No abstract

The complex interplay of physiological factors that underlies fertility in dairy cows was investigated using a mechanistic mathematical model of the dynamics of the bovine estrous cycle. The model simulates the processes of follicle and corpus luteum development and its relations with key hormones that interact to control these processes. Several factors may perturb the regular oscillatory behavior of a normal estrous cycle, and such perturbations are likely the effect of simultaneous changes in multiple parameters. The objective of this paper was to investigate how multiple parameter perturbation changes the behavior of the estrous cycle model, so as to identify biological mechanisms that could play a role in the development of cystic ovaries. Cystic ovaries are a common reason for reproductive failure in dairy cows, but much about the causes of this disorder remains unknown. We investigated in which region of the parameter space the model predicts a normal cycle, and when a progesterone pattern occurred with delayed ovulation (indicating a cystic follicle) or delayed luteolysis (indicating a persistent corpus luteum). Perturbation of the initial values for all parameters simultaneously showed 2 specific parameter configurations leading to delayed ovulation or delayed luteolysis immediately. The most important parameter changes in these 2 configurations involve the regulation of corpus luteum functioning, luteolytic signals, and GnRH synthesis, suggesting that these mechanisms are likely involved in the development of cystic ovaries. In the multidimensional parameter space, areas exist in which the parameter configurations resulted in normal cycles. These areas may be separated by areas in which irregular cycle patterns occurred. These irregular patterns thus mark the transition from one stable (normal) situation to another. Interestingly, within a series, there were some cycles with delayed ovulation and some with delayed luteolysis in these patterns. This could represent a situation of resumption of normal cyclicity (e.g., after parturition). In conclusion, the method of parameter perturbation used in the present study is an effective tool to find parameter configurations that lead to progesterone profiles associated with delayed ovulation and delayed luteolysis. Thereby, the model helps to generate hypotheses regarding the underlying cause of the development of cystic ovaries, which could be investigated in future experiments.

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.