Abstract
Synchronous probabilistic Boolean networks (PBNs) and generalized asynchronous PBNs have received significant attention over the past decade as a tool for modeling complex genetic regulatory networks. From a biological perspective, the occurrence of interactions among genes, such as transcription, translation, and degradation, may require a few milliseconds or even up to a few seconds. Such a time delay can be best characterized by generalized asynchronous PBNs. This paper attempts to study an optimal control problem in a generalized asynchronous PBN by employing the theory of average value-at-risk (AVaR) for finite horizon semi-Markov decision processes. Specifically, we first formulate a control model for a generalized asynchronous PBN as an AVaR model for finite horizon semi-Markov decision processes and then solve an optimal control problem for minimizing average value-at-risk criterion over a finite horizon. In order to illustrate the validity of our approach, a numerical example is also displayed.
Highlights
Modeling genetic regulatory networks is a core issue in system biology
The corresponding generalized asynchronous probabilistic Boolean networks (PBNs) is described as an average value-at-risk (AVaR) model in finite horizon semi-Markov decision processes with each state variable denoting a gene activity profile (GAP) at some time
We have shown that the dynamic behavior of generalized asynchronous PBN could be represented as a semi-Markov process where the state transition was completely specified by all of the possible Boolean functions
Summary
Modeling genetic regulatory networks is a core issue in system biology. Numerous models of modeling and understanding genetic regulatory networks have been proposed, which include Boolean networks, Bayesian networks, Petri net, differential equations, computational framework based on the automata and languages theory, and probabilistic Boolean networks (PBNs) [1,2,3,4,5,6,7,8]. The corresponding generalized asynchronous PBN is described as an AVaR model in finite horizon semi-Markov decision processes with each state variable denoting a GAP at some time. We will formulate a generalized asynchronous PBN as an AVaR model for finite horizon semi-Markov decision processes with specified definitions of notation as follows. We have shown that the dynamic behavior of generalized asynchronous PBN could be represented as a semi-Markov process where the state transition was completely specified by all of the possible Boolean functions. We have specified the AVaR model for the above generalized asynchronous PBN as follows: S, A x , x ∈ S , Q t, y ∣ x, a , c x, a , ηgγ t, x , g ∈ G , 9 where the state space S; the available action set A x at state x ∈ S; the semi-Markov kernel Q t, y ∣ x, a with x, y ∈ S, t ∈. As described above, using a representation of AVaR in (10), we have converted the problem of minimizing the AVaR of the finite horizon cost for a semi-Markov decision process into a bilevel optimization problem in which the outer optimization problem is an ordinary problem of minimizing a function of a single variable and the inner optimization problem is to minimize the expected positive deviation of the finite horizon cost for semi-Markov decision processes
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