Elasticity coefficient

In chemistry, the rate of a chemical reaction is influenced by many different factors, such as temperature, pH, reactant, the concentration of products, and other effectors.

The degree to which these factors change the reaction rate is described by the elasticity coefficient.

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.

Bruce Clarke[4] in the early 1970s, developed a sophisticated theory on analyzing the dynamic stability in chemical networks.

Clarke's approach relied heavily on certain structural characteristics of networks, called extreme currents (also called elementary modes in biochemical systems).

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.

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

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.

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

If then it can be easily shown than 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.

The elasticities for a reversible uni-uni enzyme catalyzed reaction was previously given by: An interesting result can be obtained by evaluating the sum

However, it is unlikely that a given enzyme will have both substrate and product concentrations much greater than their respective Kms.

This means that a substrate will have a great influence over the forward reaction rate than the corresponding product.

[6] This result has important implications for the distribution of flux control in a pathway with sub-saturated reaction steps.

In general, a perturbation near the start of a pathway will have more influence over the steady state flux than steps downstream.

This is because a perturbation that travels downstream is determined by all the substrate elasticities, whereas a perturbation downstream that has to travel upstream if determined by the product elasticities.

[7][8] The table below summarizes the extreme values for the elasticities given a reversible Michaelis-Menten rate law.

In general we can expresion this relationship as the product of the enzyme concentration and a saturation function,

Elasticities coefficient can also be computed numerically, something that is often done in simulation software.

In each case, the new reaction rate is recorded; this is called the two-point estimation method.

This quantity serves as a measure of the rate of proportional change of the function

measures the slope of the curve when plotted on a semi-logarithmic scale, that is the rate of proportional change.

Since the elasticity can be defined logarithmically, that is: differentiating in log space is an obvious approach.

[11] A more detailed examination and the rules differentiating in log space can be found at Elasticity of a function.