Co-operativity model of cellular control: Organization and complexity

Abstract

The “co-operativity model” for cellular control at the epigenetic level is introduced and solved. It enables one to define in a quantitative way the organization and the complexity of a representative “complex system”. The overall result is to provide an explanation of global properties starting from assumptions about small-scale motions and interactions. The model considers an assembly of oscillators representing the motions of biochemical reaction chains. Under certain conditions the interactions between chains take on a resonance form, thereby promoting coupling between oscillators of closely similar frequency. The dynamical framework is supplemented by a statistical hypothesis, that a complete state of a multichain system is specified by the set of probabilities with which each chain occupies its various oscillatory modes. A free energy functional is assumed to represent the major difference between “similar” biochemical systems in vivo and in vitro. General forms for energy and entropy are discussed. A parameter T 0, defined as organization temperature, multiplies the entropy term. A “self-consistent field” approximation handles the statistics. An explicit model for M reaction chains, frequency spectra such that one mode of each chain lies sufficiently close to a favored frequency called a frequency of accumulation, and attractive resonant interactions between chains, is solved variationally. The solution has the nature of a condensation of chains into a coherent state with correlated occupancies of the favored modes. The central property of the system is Δ 0, an order parameter which measures both the degree of correlation and the binding energy of the condensed chains. In sum, the organized state is characterized by Δ 0, T 0and M. M is termed the complexity of the system in the case where the components are distinguishable, as they are in a cell. The model fits in well with important features of animate behavior: the hierarchy of levels in both organization and temporal development, the stability of the system despite its extreme non-equilibrium, the advantages of passing energy and information through an open system, and the dominance of a small number of degrees of freedom. The model provides an explicit realization of a system governed by a “clock” and a “homeostat”.