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System Size Expansion
The system size expansion, also known as van Kampen's expansion or the Ω-expansion, is a technique pioneered by Nico van Kampenvan Kampen, N. G. (2007) "Stochastic Processes in Physics and Chemistry", North-Holland Personal Library used in the analysis of stochastic processes. Specifically, it allows one to find an approximation to the solution of a master equation with nonlinear transition rates. The leading order term of the expansion is given by the linear noise approximation, in which the master equation is approximated by a Fokker–Planck equation with linear coefficients determined by the transition rates and stoichiometry of the system. Less formally, it is normally straightforward to write down a mathematical description of a system where processes happen randomly (for example, radioactive atoms randomly decay in a physical system, or genes that are expressed stochastically in a cell). However, these mathematical descriptions are often too difficult to solve for the study ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nico Van Kampen
Nicolaas 'Nico' Godfried van Kampen (June 22, 1921 – October 6, 2013) was a Dutch theoretical physicist, who worked mainly on statistical mechanics and non-equilibrium thermodynamics. Van Kampen was born in Leiden, and was a nephew of Frits Zernike. He studied physics at Leiden University, where in 1952 under the direction of Hendrik Anthony Kramers he earned his PhD with thesis ''Contributions to the quantum theory of light scattering''. He showed in his thesis how to deal with singularities in quantum mechanical scattering processes, an important step in the development of renormalization, according to Kramers. Van Kampen made fundamental contributions to non-equilibrium processes (in particular on the master equation) and in many-body theory (especially in plasma physics). His work on non-equilibrium processes began in 1953 in the research group of Sybren Ruurds de Groot (the successor to Kramers) in Leiden. In 1955 Van Kampen joined the Institute of Theoretical Physics at Utr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Taylor Expansion
In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor series are equal near this point. Taylor series are named after Brook Taylor, who introduced them in 1715. A Taylor series is also called a Maclaurin series, when 0 is the point where the derivatives are considered, after Colin Maclaurin, who made extensive use of this special case of Taylor series in the mid-18th century. The partial sum formed by the first terms of a Taylor series is a polynomial of degree that is called the th Taylor polynomial of the function. Taylor polynomials are approximations of a function, which become generally better as increases. Taylor's theorem gives quantitative estimates on the error introduced by the use of such approximations. If the Taylor series of a function is convergent, its sum is the limit of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Allosteric Regulation
In biochemistry, allosteric regulation (or allosteric control) is the regulation of an enzyme by binding an effector molecule at a site other than the enzyme's active site. The site to which the effector binds is termed the ''allosteric site'' or ''regulatory site''. Allosteric sites allow effectors to bind to the protein, often resulting in a conformational change and/or a change in protein dynamics. Effectors that enhance the protein's activity are referred to as ''allosteric activators'', whereas those that decrease the protein's activity are called ''allosteric inhibitors''. Allosteric regulations are a natural example of control loops, such as feedback from downstream products or feedforward from upstream substrates. Long-range allostery is especially important in cell signaling. Allosteric regulation is also particularly important in the cell's ability to adjust enzyme activity. The term ''allostery'' comes from the Ancient Greek ''allos'' (), "other", and ''stereos' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Intrinsic Noise Analyzer
Within bioinformatics, intrinsic Noise Analyzer (iNA) is an open source software for studying reaction kinetics in living cells. The software analyzes mathematical models of intracellular reaction kinetics such as gene expression, regulatory networks or signaling pathways to quantify concentration fluctuations due to the random nature of chemical reactions. Background Under well-mixed conditions, the concentrations in living cells are often modeled by a set of deterministic reaction rate equations. This approach frequently becomes inaccurate when some molecular species are present in low molecule numbers per cell because of the randomness inherent in chemical reaction kinetics. This randomness leads to fluctuations in intracellular molecule numbers and hence to cell-to-cell variability. The more accurate stochastic description of these systems is given by the Chemical Master Equation. The latter can be easily simulated by means of Monte Carlo methods such as the stochastic s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Covariance
In probability theory and statistics, covariance is a measure of the joint variability of two random variables. If the greater values of one variable mainly correspond with the greater values of the other variable, and the same holds for the lesser values (that is, the variables tend to show similar behavior), the covariance is positive. In the opposite case, when the greater values of one variable mainly correspond to the lesser values of the other, (that is, the variables tend to show opposite behavior), the covariance is negative. The sign of the covariance therefore shows the tendency in the linear relationship between the variables. The magnitude of the covariance is not easy to interpret because it is not normalized and hence depends on the magnitudes of the variables. The normalized version of the covariance, the correlation coefficient, however, shows by its magnitude the strength of the linear relation. A distinction must be made between (1) the covariance of two random ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rate Equation
In chemistry, the rate law or rate equation for a reaction is an equation that links the initial or forward reaction rate with the concentrations or pressures of the reactants and constant parameters (normally rate coefficients and partial reaction orders). For many reactions, the initial rate is given by a power law such as :v_0\; =\; k mathrmx mathrmy where and express the concentration of the species and usually in moles per liter (molarity, ). The exponents and are the partial ''orders of reaction'' for and and the ''overall'' reaction order is the sum of the exponents. These are often positive integers, but they may also be zero, fractional, or negative. The constant is the reaction rate constant or ''rate coefficient'' of the reaction. Its value may depend on conditions such as temperature, ionic strength, surface area of an adsorbent, or light irradiation. If the reaction goes to completion, the rate equation for the reaction rate v\; =\; k cex cey applies throug ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Arithmetic Mean
In mathematics and statistics, the arithmetic mean ( ) or arithmetic average, or just the ''mean'' or the ''average'' (when the context is clear), is the sum of a collection of numbers divided by the count of numbers in the collection. The collection is often a set of results of an experiment or an observational study, or frequently a set of results from a survey. The term "arithmetic mean" is preferred in some contexts in mathematics and statistics, because it helps distinguish it from other means, such as the geometric mean and the harmonic mean. In addition to mathematics and statistics, the arithmetic mean is used frequently in many diverse fields such as economics, anthropology and history, and it is used in almost every academic field to some extent. For example, per capita income is the arithmetic average income of a nation's population. While the arithmetic mean is often used to report central tendencies, it is not a robust statistic, meaning that it is greatly influe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Protein-DNA Interaction
DNA-binding proteins are proteins that have DNA-binding domains and thus have a specific or general affinity for single- or double-stranded DNA. Sequence-specific DNA-binding proteins generally interact with the major groove of B-DNA, because it exposes more functional groups that identify a base pair. However, there are some known minor groove DNA-binding ligands such as netropsin, distamycin, Hoechst 33258, pentamidine, DAPI and others. Examples DNA-binding proteins include transcription factors which modulate the process of transcription, various polymerases, nucleases which cleave DNA molecules, and histones which are involved in chromosome packaging and transcription in the cell nucleus. DNA-binding proteins can incorporate such domains as the zinc finger, the helix-turn-helix, and the leucine zipper (among many others) that facilitate binding to nucleic acid. There are also more unusual examples such as transcription activator like effectors. Non-specific DNA-protein ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Enzyme Kinetics
Enzyme kinetics is the study of the rates of enzyme-catalysed chemical reactions. In enzyme kinetics, the reaction rate is measured and the effects of varying the conditions of the reaction are investigated. Studying an enzyme's kinetics in this way can reveal the catalytic mechanism of this enzyme, its role in metabolism, how its activity is controlled, and how a drug or a modifier ( inhibitor or activator) might affect the rate. An enzyme (E) is typically a protein molecule that promotes a reaction of another molecule, its substrate (S). This binds to the active site of the enzyme to produce an enzyme-substrate complex ES, and is transformed into an enzyme-product complex EP and from there to product P, via a transition state ES*. The series of steps is known as the mechanism: : E + S ⇄ ES ⇄ ES* ⇄ EP ⇄ E + P This example assumes the simplest case of a reaction with one substrate and one product. Such cases exist: for example, a mutase such as phosphoglucomutase ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fano Factor
In statistics, the Fano factor, like the coefficient of variation, is a measure of the dispersion of a probability distribution of a Fano noise. It is named after Ugo Fano, an Italian American physicist. The Fano factor is defined as :F=\frac, where \sigma_W^2 is the variance and \mu_W is the mean of a random process in some time window ''W''. The Fano factor can be viewed as a kind of noise-to-signal ratio; it is a measure of the reliability with which the random variable could be estimated from a time window that on average contains several random events. For a Poisson process, the variance in the count equals the mean count, so ''F'' = 1 (normalization). If the time window is chosen to be infinity, the Fano factor is similar to the variance-to-mean ratio (VMR) which in statistics is also known as the index of dispersion. Use in particle detection In particle detectors, the Fano factor results from the energy loss in a collision not being purely statistical. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coefficient Of Variation
In probability theory and statistics, the coefficient of variation (CV), also known as relative standard deviation (RSD), is a standardized measure of dispersion of a probability distribution or frequency distribution. It is often expressed as a percentage, and is defined as the ratio of the standard deviation \sigma to the mean \mu (or its absolute value, The CV or RSD is widely used in analytical chemistry to express the precision and repeatability of an assay. It is also commonly used in fields such as engineering or physics when doing quality assurance studies and ANOVA gauge R&R, by economists and investors in economic models, and in neuroscience. Definition The coefficient of variation (CV) is defined as the ratio of the standard deviation \ \sigma to the mean \ \mu , c_ = \frac. It shows the extent of variability in relation to the mean of the population. The coefficient of variation should be computed only for data measured on scales that have a meaningful zer ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
