Bulletin of Mathematical Biology

, Volume 65, Issue 6, pp 1025-1051

First online:

Identification of all steady states in large networks by logical analysis

  • Vincent DevlooAffiliated withSCMBB, CP 263, Université Libre de Bruxelles Email author 
  • , Pierre HansenAffiliated withGERAD and HEC
  • , Martine LabbéAffiliated withOptimization, ISRO, Université Libre de Bruxelles

Rent the article at a discount

Rent now

* Final gross prices may vary according to local VAT.

Get Access


The goal of generalized logical analysis is to model complex biological systems, especially so-called regulatory systems, such as genetic networks. This theory is mainly characterized by its capacity to find all the steady states of a given system and the functional positive and negative circuits, which generate multistationarity and a cycle in the state sequence graph, respectively. So far, this has been achieved by exhaustive enumeration, which severely limits the size of the systems that can be analysed. In this paper, we introduce a mathematical function, called image function, which allows the calculation of the value of the logical parameter associated with a logical variable depending on the state of the system. Thus the state table of the system is represented analytically. We then show how all steady states can be derived as solutions to a system of steady-state equations. Constraint programming, a recent method for solving constraint satisfaction problems, is applied for that purpose. To illustrate the potential of our approach, we present results from computer experiments carried out on very large randomly-generated systems (graphs) with hundreds, or even thousands, of interacting components, and show that these systems can be solved using moderate computing time. Moreover, we illustrate the approach through two published applications, one of which concerns the computation times of all steady states for a large genetic network.