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by The Mutation PublicationFebruary 22nd, 2024

This paper is available on arxiv under CC 4.0 license.

**Authors:**

(1) Macoto Kikuchi (菊池誠), Cybermedia center, Osaka University, and Department of Physics, Osaka University and [email protected].

Figure 3 shows the genotypic entropy obtained by McMC, namely, the logarithm of the number of GRNs N(f) in each bin. Since we can obtain only the relative entropy, the entropy for f ∈ [0, 0.01) is set as 0. A majority of GRNs have low fitness, and the high-fitness GRNs are rare. The figure shows that the GRNs in the whole range of fitness are sampled.

Figure 4 shows the appearance probability of the three phenotypes, the monostable GRNs, the toggle switches, and the one-way switches, obtained by McMC. The sum of the probabilities for the three phenotypes for each bin of fitness is 1. The figure shows the ratio that the three phenotypes should appear if selection bias is absent. Most GRNs are monostable at low fitness; The toggle switches increase as fitness, and finally, the one-way switches start to appear.

The appearance probabilities of the three phenotypes differ significantly for evolution from those for McMC. We

show the probabilities for three phenotypes obtained by McMC, ES0, and EST in Fig. 5(a)-(c). The sum of the probabilities for the three phenotypes for each bin of fitness is 1 for each simulation method. The results for McMC are the same as those in Fig. 4. Since most GRNs in the bin f ∈ [0.99, 1] obtained by ES0 exhibit extraordinarily high fitness, we show GRNs only of f ∈ [0.99, 0.991). The appearance probabilities of the monostable GRNs obtained by ES0 are high up to large values of f compared to McMC; This tendency is more significant for EST. It suggests that the ratio of monostable GRNs increases in evolution until the steady state is reached. The appearance of the toggle switches is delayed in evolution compared to McMC, and this tendency is more significant for EST than for ES0. The case of the one-way switches is impressive. Seeing the result of McMC, the one-way switches dominate at high fitness in the absence of the selection bias. They are suppressed by evolution, as is seen for ES0 and EST. In particular, in the steady state of evolution (EST), the appearance of the one-way switches is strongly suppressed in the observed range. Thus, the one-way switches are not favored in evolution.

To investigate the origin of the above selection bias, we counted the essential edges for obtained GRNs. Figure 6(a) shows the probability distributions P(Ne) of number of the essential edges Ne for GRNs obtained by McMC in f ∈ [0.9, 0.91). The distributions are normalized so that the sum for each phenotype is 1. The monostable GRNs have the least essential edges. Then, the toggle switches. The number distribution for the one-way switches exhibits a long

tail to the large value of Ne. The largest number of essential edges for the one-way switches observed in the present result was 110. In other words, the one-way switches include GRNs such that more than 90% of the edges are essential. We plot the same probability distributions obtained by EST in Figure 6(b). The result is significantly different from that of McMC; the distributions are strongly biased toward small values of Ne for all three phenotypes. In the case of the one-way switches, the largest number of the essential edges observed was 70. Therefore, the evolution favors mutationally robust GRNs.

We compare the essential edge distributions of three phenotypes for McMC, ES0, and EST in Fig.7(a)-(c). The fitness range is again f ∈ [0.9, 0.91). In contrast to Fig 6, they are normalized so that the sum for all three phenotypes is 1. While the ratio of the monostable GRNs increases by evolution, the essential edge distribution does not change significantly compared with McMC. In the case of the toggle switches, the distribution shifts toward small number of essential edges from McMC to ES0 and from ES0 to EST. The one-way switches exhibit a remarkable change; GRNs with many essential edges decrease largely in ES0 compared to McMC. The distribution becomes significantly different for EST, in which only GRNs with small numbers of essential edges remain.

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