Elasticities in Metabolic Control Analysis: algebraic derivation of simplified expressions

Metabolic Control Analysis is one of many disciplines that make use of scaled derivatives. In particular, 'elasticities' are used to quantify the effect of an effector or substrate concentration on an enzyme rate under locally specified conditions. Normally an algebraic expression for the elasticity of an enzyme is obtained by differentiating its rate law, multiplying by the effector concentration and dividing by the rate law itself: this results in considerable expression expansion, and when the results are subsequently simplified it is often at the expense of biological comprehensibility.

We present a novel algorithm which not only circumvents the expression expansion, but preserves an elegant separation of the components in enzyme behaviour. Easily implemented, and producing gains in both performance and numerical precision, the algorithm is potentially applicable to a number of existing packages. It also greatly assists the manual derivation and evaluation of elasticities, allowing the elasticity of even quite complex enzyme systems to be written by inspection.