Mathematical models for swimming in a viscoelastic fluid

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In this project, we will develop new mathematical models for swimming in a viscoelastic fluid. The project consists of three parts. The first is to combine a model for the sliding filaments of a sperm flagellum with resistive-force theory in the Maxwell model for a linearly viscoelastic fluid, and then to determine the swimming speed. This simple model will capture many of the features which will be present in more complicated but realistic models, and illuminate the fundamental principles of swimming in a viscoelastic fluid. The aim of the second part is to derive the slender-body equations for the forces acting on thin rods in a viscoelastic medium, analogous to the slender-body equations familiar from Stokes flow. Finally, the aim of the third part is to apply these models to the case of sperm swimming in mucus. For example, at a certain point during fertilization, sperm cells change their beat pattern to the asymmetric whip-like "hyperactivated" state. We will study how hyperactivity affects motility in the viscoelastic environments of the uterus and oviduct.This project is motivated by the fact that many swimming microorganisms encounter biological fluids with solid-like (elastic) properties as well as liquid-like (viscous) properties. Examples range from mammalian sperm in cervical fluid to the ulcer-causing bacterium Helicobacter pylori in the mucus layer lining the inside of the stomach. Despite the great importance of these examples for fertility and health, there have been few attempts to develop a quantitative theory for swimming in viscoelastic fluids; almost all previous efforts have focused on purely viscous fluids such as water. It is important to develop a theory for viscoelastic swimmers, since it will deepen our understanding of the physical constraints faced by mammalian sperm, leading to a better understanding of their biology and perhaps new clinical strategies for contraception or enhancing fertility for humans or livestock.