A discrete model of a biological regulatory network can be represented as a discrete function f that contains all available information on interactions between network components and the rules governing the evolution of the network in the discrete state space. Both the information on the structure as well as the dynamics of the system can be represented as directed graphs. Since the state space size grows exponentially with the number of network components, analysis of large networks is a complex problem.
In this paper, we introduce the notion of symbolic steady state that allows us to identify subnetworks that govern the dynamics of the original network in at least a subset of state space. We then state rules to explicitly construct attractors of the system from subnetwork attractors. A further application of the underlying concept allows us to formulate sufficient conditions for the existence of multiple attractors resp. a cyclic attractor based on the existence of positive resp. negative feedback circuits in the structure graph. All results are discussed for dynamics derived from f via the synchronous as well as the asynchronous update rule.
Asymptotic behavior is often of particular interest when analyzing asynchronous Boolean networks representing biological systems such as signal transduction or gene regulatory networks. Methods based on a generalization of the steady state notion, the so-called symbolic steady states, can be exploited to investigate attractor properties as well as for model reduction techniques conserving attractors. In this paper, we propose a novel optimization-based method for computing all maximal symbolic steady states and motivate their use. In particular, we add a new result yielding a lower bound for the number of cyclic attractors and illustrate the methods with a short study of a MAPK pathway model.
Multi-valued network models can be described by their topology and a set of parameters capturing the effects of the regulators for each component. Dynamics can then be derived and represented as state transition systems.
Different network models may lead to the same transition system, meaning dynamics analysis of a representative model covers a larger class of models. While rather clear in the Boolean case, the properties contributing to this effect become more involved for multi-valued models. We analyse these properties and present a mathematical description of the resulting model equivalence classes.