# The Connectivity Relations

### Introduction

In the previous sections both global (the control coefficients) and local (the elasticity coefficients) properties of metabolic systems were described. In this section of the MCA Web all is revealed about using the elasticity coefficients to calculate the control coefficients.

### A Metabolic Web

In each cell of a living organism there are hundreds of biochemical reactions ocurring simultaneously. These reactions are interconnected by the metabolites. Most metabolites are products of one reaction while they are also substrates of another, this links the two reactions since they are both sensitive to changes in the concentration of their common metabolite. Figure 4 illustrates this web of reactions, for a more detailed example see the acyclic fatty acid synthesis pathways or the metabolic reactions and pathways database (INRA – France).

### The Connectivity Relations

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 (Kacser and Burns 1973) that relate the control coefficients and the elasticity coefficients of steps with common intermediate metabolites.

The connectivity theorem for flux-control coefficients (Kacser and Burns 1973) states that, for a common metabolite 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 (Westerhoff and Chen 1984):

Equation 7 applies to the case in which the reference metabolite (A) is different from the perturbed metabolite (S). Equation 8 applies to the case in which the reference metabolite is the same as the perturbed metabolite.

The connectivity theorems extend MCA to be able to describe how perturbations on metabolites of a pathway propagate through the chain of enzymes (the metabolic web). The local (kinetic) properties of each enzyme effectively propagate the perturbation to and from its immediate neighbours.

### From Enzyme Kinetics to Control Analysis

For linear metabolic pathways, i.e. those in which each metabolite is never a substrate of more than one enzyme and/or a product of more than an enzyme (see figure 5 for an example), the set of connectivity relations together with the summation theorem form a system of n equations in n unknows and one can then calculate all control coefficients given all the enzyme elasticities (Heinrich & Rapoport1974, Fell & Sauro 1985). This can be done in a systematic way using matrix methods and is indeed possible for pathways with any structural complexity (see below).

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