The Influence of Canalization on the Robustness of Boolean Networks

Abstract

Time-and state-discrete dynamical systems are frequently used to model molecular networks. This paper provides a collection of mathematical and computational tools for the study of robustness in Boolean network models. The focus is on networks governed by k-canalizing functions, a recently introduced class of Boolean functions that contains the well-studied class of nested canalizing functions. The activities and sensitivity of a function quantify the impact of input changes on the function output. This paper generalizes the latter concept to c-sensitivity and provides formulas for the activities and c-sensitivity of general k-canalizing functions as well as canalizing functions with more precisely defined structure. A popular measure for the robustness of a network, the Derrida value, can be expressed as a weighted sum of the c-sensitivities of the governing canalizing functions, and can also be calculated for a stochastic extension of Boolean networks. These findings provide a computationally efficient way to obtain Derrida values of Boolean networks, deterministic or stochastic, that does not involve simulation.

Effective connectivity determines the critical dynamics of biochemical networks

“…Other measures of canalization in Boolean automata exist and have been linked to criticality, such as average sensitivity [ 57 ] and the more general c-sensitivity [ 71 ]. Effective connectivity presents several advantages over these measures.…”

“…Interestingly, both sensitivity and effective connectivity can be easily computed from our schema description methodology [ 24 ], which is available in the CANA Python package [ 63 ]. Finally, ‘ c -sensitivity’ [ 71 ] extends sensitivity to subsets of c inputs, but it results in a vector of k values for each c , which is much less amenable to the regression analysis of criticality boundaries we pursue in this study than is the scalar value measured by k e .…”

A meta-analysis of Boolean network models reveals design principles of gene regulatory networks

“…The concept of canalization, already introduced in the 1940s in the context of embryonal development [12], has been proposed as a possible explanation for the remarkable stability of GRNs in the face of ubiquitous perturbations, and accordingly, Boolean canalizing functions have been proposed as ideally suited update functions in Boolean GRN models [11]. Recently, the class of canalizing functions has been further stratified and studied [21,22]. Some smaller studies support the general hypothesis by revealing an overabundance of canalizing functions in GRN models [23,24] but a rigorous, comprehensive analysis that considers various types of canalization is still missing.…”

Symmetry Properties of Nested Canalyzing Functions

“…Many researchers have pointed out the usefulness of NCFs in modeling biological phenomena (e.g., Layne (2011); Layne et al (2012); Li et al (2011); Li and Adeyeye (2012); Li et al (2013). Properties of NCFs such as sensitivity and stability have also been studied in the literature (e.g., Kauffman et al (2004); Layne (2011); Layne et al (2012); Li et al (2011Li et al ( , 2013; Klotz et al (2013); Stearns et al (2018); Paul et al (2019); Kadelka et al (2017a)). Generalized versions of nested canalyzing functions where the variables and function values can be from a domain of size three or more have also been studied (e.g., Kadelka et al (2017b)).…”