Reduction, integration and emergence in biochemical networks

Most studies of molecular cell biology are based upon a process of decomposition of complex biological systems into their components, followed by the study of these components. The aim of the present paper is to discuss, on a physical basis, the internal logic of this process of reduction. The analysis is performed on simple biological systems, namely protein and metabolic networks. A multi-sited protein that binds two ligands x and y can be considered the simplest possible biochemical network. The organization of this network can be described through a comparison of three systems, i.e. XY, X and Y. X and Y are component sub-systems that collect states xi and yj, respectively, i.e. protein states that have bound either i molecules of x (whether or not these states have also bound y), or j molecules of y (whether or not these states have bound x). XY is a system made up of the specific association of X and Y that collects states xiyj. One can define mean self-informations per node of the network, , and . Reduction of the system XY into its components is possible if, and only if, ,is equal to the sum of and . If is smaller than the sum of and , the system is integrated, for it has less selfinformation than the set of its components X and Y. It can also occur that , be larger than the sum of and . Hence, the system XY displays negative integration and emergence of self-information relative to its components X and Y. Such a system is defined as complex. Positive or negative integration of the system implies it cannot be reduced to its components. The degree of integration can be measured by a function , called mutual information of integration. In the case of enzyme networks, emergence of self-information is associated with emergence of catalytic activity. Moreover, if the enzyme reaction is part of a metabolic sequence, its mutual information of integration can be increased by an effect of context of this sequence.