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

Koichi Kobayashi

doi:10.1155/2019/9261793

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.

Highlights

One of the central problems in systems biology is to develop control theory of gene regulatory networks
A gene regulatory network is modeled by discrete dynamical systems (e.g., Boolean networks (BNs) [6]), continuous dynamical systems, or hybrid dynamical systems
We propose a more sophisticated procedure using a minimum feedback vertex set of an interaction graph [20] obtained from a given BN

Summary

Introduction

One of the central problems in systems biology is to develop control theory of gene regulatory networks. In a BN, gene expression is modeled by a binary value (0 or 1), and interactions between genes are modeled by a set of Boolean functions While this model is quite simple, it is still useful in developing a control method for gene regulatory networks. It is shown that a feedback vertex set is significant in model reduction of BNs. Using the reduced BN, the problem of designing fixed points by external stimuli can be rewritten as an integer linear programming (ILP) problem (see Figure 1). The number of fixed points is characterized by using a minimum feedback vertex set (see Figure 1) Based on this result, a class of BNs such that the number of fixed points becomes 1 by external stimuli is clarified. We sometimes use the symbol 0 instead of 0m×n, and the symbol I instead of In

Preliminaries

Design of Fixed Points Using Model Reduction

Procedure of Model Reduction of BNs

Biological Example

Conclusion