Survival of the sparsest: robust gene networks are parsimonious

Abstract

Biological gene networks appear to be dynamically robust to mutation, stochasticity, and changes in the environment and also appear to be sparsely connected. Studies with computational models, however, have suggested that denser gene networks evolve to be more dynamically robust than sparser networks. We resolve this discrepancy by showing that misassumptions about how to measure robustness in artificial networks have inadvertently discounted the costs of network complexity. We show that when the costs of complexity are taken into account, that robustness implies a parsimonious network structure that is sparsely connected and not unnecessarily complex; and that selection will favor sparse networks when network topology is free to evolve. Because a robust system of heredity is necessary for the adaptive evolution of complex phenotypes, the maintenance of frugal network complexity is likely a crucial design constraint that underlies biological organization.

Figure 1

Denser networks dilute the costs of perturbation over more interactions. Vertical axis shows the average cost of a perturbation per interaction (CPPI), where cost measures how much the phenotype deviates from the optimal. Selecting for optimal gene expression patterns, networks were initialized at c₀=0.5 and far-from-equilibrium (ĉ=c₀±0.4) for sparse (ĉ=0.1; α=0.008, φ=0.07) and dense (ĉ=0.9; α=0.07, φ=0.008) treatments. Plot shows evolutionary response in CPPI as networks are driven to sparser (open circles) and denser (closed circles) connectivity densities (also see Supplementary Figure 3). Data are reported as expectations and 95% confidence intervals for the mean of a single optimal individual sampled from each replicate population.

Figure 2

Sparser networks evolve to be less costly. Vertical axis measures the gross cost of perturbation (GCP) on a network as the mean cost of perturbation per interaction (CPPI) multiplied by the number of interactions in the network. Plot shows the evolutionary response in the GCP for the treatments from Figure 2 driven toward more sparse (open circles) or dense (closed circles) connectivity densities (also see Supplementary Figure 3). For a network with N genes, the maximum cost is N². Sparser networks maintain a lower GCP than dense networks, even though the CPPI increases as networks become sparser. Data are reported as expectations and 95% confidence intervals for the mean of a single optimal individual sampled from each replicate population.

Figure 3

Selection systematically favors networks below their equilibrium density. Selecting for optimal gene expression patterns, 100 replicate populations were initialized with network connectivity density already at equilibrium (c₀=ĉ) for high (c₀=0.6; α=0.105, φ=0.07), intermediate (c₀=0.5; α=0.07, φ=0.07), and/or low (c₀=0.6; α=0.07, φ=0.105) treatments. Plot shows the average evolved response in connectivity density under stabilizing selection as compared to the expected equilibrium density in the absence of selection (thick dashed line). High (open diamond), intermediate (open square), and low (open triangle) systematically favor sparser-than-equilibrium networks (also see Supplementary Figure 4). Data are reported as expectations and 95% confidence intervals for the mean of a single optimal individual sampled from each replicate population.