This paper investigates questions related to the modularity in discrete models of biological interaction networks. We develop a theoretical framework based on the analysis of their asymptotic dynamics. More precisely, we exhibit formal conditions under which agents of interaction networks can be grouped into modules. As a main result, we show that the usual decomposition in strongly connected components fulfils the conditions of being a modular organisation. Furthermore, we point out that our framework enables a finer analysis providing a decomposition in elementary modules.
Understanding and exhibiting the relations between phenotypes and interactions in biological networks, i.e., the links between structures and functions [1], are among the most challenging problems at the frontier of theoretical computer science and biology. Many phenotypes can be associated to interactions of biological agents (e.g., genes or proteins) working together to guarantee some specific functions. This leads to group agents into modules and to associate to them one or more biological functions. It follows that biological networks can be seen, at a more abstract level, as modular networks in which interacting elements are modules that carry the biological information necessary to translate into precise functions.
Modularity is present in various kinds of networks including metabolic or signalling pathways and genetic or protein interaction networks. Modular organisations are notably emphasised in embryogenesis [2,3] where modules of genes are coordinated in the development process. Furthermore, methods related to modules discovery in interaction networks are generally based on both the analysis of the networks structures (a field close to graph theory) and the study of their associated dynamics [4]. Structural analysis identifies sub-networks with specific topological properties motivated either by a correspondence between topology and functionality [5,6] or by the existence of statistical biases with respect to random networks [7]. Specific topologies like cliques [8], or more generally strongly connected components (SCCs) are commonly used to reveal modules by structural analysis. Particular motifs [9,10] may also be interpreted as modules viewed as basic components. They represent over-expressed biological sub-networks with respect to random ones. Dynamical analysis lays on the hypothesis that expression profiles provide insights on the relationships between regulators, modules being possibly revealed from correlations between the biological agents expressions. For instance, using yeast gene expression data, the authors of [11,12] inferred models of co-regulated genes and the condition under which the regulation occurs. As a consequence, the discovery of a modular organisation in biological interaction networks is closely related to the influence of the expression of agents on the others and needs to investigate their expression dynamics [13,14,15,16,17].
Using a discrete model of biological interaction networks [18,19], we propose an approach that analyses the conditions of modules formation and characterises the relations between the global behaviour of a network and the local behaviours of its components. The chosen model of interaction networks is based on the assumption that phenotypes are related, at the molecular level, to characteristic states of some biological agents assimilated to equilibria3 . Thus, using dynamical analysis, we show conditions under which interaction networks can be divided into modules so that the composition of these modules behaviours matches the global behaviours of the networks.
The paper is structured as follows: First, Section 2 introduces the main definitions and notations used throughout the paper. Then, Section 3 presents the central notion of a modular organisation of a network along with its structural and dynamical properties. Section 4 defines elementary modular organisation and the conditions leading to obtain it. Some concluding remarks and perspectives are provided in Section 5.
First, we introduce basic notations. Let ⊆ S × S be a binary relation on S. Given s, s ∈ S and S ⊆ S, we denote by s s the fact that (s, s ) ∈ , by (s ) {s | s s } the image of s by , and by (S ) its generalisation to S . Similarly, we denote by ( s) and ( S ) the corresponding preimages. The composition of two binary relations will be denoted by • , the reflexive and transitive closure by * .
Let us assume that a network η is composed of a set A = {a 1 , . . . , a n } of agents. Each a i ∈ A has a local state, denoted by s a i , taking values in some nonempty finite set S a i . A state (or a configuration) of η is defined as a vector s ∈ S associating to each a i ∈ A a value in S a i , where S S a 1 × . . . × S an is the set of all possible states of the network. For any X ⊆ A and s ∈ S, we denote by s| X the restriction of s to the agents in X; this notation naturally extends to sets of states.
An evolution of η is a relation ⊆ S ×S where each s s is a transition meaning that s evolves to s by . Thus, the global evolution of η can be represented by a directed graph G = (S, ) called the state graph (or the transition graph). In this work, we pay particular attention to the notion of local evolution, since each agent a ∈ A has its own evolution a . The collection of all these local evolutions results in the asynchronous view of the global evolution of η.
Definition 1. The asynchronous dynamics (or dynamics for short) of a network η is defined as the triple A, S, ( a ) a
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