# Optimization-Based Approaches to Control of Probabilistic Boolean Networks
## 1 Introduction  
It is important to understand biological behavior as a system. For this purpose, systems biology has been established as a new research field [1; 2]. One of the practical applications in systems biology is gene therapy technologies (see, e.g., [3]). In gene therapy, the expression of certain genes must be controlled, and such a problem can be formulated as a control problem. In other words, control of gene regulatory networks corresponds to therapeutic interventions realized by radiation, chemotherapy, and so on. Thus, control theory of gene regulatory networks is important for developing medical services in the future. Furthermore, from the viewpoint of experimental study, several results on control of gene regulatory networks have been obtained in [4; 5; 6; 7]. For example, in [5], it was shown that cellular behavior can be regulated by control. These results suggest that feedback control of biological systems can be implemented. Motivated by the above background, we are developing theory of control methods for gene regulatory networks. In developing a control method of gene regulatory networks, first, it is important to consider what mathematical model is utilized. A gene regulatory network is modeled by discrete dynamical systems (e.g., Bayesian networks, Boolean networks (BNs) [8]), continuous dynamical systems (e.g., differential/difference equations), or hybrid dynamical systems (e.g., piecewise affine models) (see, e.g., [9] for further details). In control theory, BNs and piecewise affine models are frequently utilized (see, e.g., [10] for control of a biological system modeled by a piecewise affine model). The control problem of hybrid systems is generally reduced to a mixed integer programming problem, in which continuous variables and binary variables are decision variables, and the computation time to solve it is frequently too long. Hence, a class of gene regulatory networks is limited to those with a small network. A BN is a mathematical model whose interactions between genes are expressed by Booleanfunctions [8]. That is, gene expression is approximated by a binary value (0 or 1). Since a BN is too simple, there is a possibility that biological behavior is not precisely modeled (see, e.g., [11]). On the other hand, this model can be relatively applied to large-scale systems. This advantage is focused on, and a BN is utilized in the first step of developing control theory of gene regulatory networks. Furthermore, a probabilistic Boolean network (PBN) [12] and a context-sensitive PBN [13; 14] have been proposed as the extended models of BNs. In a PBN, at each discrete time, a Boolean function is randomly chosen from the candidates of Boolean functions. In a context-sensitive PBN, the time of choosing a Boolean function is also selected randomly. Hence, a context-sensitive PBN is a generalized version of a PBN. In the last decade, many results on control of BNs and PBNs have been obtained so far (see, e.g., [15; 16; 17; 18; 19; 20; 21; 22; 23; 24; 25; 26; 27; 28; 29; 30; 31; 32]). Especially, analysis and control methods using the semi-tensor product (STP) method have been widely studied (see, e.g., [33; 34; 35; 36; 37; 38; 39; 40; 41; 42; 43]). In the review paper [44], many methods for PBNs, e.g., construction and inference methods, structural intervention methods, and control methods, have been explained in a comprehensive manner. Of course, it is difficult to understand details of individual methods by reading only [44]. It is also significant to restrict topics and explain details. In this review paper, our previously proposed methods [23; 24] in optimal external control of PBNs, i.e., a polynomial optimization approach and an integer programming approach are explained. Comparing our methods with other existing methods, the main advantage of our methods is that a relatively large-scale PBN can be handled. In many existing control methods (e.g., [19; 20; 21; 30; 32; 36]), transition probability matrices with \(2^{n}\times 2^{n}\) must be manipulated (\(n\) is the number of genes).