Simple models of the chemical field around swimming plankton

International audienceThe chemical field around swimming plankton depends on the swimming style and speed of the organism and the processes affecting uptake or exudation of chemicals by the organism. Here we present a simple model for the flow field around a neutrally buoyant self-propelled organism at low Reynolds number, and numerically calculate the chemical field around the organism. We show how the concentration field close to the organism and the mass transfer rates vary with swimming speed and style for Dirichlet (diffusion limited transport) boundary conditions. We calculate how the length of the chemical wake, defined as being the distance at which the chemical field drops to 10% of the surface concentration of the organism when stationary, varies with swimming speed and style for both Dirichlet and Neumann (production limited) boundary conditions. For Dirichlet boundary conditions, the length of the chemical wake increases with increasing swimming speed, and the self-propelled organism displays a significantly longer wake than the towed-body model. For the Neumann boundary conditions the converse is true; because swimming enhances the transport of the chemical away from the organism, the surface concentration of chemical is reduced and thus the wake length is reduced

Transport of helical gyrotactic swimmers in channels

We develop a mechanistic model that describes the transport of gyrotactic cells with propulsive force and propulsive torque that are not parallel. In sufficiently weak shear this yields helical swimming trajectories, whereas in stronger shear cells can attain a stable equilibrium orientation. We obtain the stable equilibrium solution for cell orientation as a function of the shear strength and determine the feasibility region for equilibrium solutions. We compute numerically the trajectories of cells in two dimensional vertical channel flow where the shear is non-uniform. Depending on the parameter values, we show that helical swimmers may display classical gyrotactic focussing towards the centre of the channel or can display a new phenomenon of focussing away from the centre of the channel. This result can be explained by consideration of the equilibrium solution for cell orientation. In this study we consider only dilute suspensions where there is no feedback from cell swimming on the hydrodynamics, and both cell-wall and cell-cell interactions are neglected

The trapping in high-shear regions of slender bacteria undergoing chemotaxis in a channel

Recently published experimental observations of slender bacteria swimming in channel flow demonstrate that the bacteria become trapped in regions of high shear, leading to reduced concentrations near the channel's centreline. However, the commonly-used, advection-diffusion equation, formulated in macroscopic space variables and originally derived for unbounded homogeneous shear flow, predicts that the bacteria concentration is uniform across the channel in the absence of chemotactic bias. In this paper, we instead use a Smoluchowski equation to describe the probability distribution of the bacteria, in macroscopic (physical) and microscopic (orientation) space variables. We demonstrate that the Smoluchowski equation is able to predict the trapping phenomena and compare the full numerical solution of the Smoluchowski equation with the experimental results when there is no chemotactic bias and also in the presence of a uniform cross-channel chemotactic gradient. Moreover, a simple analytic approximation for the equilibrium distribution provides an excellent approximate solution for slender bacteria, suggesting that the dominant effect on equilibrium behaviour is flow-induced modification of the bacteria's swimming direction. A continuum framework is thus provided to explain how the equilibrium distribution of slender chemotactic bacteria is altered in the presence of spatially varying shear flow. In particular we demonstrate that whilst advection is an appropriate description of transport due to the mean swimming velocity, the random reorientation mechanism of the bacteria cannot be simply modelled as diffusion in physical space

Biased swimming cells do not disperse in pipes as tracers: a population model based on microscale behaviour

There is much current interest in modelling suspensions of algae and other micro-organisms for biotechnological exploitation, and many bioreactors are of tubular design. Using generalized Taylor dispersion theory, we develop a population-level swimming-advection-diffusion model for suspensions of micro-organisms in a vertical pipe flow. In particular, a combination of gravitational and viscous torques acting on individual cells can affect their swimming behaviour, which is termed gyrotaxis. This typically leads to local cell drift and diffusion in a suspension of cells. In a flow in a pipe, small amounts of radial drift across streamlines can have a major impact on the effective axial drift and diffusion of the cells. We present a Galerkin method to calculate the local mean swimming velocity and diffusion tensor based on local shear for arbitrary flow rates. This method is validated with asymptotic results obtained in the limits of weak and strong shear. We solve the resultant swimming-advection-diffusion equation using numerical methods for the case of imposed Poiseuille flow and investigate how the flow modifies the dispersion of active swimmers from that of passive scalars. We establish that generalized Taylor dispersion theory predicts an enhancement of gyrotactic focussing in pipe flow with increasing shear strength, in contrast to earlier models. We also show that biased swimming cells may behave very differently to passive tracers, drifting axially at up to twice the rate and diffusing much less.Comment: 28 pages, 4 figure

Diffusion about the mean drift location in a biased random walk

Random walks are used to model movement in a wide variety of contexts: from the movement of cells undergoing chemotaxis to the migration of animals. In a two- dimensional biased random walk, the diffusion about the mean drift position is entirely dependent on the moments of the angular distribution used to determine the movement direction at each step. Here we consider biased random walks using several different angular distributions and derive expressions for the diffusion coefficients in each direction based on either a fixed or variable movement speed, and we use these to generate a probability density function for the long-time spatial distribution. we demonstrate how diffusion is typically anisotropic around the mean drift position and illustrate these theoretical results using computer simulations. we relate these results to earlier studies of swimming microorganisms and explain how the results can be generalized to other types of animal movement. © 2010 by the Ecological Society of America

Sedimentation of elongated non-motile prolate spheroids in homogenous isotropic turbulence

Phytoplankton are the foundation of aquatic food webs. Through photosynthesis, phytoplankton draw down CO2 at magnitudes equivalent to forests and other terrestrial plants and convert it to organic material that is then consumed by other organisms of phytoplankton in higher trophic levels. Mechanisms that affect local concentrations and velocities are of primary significance to many encounter-based processes in the plankton including prey-predator interactions, fertilization and aggregate formation. We report results from simulations of sinking phytoplankton, considered as elongated spheroids, in homogenous isotropic turbulence to answer the question of whether trajectories and velocities of sinking phytoplankton are altered by turbulence. We show in particular that settling spheroids with physical characteristics similar to those of diatoms weakly cluster and preferentially sample regions of down-welling flow, corresponding to an increase of the mean settling speed with respect to the mean settling speed in quiescent fluid. We explain how different parameters can affect the settling speed and what underlying mechanisms might be involved. Interestingly, we observe that the increase in the aspect ratio of the prolate spheroids can affect the clustering and the average settling speed of particles by two mechanisms: first is the effect of aspect ratio on the rotation rate of the particles, which saturates faster than the second mechanism of increasing drag anisotropy

Modeling the dynamics of hypoxia inducible factor-1α (HIF-1α) within single cells and 3D cell culture systems

HIF (hypoxia inducible factor) is an oxygen-regulated transcription factor that mediates the intracellular response to hypoxia in human cells. There is increasing evidence that cell signaling pathways encode temporal information, and thus cell fate may be determined by the dynamics of protein levels. We have developed a mathematical model to describe the transient dynamics of the HIF-1α protein measured in single cells subjected to hypoxic shock. The essential characteristics of these data are modeled with a system of differential equations describing the feedback inhibition between HIF-1α and prolyl hydroxylases (PHD) oxygen sensors. Heterogeneity in the single-cell data is accounted through parameter variation in the model. We previously identified the PHD2 isoform as the main PHD sensor responsible for controlling the HIF-1α transient response, and make here testable predictions regarding HIF-1α dynamics subject to repetitive hypoxic pulses. The model is further developed to describe the dynamics of HIF-1α in cells cultured as 3D spheroids, with oxygen dynamics parameterized using experimental measurements of oxygen within spheroids. We show that the dynamics of HIF-1α and transcriptional targets of HIF-1α display a non-monotone response to the oxygen dynamics. Specifically we demonstrate that the dynamic transient behaviour of HIF-1α results in differential dynamics in transcriptional targets

Gyrotactic swimmer dispersion in pipe flow: testing the theory

Suspensions of microswimmers are a rich source of fascinating new fluid mechanics. Recently we predicted the active pipe flow dispersion of gyrotactic microalgae, whose orientation is biased by gravity and flow shear. Analytical theory predicts that these active swimmers disperse in a markedly distinct manner from passive tracers (Taylor dispersion). Dispersing swimmers display non-zero drift and effective diffusivity that is non-monotonic with Péclet number. Such predictions agree with numerical simulations, but hitherto have not been tested experimentally. Here, to facilitate comparison, we obtain new solutions of the axial dispersion theory accounting both for swimmer negative buoyancy and a local nonlinear response of swimmers to shear, provided by two alternative microscopic stochastic descriptions. We obtain new predictions for suspensions of the model swimming alga Dunaliella salina, whose motility and buoyant mass we parametrise using tracking video microscopy. We then present a new experimental method to measure gyrotactic dispersion using fluorescently stained D. salina and provide a preliminary comparison with predictions of a non-zero drift above the mean flow for each microscopic stochastic description. Finally, we propose further experiments for a full experimental characterisation of gyrotactic dispersion measures and discuss the implications of our results for algal dispersion in industrial photobioreactors