BB4: The role of hyperactivated motility and fluid mechanics in human fertility

Project report to follow

Background

The project focuses on three areas related to microswimmers in a low Reynolds number environment. First, we consider a sperm tethered to an epithelial surface undergoing hyperactivation, whereby the flagellum (tail) exhibits vigorous, high-amplitude asymmetric beating. Secondly, we study the flagellum in greater analytical detail with the focus on numerical implementation. The third area explores the effects of viscoelasticity on simple swimming models.

Techniques and Challenges

In order to calculate the forces generated by a tethered sperm, resistive force theory is applied in which the local flagellum velocity is assumed to be directly proportional to the viscous drag. A more accurate technique describing the Stokes flow past a beating flagellum is given by slender body theory, whereby singularities are distributed along the flagellum centreline. However, numerical challenges arise in discretising a singular centreline. To overcome these, we have developed a slender body theory, which incorporates regularised singularities and hence avoids having a singular centreline. Other work explores simple swimming models in viscoelastic media using asymptotic methods to simplify the nonlinear viscoelastic terms.

Results

The results from the tethered sperm model illustrate that hyperactivation can actually lead to forces that pull the sperm away from the point of tethering, counterintuitive to the notion that sperm are always pushed by their flagella. A parameter study highlights the importance of the increase in beat amplitude and reduction in beat wavenumber during hyperactivation in generating these tugging motions.

The Future

The calculations for the regularised slender body theory are complete, hence we are currently conducting numerical simulations to demonstrate both the ease of implementation and the accuracy. We are also continuing our analytical study into simple swimmers in a viscoelastic fluid, which will be supplemented with numerical conclusions.

References

[11/35] Curtis M.P., Kirkman-Brown J.C., Connolly T.J., Gaffney E.A.: Modelling a tethered mammalian sperm cell undergoing hyperactivation

Smith D.J., Gaffney E.A. et al: Bend propagation in the flagella of migrating human sperm, and its modulation by viscosity; Cell Motility and the Cytoskeleton, 2009

Johnson R.E.: An improved slender-body theory for Stokes flow; Journal of Fluid Mechanics, 1980

Smith D.J., Gaffney E.A., Blake J.R.: Mathematical modelling of cilia-driven transport of biological fluids. Proc. R. Soc.A, 2009

Smith D.J., Gaffney E.A. et al: Cell Motility and the Cytoskeleton, 2009

(Images reproduced from Smith, Gaffney et al: Cell Motility and the Cytoskeleton, invoking contributor rights, as defined on http://authorservices.wiley.com)