Metabolic control analysis (MCA) is a mathematical framework for describing metabolic, signaling, and genetic pathways. MCA quantifies how variables, such 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.^[1][2][3]

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^{[citation needed]}.

More recent work^[4] has shown that MCA can be mapped directly on to classical control theory and are as such equivalent.

Biochemical systems theory^[5] (BST) is a similar formalism, though with rather different objectives. Both are evolutions of an earlier theoretical analysis by Joseph Higgins.^[6]

Chemical reaction network theory is another theoretical framework that has overlap with both MCA and BST but is considerably more mathematically formal in its approach.^[7] It's emphasis is primarily on dynamic stability criteria^[8] and related theorems associated with mass-action networks. In more recent years the field has also developed ^[9] a sensitivity analysis which is similar if not identical to MCA and BST.

Control coefficients

A control coefficient^{[10] [11][12]} measures the relative steady state change in a system variable, e.g. pathway flux (J) or metabolite concentration (S), in response to a relative change in a parameter, e.g. enzyme activity or the steady-state rate (${\displaystyle v_{i}}$) of step ${\displaystyle i}$. The two main control coefficients are the flux and concentration control coefficients. Flux control coefficients are defined by

${\displaystyle C_{v_{i}}^{J}=\left({\frac {dJ}{dp}}{\frac {p}{J}}\right){\bigg /}\left({\frac {\partial v_{i}}{\partial p}}{\frac {p}{v_{i}}}\right)={\frac {d\ln J}{d\ln v_{i}}}}$

and concentration control coefficients by

${\displaystyle C_{v_{i}}^{S}=\left({\frac {dS}{dp}}{\frac {p}{S}}\right){\bigg /}\left({\frac {\partial v_{i}}{\partial p}}{\frac {p}{v_{i}}}\right)={\frac {d\ln S}{d\ln v_{i}}}}$

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Summation theorems

The flux control summation theorem was discovered independently by the Kacser/Burns group^[10] and the Heinrich/Rapoport group^[11] in the early 1970s and late 1960s. The flux control summation theorem implies that metabolic fluxes are systemic properties and that their control is shared by all reactions in the system. When a single reaction changes its control of the flux this is compensated by changes in the control of the same flux by all other reactions.

${\displaystyle \sum _{i}C_{v_{i}}^{J}=1}$

${\displaystyle \sum _{i}C_{v_{i}}^{s}=0}$

Elasticity coefficients

The elasticity coefficient measures the local response of an enzyme or other chemical reaction to changes in its environment. Such changes include factors such as substrates, products, or effector concentrations. For further information, please refer to the dedicated page at elasticity coefficients.

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Connectivity theorems

The connectivity theorems^[10][11] are specific relationships between elasticities and control coefficients. They are useful because they highlight the close relationship between the kinetic properties of individual reactions and the system properties of a pathway. Two basic sets of theorems exists, one for flux and another for concentrations. The concentration connectivity theorems are divided again depending on whether the system species ${\displaystyle S_{n}}$ is different from the local species ${\displaystyle S_{m}}$.

${\displaystyle \sum _{i}C_{i}^{J}\varepsilon _{s}^{i}=0}$

${\displaystyle \sum _{i}C_{i}^{s_{n}}\varepsilon _{s_{m}}^{i}=0\quad n\neq m}$

${\displaystyle \sum _{i}C_{i}^{s_{n}}\varepsilon _{s_{m}}^{i}=-1\quad n=m}$

Response Coefficient

Kacser and Burns^[10] introduced an additional coefficient that described how a biochemical pathway would respond the external environment. They termed this coefficient the response coefficient and designated it using the symbol R. The response coefficient is an important metric because it can be used to assess how much a nutrient or perhaps more important, how a drug can influence a pathway. This coefficient is therefore highly relevant to the pharmaceutical industry.^[15]

The response coefficient is related to the core of metabolic control analysis via the response coefficient theorem, which is stated as follows:

${\displaystyle R_{m}^{X}=C_{i}^{X}\varepsilon _{m}^{i}}$

where ${\displaystyle X}$ is a chosen observable such as a flux or metabolite concentration, ${\displaystyle i}$ is the step that the external factor targets, ${\displaystyle C_{i}^{X}}$ is the control coefficient of the target steps, and ${\displaystyle \varepsilon _{m}^{i}}$ is the elasticity of the target step with respect to the external factor ${\displaystyle m}$.

The key observation of this theorem is that an external factor such as a therapeutic drug, acts on the organism's phenotype via two influences: 1) How well the drug can affect the target itself through effective binding of the drug to the target protein and its effect on the protein activity. This effectiveness is described by the elasticity ${\displaystyle \varepsilon _{m}^{i}}$ and 2) How well do modifications of the target influence the phenotype by transmission of the perturbation to the rest of the network. This is indicated by the control coefficient ${\displaystyle C_{i}^{X}}$.

A drug action, or any external factor, is most effective when both these factors are strong. For example, a drug might be very effective at changing the activity of its target protein, however if that perturbation in protein activity is unable to be transmitted to the final phenotype then the effectiveness of the drug is greatly diminished.

If a drug or external factor, ${\displaystyle m}$, targets multiple sites of action, for example ${\displaystyle n}$ sites, then the overall response in a phenotypic factor ${\displaystyle X}$, is the sum of the individual responses:

${\displaystyle R_{m}^{X}=\sum _{i=1}^{n}C_{i}^{X}\varepsilon _{m}^{i}}$

Control equations

It is possible to combine the summation with the connectivity theorems to obtain closed expressions that relate the control coefficients to the elasticity coefficients. For example, consider the simplest non-trivial pathway:

${\displaystyle X_{o}\rightarrow S\rightarrow X_{1}}$

We assume that $$Zdroj:https://en.wikipedia.org?pojem=Metabolic_control_theory
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