Construction of a Boolean model of gene and protein regulatory network with memory

A Boolean model of gene and protein regulatory network with memory (GPBN) has recently attracted interest as a generalization of original random Boolean networks (BNs) for genetic and cellular networks. It is better suited to describe experimental data from the time-course microarray. Addressing construction problems in GPBNs may lead to a better understanding of the intrinsic dynamics in biological systems. Using the technique of the semi-tensor product (STP) of matrices, the dynamics of a GPBN can be expressed in an algebraic form and the attractors can be calculated. This paper investigates the issue of construction of GPBNs from prescribed attractors. Based on a rigorous theoretical analysis, some algebraic formulae and a computationally feasible algorithm are obtained to construct the least in-degree model with prescribed attractors. Illustrative examples are presented to show the validity of the theoretical results and the proposed algorithm.