Gene regulatory networks (GRNs) are complex systems in which many genes mutually regulate their expressions for changing the cell state adaptively to the environmental conditions. Besides the functions , the GRNs utilized by living systems possess several kinds of robustness. Here, the robustness means that the GRNs do not lose their functions when exposed to mutation or noises. Both the adaptive response and the robustness have been acquired through the evolution. In this respect, real GRNs are rare among "all the possible GRNs" . In this study, we explore the fitness landscape of GRNs and investigate how the robustness emerge in the "well-fitted" GRNs. For that purpose, we employ the Multi-Canonical Monte Carlo method, which can sample GRNs randomly in wide range of fitness. We consider a toy model of GRNs having one input gene and one output gene. The difference in the expression levels between the input states "on" and "off" is taken as the fitness. Thus the more sensitively a GRN responds to the input , the fitter it is.We show the following properties for the GRNs in the "fittest ensemble" : (1) They distinguish two different states of the input by switching the fixed points . Thus they exhibit bistability , which necessarily emerges as the fitness becomes high. (2) They are robust against noises thanks to the bistability. (3) Many GRNs in the fittest ensemble are robust against mutation. These properties are universal irrespective of the evolutionary pathway, because we did not perform evolutionary simulations .
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. 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 .
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investigate how the robustness emerge in the "well-fitted" GRNs. For that purpose, we employ the Multi-Canonical Monte Carlo method, which can sample GRNs randomly in wide range of fitness. We consider
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, the fitter it is.We show the following properties for the GRNs in the "fittest ensemble"
.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. The following properties were found
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They distinguish two different states of the input by switching the fixed points
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
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Thus they exhibit bistability , which necessarily emerges as the fitness becomes high.
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They are robust against noises thanks to the bistability.
These highly-fit GRNs were robust against noise because of their bistability. In other words, noise robustness is a byproduct of high fitness.
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Many GRNs in the fittest ensemble are robust against mutation.
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.
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we did not perform evolutionary simulations
the results do not rely on evolutionary simulation