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:

${\displaystyle \varepsilon _{s_{i}}^{v}=\left({\frac {\partial v}{\partial s_{i}}}{\frac {s_{i}}{v}}\right)_{s_{j},s_{k},\ldots }={\frac {\partial \ln v}{\partial \ln s_{i}}}\approx {\frac {v\%}{s_{i}\%}}}$

where ${\displaystyle v}$ denotes the reaction rate and ${\displaystyle s}$ denotes the substrate concentration. Be aware that the notation will use lowercase roman letters, such as ${\displaystyle s,}$ 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, enzyme, 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 control analysis and was introduced in the early 1970s and possibly earlier by Henrik Kacser and Burns^[1] in Edinburgh and Heinrich and Rapoport^[2] in Berlin.

The elasticity concept has also been described by other authors, most notably Savageau^[3] in Michigan and Clarke^[4] at Edmonton. In the late 1960s Michael Savageau^[3] developed an innovative approach called biochemical systems theory that uses power-law expansions to approximate the nonlinearities in biochemical kinetics. The theory is very similar to metabolic control analysis and has been very successfully and extensively used to study the properties of different feedback and other regulatory structures in cellular networks. The power-law expansions used in the analysis invoke coefficients called kinetic orders, which are equivalent to the elasticity coefficients.

Bruce Clarke^[4] in the early 1970s, developed a sophisticated theory on analyzing the dynamic stability in chemical networks. As part of his analysis, Clarke also introduced the notion of kinetic orders and a power-law approximation that was somewhat similar to Savageau's power-law expansions. Clarke's approach relied heavily on certain structural characteristics of networks, called extreme currents (also called elementary modes in biochemical systems). Clarke's kinetic orders are also equivalent to elasticities.

Elasticities can also be usefully interpreted as the means by which signals propagate up or down a given pathway.^[5]

The fact that different groups independently introduced the same concept implies that elasticities, or their equivalent, kinetic orders, are most likely a fundamental concept in the analysis of complex biochemical or chemical systems.

Calculating elasticity coefficients

Elasticity coefficients can be calculated either algebraically or by numerical means.

Algebraic calculation of elasticity coefficients

Given the definition of the elasticity coefficient in terms of a partial derivative, it is possible, for example, to determine the elasticity of an arbitrary rate law by differentiating the rate law by the independent variable and scaling. For example, the elasticity coefficient for a mass-action rate law such as:

${\displaystyle v=k\ s_{1}^{n_{1}}s_{2}^{n_{2}}}$

where ${\displaystyle v}$ is the reaction rate, ${\displaystyle k}$ the reaction rate constant, ${\displaystyle s_{i}}$ is the ith chemical species involved in the reaction and ${\displaystyle n_{i}}$ the ith reaction order, then the elasticity, ${\displaystyle \varepsilon _{s_{1}}^{v}}$ can be obtained by differentiating the rate law with respect to ${\displaystyle s_{1}}$ and scaling:

${\displaystyle \varepsilon _{s_{1}}^{v}={\frac {\partial v}{\partial s_{1}}}{\frac {s_{1}}{v}}=n_{1}\ k\ s_{1}^{n_{1}-1}s_{2}^{n_{2}}{\frac {s_{1}}{k\ s_{1}^{n_{1}}s_{2}^{n_{2}}}}=n_{1}}$

That is, the elasticity for a mass-action rate law is equal to the order of reaction of the species.

For example the elasticity of A in the reaction ${\displaystyle 2A\rightleftharpoons C}$ where the rate of reaction is given by: ${\displaystyle v=kA^{2}}$, the elasticity can be evaluated using:

${\displaystyle \varepsilon _{a}^{v}={\frac {\partial v}{\partial a}}{\frac {a}{v}}={\frac {2kaa}{ka^{2}}}=2}$

Elasticities can also be derived for more complex rate laws such as the Michaelis–Menten rate law. If

${\displaystyle v={\frac {V_{\max }s}{K_{m}+s}}}$

then it can be easily shown than

${\displaystyle \varepsilon _{s}^{v}={\frac {K_{m}}{K_{m}+s}}}$

This equation illustrates the idea that elasticities need not be constants (as with mass-action laws) but can be a function of the reactant concentration. In this case, the elasticity approaches unity at low reactant concentration (s) and zero at high reactant concentration.

For the reversible Michaelis–Menten rate law:

${\displaystyle v={\frac {V_{\max }/K_{m_{1}}(s-p/K_{\text{eq}})}{1+s/K_{m_{1}}+p/K_{m_{2}}}}}$

where ${\displaystyle V_{\max }}$ is the forward ${\displaystyle V_{max}}$, ${\displaystyle K_{m_{1}}}$ the forward $$Zdroj:https://en.wikipedia.org?pojem=Elasticity_coefficient
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