Most Networks in Wagner's Model Are Cycling

Abstract

In this paper we study a model of gene networks introduced by Andreas Wagner in the 1990s that has been used extensively to study the evolution of mutational robustness. We investigate a range of model features and parameters and evaluate the extent to which they influence the probability that a random gene network will produce a fixed point steady state expression pattern. There are many different types of models used in the literature, (discrete/continuous, sparse/dense, small/large network) and we attempt to put some order into this diversity, motivated by the fact that many properties are qualitatively the same in all the models. Our main result is that random networks in all models give rise to cyclic behavior more often than fixed points. And although periodic orbits seem to dominate network dynamics, they are usually considered unstable and not allowed to survive in previous evolutionary studies. Defining stability as the probability of fixed points, we show that the stability distribution of these networks is highly robust to changes in its parameters. We also find sparser networks to be more stable, which may help to explain why they seem to be favored by evolution. We have unified several disconnected previous studies of this class of models under the framework of stability, in a way that had not been systematically explored before.

The Role of Genotype in the Predictability of Dynamical Behavior in Complex Model Gene Regulatory Networks

“…Like others [44,45], we found that a majority large period lengths or aperiodicity. Our data are consistent with the conclusion that extremely complex dynamical behavior is a function of very specific patterns of interaction within dense networks.…”

“…Oscillating values of the state of expression of individual genes (S i,t ) are common features of model GRNs [44], and have also been found in some biological GRNs [5]. Like others [44,45], we found that a majority large period lengths or aperiodicity.…”

Evolution of gene regulatory networks by means of selection and random genetic drift

“…Individuals with a maturation process that concludes in (ii) or (iii) were removed from the population. Here, motivated by Pinho et al [16] who suggested that in Wagner's model most networks are cycling, we developed a circadian framework to evaluate the fitness of individuals that conclude in cyclic equilibria during the maturation step. Individuals that conclude in case (iii), or individuals that conclude in case (ii) but the period k is greater than an upper threshold (here 10,000 steps) were considered nonviable and were removed from the population.…”

Modeling the evolution of complex genetic systems: The gene network family tree

“…Implementing this model in population simulations has been a popular way to study epistasis (non linear interactions between the alleles at different loci) and complex genetic interactions and it has been used to explore the effects of mutation, recombination, genetic drift, and environmental selection in a network context (e.g., Bergman and Siegal, 2003; Masel, 2004; Azevedo et al, 2006; MacCarthy and Bergman, 2007; Borenstein et al, 2008; Draghi and Wagner, 2009; Espinosa Soto and Wagner, 2010; Espinosa Soto et al, 2011; Fierst, 2011a,b; Le Cunff and Pakdaman, 2012). Several groups (Huerta Sanchez and Durrett, 2007; Sevim and Rikvold, 2008; Pinho et al, 2012) have analyzed the model to identify how networks change when they transition from oscillatory dynamics to developmental stability. These analyses have not uncovered descriptive metrics that separate stable from unstable networks.…”