \\phi_a +{1\\over 2}F^2-{\\sqrt{\\lambda /N}\\over 2}F \\phi_a \\phi_a"
}
]
| https://en.wikipedia.org/wiki?curid=1412703 |
14128629 | SP-DEVS | SP-DEVS abbreviating "Schedule-Preserving Discrete Event System Specification" is a formalism for modeling and analyzing discrete event systems in both simulation and verification ways. SP-DEVS also provides modular and hierarchical modeling features which have been inherited from the Classic DEVS.
History.
SP-DEVS has been designed to support verification analysis of its networks by guaranteeing to obtain a finite-vertex reachability graph of the original networks, which had been an open problem of DEVS formalism for roughly 30 years. To get such a reachability graph of its networks, SP-DEVS has been imposed the three restrictions:
Thus, SP-DEVS is a sub-class of both DEVS and FD-DEVS. These three restrictions lead that SP-DEVS class is closed under coupling even though the number of states are finite. This property enables a finite-vertex graph-based verification for some qualitative properties and quantitative property, even with SP-DEVS coupled models.
Crosswalk Controller Example.
Consider a crosswalk system. Since a red light (resp. don't-walk light) behaves the opposite way of a green light (resp. walk light), for simplicity, we consider just two lights: a green light (G) and a walk light (W); and one push button as shown in Fig. 1. We want to control two lights of G and W with a set of timing constraints.
To initialize two lights, it takes 0.5 seconds to turn G on and 0.5 seconds later, W gets off. Then,
every 30 seconds, there is a chance that G becomes off and W on if someone pushed the push button. For a safety reason, W becomes on two seconds after G got off. 26 seconds later, W gets off and then two seconds later G gets back on. These behaviors repeats.
To build a controller for above requirements, we can consider one input event 'push-button' (abbreviated by ?p) and four output events 'green-on' (!g:1), 'green-off' (!g:0), 'walk-on' (!w:1) and 'walk-off (!w:0) which will be used as commands signals for the green light and the walk light. As a set of states of the controller, we considers 'booting-green' (BG), 'booting-walk' (BW), 'green-on' (G), 'green-to-red' (GR), 'red-on' (R), 'walk-on' (W), 'delay' (D). Let's design the state transitions as shown in Fig. 2. Initially, the controller starts at BG whose lifespan is 0.5 seconds. After the lifespan, it moves to BW state at this moment, it generates the 'green-on' event, too. After 0.5 seconds staying at BW, it moves to G state whose lifespan is 30 seconds. The controller can keep staying at G by looping G to G without generating any output event or can move to GR state when it receives the external input event ?p. But, the "actual staying time" at GR is the remaining time for looping at G. From GR, it moves to R state with generating an output event !g:0 and its R state last two seconds then it will move to W state with output event !w:1. 26 seconds later, it moves to D state with generating !w:0 and after staying 2 seconds at D, it moves back to G with output event !g:1.
Atomic SP-DEVS.
Formal Definition.
The above controller for crosswalk lights can be modeled by an atomic SP-DEVS model. Formally, an atomic SP-DEVS is a 7-tuple
formula_0
where
The above controller shown in Fig. 2 can be written as formula_11 where formula_12={?p}; formula_13={!g:0, !g:1, !w:0, !w:1}; formula_14={BG, BW, G, GR, R, W, D}; formula_15=BG, formula_16(BG)=0.5,formula_16(BW)=0.5, formula_16(G)=30, formula_16(GR)=30,formula_16(R)=2, formula_16(W)=26, formula_16(D)=2; formula_17(G,?p)=GR, formula_17(s,?p)=s if s formula_18G; formula_19(BG)=(!g:1, BW), formula_19(BW)=(!w:0, G),formula_19(G)=(formula_20, G), formula_19(GR)=(!g:0, R), formula_19(R)=(!w:1, W), formula_19(W)=(!w:0, D), formula_19(D)=(!g:1, G);
Behaviors of a SP-DEVS model.
To captured the dynamics of an atomic SP-DEVS, we need to introduce two variables associated to time. One is the "lifespan", the other is the "elapsed time" since the last resetting. Let formula_21 be the lifespan which is not continuously increasing but set by when a discrete event happens. Let formula_22 denote the elapsed time which is continuously increasing over time if there is no resetting.
Fig.3. shows a state trajectory associated with an event segment of the SP-DEVS model shown in Fig. 2.
The top of Fig.3. shows an event trajectory in which the horizontal axis is a time axis so it shows an event occurs at a certain time, for example, !g:1 occurs at 0.5 and !w:0 at 1.0 time unit, and so on. The bottom of Fig. 3 shows the state trajectory associated with the above event segment in which the state formula_23 is associated with its lifespan and its elapsed time in the form of formula_24. For example, (G, 30, 11) denotes that the state is G, its lifespan is and the elapsed time is 11 time units. The line segments of the bottom of Fig. 3 shows the time flow of the elapsed time which is the only one continuous variable in SP-DEVS.
One interesting feature of SF-DEVS might be the preservation of schedule the restriction (3) of SP-DEVS which is drawn at time 47 in Fig. 3. when the external event ?p happens. At this moment, even though the state can change from G to GR, the elapsed time does not change so the line segment is not broken at time 47 and formula_25 can grow up to formula_25 which is 30 in this example. Due to this preservation of the schedule from input events as well as the restriction of the time advance to the non-negative rational number (see restriction (2) above), the height of every saw can be a non-negative rational number or infinity (as shown in the bottom of Fig. 3.) in a SP-DEVS model.
A SP-DEVS model, formula_26 is DEVS formula_27 where
Advantages.
The property of non-negative rational-valued lifespans which are not changed by input events along with finite numbers of states and events guarantees that the behavior of SP-DEVS networks can be abstracted as an equivalent finite-vertex reachability graph by abstracting the infinitely-many values of the elapsed times.
To abstract the infinitely-many cases of elapsed times for each components of SP-DEVS networks, a time-abstraction method, called the "time-line abstraction" has been introduced [Hwang05],[HCZF07] in which the orders and relative difference of schedules are preserved. By using the time-line abstraction technique, the behavior of any SP-DEVS network can be abstracted as a reachability graph whose numbers of vertices and edges are finite.
As a qualitative property, safety of a SP-DEVS network is decidable by (1) generating the finite-vertex reachability graph of the given network and (2) checking whether some bad states are reachable or not [Hwang05].
As a qualitative property, liveness of a SP-DEVS network is decidable by (1) generating the finite-vertex reachability graph (RG) of the given network, (2) from RG, generating kernel directed acyclic graph (KDAG) in which a vertex is strongly connected component, and (3) checking if a vertex of KDAG contains a state transition cycle which contains a set of liveness states[Hwang05].
As a quantitative property, minimum and maximum processing time bounds from two events in SP-DEVS networks can be computed by (1) generating the finite-vertex reachability graph and (2.a) by finding the shortest paths for the minimum processing time bound and (2.b) by finding the longest paths (if available) for the maximum processing time bound [HCZF07].
Disadvantages.
Let a total state formula_39 of a SP-DEVS model be "passive" if formula_40; otherwise, it be "active".
One of known SP-DEVS's limitation is a phenomenon that "once an SP-DEVS model becomes passive, it never returns to become active (OPNA)". This phenomenon was found first at [Hwang 05b] although it was originally called ODNR ("once it dies, it never returns."). The reason why this one happens is because of the restriction (3) above in which no input event can change the schedule so the passive state can not be awaken into the active state.
For example, the toaster models drawn in Fig. 3(b) are not SP-DEVS because the total state associated with "idle" (I), is passive but it moves to an active state, "toast" (T) whose toating time is 20 seconds or 40 seconds. Actually, the model shown in Fig. 3(b) is FD-DEVS.
Tool.
There is an open source library, called DEVS# at http://xsy-csharp.sourceforge.net/DEVSsharp/, that supports some algorithms for finding safeness and liveness as well as Min/Max processing time bounds. | [
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| https://en.wikipedia.org/wiki?curid=14128629 |
14129120 | Finite & Deterministic Discrete Event System Specification | FD-DEVS (Finite & Deterministic Discrete Event System Specification) is a formalism for modeling and analyzing discrete event dynamic systems in both simulation and verification ways. FD-DEVS also provides modular and hierarchical modeling features which have been inherited from Classic DEVS.
History.
FD-DEVS was originally named as "Schedule-Controllable DEVS" [Hwang05] and designed to support verification analysis of its networks which had been an open problem of DEVS formalism for 30 years. In addition, it was also designated to resolve the so-called "OPNA" problem of SP-DEVS. From the viewpoint of Classic DEVS, FD-DEVS has three restrictions
The third restriction can be also seen as a relaxation from SP-DEVS where the schedule is always preserved by any input events. Due to this relaxation there is no longer OPNA problem, but there is also one limitation that a time-line abstraction which can be used for abstracting elapsed times of SP-DEVS networks is no longer useful for FD-DEVS network [Hwang05]. But another time abstraction method [Dill89] which was invented by Prof. D. Dill can be applicable to obtain a finite-vertex reachability graph for FD-DEVS networks.
Examples.
Ping-pong game.
Consider a single ping-pong match in which there are two players. Each player can be modeled by FD-DEVS such that the player model has an input event "?receive" and an output event "!send", and it has two states: "Send" and "Wait". Once the player gets into "Send", it will generates "!send" and go back to "Wait" after the sending time which is 0.1 time unit. When staying at "Wait" and if it gets "?receive", it changes into "Send" again. In other words, the player model stays at "Wait" forever unless it gets "?receive".
To make a complete ping-pong match, one player starts as an offender whose initial state is "Send" and the other starts as a defender whose initial state is "Wait". Thus in Fig. 1. Player A is the initial offender and Player B is the initial defender. In addition, to make the game continue, each player's "?send" event should be coupled to the other player's "?receive" as shown in Fig. 1.
Two-slot toaster.
Consider a toaster in which there are two slots that have their own start knobs as shown in Fig. 2(a). Each slot has the identical functionality except their toasting time. Initially, the knob is not pushed, but if one pushes the knob, the associated slot starts toasting for its toasting time: 20 seconds for the left slot, 40 seconds for the right slot. After the toasting time, each slot and its knobs pop up. Notice that even though one tries to push a knob when its associated slot is toasting, nothing happens.
One can model it with FD-DEVS as shown in Fig. 2(b). Two slots are modeled as atomic FD-DEVS whose input event is "?push" and output event is "!pop", states are "Idle" (I) and "Toast" (T) with the initial state is "idle". When it is "Idle" and receives "?push" (because one pushes the knob), its state changes to "Toast". In other words, it stays at "Idle" forever unless it receives "?push" event. 20 (res. 40) seconds later the left (res. right) slot returns to "Idle".
Atomic FD-DEVS.
Formal Definition.
formula_0
where
The formal representation of the player in the ping-pong example shown in Fig. 1 can be given as follows. formula_15 where formula_16={?receive}; formula_17={!send}; formula_18={Send, Wait}; formula_19=Send for Player A, Wait for Player B; formula_20(Send)=0.1,formula_20(Wait)=formula_21; formula_22(Wait,?receive)=(Send,1), formula_22(Send,?receive)=(Send,0); formula_23(Send)=(!send, Wait), formula_23(Wait)=(formula_24, Wait).
The formal representation of the slot of Two-slot Toaster Fig. 2(a) and (b) can be given as follows. formula_15 where formula_16={?push}; formula_17={!pop}; formula_18={I, T}; formula_19=I; formula_20(T)=20 for the left slot, 40 for the right slot, formula_20(I)=formula_21; formula_22(I, ?push)=(T,1), formula_22(T,?push)=(T,0); formula_23(T)=(!pop, I), formula_23(I)=(formula_24, I).
As mentioned above, FD-DEVS is an relaxation of SP-DEVS. That means, FD-DEVS is a supper class of SP-DEVS. We would give a model of FD-DEVS of a crosswalk light controller which is used for SP-DEVS in this Wikipedia. formula_15 where formula_16={?p}; formula_17={!g:0, !g:1, !w:0, !w:1}; formula_18={BG, BW, G, GR, R, W, D}; formula_19=BG, formula_20(BG)=0.5,formula_20(BW)=0.5, formula_20(G)=30, formula_20(GR)=30,formula_20(R)=2, formula_20(W)=26, formula_20(D)=2; formula_22(G,?p)=(GR,0), formula_22(s,?p)=(s,0) if s formula_25G; formula_23(BG)=(!g:1, BW), formula_23(BW)=(!w:0, G),formula_23(G)=(formula_26, G), formula_23(GR)=(!g:0, R), formula_23(R)=(!w:1, W), formula_23(W)=(!w:0, D), formula_23(D)=(!g:1, G);
Behaviors of FD-DEVS Models.
A FD-DEVS model, formula_27 is DEVS formula_28 where
formula_37
For details of DEVS behavior, the readers can refer to Behavior of Atomic DEVS
Fig. 3. shows an event segment (top) and the associated state trajectory (bottom) of Player A who plays the ping-pong game introduced in Fig. 1. In Fig. 3. the status of Player A is described as (state, lifespan, elapsed time)=(formula_41) and the line segment of the bottom of Fig. 3. denotes the value of the elapsed time. Since the initial state of Player A is "Send" and its lifetime is 0.1 seconds, the height of (Send, 0.1, formula_42) is 0.1 which is the value of formula_43. After changing into (Wait, inf, 0) when formula_44 is reset by 0, Player A doesn't know when formula_44 becomes 0 again. However, since Player B sends back the ball to Player A 0.1 second later, Player A gets back to (Send, 0.1 0) at time 0.2. From that time, 0.1 seconds later when Player A's status becomes (Send, 0.1, 0.1), Player A sends back the ball to Player B and gets into (Wait, inf, 0). Thus, this cyclic state transitions which move "Send" to "Wait" back and forth will go forever.
Fig. 4. shows an event segment (top) and the associated state trajectory (bottom) of the left slot of the two-slot toaster introduced in Fig. 2. Like Fig.3, the status of the left slot is described as (state, lifespan, elapsed time)=(formula_41) in Fig. 4. Since the initial state of the toaster is "I" and its lifetime is infinity, the height of (Wait, inf, formula_42) can be determined by when ?push occurs. Fig. 4. illustrates the case ?push happens at time 40 and the toaster changes into (T, 20, 0). From that time, 20 seconds later when its status becomes (T, 20, 20), the toaster gets back to (Wait, inf, 0) where we don't know when it gets back to "Toast" again. Fig. 4. shows the case that ?push occurs at time 90 so the toaster get into (T,20,0). Notice that even though there someone push again at time 97, that status (T, 20, 7) doesn't change at all because formula_45(T,?push)=(T,1).
Advantages.
Applicability of Time-Zone Abstraction.
The property of non-negative rational-valued lifespans which can be preserved or changed by input events along with finite numbers of states and events guarantees that the behavior of FD-DEVS networks can be abstracted as an equivalent finite-vertex reachability graph by abstracting the infinitely-many values of the elapsed times using the time abstracting technique introduced by Prof. D. Dill [Dill89]. An algorithm generating a finite-vertex reachability graph (RG)has been introduced in [HZ06a], [HZ07].
Reachability Graph.
Fig. 5. shows the reachability graph of two-slot toaster which was shown in Fig. 2. In the reachability graph, each vertex has its own discrete state and time zone which are ranges of formula_46
and formula_47. For example, for node (6) of Fig. 5, discrete state information is ((E,formula_48),(T,40)), and time zone is formula_49. Each directed arch shows how its source vertex changes into the destination vertex along with an associated event and a set of reset models. For example, the transition arc (6) to (5) is triggered by "push1" event. At that time, the set {1} of the arc denotes that elapsed time of 1 (that is formula_50 is reset by 0 when transition (6) to (5) occurs. For more detailed information, the reader can refer to [HZ07].
Decidability of Safety.
As a qualitative property, safety of a FD-DEVS network is decidable by (1) generating RG of the given network and (2) checking whether some bad states are reachable or not [HZ06b].
Decidability of Liveness.
As a qualitative property, liveness of a FD-DEVS network is decidable by (1) generating RG of the given network, (2) from RG, generating kernel directed acyclic graph (KDAG) in which a vertex is strongly connected component, and (3) checking if a vertex of KDAG contains a state transition cycle which contains a set of liveness states[HZ06b].
Disadvantages.
Weak Expressiveness for describing nondeterminism.
The features that all characteristic functions,formula_51 of FD-DEVS are deterministic can be seen as somehow a limitation to model the system that has non-deterministic behaviors. For example, if a player of the ping-pong game shown in Fig. 1. has a stochastic lifespan at "Send" state, FD-DEVS doesn't capture the non-determinism effectively.
Tool.
For Verification.
There are two open source libraries DEVS# written in C# at http://xsy-csharp.sourceforge.net/DEVSsharp/ and XSY written in Python at https://code.google.com/p/x-s-y/ that support some reachability graph-based verification algorithms for finding safeness and liveness.
For Simulation via XML.
For standardization of DEVS, especially using FDDEVS, Dr. Saurabh Mittal together with co-workers has worked on defining of XML format of FDDEVS. We can find an article at http://www.duniptechnologies.com/research/xfddevs/. This standard XML format was used for UML execution [RCMZ09] . | [
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| https://en.wikipedia.org/wiki?curid=14129120 |
14130472 | L-2-hydroxyglutarate dehydrogenase | In enzymology, an L-2-hydroxyglutarate dehydrogenase (EC 1.1.99.2) is an enzyme that catalyzes the chemical reaction
(S)-2-hydroxyglutarate + acceptor formula_0 2-oxoglutarate + reduced acceptor
Thus, the two substrates of this enzyme are (S)-2-hydroxyglutarate and acceptor, whereas its two products are 2-oxoglutarate and reduced acceptor.
Enzymes which preferentially catalyze the conversion of the (R) stereoisomer of 2-oxoglutarate also exist in both mammals and plants
and are named D-2-hydroxyglutarate dehydrogenase. L-2-hydroxyglutarate is produced by promiscuous action of malate dehydrogenase on 2-oxoglutarate; L-2-hydroxyglutarate dehydrogenase is an example of a metabolite repair enzyme that oxidizes L-2-hydroxyglutarate back to 2-oxoglutarate.
Nomenclature.
This enzyme belongs to the family of oxidoreductases, specifically those acting on the CH-OH group of donor with other acceptors. The systematic name of this enzyme class is (S)-2-hydroxyglutarate:acceptor 2-oxidoreductase. Other names in common use include:
<templatestyles src="Div col/styles.css"/>
Clinical significance.
Deficiency in this enzyme in humans (L2HGDH) or in the model plant "Arabidopsis thaliana" (At3g56840) leads to accumulation of L-2-hydroxyglutarate. In humans this results in the fatal neurometabolic disorder 2-Hydroxyglutaric aciduria whereas plants seem to be unaffected by elevated cellular concentrations of this compound
References.
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Further reading.
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| https://en.wikipedia.org/wiki?curid=14130472 |
14130492 | 2-oxo-acid reductase | Class of enzymes
In enzymology, a 2-oxo-acid reductase (EC 1.1.99.30) is an enzyme that catalyzes the chemical reaction
a (2R)-hydroxy-carboxylate + acceptor formula_0 a 2-oxo-carboxylate + reduced acceptor
Thus, the two substrates of this enzyme are (2R)-hydroxy-carboxylate and acceptor, whereas its two products are 2-oxo-carboxylate and reduced acceptor.
This enzyme belongs to the family of oxidoreductases, specifically those acting on the CH-OH group of donor with other acceptors. The systematic name of this enzyme class is (2R)-hydroxy-carboxylate:acceptor oxidoreductase. Other names in common use include (2R)-hydroxycarboxylate-viologen-oxidoreductase, HVOR, and 2-oxoacid reductase.
References.
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| https://en.wikipedia.org/wiki?curid=14130492 |
14130514 | 3-hydroxycyclohexanone dehydrogenase | Class of enzymes
In enzymology, a 3-hydroxycyclohexanone dehydrogenase (EC 1.1.99.26) is an enzyme that catalyzes the chemical reaction
3-hydroxycyclohexanone + acceptor formula_0 cyclohexane-1,3-dione + reduced acceptor
Thus, the two substrates of this enzyme are 3-hydroxycyclohexanone and acceptor, whereas its two products are cyclohexane-1,3-dione and reduced acceptor.
This enzyme belongs to the family of oxidoreductases, specifically those acting on the CH-OH group of donor with other acceptors. The systematic name of this enzyme class is 3-hydroxycyclohexanone:acceptor 1-oxidoreductase.
References.
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| https://en.wikipedia.org/wiki?curid=14130514 |
14130543 | 4-hydroxymandelate oxidase | Class of enzymes
In enzymology, a 4-hydroxymandelate oxidase (EC 1.1.3.19) is an enzyme that catalyzes the chemical reaction
(S)-2-hydroxy-2-(4-hydroxyphenyl)acetate + O2 formula_0 4-hydroxybenzaldehyde + CO2 + H2O2
Thus, the two substrates of this enzyme are (S)-2-hydroxy-2-(4-hydroxyphenyl)acetate and O2, whereas its 3 products are 4-hydroxybenzaldehyde, CO2, and H2O2.
This enzyme belongs to the family of oxidoreductases, specifically those acting on the CH-OH group of donor with oxygen as acceptor. The systematic name of this enzyme class is (S)-2-hydroxy-2-(4-hydroxyphenyl)acetate:oxygen 1-oxidoreductase. This enzyme is also called L-4-hydroxymandelate oxidase (decarboxylating). It has 2 cofactors: FAD, and Manganese.
References.
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| https://en.wikipedia.org/wiki?curid=14130543 |
14130570 | Alcohol dehydrogenase (acceptor) | In enzymology, an alcohol dehydrogenase (acceptor) (EC 1.1.99.8) is an enzyme that catalyzes the chemical reaction
a primary alcohol + acceptor formula_0 an aldehyde + reduced acceptor
Thus, the two substrates of this enzyme are primary alcohol and acceptor, whereas its two products are aldehyde and reduced acceptor.
This enzyme belongs to the family of oxidoreductases, specifically those acting on the CH-OH group of donor with other acceptors. The systematic name of this enzyme class is alcohol:acceptor oxidoreductase. Other names in common use include primary alcohol dehydrogenase, MDH, quinohemoprotein alcohol dehydrogenase, quinoprotein alcohol dehydrogenase, quinoprotein ethanol dehydrogenase, and alcohol:(acceptor) oxidoreductase. This enzyme participates in 5 metabolic pathways: glycolysis / gluconeogenesis, 1,2-dichloroethane degradation, propanoate metabolism, butanoate metabolism, and methane metabolism. It employs one cofactor, PQQ.
Structural studies.
As of late 2007, 11 structures have been solved for this class of enzymes, with PDB accession codes 1G72, 1H4I, 1H4J, 1LRW, 1W6S, 2AD6, 2AD7, 2AD8, 2D0V, 4AAH, and 8ADH.
References.
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14130594 | Alcohol oxidase | Class of enzymes
In enzymology, an alcohol oxidase (EC 1.1.3.13) is an enzyme that catalyzes the chemical reaction
a primary alcohol + O2 formula_0 an aldehyde + H2O2
Thus, the two substrates of this enzyme are primary alcohol and O2, whereas its two products are aldehyde and H2O2.
This enzyme belongs to the family of oxidoreductases, specifically those acting on the CH-OH group of the donor with oxygen as the acceptor. It employs one cofactor, FAD.
Name.
The systematic name of this enzyme class is alcohol:oxygen oxidoreductase. This enzyme is also called methanol oxidase and ethanol oxidase. Sometimes, this enzyme is called short-chain alcohol oxidase (SCAO) to differentiate it from long-chain-alcohol oxidase (LCAO), aryl-alcohol oxidase (AAO) and secondary-alcohol oxidase (SAO).
Reaction.
Alcohol oxidases catalyzes the oxidation of primary alcohols to their corresponding aldehydes. Unlike alcohol dehydrogenases, they are unable to catalyze the reverse reaction. This is reflected in their cofactor as well—unlike alcohol dehydrogenases, which use NAD+, alcohol oxidases use FAD. SCAO is capable of oxidizing alcohols with up to 8 carbons, but their primary substrates are methanol and ethanol.
Known inhibitors of this enzyme include H2O2, Cu2+, phenanthroline, acetamide, potassium cyanide, or cyclopropanone.
Properties.
SCAO is an intracellular enzyme. Its common source are fungi and yeasts, but it has also been shown to be present in mollusks.
It appears as an octameric protein, except for SCAO from "A. ochraceus", which has been shown to be a tetramer. Each of the subunites is 65-80 kDa. Nine SCAO isoforms have been found in "Ogataea methanolica", which has been shown to be a result of different combinations of two different subunits in the octamers, each coded by a different gene.
Structural studies.
As of late 2007, 9 structures have been solved for this class of enzymes, with PDB accession codes 1AHU, 1AHV, 1AHZ, 1VAO, 1W1J, 1W1K, 1W1L, 1W1M, and 2VAO.
Potential use.
Several potential uses have been suggested for SCAO:
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14130606 | Alkan-1-ol dehydrogenase (acceptor) | In enzymology, an alkan-1-ol dehydrogenase (acceptor) (EC 1.1.99.20) is an enzyme that catalyzes the chemical reaction
primary alcohol + acceptor formula_0 aldehyde + reduced acceptor
Thus, the two substrates of this enzyme are primary alcohol and acceptor, whereas its two products are aldehyde and reduced acceptor.
This enzyme belongs to the family of oxidoreductases, specifically those acting on the CH-OH group of donor with other acceptors. The systematic name of this enzyme class is alkan-1-ol:acceptor oxidoreductase. Other names in common use include polyethylene glycol dehydrogenase, and alkan-1-ol:(acceptor) oxidoreductase. This enzyme participates in fatty acid metabolism. It employs one cofactor, PQQ.
References.
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14130622 | Aryl-alcohol oxidase | In enzymology, an aryl-alcohol oxidase (EC 1.1.3.7) is an enzyme that catalyzes the chemical reaction
an aromatic primary alcohol + O2 formula_0 an aromatic aldehyde + H2O2
Thus, the two substrates of this enzyme are aromatic primary alcohol and O2, whereas its two products are aromatic aldehyde and H2O2.
This enzyme belongs to the family of oxidoreductases, specifically those acting on the CH-OH group of donor with oxygen as acceptor. The systematic name of this enzyme class is aryl-alcohol:oxygen oxidoreductase. Other names in common use include aryl alcohol oxidase, veratryl alcohol oxidase, and arom. alcohol oxidase.
Structural studies.
As of late 2007, 4 structures have been solved for this class of enzymes, with PDB accession codes 1E8F, 1E8H, 1QLT, and 1QLU.
References.
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14130635 | Catechol oxidase (dimerizing) | Enzyme
In enzymology, a catechol oxidase (dimerizing) (EC 1.1.3.14) is an enzyme that catalyzes the chemical reaction
4 catechol + 3 O2 formula_0 2 dibenzo[1,4]dioxin-2,3-dione + 6 H2O
Thus, the two substrates of this enzyme are catechol and O2, whereas its two products are dibenzo[1,4]dioxin-2,3-dione and H2O.
This enzyme belongs to the family of oxidoreductases, specifically those acting on the CH-OH group of donor with oxygen as acceptor. The systematic name of this enzyme class is catechol:oxygen oxidoreductase (dimerizing).
References.
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14130647 | Cellobiose dehydrogenase (acceptor) | In enzymology, a cellobiose dehydrogenase (acceptor) (EC 1.1.99.18) is an enzyme that catalyzes the chemical reaction
cellobiose + acceptor formula_0 cellobiono-1,5-lactone + reduced acceptor
Thus, the two substrates of this enzyme are cellobiose and acceptor, whereas its two products are cellobiono-1,5-lactone and reduced acceptor.
This enzyme belongs to the family of oxidoreductases, to be specific those acting on the CH-OH group of donor with other acceptors. The systematic name of this enzyme class is cellobiose:acceptor 1-oxidoreductase. Other names in common use include cellobiose dehydrogenase, cellobiose oxidoreductase, Phanerochaete chrysosporium cellobiose oxidoreductase, CBOR, cellobiose oxidase, cellobiose:oxygen 1-oxidoreductase, CDH, and cellobiose:(acceptor) 1-oxidoreductase. It employs sometimes one cofactor, FAD, but in most cases both a heme and a FAD located in separate domains. It makes the enzyme to one of the more complex extracellular oxidoreductases. It is produced by wood degrading organisms.
Structural studies.
To date, structures of the separated dehydrogenase (DH) and cytochrome (CYT) domains were reported (PDB accession codes 1NAA and 1PL3). In 2015, full-length structures of the enzyme were resolved for CDH from "Crassicarpon hotsonii" syn. "Myriococcum thermophilum" ("Ch"CDH, PDB accession code 4QI6) and CDH from "Neurospora crassa" ("Nc"CDH, PDB accession code 4QI7).
The mobility of the CYT domain prevented for a long time the crystallization and X-ray structure elucidation of CDH. However, the crystal structures of the individual CYT and DH domains already showed structural complementarity and the position of the domain interface and provided a structural basis for the interdomain electron transfer observed in biochemical and bioelectrochemical studies.
The crystal structure of the "Phanerochaete chrysosporium" DH domain was described by Hallberg et al. as peanut-shaped with dimensions of ~72 x 57 x 45 Å. The observed GMC-oxidoreductase protein fold features an FAD-binding and a substrate-binding subdomain. The FAD-binding subdomain primarily consists of the Rossmann fold, which is typical for NAD(P)- and FAD-dependent enzymes whereas the substrate-binding subdomain consists of a central twisted, seven-stranded β-sheet with three α-helices on one side of the sheet and the active-site on the other side. The substrate-binding subdomain hosts the active-site that consists of a substrate binding-site (B-site) and the catalytic-site (C-site). The B-site holds the non-reducing end of cellobiose in position whereas at the C-site oxidizes the reducing end of cellobiose.
The CYT domain consist of an ellipsoidal antiparallel β-sandwich with dimensions of 30 x 36 x 47 Å and a topology resembling the variable heavy chain of the antibody Fab fragment with a five-stranded inner β-sheet and a six-stranded outer β-sheet . This protein fold was a new fold observed for a cytochrome. Both individual crystal structures provided mechanistic insights. The DH domain was soaked with the inhibitor cellobionolactam which resolved the binding position of the substrate and gave a mechanistic explanation for the reductive half-reaction [52]. The CYT domain structure revealed an unusual axial haem b iron coordination by a His and a Met residue, which is the basis for intramolecular electron transfer between the two domains. A mutational study on the iron-coordinating haem ligands performed by Rotseart et al. showed that either the replacement of Met by His, or the switching of both ligands resulted in an IET inactive CYT domain.
In 2015, the elucidation of the full-length structures of CDH from "Crassicarpon hotsonii" (syn. "Myriococcum thermophilum") (PDB ID: 4QI6) and "Neurospora crassa" (PDB ID: 4QI7) increased the understanding of the domain mobility for the role of CYT as built-in redox mediator. Tan and Kracher et al. showed that CDH exists in a closed-state and an open-state conformation. In the closed-state the haem b accepts an electron from the FAD after substrate oxidation and it donates the electron to an external electron acceptor in the open-state. It was found that IET does not depend on the prominent Trp residue positioned directly between the FAD and the haem cofactor, but on a surface exposed Arg in the substrate channel, which stabilises the interaction with CYT through a haem propionate group. The heme b propionate-D is folded away, and a hydrogen bond to Tyr99 in the CYT domain prevents it from interacting directly with the DH active site. The closest edge-to-edge distance between heme b and FAD is 9 Å, which is well within the 14-Å limit for efficient electron transfer. The mobile CYT domain of CDH is a unique feature among GMC-oxidoreductases and acquired at a later stage of evolution. The CYT domain mobility is regulated by a flexible peptide linker of varying length and composition ("e.g.", "N. crassa" CDH IIA = 17 amino acids, "N. crassa" CDH IIB = 33 amino acids), which connects both domains. The movement of CYT has been observed by AFM and SAXS studies.
Phylogenetic classification.
The CDH sequences evolved into four phylogenetic branches: Class I CDH sequences are found exclusively in Basidiomycota whereas CDH sequences from Class II, Class III, and Class IV are found only in Ascomycota. Class I CDHs have a strong affinity for cellulose and bind presumably via a cellulose binding-site on the surface of the DH domain that is different from the active-site. This cellulose binding-site is not present in other CDH classes. The binding to cellulose in Class II CDHs depends on the presence of a C-terminal family 1 carbohydrate binding module (CBM1). Some Class II CDH feature a C-terminal CBM1 and are classified as Class IIA CDHs, whereas Class IIB CDHs have no CBM1 and do not bind to cellulose. Other differences between Class I and Class II CDHs are in their substrate specificity and the pH optimum of the IET. Little can be said about Class III and Class IV CDHs, which have not yet been expressed and characterised. From sequence alignments a different active-site geometry can be deduced, but the substrate is unknown. Class IV CDHs are evolutionary furthest related to the other CDH classes and do not feature an electron transferring CYT domain, which suggests a different physiological role and the loss of its ability for DET.
Catalysis.
CDH catalyses the 2e-/2H+ oxidation of the anomeric carbon atom (C1) of the disaccharide cellobiose to the cellobiono-δ-lactone hydrolyses further to cellobionic acid in water. Besides the natural substrate cellobiose and the cellodextrins cellotriose, cellotetraose, cellopentaose, and cellohexaose also the cellulose building unit glucose, as well as the hemicellulose building monomers or breakdown products galactose, mannose, mannobiose, mannopentaose, xylose, xylobiose and xylotriose, or starch derived maltose, maltotriose and maltotetraose have been reported as substrates of greatly varying catalytic efficiency for CDH. A very good substrate not related to plant polysaccharides is lactose, because of its structural similarity to cellobiose. Cellobiose, cello-oligosaccharides, and lactose are the substrates for which CDH exhibits the highest catalytic efficiency, whereas monosaccharides are bad substrates with a very low affinity (high KM) to only the catalytic C-site of CDH. The extreme discrepancy of the catalytic efficiencies of "P. chrysosporium" CDH for cellobiose over glucose (87500 : 1) is being connected to the physiological role of the white-rot basidiomycete enzyme. In ascomycete CDHs the discrimination of glucose is less strict and a wider spectrum of mono- and oligosaccharides are converted. Of course, the affinity of CDH to di- and oligosaccharide substrates is much higher than for monosaccharides. However, Class II CDHs have KM-values for glucose of 10–100 mM, which is in the range of FAD-dependent glucose oxidase, also a GMC-oxidoreductase (5–100 mM). The lower catalytic rate of Class I and Class II CDHs (below 50 s−1) compared to glucose oxidase (300–2000 s−1) is probably an evolutionary adaption to the acquired CYT domain to optimize IET and prevent futile reactions of the substrate in an idle active-site.
CDH as a bioelectrocatalyst.
CDH exhibits various properties that makes it a suitable electrocatalyst for biosensors or biofuel cells. Unlike any other carbohydrate converting GMC-oxidoreductase it can be contacted via mediated electron transfer (MET) or direct electron transfer (DET). CDH-based 2nd and 3rd generation biosensors for the quantitation of commercially or medically relevant molecules such as glucose or lactose have been developed. The advantage 2nd generation biosensors is their high sensitivity, the advantage of 3rd generation biosensors is the avoidance of redox mediators in implantable blood glucose monitoring systems. CDH-based 3rd generation biosensors have a robust performance because of the good thermal and turnover stability of CDH and the mobile CYT domain, which can interact with many electrode materials and surface modifications and deliver reasonably high current densities.
CDH-based biosensors.
CDH has been used as biosensor element for the detection of a number of different analytes ever since 1992, when Elmgren et al. reported on a new biosensor sensitive towards cello-oligosaccharides with degrees of polymerization of 2 to 6, lactose and maltose. Research on CDH employed as biosensor since then until 2013 has been reviewed, providing lists of CDH-based biosensors in both DET and MET mode for phenols (catechol, dopamine, noradrenaline, hydroquinone, aminophenol) and carbohydrates (cellobiose, lactose, maltose, glucose).
Both Class I and Class II CDHs have been employed as biocatalysts for biosensors. In accordance with their preferred substrates, Class I CDHs were mainly applied for the detection and quantitation of phenols, cellobiose, and lactose, whereas Class II CDHs have been applied for the detection of cellobiose, lactose, maltose and glucose. Among Class I CDHs, "Phanerochaete chrysosporium" CDH is the most comprehensively studied enzyme was employed for the detection of various phenol derivates, cellobiose and lactose. "Phanerochaete sordida" and "Trametes villosa" CDH have been used for the detection of lactose and Sclerotium rolfsii CDH for the detection of dopamine.
Among Class II CDHs, "Crassicarpon hotsonii" and "Crassicarpon thermophilum" CDH were employed as recognition element for cellobiose, lactose, glucose and maltose and "Humicola insolens" CDH for glucose and maltose. Detection of maltose was in all cases only reported to show that maltose is not interfering with glucose detection, which is important for blood glucose testing.
In 2017, DirectSens GmbH launched the first lactose biosensor based on a CDH for very low lactose concentrations in lactose reduced milk products. LactoSens is the only third-generation biosensor on the market and distributed globally to dairy companies.
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14130663 | Cholesterol oxidase | Class of enzymes
In enzymology, a cholesterol oxidase (EC 1.1.3.6) is an enzyme that catalyzes the chemical reaction
cholesterol + O2 formula_0 cholest-4-en-3-one + H2O2
Thus, the two substrates of this enzyme are cholesterol and O2, whereas its two products are cholest-4-en-3-one and H2O2.
This enzyme belongs to the family of oxidoreductases, specifically those acting on the CH-OH group of donor with oxygen as acceptor. The systematic name of this enzyme class is cholesterol:oxygen oxidoreductase. Other names in common use include cholesterol- O2 oxidoreductase, 3beta-hydroxy steroid oxidoreductase, and 3beta-hydroxysteroid:oxygen oxidoreductase. This enzyme participates in bile acid biosynthesis.
The substrate-binding domain found in some bacterial cholesterol oxidases is composed of an eight-stranded mixed beta-pleated sheet and six alpha-helices. This domain is positioned over the isoalloxazine ring system of the FAD cofactor bound by the FAD-binding domain and forms the roof of the active site cavity, allowing for catalysis of oxidation and isomerisation of cholesterol to cholest-4-en-3-one.
Structural studies.
As of late 2007, 14 structures have been solved for this class of enzymes, with PDB accession codes 1B4V, 1B8S, 1CBO, 1CC2, 1COY, 1I19, 1IJH, 1MXT, 1N1P, 1N4U, 1N4V, 1N4W, 2GEW, and 3COX.
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14130680 | Choline dehydrogenase | Class of enzymes
In enzymology, a choline dehydrogenase (EC 1.1.99.1) is an enzyme that catalyzes the chemical reaction
choline + acceptor formula_0 betaine aldehyde + reduced acceptor
Thus, the two substrates of this enzyme are choline and acceptor, whereas its two products are betaine aldehyde and reduced acceptor.
This enzyme belongs to the family of oxidoreductases, specifically those acting on the CH-OH group of donor with other acceptors. The systematic name of this enzyme class is choline:acceptor 1-oxidoreductase. Other names in common use include choline oxidase, choline-cytochrome c reductase, choline:(acceptor) oxidoreductase, and choline:(acceptor) 1-oxidoreductase. This enzyme participates in glycine, serine and threonine metabolism. It employs one cofactor, PQQ.
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14130703 | Choline oxidase | In enzymology, a choline oxidase (EC 1.1.3.17) is an enzyme that catalyzes the chemical reaction
choline + O2 formula_0 betaine aldehyde + H2O2
Thus, the two substrates of this enzyme are choline and O2, whereas its two products are betaine aldehyde and H2O2.
This enzyme belongs to the family of oxidoreductases, specifically those acting on the CH-OH group of donor with oxygen as acceptor. The systematic name of this enzyme class is choline:oxygen 1-oxidoreductase. This enzyme participates in glycine, serine, and threonine metabolism. It employs one cofactor, FAD.
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14130725 | D-2-hydroxy-acid dehydrogenase | In enzymology, a D-2-hydroxy-acid dehydrogenase (EC 1.1.99.6) is an enzyme that catalyzes the chemical reaction
(R)-lactate + acceptor formula_0 pyruvate + reduced acceptor
Thus, the two substrates of this enzyme are (R)-lactate and acceptor, whereas its two products are pyruvate and reduced acceptor.
This enzyme belongs to the family of oxidoreductases, specifically those acting on the CH-OH group of donor with other acceptors. The systematic name of this enzyme class is (R)-2-hydroxy-acid:acceptor 2-oxidoreductase. Other names in common use include D-2-hydroxy acid dehydrogenase, and (R)-2-hydroxy-acid:(acceptor) 2-oxidoreductase. It has 2 cofactors: FAD, and Zinc.
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14130734 | D-arabinono-1,4-lactone oxidase | In enzymology, a D-arabinono-1,4-lactone oxidase (EC 1.1.3.37) is an enzyme that catalyzes the chemical reaction
D-arabinono-1,4-lactone + O2 formula_0 D-erythro-ascorbate + H2O2
Thus, the two substrates of this enzyme are D-arabinono-1,4-lactone and O2, whereas its two products are D-erythro-ascorbate and H2O2.
This enzyme belongs to the family of oxidoreductases, specifically those acting on the CH-OH group of donor with oxygen as acceptor. The systematic name of this enzyme class is D-arabinono-1,4-lactone:oxygen oxidoreductase. It employs one cofactor, FAD.
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14130752 | Dehydrogluconate dehydrogenase | In enzymology, a dehydrogluconate dehydrogenase (EC 1.1.99.4) is an enzyme that catalyzes the chemical reaction
2-dehydro-D-gluconate + acceptor formula_0 2,5-didehydro-D-gluconate + reduced acceptor
Thus, the two substrates of this enzyme are 2-dehydro-D-gluconate and acceptor, whereas its two products are 2,5-didehydro-D-gluconate and reduced acceptor.
This enzyme belongs to the family of oxidoreductases, specifically those acting on the CH-OH group of donor with other acceptors. The systematic name of this enzyme class is 2-dehydro-D-gluconate:acceptor 2-oxidoreductase. Other names in common use include ketogluconate dehydrogenase, alpha-ketogluconate dehydrogenase, 2-keto-D-gluconate dehydrogenase, and 2-oxogluconate dehydrogenase. It has 2 cofactors: FAD, and Flavoprotein.
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14130770 | D-lactate dehydrogenase (cytochrome) | Class of enzymes
In enzymology, a D-lactate dehydrogenase (cytochrome) (EC 1.1.2.4) is an enzyme that catalyzes the chemical reaction
(D)-lactate + 2 ferricytochrome c formula_0 pyruvate + 2 ferrocytochrome c
Thus, the two substrates of this enzyme are (D)-lactate and ferricytochrome c, whereas its two products are pyruvate and ferrocytochrome c.
This enzyme belongs to the family of oxidoreductases, specifically those acting on the CH-OH group of donor with a cytochrome as acceptor. The systematic name of this enzyme class is (D)-lactate:ferricytochrome-c 2-oxidoreductase. Other names in common use include lactic acid dehydrogenase, D-lactate (cytochrome) dehydrogenase, cytochrome-dependent D-(−)-lactate dehydrogenase, D-lactate-cytochrome c reductase, and D-(−)-lactic cytochrome c reductase. This enzyme participates in pyruvate metabolism. It employs one cofactor, FAD. This type of enzyme has been characterized in animals, fungi, bacteria and recently in plants
. It is believed to be important in the detoxification of methylglyoxal through the glyoxylase pathway
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14130785 | D-lactate dehydrogenase (cytochrome c-553) | Class of enzymes
In enzymology, a D-lactate dehydrogenase (cytochrome c-553) (EC 1.1.2.5) is an enzyme that catalyzes the chemical reaction
(R)-lactate + 2 ferricytochrome c-553 formula_0 pyruvate + 2 ferrocytochrome c-553
Thus, the two substrates of this enzyme are (R)-lactate and ferricytochrome c-553, whereas its two products are pyruvate and ferrocytochrome c-553.
This enzyme belongs to the family of oxidoreductases, to be specific those acting on the CH-OH group of donor with a cytochrome as acceptor. The systematic name of this enzyme class is (R)-lactate:ferricytochrome-c-553 2-oxidoreductase. This enzyme participates in pyruvate metabolism.
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14130801 | D-mannitol oxidase | In enzymology, a d-mannitol oxidase (EC 1.1.3.40) is an enzyme that catalyzes the chemical reaction
mannitol + O2 formula_0 mannose + H2O2
Thus, the two substrates of this enzyme are mannitol and O2, whereas its two products are mannose and H2O2.
This enzyme belongs to the family of oxidoreductases, specifically those acting on the CH-OH group of donor with oxygen as acceptor. The systematic name of this enzyme class is mannitol:oxygen oxidoreductase (cyclizing). Other names in common use include mannitol oxidase, and D-arabitol oxidase.
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14130821 | D-sorbitol dehydrogenase (acceptor) | In enzymology, a D-sorbitol dehydrogenase (acceptor) (EC 1.1.99.21) is an enzyme that catalyzes the chemical reaction
D-sorbitol + acceptor formula_0 L-sorbose + reduced acceptor
Thus, the two substrates of this enzyme are D-sorbitol and acceptor, whereas its two products are L-sorbose and reduced acceptor.
This enzyme belongs to the family of oxidoreductases, specifically those acting on the CH-OH group of donor with other acceptors. The systematic name of this enzyme class is D-sorbitol:acceptor 1-oxidoreductase. This enzyme is also called D-sorbitol:(acceptor) 1-oxidoreductase. This enzyme participates in fructose and mannose metabolism. It employs one cofactor, FAD.
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14130838 | Ecdysone oxidase | Class of enzymes
In enzymology, an ecdysone oxidase (EC 1.1.3.16) is an enzyme that catalyzes the chemical reaction
ecdysone + O2 formula_0 3-dehydroecdysone + H2O2
Thus, the two substrates of this enzyme are ecdysone and O2, whereas its two products are 3-dehydroecdysone and H2O2.
This enzyme may or may not belong to the family of oxidoreductases, specifically those acting on the CH-OH group of donor with oxygen as acceptor. The systematic name of this enzyme class is ecdysone:oxygen 3-oxidoreductase. This enzyme might also be called beta-ecdysone oxidase.
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14130848 | Fructose 5-dehydrogenase | In enzymology, a fructose 5-dehydrogenase (EC 1.1.99.11) is an enzyme that catalyzes the chemical reaction
D-fructose + acceptor formula_0 5-dehydro-D-fructose + reduced acceptor
Thus, the two substrates of this enzyme are D-fructose and acceptor, whereas its two products are 5-dehydro-D-fructose and reduced acceptor.
This enzyme belongs to the family of oxidoreductases, specifically those acting on the CH-OH group of donor with other acceptors. The systematic name of this enzyme class is D-fructose:acceptor 5-oxidoreductase. Other names in common use include fructose 5-dehydrogenase (acceptor), D-fructose dehydrogenase, and D-fructose:(acceptor) 5-oxidoreductase.
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14130887 | Gluconate 2-dehydrogenase (acceptor) | In enzymology, a gluconate 2-dehydrogenase (acceptor) (EC 1.1.99.3) is an enzyme that catalyzes the chemical reaction
D-gluconate + acceptor formula_0 2-dehydro-D-gluconate + reduced acceptor
Thus, the two substrates of this enzyme are D-gluconate and acceptor, whereas its two products are 2-dehydro-D-gluconate and reduced acceptor.
This enzyme belongs to the family of oxidoreductases, specifically those acting on the CH-OH group of donor with other acceptors. The systematic name of this enzyme class is D-gluconate:acceptor 2-oxidoreductase. Other names in common use include gluconate oxidase, gluconate dehydrogenase, gluconic dehydrogenase, D-gluconate dehydrogenase, gluconic acid dehydrogenase, 2-ketogluconate reductase, D-gluconate dehydrogenase, 2-keto-D-gluconate-yielding, and D-gluconate:(acceptor) 2-oxidoreductase. This enzyme participates in pentose phosphate pathway. It employs one cofactor, FAD.
References.
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| https://en.wikipedia.org/wiki?curid=14130887 |
14130909 | Glucose 1-dehydrogenase (FAD, quinone) | In enzymology, a glucose 1-dehydrogenase (FAD, quinone) (EC 1.1.5.9) is an enzyme that catalyzes the chemical reaction
D-glucose + a quinone formula_0 D-glucono-1,5-lactone + a quinol
Thus, the two substrates of this enzyme are D-glucose and a quinone, whereas its two products are D-glucono-1,5-lactone and a quinol.
This enzyme belongs to the family of oxidoreductases, specifically those acting on the CH-OH group of donor with other acceptors. The systematic name of this enzyme class is D-glucose:acceptor 1-oxidoreductase. Other names in common use include glucose dehydrogenase ("Aspergillus"), glucose dehydrogenase (decarboxylating), and D-glucose:(acceptor) 1-oxidoreductase. This enzyme participates in pentose phosphate pathway. It employs one cofactor, FAD.
References.
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14130932 | Glucose-fructose oxidoreductase | In enzymology, a glucose-fructose oxidoreductase (EC 1.1.99.28) is an enzyme that catalyzes the chemical reaction
D-glucose + D-fructose formula_0 D-gluconolactone + D-glucitol
Thus, the two substrates of this enzyme are D-glucose and D-fructose, whereas its two products are D-gluconolactone and D-glucitol.
This enzyme belongs to the family of oxidoreductases, specifically those acting on the CH-OH group of donor with other acceptors. The systematic name of this enzyme class is D-glucose:D-fructose oxidoreductase.
Structural studies.
As of late 2007, 7 structures have been solved for this class of enzymes, with PDB accession codes 1H6A, 1H6B, 1H6C, 1H6D, 1OFG, 1RYD, and 1RYE.
References.
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14130953 | Glucoside 3-dehydrogenase | In enzymology, a glucoside 3-dehydrogenase (EC 1.1.99.13) is an enzyme that catalyzes the chemical reaction
sucrose + acceptor formula_0 3-dehydro-alpha-D-glucosyl-beta-D-fructofuranoside + reduced acceptor
Thus, the two substrates of this enzyme are sucrose and acceptor, whereas its two products are 3-dehydro-alpha-D-glucosyl-beta-D-fructofuranoside and reduced acceptor.
This enzyme participates in galactose metabolism and starch and sucrose metabolism. It employs one cofactor, FAD.
Nomenclature.
This enzyme belongs to the family of oxidoreductases, specifically those acting on the CH-OH group of donor with other acceptors. The systematic name of this enzyme class is D-aldohexoside:acceptor 3-oxidoreductase. Other names in common use include D-glucoside 3-dehydrogenase, D-aldohexopyranoside dehydrogenase, D-aldohexoside:cytochrome c oxidoreductase, D-glucoside 3-dehydrogenase, hexopyranoside-cytochrome c oxidoreductase, and D-aldohexoside:(acceptor) 3-oxidoreductase.
References.
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| https://en.wikipedia.org/wiki?curid=14130953 |
14130973 | Glycerol-3-phosphate oxidase | In enzymology, a glycerol-3-phosphate oxidase (EC 1.1.3.21) is an enzyme that catalyzes the chemical reaction
sn-glycerol 3-phosphate + O2 formula_0 glycerone phosphate + H2O2
Thus, the two substrates of this enzyme are sn-glycerol 3-phosphate and O2, whereas its two products are dihydroxyacetone phosphate (DHAP) and H2O2.
This enzyme belongs to the family of oxidoreductases, specifically those acting on the CH-OH group of donor with oxygen as acceptor. The systematic name of this enzyme class is sn-glycerol-3-phosphate:oxygen 2-oxidoreductase. Other names in common use include glycerol phosphate oxidase, glycerol-1-phosphate oxidase, glycerol phosphate oxidase, L-alpha-glycerophosphate oxidase, alpha-glycerophosphate oxidase, and L-alpha-glycerol-3-phosphate oxidase. This enzyme participates in glycerophospholipid metabolism. It employs one cofactor, FAD.
References.
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14130988 | Glycerol dehydrogenase (acceptor) | In enzymology, a glycerol dehydrogenase (acceptor) (EC 1.1.99.22) is an enzyme that catalyzes the chemical reaction
glycerol + acceptor formula_0 glycerone + reduced acceptor
Thus, the two substrates of this enzyme are glycerol and acceptor, whereas its two products are glycerone and reduced acceptor.
This enzyme belongs to the family of oxidoreductases, specifically those acting on the CH-OH group of donor with other acceptors. The systematic name of this enzyme class is glycerol:acceptor 1-oxidoreductase. This enzyme is also called glycerol:(acceptor) 1-oxidoreductase. It employs one cofactor, PQQ.
References.
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14131003 | Glycolate dehydrogenase | In enzymology, a glycolate dehydrogenase (EC 1.1.99.14) is an enzyme that catalyzes the chemical reaction
glycolate + acceptor formula_0 glyoxylate + reduced acceptor
Thus, the two substrates of this enzyme are glycolate and acceptor, whereas its two products are glyoxylate and reduced acceptor.
This enzyme belongs to the family of oxidoreductases, specifically those acting on the CH-OH group of donor with other acceptors. The systematic name of this enzyme class is glycolate:acceptor 2-oxidoreductase. Other names in common use include glycolate oxidoreductase, glycolic acid dehydrogenase, and glycolate:(acceptor) 2-oxidoreductase. This enzyme participates in glyoxylate and dicarboxylate metabolism.
References.
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14131016 | Hexose oxidase | In enzymology, a hexose oxidase (EC 1.1.3.5) is an enzyme that catalyzes the chemical reaction
D-glucose + O2 formula_0 D-glucono-1,5-lactone + H2O2
Thus, the two substrates of this enzyme are D-glucose and O2, whereas its two products are D-glucono-1,5-lactone and H2O2.
This enzyme belongs to the family of oxidoreductases, specifically those acting on the CH-OH group of donor with oxygen as acceptor. The systematic name of this enzyme class is D-hexose:oxygen 1-oxidoreductase. This enzyme participates in pentose phosphate pathway. It employs one cofactor, copper.
References.
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14131040 | Hydroxyacid-oxoacid transhydrogenase | In enzymology, a hydroxyacid-oxoacid transhydrogenase (EC 1.1.99.24) is an enzyme that catalyzes the chemical reaction
(S)-3-hydroxybutanoate + 2-oxoglutarate formula_0 acetoacetate + (R)-2-hydroxyglutarate
Thus, the two substrates of this enzyme are (S)-3-hydroxybutanoate and 2-oxoglutarate, whereas its two products are acetoacetate and (R)-2-hydroxyglutarate.
This enzyme belongs to the family of oxidoreductases, specifically those acting on the CH-OH group of donor with other acceptors. The systematic name of this enzyme class is (S)-3-hydroxybutanoate:2-oxoglutarate oxidoreductase. This enzyme is also called transhydrogenase, hydroxy acid-oxo acid.
References.
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14131062 | Hydroxyphytanate oxidase | In enzymology, a hydroxyphytanate oxidase (EC 1.1.3.27) is an enzyme that catalyzes the chemical reaction
L-2-hydroxyphytanate + O2 formula_0 2-oxophytanate + H2O2
Thus, the two substrates of this enzyme are L-2-hydroxyphytanate and O2, whereas its two products are 2-oxophytanate and H2O2.
This enzyme belongs to the family of oxidoreductases, specifically those acting on the CH-OH group of donor with oxygen as acceptor. The systematic name of this enzyme class is L-2-hydroxyphytanate:oxygen 2-oxidoreductase. This enzyme is also called L-2-hydroxyphytanate:oxygen 2-oxidoreductase.
References.
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14131081 | Lactate—malate transhydrogenase | In enzymology, a lactate—malate transhydrogenase (EC 1.1.99.7) is an enzyme that catalyzes the chemical reaction
(S)-lactate + oxaloacetate formula_0 pyruvate + malate
Thus, the two substrates of this enzyme are (S)-lactate and oxaloacetate, whereas its two products are pyruvate and malate.
This enzyme belongs to the family of oxidoreductases, specifically those acting on the CH-OH group of donor with other acceptors. The systematic name of this enzyme class is (S)-lactate:oxaloacetate oxidoreductase. This enzyme is also called malate-lactate transhydrogenase. This enzyme participates in pyruvate metabolism. It employs one cofactor, nicotinamide D-ribonucleotide.
References.
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Further reading.
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14131097 | Long-chain-alcohol oxidase | Long-chain alcohol oxidase is one of two enzyme classes that oxidize long-chain or fatty alcohols to aldehydes. It has been found in certain "Candida" yeast, where it participates in omega oxidation of fatty acids to produce acyl-CoA for energy or industrial use, as well as in other fungi, plants, and bacteria.
Long-chain alcohol oxidase catalyzes the chemical reaction
long-chain alcohol + O2 formula_0 2 long-chain aldehyde + 2 H2O2
Thus, the two substrates of this enzyme are long-chain/fatty alcohol and O2, whereas its two products are long-chain/fatty aldehyde and hydrogen peroxide.
Mechanism.
This enzyme belongs to the family of oxidoreductases, specifically those acting on the CH-OH group of donor with oxygen as acceptor. The systematic name of this enzyme class is long-chain-alcohol:oxygen oxidoreductase. Other names in common use include long-chain fatty alcohol oxidase, fatty alcohol oxidase, fatty alcohol:oxygen oxidoreductase, and long-chain fatty acid oxidase.
Structure.
The enzyme is an octamer of ~46kD subunits (except in "C. tropicalis", in which it is a dimer of subunits ~70kD). It is a Cytochrome c oxidase containing a covalently-bound heme group using the Cys-X-X-Cys-His motif. It also contains flavin to assist in oxidation-reduction. The enzyme is bound to the endoplasmic reticulum membrane.
Long-chain fatty alcohol oxidases vary between species in their specificity; some species have multiple different alcohol oxidases. They generally have a broad range of substrates, ranging from short chain alcohols starting at 4 carbons to the longest long-chain alcohols at 22 carbons. Some can also oxidize select diols, secondary alcohols, hydroxy fatty acids, and even long-chain aldehydes. However, each enzyme is optimized to function for specific alcohol, often between 10 and 16 carbons. In at least one species, the enzyme was stereoselective for the R(-) entantiomer.
Function.
This enzyme can be induced in many Candida yeast strains by growing them on long-chain alkanes as the major food source. Long-chain fatty alcohol oxidases participate in omega-oxidation of long chain alkanes or fatty acids. The alkane is first oxidized to an alcohol by an enzyme of the Cytochrome P450 family using NADPH. This alcohol is oxidized by long-chain fatty alcohol oxidase in yeast.
The long-chain alcohol is then oxidized by long-chain fatty aldehyde dehydrogenase to a carboxylic acid, also producing NADH from NAD+. Fatty acids can be oxidized again to make dicarboxylic species that join with coenzyme A and enter the beta oxidation pathway in the peroxisome.
Long-chain alcohol oxidase is also used in germinating seedlings of jojoba ("Simmondsia chinensis") to degrade esterified long-chain fatty alcohols stored as wax.
Species.
This enzyme has been found in the following organisms:
Yeast
"Candida cloacae"
"Candida tropicalis"
"Starmerella bombicola"
"Yarrowia lipolytica"
Other Fungi
"Aspergillus terreus"
"Mucor circinelloides"
Plants
"Arabidopsis thaliana" (thale cress)
"Lotus japonicus"
"Simmondsia chinensis" (jojoba)
"Tanacetum vulgare" (common tansy)
Archaea
"Uncultured marine euryarchaeota"
Industrial use.
This enzyme is required for production of dicarboxylic acids by industrial Candida yeast, which have nonfunctional beta oxidation pathways. They can thus produce relatively pure saturated and unsaturated dicarboxylic acids in high yield, which is not possible using chemical synthesis. The dicarboxylic acids are used to produce fragrances, polyamides, polyesters, adhesives, and antibiotics.
References.
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14131120 | L-sorbose oxidase | In enzymology, a L-sorbose oxidase (EC 1.1.3.11) is an enzyme that catalyzes the chemical reaction
L-sorbose + O2 formula_0 5-dehydro-D-fructose + H2O2
Thus, the two substrates of this enzyme are L-sorbose and O2, whereas its two products are 5-dehydro-D-fructose and H2O2.
This enzyme belongs to the family of oxidoreductases, specifically those acting on the CH-OH group of donor with oxygen as acceptor. The systematic name of this enzyme class is L-sorbose:oxygen 5-oxidoreductase.
References.
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14131148 | Malate dehydrogenase (quinone) | Enzyme class
In enzymology, a malate dehydrogenase (quinone) (EC 1.1.5.4), formerly malate dehydrogenase (acceptor) (EC 1.1.99.16), is an enzyme that catalyzes the chemical reaction
(S)-malate + a quinone formula_0 oxaloacetate + reduced quinone
Thus, the two substrates of this enzyme are (S)-malate and a quinone, whereas its two products are oxaloacetate and reduced quinone.
This enzyme belongs to the family of oxidoreductases, specifically those acting on the CH-OH group of donor with a quinone as acceptor. The systematic name of this enzyme class is (S)-malate:quinone oxidoreductase. Other names in common use include FAD-dependent malate-vitamin K reductase, malate-vitamin K reductase, and (S)-malate:(quinone) oxidoreductase. This enzyme participates in pyruvate metabolism. It employs one cofactor, FAD.
References.
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14131172 | Malate oxidase | In enzymology, a malate oxidase (EC 1.1.3.3) is an enzyme that catalyzes the chemical reaction
(S)-malate + O2 formula_0 oxaloacetate + H2O2
Thus, the two substrates of this enzyme are (S)-malate and O2, whereas its two products are oxaloacetate and H2O2.
This enzyme belongs to the family of oxidoreductases, specifically those acting on the CH-OH group of donor with oxygen as acceptor. The systematic name of this enzyme class is (S)-malate:oxygen oxidoreductase. Other names in common use include FAD-dependent malate oxidase, malic oxidase, and malic dehydrogenase II. This enzyme participates in pyruvate metabolism. It employs one cofactor, FAD. The enzyme is commonly localized on the inner surface of the cytoplasmic membrane although another family member (malate dehydrogenase 2 (NAD)) is found in the mitochondrial matrix.
Mechanisms.
Malate oxidase belongs to the family of malate dehydrogenases (EC 1.1.1.37) (MDH) that reversibly catalyze the oxidation of malate to oxaloacetate by means of the reduction of a cofactor. The most common isozymes of malate dehydrogenase use NAD+ or NADP+ as a cofactor to accept electrons and protons.
However, the main difference of malate oxidase is that it normally employs FAD as redox partner as alternative. Contrary to pyridine based NAD+/NADP+, FAD comprises a quinone moiety, which is reduced by the forward reaction. FAD is thereby converted to FADH2. In this case, malate oxidase is qualified as malate dehydrogenase (quinone).
In mutant strains of "Escherichia coli" lacking the activity of NAD-dependent malate dehydrogenase, malate oxidase is expressed. It is suggested that products of malate dehydrogenase could be responsible for repression of malate oxidase. This would confirm the existence of a family of structurally different malate dehydrogenases. Malate oxidase is induced only in cells, which completely lack the activity of NAD-specific malate dehydrogenase.
Irradiation of cytoplasm membranes of "Mycobacterium smegmatis" with ultraviolet light (360 nm) for 10 minutes resulted in about a 50% loss of malate oxidase activity. The addition of vitamin K, containing a functional naphthoquinone ring, restores the oxidation activity of malate oxidase. The quinone functionality of vitamin K can hence act as an alternative for FAD.
Biological Reference.
However, instead of using NAD+, NADP+ or FAD as cofactors, malate oxidase can also shift to oxygen as oxidant and proton acceptor.
(S)-malate + O2 ⇌ oxaloacetate + H2O2
Although seemingly unlikely because of its reactive oxidative character, hydrogen peroxide is found in biological systems including the human body. It signals oxidative stress from wounds to the immune system to recruit white blood cells for the healing process.
A study in "Nature" suggested that asthma sufferers have higher levels of hydrogen peroxide in their lungs than healthy people, which would explain why these patients also have inappropriate levels of white blood cells in their lungs. Asthma sufferers might have certain variations in cellular levels of NAD+/NADP+ or FAD, which causes malate oxidase to shift to oxygen as its oxidant, due to its high abundancy in the lungs. This could be a possible explanation for the elevated levels of hydrogen peroxide in their lungs.
Uses.
Topical compositions of malate oxidase combined with suitable disease-detecting biomarkers and a chemiluminescent dye are used in disease detecting systems. The biomarker activates the malate oxidase to generate hydrogen peroxide that excites the light-emitting dye, which exhibits chemiluminescence in the presence of the peroxide. Such contemporary compositions are thus used as a diagnostic tool for detecting diseases.
In a similar method, malate oxidase is used in the transcutaneous measurement of the amount of a substrate in blood. The method is conducted by contacting the skin with the enzyme, reacting the substrate with the enzyme and directly detecting the amount of H2O2 produced as a measure of the amount of substrate in the blood, with use of a hydrogen peroxide electrode. Further dermatological applications are in drugs or cosmetic agents, comprising a suitable substrate and malate oxidase as hydrogen peroxide producing enzyme for skin lightening and age spots or freckles.
Other illustrative uses that employ the capacity of malate oxidase to yield hydrogen peroxide in the presence of a suitable substrate, including malate, are found in toothpaste to remove bacterial plaque, cleaning compositions for removing blood stains and the like, and in the removal of chewing gum lumps stuck on surfaces by enzymatic degradation.
Malate oxidase is also employed in the inhibition of corrosion by dissolved oxygen in water by converting it to hydrogen peroxide, which is subsequently broken down into water and oxygen by catalase.
References.
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14131196 | Mannitol dehydrogenase (cytochrome) | Enzyme class
In enzymology, a mannitol dehydrogenase (cytochrome) (EC 1.1.2.2) is an enzyme that catalyzes the chemical reaction
D-mannitol + ferricytochrome c formula_0 D-fructose + ferrocytochrome c
Thus, the two substrates of this enzyme are D-mannitol and ferricytochrome c, whereas its two products are D-fructose and ferrocytochrome c.
This enzyme belongs to the family of oxidoreductases, to be specific those acting on the CH-OH group of donor with a cytochrome as acceptor. The systematic name of this enzyme class is D-mannitol:ferricytochrome-c 2-oxidoreductase. This enzyme is also called polyol dehydrogenase. This enzyme participates in pentose and glucuronate interconversions and fructose and mannose metabolism
References.
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14131224 | N-acylhexosamine oxidase | In enzymology, a N-acylhexosamine oxidase (EC 1.1.3.29) is an enzyme that catalyzes the chemical reaction
N-acetyl-D-glucosamine + O2 formula_0 N-acetyl-D-glucosaminate + H2O2
Thus, the two substrates of this enzyme are N-acetyl-D-glucosamine and O2, whereas its two products are N-acetyl-D-glucosaminate and H2O2.
This enzyme belongs to the family of oxidoreductases, specifically those acting on the CH-OH group of donor with oxygen as acceptor. The systematic name of this enzyme class is N-acyl-D-hexosamine:oxygen 1-oxidoreductase. Other names in common use include N-acyl-D-hexosamine oxidase, and N-acyl-beta-D-hexosamine:oxygen 1-oxidoreductase.
References.
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14131247 | Polyvinyl-alcohol dehydrogenase (acceptor) | In enzymology, a polyvinyl-alcohol dehydrogenase (acceptor) (EC 1.1.99.23) is an enzyme that catalyzes the chemical reaction
polyvinyl alcohol + acceptor formula_0 oxidized polyvinyl alcohol + reduced acceptor
Thus, the two substrates of this enzyme are polyvinyl alcohol and acceptor, whereas its two products are oxidized polyvinyl alcohol and reduced acceptor.
This enzyme belongs to the family of oxidoreductases, specifically those acting on the CH-OH group of donor with other acceptors. The systematic name of this enzyme class is polyvinyl-alcohol:acceptor oxidoreductase. Other names in common use include PVA dehydrogenase, and polyvinyl-alcohol:(acceptor) oxidoreductase. It employs one cofactor, PQQ.
References.
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14131264 | Polyvinyl-alcohol oxidase | In enzymology, a polyvinyl-alcohol oxidase (EC 1.1.3.30) is an enzyme that catalyzes the chemical reaction
polyvinyl alcohol + O2 formula_0 oxidized polyvinyl alcohol + H2O2
Thus, the two substrates of this enzyme are polyvinyl alcohol and O2, whereas its two products are oxidized polyvinyl alcohol and H2O2.
This enzyme belongs to the family of oxidoreductases, specifically those acting on the CH-OH group of donor with oxygen as acceptor. The systematic name of this enzyme class is polyvinyl-alcohol:oxygen oxidoreductase. Other names in common use include dehydrogenase, polyvinyl alcohol, and PVA oxidase.
References.
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14131276 | Pyranose oxidase | In enzymology, a pyranose oxidase (EC 1.1.3.10) is an enzyme that catalyzes the chemical reaction
D-glucose + O2 formula_0 2-dehydro-D-glucose + H2O2
Thus, the two substrates of this enzyme are D-glucose and O2, whereas its two products are 2-dehydro-D-glucose and H2O2.
Pyranose oxidase is able to oxidize D-xylose, L-sorbose, D-galactose, and D-glucono-1,5-lactone, which have the same ring conformation and configuration at C-2, C-3 and C-4.
This enzyme belongs to the family of oxidoreductases, specifically those acting on the CH-OH group of donor with oxygen as acceptor. The systematic name of this enzyme class is pyranose:oxygen 2-oxidoreductase. Other names in common use include glucose 2-oxidase, and pyranose-2-oxidase. This enzyme participates in pentose phosphate pathway. It employs one cofactor, FAD.
Structural studies.
As of late 2007, 8 structures have been solved for this class of enzymes, with PDB accession codes 1TT0, 1TZL, 2F5V, 2F6C, 2IGK, 2IGM, 2IGN, and 2IGO.
Use in biosensors.
Recently, pyranose oxidase has been gaining on popularity within biosensors. Unlike glucose oxidase, it can produce higher power output, given that it is not glycosylated, has more favorable value of Michaelis-Menten constants, and can catalytically convert both anomers of glucose. It reacts with a wider range of substrates. Pyranose oxidase does not cause an unwanted pH shift. It is also possible to easily express and produce it in high yields using "E. coli".
References.
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Further reading.
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14131298 | Pyridoxine 4-oxidase | In enzymology, a pyridoxine 4-oxidase (EC 1.1.3.12) is an enzyme that catalyzes the chemical reaction
pyridoxine + O2 formula_0 pyridoxal + H2O2
Thus, the two substrates of this enzyme are pyridoxine and O2, whereas its two products are pyridoxal and H2O2.
This enzyme belongs to the family of oxidoreductases, specifically those acting on the CH-OH group of donor with oxygen as acceptor. The systematic name of this enzyme class is pyridoxine:oxygen 4-oxidoreductase. Other names in common use include pyridoxin 4-oxidase, and pyridoxol 4-oxidase. This enzyme participates in vitamin B6 metabolism. It employs one cofactor, FAD.
References.
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14131317 | Pyridoxine 5-dehydrogenase | In enzymology, a pyridoxine 5-dehydrogenase (EC 1.1.99.9) is an enzyme that catalyzes the chemical reaction
pyridoxine + acceptor formula_0 isopyridoxal + reduced acceptor
Thus, the two substrates of this enzyme are pyridoxine and acceptor, whereas its two products are isopyridoxal and reduced acceptor.
This enzyme belongs to the family of oxidoreductases, specifically those acting on the CH-OH group of donor with other acceptors. The systematic name of this enzyme class is pyridoxine:acceptor 5-oxidoreductase. Other names in common use include pyridoxal-5-dehydrogenase, pyridoxol 5-dehydrogenase, pyridoxin 5-dehydrogenase, pyridoxine dehydrogenase, pyridoxine 5'-dehydrogenase, and pyridoxine:(acceptor) 5-oxidoreductase. This enzyme participates in vitamin B6 metabolism. It has 2 cofactors: FAD, and PQQ.
References.
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14131358 | Quinoprotein glucose dehydrogenase | Enzyme
In enzymology, a quinoprotein glucose dehydrogenase (EC 1.1.5.2) is an enzyme that catalyzes the chemical reaction
D-glucose + ubiquinone formula_0 D-glucono-1,5-lactone + ubiquinol
Thus, the two substrates of this enzyme are D-glucose and ubiquinone, whereas its two products are D-glucono-1,5-lactone and ubiquinol.
This enzyme belongs to the family of oxidoreductases, specifically those acting on the CH-OH group of donor with a quinone or similar compound as acceptor. The systematic name of this enzyme class is D-glucose:ubiquinone oxidoreductase. Other names in common use include D-glucose:(pyrroloquinoline-quinone) 1-oxidoreductase, glucose dehydrogenase (PQQ-dependent), glucose dehydrogenase (pyrroloquinoline-quinone), and quinoprotein D-glucose dehydrogenase. This enzyme participates in pentose phosphate pathway. It employs one cofactor, PQQ.
References.
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Further reading.
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14131376 | (R)-pantolactone dehydrogenase (flavin) | Class of enzymes
In enzymology, a (R)-pantolactone dehydrogenase (flavin) (EC 1.1.99.27) is an enzyme that catalyzes the chemical reaction
(R)-pantolactone + acceptor formula_0 2-dehydropantolactone + reduced acceptor
Thus, the two substrates of this enzyme are (R)-pantolactone and acceptor, whereas its two products are 2-dehydropantolactone and reduced acceptor.
This enzyme belongs to the family of oxidoreductases, specifically those acting on the CH-OH group of donor with other acceptors. The systematic name of this enzyme class is (R)-pantolactone:acceptor oxidoreductase (flavin-containing). Other names in common use include 2-dehydropantolactone reductase (flavin), 2-dehydropantoyl-lactone reductase (flavin), and (R)-pantoyllactone dehydrogenase (flavin).
References.
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14131390 | (S)-2-hydroxy-acid oxidase | Class of enzymes
In enzymology, an (S)-2-hydroxy-acid oxidase (EC 1.1.3.15) is an enzyme that catalyzes the chemical reaction
(S)-2-hydroxy acid + O2 formula_0 2-oxo acid + H2O2
Thus, the two substrates of this enzyme are (S)-2-hydroxy acid and O2, whereas its two products are 2-oxo acid and H2O2.
This enzyme belongs to the family of oxidoreductases, specifically those acting on the CH-OH group of donor with oxygen as acceptor. The systematic name of this enzyme class is (S)-2-hydroxy-acid:oxygen 2-oxidoreductase. Other names in common use include glycolate oxidase, hydroxy-acid oxidase A, hydroxy-acid oxidase B, oxidase, L-2-hydroxy acid, hydroxyacid oxidase A, L-alpha-hydroxy acid oxidase, and L-2-hydroxy acid oxidase. This enzyme participates in glyoxylate and dicarboxylate metabolism. It employs one cofactor, FMN.
Structural studies.
As of late 2007, 5 structures have been solved for this class of enzymes, with PDB accession codes 1AL7, 1AL8, 1GYL, 1TB3, and 2NZL.
References.
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| https://en.wikipedia.org/wiki?curid=14131390 |
14131410 | Secondary-alcohol oxidase | In enzymology, a secondary-alcohol oxidase (EC 1.1.3.18) is an enzyme that catalyzes the chemical reaction
a secondary alcohol + O2 formula_0 a ketone + H2O2
Thus, the two substrates of this enzyme are secondary alcohol and O2, whereas its two products are ketone and H2O2.
This enzyme belongs to the family of oxidoreductases, specifically those acting on the CH-OH group of donor with oxygen as acceptor. The systematic name of this enzyme class is secondary-alcohol:oxygen oxidoreductase. Other names in common use include polyvinyl alcohol oxidase, and secondary alcohol oxidase.
References.
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| https://en.wikipedia.org/wiki?curid=14131410 |
14131434 | Thiamine oxidase | In enzymology, a thiamine oxidase (EC 1.1.3.23) is an enzyme that catalyzes the chemical reaction
thiamine + 2 O2 + H2O formula_0 thiamine acetic acid + 2 H2O2
The 3 substrates of this enzyme are thiamine, O2, and H2O, whereas its two products are thiamine acetic acid and H2O2.
This enzyme belongs to the family of oxidoreductases, specifically those acting on the CH-OH group of donor with oxygen as acceptor. The systematic name of this enzyme class is thiamine:oxygen 5-oxidoreductase. Other names in common use include thiamin dehydrogenase, thiamine dehydrogenase, and thiamin:oxygen 5-oxidoreductase. This enzyme participates in thiamine metabolism. It employs one cofactor, FAD.
References.
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| https://en.wikipedia.org/wiki?curid=14131434 |
14131450 | Vanillyl-alcohol oxidase | In enzymology, a vanillyl-alcohol oxidase (EC 1.1.3.38) is an enzyme that catalyzes the chemical reaction
+ O2 formula_0 + H2O2
Thus, the two substrates of this enzyme are vanillyl alcohol and O2, whereas its two products are vanillin and H2O2.
This enzyme belongs to the family of oxidoreductases, specifically those acting on the CH-OH group of donor with oxygen as acceptor. The systematic name of this enzyme class is vanillyl alcohol:oxygen oxidoreductase. This enzyme is also called 4-hydroxy-2-methoxybenzyl alcohol oxidase. This enzyme participates in 2,4-dichlorobenzoate degradation. It employs one cofactor, FAD.
Structural studies.
As of late 2007, 3 structures have been solved for this class of enzymes, with PDB accession codes 1DZN, 1E0Y, and 1E8G.
References.
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| https://en.wikipedia.org/wiki?curid=14131450 |
14131468 | Vitamin-K-epoxide reductase (warfarin-insensitive) | Class of enzymes
In enzymology, a vitamin-K-epoxide reductase (warfarin-insensitive) (EC 1.17.4.5) is an enzyme that catalyzes the chemical reaction
3-hydroxy-2-methyl-3-phytyl-2,3-dihydronaphthoquinone + oxidized dithiothreitol formula_0 2,3-epoxy-2,3-dihydro-2-methyl-3-phytyl-1,4-naphthoquinone + 1,4-dithiothreitol
Thus, the two substrates of this enzyme are 3-hydroxy-2-methyl-3-phytyl-2,3-dihydronaphthoquinone and oxidized dithiothreitol, whereas its two products are 2,3-epoxy-2,3-dihydro-2-methyl-3-phytyl-1,4-naphthoquinone and 1,4-dithiothreitol.
This enzyme belongs to the family of oxidoreductases, specifically those acting on the CH or CH2 groups of donor with a disulfide as acceptor. The systematic name of this enzyme class is 3-hydroxy-2-methyl-3-phytyl-2,3-dihydronaphthoquinone:oxidized-dithi othreitol oxidoreductase.
References.
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| https://en.wikipedia.org/wiki?curid=14131468 |
14131489 | Vitamin-K-epoxide reductase (warfarin-sensitive) | In enzymology, a vitamin-K-epoxide reductase (warfarin-sensitive) (EC 1.17.4.4) is an enzyme that catalyzes the chemical reaction
2-methyl-3-phytyl-1,4-naphthoquinone + oxidized dithiothreitol formula_0 2,3-epoxy-2,3-dihydro-2-methyl-3-phytyl-1,4-naphthoquinone + 1,4-dithiothreitol
Thus, the two substrates of this enzyme are 2-methyl-3-phytyl-1,4-naphthoquinone and oxidized dithiothreitol, whereas its two products are 2,3-epoxy-2,3-dihydro-2-methyl-3-phytyl-1,4-naphthoquinone and 1,4-dithiothreitol.
This enzyme belongs to the family of oxidoreductases, specifically those acting on the CH or CH2 groups of donor with a disulfide as acceptor. The systematic name of this enzyme class is 2-methyl-3-phytyl-1,4-naphthoquinone:oxidized-dithiothreitol oxidoreductase. This enzyme participates in biosynthesis of steroids. At least one compound, Warfarin is known to inhibit this enzyme.
References.
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| https://en.wikipedia.org/wiki?curid=14131489 |
14131507 | Xylitol oxidase | In enzymology, a xylitol oxidase (EC 1.1.3.41) is an enzyme that catalyzes the chemical reaction
xylitol + O2 formula_0 xylose + H2O2
Thus, the two substrates of this enzyme are xylitol and O2, whereas its two products are xylose and H2O2.
This enzyme belongs to the family of oxidoreductases, specifically those acting on the CH-OH group of donor with oxygen as acceptor. The systematic name of this enzyme class is xylitol:oxygen oxidoreductase.
References.
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| https://en.wikipedia.org/wiki?curid=14131507 |
14131863 | 2,5-dioxovalerate dehydrogenase | Class of enzymes
In enzymology, a 2,5-dioxovalerate dehydrogenase (EC 1.2.1.26) is an enzyme that catalyzes the chemical reaction
2,5-dioxopentanoate + NADP+ + H2O formula_0 2-oxoglutarate + NADPH + 2 H+
The 3 substrates of this enzyme are 2,5-dioxopentanoate, NADP+, and H2O, whereas its 3 products are 2-oxoglutarate, NADPH, and H+.
This enzyme belongs to the family of oxidoreductases, specifically those acting on the aldehyde or oxo group of donor with NAD+ or NADP+ as acceptor. The systematic name of this enzyme class is 2,5-dioxopentanoate:NADP+ 5-oxidoreductase. Other names in common use include 2-oxoglutarate semialdehyde dehydrogenase, and alpha-ketoglutaric semialdehyde dehydrogenase. This enzyme participates in ascorbate and aldarate metabolism.
References.
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| https://en.wikipedia.org/wiki?curid=14131863 |
14131878 | 2-oxoaldehyde dehydrogenase (NAD+) | Enzyme
In enzymology, a 2-oxoaldehyde dehydrogenase (NAD+) (EC 1.2.1.23) is an enzyme that catalyzes the chemical reaction
a 2-oxoaldehyde + NAD+ + H2O formula_0 a 2-oxo acid + NADH + H+
The 3 substrates of this enzyme are 2-oxoaldehyde, NAD+, and H2O, whereas its 3 products are 2-oxo acid, NADH, and H+.
This enzyme participates in pyruvate metabolism.
Nomenclature.
This enzyme belongs to the family of oxidoreductases, specifically those acting on the aldehyde or oxo group of donor with NAD+ or NADP+ as acceptor. The systematic name of this enzyme class is 2-oxoaldehyde:NAD+ 2-oxidoreductase.
Other names in common use include:
References.
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| https://en.wikipedia.org/wiki?curid=14131878 |
14131898 | 2-oxoaldehyde dehydrogenase (NADP+) | Class of enzymes
In enzymology, a 2-oxoaldehyde dehydrogenase (NADP+) (EC 1.2.1.49) is an enzyme that catalyzes the chemical reaction
a 2-oxoaldehyde + NADP+ + H2O formula_0 a 2-oxo acid + NADPH + H+
The 3 substrates of this enzyme are 2-oxoaldehyde, NADP+, and H2O, whereas its 3 products are 2-oxo acid, NADPH, and H+.
This enzyme belongs to the family of oxidoreductases, specifically those acting on the aldehyde or oxo group of donor with NAD+ or NADP+ as acceptor. The systematic name of this enzyme class is 2-oxoaldehyde:NADP+ 2-oxidoreductase. Other names in common use include alpha-ketoaldehyde dehydrogenase, methylglyoxal dehydrogenase, NADP+-linked alpha-ketoaldehyde dehydrogenase, 2-ketoaldehyde dehydrogenase, NADP+-dependent alpha-ketoaldehyde dehydrogenase, and 2-oxoaldehyde dehydrogenase (NADP+). This enzyme participates in pyruvate metabolism.
References.
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| https://en.wikipedia.org/wiki?curid=14131898 |
14131914 | 2-oxobutyrate synthase | Class of enzymes
In enzymology, a 2-oxobutyrate synthase (EC 1.2.7.2) is an enzyme that catalyzes the chemical reaction
2-oxobutanoate + CoA + oxidized ferredoxin formula_0 propanoyl-CoA + CO2 + reduced ferredoxin
The 3 substrates of this enzyme are 2-oxobutanoate, CoA, and oxidized ferredoxin, whereas its 3 products are propanoyl-CoA, CO2, and reduced ferredoxin.
This enzyme belongs to the family of oxidoreductases, specifically those acting on the aldehyde or oxo group of donor with an iron-sulfur protein as acceptor. The systematic name of this enzyme class is 2-oxobutanoate:ferredoxin 2-oxidoreductase (CoA-propanoylating). Other names in common use include alpha-ketobutyrate-ferredoxin oxidoreductase, 2-ketobutyrate synthase, alpha-ketobutyrate synthase, 2-oxobutyrate-ferredoxin oxidoreductase, and 2-oxobutanoate:ferredoxin 2-oxidoreductase (CoA-propionylating). This enzyme participates in propanoate metabolism.
References.
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| https://en.wikipedia.org/wiki?curid=14131914 |
14131934 | 2-oxoglutarate synthase | Class of enzymes
In enzymology, a 2-oxoglutarate synthase (EC 1.2.7.3) is an enzyme that catalyzes the chemical reaction
2-oxoglutarate + CoA + 2 oxidized ferredoxin formula_0 succinyl-CoA + CO2 + 2 reduced ferredoxin
The 3 substrates of this enzyme are 2-oxoglutarate, CoA, and oxidized ferredoxin, whereas its 3 products are succinyl-CoA, CO2, and reduced ferredoxin.
This enzyme belongs to the family of oxidoreductases, specifically those acting on the aldehyde or oxo group of donor with an iron-sulfur protein as acceptor. The systematic name of this enzyme class is 2-oxoglutarate:ferredoxin oxidoreductase (decarboxylating). Other names in common use include 2-ketoglutarate ferredoxin oxidoreductase, 2-oxoglutarate:ferredoxin oxidoreductase, KGOR, 2-oxoglutarate ferredoxin oxidoreductase, and 2-oxoglutarate:ferredoxin 2-oxidoreductase (CoA-succinylating). This enzyme participates in the Citric acid cycle. Some forms catalyze the reverse reaction within the Reverse Krebs cycle, as a means of carbon fixation.
References.
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| https://en.wikipedia.org/wiki?curid=14131934 |
14131951 | 2-oxoisovalerate dehydrogenase (acylating) | Class of enzymes
In enzymology, a 2-oxoisovalerate dehydrogenase (acylating) (EC 1.2.1.25) is an enzyme that catalyzes the chemical reaction
3-methyl-2-oxobutanoate + CoA + NAD+ formula_0 2-methylpropanoyl-CoA + CO2 + NADH
The 3 substrates of this enzyme are 3-methyl-2-oxobutanoate, CoA, and NAD+, whereas its 3 products are 2-methylpropanoyl-CoA, CO2, and NADH.
This enzyme belongs to the family of oxidoreductases, specifically those acting on the aldehyde or oxo group of donor with NAD+ or NADP+ as acceptor. The systematic name of this enzyme class is 3-methyl-2-oxobutanoate:NAD+ 2-oxidoreductase (CoA-methyl-propanoylating). Other names in common use include 2-oxoisovalerate dehydrogenase, and alpha-ketoisovalerate dehydrogenase. This enzyme participates in valine, leucine and isoleucine degradation.
References.
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14131972 | 3alpha,7alpha,12alpha-trihydroxycholestan-26-al 26-oxidoreductase | Enzyme
In enzymology, a 3alpha,7alpha,12alpha-trihydroxycholestan-26-al 26-oxidoreductase is an enzyme that catalyzes the chemical reaction:
(25R)-3alpha,7alpha,12alpha-trihydroxy-5beta-cholestan-26-al + NAD+ + H2O formula_0 (25R)-3alpha,7alpha,12alpha-trihydroxy-5beta-cholestan-26-oate + NADH + 2 H+
The 3 substrates of this enzyme are (25R)-3alpha,7alpha,12alpha-trihydroxy-5beta-cholestan-26-al, NAD+, and H2O, whereas its 2 products are (25R)-3alpha,7alpha,12alpha-trihydroxy-5beta-cholestan-26-oate, NADH, and H+. This enzyme participates in bile acid biosynthesis.
Nomenclature.
This enzyme belongs to the family of oxidoreductases, specifically those acting on the aldehyde or oxo group of donor with NAD+ or NADP+ as acceptor. The systematic name of this enzyme class is (25R)-3alpha,7alpha,12alpha-trihydroxy-5beta-cholestan-26-al:NAD+ 26-oxidoreductase. Other names in common use include:
References.
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| https://en.wikipedia.org/wiki?curid=14131972 |
14131996 | 3-methyl-2-oxobutanoate dehydrogenase | Class of enzymes
In enzymology, a 3-methyl-2-oxobutanoate dehydrogenase (EC 1.2.4.4) is an enzyme that catalyzes the chemical reaction
3-methyl-2-oxobutanoate + [dihydrolipoyllysine-residue (2-methylpropanoyl)transferase] lipoyllysine formula_0 [dihydrolipoyllysine-residue (2-methylpropanoyl)transferase] S-(2-methylpropanoyl)dihydrolipoyllysine + CO2
The 3 substrates of this enzyme are 3-methyl-2-oxobutanoate, dihydrolipoyllysine-residue (2-methylpropanoyl)transferase, and lipoyllysine, whereas its 3 products are dihydrolipoyllysine-residue (2-methylpropanoyl)transferase, S-(2-methylpropanoyl)dihydrolipoyllysine, and CO2.
This enzyme belongs to the family of oxidoreductases, specifically those acting on the aldehyde or oxo group of donor with a disulfide as acceptor.
This enzyme participates in valine, leucine and isoleucine degradation. It employs one cofactor, thiamin diphosphate. It is the E1 subunit of a catalytic complex.
Structural studies.
As of late 2007, twenty-nine structures have been solved for this class of enzymes, with PDB accession codes 1DTW, 1OLS, 1OLU, 1OLX, 1U5B, 1UM9, 1UMB, 1UMC, 1UMD, 1V11, 1V16, 1V1M, 1V1R, 1WCI, 1X7W, 1X7X, 1X7Y, 1X7Z, 1X80, 2BEU, 2BEV, 2BEW, 2BFB, 2BFC, 2BFD, 2BFE, 2BFF, 2BP7, and 2J9F.
References.
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14132015 | 3-methyl-2-oxobutanoate dehydrogenase (ferredoxin) | Class of enzymes
In enzymology, a 3-methyl-2-oxobutanoate dehydrogenase (ferredoxin) (EC 1.2.7.7) is an enzyme that catalyzes the chemical reaction
3-methyl-2-oxobutanoate + CoA + oxidized ferredoxin formula_0 S-(2-methylpropanoyl)-CoA + CO2 + reduced ferredoxin
The 3 substrates of this enzyme are 3-methyl-2-oxobutanoate, CoA, and oxidized ferredoxin, whereas its 3 products are S-(2-methylpropanoyl)-CoA, CO2, and reduced ferredoxin.
This enzyme belongs to the family of oxidoreductases, specifically those acting on the aldehyde or oxo group of donor with an iron-sulfur protein as acceptor. The systematic name of this enzyme class is . Other names in common use include 2-ketoisovalerate ferredoxin reductase, 3-methyl-2-oxobutanoate synthase (ferredoxin), VOR, branched-chain ketoacid ferredoxin reductase, branched-chain oxo acid ferredoxin reductase, keto-valine-ferredoxin oxidoreductase, ketoisovalerate ferredoxin reductase, and 2-oxoisovalerate ferredoxin reductase.
References.
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14132060 | 4-formylbenzenesulfonate dehydrogenase | Class of enzymes
In enzymology, a 4-formylbenzenesulfonate dehydrogenase (EC 1.2.1.62) is an enzyme that catalyzes the chemical reaction
4-formylbenzenesulfonate + NAD+ + H2O formula_0 4-sulfobenzoate + NADH + 2 H+
The 3 substrates of this enzyme are 4-formylbenzenesulfonate, NAD+, and H2O, whereas its 3 products are 4-sulfobenzoate, NADH, and H+.
This enzyme belongs to the family of oxidoreductases, specifically those acting on the aldehyde or oxo group of donor with NAD+ or NADP+ as acceptor. The systematic name of this enzyme class is 4-formylbenzenesulfonate:NAD+ oxidoreductase. This enzyme participates in 2,4-dichlorobenzoate degradation.
References.
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| https://en.wikipedia.org/wiki?curid=14132060 |
14132082 | 4-hydroxybenzaldehyde dehydrogenase | Class of enzymes
In enzymology, a 4-hydroxybenzaldehyde dehydrogenase (EC 1.2.1.64) is an enzyme that catalyzes the chemical reaction
4-hydroxybenzaldehyde + NAD+ + H2O formula_0 4-hydroxybenzoate + NADH + 2 H+
The 3 substrates of this enzyme are 4-hydroxybenzaldehyde, NAD+, and H2O, whereas its 3 products are 4-hydroxybenzoate, NADH, and H+.
This enzyme belongs to the family of oxidoreductases, specifically those acting on the aldehyde or oxo group of donor with NAD+ or NADP+ as acceptor. The systematic name of this enzyme class is 3-hydroxybenzaldehyde:NAD+ oxidoreductase. This enzyme is also called p-hydroxybenzaldehyde dehydrogenase. This enzyme participates in toluene and xylene degradation in bacteria. It is also found in carrots ("Daucus carota").
References.
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14132107 | 4-hydroxymuconic-semialdehyde dehydrogenase | Class of enzymes
In enzymology, a 4-hydroxymuconic-semialdehyde dehydrogenase (EC 1.2.1.61) is an enzyme that catalyzes the chemical reaction
4-hydroxymuconic semialdehyde + NAD+ + H2O formula_0 maleylacetate + NADH + 2 H+
The 3 substrates of this enzyme are 4-hydroxymuconic semialdehyde, NAD+, and H2O, whereas its 3 products are maleylacetate, NADH, and H+.
This enzyme belongs to the family of oxidoreductases, specifically those acting on the aldehyde or oxo group of donor with NAD+ or NADP+ as acceptor. The systematic name of this enzyme class is 4-hydroxymuconic-semialdehyde:NAD+ oxidoreductase. This enzyme participates in gamma-hexachlorocyclohexane degradation.
References.
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| https://en.wikipedia.org/wiki?curid=14132107 |
14132145 | 4-hydroxyphenylacetaldehyde dehydrogenase | Class of enzymes
In enzymology, a 4-hydroxyphenylacetaldehyde dehydrogenase (EC 1.2.1.53) is an enzyme that catalyzes the chemical reaction
4-hydroxyphenylacetaldehyde + NAD+ + H2O formula_0 4-hydroxyphenylacetate + NADH + 2 H+
The 3 substrates of this enzyme are 4-hydroxyphenylacetaldehyde, NAD+, and H2O, whereas its 3 products are 4-hydroxyphenylacetate, NADH, and H+.
This enzyme belongs to the family of oxidoreductases, specifically those acting on the aldehyde or oxo group of donor with NAD+ or NADP+ as acceptor. The systematic name of this enzyme class is 4-hydroxyphenylacetaldehyde:NAD+ oxidoreductase. This enzyme is also called 4-HPAL dehydrogenase. This enzyme participates in tyrosine metabolism.
References.
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| https://en.wikipedia.org/wiki?curid=14132145 |
14132160 | 4-hydroxyphenylpyruvate oxidase | Class of enzymes
In enzymology, a 4-hydroxyphenylpyruvate oxidase (EC 1.2.3.13) is an enzyme that catalyzes the chemical reaction
4-hydroxyphenylpyruvate + 1/2 O2 formula_0 4-hydroxyphenylacetate + CO2
Thus, the two substrates of this enzyme are 4-hydroxyphenylpyruvate and O2, whereas its two products are 4-hydroxyphenylacetate and CO2.
This enzyme belongs to the family of oxidoreductases, specifically those acting on the aldehyde or oxo group of donor with oxygen as acceptor. The systematic name of this enzyme class is 4-hydroxyphenylpyruvate:oxygen oxidoreductase (decarboxylating). This enzyme participates in tyrosine metabolism.
References.
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| https://en.wikipedia.org/wiki?curid=14132160 |
14132181 | 4-trimethylammoniobutyraldehyde dehydrogenase | Class of enzymes
In enzymology, a 4-trimethylammoniobutyraldehyde dehydrogenase (EC 1.2.1.47) is an enzyme that catalyzes the chemical reaction
4-trimethylammoniobutanal + NAD+ + H2O formula_0 4-trimethylammoniobutanoate + NADH + 2 H+
The 3 substrates of this enzyme are 4-trimethylammoniobutanal, NAD+, and H2O, whereas its 3 products are 4-trimethylammoniobutanoate, NADH, and H+.
This enzyme belongs to the family of oxidoreductases, specifically those acting on the aldehyde or oxo group of donor with NAD+ or NADP+ as acceptor. The systematic name of this enzyme class is 4-trimethylammoniobutanal:NAD+ 1-oxidoreductase. Other names in common use include 4-trimethylaminobutyraldehyde dehydrogenase, and 4-N-trimethylaminobutyraldehyde dehydrogenase. This enzyme participates in lysine degradation and carnitine biosynthesis.
References.
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| https://en.wikipedia.org/wiki?curid=14132181 |
14132202 | 5-carboxymethyl-2-hydroxymuconic-semialdehyde dehydrogenase | InterPro Family
In enzymology, a 5-carboxymethyl-2-hydroxymuconic-semialdehyde dehydrogenase (EC 1.2.1.60) is an enzyme that catalyzes the chemical reaction
5-carboxymethyl-2-hydroxymuconate semialdehyde + H2O + NAD+ formula_0 5-carboxymethyl-2-hydroxymuconate + NADH + 2 H+
The 3 substrates of this enzyme are 5-carboxymethyl-2-hydroxymuconate semialdehyde, H2O, and NAD+, whereas its 3 products are 5-carboxymethyl-2-hydroxymuconate, NADH, and H+.
This enzyme belongs to the family of oxidoreductases, specifically those acting on the aldehyde or oxo group of donor with NAD+ or NADP+ as acceptor. The systematic name of this enzyme class is 5-carboxymethyl-2-hydroxymuconic-semialdehyde:NAD+ oxidoreductase. This enzyme is also called carboxymethylhydroxymuconic semialdehyde dehydrogenase. 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 2D4E.
References.
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| https://en.wikipedia.org/wiki?curid=14132202 |
14132219 | 6-oxohexanoate dehydrogenase | Class of enzymes
In enzymology, a 6-oxohexanoate dehydrogenase (EC 1.2.1.63) is an enzyme that catalyzes the chemical reaction
6-oxohexanoate + NADP+ + H2O formula_0 adipate + NADPH + 2 H+
The 3 substrates of this enzyme are 6-oxohexanoate, NADP+, and H2O, whereas its 3 products are adipate, NADPH, and H+.
This enzyme belongs to the family of oxidoreductases, specifically those acting on the aldehyde or oxo group of donor with NAD+ or NADP+ as acceptor. The systematic name of this enzyme class is 6-oxohexanoate:NADP+ oxidoreductase. This enzyme participates in caprolactam degradation.
References.
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| https://en.wikipedia.org/wiki?curid=14132219 |
14132238 | Abscisic-aldehyde oxidase | Class of enzymes
In enzymology, an abscisic-aldehyde oxidase (EC 1.2.3.14) is an enzyme that catalyzes the chemical reaction
abscisic aldehyde + H2O + O2 formula_0 abscisate + H2O2
The 3 substrates of this enzyme are abscisic aldehyde, H2O, and O2, whereas its two products are abscisate and H2O2.
This enzyme belongs to the family of oxidoreductases, specifically those acting on the aldehyde or oxo group of donor with oxygen as acceptor. The systematic name of this enzyme class is abscisic-aldehyde:oxygen oxidoreductase. Other names in common use include abscisic aldehyde oxidase, AAO3, AOd, and AOdelta. This enzyme participates in carotenoid biosynthesis.
References.
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| https://en.wikipedia.org/wiki?curid=14132238 |
14132257 | Aldehyde dehydrogenase (FAD-independent) | Enzyme in the family of oxidoreductases
In enzymology, an aldehyde dehydrogenase (FAD-independent) (EC 1.2.99.7) is an enzyme that catalyzes the chemical reaction
an aldehyde + H2O + acceptor formula_0 a carboxylate + reduced acceptor
The 3 substrates of this enzyme are aldehyde, H2O, and acceptor, whereas its two products are carboxylate and reduced acceptor.
This enzyme belongs to the family of oxidoreductases, specifically those acting on the aldehyde or oxo group of donor with other acceptors. The systematic name of this enzyme class is aldehyde:acceptor oxidoreductase (FAD-independent). Other names in common use include aldehyde oxidase, aldehyde oxidoreductase, Mop, and AORDd.
Structural studies.
As of late 2007, only one structure has been solved for this class of enzymes, with the PDB accession code 1ZCS.
References.
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| https://en.wikipedia.org/wiki?curid=14132257 |
14132274 | Aldehyde dehydrogenase (NAD+) | In enzymology, an aldehyde dehydrogenase (NAD+) (EC 1.2.1.3) is an enzyme that catalyzes the chemical reaction
an aldehyde + NAD+ + H2O formula_0 an acid + NADH + H+
The 3 substrates of this enzyme are aldehyde, NAD+, and H2O, whereas its 3 products are acid, NADH, and H+.
This enzyme belongs to the family of oxidoreductases, specifically those acting on the aldehyde or oxo group of donor with NAD+ or NADP+ as acceptor. The systematic name of this enzyme class is aldehyde:NAD+ oxidoreductase. Other names in common use include CoA-independent aldehyde dehydrogenase, m-methylbenzaldehyde dehydrogenase, NAD-aldehyde dehydrogenase, NAD-dependent 4-hydroxynonenal dehydrogenase, NAD-dependent aldehyde dehydrogenase, NAD-linked aldehyde dehydrogenase, propionaldehyde dehydrogenase, and aldehyde dehydrogenase (NAD). This enzyme participates in 17 metabolic pathways: glycolysis / gluconeogenesis, ascorbate and aldarate metabolism, fatty acid metabolism, bile acid biosynthesis, urea cycle and metabolism of amino groups, valine, leucine and isoleucine degradation, lysine degradation, histidine metabolism, tryptophan metabolism, beta-alanine metabolism, glycerolipid metabolism, pyruvate metabolism, 1,2-dichloroethane degradation, propanoate metabolism, 3-chloroacrylic acid degradation, butanoate metabolism, and limonene and pinene degradation.
References.
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| https://en.wikipedia.org/wiki?curid=14132274 |
14132288 | Aldehyde dehydrogenase (NAD(P)+) | In enzymology, an aldehyde dehydrogenase [NAD(P)+] (EC 1.2.1.5) is an enzyme that catalyzes the chemical reaction
an aldehyde + NAD(P)+ + H2O formula_0 an acid + NAD(P)H + H+
The 4 substrates of this enzyme are aldehyde, NAD+, NADP+, and H2O, whereas its 4 products are acid, NADH, NADPH, and H+.
This enzyme belongs to the family of oxidoreductases, specifically those acting on the aldehyde or oxo group of donor with NAD+ or NADP+ as acceptor. The systematic name of this enzyme class is aldehyde:NAD(P)+ oxidoreductase. Other names in common use include aldehyde dehydrogenase [NAD(P)+], and ALDH. This enzyme participates in 5 metabolic pathways: glycolysis / gluconeogenesis, histidine metabolism, tyrosine metabolism, phenylalanine metabolism, and metabolism of xenobiotics by cytochrome p450.
Structural studies.
As of late 2007, 4 structures have been solved for this class of enzymes, with PDB accession codes 1AD3, 1EYY, 1EZ0, and 2AMF.
References.
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| https://en.wikipedia.org/wiki?curid=14132288 |
14132308 | Aldehyde dehydrogenase (NADP+) | In enzymology, an aldehyde dehydrogenase (NADP+) (EC 1.2.1.4) is an enzyme that catalyzes the chemical reaction
an aldehyde + NADP+ + H2O formula_0 an acid + NADPH + H+
The 3 substrates of this enzyme are aldehyde, NADP+, and H2O, whereas its 3 products are acid, NADPH, and H+.
This enzyme belongs to the family of oxidoreductases, specifically those acting on the aldehyde or oxo group of donor with NAD+ or NADP+ as acceptor. The systematic name of this enzyme class is aldehyde:NADP+ oxidoreductase. Other names in common use include NADP+-acetaldehyde dehydrogenase, NADP+-dependent aldehyde dehydrogenase, and aldehyde dehydrogenase (NADP+). This enzyme participates in caprolactam degradation.
References.
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| https://en.wikipedia.org/wiki?curid=14132308 |
14132318 | Aldehyde dehydrogenase (pyrroloquinoline-quinone) | In enzymology, an aldehyde dehydrogenase (pyrroloquinoline-quinone) (EC 1.2.99.3) is an enzyme that catalyzes the chemical reaction
an aldehyde + acceptor + H2O formula_0 a carboxylate + reduced acceptor
The 3 substrates of this enzyme are aldehyde, acceptor, and H2O, whereas its two products are carboxylate and reduced acceptor.
This enzyme belongs to the family of oxidoreductases, specifically those acting on the aldehyde or oxo group of donor with other acceptors. The systematic name of this enzyme class is aldehyde:(pyrroloquinoline-quinone) oxidoreductase. This enzyme is also called aldehyde dehydrogenase (acceptor). This enzyme participates in 4 metabolic pathways: fatty acid metabolism, pyruvate metabolism, propanoate metabolism, and butanoate metabolism. It employs one cofactor, PQQ.
References.
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| https://en.wikipedia.org/wiki?curid=14132318 |
14132355 | Aminobutyraldehyde dehydrogenase | In enzymology, an aminobutyraldehyde dehydrogenase (EC 1.2.1.19) is an enzyme that catalyzes the chemical reaction
4-aminobutanal + NAD+ + H2O formula_0 4-aminobutanoate + NADH + 2 H+
The 3 substrates of this enzyme are 4-aminobutanal, NAD+, and H2O, whereas its 3 products are 4-aminobutanoate, NADH, and H+.
This enzyme belongs to the family of oxidoreductases, specifically those acting on the aldehyde or oxo group of donor with NAD+ or NADP+ as acceptor. The systematic name of this enzyme class is 4-aminobutanal:NAD+ 1-oxidoreductase. Other names in common use include gamma-guanidinobutyraldehyde dehydrogenase (ambiguous), ABAL dehydrogenase, 4-aminobutyraldehyde dehydrogenase, 4-aminobutanal dehydrogenase, gamma-aminobutyraldehyde dehydrogenase, 1-pyrroline dehydrogenase, ABALDH, and YdcW. This enzyme participates in the urea cycle and the metabolism of amino groups and beta-alanine.
References.
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| https://en.wikipedia.org/wiki?curid=14132355 |
14132368 | Aminomuconate-semialdehyde dehydrogenase | In enzymology, an aminomuconate-semialdehyde dehydrogenase (EC 1.2.1.32) is an enzyme that catalyzes the chemical reaction
2-aminomuconate 6-semialdehyde + NAD+ + H2O formula_0 2-aminomuconate + NADH + 2 H+
The 3 substrates of this enzyme are 2-aminomuconate 6-semialdehyde, NAD+, and H2O, whereas its 3 products are 2-aminomuconate, NADH, and H+.
This enzyme belongs to the family of oxidoreductases, specifically those acting on the aldehyde or oxo group of donor with NAD+ or NADP+ as acceptor. This enzyme participates in 3 metabolic pathways: benzoic acid degradation via hydroxylation, tryptophan metabolism, and the degradation pathway for toluene and xylene.
Nomenclature.
The systematic name of this enzyme class is 2-aminomuconate-6-semialdehyde:NAD+ 6-oxidoreductase. Other names in common use include 2-aminomuconate semialdehyde dehydrogenase, 2-hydroxymuconic acid semialdehyde dehydrogenase, 2-hydroxymuconate semialdehyde dehydrogenase, alpha-aminomuconic epsilon-semialdehyde dehydrogenase, alpha-hydroxymuconic epsilon-semialdehyde dehydrogenase, and 2-hydroxymuconic semialdehyde dehydrogenase.
References.
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Further reading.
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| https://en.wikipedia.org/wiki?curid=14132368 |
14132386 | Aryl-aldehyde dehydrogenase | In enzymology, an aryl-aldehyde dehydrogenase (EC 1.2.1.29) is an enzyme that catalyzes the chemical reaction
an aromatic aldehyde + NAD+ + H2O formula_0 an aromatic acid + NADH + H+
The 3 substrates of this enzyme are aromatic aldehyde, NAD+, and H2O, whereas its 3 products are aromatic acid, NADH, and H+.
This enzyme belongs to the family of oxidoreductases, specifically those acting on the aldehyde or oxo group of donor with NAD+ or NADP+ as acceptor. The systematic name of this enzyme class is aryl-aldehyde:NAD+ oxidoreductase. This enzyme participates in tyrosine metabolism and biphenyl degradation.
References.
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| https://en.wikipedia.org/wiki?curid=14132386 |
14132401 | Aryl-aldehyde dehydrogenase (NADP+) | In enzymology, an aryl-aldehyde dehydrogenase (NADP+) (EC 1.2.1.30) is an enzyme that catalyzes the chemical reaction
an aromatic aldehyde + NADP+ + AMP + diphosphate + H2O formula_0 an aromatic acid + NADPH + H+ + ATP
The 5 substrates of this enzyme are aromatic aldehyde, NADP+, AMP, diphosphate, and H2O, whereas its 4 products are aromatic acid, NADPH, H+, and ATP.
This enzyme belongs to the family of oxidoreductases, specifically those acting on the aldehyde or oxo group of donor with NAD+ or NADP+ as acceptor. The systematic name of this enzyme class is aryl-aldehyde:NADP+ oxidoreductase (ATP-forming). Other names in common use include aromatic acid reductase, and aryl-aldehyde dehydrogenase (NADP+).
References.
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| https://en.wikipedia.org/wiki?curid=14132401 |
14132415 | Aryl-aldehyde oxidase | In enzymology, an aryl-aldehyde oxidase (EC 1.2.3.9) is an enzyme that catalyzes the chemical reaction
an aromatic aldehyde + O2 + H2O formula_0 an aromatic carboxylic acid + H2O2
The 3 substrates of this enzyme are aromatic aldehyde, O2, and H2O, whereas its two products are aromatic carboxylic acid and H2O2.
This enzyme belongs to the family of oxidoreductases, specifically those acting on the aldehyde or oxo group of donor with oxygen as acceptor. The systematic name of this enzyme class is aryl-aldehyde:oxygen oxidoreductase.
References.
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| https://en.wikipedia.org/wiki?curid=14132415 |
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