Developing control theory of gene regulatory networks is one of the significant topics in the field of systems biology, and it is expected to apply the obtained results to gene therapy technologies in the future. In this paper, a control method using a Boolean network (BN) is studied. A BN is widely used as a model of gene regulatory networks, and gene expression is expressed by a binary value (0 or 1). In the control problem, we assume that the concentration level of a part of genes is arbitrarily determined as the control input. However, there are cases that no gene satisfying this assumption exists, and it is important to consider structural control via external stimuli. Furthermore, these controls are realized by multiple drugs, and it is also important to consider multiple effects such as duration of effect and side effects. In this paper, we propose a BN model with two types of the control inputs and an optimal control method with duration of drug effectiveness. First, a BN model and duration of drug effectiveness are discussed. Next, the optimal control problem is formulated and is reduced to an integer linear programming problem. Finally, numerical simulations are shown.

In the field of systems biology, there have been a lot of studies on modeling, analysis, and control of gene regulatory networks. Especially, control of gene regulatory networks corresponds to therapeutic interventions, which are realized by radiation, chemotherapy, and so on. In order to develop gene therapy technologies (see, e.g., [

Gene regulatory networks are in general expressed by ordinary/partial differential equations with high nonlinearity and high dimensionality. In order to deal with such a system, it is important to consider a simple model, and various models such as Bayesian networks, Boolean networks (BNs) [

Furthermore, in the control theory of gene regulatory networks, the control input is given by the concentration level of a part of genes; that is, we assume that the concentration level of a part of genes can be arbitrarily determined. However, in the case where this assumption is not satisfied, it is important to consider structural control via external stimuli [

Thus, in this paper, we propose a BN model with two types of the control inputs and an optimal control method with duration of drug effectiveness. The first control input is the control input satisfying the assumption that the binary value is arbitrarily determined. The second control input is called a structural control input herein, and the dynamics, that is, the Boolean functions, are selected among the candidates of the dynamics. However, it is difficult to uniquely select one Boolean function. Hence, we suppose that one Boolean function is selected probabilistically, and the probability distribution is switched by using the structural control input. A structural control method has been discussed in [

In optimal control of PBNs and CS-PBNs, many results have been obtained so far (see, e.g., [

In this paper, based on our previously proposed method [

This paper is organized as follows. In Section

Let

A Boolean network (BN) with

where

Next, the control inputs are added to a BN (

where

Then, consider the structural control input. Suppose that the candidates of

hold. In addition,

is imposed. Here, we show a simple example.

As a simple example, consider the simplified model of an apoptosis network in Figure

where the concentration level (high or low) of the inhibitor of apoptosis proteins (IAPs) is denoted by

Suppose that

We suppose that the Boolean function

A simplified model of an apoptosis network: activation (solid) and inhibition (broken).

A BN with two types of the control inputs includes the probabilistic behavior, and we assume that the probability distribution can be controlled. From these facts, a BN studied in this paper can be regarded as a generalized form of a probabilistic Boolean network (PBN). To explain the relation between the proposed BN model and a PBN, we show a simple example.

As a simple example, consider the PBN with three states and one control input. Suppose that the Boolean functions are given as follows:

This PBN corresponds to the cases of

Next, consider the state orbit of this PBN. In PBNs, one Boolean function is probabilistically selected at each time. Then, for

In this example, the cardinality of the finite state set

The state transition diagram with

As shown in this example, computing state transition diagrams with

By adding the candidates of the Boolean functions, BNs with two types of the control inputs can be transformed into BNs with only the structural control input. That is, the control input

The control input

The parameter

First, suppose that, for the control input

Another example is shown. Suppose that, for the control input

By using the three parameters

First, the following two notations are defined. Let

For simplicity of notation,

Next, for the Boolean networks with

Suppose that, for the Boolean network with

find two control input sequences

subject to the constraint on duration of drug effectiveness, where

For simplicity of discussion, a linear function with respect to

For a binary variable

In control of gene regulatory networks, the expression of a certain gene is frequently focused (see, e.g., [

Furthermore, in many existing methods on optimal control of PBNs, the expected value of a nonnegative function is frequently used as a cost function (see, e.g., [

We show an example for setting weighting vectors from the biological viewpoint.

Consider the Boolean network expressing an apoptosis network in Example

By the coordinate transformation of

We propose a solution method for Problem

As preparations, two lemmas are introduced. To reduce Problem

Consider the two binary variables

For example,

Suppose that the binary variables

where

From Lemmas

Now, we consider reducing Problem

By using Lemma

where

The binary vector

Then, by using the natural logarithm,

In this expression, one probability distribution is selected by using

By using Lemma

where

Here in after, for simplicity of notation,

Now, we consider transforming Problem A by using (

obtained from the state equation in (

where

Next, the inequality constraint

where

where

Next, consider the constraint on duration of drug effectiveness. This constraint can be expressed as a Boolean function. Then, by using Lemmas

where

We show two examples.

Consider Example

and in this case, these are equivalent to

We explain

where

Next, consider the case of

and this case is one of the simplest cases. In the case of

where

Finally, the cost function (

where

Problem B can be solved by a suitable solver such as IBM ILOG CPLEX [

In this section, we show numerical simulations. First, we consider the WNT5A network [

The gene regulatory network with the gene WNT5A is related to melanoma, and it has been extensively studied (see, e.g., [

where the concentration level (high or low) of the gene WNT5A is denoted by

The optimal control problem is formulated. For simplicity, we consider only the structural control input. Then, suppose that the number of the structural control inputs is two. If

If

The case of

We show the computational result. Let

Noting that

Next, consider the case of

From the obtained inputs, we see that the system is controlled by switching two discrete probability distributions, and the obtained inputs satisfy the constraint on duration of drug effectiveness. Noting that the trivial value of

Finally, we discuss the computation time for solving Problem

In order to evaluate the computation time for solving Problem

The optimal control problem is formulated. In this example, we consider 3 control inputs and 2 structural control inputs. If

If

The Boolean function

The Boolean function

From the previous setting,

Next, we discuss the computation time. Consider the two cases of

In this paper, we have proposed a Boolean network (BN) model with two types of the control inputs and an optimal control method. By using this model, several situations in control of gene regulatory networks can be modeled. To model more realistic situations, duration of drug effectiveness has also been introduced. Since duration is given for each control input, effectiveness of multiple drugs can be evaluated. Furthermore, for this BN model, the optimal control problem has been formulated, and this problem is reduced to an integer linear programming problem. Finally, numerical simulations have been shown. The proposed method provides us with a basic in the control theory of gene regulatory networks.

Recently, to simplify state transition diagrams such as that in Figure

This work was partially supported by the Grant-in-Aid for Young Scientists (B) no. 23760387.