Beyond Boolean networks: new tools for the steady state analysis of multivalued networks

Abstract

Boolean networks can be viewed as functions on the set of binary strings of a given length, described via logical rules. They were introduced as dynamic models into biology, in particular as logical models of intracellular regulatory networks involving genes, proteins, and metabolites. Since genes can have several modes of action depending on their expression levels, binary variables are often not sufficiently rich, requiring the use of multivalued networks instead. In this paper, we explore the multivalued generalization of Boolean networks by writing the standard $(\wedge, \vee, \lnot)$ operations on $\{0, 1\}$ in terms of the operations $(\odot, \oplus, \neg)$ on $\big\{0,\frac{1}{m}, \frac{2}{m}, \dots, \frac{m-1}{m}, 1\big\}$ from multivalued logic. We recall the basic theory of this mathematical framework, and give a novel algorithm for computing the fixed points that in many cases has essentially the same complexity as in the binary case. Our approach provides a biologically intuitive representation of the network. Furthermore, it uses tools to compute lattice points in rational polytopes, tapping a rich area of algebraic combinatorics as a source for combinatorial algorithms for network analysis. An implementation of the algorithm is provided.