An Abstraction Theory for Qualitative Models of Biological Systems

Abstract

Multi-valued network models are an important qualitative modelling approach used widely by the biological community. In this paper we consider developing an abstraction theory for multi-valued network models that allows the state space of a model to be reduced while preserving key properties of the model. This is important as it aids the analysis and comparison of multi-valued networks and in particular, helps address the well-known problem of state space explosion associated with such analysis. We also consider developing techniques for efficiently identifying abstractions and so provide a basis for the automation of this task. We illustrate the theory and techniques developed by investigating the identification of abstractions for two published MVN models of the lysis-lysogeny switch in the bacteriophage lambda

A Formal Framework for Composing Qualitative Models of Biological Systems

“…The state of a Boolean network can be updated either synchronously [9,16], where the state of all entities is updated simultaneously in a single update step, or asynchronously [6], where entities update their state independently. In the following we focus on the synchronous update semantics which has received considerable attention in the literature (see for example [8,9,1,16,12,2]). Given two states S 1 , S 2 ∈ S BN , let S 1 → S 2 represent a (synchronous) update step such that S 2 is the state that results from simultaneously updating the state of each entity g i using its associated update function F (g i ) and the appropriate neighbourhood of states from S 1 .…”

Abstracting Asynchronous Multi-Valued Networks: An Initial Investigation

“…The idea is to formulate what it means for an MVN to be correctly abstracted by a simpler MVN with the same network structure but smaller state space. We take as our starting point the abstraction techniques developed for synchronous MVNs [5] and investigate extending these to the asynchronous case. We show that our approach allows sound analysis inferences about positive reachability properties and that all attractors of an asynchronous abstraction correspond to attractors in the original MVN.…”

“…We begin by recalling the notion of a state mapping and abstraction mapping [5] used to reduce an entity's state space. Definition 5.…”

“…The abstraction theory we present for asynchronous MVNs is based on extending the synchronous abstraction theory presented in [5]. We formulate a notion of what it means for an MVN to be correctly abstracted by a simpler MVN with the same network structure but smaller state space.…”

On the Stability and Instability of Finite Dynamical Systems with Prescribed Interaction Graphs

“…Since entities influence each other, this local update function depends on the states of some of the entities. Such a network is called a Finite Dynamical System (FDS), with special cases or variants appearing under different names, such as Boolean Networks [11,18], Boolean Automata Networks [13], Multi-Valued Networks [3], etc. The main problem when studying an FDS is to determine its dynamics, such as the number of its fixed points, or how the trajectory of a state depends on the initial state.…”