Sarah D. Olson

Coupling biochemistry and hydrodynamics captures hyperactivated sperm motility in a simple flagellar model

Hyperactivation in mammalian sperm is characterized by highly asymmetrical waveforms and an increase in the amplitude of flagellar bends. It is important for the sperm to be able to achieve hyperactivated motility in order to reach and fertilize the egg. Calcium (Ca(2+)) dynamics are known to play a large role in the initiation and maintenance of hyperactivated motility. Here we present an integrative model that couples the CatSper channel mediated Ca(2+) dynamics of hyperactivation to a mechanical model of an idealized sperm flagellum in a 3-d viscous, incompressible fluid. The mechanical forces are due to passive stiffness properties and active bending moments that are a function of the local Ca(2+) concentration along the length of the flagellum. By including an asymmetry in bending moments to reflect an asymmetry in the axonemes response to Ca(2+), we capture the transition from activated motility to hyperactivated motility. We examine the effects of elastic properties of the flagellum and the Ca(2+) dynamics on the overall swimming patterns. The swimming velocities of the model flagellum compare well with data for hyperactivated mouse sperm.

Modeling the dynamics of an elastic rod with intrinsic curvature and twist using a regularized Stokes formulation

We develop a Lagrangian numerical algorithm for an elastic rod immersed in a viscous, incompressible fluid at zero Reynolds number. The elasticity of the rod is described by a version of the Kirchhoff rod model, where intrinsic curvature and twist are prescribed, and the fluid is governed by the Stokes equations in R^3. The elastic rod is represented by a space curve corresponding to the centerline of the rod and an orthonormal triad, which encodes the bend and twist of the rod. In this method, the differences between the rod configuration and its intrinsic shape generate force and torque along the centerline. The coupling to the fluid is accomplished by the use of the method of regularized Stokeslets for the force and regularized rotlets for the torque. This technique smooths out the singularity in the fundamental solutions of the Stokes equations for the computation of the velocity of the rod centerline. In addition, the computation of the angular velocity of the rod requires the use of regularized (potential) dipoles. As a benchmark problem, we consider open and closed rods with intrinsic curvature and twist in a viscous fluid. Equilibrium configurations and dynamic instabilities are compared with known results in elastic rod theory. For cases when the exact solution is unknown, the numerical results are compared to those produced by the generalized immersed boundary (gIB) method, where the fluid is governed by the Navier-Stokes equations with small Reynolds number on a finite (periodic) domain. It is shown that the regularization method combined with Kirchhoff rod theory contributes substantially to the reduction of computation time and efficient memory usage in comparison to the gIB method. We also illustrate how the regularized method can be used to model microorganism motility where the organism is propelled by a flagellum propagating sinusoidal waves. The swimming speeds of this flagellum using the regularized Stokes formulation are matched well with classical asymptotic results of Taylors infinite cylinder in terms of frequency and amplitude of the undulation.

Mathematical modeling of calcium signaling during sperm hyperactivation.

Mammalian sperm must hyperactivate in order to fertilize oocytes. Hyperactivation is characterized by highly asymmetrical flagellar bending. It serves to move sperm out of the oviductal reservoir and to penetrate viscoelastic fluids, such as the cumulus matrix. It is absolutely required for sperm penetration of the oocyte zona pellucida. In order for sperm to hyperactivate, cytoplasmic Ca(2+) levels in the flagellum must increase. The major mechanism for providing Ca(2+) to the flagellum, at least in mice, are CatSper channels in the plasma membrane of the principal piece of the flagellum, because sperm from CatSper null males are unable to hyperactivate. There is some evidence for the existence of other types of Ca(2+) channels in sperm, but their roles in hyperactivation have not been clearly established. Another Ca(2+) source for hyperactivation is the store in the redundant nuclear envelope of sperm. To stabilize levels of cytoplasmic Ca(2+), sperm contain Ca(2+) ATPase and exchangers. The interactions between channels, Ca(2+) ATPases, and exchangers are poorly understood; however, mathematical modeling can help to elucidate how they work together to produce the patterns of changes in Ca(2+) levels that have been observed in sperm. Mathematical models can reveal interesting and unexpected relationships, suggesting experiments to be performed in the laboratory. Mathematical analysis of Ca(2+) dynamics has been used to develop a model for Ca(2+) clearance and for CatSper-mediated Ca(2+) dynamics. Models may also be used to understand how Ca(2+) patterns produce flagellar bending patterns of sperm in fluids of low and high viscosity and elasticity.

A Model of CatSper Channel Mediated Calcium Dynamics in Mammalian Spermatozoa

CatSpers are calcium (Ca2+) channels that are located along the principal piece of mammalian sperm flagella and are directly linked to sperm motility and hyperactivation. It has been observed that Ca2+ entry through CatSper channels triggers a tail to head Ca2+ propagation in mouse sperm, as well as a sustained increase of Ca2+ in the head. Here, we develop a mathematical model to investigate this propagation and sustained increase in the head. A 1-d reaction-diffusion model tracking intracellular Ca2+ with flux terms for the CatSper channels, a leak flux, and plasma membrane Ca2+ clearance mechanism is studied. Results of this simple model exhibit tail to head Ca2+ propagation, but no sustained increase in the head. Therefore, in this model, a simple plasma membrane pump-leak system with diffusion in the cytosol cannot account for these experimentally observed results. It has been proposed that Ca2+ influx from the CatSper channels induce additional Ca2+ release from an internal store. We test this hypothesis by examining the possible role of Ca2+ release from the redundant nuclear envelope (RNE), an inositol 1,4,5-trisphosphate (IP3) gated Ca2+ store in the neck. The simple model is extended to include an equation for IP3 synthesis, degradation, and diffusion, as well as flux terms for Ca2+ in the RNE. When IP3 and the RNE are accounted for, the results of the model exhibit a tail to head Ca2+ propagation as well as a sustained increase of Ca2+ in the head.

The dynamics of sperm detachment from epithelium in a coupled fluid-biochemical model of hyperactivated motility

Hyperactivation in mammalian sperm is characterized by a high-amplitude, asymmetric flagellar waveform. A mechanical advantage of this hyperactivated waveform has been hypothesized to be the promotion of flagellar detachment from oviductal epithelium. In order to investigate the dynamics of a free-swimming sperm׳s binding and escaping from a surface, we present an integrative model that couples flagellar force generation and a viscous, incompressible fluid. The elastic flagellum is actuated by a preferred curvature model that depends upon an evolving calcium profile along its length. In addition, forces that arise due to elastic bonds that form and break between the flagellar head and the surface are accounted for. As in recent laboratory experiments, we find that a hyperactive waveform does result in frequent detaching and binding dynamics that is not observed for symmetric flagellar beats. Moreover, we demonstrate that flagellar behavior depends strongly on the assumptions of the bond model, suggesting the need for more experimental investigation of the biochemistry of epithelial bonding and the shedding of binding proteins on the sperm head.

Fluid dynamic model of invertebrate sperm chemotactic motility with varying calcium inputs.

In a marine environment, invertebrate sperm are able to adjust their trajectory in response to a gradient of chemical factors released by the egg in a process called chemotaxis. In response to this chemical factor, a signaling cascade is initiated that causes an increase in intracellular calcium (Ca(2+)). This increase in Ca(2+) causes the sperm flagellar curvature to change, and a change in swimming direction ensues. In previous experiments, sperm swimming in a gradient of chemoattractant have exhibited Ca(2+) oscillations of varying peaks and frequency. Here, we model a simplified sperm flagellum with mechanical forces, including a passive stiffness component and an active bending component that is coupled to the time varying Ca(2+) input. The flagellum is immersed in a viscous, incompressible fluid and we use a fluid dynamic model to investigate emergent trajectories. We investigate the sensitivity of the model to the frequency of Ca(2+) oscillations. In this coupled model, we observe that longer periods of Ca(2+) oscillation corresponds to circular paths with greater drift. In contrast, shorter periods of Ca(2+) oscillations corresponded to tighter search patterns. These outcomes shed light on the relation between Ca(2+) oscillations and different searching trajectories and strategies that invertebrate sperm may utilize to reach and fertilize the egg in a marine environment.

Hydrodynamic interactions of sheets vs filaments: Synchronization, attraction, and alignment

The synchronization of nearby sperm flagella as they swim in a viscousfluid was observed nearly a century ago. In the early 1950s, in an effort to shed light on this intriguing phenomenon, Taylor initiated the mathematical analysis of the fluid dynamics of microorganism motility. Since then, models have investigated sperm hydrodynamics where the flagellum is treated as a waving sheet (2D) or as a slender waving filament (3D). Here, we study the interactions of two finite length, flexible filaments confined to a plane in a 3D fluid and compare these to the interactions of the analogous pair of finite, flexiblesheets in a 2D fluid. Within our computational framework using regularized Stokeslets, this comparison is easily achieved by choosing either the 2D or 3D regularized kernel to compute fluid velocities induced by the actuated structures. We find, as expected, that two flagella swimming with a symmetric beatform will synchronize (phase-lock) on a fast time scale and attract towards each other on a longer time scale in both 2D and 3D. For a symmetric beatform, synchronization occurs faster in 2D than 3D for sufficiently stiff swimmers. In 3D, a greater enhancement in efficiency and swimming velocity is observed for attracted swimmers relative to the 2D case. We also demonstrate the tendency of two asymmetrically beating filaments in a 3D fluid to align — in tandem — exhibiting an efficiency boost for the duration of their sustained alignment.

Swimming in a two-dimensional Brinkman fluid: Computational modeling and regularized solutions

The incompressible Brinkman equation represents the homogenized fluid flow past obstacles that comprise a small volume fraction. In nondimensional form, the Brinkman equation can be characterized by a single parameter that represents the friction or resistance due to the obstacles. In this work, we derive an exact fundamental solution for 2D Brinkman flow driven by a regularized point force and describe the numerical method to use it in practice. To test our solution and method, we compare numerical results with an analytic solution of a stationary cylinder in a uniform Brinkman flow. Our method is also compared to asymptotic theory; for an infinite-length, undulating sheet of small amplitude, we recover an increasing swimming speed as the resistance is increased. With this computational framework, we study a model swimmer of finite length and observe an enhancement in propulsion and efficiency for small to moderate resistance. Finally, we study the interaction of two swimmers where attraction does not occur when the initial separation distance is larger than the screening length.

Radial basis function (RBF)-based parametric models for closed and open curves within the method of regularized stokeslets

Summary The method of regularized Stokeslets (MRS) is a numerical approach using regularized fundamental solutions to compute the flow due to an object in a viscous fluid where inertial effects can be neglected. The elastic object is represented as a Lagrangian structure, exerting point forces on the fluid. The forces on the structure are often determined by a bending or tension model, previously calculated using finite difference approximations. In this paper, we study spherical basis function (SBF), radial basis function (RBF), and Lagrange–Chebyshev parametric models to represent and calculate forces on elastic structures that can be represented by an open curve, motivated by the study of cilia and flagella. The evaluation error for static open curves for the different interpolants, as well as errors for calculating normals and second derivatives using different types of clustered parametric nodes, is given for the case of an open planar curve. We determine that SBF and RBF interpolants built on clustered nodes are competitive with Lagrange–Chebyshev interpolants for modeling twice-differentiable open planar curves. We propose using SBF and RBF parametric models within the MRS for evaluating and updating the elastic structure. Results for open and closed elastic structures immersed in a 2D fluid are presented, showing the efficacy of the RBF–Stokeslets method. Copyright

Swimming speeds of filaments in viscous fluids with resistance.

Many microorganisms swim in a highly heterogeneous environment with obstacles such as fibers or polymers. To better understand how this environment affects microorganism swimming, we study propulsion of a cylinder or filament in a fluid with a sparse, stationary network of obstructions modeled by the Brinkman equation. The mathematical analysis of swimming speeds is investigated by studying an infinite-length cylinder propagating lateral or spiral displacement waves. For fixed bending kinematics, we find that swimming speeds are enhanced due to the added resistance from the fibers. In addition, we examine the work and the torque exerted on the cylinder in relation to the resistance. The solutions for the torque, swimming speed, and work of an infinite-length cylinder in a Stokesian fluid are recovered as the resistance is reduced to zero. Finally, we compare the asymptotic solutions with numerical results for the Brinkman flow with regularized forces. The swimming speed of a finite-length filament decreases as its length decreases and planar bending induces an angular velocity that increases linearly with added resistance. The comparisons between the asymptotic analysis and computation give insight on the effect of the length of the filament, the permeability, and the thickness of the cylinder in terms of the overall performance of planar and helical swimmers.