Advanced Modeling and Applied Computing, Department of Mathematics, The University of Hong Kong, Hong Kong
Abstract
Background
Probabilistic Boolean Networks (PBNs) provide a convenient tool for studying genetic regulatory networks. There are three major approaches to develop intervention strategies: (1) resetting the state of the PBN to a desirable initial state and letting the network evolve from there, (2) changing the steadystate behavior of the genetic network by minimally altering the rulebased structure and (3) manipulating external control variables which alter the transition probabilities of the network and therefore desirably affects the dynamic evolution. Many literatures study various types of external control problems, with a common drawback of ignoring the number of times that external control(s) can be applied.
Results
This paper studies the intervention problem by manipulating multiple external controls in a finite time interval in a PBN. The maximum numbers of times that each control method can be applied are given. We treat the problem as an optimization problem with multiconstraints. Here we introduce an algorithm, the "Reserving Place Algorithm'', to find all optimal intervention strategies. Given a fixed number of times that a certain control method is applied, the algorithm can provide all the suboptimal control policies. Theoretical analysis for the upper bound of the computational cost is also given. We also develop a heuristic algorithm based on Genetic Algorithm, to find the possible optimal intervention strategy for networks of large size.
Conclusions
Studying the finitehorizon control problem with multiple hardconstraints is meaningful. The problem proposed is NPhard. The Reserving Place Algorithm can provide more than one optimal intervention strategies if there are. Moreover, the algorithm can find all the suboptimal control strategies corresponding to the number of times that certain control method is conducted. To speed up the computational time, a heuristic algorithm based on Genetic Algorithm is proposed for genetic networks of large size.
Background
The major goal of functional genomics is screening genes that determine specific cellular phenotypes (diseases) and modeling their activities in such a way that normal and abnormal behaviors can be differentiated. The pragmatic manifestation of the above goal is developing therapies based on the disruption or mitigation of aberrant gene function contributing to the pathology of certain disease. Mitigation would be accomplished by the use of drugs to act on the gene products. There are three things involved in engineering therapeutic tools, namely, synthesizing nonlinear dynamical networks, characterizing gene regulation and developing intervention strategies to modify dynamical behavior. In this paper, we introduce a new optimization finitehorizon control problem with multiple hardconstraints. First we review some models for studying genetic regulatory networks, Boolean networks (BNs) and Probabilistic Boolean networks (PBNs). A brief review of intervention strategies is also given. We then introduce our mathematical formulation of the problem and the Reserving Place Algorithm. The upper bound for the computational cost is also estimated. We report the results of computational experiments for different genetic regulatory networks by Reserving Place Algorithm and (or) the Genetic Algorithm. Finally, conclusions are given at the end of the paper.
A Review on Boolean Networks (BNs) and Probabilistic Boolean Networks (PBNs)
In computational systems biology, building mathematical models and efficient numerical algorithms to study regulatory interactions among DNA, RNA, proteins and small molecules are important issues
Among these models, BNs and its extension PBNs have received much attention as they can capture the switching behavior of the biological process
However, genetic regulation process exhibits uncertainty and microarray data sets used to infer the model have errors due to experimental noise in the complex measurement process. Thus it is more realistic to consider stochastic models. To extend BNs to PBNs, the main idea is as follows. To determine
Review of intervention strategies
Intervention strategies are applied to drive the whole genetic network into a desirable steadystate probability distribution. The intervention studies used three different approaches: (1) resetting the state of the PBN to a more desirable initial state and letting the network evolve from there, (2) changing the steadystate(longrun) behavior of the genetic network by minimally altering the rulebased structure and (3) manipulating external control variables which alter the transition probabilities of the network and therefore desirably affect the dynamic evolution. In
Our contribution
All the above optimal control formulations did not consider the realistic hard constraints that the number of times of applying controls are bounded. In case of disease such as cancer, control inputs can be medication or radiation, etc. They are typically applied during a time period. Certain treatments such as radiation can not be applied too many times. The study in
Methods
We first give some necessary notations to introduce the mathematical formulation of our optimization control problem. We then describe an algorithm, the Reserving Place Algorithm, for obtaining all the optimal solutions. The upper bound of computational cost is also estimated. Based on this, the drawbacks of the Reserving Place Algorithm is stated and we apply the Genetic Algorithm to networks of large size. Here we study an optimization control problem with multiple hardconstraints. Our goal is to find an optimal strategy for manipulating external control variables to desirably affect the dynamic evolution of a random PBN over a finite time horizon with minimum corresponding cost. Without loss of generality, here we only consider two control methods. At each time point
Here
Optimal solution(s) exists(exist) if
Algorithms
Our proposed problem is NPhard. Here we develop an algorithm for computing all the optimal solutions. In order to find the feasible solution set for the optimal control problem with hardconstraint,
We first assume that the number of times that Control 2 is applied is fixed as
Here we provide an upper bound of the computational cost for our Reserving Place Algorithm.
Theorem 1
Proof: The main computational cost of the algorithm comes from the matrixvector multiplication. For each control strategy, the number of matrixvector multiplication is
It has been shown that finding a control strategy for a BN to a global state is actually NPhard
Results
This section is organized into three parts. First, we provide a 2gene hypothetical genetic network. Both the Reserving Place Algorithm and Genetic Algorithm are applied. The contrast in computational time is also given. Then both algorithms are applied to a 3gene hypothetical genetic network. Finally, the comparison of the two algorithm is conducted.
2gene network
We start with a 2gene hypothetical genetic network. The network consists of two genes denoted by
States of Genes in the 2gene network
States
1
2
3
4
Off
Off
On
On
Off
On
Off
On
Our objective is to find a control strategy that ensures after time length
Suboptimal solutions for 2gene example (
Control Strategy
Cost
Computing Time
0
0
0
1
0
0
1
1
1
1
0
0
0
0
1
0
1
1
1
1
12.5
0.156
0
0
0
0
0
1
1
1
1
1
0
0
0
0
0
0
2
1
1
1
11.5
0.718
0
0
0
0
0
2
2
1
1
1
0
0
0
0
2
0
2
1
1
1
0
0
0
2
0
0
2
1
1
1
15.5
9.375
0
0
2
0
0
0
2
1
1
1
0
2
0
0
0
0
2
1
1
1
2
0
0
0
0
0
2
1
1
1
Optimal solutions for 2gene example under various T by Genetic Algorithm
Control Strategy
Cost
0
0
0
0
0
0
2
1
1
1
11.5
0
0
0
0
0
0
0
2
1
1
1
11.5
0
0
0
0
0
0
0
0
2
1
1
1
11.5
0
0
0
0
0
0
0
0
0
2
1
1
1
11.5
0
0
0
0
0
0
0
0
0
0
2
1
1
1
11.5
0
0
0
0
0
0
0
0
0
0
0
2
1
1
1
11.5
Comparison of computing time of the two algorithms
Reserving Place Algorithm(sec)
Genetic Algorithm(sec)
10.3
29.6
68.8
29.9
315.2
30.1
1177.0
28.9
4017.9
30.0
10796.0
29.5
3gene network
Here we consider a hypothetical network consisting of 3 genes A, B, C. The states of genes
States of Genes in the 3gene network
States
1
2
3
4
5
6
7
8
Off
Off
Off
Off
On
On
On
On
Off
Off
On
On
Off
Off
On
On
Off
On
Off
On
Off
On
Off
On
Our objective is to find a control strategy that ensures the total probability of gene
Table
Suboptimal solutions for 3gene example (
Control Strategy
Cost
Computing Time


0.08
0
0
0
0
0
0
0
1
0
2
6.5
0.644
0
0
0
0
0
0
0
0
2
1
0
0
0
0
0
0
0
2
0
2
8
6.22
Optimal solutions for 3gene example under various
Control Strategy
Cost
Average Computing Time(sec)
0
0
0
0
0
0
0
0
2
1
6.5
32.25
0
0
0
0
0
0
0
0
0
0
0
0
0
0
2
4
30.23
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
2
4
27.7
A comparison of the two algorithms
Based on the numerical experiments, we draw the following remarks for the comparison of the Reserving Place Algorithm and the Genetic Algorithm. The Reserving Place Algorithm obtains all the optimal control strategies, meanwhile the Genetic Algorithm provides one possible optimal solution. Moreover, the Reserving Place Algorithm can give all the suboptimal control strategies for a fixed number of times that certain control method is applied. This is useful in practice as the results can provide more applicable control strategies to be chosen and more information about the effects of combining various control methods. In the aspect of computing time, the computing time of the Reserving Place Algorithm is closely corresponding to the length of time interval
Conclusions
In this paper, we introduced a new optimal finitehorizon control problem with multiple hardconstraints. We proposed an algorithm, the Reserving Place Algorithm, to generate all optimal solutions. The upper bound for the computational cost was also estimated. We remark that our formulation can be applied to both perturbed and contextsensitive PBNs.
Competing interests
The authors declare that they have no competing interests.
Authors' contributions
WaiKi proposed the optimization problem. Yang designed and analyzed the Reserving Place Algorithm, performed the numerical experiments. Yang, WaiKi and NamKiu wrote the manuscript. WaiKi and HoYin contributed to the numerical experiment analysis and modification of the manuscript. All authors have read and approved the final version of the manuscript.
Acknowledgements
The preliminary version of the paper has been appeared in
This article has been published as part of