Canalization reduces the nonlinearity of regulation in biological networks

TL;DR

It is shown that the abundance of canalization in biological networks can almost completely explain their recently postulated high approximability, and an analysis of random N–K Kauffman models reveals a strong dependence of approximability on the dynamical robustness of a network.

Abstract

Biological networks such as gene regulatory networks possess desirable properties. They are more robust and controllable than random networks. This motivates the search for structural and dynamical features that evolution has incorporated in biological networks. A recent meta-analysis of published, expert-curated Boolean biological network models has revealed several such features, often referred to as design principles. Among others, the biological networks are enriched for certain recurring network motifs, the dynamic update rules are more redundant, more biased and more canalizing than expected, and the dynamics of biological networks are better approximable by linear and lower-order approximations than those of comparable random networks. Since most of these features are interrelated, it is paramount to disentangle cause and effect, that is, to understand which features evolution actively selects for, and thus truly constitute evolutionary design principles. Here, we show that approximability is strongly dependent on the dynamical robustness of a network, and that increased canalization in biological networks can almost completely explain their recently postulated high approximability.

Figures (8)

• Figure 1: Canalization explains the high approximability of biological networks.. The distribution of mean approximation errors is shown for the biological networks (orange) and three different types of random null networks (shades of blue), which match different characteristics (bias and/or canalizing depth) of the biological network. Each box depicts the interquartile range (IQR), each whisker extends to the most extreme value within 1.5 * IQR from the box, and each horizontal line within a box depicts the median. For a fixed approximation order (1-3, x-axis), differences between the MAE distribution of the biological and the random networks are assessed using the two-sided Wilcoxon signed-rank test. Fig. \ref{['fig:bio_networks_approximability_detailed']} contains scatterplots showing the MAE values of all biological networks and their random null models.
• Figure 2: Mean approximation errors of biological networks and their random null models. For a fixed approximation order (1-3, columns), differences between the MAE values of the 110 biological and the different random networks (rows) are shown, in addition to the Spearman correlation coefficient, $\rho$. A summary of this data is shown in Fig. \ref{['fig:bio_networks_approximability']}.
• Figure 3: Predictors of approximability of biological networks. Pairwise Spearman correlation between the first-, second- and third-order mean approximation errors and various network properties across the 110 published biological networks, ordered by the mean correlation. $<\cdot>$ denotes the mean, $p =$ output bias, $K =$ number of variables, $K_e =$ effective connectivity, Cov = covariance of $p(1 - p)$ and $K$. The pairwise Spearman correlations between all shown properties are in Fig. \ref{['fig:spearman_detailed']}.
• Figure 4: Effect of bias and in-degree on the approximability of the dynamics of Boolean networks. For strongly connected $15$-node Boolean networks with a constant in-degree (y-axis) governed by random update functions generated with a certain bias (x-axis), the mean error is shown when approximating their dynamics using different order Taylor polynomials (subplots). Each cell depicts the MAE across $50$ networks, and the same networks were used to estimate the MAE using first-order to fourth-order Taylor polynomials. Results from an equivalent analysis where the functions are required to be essential in all its variables are shown in Fig. \ref{['fig:p_vs_k_nondeg']}.
• Figure 5: Predictors of approximability of random networks. Pairwise Spearman correlation between the first-, second- and third-order mean approximation errors and network properties related explicitly to dynamics, across 2000 random strongly-connected Boolean networks with fixed degree $K\in\{2,3,4,5\}$ and bias $p\in\{0.1,0.2,0.3,0.4,0.5\}$ (100 for each combination).