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:

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

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.