Efficient Estimation of the Robustness Region of Biological Models with Oscillatory Behavior

Abstract

Robustness is an essential feature of biological systems, and any mathematical model that describes such a system should reflect this feature. Especially, persistence of oscillatory behavior is an important issue. A benchmark model for this phenomenon is the Laub-Loomis model, a nonlinear model for cAMP oscillations in Dictyostelium discoideum. This model captures the most important features of biomolecular networks oscillating at constant frequencies. Nevertheless, the robustness of its oscillatory behavior is not yet fully understood. Given a system that exhibits oscillating behavior for some set of parameters, the central question of robustness is how far the parameters may be changed, such that the qualitative behavior does not change. The determination of such a “robustness region” in parameter space is an intricate task. If the number of parameters is high, it may be also time consuming. In the literature, several methods are proposed that partially tackle this problem. For example, some methods only detect particular bifurcations, or only find a relatively small box-shaped estimate for an irregularly shaped robustness region. Here, we present an approach that is much more general, and is especially designed to be efficient for systems with a large number of parameters. As an illustration, we apply the method first to a well understood low-dimensional system, the Rosenzweig-MacArthur model. This is a predator-prey model featuring satiation of the predator. It has only two parameters and its bifurcation diagram is available in the literature. We find a good agreement with the existing knowledge about this model. When we apply the new method to the high dimensional Laub-Loomis model, we obtain a much larger robustness region than reported earlier in the literature. This clearly demonstrates the power of our method. From the results, we conclude that the biological system underlying is much more robust than was realized until now.

Regulation of oscillation dynamics in biochemical systems with dual negative feedback loops

“…Stochastic simulations showed that noise can extend the Hopf bifurcation point in both single-(F 31 ) and dual-feedback systems (F 31,32 ), producing oscillations at parameter values that does not lead to oscillations in a deterministic setting. Comparing the bifurcation points in F 31 and F 31,32 at different levels of the inner loop showed that the points were not altered, suggesting that addition of the second loop in the nested arrangement had no effect on the stochastic bifurcation points. Further understanding concerning the interplay between noise and oscillations in coupled-feedback settings should be among the focuses of future systems biology studies.…”

“…To facilitate analytical treatment, we instead employ the Routh -Hurwitz theorem [29,30] which states that the eigenvalues, essentially the roots of the characteristic polynomial jJ 2 lIj 31,21 . To verify that (2.4) also constitute the sufficient condition for oscillations in these systems, we employed analysis based on the Floquet theory that globally assesses the stability of the obtained periodic solutions [31][32][33]. The mathematical and computational background of the Floquet analysis is given in the electronic supplementary material.…”

“…This method was numerically implemented in MATHEMATICA v. 8.0 (code can be provided on request). In the following sections, we will present first the analytical analysis based on (2.4) for the single-loop system F 31 followed by the exemplified dual-loop motif F 31,32 . Detailed derivations for the remaining motifs are provided in the electronic supplementary material.…”

Inverse problem studies of biochemical systems with structure identification of S-systems by embedding training functions in a genetic algorithm

“…During aggregation stage of development, Dictyostelium cells show spontaneous oscillations and chemotactic movement that are correlated with periodic changes in the extracellular cAMP concentrations. The molecular network regulating these oscillations has been studied [41,42] . The oscillations in external cAMP ( X 6 ) concentration and intracellular cAMP ( X 5 ) concentration are maintained by periodic activation and deactivation of regulatory units, viz.…”

“…, regulation of cAMP oscillations in Dictyostelium discoideum ( Section 3.2 ) and regulation of circadian oscillations in Drosophila ( Section 3.3 ) are next described. The cAMP oscillation model used in Section 3.2 has a known S-system representation [41,42] while the circadian oscillations model used in Section 3.3 follows a Michaelis-Menten and Hill type kinetics [43] . It would be interesting to see if an acceptable S-system model with an identified structure can be obtained by ETFGA especially when the true system phenomenology is different.…”