_{1}

Design of control strategies for gene regulatory networks is a challenging and important topic in systems biology. In this paper, the problem of finding both a minimum set of control nodes (control inputs) and a controller is studied. A control node corresponds to a gene that expression can be controlled. Here, a Boolean network is used as a model of gene regulatory networks, and control specifications on attractors, which represent cell types or states of cells, are imposed. It is important to design a gene regulatory network that has desired attractors and has no undesired attractors. Using a matrix-based representation of BNs, this problem can be rewritten as an integer linear programming problem. Finally, the proposed method is demonstrated by a numerical example on a WNT5A network, which is related to melanoma.

Modeling, analysis, and control of gene regulatory networks are one of the fundamental problems in the field of systems biology. Control of gene regulatory networks is closely related to therapeutic interventions, which are realized by radiation, chemotherapy, and so on. Hence, developing control theory of gene regulatory networks is important for gene therapy technologies (see, e.g., [

In this paper, we study simultaneous design of a minimum set of control nodes and controllers in a Boolean networks (BN). A BN [

As a related problem, the conventional controllability problem of BNs has been widely studied (see, e.g., [

In order to solve the simultaneous design problem, a matrix-based representation of BNs [

Notation: For the finite set

where

Consider the following Boolean network (BN):

where

crete time. The set

sponding to genes. The function

Next, the notion of attractors is defined as follows.

Definition 1 The state

Definition 2 The set of states

We present a simple example.

Example 1 Consider the following BN:

In this BN,

In this paper, we focus on only a set of singleton attractors. Hereafter, let

The states

where

Then, consider the following problem.

Problem 1 For the BN (3), suppose that Boolean functions

and

respectively. Then, find a minimum set of control nodes and a controller

and

If this problem is infeasible, then there exists no controller that realizes specifications on singleton attractors. This fact implies that the BN (3) is uncontrollable from the viewpoint of singleton attractors, and the change of the graph structure

If the set

In order to solve Problem 1, we use a matrix-based representation of BNs [

In this representation, one state

Then, the matrix-based representation for

where

Example 2 Consider the BN (2) again. Then, we can obtain the truth table for each

x_{3}(k) | x_{1}(k + 1) | x_{1}(k) | x_{3}(k) | x_{2}(k + 1) | x_{1}(k) | x_{2}(k) | x_{3}(k + 1) | ||
---|---|---|---|---|---|---|---|---|---|

0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | ||

1 | 1 | 0 | 1 | 0 | 0 | 1 | 0 | ||

1 | 0 | 1 | 1 | 0 | 1 | ||||

1 | 1 | 0 | 1 | 1 | 1 |

where each element of

See [

The matrix-based representation is useful for considering Problem 1. Using the matrix-based representation, the problem of finding a Boolean function is reduced to the problem of finding

Remark 1 Recently, there have been many results about a matrix-based representation of Boolean functions using the semi-tensor product (STP) of matrices (see, e.g., [

Based on the matrix-based representation of BNs explained in the previous section, we consider a solution method for Problem 1.

As preparations, the notation is defined. If

Since only singleton attractors are focused in Problem 1, Problem 1 is equivalently transformed into the following problem.

Problem 2 Find matrices

(ii) For each

In addition, each element of

In this problem, the condition (i) guarantees that

Problem 2 can be rewritten as an integer linear programming (ILP) problem, and can be solved by a suitable solver. An SMT (Satisfiability Modulo Theories) solver such as the Yices SMT Solver [

Here, using a simple example, we explain a method to rewrite Problem 2 as an ILP problem. Consider the BN in Example 2 of Section 4. Then, the constraints appeared in Problem 2 are given by

and

where

From (12)-(14), we can obtain

From the condition (ii) in Problem 2, at least one of (17)-(19) must be satisfied. Noting this fact, from (17)-(19) we can obtain

where

Under these conditions, at least one of

subject to (14)-(16), (20)-(22), and (23).

Finally, the product of binary variables such as

Lemma 1 Suppose that binary variables

where

Using this lemma, Problem 2 for the BN in Example 2 can be rewritten as an ILP problem. From the above, we see that Problem 2 for a general BN (3) can be rewritten as an ILP problem. In general, an ILP problem is NP-hard, but several free/commercial solvers were developed. Hence, we can solve the ILP problem obtained.

In this section, as an example, we consider the gene regulatory network with the gene WNT5A, which is related to melanoma. A BN model is given by

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

For this BN model, consider solving Problem 1. We assume that

respectively. This setting is artificially given.

First, we derive a matrix-based representation. Then, we can obtain the following matrix-based representation:

Next, we explain the solution for Problem 2. By solving this problem, we can obtain

Finally, we observe the set of singleton attractors. For the BN model obtained, the set of singleton attractors is given by

From this set, we see that specifications on singleton attractors (

For Boolean networks, we studied the problem of finding both a minimum set of control nodes and a controller, where specifications on singleton attractors are imposed. In biological networks, control nodes (control input) are not necessarily given. Hence, this problem is important in both systems biology and synthetic biology. Using a matrix-based representation of BNs, this problem can be rewritten as an integer linear programming problem. The proposed method is effective as one of the fundamental methods for control theory of gene regulatory networks.

There are several open problems. In this paper, we focused on only the set of singleton attractors. An extension of the proposed method to the case of the set of periodic attractors is important, but it will be a challenging topic. Utilizing a probabilistic BN as a model of gene regulatory networks is also important. Then, a discrete probability distribution of attractors must be considered. Finally, in order to clarify the effectiveness of the proposed method, it is important to consider applying the proposed method to several biological systems.

This research was partly supported by JSPS Grant-in-Aid for Scientific Research (C) 26420412.

Koichi Kobayashi, (2016) Attractor-Based Simultaneous Design of the Minimum Set of Control Nodes and Controllers in Boolean Networks. Applied Mathematics,07,1510-1520. doi: 10.4236/am.2016.714131