Evolving Always-Critical Networks

Abstract

Living beings share several common features at the molecular level, but there are very few large-scale “operating principles” which hold for all (or almost all) organisms. However, biology is subject to a deluge of data, and as such, general concepts such as this would be extremely valuable. One interesting candidate is the “criticality” principle, which claims that biological evolution favors those dynamical regimes that are intermediaries between ordered and disordered states (i.e., “at the edge of chaos”). The reasons why this should be the case and experimental evidence are briefly discussed, observing that gene regulatory networks are indeed often found on, or close to, the critical boundaries. Therefore, assuming that criticality provides an edge, it is important to ascertain whether systems that are critical can further evolve while remaining critical. In order to explore the possibility of achieving such “always-critical” evolution, we resort to simulated evolution, by suitably modifying a genetic algorithm in such a way that the newly-generated individuals are constrained to be critical. It is then shown that these modified genetic algorithms can actually develop critical gene regulatory networks with two interesting (and quite different) features of biological significance, involving, in one case, the average gene activation values and, in the other case, the response to perturbations. These two cases suggest that it is often possible to evolve networks with interesting properties without losing the advantages of criticality. The evolved networks also show some interesting features which are discussed.

Exploring the Dynamic Organization of Random and Evolved Boolean Networks

“…This observation is compatible with the fact that the asymptotic states of these systems must have a high number of active genes. (This observation is compatible with a remark present in [34], albeit this latter remark provides a lower evidence because of the smaller number of analyzed systems. )…”

“…Given that criticality is supposed to be an important property, it is particularly interesting to consider a modified version of the GA, presented in [34], which preserves the bias of the Boolean functions during inheritance from parents to children. This GA maintains the usual selection procedure (a roulette wheel) and has a slightly modified mutation and crossover.…”

“…In particular, we use the single-point crossover (where the cut-off point is randomly chosen among those that keep the parental bias as much as possible), while the mutation hits randomly, but in a way to again get closer to the bias of the parents. It has been verified that this modified GA preserves the bias of the initial individuals: additionally, in case of populations composed of dynamically critical individuals, the property of criticality is maintained over the generations [34]. (In this article, we follow the classic criticality measure for RBNs, represented by the Derrida procedure carried out on a high number of random initial conditions [20,47].…”

“…We have therefore introduced a modified GA which tends to keep the evolved networks critical. It has been presented elsewhere [34], where we have also shown that it is successful in maintaining criticality while achieving its goal, in the case where the goal is a fixed fraction of active nodes (as in the example just discussed), as well as in the case where the target is that of giving rise to avalanches with a predefined fraction of "up" nodes, i.e., nodes whose activation is higher in the perturbed network than in the wild type. The modified GA is briefly recalled in Section 2.3.…”

Attractor-Specific and Common Expression Values in Random Boolean Network Models (with a Preliminary Look at Single-Cell Data)

“…Random Boolean Networks (RBNs for short) are strongly simplified models of gene regulatory networks (GRNs), proposed by one of us (Kauffman) more than 50 years ago, which have also been widely studied as abstract models of complex systems, thanks to the fact that their dynamical behavior can be tuned from ordered to disordered by modifying a few key parameters. They have also been applied to different biological phenomena, such as, e.g., cell differentiation [1][2][3], as well as to different fields, including robotics [4][5][6], the study of evolutionary processes [7][8][9] and the simulation of social systems [10][11][12][13].…”