Although outdegree distributions of gene regulatory networks have scale-free characteristics similar to other biological networks, indegree distributions have single-scale characteristics with significantly lower variance than that of outdegree distributions. In this study, we mathematically explain that such asymmetric characteristics arise from dynamical robustness, which is the property of maintaining an equilibrium state of gene expressions against inevitable perturbations to the networks, such as gene dysfunction and mutation of promoters. We reveal that the expression of a single gene is robust to a perturbation for a large number of inputs and a small number of outputs. Applying these results to the networks, we also show that an equilibrium state of the networks is robust if the variance of the indegree distribution is low (i.e., single-scale characteristics) and that of the outdegree distribution is high (i.e., scale-free characteristics). These asymmetric characteristics are conserved across a wide range of species, from bacteria to humans.
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