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Biochemical Systems Equation
The biochemical systems equation is a compact equation of nonlinear differential equations for describing a kinetic model for any network of coupled biochemical reactions and transport processes. The equation is expressed in the following form: \dfrac = ( (p), p) The notation for the dependent variable x varies among authors. For example, some authors use s, indicating species. x is used here to match the state space notation used in control theory but either notation is acceptable. \bf N is the stoichiometry matrix which is an m by n matrix of stoichiometry coefficient. m is the number of species and n the number of biochemical reactions. The notation for \bf N is also variable. In constraint-based modeling the symbol \bf N tends to be used to indicate 'stoichiometry'. However in biochemical dynamic modeling and sensitivity analysis, \bf N tends to be in more common use to indicate 'number'. In the chemistry domain, the symbol used for the stoichiometry ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Differential Equation
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, and the differential equation defines a relationship between the two. Such relations are common; therefore, differential equations play a prominent role in many disciplines including engineering, physics, economics, and biology. Mainly the study of differential equations consists of the study of their solutions (the set of functions that satisfy each equation), and of the properties of their solutions. Only the simplest differential equations are solvable by explicit formulas; however, many properties of solutions of a given differential equation may be determined without computing them exactly. Often when a closed-form expression for the solutions is not available, solutions may be approximated numerically using computers. The theory of d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
State-space Representation
In control engineering, a state-space representation is a mathematical model of a physical system as a set of input, output and state variables related by first-order differential equations or difference equations. State variables are variables whose values evolve over time in a way that depends on the values they have at any given time and on the externally imposed values of input variables. Output variables’ values depend on the values of the state variables. The "state space" is the Euclidean space in which the variables on the axes are the state variables. The state of the system can be represented as a ''state vector'' within that space. To abstract from the number of inputs, outputs and states, these variables are expressed as vectors. If the dynamical system is linear, time-invariant, and finite-dimensional, then the differential and algebraic equations may be written in matrix form. The state-space method is characterized by significant algebraization of general system ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Control Theory
Control theory is a field of mathematics that deals with the control of dynamical systems in engineered processes and machines. The objective is to develop a model or algorithm governing the application of system inputs to drive the system to a desired state, while minimizing any ''delay'', ''overshoot'', or ''steady-state error'' and ensuring a level of control stability; often with the aim to achieve a degree of optimality. To do this, a controller with the requisite corrective behavior is required. This controller monitors the controlled process variable (PV), and compares it with the reference or set point (SP). The difference between actual and desired value of the process variable, called the ''error'' signal, or SP-PV error, is applied as feedback to generate a control action to bring the controlled process variable to the same value as the set point. Other aspects which are also studied are controllability and observability. Control theory is used in control system eng ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stoichiometry Matrix
Stoichiometry refers to the relationship between the quantities of reactants and products before, during, and following chemical reactions. Stoichiometry is founded on the law of conservation of mass where the total mass of the reactants equals the total mass of the products, leading to the insight that the relations among quantities of reactants and products typically form a ratio of positive integers. This means that if the amounts of the separate reactants are known, then the amount of the product can be calculated. Conversely, if one reactant has a known quantity and the quantity of the products can be empirically determined, then the amount of the other reactants can also be calculated. This is illustrated in the image here, where the balanced equation is: : Here, one molecule of methane reacts with two molecules of oxygen gas to yield one molecule of carbon dioxide and two molecules of water. This particular chemical equation is an example of complete combustion. St ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Metabolic Control Analysis
Metabolic control analysis (MCA) is a mathematical framework for describing metabolic, signaling, and genetic pathways. MCA quantifies how variables,elastsuch as fluxes and species concentrations, depend on network parameters. In particular, it is able to describe how network-dependent properties, called control coefficients, depend on local properties called elasticities or Elasticity Coefficients. MCA was originally developed to describe the control in metabolic pathways but was subsequently extended to describe signaling and genetic networks. MCA has sometimes also been referred to as ''Metabolic Control Theory,'' but this terminology was rather strongly opposed by Henrik Kacser, one of the founders. More recent work has shown that MCA can be mapped directly on to classical control theory and are as such equivalent. Biochemical systems theory is a similar formalism, though with rather different objectives. Both are evolutions of an earlier theoretical analysis by Joseph Higg ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linearization
In mathematics, linearization is finding the linear approximation to a function at a given point. The linear approximation of a function is the first order Taylor expansion around the point of interest. In the study of dynamical systems, linearization is a method for assessing the local stability of an equilibrium point of a system of nonlinear differential equations or discrete dynamical systems. This method is used in fields such as engineering, physics, economics, and ecology. Linearization of a function Linearizations of a function are lines—usually lines that can be used for purposes of calculation. Linearization is an effective method for approximating the output of a function y = f(x) at any x = a based on the value and slope of the function at x = b, given that f(x) is differentiable on , b/math> (or , a/math>) and that a is close to b. In short, linearization approximates the output of a function near x = a. For example, \sqrt = 2. However, what would be a good appro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elasticity Coefficient
The rate of a chemical reaction is influenced by many different factors, such as temperature, pH, reactant, and product concentrations and other effectors. The degree to which these factors change the reaction rate is described by the elasticity coefficient. This coefficient is defined as follows: : \varepsilon_^v=\left(\frac \frac\right)_=\frac \approx \frac where v denotes the reaction rate and s denotes the substrate concentration. Be aware that the notation will use lowercase roman letters, such as s, to indicate concentrations. The partial derivative in the definition indicates that the elasticity is measured with respect to changes in a factor S while keeping all other factors constant. The most common factors include substrates, products, and effectors. The scaling of the coefficient ensures that it is dimensionless and independent of the units used to measure the reaction rate and magnitude of the factor. The elasticity coefficient is an integral part of metabolic con ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Metabolic Control Analysis
Metabolic control analysis (MCA) is a mathematical framework for describing metabolic, signaling, and genetic pathways. MCA quantifies how variables,elastsuch as fluxes and species concentrations, depend on network parameters. In particular, it is able to describe how network-dependent properties, called control coefficients, depend on local properties called elasticities or Elasticity Coefficients. MCA was originally developed to describe the control in metabolic pathways but was subsequently extended to describe signaling and genetic networks. MCA has sometimes also been referred to as ''Metabolic Control Theory,'' but this terminology was rather strongly opposed by Henrik Kacser, one of the founders. More recent work has shown that MCA can be mapped directly on to classical control theory and are as such equivalent. Biochemical systems theory is a similar formalism, though with rather different objectives. Both are evolutions of an earlier theoretical analysis by Joseph Higg ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stoichiometry Matrix
Stoichiometry refers to the relationship between the quantities of reactants and products before, during, and following chemical reactions. Stoichiometry is founded on the law of conservation of mass where the total mass of the reactants equals the total mass of the products, leading to the insight that the relations among quantities of reactants and products typically form a ratio of positive integers. This means that if the amounts of the separate reactants are known, then the amount of the product can be calculated. Conversely, if one reactant has a known quantity and the quantity of the products can be empirically determined, then the amount of the other reactants can also be calculated. This is illustrated in the image here, where the balanced equation is: : Here, one molecule of methane reacts with two molecules of oxygen gas to yield one molecule of carbon dioxide and two molecules of water. This particular chemical equation is an example of complete combustion. St ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chemical Reaction Network Theory
Chemical reaction network theory is an area of applied mathematics that attempts to model the behaviour of real-world chemical systems. Since its foundation in the 1960s, it has attracted a growing research community, mainly due to its applications in biochemistry and theoretical chemistry. It has also attracted interest from pure mathematicians due to the interesting problems that arise from the mathematical structures involved. History Dynamical properties of reaction networks were studied in chemistry and physics after the invention of the law of mass action. The essential steps in this study were introduction of detailed balance for the complex chemical reactions by Rudolf Wegscheider (1901), development of the quantitative theory of chemical chain reactions by Nikolay Semyonov (1934), development of kinetics of catalytic reactions by Cyril Norman Hinshelwood, and many other results. Three eras of chemical dynamics can be revealed in the flux of research and publ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
List Of Systems Biology Modeling Software
Systems biology relies heavily on building mathematical models to help understand and make predictions of biological processes. Specialized software to assist in building models has been developed since the arrival of the first digital computers. The following list gives the currently supported software applications available to researchers. The vast majority of modern systems biology modeling software support SBML, which is the de facto standard for exchanging models of biological cellular processes. Some tools also support CellML, a standard used for representing physiological processes. The advantage of using standard formats is that even though a particular software application may eventually become unsupported and even unusable, the models developed by that application can be easily transferred to more modern equivalents. This allows scientific research to be reproducible long after the original publication of the work. Actively supported open-source software applications ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Biochemistry Methods
The following outline is provided as an overview of and topical guide to biochemistry: Biochemistry – study of chemical processes in living organisms, including living matter. Biochemistry governs all living organisms and living processes. Applications of biochemistry * Testing ** Ames test – salmonella bacteria is exposed to a chemical under question (a food additive, for example), and changes in the way the bacteria grows are measured. This test is useful for screening chemicals to see if they mutate the structure of DNA and by extension identifying their potential to cause cancer in humans. ** Pregnancy test – one uses a urine sample and the other a blood sample. Both detect the presence of the hormone human chorionic gonadotropin (hCG). This hormone is produced by the placenta shortly after implantation of the embryo into the uterine walls and accumulates. ** Breast cancer screening – identification of risk by testing for mutations in two genes&md ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
