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:
Contents
- Calculating elasticity coefficients
- Algebraic calculation of elasticity coefficients
- Differentiating in log space
- Numerical calculation of elasticity coefficients
- Elasticity matrix
- References
where
The elasticity concept has also been described by other authors, most notably Savageau in Michigan and Clarke at Edmonton. In the late 1960s Michael Savageau 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 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.
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 in various ways, either numerically or algebraically.
Algebraic calculation of elasticity coefficients
Given the definition of the elasticity 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:
where
That is the elasticity for a mass-action rate law is equal to the order of reaction of the species.
Elasticities can also be derived for more complex rate laws such as the Michaelis-Menten rate law. 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. 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:
where
where
As a final example, consider the Hill equation:
where n is the Hill coefficient and
Note that at low S the elasticity approaches n. At high S the elasticity approaches zero. This means the elasticity is bounded between zero and the Hill coefficient.
Differentiating in log space
An approach that is amenable to algebraic calculation by computer algebra methods is to differentiate in log space. Since the elasticity can be defined logarithmically, that is:
differentiating in log space is an obvious approach. Logarithmic differentiation is particularly convenient in algebra software such as Mathematica or Maple, where logarithmic differentiation rules can be defined.
Numerical calculation of elasticity coefficients
Elasticities coefficient can also be computed numerically, something that is often done in simulation software.
Elasticity matrix
The unscaled elasticities are often depicted in matrix form, called the elasticity matrix. Given a network with m molecular species and n reactions, then the elasticity matrix is defined as: