Despite the introduction of millions of sperm into the female mammalian reproductive tract, fewer than 100 actually arrive at and penetrate the egg, enabling fertilization. A tenth of every ten million sperm reach the cervix, where they encounter a complex fluid environment containing embedded polymer structures. Only a tenth of those make it through the uterus. To reach the egg, the remaining sperm then must pass through the contracting oviduct. Oviducts are lined with mucosal folds and coordinated beating cilia that contribute to sperm transport. Some sperm may adhere to oviductal epithelia, requiring a change in their oscillation pattern to escape.
Mammalian fertilization involves many components, including sperm motility, female reproductive tract environment, biochemical signaling, and complex viscoelastic fluids. Successful reproduction in mammals relies on interactions of elastic structures with a fluid environment. Lisa Fauci (Tulane University) models these interactions using an immersed boundary (IB) framework to address fundamental questions about the biology of reproduction.
Why Biofluid Models Are Helpful
Sperm models help researchers understand various aspects of fertilization (see Figure 1). Sperm physiologists want to answer basic questions about the possibility of increasing sperm motility via treatment of chemical environments, which is beneficial to many industries.
Figure 1. Sperm-egg penetration model. Image credit: Jacek Wróbel, Julie Simons, Ricardo Cortez, and Lisa Fauci.
Recognizing how non-Newtonian viscoelastic fluids help or hinder fertilization is also important. And a better understanding of how fertilized ova implant in the uterus can lead to improvements in in vitro fertilization and clinical practices. For instance, should the injection of a fertilized ovum be timed with uterine contractions?
The rise of technologies such as micro-fluidic devices, labs-on-a-chip, and the ability to manipulate bacteria and flagellated organisms has triggered a surge in research activity surrounding the fluid dynamics of microorganism motility. For example, the creation of non-biological microrobots might facilitate drug delivery. Understanding how sperm deliver their payloads could help guide fabricated microswimmers towards tumors.
Fauci uses computational methods that couple both mechanical and biochemical systems with fluid dynamics in order to model fertilization and reproduction. As an alternative to continuum models, her group models a complex mucosal network as discrete nodes connected by viscoelastic elements. Using discrete networks is advantageous in that a network’s connectivity can evolve to accompany structures’ inhomogeneous material properties.
Motility: Beating Through Viscoelastic Fluids
Complex geometries, non-Newtonian fluids, and moving elastic and actuated interfaces all complicate sperm motility models. “Choices have to be made when choosing the types of models and what to include,” Fauci says. Paraphrasing a famous quote by George E.P. Box, she adds, “All models are wrong, but some of them are informative.”
Preliminary models, which initially looked at only one interface and one sperm or cilium, were 2D with simple domain geometries, like a periodic box rather than a more complicated pulsing tube. Researchers had to decide whether to use prescribed kinematic models of flagellar motion, or elastic rod models in which the flagellar shape emerges from the elastohydrodynamic coupling.
Elastic rod models assume that sperm flagella have tensile and bending energy that is minimized when the flagellar shape meets a preferred curvature. Forces derived from these energies also depend on stiffness coefficients. In models with biochemical properties, this preferred curvature is a function of the evolving calcium profile along the flagellum. Flagellar forces are coupled to a surrounding fluid. And at the microscale, inertia is negligible and Stokes equations are used. The coupled system is solved using the method of regularized Stokeslets.
Fauci’s group also studies whether sperm gain any advantage by swimming through a polymeric network rather than a Newtonian Stokesian fluid. The model of a fluid coupled with an elastic network accounts for discrete links. It overlays a polymeric network onto a Stokes 3D fluid, with nodes connected by springs or Maxwell elements that transmit forces.
The group first looked at the rheology of viscoelastic structures in fluid to characterize the structures’ properties when forces and shear are applied as single or periodic pulses. Then they introduced a flagellum with specified kinematics and particular waveform into the network (see Figure 2). The waveform makes it through the mesh without feedback from it, and deforms the compliant network. As flagella encounter denser networks they still pass through, but with lower velocities than in free space for most of the journey; however, a velocity boost is visible as the flagella exit the object. Even though the stored elastic energy in the network increases the velocity, the flagella require more power to maintain their waveforms in the denser environment, so their efficiency is seen to be less than in free Stokes flow.
Figure 2. Swimmer affects compliant viscoelastic network. Image credit: .
Though there is still much analysis to be done, thus far the models reveal that a viscoelastic network can enhance swimming. Future models will include two-way coupling and won’t prescribe the kinematics, preventing sperm from swimming with fixed amplitudes. As motors face more resistance, lower amplitude waveforms may result.
Encountering a “Wall”
Using an elastic model of a flagellum, Fauci’s group also studies the elastic connections and binding proteins that attach sperm to an oviductal epithelial “wall.” Shapes evolve based on other forces, and simulations help the researchers determine if the hyperactivated form allows flagella to more easily escape when attached to a wall.
Figure 3. Model of flagellar adhesion to a planar wall. Image credit: .
As sperm approach a wall, elastic connections are created with forces able to attach it (see Figure 3). Simulations help visualize this interplay under different scenarios. With no elastic linkages, sperm get close to the wall but don’t attach. With no-slip boundary conditions, they don’t escape. When sperm enter hyperactived mode, they attach, detach, and reattach, consistent with experimental observations.
These simulations demonstrate that bond behavior can enable sperm movement away from an epithelial cell, and suggest the necessity of more experiments for further bonding characterization. Fauci’s group also developed a system to simulate sperm’s encounter with a network surrounding a solid sphere, and plans to study actual penetration mechanisms in more detail.
Using mechanical forces, sperm can push themselves through viscoelastic networks to penetrate an egg. During symmetric flagellar bending, they take on linear trajectories and are surrounded by a resting level of calcium ions. Sperm swim straight; the moderate-amplitude beating of their tails has the quintessential sinusoidal wave form, with increasing amplitude towards the tail.
When activated by high levels of calcium, sperm enter a hyperactive state with a different motility pattern characterized by repeated, high-amplitude asymmetric beating. Sperm in this state move in circles and appear ‘confused.’ Hyperactivity is an important process that provides the higher mechanical forces necessary to unstick sperm from mucosal folds in the oviduct and prevent reattachment. These higher forces also allow sperm to penetrate the viscoelastic layer surrounding the ovum right before fertilization. Clear biochemical pathways initiate this state.
Studying the mechanics, forces, and calcium dependence helps elucidate the functional implications of the hyperactivated mode of motility, and begs the following questions:
- What biochemical pathways initiate hyperactivation?
- What happens to sperm when exposed to sufficient calcium?
- What force-generating mechanisms change the beat?
- Compared to the force of moderate amplitude beating, how much more force does asymmetric bending generate to prevent the flagellum from sticking?
Sperm can also penetrate an egg by dissolving the network in front of it. When ready to fertilize the egg, a sperm cell’s body releases enzymes that dissolve some of the links in the viscoelastic mesh.
Motivated by this occurrence, Fauci’s group is currently working on computational experiments—applying forces and dissolving links—with these viscoelastic webs. They are attempting to couple the biochemical part of the puzzle with different reduced models of sperm motility. The group plans to study the penetration process in more detail to understand the significance of sperm-egg penetration mechanics in relation to its biochemistry and the interplay between the two.
This article is based on Lisa Fauci’s AWM-SIAM Sonia Kovalevsky Lecture at the SIAM Annual Meeting, which was held in Boston this July.
 Simons, J., Olson, S., Cortez, R., & Fauci, L. (2014). The dynamics of sperm detachment from epithelium in a coupled fluid-biochemical model of hyperactivated motility. J. Theor. Biol., 354, 81-94.
 Wróbel, J.K., Lynch, S., Barrett, A., Fauci, L., & Cortez, R. (2016, March). Enhanced flagellar swimming through a compliant viscoelastic network in Stokes flow. J. Fluid Mech., 792, 775-797.
 Olson, S.D., Suarez, S.S., & Fauci, L. (2011). Coupling biochemistry and hydrodynamics captures hyperactivated sperm motility in a simple flagellar model. J. Theor. Biol. 283, 203-216.