Biologically meaningful regulatory logic enhances the convergence rate in Boolean networks and bushiness of their state transition graph

Abstract

Boolean network (BN) models of gene regulatory networks (GRNs) have gained widespread traction as they can easily recapitulate cellular phenotypes via their attractor states. The overall dynamics of such models are embodied in the systems state transition graph (STG) which is highly informative. Indeed, even if two BN models have the same network structure and recover the same attractors, their STGs can be drastically different depending on the type of regulatory logic rules or Boolean functions (BFs) employed. A key objective of the present work is to systematically delineate the effects of different classes of regulatory logic rules on the structural features of the STG of reconstructed Boolean GRNs, while keeping BN structure and biological attractors fixed. Furthermore, we ask how such global features might be driven by characteristics of the underlying BFs. For that, we draw from ideas and concepts proposed in cellular automata for both the structural features and their associated proxies. We use the network of 10 reconstructed Boolean GRNs to generate ensembles that differ in the type of logic used while keeping their structure fixed and recovering their biological attractors, and compute quantities associated with the structural features of the STG: bushiness and convergence, that are based on the number of garden-of-Eden (GoE) states and transient times to reach attractor states when originating at them. We find that ensembles employing biologically meaningful BFs have higher bushiness and convergence than those employing random ones. Computing these global measures gets expensive with larger network sizes, stressing the need for more feasible proxies. We thus adapt Wuensches Z-parameter to BFs in BNs and provide 4 natural variants, which along with the network sensitivity, comprise our descriptors of local dynamics. One variant of the network Z-parameter as well as the network sensitivity correlate particularly very well with the bushiness, serving as a good proxy for the same. Finally, we provide an excellent proxy for the convergence based on computing transient lengths originating at random states rather than GoE states.