The Orientation of Swimming Biflagellates in Shear Flows

The three-dimensional coarse-graining formulation of interacting elastohydrodynamic filaments and multi-body microhydrodynamics

Elastic filaments are vital to biological, physical and engineering systems, from cilia driving fluid in the lungs to artificial swimmers and micro-robotics. Simulating slender structures requires intricate balance of elastic, body, active and hydrodynamic moments, all in three dimensions. Here, we present a generalized three-dimensional (3D) coarse-graining formulation that is efficient, simple-to-implement, readily extendable and usable for a wide array of applications. Our method allows for simulation of collections of 3D elastic filaments, capable of full flexural and torsional deformations, coupled non-locally via hydrodynamic interactions, and including multi-body microhydrodynamics of structures with arbitrary geometry. The method exploits the exponential mapping of quaternions for tracking 3D rotations of each interacting element in the system, allowing for computation times up to 150 times faster than a direct quaternion implementation. Spheres are used as a 'building block' of both filaments and solid microstructures for straightforward and intuitive construction of arbitrary three-dimensional geometries present in the environment. We highlight the strengths of the method in a series of non-trivial applications including bi-flagellated swimming, sperm-egg scattering and particle transport by cilia arrays. Applications to lab-on-a-chip devices, multi-filaments, mono-to-multi flagellated microorganisms, Brownian polymers, and micro-robotics are straightforward. A Matlab code is provided for further customization and generalizations.

The Transport Behavior of a Biflagellated Microswimmer before and after Cargo Loading

The changes in the transport behavior of a microswimmer before and after cargo loading are crucial to understanding and control of the motion of a biohybrid microbot. In this work, we show the change in swimming behavior of biflagellated microalgae Chlamydomonas reinhardtii picking up a 4.5 μm polystyrene microbead upon collision. The microswimmer changed from linear forward motion into helical motion upon the attachment of the cargo and swam with a decreased swimming velocity. We revealed the helical motion of the microswimmer upon cargo loading due to suppression of flagella by image analysis of magnified time-lapse images of C. reinhardtii with one microbead attached at the anterior end (between the flagella). Furthered suppression on the flagellum imposed by the loading of the second cargo has led to increased oscillation per displacement traveled and decreased swimming velocity. Moreover, the microswimmer with a microbead attached at the posterior end swam with swimming velocity close to free swimming microalgae and did not exhibit helical swimming behavior. The experimental results and analysis showed that the loading location of the cargo has a great influence over the swimming behavior of the microswimmer. Furthermore, the work balance calculation and mathematical analysis based on Lighthill's model are well consistent with our experimental findings.

The shape effect of flagella is more important than bottom-heaviness on passive gravitactic orientation in Chlamydomonas reinhardtii

The way the unicellular, biflagellated, green alga Chlamydomonas orients upward has long been discussed in terms of both mechanics and physiology. In this study, we focus on the mechanics, i.e. the 'passive' mechanisms, of gravitaxis. To rotate the body upwards, cellular asymmetry is critical. Chlamydomonas can be depicted as a nearly spherical cell body with two anterior, symmetric flagella. The present study looks at the question of whether the existence of the flagella significantly affects torque generation in upward reorientation. The 'density asymmetry model' assumes that the cell is spherical and bottom-heavy and that the shape and weight of the flagella are negligible, while the 'shape asymmetry model' considers the shape of the flagella. Both our experimental and simulation results revealed a considerable contribution from shape asymmetry to the upward orientation of Chlamydomonas reinhardtii, which was several times larger than that of density asymmetry. From the experimental results, we also quantified the extent of bottom-heaviness, i.e. the distance between the centers of gravity and the figure when the cell body is assumed to be spherical. Our estimation was approximately 30 nm, only one-third of previous assumptions. These findings indicate the importance of the viscous drag of the flagella to the upward orientation, and thus negative gravitaxis, in Chlamydomonas.

Chain formation can enhance the vertical migration of phytoplankton through turbulence

Many species of motile phytoplankton can actively form long multicellular chains by remaining attached to one another after cell division. While chains swim more rapidly than single cells of the same species, chain formation also markedly reduces phytoplankton's ability to maintain their bearing. This suggests that turbulence, which acts to randomize swimming direction, could sharply attenuate a chain's ability to migrate between well-lit surface waters during the day and deeper nutrient-rich waters at night. Here, we use numerical models to investigate how chain formation affects the migration of phytoplankton through a turbulent water column. Unexpectedly, we find that the elongated shape of chains helps them travel through weak to moderate turbulence much more effectively than single cells, and isolate the physical processes that confer chains this ability. Our findings provide a new mechanistic understanding of how turbulence can select for phytoplankton with elongated morphologies and may help explain why turbulence triggers chain formation.

Physical flow effects can dictate plankton population dynamics

Oceanic flows do not necessarily mix planktonic species. Differences in individual organisms’ physical and hydrodynamic properties can cause changes in drift normal to the mean flow, leading to segregation between species. This physically driven heterogeneity may have important consequences at the scale of population dynamics. Here, we describe how one form of physical forcing, circulating flows with different inertia effects between phytoplankton and zooplankton, can dramatically alter excitable plankton bloom dynamics. This may impact our understanding of the initiation and development of harmful algal blooms (HABs), which have significant negative ecological and socio-economic consequences. We study this system in detail, providing spatio-temporal dynamics for particular scenarios and summarizing large-scale behaviour via spatially averaged bifurcation diagrams. The key message is that, across a large range of parameter values, fluid flow can induce plankton blooms and mean-field population dynamics that are distinct from those predicted for well-mixed systems. The implications for oceanic population dynamic studies are manifest: we argue that the formation of HABs will depend strongly on the physical and biological state of the ecosystem, and that local increases in zooplankton heterogeneity are likely to precede phytoplankton blooms.

Flagellar swimming in viscoelastic fluids: role of fluid elastic stress revealed by simulations based on experimental data

Many important biological functions depend on microorganisms' ability to move in viscoelastic fluids such as mucus and wet soil. The effects of fluid elasticity on motility remain poorly understood, partly because the swimmer strokes depend on the properties of the fluid medium, which obfuscates the mechanisms responsible for observed behavioural changes. In this study, we use experimental data on the gaits of Chlamydomonas reinhardtii swimming in Newtonian and viscoelastic fluids as inputs to numerical simulations that decouple the swimmer gait and fluid type in order to isolate the effect of fluid elasticity on swimming. In viscoelastic fluids, cells employing the Newtonian gait swim faster but generate larger stresses and use more power, and as a result the viscoelastic gait is more efficient. Furthermore, we show that fundamental principles of swimming based on viscous fluid theory miss important flow dynamics: fluid elasticity provides an elastic memory effect that increases both the forward and backward speeds, and (unlike purely viscous fluids) larger fluid stress accumulates around flagella moving tangent to the swimming direction, compared with the normal direction.

Coarse-Graining the Fluid Flow around a Human Sperm

The flagellar beat is extracted from human sperm digital imaging microscopy and used to determine the flow around the cell and its trajectory, via boundary element simulation. Comparison of the predicted cell trajectory with observation demonstrates that simulation can predict fine-scale sperm dynamics at the qualitative level. The flow field is also observed to reduce to a time-dependent summation of regularized Stokes flow singularities, approximated at leading order by a blinking force triplet. Such regularized singularity decompositions may be used to upscale cell level detail into population models of human sperm motility.

Scale-dependent colocalization in a population of gyrotactic swimmers

We study the small scale clustering of gyrotactic swimmers transported by a turbulent flow, when the intrinsic variability of the swimming parameters within the population is considered. By means of extensive numerical simulations, we find that the variety of the population introduces a characteristic scale R^{*} in its spatial distribution. At scales smaller than R^{*} the swimmers are homogeneously distributed, while at larger scales an inhomogeneous distribution is observed with a fractal dimension close to what observed for a monodisperse population characterized by mean parameters. The scale R^{*} depends on the dispersion of the population and it is found to scale linearly with the standard deviation both for a bimodal and for a Gaussian distribution. Our numerical results, which extend recent findings for a monodisperse population, indicate that in principle it is possible to observe small scale, fractal clustering in a laboratory experiment with gyrotactic cells.

Shear-induced orientational dynamics and spatial heterogeneity in suspensions of motile phytoplankton

Fluid flow, ubiquitous in natural and man-made environments, has the potential to profoundly impact the transport of microorganisms, including phytoplankton in aquatic habitats and bioreactors. Yet, the effect of ambient flow on the swimming behaviour of phytoplankton has remained poorly understood, largely owing to the difficulty of observing cell-flow interactions at the microscale. Here, we present microfluidic experiments where we tracked individual cells for four species of motile phytoplankton exposed to a spatially non-uniform fluid shear rate, characteristic of many flows in natural and artificial environments. We observed that medium-to-high mean shear rates (1-25 s(-1)) produce heterogeneous cell concentrations in the form of regions of accumulation and regions of depletion. The location of these regions relative to the flow depends on the cells' propulsion mechanism, body shape and flagellar arrangement, as captured by an effective aspect ratio. Species having a large effective aspect ratio accumulated in the high-shear regions, owing to shear-induced alignment of the swimming orientation with the fluid streamlines. Species having an effective aspect ratio close to unity exhibited little preferential accumulation at low-to-moderate flow rates, but strongly accumulated in the low-shear regions under high flow conditions, potentially owing to an active, behavioural response of cells to shear. These observations demonstrate that ambient fluid flow can strongly affect the motility and spatial distribution of phytoplankton and highlight the rich dynamics emerging from the interaction between motility, morphology and flow.

Centripetal focusing of gyrotactic phytoplankton

A suspension of gyrotactic microalgae Chlamydomonas augustae swimming in a cylindrical water vessel in solid-body rotation is studied. Our experiments show that swimming algae form an aggregate around the axis of rotation, whose intensity increases with the rotation speed. We explain this phenomenon by the centripetal orientation of the swimming direction towards the axis of rotation. This centripetal focusing is contrasted by diffusive fluxes due to stochastic reorientation of the cells. The competition of the two effects lead to a stationary distribution, which we analytically derive from a refined mathematical model of gyrotactic swimmers. The temporal evolution of the cell distribution, obtained via numerical simulations of the stochastic model, is in quantitative agreement with the experimental measurements in the range of parameters explored.

Individual Flagellar Waveform Affects Collective Behavior of Chlamydomonas reinhardtii

Bioconvection is a form of collective motion that occurs spontaneously in the suspension of swimming microorganisms. In a previous study, we quantitatively described the "pattern transition," a phase transition phenomenon that so far has exclusively been observed in bioconvection of the unicellular green alga Chlamydomonas. We suggested that the transition could be induced by changes in the balance between the gravitational and shear-induced torques, both of which act to determine the orientation of the organism in the shear flow. As both of the torques should be affected by the geometry of the Chlamydomonas cell, alteration in the flagellar waveform might change the extent of torque generation by altering overall geometry of the cell. Based on this working hypothesis, we examined bioconvection behavior of two flagellar mutants of Chlamydomonas reinhardtii, ida1 and oda2, making reference to the wild type. Flagella of ida1 beat with an abnormal waveform, while flagella of oda2 show a normal waveform but lower beat frequency. As a result, both mutants had swimming speed of less than 50% of the wild type. ida1 formed bioconvection patterns with smaller spacing than those of wild type and oda2. Two-axis view revealed the periodic movement of the settling blobs of ida1, while oda2 showed qualitatively similar behavior to that of wild type. Unexpectedly, ida1 showed stronger negative gravitaxis than did wild type, while oda2 showed relatively weak gravitaxis. These findings suggest that flagellar waveform, not swimming speed or beat frequency, strongly affect bioconvection behavior in C. reinhardtii.

Phytoplankton's motion in turbulent ocean

We study the influence of turbulence on upward motion of phytoplankton. Interaction with the flow is described by the Pedley-Kessler model considering spherical microorganisms. We find a range of parameters when the upward drift is only weakly perturbed or when turbulence completely randomizes the drift direction. When the perturbation is small, the drift is either determined by the local vorticity or is Gaussian. We find a range of parameters where the phytoplankton interaction with the flow can be described consistently as diffusion of orientation in effective potential. By solving the corresponding Fokker-Planck equation we find exponential steady-state distribution of phytoplankton's propulsion orientation. We further identify the range of parameters where phytoplankton's drift velocity with respect to the flow is determined uniquely by its position. In this case, one can describe phytoplankton's motion by a smooth flow and phytoplankton concentrates on fractal. We find fractal dimensions and demonstrate that phytoplankton forms vertical stripes in space with a nonisotropic pair-correlation function of concentration increased in the vertical direction. The probability density function of the distance between two particles obeys power law with the negative exponent given by the ratio of integrals of the turbulent energy spectrum. We find the regime of strong clustering where the exponent is of order one so that turbulence increases the rate of collisions by a large factor. The predictions hold for Navier-Stokes turbulence and stand for testing.

Wavelength Selection in Gyrotactic Bioconvection

We investigate pattern formation by swimming micro-organisms (bioconvection), when their orientation is determined by balance between gravitational and viscous torques (gyrotaxis), due to being bottom heavy. The governing equations, which consist of the Navier-Stokes equations for an incompressible fluid coupled with a micro-organism conservation equation, are solved numerically in a large cross section chamber with periodic boundary conditions in the horizontal directions. The influence of key parameters on wavelength selection in bioconvection patterns is investigated numerically. For realistic ranges of parameter values, the computed wavelengths are in good agreement with the experimental observations provided that the diffusion due to randomness in cell swimming behaviour is small, refuting a recently published claim that the mathematical model becomes inaccurate at long times. We also provide the first computational evidence of "bottom-standing" plumes in a three-dimensional simulation.

Cell-body rocking is a dominant mechanism for flagellar synchronization in a swimming alga

The unicellular green alga Chlamydomonas swims with two flagella that can synchronize their beat. Synchronized beating is required to swim both fast and straight. A long-standing hypothesis proposes that synchronization of flagella results from hydrodynamic coupling, but the details are not understood. Here, we present realistic hydrodynamic computations and high-speed tracking experiments of swimming cells that show how a perturbation from the synchronized state causes rotational motion of the cell body. This rotation feeds back on the flagellar dynamics via hydrodynamic friction forces and rapidly restores the synchronized state in our theory. We calculate that this "cell-body rocking" provides the dominant contribution to synchronization in swimming cells, whereas direct hydrodynamic interactions between the flagella contribute negligibly. We experimentally confirmed the two-way coupling between flagellar beating and cell-body rocking predicted by our theory.

Dispersion of swimming algae in laminar and turbulent channel flows: consequences for photobioreactors

Shear flow significantly affects the transport of swimming algae in suspension. For example, viscous and gravitational torques bias bottom-heavy cells to swim towards regions of downwelling fluid (gyrotaxis). It is necessary to understand how such biases affect algal dispersion in natural and industrial flows, especially in view of growing interest in algal photobioreactors. Motivated by this, we here study the dispersion of gyrotactic algae in laminar and turbulent channel flows using direct numerical simulation (DNS) and a previously published analytical swimming dispersion theory. Time-resolved dispersion measures are evaluated as functions of the Péclet and Reynolds numbers in upwelling and downwelling flows. For laminar flows, DNS results are compared with theory using competing descriptions of biased swimming cells in shear flow. Excellent agreement is found for predictions that employ generalized Taylor dispersion. The results highlight peculiarities of gyrotactic swimmer dispersion relative to passive tracers. In laminar downwelling flow the cell distribution drifts in excess of the mean flow, increasing in magnitude with Péclet number. The cell effective axial diffusivity increases and decreases with Péclet number (for tracers it merely increases). In turbulent flows, gyrotactic effects are weaker, but discernable and manifested as non-zero drift. These results should have a significant impact on photobioreactor design.

Differential dynamic microscopy: a high-throughput method for characterizing the motility of microorganisms

We present a fast, high-throughput method for characterizing the motility of microorganisms in three dimensions based on standard imaging microscopy. Instead of tracking individual cells, we analyze the spatiotemporal fluctuations of the intensity in the sample from time-lapse images and obtain the intermediate scattering function of the system. We demonstrate our method on two different types of microorganisms: the bacterium Escherichia coli (both smooth swimming and wild type) and the biflagellate alga Chlamydomonas reinhardtii. We validate the methodology using computer simulations and particle tracking. From the intermediate scattering function, we are able to extract the swimming speed distribution, fraction of motile cells, and diffusivity for E. coli, and the swimming speed distribution, and amplitude and frequency of the oscillatory dynamics for C. reinhardtii. In both cases, the motility parameters were averaged over ∼10(4) cells and obtained in a few minutes.