Structure and behavior in Boolean monotonic model pools.

One of the main theoretical questions in the field of discrete regulatory networks is the question what aspects of the dynamics - the structure of the state transition graph - are already imposed by structural descriptions of the network such as the interaction graph. For Boolean networks, prior work has concentrated on different versions of the Thomas conjectures that link feedback cycles in the network structure to attractor properties. Other approaches check algorithmically whether certain properties hold true for all models sharing specific structural constraints, e.g. by using model checking techniques. In this work we investigate the behavior of the pool of Boolean networks in agreement with a given interaction graph using a different approach. Grouping together states that are updated consistently across the pool we derive an equivalence relation and analyze a corresponding quotient graph on the state space. By construction this graph yields information about the dynamics of all functions in the pool. Our main result is that this graph can be computed efficiently without enumerating and analyzing all individual functions. This opens up new possibilities for applications, where such model pools arise when modeling under uncertainty.

Follicular competition in cows: the selection of dominant follicles as a synergistic effect.

The reproductive cycle of mono-ovulatory species such as cows or humans is known to show two or more waves of follicular growth and decline between two successive ovulations. Within each wave, there is one dominant follicle escorted by subordinate follicles of varying number. Under the surge of the luteinizing hormone a growing dominant follicle ovulates. Rarely the number of ovulating follicles exceeds one. In the biological literature, the change of hormonal concentrations and individually varying numbers of follicular receptors are made responsible for the selection of exactly one dominant follicle, yet a clear cause has not been identified. In this paper, we suggest a synergistic explanation based on competition, formulated by a parsimoniously defined system of ordinary differential equations (ODEs) that quantifies the time evolution of multiple follicles and their competitive interaction during one wave. Not discriminating between follicles, growth and decline are given by fixed rates. Competition is introduced via a growth-suppressing term, equally supported by all follicles. We prove that the number of dominant follicles is determined exclusively by the ratio of follicular growth and competition. This number turns out to be independent of the number of subordinate follicles. The asymptotic behavior of the corresponding dynamical system is investigated rigorously, where we demonstrate that the [Formula: see text]-limit set only contains fixed points. When also including follicular decline, our ODEs perfectly resemble ultrasound data of bovine follicles. Implications for the involved but not explicitly modeled hormones are discussed.