Control Of Probabilistic Boolean Networks Research Articles

Local set stability and target control of probabilistic Boolean networks

This paper analyzes the local set stability of probabilistic Boolean networks (PBNs), including local finite-time set stability with probability one (FSSPO) and local asymptotic set stability with probability one (ASSPO). Firstly, a necessary and sufficient condition is proposed for the local FSSPO of PBNs. Then, a reachable set with probability one is constructed, based on which, the largest domain of attraction for the local FSSPO of PBNs is determined. Secondly, by constructing two parameterized sets, the transition probability matrix of PBNs is properly partitioned, and the largest domain of attraction for the ASSPO is depicted. Finally, for PBNs which are locally stable, the target control is introduced to achieve the global FSSPO and global ASSPO of PBNs, respectively.

Policy Iteration Approach to the Infinite Horizon Average Optimal Control of Probabilistic Boolean Networks

This article studies the optimal control of probabilistic Boolean control networks (PBCNs) with the infinite horizon average cost criterion. By resorting to the semitensor product (STP) of matrices, a nested optimality equation for the optimal control problem of PBCNs is proposed. The Laurent series expression technique and the Jordan decomposition method derive a novel policy iteration-type algorithm, where finite iteration steps can provide the optimal state feedback law, which is presented. Finally, the intervention problem of the probabilistic Ara operon in E. coil, as a biological application, is solved to demonstrate the effectiveness and feasibility of the proposed theoretical approach and algorithms.

Optimal control for probabilistic Boolean networks using discrete-time Markov decision processes

A new mathematical model, which is called probabilistic Boolean networks (PBNs), has been developed to model complex genetic regulatory networks during the past decade. An important problem is to find optimal policies for a PBN so as to avoid undesirable states. Although several models of discrete time Markov decision processes (DTMDPs) have been applied to find optimal control policies, the existing literatures show that the objective functions focus on the long term and short term expected rewards/costs. Compared with previous works, a probability criterion for DTMDPs has been proposed to make the dynamic comparison of different treatment in the framework of PBNs. More precisely, this paper concentrates on the probability that the total loss incurred before the network first visit to desirable states. The application of our approach is also exhibited via two numerical examples.

Optimization-Based Approaches to Control of Probabilistic Boolean Networks

Control of gene regulatory networks is one of the fundamental topics in systems biology. In the last decade, control theory of Boolean networks (BNs), which is well known as a model of gene regulatory networks, has been widely studied. In this review paper, our previously proposed methods on optimal control of probabilistic Boolean networks (PBNs) are introduced. First, the outline of PBNs is explained. Next, an optimal control method using polynomial optimization is explained. The finite-time optimal control problem is reduced to a polynomial optimization problem. Furthermore, another finite-time optimal control problem, which can be reduced to an integer programming problem, is also explained.

State feedback based output tracking control of probabilistic Boolean networks

This paper studies the state feedback based output tracking control of probabilistic Boolean control networks (PBCNs) by using the semi-tensor product of matrices. A series of reachable sets (with probability 1) is defined inductively for PBCNs, and some useful properties are obtained for the reachable sets. Based on the properties of the reachable sets, a necessary and sufficient condition is presented for the state feedback based output tracking control of PBCNs. Meanwhile, a constructive procedure is proposed to design state feedback laws for PBCNs to track a constant reference signal. The study of two illustrative examples shows that the obtained new results are very effective.

Structural Control of Probabilistic Boolean Networks and Its Application to Design of Real-Time Pricing Systems

In this paper, a new control method for a probabilistic Boolean network (PBN) is proposed. A PBN is widely used as a model of complex systems such as gene regulatory networks. For a PBN, the structural control problem is newly formulated. In this problem, a discrete probability distribution appeared in a PBN is controlled by the continuous-valued input. In the proposed solution method, using a matrix-based representation for a PBN, the problem is approximated by a linear programming problem. Furthermore, design of real-time pricing systems of electricity is considered as an application. By appropriately designing realtime pricing systems, electricity conservation is achieved. The effectiveness of the proposed method is presented by a numerical example on real-time pricing systems.

Optimal control for probabilistic Boolean networks

Aberrant gene functions usually contribute to the pathology or diseases. Avoiding undesirable cellular phenotypes as many as possible is a major purpose of external control for gene regulatory networks. An interesting question is how to control a gene network subjected to the condition that the genes reach some undesirable states with minimal probability during a cell cycle. In this paper, we make use of the theory of the first passage model for discrete-time Markov decision processes to determine the optimal control for a gene intervention model. Specifically, we first use a control model for a probabilistic Boolean network to model interactions among genes and then solve an optimal control problem for maximising the probability of the first arrival time to desirable gene states. In order to illustrate the validity of our approach, examples are also displayed.

Sampling-rate-dependent probabilistic Boolean networks

External control of a genetic regulatory network is used for the purpose of avoiding undesirable states, such as those associated with a disease. To date, intervention has mainly focused on the external control of probabilistic Boolean networks via the associated discrete-time discrete-space Markov processes. Implementation of an intervention policy derived for probabilistic Boolean networks requires nearly continuous observation of the underlying biological system since precise application requires the observation of all transitions. In medical applications, as in many engineering problems, the process is sampled at discrete time intervals and a decision to intervene or not must be made at each sample point. In this work, sampling-rate-dependent probabilistic Boolean network is proposed as an extension of probabilistic Boolean network. The proposed framework is capable of capturing the sampling rate of the underlying system.

On approximate stochastic control in genetic regulatory networks

The control of probabilistic Boolean networks as a model of genetic regulatory networks is formulated as an optimal stochastic control problem and has been solved using dynamic programming; however, the proposed methods fail when the number of genes in the network goes beyond a small number. There are two dimensionality problems. First, the complexity of optimal stochastic control exponentially increases with the number of genes. Second, the complexity of estimating the probability distributions specifying the model increases exponentially with the number of genes. We propose an approximate stochastic control method based on reinforcement learning that mitigates the curses of dimensionality and provides polynomial time complexity. Using a simulator, the proposed method eliminates the complexity of estimating the probability distributions and, because the method is a model-free method, it eliminates the impediment of model estimation. The method can be applied on networks for which dynamic programming cannot be used owing to computational limitations. Experimental results demonstrate that the performance of the method is close to optimal stochastic control.

Optimal infinite-horizon control for probabilistic Boolean networks

External control of a genetic regulatory network is used for the purpose of avoiding undesirable states, such as those associated with disease. Heretofore, intervention has focused on finite-horizon control, i.e., control over a small number of stages. This paper considers the design of optimal infinite-horizon control for context-sensitive probabilistic Boolean networks (PBNs). It can also be applied to instantaneously random PBNs. The stationary policy obtained is independent of time and dependent on the current state. This paper concentrates on discounted problems with bounded cost per stage and on average-cost-per-stage problems. These formulations are used to generate stationary policies for a PBN constructed from melanoma gene-expression data. The results show that the stationary policies obtained by the two different formulations are capable of shifting the probability mass of the stationary distribution from undesirable states to desirable ones.