The Dynamics of Conjunctive and Disjunctive Boolean Network Models

Abstract

For many biological networks, the topology of the network constrains its dynamics. In particular, feedback loops play a crucial role. The results in this paper quantify the constraints that (unsigned) feedback loops exert on the dynamics of a class of discrete models for gene regulatory networks. Conjunctive (resp. disjunctive) Boolean networks, obtained by using only the AND (resp. OR) operator, comprise a subclass of networks that consist of canalyzing functions, used to describe many published gene regulation mechanisms. For the study of feedback loops, it is common to decompose the wiring diagram into linked components each of which is strongly connected. It is shown that for conjunctive Boolean networks with strongly connected wiring diagram, the feedback loop structure completely determines the long-term dynamics of the network. A formula is established for the precise number of limit cycles of a given length, and it is determined which limit cycle lengths can appear. For general wiring diagrams, the situation is much more complicated, as feedback loops in one strongly connected component can influence the feedback loops in other components. This paper provides a sharp lower bound and an upper bound on the number of limit cycles of a given length, in terms of properties of the partially ordered set of strongly connected components.

Controllability of Conjunctive Boolean Networks With Application to Gene Regulation

“…Consider the induced dynamics on G j : First, from the value update rule and the first item of Proposition 3, if x Uj (0) contains an entry of value 0, then so does x Uj (tp * ) for all t ≥ 0. Second, since G 0 is irreducible, a periodic orbit of the induced dynamics has to be a fixed point [19], [48]. Combining these two facts, we know that there is a time t 0 ≥ 0 such that x Uj (tp * ) = 0 for all t ≥ t 0 .…”

“…Roughly speaking, a Boolean network is monotonic if its Boolean function has the property that the output value of the function for each variable is non-decreasing if the number of "1"s in the inputs increases. For example, Boolean networks whose Boolean functions are monomials [16]- [19] are monotonic. For other types of monotonic Boolean networks, we refer the reader to [20]- [23] and the references therein.…”

A Polynomial-Time Algorithm for Solving the Minimal Observability Problem in Conjunctive Boolean Networks

“…There is a one-to-one correspondence between a CBN and its dependency graph, which enables a graph-theoretic analysis of CBNs. This has been used to analyze various properties of CBNs including: characterization of the periodic orbits [24], [28], robustness of these orbits to single bit perturbations [29], and controllability of CBNs [30], [31].…”

The number of fixed points of AND-OR networks with chain topology

“…Boolean networks, f : {0, 1} n → {0, 1} n , have been used to study problems arising from areas such as mathematics, computer science, and biology (Akutsu et al, 1998;Albert and Othmer, 2003;Mendoza and Xenarios, 2006;Jarrah et al, 2010;Veliz-Cuba and Stigler, 2011). A particular problem of interest is counting the number of fixed points (x such that f (x) = x).…”

“…A particular problem of interest is counting the number of fixed points (x such that f (x) = x). To simplify this problem one can restrict the class of Boolean functions or the topology of the network (Agur et al, 1988;Aracena et al, 2004;Jarrah et al, 2007;Aracena, 2008;Murrugarra and Laubenbacher, 2011;Veliz-Cuba and Laubenbacher, 2011;Jarrah et al, 2010;Veliz-Cuba et al, 2013, 2014, 2015, which in some cases allows to find effective algorithms or formulas in closed form.…”