Degeneracy measures in biologically plausible random Boolean networks

Abstract

Biological systems show diversity in terms of the underlying network structure and the governing rules of such networks. Yet, different types of biological networks may develop similar adaptation strategies in face of environmental changes. Degeneracy refers to the ability to compensate for compromised function without the need for a redundant component in the system. Highly degenerate systems show resilience to perturbations and damage because the system can compensate for compromised function due to reconfiguration of the underlying network dynamics. Although formal definitions of degeneracy have been proposed, these definitions have only been tested in relatively simple networks involving weighted connections between network nodes. In this study, we test an information theoretic definition of degeneracy on random Boolean networks, frequently used to model gene regulatory networks. Random Boolean networks are discrete dynamical systems with binary connectivity and thus, these networks are well-suited for tracing information flow and the causal effects. By generating networks with random binary wiring diagrams, we test the effects of systematic lesioning of connections and perturbations of the network nodes on the degeneracy measure. Our analysis shows that degeneracy, on average, is the highest in networks in which ~20% of the connections are lesioned while 50% of the nodes are perturbed. Moreover, our results for the networks with no lesions and the fully-lesioned networks are comparable to the degeneracy measures from weighted networks, thus we show that the degeneracy measure is applicable to different networks. Such a generalized applicability implies that degeneracy can be used to make predictions about the variety of systems’ ability to recover function. Author Summary Degeneracy – the ability of structurally different elements to perform similar functions – is a property of many biological systems. Systems exhibiting a high degree of degeneracy continue to exhibit the same macroscopic behavior following a lesion even though the underlying network dynamics are significantly different. Degeneracy thus suggests how biological systems can thrive despite changes to internal and external demands. Although degeneracy is a feature of network topologies and seems to be implicated in a wide variety of biological processes, research on degeneracy in biological networks is mostly limited to weighted networks (e.g., neural networks). To date, there has been no extensive investigation of information theoretic measures of degeneracy in other types of biological networks. In this paper, we apply existing approaches for quantifying degeneracy to random Boolean networks used for modeling biological gene regulatory networks. Using random Boolean networks with randomly generated rulesets to generate synthetic gene expression data sets, we systematically investigate the effect of network lesions on measures of degeneracy. Our results are comparable to measures of degeneracy using weighted networks, and this suggests that degeneracy measures may be a useful tool for investigating gene regulatory networks.