The properties of random Boolean networks have been investigated extensively as models of regulation in biological systems. However, the Boolean functions (BFs) specifying the associated logical update rules should not be expected to be random. In this contribution, we focus on biologically meaningful types of BFs, and perform a systematic study of their preponderance in a compilation of 2,687 functions extracted from published models. A surprising feature is that most of these BFs have odd “bias”, that is they produce “on” outputs for a total number of input combinations that is odd. Upon further analysis, we are able to explain this observation, along with the enrichment of read-once functions (RoFs) and its nested canalyzing functions (NCFs) subset, in terms of 2 complexity measures: Boolean complexity based on string lengths in formal logic, which is yet unexplored in biological contexts, and the so-called average sensitivity. RoFs minimize Boolean complexity and all such functions have odd bias. Furthermore, NCFs minimize not only the Boolean complexity but also the average sensitivity. These results reveal the importance of minimum complexity in the regulatory logic of biological networks.

Boolean models are a well-established framework to model developmental gene regulatory networks (DGRN) for acquisition of cellular identity. During the reconstruction of Boolean DGRNs, even if the network structure is given, there is generally a very large number of combinations of Boolean functions (BFs) that will reproduce the different cell fates (biological attractors). Here we leverage the developmental landscape to enable model selection on such ensembles using the relative stability of the attractors. First we show that 5 previously proposed measures of relative stability are strongly correlated and we stress the usefulness of the one that captures best the cell state transitions via the mean first passage time (MFPT) as it also allows the construction of a cellular lineage tree. A property of great computational convenience is the relative insensitivity of the different measures to changes in noise intensities. That allows us to use stochastic approaches to estimate the MFPT and thus to scale up the computations to large networks. Given this methodology, we study the landscape of 3 Boolean models of Arabidopsis thaliana root development and find that the latest one (a 2020 model) does not respect the biologically expected hierarchy of cell states based on their relative stabilities. Therefore we developed an iterative greedy algorithm that searches for models which satisfy the expected hierarchy of cell states. By applying our algorithm to the 2020 model, we find many Boolean models that do satisfy the expected hierarchy. Our methodology thus provides new tools that can enable reconstruction of more realistic and accurate Boolean models of DGRNs.

Boolean models are a well-established framework to model developmental gene regulatory networks (DGRNs) for acquisition of cellular identities. During the reconstruction of Boolean DGRNs, even if the network structure is given, there is generally a large number of combinations of Boolean functions that will reproduce the different cell fates (biological attractors). Here we leverage the developmental landscape to enable model selection on such ensembles using the relative stability of the attractors. First we show that previously proposed measures of relative stability are strongly correlated and we stress the usefulness of the one that captures best the cell state transitions via the mean first passage time (MFPT) as it also allows the construction of a cellular lineage tree. A property of great computational importance is the insensitivity of the different stability measures to changes in noise intensities. That allows us to use stochastic approaches to estimate the MFPT and thereby scale up the computations to large networks. Given this methodology, we revisit different Boolean models of Arabidopsis thaliana root development, showing that a most recent one does not respect the biologically expected hierarchy of cell states based on relative stabilities. We therefore developed an iterative greedy algorithm that searches for models which satisfy the expected hierarchy of cell states and found that its application to the root development model yields many models that meet this expectation. Our methodology thus provides new tools that can enable reconstruction of more realistic and accurate Boolean models of DGRNs.