Sensitive dependence on initial conditions in gene networks

Abstract

Active regulation in gene networks poses mathematical challenges that have led to conflicting approaches to analysis. Competing regulation that keeps concentrations of some transcription factors at or near threshold values leads to so-called singular dynamics when steeply sigmoidal interactions are approximated by step functions. An extension, due to Artstein and coauthors, of the classical singular perturbation approach was suggested as an appropriate way to handle the complex situation where non-trivial dynamics, such as a limit cycle, of fast variables occur in switching domains. This non-trivial behaviour can occur when a gene regulates multiple other genes at the same threshold. Here, it is shown that it is possible for nonuniqueness to arise in such a system in the case of limiting step-function interactions. This nonuniqueness is reminiscent of but not identical to the nonuniqueness of Filippov solutions. More realistic gene network models have sigmoidal interactions, however, and in the example considered here, it is shown numerically that the corresponding phenomenon in smooth systems is a sensitivity to initial conditions that leads in the limit to densely interwoven basins of attraction of different fixed point attractors.

Introduction to Focus Issue: Quantitative Approaches to Genetic Networks

“…66,70,71 In the current issue, Machina et al examine a toy model demonstrating that rapid cycling in the neighborhood of thresholds can lead to densely interwoven basins of attraction of different attractors, and thus to sensitivity of asymptotic dynamics to the initial condition. 72 In order to develop general purpose software for qualitative analysis of gene network dynamics, these technical problems must be recognized and appropriately handled. 71,73,74 An interesting question is the extent to which different classes of models might offer differing perspectives and insights on a given network.…”

“…As suggested by the quotations at the start of this article, analogies between logical circuits and genetic networks now provide a main direction for theoretical analysis of genetic networks and several articles in this issue consider different aspects of this approach. 39,40,48,53,72,75 However, as these articles make clear, current approaches are not adequate to deal with many of the practical problems confronted by real networks. In particular, better methods are needed to understand the relationships between structure and dynamics 025001- 5 Albert, Collins, and Glass Chaos 23, 025001 (2013) in very large networks; 48,53 time delays due to physical processes of synthesis or diffusion can play a major role in determining the dynamics, 39,40 transcription factors do not have on-off control and may have different thresholds for different processes so that strictly Boolean logic may not be suitable.…”

Zeno breaking, the Contact effect and sensitive behaviour in piecewise-linear systems

“…Now, Z 4 = 0 while y 4 < 0, and during intervals of time when this holds, y 1 , y 2 and y 3 converge to their threshold intersection in finite time, as shown in [17]. When y 4 reaches 0, we enter a codimension-4 switching region, and with our choice of parameter values, a, b and c, the trajectory can exit the (Z 1 , Z 2 , Z 3 , Z 4 ) box in two places, (0, 0, b/(10 + c), 1) or (1, 1, 1, 1), as can be shown by an analysis of the boundary layer equations in fast time [18]. The choice of exit points is not determined in the q → 0 limit.…”

“…It is also important to recognize that the classical singular perturbation theory, based on Tikhonov's theorem, relies on asymptotic convergence of fast variables to a stable equilibrium, in order to continue the evolution of the slow variables in a way that ensures that behaviour of nearby smooth systems is still well approximated (at least for arbitrary finite time intervals) [21]. However, in [17,18], it was shown how an extension of this method due to Artstein and collaborators [3][4][5] allows continuation of solutions even when the attractor in the fast variables is not a fixed point.…”

“…In that example, all three variables reach their thresholds in finite time, and remain there, but the fast variables in the codimension-3 switching domain converge to a periodic orbit, called a 'quasi-Rössler' attractor. In [18], this example was extended to a four-dimensional system, in which a fourth variable continued to evolve slowly, while the other three executed their microscopic periodic orbit within their triple threshold intersection. The subsequent motion depended sensitively on the location on the periodic orbit of the first three variables at the moment the fourth hit its threshold.…”