Generating probabilistic Boolean networks from a prescribed stationary distribution

Modeling gene regulation is an important problem in genomic research. Boolean networks (BN) and its generalization Probabilistic Boolean networks (PBNs) have been proposed to model genetic regulatory interactions. BN is a deterministic model while PBN is a stochastic model. In a PBN, on one hand, its stationary distribution gives important information about the long-run behavior of the network. On the other hand, one may be interested in system synthesis which requires the construction of networks from the observed stationary distribution. This results in an inverse problem which is ill-posed and challenging. Because there may be many networks or no network having the given properties and the size of the inverse problem is huge. In this paper, we consider the problem of constructing PBNs from a given stationary distribution and a set of given Boolean Networks (BNs). We first formulate the inverse problem as a constrained least squares problem. We then propose a heuristic method based on Conjugate Gradient (CG) algorithm, an iterative method, to solve the resulting least squares problem. We also introduce an estimation method for the parameters of the PBNs. Numerical examples are then given to demonstrate the effectiveness of the proposed methods.