Numerous modelers have inferred Boolean gene regulatory networks in different biological systems, taking genes to be either on or off depending on their inputs according to associated logic rules. When examining over 2600 such rules in previously published models, we find that almost all fall into two very particular classes: Nested Canalyzing Functions (NCF) and Read once Functions (RoFs). We interpret this striking fact in terms of two complexity measures: Boolean complexity based on string lengths in formal logic, and the so-called average sensitivity which corresponds to robustness of the input-output function to noise. We show that RoFs minimize Boolean complexity while NCFs minimize both Boolean complexity and average sensitivity, from which we infer that selection under evolution strongly favors low complexity rules [1]. Such a "design principle" also affects the collective dynamics of the associated networks. Indeed, we find that networks with these two classes of logic provide a good compromise between stability and responsiveness, in line with the paradigm that adaptive and controlable systems sit "at the edge of chaos".

These insights can be used to provide selection criteria for modelers who seek to infer the GRN of a biological system. To illustrate this, we have considered developmental systems where the steady states of the dynamical network correspond to different cell fates (biological attractors). In these dynamical systems, even if the network connectivity is specified, there is generally a huge number of combinations of Boolean functions that will reproduce the different attractors. Imposing that each function be of the NCF or RoF type reduces exponentially the space of allowable models. In addition, one can leverage the "developmental landscape" to further select dynamical GRNs that respect known constraints on cell lineages, i.e., which cell type derives from which other (less differentiated) cell type. We do this based on the mean first passage times between the different attractors when noise is added. We have implemented this methodology on a developmental system having a hierarchy of states, going from stem cells to terminally differentiated cells. To automatize the search for GRNs satisfying the given hierarchy of cell states, we developed an iterative greedy algorithm that samples the many logic rules for each gene. Applied to a 2020 model of the root stem cell niche in Arabidopsis, that search led to hundreds of models improving over the published one. Overall, our methodology can be applied to any regulatory system and it will accelerate modeling efforts to reconstruct dynamical gene regulatory networks.

[1] A. Subbaroyan, O.C. Martin and A. Samal. Minimum complexity drives regulatory logic in Boolean models of living systems. PNAS Nexus, 1(1): pgac017 (2022).
[2] A. Subbaroyan, P. Sil, O.C.Martin and A. Samal. Leveraging developmental landscapes for model selection in Boolean gene regulatory networks. bioRxiv
2023.01.08.523151 (2023).