Finding and Analyzing the Minimum Set of Driver Nodes in Control of Boolean Networks

Abstract

We study the minimum number of driver nodes control of which leads a Boolean network (BN) from an initial state to a target state in a specified number of time steps. We show that the problem is NP-hard and present an integer linear programming-based method that solves the problem exactly. We mathematically analyze the average size of the minimum set of driver nodes for random Boolean networks with bounded in-degree and with a small number of time steps. The results of computational experiments using randomly generated BNs show good agreements with theoretical analyses. A further examination in realistic BNs demonstrates the efficiency and generality of our theoretical analyses.

On the number of driver nodes for controlling a Boolean network when the targets are restricted to attractors

“…Concerning the scaling behavior of the minimum control cost of BNs without structural or temporal constraints, however, there is not yet a conclusive results in literature. While simulation results suggest that a small number of driver nodes are enough if the target states are restricted to attractors [38,41], results from [14,20] imply that O(N ) driver nodes are required if an arbitrary state is specified as the target. both initial and target states are specified (resp., only a target state is specified).…”

“…Since a BN with N nodes has 2 N possible states, it will eventually reach a previously visited state, thus stay in that state circle, called an attractor. Among various problems on BNs, control of a BN is particularly important in which the values of a subset of nodes or external signals are manipulated so as to drive the BN to a desired state [9,[12][13][14][15][16]. For example, in disease treatment, one may need to conduct therapeutic intervention that drives the cell state of a patient from a current state to a desired state such as a benign state, and keep this state afterward.…”

Critical controllability analysis of directed biological networks using efficient graph reduction

“…Since most of the observed complex networks in nature are scale-free networks and the MDS approach can cope with non-linear systems 4 , 5 , analysing the controllability of real networks using the MDS approach appears reasonable. Actually, the MDS approach has been applied to analysis of various biological networks and reported to be useful for finding important genes and molecules not only by our group but also by other groups 6 – 19 .…”

On the Number of Driver Nodes for Controlling a Boolean Network to Attractors

“…Among various problems on BNs, control of a BN is particularly important in which the values of a subset of nodes or external signals are manipulated so as to drive the BN to a desired state [9,[12][13][14][15][16]. For example, in disease treatment, one may need to conduct therapeutic intervention that drives the cell state of a patient from a current state to a desired state such as a benign state, and keep this state afterward.…”

“…Concerning the scaling behavior of the minimum control cost of BNs without structural or temporal constraints, however, there is not yet a conclusive results in literature. While simulation results suggest that a small number of driver nodes are enough if the target states are restricted to attractors [38,41], results from [14,20] imply that O(N ) driver nodes are required if an arbitrary state is specified as the target. In this paper, we mathematically prove that the expected number of driver nodes is only log 2 (N ) + log 2 (M ) + 2 if the targets are restricted to attractors, under a reasonable assumption.…”