Phenotype Selection due to Mutational Robustness: Model and Methods

Table of Links

Introduction

Model and Methods

Results

Discussion and Acknowledgment

II. MODEL AND METHODS

Many genes regulate their expressions each other in living cells and form complex networks called gene regulatory networks (GRNs). We employ an abstract connectionist-type model of GRNs, in which we express a GRN by a directed graph, ignoring the details of gene expression.[16, 21–29] The nodes represent genes, and the edges represent the regulatory interactions. Hereafter, we consider one-in-one-out GRNs that have one input gene that accepts the input signal from the outside world and one output gene that expresses the target protein. Fig.1 shows an example of a small model. In the present study, we restrict ourselves to GRNs with the number of nodes N = 40 and the number of edges K = 120.

For this network model, we assume the following discrete-time dynamics:

Namely, we require that the output level of the steady state for I = 0 is as low as possible and that for I = 1 is as high as possible. Using the steady state of I = 0 as the initial state for I = 1 is important, and the value of f reflects this history. In case that f becomes negative, we regard f = 0.

By investigating the stability of each GRN using the procedure described below, we found that GRNs with the following three distinct stabilities appear at high fitness under this definition of the fitness function: (1) Monostable: When I is increased gradually from 0 to 1 and after that is decreased gradually from 1 to 0, xout(I) follows the same trace in both upward and downward change. (2) Toggle switch: A Hysteresis appears in some range of I between upward and downward change. (3) One-way switch: xout(0) does not return to the initial value and keeps a high value after the downward change. Figs.2(a)-(c) show examples of three stabilities. In terms of dynamical systems, the change of the input signal induces a transition between two fixed points for the bistability to occur. In the case of the toggle switch, the saddle-node bifurcations take place twice in the parameter range, which gives rise to hysteresis in the transition between two fixed points. In contrast, the saddle-node bifurcation takes place only once in the allowed range of the parameter in the case of the one-way switch and thus the transition between two fixed points is irreversible. Let us regard these three stabilities as three distinct phenotypes. An important point is that these three phenotypes appear for the same fitness value in some fitness range.