Complexity of the BN and the PBN Models of GRNs and Mappings for Complexity Reduction

Introduction

One can think of a Gene Regulatory Network (GRN) as a network of relations among strands of DNA (genes) and the regulatory activities associated with those genes (Dougherty and Braga-Neto, 2006). This general definition allows for many mathematical (usually dynamical) systems to be called GRNs. The goodness of each such model is evaluated using several important criteria: the level of description of the biochemical reactions involved, complexity of the model, model parameter estimation, and the predictive power of the model. There have been many attempts to model the structure and dynamical behavior of GRNs, ranging from deterministic with discrete time space to fully stochastic with continuous time space. One can find a good review of such attempts in (de Jong, 2002). The so called central ‘dogma’ of molecular biology (Crick, 1970) implies that genes communicate via the proteins they encode. Both stages of protein production, transcription and translation, are controlled by a multitude of biochemical reactions, and are influenced by both internal and external to the cell factors. This perspective suggests that the expression of a given gene i, i.e. the quantity of either protein or messenger RNA, should be considered as a random function X_i(t) of the cell’s internal and external environments. Thus, if one wants to study the dynamical behavior of a GRN, one must design a mathematical model for the gene-expression vector X(t) = (X₁(t), X₂(t), …, X_n(t)) for the n genes that form the network. The stochastic differential equation model appears to provide the most detailed description of the dynamics of X(t). In principle, it could include all of the information about the biochemical processes involved in gene regulation. At the same time, the estimation of its parameters cannot be done without large amount of reliable time-series data. Thus, one is forced to take a more pragmatic approach and look for simpler models for the dynamics of the gene-expression vector. One of the most extreme simplifications is the Boolean network (BN) model, originally proposed by Kauffman (1969a). The BN model is based on the observation that during the regulation of its functional states the cell often exhibits switch-like behavior. Recent work using the NCI 60 Anti-Cancer Drug Screen has demonstrated that Boolean logic type interactions can be detected in gene expression data (Pal at al., 2005). While there are instances in gene regulation where the Boolean logic is the appropriate level of description of the interactions – for instance, when transcription factors have to form a complex that binds to the cis-regulatory DNA to activate transcription, one should keep in mind that discrete models cannot capture the details of the biochemical reactions involved in those processes. It is not the binary nature of the BN model that is its greatest weakness, one even more important deficiency is its determinism. Deterministic models, such as the BN, cannot represent the consequential perturbations due to external latent variables. In addition, the BN model cannot be used to represent biologically meaningful events, such as gene mutations. The stochastic extension of the BN model - probabilistic Boolean network (PBN), was introduced by Shmulevich at al. (2002b) in an attempt to account for those latent variables and gene perturbations while keeping the Boolean logic as the model for the gene-gene interactions. As a collection of BNs with a probability structure, the PBN model could be viewed as a minimal extension of the BN which allows for modeling of the stochastic nature of complex systems with lots of latent variables and random experimental effects. However, even such a minimal extension of the deterministic model exhibits high complexity which impedes its practical applications to model GRN of more than 20 genes. Hence, there is a need for constructing size reducing mappings that produce new and more tractable models that share some of the biologically meaningful properties of the larger-scale models. In addition, one needs to develop methods for complexity estimation of both the model and the mapping used to reduce its size.