Complete integrability of information processing by biochemical reactions

p-star models, mean-field random networks, and the heat hierarchy

We consider the mean-field analog of the p-star model for homogeneous random networks, and we compare its behavior with that of the p-star model and its classical mean-field approximation in the thermodynamic regime. We show that the partition function of the mean-field model satisfies a sequence of partial differential equations known as the heat hierarchy, and the models connectance is obtained as a solution of a hierarchy of nonlinear viscous PDEs. In the thermodynamic limit, the leading-order solution develops singularities in the space of parameters that evolve as classical shocks regularized by a viscous term. Shocks are associated with phase transitions and stable states are automatically selected consistently with the Maxwell construction. The case p=3 is studied in detail. Monte Carlo simulations show an excellent agreement between the p-star model and its mean-field analog at the macroscopic level, although significant discrepancies arise when local features are compared.

Exact analysis of phase transitions in mean-field Potts models

We construct the exact partition function of the Potts model on a complete graph subject to external fields with linear and nematic type couplings. The partition function is obtained as a solution to a linear diffusion equation and the free energy, in the thermodynamic limit, follows from its semiclassical limit. Analysis of singularities of the equations of state reveals the occurrence of phase transitions of nematic type at not zero external fields and allows for an interpretation of the phase transitions in terms of shock dynamics in the space of thermodynamic variables. The approach is shown at work in the case of a q-state model for q=3 but the method generalizes to arbitrary q. Critical asymptotics of magnetization, susceptibility, specific heat and relative critical exponents β,γ, and α are also provided.

Graphical Classification in Multi-Centrality-Index Diagrams for Complex Chemical Networks

Various sizes of chemical reaction network exist, from small graphs of linear networks with several inorganic species to huge complex networks composed of protein reactions or metabolic systems. Huge complex networks of organic substrates have been well studied using statistical properties such as degree distributions. However, when the size is relatively small, statistical data suffers from significant errors coming from irregular effects by species, and a macroscopic analysis is frequently unsuccessful. In this study, we demonstrate a graphical classification method for chemical networks that contain tens of species. Betweenness and closeness centrality indices of a graph can create a two-dimensional diagram with information of node distribution for a complex chemical network. This diagram successfully reveals systematic sharing of roles among species as a semi-statistical property in chemical reactions, and distinguishes it from the ones in random networks, which has no functional node distributions. This analytical approach is applicable for rapid and approximate understanding of complex chemical network systems such as plasma-enhanced reactions as well as visualization and classification of other graphs.