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Gene regulatory networks (GRNs) are complex systems in which many genes regulate mutually to adapt the cell state to environmental conditions. In addition to function, the GRNs possess several kinds of robustness. This robustness means that systems do not lose their functionality when exposed to disturbances such as mutations or noise, and is widely observed at many levels in living systems. Both function and robustness have been acquired through evolution. In this respect, GRNs utilized in living systems are rare among all possible GRNs. In this study, we explored the fitness landscape of GRNs and investigated how robustness emerged in highly-fit GRNs. We considered a toy model of GRNs with one input gene and one output gene. The difference in the expression level of the output gene between two input states, “on” and “off”, was considered as fitness. Thus, the determination of the fitness of a GRN was based on how sensitively it responded to the input. We employed the multicanonical Monte Carlo method, which can sample GRNs randomly in a wide range of fitness levels, and classified the GRNs according to their fitness. As a result, the following properties were found: (1) Highly-fit GRNs exhibited bistability for intermediate input between “on” and “off”. This means that such GRNs responded to two input states by using different fixed points of dynamics. This bistability emerges necessarily as fitness increases. (2) These highly-fit GRNs were robust against noise because of their bistability. In other words, noise robustness is a byproduct of high fitness. (3) GRNs that were robust against mutations were not extremely rare among the highly-fit GRNs. This implies that mutational robustness is readily acquired through the evolutionary process. These properties are universal irrespective of the evolutionary pathway, because the results do not rely on evolutionary simulation., Author summary Living systems have developed through a long history of Darwinian evolution. They acquired characteristic properties distinct from other physical systems; one is biological function. Another important property, which is overlooked by non-experts, is robustness to noise and mutation. Here, robustness means that a system does not lose its functionality when exposed to disturbances. Then, how do they relate to each other? In this paper, we explored this question using a toy model of gene regulatory networks (GRNs). While evolutionary simulations are usually used for such purposes, we instead generated GRNs randomly and classified them according to functionality. By requiring sensitive responses to environmental change as a function, we found that bistability emerges as a common property of highly-functional GRNs. Since this property does not depend on a particular evolutionary pathway, if the evolution was rewound and repeated over and over again, phenotypes with the same property would always evolve. At the same time, such bistable GRNs were robust to noise. We also found that GRNs robust to mutation were not extremely rare among the highly-functional GRNs. This implies that mutational robustness would be readily acquired through evolution.

We successfully synthesized verdazyl biradical crystals of $p$-BIP-V${}_{2}$ [4,4${}^{\ensuremath{'}}$-bis(1,5-diphenylverdazyl-3-yl)biphenyl]. Two types of intermolecular interactions are deduced from the spin-density distribution and molecular packing in the crystal, and these interactions are considered to result in the formation of a honeycomb lattice. The $\mathit{ab}$ $\mathit{initio}$ molecular orbital calculation also indicated such an exchange network of intermolecular interactions. The intramolecular interaction acts as a diagonal bridge across the honeycomb lattice to form a square lattice. We evaluated the intramolecular antiferromagnetic (AF) interaction ${J}_{1}$ to be about 8.8 K from an analysis of the magnetic susceptibility of isolated $p$-BIP-V${}_{2}$ molecules in terms of an isolated $S$ $=$ 1/2 AF dimer. We analyzed the magnetic susceptibility of the crystals using the quantum Monte Carlo method. Consequently, two types of intermolecular AF interactions, ${J}_{2}$ and ${J}_{3}$, were found to be almost comparable and about 4.5 times larger than the intramolecular interaction ${J}_{1}$. Thus, we confirmed that $p$-BIP-V${}_{2}$ has a slightly distorted $S=1/2$ honeycomb-like lattice.