A Computational Model of the Fluid Dynamics of Undulatory and Flagellar Swimming

We present a mathematical model and numerical method designedto study the fluid dynamics of swimming organisms. The fullNavier— Stokes equations are solved in a domain of fluidwithin which an organism undergoing time—dependent motionsis immersed. Of interest are both the dynamics of a single organismand the relationship of its morphology to its motility properties,as well as the collective hydrodynamic interactions of groupsof swimmers with each other and their environment. Biologicalapplications include spermatozoa motility in the reproductivetract, swimming of non-smooth filaments, and collective swimmingof algal cells.     

Swimming in circles: motion of bacteria near solid boundaries

Near a solid boundary, Escherichia coli swims in clockwise circular motion. We provide a hydrodynamic model for this behavior. We show that circular trajectories are natural consequences of force-free and torque-free swimming and the hydrodynamic interactions with the boundary, which also leads to a hydrodynamic trapping of the cells close to the surface. We compare the results of the model with experimental data and obtain reasonable agreement. In particular, the radius of curvature of the trajectory is observed to increase with the length of the bacterium body.     

In ovo time-lapse analysis after dorsal neural tube ablation shows rerouting of chick hindbrain neural crest

Previous analyses of single neural crest cell trajectories have suggested important roles for interactions between neural crest cells and the environment, and amongst neural crest cells. To test the relative contribution of intrinsic versus extrinsic information in guiding cells to their appropriate sites, we ablated subpopulations of premigratory chick hindbrain neural crest and followed the remaining neural crest cells over time using a new in ovo imaging technique. Neural crest cell migratory behaviors are dramatically different in ablated compared with unoperated embryos. Deviations from normal migration appear either shortly after cells emerge from the neural tube or en route to the branchial arches, areas where cell-cell interactions typically occur between neural crest cells in normal embryos. Unlike the persistent, directed trajectories in normal embryos, neural crest cells frequently change direction and move somewhat chaotically after ablation. In addition, the migration of neural crest cells in collective chains, commonly observed in normal embryos, was severely disrupted. Hindbrain neural crest cells have the capacity to reroute their migratory pathways and thus compensate for missing neural crest cells after ablation of neighboring populations. Because the alterations in neural crest cell migration are most dramatic in regions that would normally foster cell-cell interactions, the trajectories reported here argue that cell-cell interactions have a key role in the shaping of the neural crest migration.     

Estimating 3D Movements from 2D Observations Using a Continuous Model of Helical Swimming

Helical swimming is among the most common movement behaviors in a wide range of microorganisms, and these movements have direct impacts on distributions, aggregations, encounter rates with prey, and many other fundamental ecological processes. Microscopy and video technology enable the automated acquisition of large amounts of tracking data; however, these data are typically two-dimensional. The difficulty of quantifying the third movement component complicates understanding of the biomechanical causes and ecological consequences of helical swimming. We present a versatile continuous stochastic model—the correlated velocity helical movement (CVHM) model—that characterizes helical swimming with intrinsic randomness and autocorrelation. The model separates an organism’s instantaneous velocity into a slowly varying advective component and a perpendicularly oriented rotation, with velocities, magnitude of stochasticity, and autocorrelation scales defined for both components. All but one of the parameters of the 3D model can be estimated directly from a two-dimensional projection of helical movement with no numerical fitting, making it computationally very efficient. As a case study, we estimate swimming parameters from videotaped trajectories of a toxic unicellular alga, Heterosigma akashiwo (Raphidophyceae). The algae were reared from five strains originally collected from locations in the Atlantic and Pacific Oceans, where they have caused Harmful Algal Blooms (HABs). We use the CVHM model to quantify cell-level and strain-level differences in all movement parameters, demonstrating the utility of the model for identifying strains that are difficult to distinguish by other means.     

Collective Bacterial Dynamics Revealed Using a Three-Dimensional Population-Scale Defocused Particle Tracking Technique

An ability to monitor bacterial locomotion and collective dynamics is crucial to our understanding of a number of well-characterized phenotypes including biofilm formation, chemotaxis, and virulence. Here, we report the tracking of multiple swimming Escherichia coli cells in three spatial dimensions and at single-cell resolution using a novel three-dimensional (3D) defocused particle tracking (DPT) method. The 3D trajectories were generated for wild-type Escherichia coli strain RP437 as well as for isogenic derivatives that display smooth swimming due to a cheA deletion (strain RP9535) or incessant tumbling behavior due to a cheZ deletion (strain RP1616). The 3D DPT method successfully differentiated these three modes of locomotion and allowed direct calculation of the diffusion coefficient for each strain. As expected, we found that the smooth swimmer diffused more readily than the wild type, and both the smooth swimmer and the wild-type cells exhibited diffusion coefficients that were at least two orders of magnitude larger than that of the tumbler. Finally, we found that the diffusion coefficient increased with increasing cell density, a phenomenon that can be attributed to the hydrodynamic disturbances caused by neighboring bacteria.     

Stoichiometry in producer-grazer systems: Linking energy flow with element cycling

All organisms are composed of multiple chemical elements such as carbon, nitrogen and phosphorus. While energy flow and element cycling are two fundamental and unifying principles in ecosystem theory, population models usually ignore the latter. Such models implicitly assume chemical homogeneity of all trophic levels by concentrating on a single constituent, generally an equivalent of energy. In this paper, we examine ramifications of an explicit assumption that both producer and grazer are composed of two essential elements: carbon and phosphorous. Using stoichiometric principles, we construct a two-dimensional Lotka-Volterra type model that incorporates chemical heterogeneity of the first two trophic levels of a food chain. The analysis shows that indirect competition between two populations for phosphorus can shift predator—prey interactions from a (+, −) type to an unusual (−, −) class. This leads to complex dynamics with multiple positive equilibria, where bistability and deterministic extinction of the grazer are possible. We derive simple graphical tests for the local stability of all equilibria and show that system dynamics are confined to a bounded region. Numerical simulations supported by qualitative analysis reveal that Rosenzweig’s paradox of enrichment holds only in the part of the phase plane where the grazer is energy limited; a new phenomenon, the paradox of energy enrichment, arises in the other part, where the grazer is phosphorus limited. A bifurcation diagram shows that energy enrichment of producer—grazer systems differs radically from nutrient enrichment. Hence, expressing producer—grazer interactions in stoichiometrically realistic terms reveals qualitatively new dynamical behavior.     

Collective bacterial dynamics revealed using a three-dimensional population-scale defocused particle tracking technique

An ability to monitor bacterial locomotion and collective dynamics is crucial to our understanding of a number of well-characterized phenotypes including biofilm formation, chemotaxis, and virulence. Here, we report the tracking of multiple swimming Escherichia coli cells in three spatial dimensions and at single-cell resolution using a novel three-dimensional (3D) defocused particle tracking (DPT) method. The 3D trajectories were generated for wild-type Escherichia coli strain RP437 as well as for isogenic derivatives that display smooth swimming due to a cheA deletion (strain RP9535) or incessant tumbling behavior due to a cheZ deletion (strain RP1616). The 3D DPT method successfully differentiated these three modes of locomotion and allowed direct calculation of the diffusion coefficient for each strain. As expected, we found that the smooth swimmer diffused more readily than the wild type, and both the smooth swimmer and the wild-type cells exhibited diffusion coefficients that were at least two orders of magnitude larger than that of the tumbler. Finally, we found that the diffusion coefficient increased with increasing cell density, a phenomenon that can be attributed to the hydrodynamic disturbances caused by neighboring bacteria.     

Hydrodynamic interactions between two swimming bacteria

This article evaluates the hydrodynamic interactions between two swimming bacteria precisely. We assume that each bacterium is force free and torque free, with a Stokes flow field around it. The geometry of each bacterium is modeled as a spherical or spheroidal body with a single helical flagellum. The movements of two interacting bacteria in an infinite fluid otherwise at rest are computed using a boundary element method, and the trajectories of the two interacting bacteria and the stresslet are investigated. The results show that as the two bacteria approach each other, they change their orientations considerably in the near field. The bacteria always avoided each other; no stable pairwise swimming motion was observed in this study. The effects of the hydrodynamic interactions between two bacteria on the rheology and diffusivity of a semidilute bacterial suspension are discussed.     

Integrating behavioral and neural data in a model of zebrafish network interaction

The spinal neural networks of larval zebrafish (Danio rerio) generate a variety of movements such as escape, struggling, and swimming. Various mechanisms at the neural and network levels have been proposed to account for switches between these behaviors. However, there are currently no detailed demonstrations of such mechanisms. This makes determining which mechanisms are plausible extremely difficult. In this paper, we propose a detailed biologically plausible model of the interactions between the swimming and escape networks in the larval zebrafish, while taking into account anatomical and physiological evidence. We show that the results of our neural model generate the expected behavior when used to control a hydrodynamic model of carangiform locomotion. As a result, the model presented here is a clear demonstration of a plausible mechanism by which these distinct behaviors can be controlled. Interestingly, the networks are anatomically overlapping, despite clear differences in behavioral function and physiology.     

Hydrodynamic interaction of two unsteady model microorganisms

The study of pair-wise interactions between swimming microorganisms is fundamental to the understanding of the rheological and transport properties of semi-dilute suspensions. In this paper, the hydrodynamic interaction of two ciliated microorganisms is investigated numerically using a boundary-element method, and the microorganisms are modeled as spherical squirmers that swim by time-dependent surface deformations. The results show that the inclusion of the unsteady terms in the ciliary propulsion model has a large impact on the trajectories of the interacting cells, and causes a significant change in scattering angles with potential important consequences on the diffusion properties of semi-dilute suspensions. Furthermore, the analysis of the shear stress acting on the surface of the microorganisms revealed that the duration and the intensity of the near-field interaction are significantly modified by the presence of unsteadiness. This observation may account for the hydrodynamic nature of randomness in some biological reactions, and supersedes the distinction between intrinsic randomness and hydrodynamic interactions, adding a further element to the understanding and modeling of interacting microorganisms.     

Effects of osmolality, morphology perturbations and intracellular nucleotide content during the movement of sea bass (Dicentrarchus labrax) spermatozoa.

Sea bass spermatozoa are maintained immotile in the seminal fluid, but initiate swimming for 45 s at 20 degrees C, immediately after dispersion in a hyperosmotic medium (1100 mOsm kg-1). The duration of this motile period could be extended by a reduction of the amplitude of the hyperosmotic shock. Five seconds after the initiation of motility, 94.4 +/- 1.8% of spermatozoa were motile with a swimming velocity of 141.8 +/- 1.2 microns s-1, a flagellar beat frequency of 60 Hz and a symmetric type of flagellar swimming, resulting in linear tracks. Velocity, flagellar beat frequency, percentage of motile cells and trajectory diameter decreased concomitantly throughout the swimming phase. After 30 s of motility, the flagellar beat became asymmetric, leading to circular trajectories. Ca2+ modulated the swimming pattern of demembranated spermatozoa, suggesting that the asymmetric waves produced by intact spermatozoa after 30 s of motility were induced by an accumulation of intracellular Ca2+. Moreover, increased ionic strength in the reactivation medium induced a dampening of waves in the distal portion of the flagellum and, at high values, resulted in an arrest of wave generation in demembranated spermatozoa. In non-demembranated cells, the intracellular ATP concentration fell immediately after transfer to sea water. In contrast, the AMP content increased during the same period, while the ADP content increased slightly. In addition, several morphological changes affected the mitochondria, chromatin and midpiece. These results indicate that the short swimming period of sea bass spermatozoa is controlled by energetic and cytoplasmic ionic conditions and that it is limited by osmotic stress, which induces marked changes in cell morphology.     

Some views on the role of noise in “self”-organizing systems

This paper deals with information transfer from the environment and “self”-organization in open, nonlinear systems far from thermodynamic equilibrium — in the presence of either non-stationary phase jitter noise, or amplitude stationary noise. By “self”-organization we mean here the progressive formation within the system of sequential, ordered (coherent) relationships between appropriate dynamical variables-like for example, the phase differences between the oscillating components of the system. We take up (in Section II) the classical Laser as a specific example and examine in detail the influence of phase jitter noise in the mode (phase) locking process. We find—as expected—that phase fluctuations in the cavity cause degradation of the coherent behaviour (i.e. increase the entropy) of the system — which, however, levels off, or saturates with time. Further (in Section III) we examine systems where the number of self-sustained oscillating components may vary with time in such a way that the maximum entropy of the system increases faster than the overall instantaneous entropy. We put forth the hypothesis that in such cases — because of the increase of the redundancy — the system gets organized not just in spite of, but merely because of the presence of Noise. Possible applications in biological systems (especially concerning a model of cerebral organization) are briefly discussed. It is understood here, that the system has to display some preliminary dynamical structure before the organizing procedure takes over. What happens afterwards is the subject of this paper.     

Generalized Voronoi Tessellation as a Model of Two-dimensional Cell Tissue Dynamics

Voronoi tessellations have been used to model the geometric arrangement of cells in morphogenetic or cancerous tissues, however, so far only with flat hyper-surfaces as cell-cell contact borders. In order to reproduce the experimentally observed piecewise spherical boundary shapes, we develop a consistent theoretical framework of multiplicatively weighted distance functions, defining generalized finite Voronoi neighborhoods around cell bodies of varying radius, which serve as heterogeneous generators of the resulting model tissue. The interactions between cells are represented by adhesive and repelling force densities on the cell contact borders. In addition, protrusive locomotion forces are implemented along the cell boundaries at the tissue margin, and stochastic perturbations allow for non-deterministic motility effects. Simulations of the emerging system of stochastic differential equations for position and velocity of cell centers show the feasibility of this Voronoi method generating realistic cell shapes. In the limiting case of a single cell pair in brief contact, the dynamical nonlinear Ornstein–Uhlenbeck process is analytically investigated. In general, topologically distinct tissue conformations are observed, exhibiting stability on different time scales, and tissue coherence is quantified by suitable characteristics. Finally, an argument is derived pointing to a tradeoff in natural tissues between cell size heterogeneity and the extension of cellular lamellae.     

Neighbor-enhanced diffusivity in dense,cohesive cell populations

The dispersal or mixing of cells within cellular tissue is a crucial property for diverse biological processes, ranging from morphogenesis, immune action, to tumor metastasis. With the phenomenon of ‘contact inhibition of locomotion,’ it is puzzling how cells achieve such processes within a densely packed cohesive population. Here we demonstrate that a proper degree of cell-cell adhesiveness can, intriguingly, enhance the super-diffusive nature of individual cells. We systematically characterize the migration trajectories of crawling MDA-MB-231 cell lines, while they are in several different clustering modes, including freely crawling singles, cohesive doublets of two cells, quadruplets, and confluent population on two-dimensional substrate. Following data analysis and computer simulation of a simple cellular Potts model, which faithfully recapitulated all key experimental observations such as enhanced diffusivity as well as periodic rotation of cell-doublets and cell-quadruplets with mixing events, we found that proper combination of active self-propelling force and cell-cell adhesion is sufficient for generating the observed phenomena. Additionally, we found that tuning parameters for these two factors covers a variety of different collective dynamic states.     

Is the hook of muroid rodent's sperm related to sperm train formation?

Competition between spermatozoa of rival males to gain fertilizations has led to a wide array of modifications in sperm structure and function. Sperm cells of most muroid rodents have hook‐shaped extensions in the apical–ventral tip of the head, but the function of this structure is largely unknown. These ‘hooks’ may facilitate aggregation of spermatozoa in so‐called ‘trains’, as an adaptation to sperm competition, because sperm in trains may swim faster than free‐swimming cells. However, there is controversy regarding the role of the hook in train formation, and in relation to whether it is selected by sperm competition. We examined spermatozoa from muroid rodents with varying levels of sperm competition to assess whether (i) sperm aggregates are common in these taxa, (ii) presence of a hook relates to the formation of sperm aggregations, and (iii) formation of sperm aggregations is explained by sperm competition. Our analyses in 25 muroid species revealed that > 92% of spermatozoa swim individually in all species, with the exception of the wood mouse, Apodemus sylvaticus, which has ~50% spermatozoa swimming freely. Species with hooked spermatozoa had higher sperm competition levels and longer sperm than species whose sperm lack a hook. Neither the presence of hook nor sperm competition levels were related to the percentage of sperm in aggregations. Thus, (i) sperm aggregates in muroid rodents are an exceptional trait found only in a few species, (ii) evolution of the sperm hook is associated to sperm competition levels, but (iii) the hook is unlikely to be related to the formation of sperm aggregates. The evolutionary significance of the sperm head hook thus remains elusive, and future studies should examine potential roles of this pervasive structure in sperm's hydrodynamic efficiency and sperm–female tract interactions.     

Microscopic diffusion and hydrodynamic interactions of hemoglobin in red blood cells

The cytoplasm of red blood cells is congested with the oxygen storage protein hemoglobin occupying a quarter of the cell volume. The high protein concentration leads to a reduced mobility; the self-diffusion coefficient of hemoglobin in blood cells is six times lower than in dilute solution. This effect is generally assigned to excluded volume effects in crowded media. However, the collective or gradient diffusion coefficient of hemoglobin is only weakly dependent on concentration, suggesting the compensation of osmotic and friction forces. This would exclude hydrodynamic interactions, which are of dynamic origin and do not contribute to the osmotic pressure. Hydrodynamic coupling between protein molecules is dominant at short time- and length scales before direct interactions are fully established. Employing neutron spin-echo-spectroscopy, we study hemoglobin diffusion on a nanosecond timescale and protein displacements on the scale of a few nanometers. A time- and wave-vector dependent diffusion coefficient is found, suggesting the crossover of self- and collective diffusion. Moreover, a wave-vector dependent friction function is derived, which is a characteristic feature of hydrodynamic interactions. The wave-vector and concentration dependence of the long-time self-diffusion coefficient of hemoglobin agree qualitatively with theoretical results on hydrodynamics in hard spheres suspensions. Quantitative agreement requires us to adjust the volume fraction by including part of the hydration shell: Proteins exhibit a larger surface/volume ratio compared to standard colloids of much larger size. It is concluded that hydrodynamic and not direct interactions dominate long-range molecular transport at high concentration.     

Conditions Under Which Nitrogen Can Limit Steady-State Net Primary Production in a General Class of Ecosystem Models

Human activity is drastically altering global nitrogen (N) availability. The extent to which ecosystems absorb additional N—and with it, additional CO₂—depends on whether net primary production (NPP) is N-limited, so it is important to understand conditions under which N can limit NPP. Here I use a general dynamical model to show that N limitation at steady-state—such as in old-growth forests—depends on the balance of biotically controllable versus uncontrollable N inputs and losses. Steady-state N limitation is only possible when uncontrollable inputs (for example, atmospheric deposition) exceed controllable losses (for example, leaching of plant-available soil N), which is the same as when uncontrollable losses (for example, leaching of plant-unavailable soil N) exceed controllable inputs (biological N fixation). These basic results are robust to many model details, such as the number of plant-unavailable soil N pools and the number and type of N fixers. Empirical data from old-growth tropical (Hawai’i) and temperate (Oregon, Washington, Chile) forests support the model insights. Practically, this means that any N fixer—symbiotic or not—could overcome ecosystem N limitation, so understanding N limitation requires understanding controls on all N fixers. Further, comparing losses of plant-available N to abiotic inputs could offer a rapid diagnosis of whether ecosystems can be N-limited, although the applicability of this result is constrained to ecosystems with a steady-state N cycle such as old-growth forests largely devoid of disturbance.     

Study of local hydrodynamic environment in cell-substrate adhesion using side-view μPIV technology

Tumor cell adhesion to the endothelium under shear flow conditions is a critical step that results in circulation-mediated tumor metastasis. This study presents experimental and computational techniques for studying the local hydrodynamic environment around adherent cells and how local shear conditions affect cell-cell interactions on the endothelium in tumor cell adhesion. To study the local hydrodynamic profile around heterotypic adherent cells, a side-view flow chamber assay coupled with micro particle imaging velocimetry (μPIV) technique was developed, where interactions between leukocytes and tumor cells in the near-endothelial wall region and the local shear flow environment were characterized. Computational fluid dynamics (CFD) simulations were also used to obtain quantitative flow properties around those adherent cells. Results showed that cell dimension and relative cell-cell positions had strong influence on local shear rates. The velocity profile above leukocytes and tumor cells displayed very different patterns. Larger cell deformations led to less disturbance to the flow. Local shear rates above smaller cells were observed to be more affected by relative positions between two cells.     

Shear modulation of intercellular contact area between two deformable cells colliding under flow

Shear rate has been shown to critically affect the kinetics and receptor specificity of cell-cell interactions. In this study, the collision process between two modeled cells interacting in a linear shear flow is numerically investigated. The two identical biological or artificial cells are modeled as deformable capsules composed of an elastic membrane. The cell deformation and trajectories are computed using the immersed boundary method (IBM) for shear rates of 100-400s(-1). As the two cells collide under hydrodynamic shear, large local cell deformations develop. The effective contact area between the two cells is modulated by the shear rate, and reaches a maximum value at intermediate levels of shear. At relatively low shear rate, the contact area is an enclosed region. As the shear rate increases, dimples form on the membrane surface, and the contact region becomes annular. The nonmonotonic increase of the contact area with the increase of shear rate from computational results implies that there is a maximum effective receptor-ligand binding area for cell adhesion. This finding suggests the existence of possible hydrodynamic mechanism that could be used to interpret the observed maximum leukocyte aggregation in shear flow. The critical shear rate for maximum intercellular contact area is shown to vary with cell properties such as radius and membrane elastic modulus.