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 IV. DISCUSSION The results presented in this paper suggest that the one-way switches are hardly selected in evolution because mutationally robust GNRs are favored. This scenario of the phenotype selection was confirmed by comparing the outcomes of the multicanonical ensemble method and the steady state of stochastic evolutionary simulations because the steady state does not depend on the evolutionary history. We consider this selection mechanism to be widely valid and not restricted to the case studied in this paper. Since this phenotype selection can occur whenever more than one phenotype exists for the same fitness value, we expect this phenomenon to occur in real living systems. We may call it the third-order selection because it is a consequence of the second-order selection. Since the second-order selection is a weak effect, the above phenotypic selection may be a further weak effect. However, as the present results suggest, the phenomenon may become significant if the steady state of evolution is maintained. Such situations can be realized experimentally. For example, it will be possible for the effect to be observed for bacteria under constant stress. It should be noted that what is the matter is not the absolute strength of mutational robustness but its relative strength. Thus, phenotype selection can occur even between mutationally not-so-robust phenotypes. Next, let us consider an implication of the evolution of the model dealt with observed in this paper. The result of McMC shows that most GRNs are monostable at low fitness; then as fitness increases the toggle switches dominate, and eventually most GRNs become one-way switches. A one-way switch of GRN realizes an irreversible change in the cell states. We note that irreversibility was not required in the definition of fitness. Thus, the irreversible switches can emerge even without it being required explicitly as fitness. It suggests the possibility that the irreversibility observed for real cells has evolved from the monostable reversible GRNs. Finally, the present study shows that McMC is an effective method for investigating characteristic properties of evolution. McMC clarifies what kind of phenotypes can exist if the evolution process is not considered. The fact that the outcomes of McMC and evolutionary simulation differ considerably implies a possibility that some phenotypes are not reached by evolution. Considering the experimental situation, such phenotypes may be accessible by direct genome engineering other than evolutionary experiments. 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