Maximum Power Efficiency and Criticality in Random Boolean Networks

Random Boolean networks are models of disordered causal systems that can occur in cells and the biosphere. These are open thermodynamic systems exhibiting a flow of energy that is dissipated at a finite rate. Life does work to acquire more energy, then uses the available energy it has gained to perform more work. It is plausible that natural selection has optimized many biological systems for power efficiency: useful power generated per unit fuel. In this letter we begin to investigate these questions for random Boolean networks using Landauers erasure principle, which defines a minimum entropy cost for bit erasure. We show that critical Boolean networks maximize available power efficiency, which requires that the system have a finite displacement from equilibrium. Our initial results may extend to more realistic models for cells and ecosystems.

Mutual information in random Boolean models of regulatory networks

The amount of mutual information contained in time series of two elements gives a measure of how well their activities are coordinated. In a large, complex network of interacting elements, such as a genetic regulatory network within a cell, the average of the mutual information over all pairs <I> is a global measure of how well the system can coordinate its internal dynamics. We study this average pairwise mutual information in random Boolean networks (RBNs) as a function of the distribution of Boolean rules implemented at each element, assuming that the links in the network are randomly placed. Efficient numerical methods for calculating <I> show that as the number of network nodes N approaches infinity, the quantity N<I> exhibits a discontinuity at parameter values corresponding to critical RBNs. For finite systems it peaks near the critical value, but slightly in the disordered regime for typical parameter variations. The source of high values of N<I> is the indirect correlations between pairs of elements from different long chains with a common starting point. The contribution from pairs that are directly linked approaches zero for critical networks and peaks deep in the disordered regime.