The talk begins with a brief discussion of the pheromone response signaling pathway in yeast and the goal of the Alpha Project research which is to make quantitative predictions about yeast response to pheromone and defined experimental perturbations. The pheromone pathway is well-studied by biologists. It is qualitatively understood, i.e. scientists agree on the components of the pathway and how they interact. However, quantitative information about the rates of reactions and numbers of molecules is lacking due to experimental and measurement limitations. I introduce some ideas on how to use methods from dynamical systems theory to model this complex biological system.
The second portion of the talk regards the shepherd moon phenomena (observed on Saturn and other planets) and provides a dynamical systems approach to explain how a planetary ring can have a sharp boundary if a small shepherding moon is near to it. Rings with sharp boundaries are unexpected since rings are made of particles that tend to diffuse causing fuzzy ring edges. A mathematical model is developed to describe the interactions of the ring particles with the shepherd moon under the assumptions in the literature that the shepherd moon is small enough to have only a local effect and yet close enough to perturb orbits of ring particles and therefore the dynamics of the system.
A particle’s orbit is determined by two parameters: the moon’s mass and the radial distance between the moon and particle’s orbits. Classically, orbits are studied in relation to when the moon’s mass limits to zero. Here we extend upon the literature and allow both parameters to limit to zero simultaneously. The model is a special case of the Three-Body problem and is describe by a four-dimensional system of coupled nonlinear differential equations. Predictions are made about the qualitative structure of ring particle orbits based on numerical and analytic techniques. Bifurcations are found for the aforementioned parameters.
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