Global analysis of steady points for systems of differential equations with sigmoid interactions

Abstract

A new method to investigate asymptotic properties of linear differential equations with strong threshold and switching effects is presented. The method is applied to systems of equations of the form d x / d t = F(x) -yx, where y = constant and the dependence of F on x is mediated by sigmoid functions. Using a special sigmoid function called a logoid, which rises monotonically from zero to one in a narrow interval surrounding the threshold value, exact analytical expressions for the limiting value of all steady points can be found in the limit when the logoid approaches a step function. The limiting values are independent of the shape of the logoid for a large class of logoids. Relations between steady points and limit cycles of the equations with logoids, their step function limit and the corresponding piecewise linear equations are derived. It is found that the approximation of sigmoids by the step function idealization is not always warranted. The results strongly suggest the use of logoids instead of other sigmoids hitherto employed.

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

“…For these equations, the steady states can also be determined analytically (Mestl et al, 1995a) and progress has been made in locating periodic solutions (Mestl et al, 1995b). For differential equations using steep sigmoids, an analytic method for determining the steady states has recently been developed (Plahte et al, 1994. However, few results have been obtained for determining periodic solutions of these equations, or for characterizing more complex dynamics.…”

“…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.…”

Qualitative simulation of genetic regulatory networks using piecewise-linear models

“…The behavior of systems described by a PL model of the form (2) has been wellcharacterized in the regulatory domains of [25,52,60]. …”

“…However, if solution trajectories in different regulatory domains evolve towards the same switching domain, as is the case for trajectories arriving at The underlying cause of this problem is the occurrence of discontinuities in the right-hand side of (2), due to the use of step functions. The discontinuities occur at c In [47,52] threshold hyperplanes, which separate regulatory domains in which the dynamics of the regulatory network is described by a different system of differential equations (11). In order to deal with these discontinuities in a general and rigorous way, we will use a method originally proposed by Filippov [19].…”

“…Furthermore, the models have mathematical properties that favour qualitative analysis of the steady-state and transient behavior of regulatory systems [14,15,16,24,25,28,29,31,41,47,48,52,60,61]. On a formal level, the PL models are related to a class of asynchronous logical models proposed by Thomas and colleagues [66,67].…”