a control coefficient is a relative measure of how much a perturbation on a parameter affects a system variable (e.g. fluxes or concentrations). It is defined [Kacser73] [Heinrich74] [Burns85] as:

where A is the variable, i the step (enzyme) and v the steady-state rate of the perturbed step. The most common control coefficients are those for fluxes and species concentrations, but any variable of the system can be analyzed with MCA and have control coefficients defined by equations analogous to equation 1. In fact, there is no need even for the system to be in a steady state. Gepasi only calculates directly the steady-state concentration- and flux-control coefficients, those for other variables can still be estimated by simulating small perturbations.

a very important property of steady-state metabolic systems was uncovered with the MCA formalism. This concerns the summation of all the flux control coefficients of a pathway. By various procedures [Kacser73] [Heinrich75] [Giersch88] [Reder88] it can be demonstrated that for a given reference flux the sum of all flux-control coefficients (of all steps) is equal to unity:

For a given reference species concentration the sum of all concentration-control coefficients is zero:

where the summations are over all the steps of the system.

According to the first summation theorem, increases in some of the flux-control coefficients imply decreases in the others so that the total remains unity. As a consequence of the summation theorems, one concludes that the control coefficients are global properties and that in metabolic systems, control is a systemic property, dependent on all of the system's elements (steps).

In enzyme kinetics the behavior of isolated enzymes is studied through the dependence of the initial rates of reaction with the concentration of the substrate(s). Enzyme kinetic studies are centered on derivation of rate equations and the determination of their kinetic constants such as Michaelis constants or limiting-rates or even on the elementary rate constants of a specific reaction mechanism.

In metabolic control analysis the properties of each (isolated) enzyme are measured in a way very similar to the flux-control properties: using a sensitivity, known as the elasticity coefficient [Kacser73] [Heinrich74] [Burns85]. In this case, one has to consider the effect of perturbations of a reaction parameter on the local reaction rate. By local one means that this sensitivity refers to the isolated reaction which has the same characteristics (effector and enzyme concentrations, temperature, and so on) as in the whole system at the operating point (steady state) of interest. The elasticity coefficients are defined as the ratio of relative change in local rate to the relative change in one parameter (normally the concentration of an effector). Infinitesimally, this is written as:

where v is the rate of the enzyme in question and p is the parameter of the perturbation. Each enzyme has as many elasticity coefficients as the number of parameters that affect it. One can immediately recognize the concentration of the reaction substrates, products and modifiers as parameters of the reaction. Unlike control coefficients, elasticity coefficients are not systemic properties but rather measure how isolated enzymes are sensitive to changes in their parameters. The elasticity coefficients can be obtained from the kinetic functions by partial derivation. Again like the control coefficients, the elasticity coefficients are not constants, they are dependent on the value of the relevant parameter and so are different for each steady-state.

a particularly useful and important feature of MCA is that it can relate the kinetic properties of the individual reactions (local properties) with (global) properties of the whole intact pathway. This is done through the connectivity theorems [Kacser73] that relate the control coefficients and the elasticity coefficients of steps with common intermediate species.

The connectivity theorem for flux-control coefficients [Kacser73] states that, for a common species S, the sum of the products of the flux-control coefficient of all (i) steps affected by S and its elasticity coefficients towards S, is zero:

For the concentration-control coefficients, the following two equations apply [Westerhoff84]:

The first equation applies to the case in which the reference species (A) is different from the perturbed species (S). Whereas the second applies to the case in which the reference species is the same as the perturbed species.

The connectivity theorems allow MCA to describe how perturbations on species of a pathway propagate through the chain of enzymes. The local (kinetic) properties of each enzyme effectively propagate the perturbation to and from its immediate neighbors.

COPASI calculates (non-normalized) elasticity coefficients by numerical derivation with finite differences. To calculate control coefficients from steady-state data, COPASI applies the method described in [Reder88]. This method works with the reduced system where some variables are eliminated using conservation relations (see Deterministic Model). All coefficients are obtained unscaled by this method and are scaled with the appropriate steady state concentrations and fluxes (the same with the elasticities). Both scaled and unscaled coefficients and elasticities are displayed and available for output.

The rest of the options is described in the sections for Steady State and Time Course.