This paper is under CC 4.0 license. available on arxiv Authors: (1) Macoto Kikuchi (菊池誠), Cybermedia center, Osaka University, and Department of Physics, Osaka University and kikuchi.macoto.cmc@osaka-u.ac.jp. Table of Links Introduction Model and Methods Results Discussion and Acknowledgment The mutation-selection mechanism of Darwinian evolution gives rise not only to the adaptation to environmental conditions but also to the enhancement of the robustness against mutation. Suppose more than one phenotypes share the same fitness value. The robustness distribution should differ for different phenotypes. Thus, we expect that some phenotype is favored in evolution and some is hardly selected as a consequence of the selection bias for mutational robustness. In this paper, we investigated this selection bias on phenotypes for a model of gene regulatory networks (GRNs) by numerical simulations. The model we used exhibits three types of response to the change of input signal; monostable, toggle switch, and one-way switch. We regarded these three response types as three phenotypes. We constructed the randomly generated set of GRNs using the multicanonical Monte Carlo method and compared it to the outcomes of evolutionary simulations. The results suggest that the one-way switches were strongly suppressed in evolution because a majority of the one-way switches were not mutationally robust. I. INTRODUCTION Living systems have developed through Darwinian evolution, a process in which a genotype changes so that the phenotype fits the environmental conditions. This process consists of the repetition of a mutation of genotype and a selection of individuals. This mutation-selection mechanism of evolution not only gives rise to the adaptation to the environment; it also causes a selection bias in evolution. One of the best-known such biases is the bias for mutational robustness.[1–3] Mutational robustness is a characteristic trait of living systems that they do not readily lose their functionality by a genotype mutation. The fact that the existing organisms possess this trait can be confirmed experimentally, especially for microorganisms, for example, by the comprehensive gene knock-out or rewiring of the gene regulatory networks.[4– 7] However, how such robustness has developed through evolution is difficult to investigate by experiments. Thus, theoretical studies or computational experiments provide indispensable information. A theory based on population dynamics has shown that mutational robustness evolves in neutral evolution.[8] Since this selection of mutationally robust genotypes is a selection irrespective of fitness, it is called “second-order”.[2] The second-order selection is considered to enhance the evolvability in the cases of non-neutral evolution because the mutationally robust genotypes have a higher possibility of having offspring with high fitness. We note here the usage of the word “fitness.” Conventionally, fitness is defined based on the number of offspring, and the effects of all the selection biases are taken into account. In this paper, however, we define fitness by a value calculated from a specific fitness function and consider selection biases separately. Suppose there are more than one phenotypes that take similar values of fitness. Their robustness distributions of the corresponding genotypes should be different in general. Thus, we expect that second-order selection affects the appearance probabilities of these phenotypes; the phenotypes with relatively robust genotypes will be favored in evolution; in contrast, the phenotypes with relatively fragile genotypes will be suppressed. We expect this phenotype selection that originates from the difference in mutational robustness of genotypes to occur in general situations in evolution. In this paper, we deal with a model of the gene regulatory networks (GRNs) as an example and investigate the enhancement of mutational robustness in evolution and the phenotype selection induced by the robustness using computational methods. We usually conduct evolutionary simulations (ES) to investigate Darwinian evolution numerically. However, ES alone is not enough for discussing selection biases in evolution because the outcomes of ESs already involve the effects of the selection biases. For exploring characteristic properties of evolution, we need a reference to compare with ES. In the case of GRNs, an appropriate reference system is a randomly generated set of GRNs. If there is no selection bias, evolutionary obtained GRNs should coincide with that picked up from this set having the same fitness. Ciliberti et al. conducted a random sampling of GRNs in a different context to investigate the structure of the neutral space.[9, 10] However, simple random sampling does not work for the present purpose because GRNs with high fitness are rare. Thus, we need some rare-event sampling method. We have proposed to use the multicanonical Monte Carlo (McMC) method for this purpose. McMC has been proposed originally for investigating phase transitions of spin systems.[11, 12] In those cases, McMC enables us to sample energy evenly in its entire range. After that, it has been recognized that McMC can also be used for rare-event sampling of non-physical systems by regarding any quantity as energy.[13] Counting the number of large magic squares is an example.[14] It was used to investigate biological networks by Saito and the author.[15] Nagata and the author applied it to GRNs by regarding fitness as energy.[16] In this case, McMC enables us to sample GRNs evenly in the entire range of fitness. The method of comparing the outcomes of ES and the reference set obtained by McMC has been proposed by Kaneko and the author.[17]. A comparison of ES and McMC was conducted recently to discuss the evolution of the genetic code.[18] In the previous paper, we showed for a model of GRN that the mutational robustness is enhanced by evolution, and the appearance of bistability is delayed at the same time.[17] These two phenomena are closely related to each other; our analysis suggested that the reason for the delay in the appearance of bistability was attributed to the fact that the bistable GRNs were less mutationally robust than the monostable GRNs. In the present paper, we further explore this phenomenon of phenotype selection due to mutational robustness. We use a different definition of fitness in the present work. In this definition, the GRNs exhibit two types of bistability at high fitness adding to monostability: the toggle switch and the one-way switch.[19] The one-way switches of GRNs realize irreversible changes in the cell states in living cells and are relevant to processes such as cell maturations and cell differentiations; an example is the maturation of Xenopus oocytes.[20] Although each fixed point of GRN corresponds to a cell state in living systems, in this paper, we regard three distinguishable stabilities, monostable, the toggle switch, and the one-way switch, as three phenotypes for discussing phenotype selection. These three stabilities exhibit different robustness distributions. We will show that the appearance of the one-way switch, which is the least robust among the three, is suppressed in the evolution.

Phenotype Selection due to Mutational Robustness: Introduction

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