Eamonn A. Gaffney

Human sperm accumulation near surfaces: a simulation study

A hybrid boundary integral/slender body algorithm for modelling flagellar cell motility is presented. The algorithm uses the boundary element method to represent the ‘wedge-shaped’ head of the human sperm cell and a slender body theory representation of the flagellum. The head morphology is specified carefully due to its significant effect on the force and torque balance and hence movement of the free-swimming cell. The technique is used to investigate the mechanisms for the accumulation of human spermatozoa near surfaces. Sperm swimming in an infinite fluid, and near a plane boundary, with prescribed planar and three-dimensional flagellar waveforms are simulated. Both planar and ‘elliptical helicoid’ beating cells are predicted to accumulate at distances of approximately 8.5–22 μm from surfaces, for flagellar beating with angular wavenumber of 3π to 4π. Planar beating cells with wavenumber of approximately 2.4π or greater are predicted to accumulate at a finite distance, while cells with wavenumber of approximately 2π or less are predicted to escape from the surface, likely due to the breakdown of the stable swimming configuration. In the stable swimming trajectory the cell has a small angle of inclination away from the surface, no greater than approximately 0.5°. The trapping effect need not depend on specialized non-planar components of the flagellar beat but rather is a consequence of force and torque balance and the physical effect of the image systems in a no-slip plane boundary. The effect is relatively weak, so that a cell initially one body length from the surface and inclined at an angle of 4°–6° towards the surface will not be trapped but will rather be deflected from the surface. Cells performing rolling motility, where the flagellum sweeps out a ‘conical envelope’, are predicted to align with the surface provided that they approach with sufficiently steep angle. However simulation of cells swimming against a surface in such a configuration is not possible in the present framework. Simulated human sperm cells performing a planar beat with inclination between the beat plane and the plane-of-flattening of the head were not predicted to glide along surfaces, as has been observed in mouse sperm. Instead, cells initially with the head approximately 1.5–3 μm from the surface were predicted to turn away and escape. The simulation model was also used to examine rolling motility due to elliptical helicoid flagellar beating. The head was found to rotate by approximately 240° over one beat cycle and due to the time-varying torques associated with the flagellar beat was found to exhibit ‘looping’ as has been observed in cells swimming against coverslips.

Predicted Phenotypes of Dry Eye: Proposed Consequences of Its Natural History

This paper reviews current knowledge of the pathophysiology of dry eye and predicts that the clinical picture in late disease differs in both severity and quality from that in early disease. It is hypothesized that hybrid forms evolve, in which aqueous-deficient dry eye (ADDE) takes on features of evaporative dry eye (EDE) and vice versa. As a consequence, early and late forms may require different diagnostic criteria and respond to different therapeutic regimes. Tear hyperosmolarity plays a key role in the damage mechanism of dry eye, and ADDE is recognized to be a low-volume, hyperosmolar state. As ADDE advances, a progressive decrease in lacrimal secretion occurs, exacerbated by loss of the corneal reflex. This causes a decrease in tear volume, thinning of the aqueous tear film, and retarded spreading of the tear film lipid layer. The latter is hypothesized to cause an increase in evaporative water loss and an added evaporative component to the dry eye. Thus, in advanced disease, the hybrid state would be an organic ADDE, accompanied by a functional EDE in the absence of meibomian gland dysfunction. This functional EDE would respond to agents that expand the tear volume, restore corneal sensitivity, or provide an artificial tear film lipid layer.

Bend Propagation in the Flagella of Migrating Human Sperm, and Its Modulation by Viscosity

A pre-requisite for sexual reproduction is successful unification of the male and female gametes; in externally-fertilising echinoderms the male gamete is brought into close proximity to the female gamete through chemotaxis, the associated signalling and flagellar beat changes being elegantly characterised in several species. In the human, sperm traverse a relatively high-viscosity mucus coating the tract surfaces, there being a tantalising possible role for chemotaxis. To understand human sperm migration and guidance, studies must therefore employ similar viscous in vitro environments. High frame rate digital imaging is used for the first time to characterise the flagellar movement of migrating sperm in low and high viscosities. While qualitative features have been reported previously, we show in precise spatial and temporal detail waveform evolution along the flagellum. In low viscosity the flagellum continuously moves out of the focal plane, compromising the measurement of true curvature, nonetheless the presence of torsion can be inferred. In high viscosities curvature can be accurately determined and we show how waves propagate at approximately constant speed. Progressing waves increase in curvature approximately linearly except for a sharper increase over a distance approximately 20-27 microm from the head/midpiece junction. Curvature modulation, likely influenced by the outer dense fibres, creates the characteristic waveforms of high viscosity swimming, with remarkably effective cell progression against greatly increased resistance, even in high viscosity liquids. Assessment of motility in physiological viscosities will be essential in future basic research, studies of chemotaxis and novel diagnostics.

A mass and solute balance model for tear volume and osmolarity in the normal and the dry eye

Tear hyperosmolarity is thought to play a key role in the mechanism of dry eye, a common symptomatic condition accompanied by visual disturbance, tear film instability, inflammation and damage to the ocular surface. We have constructed a model for the mass and solute balance of the tears, with parameter estimation based on extensive data from the literature which permits the influence of tear evaporation, lacrimal flux and blink rate on tear osmolarity to be explored. In particular the nature of compensatory events has been estimated in aqueous-deficient (ADDE) and evaporative (EDE) dry eye. The model reproduces observed osmolarities of the tear meniscus for the healthy eye and predicts a higher concentration in the tear film than meniscus in normal and dry eye states. The differential is small in the normal eye, but is significantly increased in dry eye, especially for the simultaneous presence of high meniscus concentration and low meniscus radius. This may influence the interpretation of osmolarity values obtained from meniscus samples since they need not fully reflect potential damage to the ocular surface caused by tear film hyperosmolarity. Interrogation of the model suggests that increases in blink rate may play a limited role in compensating for a rise in tear osmolarity in ADDE but that an increase in lacrimal flux, together with an increase in blink rate, may delay the development of hyperosmolarity in EDE. Nonetheless, it is predicted that tear osmolarity may rise to much higher levels in EDE than ADDE before the onset of tear film breakup, in the absence of events at the ocular surface which would independently compromise tear film stability. Differences in the predicted responses of the pre-ocular tears in ADDE compared to EDE or hybrid disease to defined conditions suggest that no single, empirically-accessible variable can act as a surrogate for tear film concentration and the potential for ocular surface damage. This emphasises the need to measure and integrate multiple diagnostic indicators to determine outcomes and prognosis. Modelling predictions in addition show that further studies concerning the possibility of a high lacrimal flux phenotype in EDE are likely to be profitable.

Modelling bacterial behaviour close to a no-slip plane boundary: the influence of bacterial geometry

We describe a boundary-element method used to model the hydrodynamics of a bacterium propelled by a single helical flagellum. Using this model, we optimize the power efficiency of swimming with respect to cell body and flagellum geometrical parameters, and find that optima for swimming in unbounded fluid and near a no-slip plane boundary are nearly indistinguishable. We also consider the novel optimization objective of torque efficiency and find a very different optimal shape. Excluding effects such as Brownian motion and electrostatic interactions, it is demonstrated that hydrodynamic forces may trap the bacterium in a stable, circular orbit near the boundary, leading to the empirically observable surface accumulation of bacteria. Furthermore, the details and even the existence of this stable orbit depend on geometrical parameters of the bacterium, as described in this article. These results shed some light on the phenomenon of surface accumulation of micro-organisms and offer hydrodynamic explanations as to why some bacteria may accumulate more readily than others based on morphology.

Fluid mechanics of nodal flow due to embryonic primary cilia

Breaking of left–right symmetry is crucial in vertebrate development. The role of cilia-driven flow has been the subject of many recent publications, but the underlying mechanisms remain controversial. At approximately 8 days post-fertilization, after the establishment of the dorsal–ventral and anterior–posterior axes, a depressed structure is found on the ventral side of mouse embryos, termed the ventral node. Within the node, ‘whirling’ primary cilia, tilted towards the posterior, drive a flow implicated in the initial left–right signalling asymmetry. However, the underlying fluid mechanics have not been fully and correctly explained until recently and accurate characterization is required in determining how the flow triggers the downstream signalling cascades. Using the approximation of resistive force theory, we show how the flow is produced and calculate the optimal configuration to cause maximum flow, showing excellent agreement with in vitro measurements and numerical simulation, and paralleling recent analogue experiments. By calculating numerical solutions of the slender body theory equations, we present time-dependent physically based fluid dynamics simulations of particle pathlines in flows generated by large arrays of beating cilia, showing the far-field radial streamlines predicted by the theory.

Turing's model for biological pattern formation and the robustness problem

One of the fundamental questions in developmental biology is how the vast range of pattern and structure we observe in nature emerges from an almost uniformly homogeneous fertilized egg. In particular, the mechanisms by which biological systems maintain robustness, despite being subject to numerous sources of noise, are shrouded in mystery. Postulating plausible theoretical models of biological heterogeneity is not only difficult, but it is also further complicated by the problem of generating robustness, i.e. once we can generate a pattern, how do we ensure that this pattern is consistently reproducible in the face of perturbations to the domain, reaction time scale, boundary conditions and so forth. In this paper, not only do we review the basic properties of Turings theory, we highlight the successes and pitfalls of using it as a model for biological systems, and discuss emerging developments in the area.

Stability analysis of non-autonomous reaction-diffusion systems: The effects of growing domains

By using asymptotic theory, we generalise the Turing diffusively-driven instability conditions for reaction-diffusion systems with slow, isotropic domain growth. There are two fundamental biological differences between the Turing conditions on fixed and growing domains, namely: (i) we need not enforce cross nor pure kinetic conditions and (ii) the restriction to activator-inhibitor kinetics to induce pattern formation on a growing biological system is no longer a requirement. Our theoretical findings are confirmed and reinforced by numerical simulations for the special cases of isotropic linear, exponential and logistic growth profiles. In particular we illustrate an example of a reaction-diffusion system which cannot exhibit a diffusively-driven instability on a fixed domain but is unstable in the presence of slow growth.

Age‐Related Changes in Speed and Mechanism of Adult Skeletal Muscle Stem Cell Migration

Skeletal muscle undergoes a progressive age‐related loss in mass and function. Preservation of muscle mass depends in part on satellite cells, the resident stem cells of skeletal muscle. Reduced satellite cell function may contribute to the age‐associated decrease in muscle mass. Here, we focused on characterizing the effect of age on satellite cell migration. We report that aged satellite cells migrate at less than half the speed of young cells. In addition, aged cells show abnormal membrane extension and retraction characteristics required for amoeboid‐based cell migration. Aged satellite cells displayed low levels of integrin expression. By deploying a mathematical model approach to investigate mechanism of migration, we have found that young satellite cells move in a random “memoryless” manner, whereas old cells demonstrate superdiffusive tendencies. Most importantly, we show that nitric oxide, a key regulator of cell migration, reversed the loss in migration speed and reinstated the unbiased mechanism of movement in aged satellite cells. Finally, we found that although hepatocyte growth factor increased the rate of aged satellite cell movement, it did not restore the memoryless migration characteristics displayed in young cells. Our study shows that satellite cell migration, a key component of skeletal muscle regeneration, is compromised during aging. However, we propose clinically approved drugs could be used to overcome these detrimental changes. STEM CELLS2012;30:1182–1195

Partial differential equations for self-organization in cellular and developmental biology

Understanding the mechanisms governing and regulating the emergence of structure and heterogeneity within cellular systems, such as the developing embryo, represents a multiscale challenge typifying current integrative biology research, namely, explaining the macroscale behaviour of a system from microscale dynamics. This review will focus upon modelling how cell-based dynamics orchestrate the emergence of higher level structure. After surveying representative biological examples and the models used to describe them, we will assess how developments at the scale of molecular biology have impacted on current theoretical frameworks, and the new modelling opportunities that are emerging as a result. We shall restrict our survey of mathematical approaches to partial differential equations and the tools required for their analysis. We will discuss the gap between the modelling abstraction and biological reality, the challenges this presents and highlight some open problems in the field.