Design of Fixed Points in Boolean Networks Using Feedback Vertex Sets and Model Reduction

Abstract

Fixed points in Boolean networks (BNs) represent cell types or states of cells and are important to decide characteristics of cells. As the control problem on fixed points, it is important to consider the problem of changing fixed points by using external stimuli (i.e., control inputs). In this paper, we propose two methods for designing fixed points. First, a design method using model reduction is proposed. Using the reduced model, the problem of placing fixed points can be rewritten as an integer linear programming problem. Next, we consider the design problem using only the graph structure of a given BN and derive some results. In both methods, a feedback vertex set of a directed graph plays an important role. Finally, a biological example is presented.

A logical method for the synthesis of periodic trajectory in a binary dynamical system

“…In [22], to solve the synthesis problem, a method of reduction (simplification) of a Boolean network graph was proposed. Additionally, this method was demonstrated on a biological example of an apoptosis gene regulatory network.…”

Fixed Point Preserving Model Reduction of Boolean Networks Focusing on Complement and Absorption Laws

“…In [15], the model reduction method using 𝑙 1 -gain analysis has been proposed. In [16], [17], the model reduction methods that the information on fixed points (singleton attractors) is preserved have been proposed. In [16]- [18], a feedback vertex set (FVS), which is a subset of the vertex set of the graph expressing interactions between elements of the state, plays an important role (see Sect.…”

“…2.2 for details of an FVS). By applying the methods in [16], [17] to a given BN, the dimension of the state can be reduced to the number of vertices of an FVS. However, the number of vertices of an FVS is in general not a minimal dimension of the state of the reduced model (see, e.g., [19]).…”

“…However, the number of vertices of an FVS is in general not a minimal dimension of the state of the reduced model (see, e.g., [19]). The reduced BN obtained by using the methods in [16], [17] may be possible to be further reduced.…”

Construction Method of Probabilistic Boolean Networks Based on Imperfect Information

“…For a BN, controllability/reachability analysis [4][5][6][7] and optimal control [8][9][10] have been studied. Model reduction has also been studied in [11][12][13]. For a PBN, controllability/reachability analysis [14][15][16][17] and optimal control [18][19][20][21][22] have been studied (see also the survey paper [23]).…”