Asymmetry in indegree and outdegree distributions of gene regulatory networks arising from dynamical robustness

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.

Figure 1

Density distribution of $u_i$. The solid lines indicate the simulated distributions, and the dashed lines indicate the theoretical distributions by Eq. (10). Each simulated distribution is calculated by 1000 networks. Here, $N=50$, where $N$ is the number of genes. We show the distributions for the following numbers of input connections $k_{\text{in}}=10$, 25, and 49. The correlation $\rho$ is estimated by the mode of the simulated distribution with $k_{\text{in}}=49$ ($\rho =0.539$).

Figure 2

Robustness $r_{\text{in}}\left(k_{\text{in}}\right)$ for the number of input connections $k_{\text{in}}$. The solid line indicates the simulation results and the dashed line indicates the theoretical values of Eq. (18). Each measure is calculated by 1000 networks ($N=50$, $\rho =0.539$).

Figure 3

Robustness $r_{\text{out}}\left(k_{\text{out}}\right)$ for the number of output connections $k_{\text{out}}$. The solid line indicates the simulation results and the dashed line indicates the theoretical values of Eq. (20). All measures are calculated by 10 000 networks ($N=50$, $k_{\text{in}}=20$, $r_{\text{in}}\left(20\right)=0.880$).

Figure 4

Degree distributions of E. coli (RegulonDB). The solid lines indicate the degree distributions and the dashed lines indicate the theoretical distributions with the best fit. (a) Indegree distribution. The exponential distribution has the best fit ($\gamma =0.509$, $k_{\text{max}}=14$). (b) Outdegree distribution. The power-law distribution has the best fit ($\alpha =1.41$, $k_{\text{max}}=281$).

Figure 5

Degree distributions of fruit fly (DroID). The solid lines indicate the degree distributions and the dashed lines indicate the theoretical distributions with the best fit. (a) Indegree distributions. The normal distribution has the best fit ($\mu =23.3$, ${\sigma }^2=80.8$, $k_{\text{max}}=55$). (b) Outdegree distributions. The power-law distribution has the best fit ($\alpha =0.811$, $k_{\text{max}}=1,150$).

Figure 6

Degree distributions of human (ENCODE). The solid lines indicate the degree distributions and the dashed lines indicate the theoretical distributions with the best fit. (a) Indegree distributions. The normal distribution has the best fit ($\mu =4.45$, ${\sigma }^2=123$, $k_{\text{max}}=53$). (b) Outdegree distributions. The power-law distribution has the best fit ($\alpha =0.923$, $k_{\text{max}}=6,246$).

Figure 7

Indegree distribution estimated by nonredundant data of TRANSFAC. The solid line indicates the indegree distribution and the dashed line indicates the Poisson distribution ($\lambda =31.2$).