Correcting hidden sampling biases in null models of canalizing Boolean networks

Abstract

Boolean networks are widely used to model gene regulatory systems. Their structural and dynamical properties are commonly interpreted by comparison with ensembles of random Boolean networks generated by sampling Boolean functions for individual nodes. Canalizing and nested canalizing functions, in which one or more regulatory inputs dominate the output, capture an important feature of gene regulation. These functions are typically generated by sampling their defining parameters uniformly at random. Because multiple parameterizations can represent the same Boolean function, however, this procedure induces a biased distribution over functions and consequently over null models. We develop efficient algorithms for uniformly sampling Boolean functions with prescribed canalizing depth, thereby correcting this systematic bias. Using these unbiased null models, we show that the sampling measure substantially alters function- and network-level properties. Whereas parameter-uniform sampling yields nested canalizing functions with expected average sensitivity one, these sensitivities increase with degree under function-uniform sampling and approach 1.183. These differences alter expectations for robustness, attractor structure, and stability. Reanalysis of 122 published Boolean gene regulatory network models reveals substantially stronger enrichment of low-sensitivity canalizing architectures than previously recognized. Widely used parameter-based null models therefore systematically underestimate baseline sensitivity and overestimate the stabilizing role of canalization.

12 pages, 4 figures