Proprioceptive feedback amplification restores effective locomotion in a neuromechanical model of lampreys with spinal injuries.

Spinal injuries in many vertebrates can result in partial or complete loss of locomotor ability. While mammals often experience permanent loss, some nonmammals, such as lampreys, can regain swimming function, though the exact mechanism is not well understood. One hypothesis is that amplified proprioceptive (body-sensing) feedback can allow an injured lamprey to regain functional swimming even if the descending signal is lost. This study employs a multiscale, integrative, computational model of an anguilliform swimmer fully coupled to a viscous, incompressible fluid and examines the effects of amplified feedback on swimming behavior. This represents a model that analyzes spinal injury recovery by combining a closed-loop neuromechanical model with sensory feedback coupled to a full Navier-Stokes model. Our results show that in some cases, feedback amplification below a spinal lesion is sufficient to partially or entirely restore effective swimming behavior.

Resilience of neural networks for locomotion.

Locomotion is an essential behaviour for the survival of all animals. The neural circuitry underlying locomotion is therefore highly robust to a wide variety of perturbations, including injury and abrupt changes in the environment. In the short term, fault tolerance in neural networks allows locomotion to persist immediately after mild to moderate injury. In the longer term, in many invertebrates and vertebrates, neural reorganization including anatomical regeneration can restore locomotion after severe perturbations that initially caused paralysis. Despite decades of research, very little is known about the mechanisms underlying locomotor resilience at the level of the underlying neural circuits and coordination of central pattern generators (CPGs). Undulatory locomotion is an ideal behaviour for exploring principles of circuit organization, neural control and resilience of locomotion, offering a number of unique advantages including experimental accessibility and modelling tractability. In comparing three well-characterized undulatory swimmers, lampreys, larval zebrafish and Caenorhabditis elegans, we find similarities in the manifestation of locomotor resilience. To advance our understanding, we propose a comparative approach, integrating experimental and modelling studies, that will allow the field to begin identifying shared and distinct solutions for overcoming perturbations to persist in orchestrating this essential behaviour.

Effects of cell morphology and attachment to a surface on the hydrodynamic performance of unicellular choanoflagellates.

Choanoflagellates, eukaryotes that are important predators on bacteria in aquatic ecosystems, are closely related to animals and are used as a model system to study the evolution of animals from protozoan ancestors. The choanoflagellate Salpingoeca rosetta has a complex life cycle with different morphotypes, some unicellular and some multicellular. Here we use computational fluid dynamics to study the hydrodynamics of swimming and feeding by different unicellular stages of S. rosetta: a swimming cell with a collar of prey-capturing microvilli surrounding a single flagellum, a thecate cell attached to a surface and a dispersal-stage cell with a slender body, long flagellum and short collar. We show that a longer flagellum increases swimming speed, longer microvilli reduce speed and cell shape only affects speed when the collar is very short. The flux of prey-carrying water into the collar capture zone is greater for swimming than sessile cells, but this advantage decreases with collar size. Stalk length has little effect on flux for sessile cells. We show that ignoring the collar, as earlier models have done, overestimates flux and greatly overestimates the benefit to feeding performance of swimming versus being attached, and of a longer stalk for attached cells.

The role of curvature feedback in the energetics and dynamics of lamprey swimming: A closed-loop model.

Like other animals, lampreys have a central pattern generator (CPG) circuit that activates muscles for locomotion and also adjusts the activity to respond to sensory inputs from the environment. Such a feedback system is crucial for responding appropriately to unexpected perturbations, but it is also active during normal unperturbed steady swimming and influences the baseline swimming pattern. In this study, we investigate different functional forms of body curvature-based sensory feedback and evaluate their effects on steady swimming energetics and kinematics, since little is known experimentally about the functional form of curvature feedback. The distributed CPG is modeled as chains of coupled oscillators. Pairs of phase oscillators represent the left and right sides of segments along the lamprey body. These activate muscles that flex the body and move the lamprey through a fluid environment, which is simulated using a full Navier-Stokes model. The emergent curvature of the body then serves as an input to the CPG oscillators, closing the loop. We consider two forms of feedback, each consistent with experimental results on lamprey proprioceptive sensory receptors. The first, referred to as directional feedback, excites or inhibits the oscillators on the same side, depending on the sign of a chosen gain parameter, and has the opposite effect on oscillators on the opposite side. We find that directional feedback does not affect beat frequency, but does change the duration of muscle activity. The second feedback model, referred to as magnitude feedback, provides a symmetric excitatory or inhibitory effect to oscillators on both sides. This model tends to increase beat frequency and reduces the energetic cost to the lamprey when the gain is high and positive. With both types of feedback, the body curvature has a similar magnitude. Thus, these results indicate that the same magnitude of curvature-based feedback on the CPG with different functional forms can cause distinct differences in swimming performance.

A Model for the Acrosome Reaction in Mammalian Sperm.

The acrosome reaction is a complex, calcium-dependent reaction that results in an exocytotic event required for successful fertilization of the egg. It has long been thought that the acrosome reaction occurs upon sperm binding to the zona pellucida, a viscoelastic layer surrounding the oocyte. Recent studies have suggested that the reaction may even occur before the sperm encounters the zona, perhaps mediated by progesterone or some other agonist. It has been particularly difficult to understand differences between progesterone-induced and zona-induced reactions experimentally and whether one substance is the more biologically relevant trigger. Until this present work, there has been little effort to mathematically model the acrosome reaction in sperm as a whole. Instead, attention has been paid to modeling portions of the pathways involved in other cell types. Here we present a base model for the acrosome reaction which characterizes the known biochemical reactions and behaviors of the system. Our model allows us to analyze several pathways that may act as a stabilizing mechanism for avoiding sustained oscillatory calcium responses often observed in other cell types. Such an oscillatory regime might otherwise prevent acrosomal exocytosis and therefore inhibit fertilization. Results indicate that the acrosome reaction may rely upon multiple redundant mechanisms to avoid entering an oscillatory state and instead maintain a high resting level of calcium, known to be required for successful acrosomal exocytosis and, ultimately, fertilization of the oocyte.

Mixing and pumping by pairs of helices in a viscous fluid.

Here, we study the fluid dynamics of a pair of rigid helices rotating at a constant velocity, tethered at their bases, in a viscous fluid. Our computations use a regularized Stokeslet framework, both with and without a bounding plane, so we are able to discern precisely what flow features are unaccounted for in studies that ignore the surface from which the helices emanate. We examine how the spacing and phase difference between identical rotating helices affects their pumping ability, axial thrust, and power requirements. We also find that optimal mixing of the fluid around two helices is achieved when they rotate in opposite phase, and that the mixing is enhanced as the distance between the helices decreases.

Interaction of toroidal swimmers in Stokes flow.

A doughnut-shaped object supporting surface rotations was a hypothetical construct proposed by both Taylor and Purcell as a swimmer that would be able to propel itself in a Stokesian fluid because of the irreversibility of its stroke. Here we numerically examine the hydrodynamic interaction of pairs and trios of these free toroidal swimmers. First, we study the axisymmetric case of two toroidal swimmers placed in tandem, and show that a single torus of a corotating pair is more efficient than when it swims alone, but less efficient when paired with a counterrotating partner. Using a regularized Stokeslet framework, we study the nonaxisymmetric case of toroidal swimmers whose axes are initially parallel, but not collinear. These perturbed in tandem swimmers can exhibit qualitatively different trajectories that may, for instance, repel the swimmers or have them settle into a periodic state. We also illustrate interesting dynamics that occur for different initial configurations of three tori.

The effect of intrinsic muscular nonlinearities on the energetics of locomotion in a computational model of an anguilliform swimmer.

Animals move through their environments using muscles to produce force. When an animal׳s nervous system activates a muscle, the muscle produces different amounts of force depending on its length, its shortening velocity, and its time history of force production. These muscle forces interact with forces from passive tissue properties and forces from the external environment. Using an integrative computational model that couples an elastic, actuated model of an anguilliform, lamprey-like swimmer with a surrounding Navier-Stokes fluid, we study the effects of this coupling between the muscle force and the body motion. Swimmers with different forms of this coupling can achieve similar motions, but use different amounts of energy. The velocity dependence is the most important property of the ones we considered for reducing energy costs and helping us to stabilize oscillations. These effects are strongly influenced by how rapidly the muscle deactivates; if force decays too slowly, muscles on opposite sides of the body end up fighting each other, increasing energy cost. Work-dependent deactivation, an effect that causes a muscle to deactivate more rapidly if it has recently produced mechanical work, works together with the velocity dependence to reduce the energy cost of swimming.

A model of Stokesian peristalsis and vesicle transport in a three-dimensional closed cavity.

The complexity of the mechanics involved in the mammalian reproductive process is evident. Neither an ovum nor an embryo is self-propelled, but move through the oviduct or uterus due to the peristaltic action of the tube walls, imposed pressure gradients, and perhaps ciliary motion. Here we use the method of regularized Stokeslets to model the transport of an ovum or an embryo within a peristaltic tube. We represent the ovum or the embryo as a spherical vesicle of finite volume - not a massless point particle. The outer membrane of the neutrally buoyant vesicle is discretized by nodes that are joined by a network of springs. The elastic moduli of these springs are chosen large enough so that a spherical shape is maintained. For simplicity, here we choose an axisymmetric tube where the geometry of the two-dimensional cross-section along the tube axis reflects that of the sagittal cross-section of the uterine cavity. Although the tube motion is axisymmetric, the presence of the vesicle within the tube requires a fully three-dimensional model. As was found in Yaniv et al. (2009, 2012) for a 2D closed channel, we find that the flow dynamics in a 3D peristaltic tube are strongly influenced by the closed end and the manner in which the peristaltic wave damps out towards the closure. In addition, we demonstrate that the trajectory of a vesicle of finite volume can greatly differ from the trajectory of a massless fluid particle initially placed at the vesicle׳s centroid.

A fully three-dimensional model of the interaction of driven elastic filaments in a Stokes flow with applications to sperm motility.

In many animals, sperm flagella exhibit primarily planar waveforms. An isolated sperm with a planar flagellar beat in a three-dimensional unbounded fluid domain would remain in a plane. However, because sperm must navigate through complex, three-dimensional confined spaces along with other sperm, forces that bend or move the flagellum out of its current beat plane develop. Here we present an extension of previous models of an elastic sperm flagellar filament whose shape change is driven by the pursuit of a preferred curvature wave. In particular, we extend the energy of the generalized elastica to include a term that penalizes out-of-plane motion. We are now able to study the interaction of free-swimmers in a 3D Stokes flow that do not start out beating in the same plane. We demonstrate the three-dimensional nature of swimming behavior as neighboring sperm swim close to each other and affect each others' trajectories via fluid-structure coupling.

Hydrodynamics of diatom chains and semiflexible fibres.

Diatoms are non-motile, unicellular phytoplankton that have the ability to form colonies in the form of chains. Depending upon the species of diatoms and the linking structures that hold the cells together, these chains can be quite stiff or very flexible. Recently, the bending rigidities of some species of diatom chains have been quantified. In an effort to understand the role of flexibility in nutrient uptake and aggregate formation, we begin by developing a three-dimensional model of the coupled elastic-hydrodynamic system of a diatom chain moving in an incompressible fluid. We find that simple beam theory does a good job of describing diatom chain deformation in a parabolic flow when its ends are tethered, but does not tell the whole story of chain deformations when they are subjected to compressive stresses in shear. While motivated by the fluid dynamics of diatom chains, our computational model of semiflexible fibres illustrates features that apply widely to other systems. The use of an adaptive immersed boundary framework allows us to capture complicated buckling and recovery dynamics of long, semiflexible fibres in shear.

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.

The role of mechanical resonance in the neural control of swimming in fishes.

The bodies of many fishes are flexible, elastic structures; if you bend them, they spring back. Therefore, they should have a resonant frequency: a bending frequency at which the output amplitude is maximized for a particular input. Previous groups have hypothesized that swimming at this resonant frequency could maximize efficiency, and that a neural circuit called the central pattern generator might be able to entrain to a mechanical resonance. However, fishes swim in water, which may potentially damp out many resonant effects. Additionally, their bodies are elongated, which means that bending can occur in complicated ways along the length of the body. We review previous studies of the mechanical properties of fish bodies, and then present new data that demonstrate complex bending properties of elongated fish bodies. Resonant peaks in amplitude exist, but there may be many of them depending on the body wavelength. Additionally, they may not correspond to the maximum swimming speed. Next, we describe experiments using a closed-loop preparation of the lamprey, in which a preparation of the spinal cord is linked to a real-time simulation of the muscle and body properties, allowing us to examine resonance entrainment as we vary the simulated resonant frequency. We find that resonance entrainment does occur, but is rare. Gain had a significant, though weak, effect, and a nonlinear muscle model produced resonance entrainment more often than a linear filter. We speculate that resonance may not be a critical effect for efficient swimming in elongate, anguilliform swimmers, though it may be more important for stiffer carangiform and thunniform fishes.

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 axoneme's 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.

Interactions between internal forces, body stiffness, and fluid environment in a neuromechanical model of lamprey swimming.

Animal movements result from a complex balance of many different forces. Muscles produce force to move the body; the body has inertial, elastic, and damping properties that may aid or oppose the muscle force; and the environment produces reaction forces back on the body. The actual motion is an emergent property of these interactions. To examine the roles of body stiffness, muscle activation, and fluid environment for swimming animals, a computational model of a lamprey was developed. The model uses an immersed boundary framework that fully couples the Navier-Stokes equations of fluid dynamics with an actuated, elastic body model. This is the first model at a Reynolds number appropriate for a swimming fish that captures the complete fluid-structure interaction, in which the body deforms according to both internal muscular forces and external fluid forces. Results indicate that identical muscle activation patterns can produce different kinematics depending on body stiffness, and the optimal value of stiffness for maximum acceleration is different from that for maximum steady swimming speed. Additionally, negative muscle work, observed in many fishes, emerges at higher tail beat frequencies without sensory input and may contribute to energy efficiency. Swimming fishes that can tune their body stiffness by appropriately timed muscle contractions may therefore be able to optimize the passive dynamics of their bodies to maximize peak acceleration or swimming speed.

Viscoelastic fluid response can increase the speed and efficiency of a free swimmer.

Microorganisms navigate through complex environments such as biofilms and mucosal tissues and tracts. To understand the effect of a complex medium upon their locomotion, we investigate numerically the effect of fluid viscoelasticity on the dynamics of an undulating swimming sheet. First, we recover recent small-amplitude results for infinite sheets that suggest that viscoelasticity impedes locomotion. We find the opposite result when simulating free swimmers with large tail undulations, with both velocity and mechanical efficiency peaking for Deborah numbers near one. We associate this with regions of highly stressed fluid aft of the undulating tail.

Using Lagrangian coherent structures to analyze fluid mixing by cilia.

Motivated by the desire to understand the fluid flow within the airway surface liquid of the lung, we consider the flow generated by a computational model of a motile, internally actuated cilium. The cilium, along with a mucus layer modeled by linear elastic elements, is coupled to a viscous, incompressible fluid. The evolution of this coupled system is captured using an immersed boundary method. The Eulerian velocity field computed on a grid is used to compute finite-time Lyapunov exponent fields, whose maximal ridges identify Lagrangian coherent structures (LCSs). The computed LCS uncovers a barrier that separates a recirculation region of fluid that remains near the beating cilium from fluid that is advected downstream. Moreover, periodic stretching and folding of this region gives rise to complex mixing. Flow structures around a cilium propelling a mucus layer are compared to flow structures around a cilium with no mucus load.

A model of CatSper channel mediated calcium dynamics in mammalian spermatozoa.

CatSpers are calcium (Ca(2+)) 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 Ca(2+) entry through CatSper channels triggers a tail to head Ca(2+) propagation in mouse sperm, as well as a sustained increase of Ca(2+) 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 Ca(2+) with flux terms for the CatSper channels, a leak flux, and plasma membrane Ca(2+) clearance mechanism is studied. Results of this simple model exhibit tail to head Ca(2+) 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 Ca(2+) influx from the CatSper channels induce additional Ca(2+) release from an internal store. We test this hypothesis by examining the possible role of Ca(2+) release from the redundant nuclear envelope (RNE), an inositol 1,4,5-trisphosphate (IP(3)) gated Ca(2+) store in the neck. The simple model is extended to include an equation for IP(3) synthesis, degradation, and diffusion, as well as flux terms for Ca(2+) in the RNE. When IP(3) and the RNE are accounted for, the results of the model exhibit a tail to head Ca(2+) propagation as well as a sustained increase of Ca(2+) in the head.

EVALUATION OF INTERFACIAL FLUID DYNAMICAL STRESSES USING THE IMMERSED BOUNDARY METHOD.

The goal of this paper is to examine the evaluation of interfacial stresses using a standard, finite difference based, immersed boundary method (IMBM). This calculation is not trivial for two fundamental reasons. First, the immersed boundary is represented by a localized boundary force which is distributed to the underlying fluid grid by a discretized delta function. Second, this discretized delta function is used to impose a spatially averaged no-slip condition at the immersed boundary. These approximations can cause errors in interpolating stresses near the immersed boundary.To identify suitable methods for evaluating stresses, we investigate three model flow problems at very low Reynolds numbers. We compare the results of the immersed boundary calculations to those achieved by the boundary element method (BEM). The stress on an immersed boundary may be calculated either by direct evaluation of the fluid stress (FS) tensor or, for the stress jump, by direct evaluation of the locally distributed boundary force (wall stress or WS). Our first model problem is Poiseuille channel flow. Using an analytical solution of the immersed boundary formulation in this simple case, we demonstrate that FS calculations should be evaluated at a distance of approximately one grid spacing inward from the immersed boundary. For a curved immersed boundary we present a procedure for selecting representative interfacial fluid stresses using the concepts from the Poiseuille flow test problem. For the final two model problems, steady state flow over a bump in a channel and unsteady peristaltic pumping, we present an 'exclusion filtering' technique for accurately measuring stresses. Using this technique, these studies show that the immersed boundary method can provide reliable approximations to interfacial stresses.

An integrative computational model of multiciliary beating.

The coordinated beating of motile cilia is responsible for ovum transport in the oviduct, transport of mucus in the respiratory tract, and is the basis of motility in many single-celled organisms. The beating of a single motile cilium is achieved by the ATP-driven activation cycles of thousands of dynein molecular motors that cause neighboring microtubule doublets within the ciliary axoneme to slide relative to each other. The precise nature of the spatial and temporal coordination of these individual motors is still not completely understood. The emergent geometry and dynamics of ciliary beating is a consequence of the coupling of these internal force-generating motors, the passive elastic properties of the axonemal structure, and the external viscous, incompressible fluid. Here, we extend our integrative model of a single cilium that couples internal force generation with the surrounding fluid to the investigation of multiciliary interaction. This computational model allows us to predict the geometry of beating, along with the detailed description of the time-dependent flow field both near and away from the cilia. We show that synchrony and metachrony can, indeed, arise from hydrodynamic coupling. We also investigate the effects of viscosity and neighboring cilia on ciliary beat frequency. Moreover, since we have precise flow information, we also measure the dependence of the total flow pumped per cilium per beat upon parameters such as viscosity and ciliary spacing.