A methodological basis for description and analysis of systems with complex switch-like interactions

Abstract

A wide range of complex systems appear to have switch-like interactions, i.e. below (or above) a certain threshold x has no or little influence on y, while above (or below) this threshold the effect of x on y saturates rapidly to a constant level. Switching functions are frequently described by sigmoid functions or combinations of these. Within the context of ordinary differential equations we present a very general methodological basis for designing and analysing models involving complicated switching functions together with any other non-linearities. A procedure to determine position and stability properties of all stationary points lying close to a threshold for one or several variables, so-called singular stationary points, is developed. Such points may represent homeostatic states in models, and are therefore of considerable interest. The analysis provides a profound insight into the generic effects of steep sigmoid interactions on the dynamics around homeostatic points. It leads to qualitative as well as quantitative predictions without using advanced mathematical methods. Thus, it may have an important heuristic function in connection with numerical simulations aimed at unfolding the predictive potential of realistic models.

Modeling Transcriptional Control in Gene Networks—Methods, Recent Results, and Future Directions

“…Another frequently used type of model (Snoussi and Thomas, 1993;Mestl et al, 1995aMestl et al, , 1996Plahte et al, 1998) that describes gene networks is:…”

“…However, few results have been obtained for determining periodic solutions of these equations, or for characterizing more complex dynamics. Recently, the methods of Plahte et al (1994Plahte et al ( , 1998 have been applied to analyse the existence and location of equilibria in a model of cellular iron homeostasis . This model is based on translational rather than transcriptional control, but is still cast as a set of differential equations using steep sigmoids.…”

“…In the future, computerized versions of the methods of Plahte et al (1998), Mestl et al (1996), Mestl et al (1995a), or Snoussi and Thomas (1993) could prove a valuable adjunct to numerical simulations. These methods could make tentative identifications, subject to experimental confirmation, of steady and periodic dynamic solutions for gene product concentrations in specific genetic regulatory systems.…”

Qualitative simulation of genetic regulatory networks using piecewise-linear models

“…State equation for gene a: A regulation function ¤ describes the logic of gene regulation [54,60]. More precisely, it describes the conditions under which the protein encoded by gene…”

An Algorithm for Qualitative Simulation of Gene Regulatory Networks with Steep Sigmoidal Response Functions

“…The thresholds associated with each x i are ordered according to θ ij < θ ik if j < k. Finally, x(t, x 0 , q) is the solution satisfying the initial condition x(0, x 0 , q) = x 0 . The differentiable functions f i are regulatory production functions, frequently composed by algebraic equivalents of Boolean functions [9]. Eq.…”