Effective Connectivity and Bias Entropy Improve Prediction of Dynamical Regime in Automata Networks

Biomolecular network dynamics are thought to operate near the critical boundary between ordered and disordered regimes, where large perturbations to a small set of elements neither die out nor spread on average. A biomolecular automaton (e.g., gene, protein) typically has high regulatory redundancy, where small subsets of regulators determine activation via collective canalization. Previous work has shown that effective connectivity, a measure of collective canalization, leads to improved dynamical regime prediction for homogeneous automata networks. We expand this by (i) studying random Boolean networks (RBNs) with heterogeneous in-degree distributions, (ii) considering additional experimentally validated automata network models of biomolecular processes, and (iii) considering new measures of heterogeneity in automata network logic. We found that effective connectivity improves dynamical regime prediction in the models considered; in RBNs, combining effective connectivity with bias entropy further improves the prediction. Our work yields a new understanding of criticality in biomolecular networks that accounts for collective canalization, redundancy, and heterogeneity in the connectivity and logic of their automata models. The strong link we demonstrate between criticality and regulatory redundancy provides a means to modulate the dynamical regime of biochemical networks.To get more news about costra-fx review, you can visit wikifx.com official website.
Introduction

The collective behavior of coupled automata is governed by the interplay between structural and dynamical parameters [1,2,3,4,5,6]. Tuning a small number of these parameters can lead to dramatic changes in the emergent properties of interlinked automata. A foundational example that illustrates this is the random Boolean network (RBN) models of gene regulation introduced by Kauffman [7], which have sustained interest over the intervening five decades (reviewed in [8,9]). In the Kauffman model, each of N Boolean automata (nodes) receives inputs from exactly K other nodes, chosen uniformly at random. An update function for each node is randomly generated by independently and randomly assigning an output value to each of the 2K possible input configurations, such that the output is 1 with probability P. The probability of activation of each input, P, is shared among all nodes in a network and is known as bias.
At each time-step, the vector of node variable values, called the network configuration, is synchronously updated according to these update functions.
The response of RBNs to perturbations has been of particular interest and is traditionally measured by the Derrida coefficient, δ. This parameter is defined as the separation (Hamming distance) after one time-step between two network configurations that initially differ in only one node value [10,11]. In the thermodynamic limit, N→∞, RBNs undergo an order to chaos phase transition characterized by the critical boundary δ=1. In the ordered regime, when δ is below this threshold, trajectories are characterized, on average, by short transient lengths and quickly vanishing perturbations. In the chaotic regime, when δ is above this threshold, transient lengths are long and perturbations grow in time, on average. Along the critical boundary, δ=1, on average, perturbations neither grow nor decay.
Contributions to the Derrida coefficient from an individual automaton can be measured using its sensitivity, which is defined as the number of inputs that can individually toggle the output of the automaton, averaged over all possible input configurations . The average sensitivity of the automata in a Boolean network gives the Derrida coefficient. In the thermodynamic limit, sensitivity can be computed as 2KP(1−P), which gives rise to the classical critical boundary:
A particularly relevant interpretation of Equation (1) is that it decomposes the Derrida coefficient into two contributions: average in-degree (K), which describes the average number of inputs nodes have, and bias-variance (P(1−P)), which describes how much spread there is in the distribution of activation probability (for all automata nodes in the network or ensemble.) The infinite-size limit in which the thermodynamic theory applies is an idealization, nevertheless, characteristics of the order to chaos transition can be observed in networks of eukaryotic cells [13], gene transcription [14], and other empirical databases [15,16] that have many fewer nodes than the typical number of protein-coding genes in an organism.
Various extensions of the Kauffman model have been studied to examine features of biomolecular networks that are not emphasized in the traditional model. For instance, gene regulatory networks tend to exhibit high modularity and power-law degree distributions. As such, modifications to the network structure of the Kauffman model have been considered for any in-degree distribution [17], power-law in-degree structure [8,18], and others [19]. Furthermore, in the Kauffman model, all update functions with the same activation bias are equally likely, but the regulatory logic of real biological networks is known to have a highly non-random structure [20]. To account for this, random Boolean models that use alternate methods for generating update functions, such as nested canalizing Boolean functions [21,22] and random threshold networks [23,24], have been proposed.
Here, we take structural heterogeneity into account directly by constructing RBNs with a truncated power-law in-degree distribution. Additionally, we consider the dynamical impact of regulatory logic through the lens of collective canalization. Broadly, the term canalization, coined by Waddington [25], refers to the ability of a small subset of variables (sometimes just a single variable) to determine the outcome of a regulatory process. Various measures have been proposed to quantify this behavior [26,27,28,29]. These measures are not necessarily in agreement about which Boolean functions are more or less canalizing than others. It is generally agreed, however, that the concept of canalization is closely related to robustness to genetic perturbations, which has been shown to play a crucial role in the ensemble properties of RBNs [7,12].
Collective canalization [20,26,28,30] refers to the degree to which a small subset of jointly activated inputs renders other inputs redundant. Effective connectivity, ke, has been proposed to measure this effect by computing the average size of the subset of inputs necessary to determine the output of an automaton [20,28]. It is obtained by computing the set of all prime implicants of a Boolean function (or the automaton’s look-up-table), which yields a maximal set of irreducible conditions for dynamical transition (see Appendix A for formal definition). This is equivalent to identifying and removing dynamical redundancy [28]. In other words, effective connectivity is the dual concept of dynamical redundancy in the logic of (collectively) canalized automata transitions. Bounded from above by in-degree, k, ke attains this maximum only when every input state must be known to determine the automaton’s next logical state. This only occurs for the parity functions (such as the case of a non-constant function of one variable or the XOR function of two variables). These are situations without any logical redundancy (or collective canalization). In the case of tautologies or contradictions (i.e., constant Boolean functions), ke=0 by definition, which denotes that all inputs are fully redundant.
Removing dynamical redundancy has already been used to reveal an alternative dynamically effective structure that includes collective canalization effects and is useful to characterize control in biochemical signaling and regulatory pathways [4,20]. Certainly, network controllability is an important aspect of automata models of biochemical regulation [31,32]. It is equally important to understand how perturbations spread in such models. Therefore, we focus here on the relevance of effective connectivity in determining the dynamical regime of Boolean networks and characterizing the critical boundary between order and chaos. Revising Equation (1) to utilize effective connectivity (ke) instead of in-degree (k), previous work has shown a significant improvement in dynamical regime prediction (as chaotic, critical, or ordered) of finite-size RBNs with homogeneous in-degree [33]. In other words, collective canalization (as measured by effective connectivity) explains the dynamical regime better than the apparent (structural) connectivity of such networks.
Here, we build upon that work to study RBNs with power-law in-degree distributions and study a larger set of experimentally validated Boolean network models of biomolecular processes. We show that in finite random networks and experimentally validated models, effective connectivity and bias-variance provide a better prediction of the dynamical regime—as measured by the Derrida coefficient and sensitivity—than the classical boundary of Equation (1) defined by the in-degree and bias-variance in the thermodynamic limit. We also show that the prediction of the Derrida coefficient is further improved in random networks by measuring the spread in bias using the entropy instead of the variance. In empirical models, the difference between the entropy and the bias is less pronounced, and the two measures perform similarly in predicting the dynamical regime.