Abstract
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.
Highlights
Living systems have advanced through a long history of Darwinian evolution
The sum of O(f) is normalized to unity. This figure does not represent a conventional fitness landscape in which the fitness is drawn in the genotypic space, we may denote it as a “fitness landscape” in the same sense as the energy landscape in the protein folding problem, in which the entropy is drawn against the energy
The fitness landscape bends at f ’ 0.2 and the gene regulatory networks (GRNs) become exponentially rare as f increases
Summary
Living systems have advanced through a long history of Darwinian evolution As a result, they have acquired properties distinct from other physical systems. The cells have developed to undergo metabolism and proliferation Another significant property of living systems is the existence of several types of robustness [1,2,3]. If we regard the process of developing functions as an optimization process, highly optimized systems are intuitively considered as fragile against disturbances, and it is natural to consider that they readily lose their functions by mutation In this respect, the process of evolution should be something different from a simple optimization process. Robustness against several types of noise, such as disturbance from the environment and the noise that occurs within the cell, and in particular, the fluctuation due to the finiteness of the number of molecules, is important, because living systems function stochastically in the noisy world in which we live
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