50-year history and perspective on biomechanics of swimming microorganisms: Part II. Collective behaviours.

The paired review papers in Parts I and II describe the 50-year history of research on the biomechanics of swimming microorganisms and its prospects in the next 50 years. Parts I and II are divided not by the period covered, but by the content of the research: Part I explains the behaviours of individual microorganisms, and Part II explains collective behaviour. In the 1990s, the description of microbial suspensions as a continuum progressed, and macroscopic flow structures such as bioconvection were analysed. The continuum model was later extended to analyse various phenomena such as flow induced trapping of microorganisms and accumulation of cells at interfaces. In the 2000s, the collective behaviour of swimming microorganisms came into the limelight, and physicists as well as biomechanics researchers carried out many studies probing microorganism collectivity. In particular, research on the turbulence-like flow structure of dense bacterial suspensions has led to dramatic developments in the field of microbial biomechanics. Efforts to bridge the cellular scale to the macroscopic scale by extracting macroscopic physical quantities from the microstructure of cell suspensions are also underway. This Part II reviews these collective behaviours of swimming microorganisms and discusses future prospects of the field.

50-year History and perspective on biomechanics of swimming microorganisms: Part I. Individual behaviours.

The paired review papers in Parts I and II describe the 50-year history of research on the biomechanics of swimming microorganisms and its prospects in the next 50 years: Part I explains the behaviour of individual microorganisms, and Part II explains collective behaviour. Since the discovery of microorganisms by van Leeuwenhoek in the 17th century, many natural scientists have been interested in their motility because it is directly associated with biological function. A research upsurge occurred in the 1970s, with the elucidation of swimming mechanisms among individual microorganisms and the theoretical derivation of swimming speeds. Various swimming strategies of three types of microorganisms, i.e. bacteria, ciliates and microalgae, are explained in this Part I. We show that some of the behaviours of microorganisms can be described by biomechanical equations and are to some extent predictable. Recent researches have revealed the behaviour of microorganisms in more complex environments and more realistic settings, which are also reviewed in the paper. Last, we provide future prospects for research on microbial behaviour.

Squirmers with swirl: a model for Volvox swimming.

Colonies of the green alga Volvox are spheres that swim through the beating of pairs of flagella on their surface somatic cells. The somatic cells themselves are mounted rigidly in a polymeric extracellular matrix, fixing the orientation of the flagella so that they beat approximately in a meridional plane, with axis of symmetry in the swimming direction, but with a roughly [Formula: see text] azimuthal offset which results in the eponymous rotation of the colonies about a body-fixed axis. Experiments on colonies of Volvox carteri held stationary on a micropipette show that the beating pattern takes the form of a symplectic metachronal wave (Brumley et al. Phys. Rev. Lett., vol. 109, 2012, 268102). Here we extend the Lighthill/Blake axisymmetric, Stokes-flow model of a free-swimming spherical squirmer (Lighthill Commun. Pure Appl. Maths, vol. 5, 1952, pp. 109-118; Blake J. Fluid Mech., vol. 46, 1971b, pp. 199-208) to include azimuthal swirl. The measured kinematics of the metachronal wave for 60 different colonies are used to calculate the coefficients in the eigenfunction expansions and hence predict the mean swimming speeds and rotation rates, proportional to the square of the beating amplitude, as functions of colony radius. As a test of the squirmer model, the results are compared with measurements (Drescher et al. Phys. Rev. Lett., vol. 102, 2009, 168101) of the mean swimming speeds and angular velocities of a different set of 220 colonies, also given as functions of colony radius. The predicted variation with radius is qualitatively correct, but the model underestimates both the mean swimming speed and the mean angular velocity unless the amplitude of the flagellar beat is taken to be larger than previously thought. The reasons for this discrepancy are discussed.

Dispersion of model microorganisms swimming in a nonuniform suspension.

Although diffusion properties of a suspension of swimming microorganisms in equilibrium have been studied intensively, those under nonequilibrium conditions remain unclear. In this study, we investigate the spreading of model microorganisms from high concentration to low by the Stokesian dynamics method. The results reveal that the spreading is neither purely diffusive nor ballistic. When the dipole component of the swimming velocity is small, the cells actively direct themselves towards lower concentrations. The concentration distribution shows stronger oscillations than would be expected for ballistic swimmers with constant orientations. The mechanism can be explained by the near-field hydrodynamic interactions between cells and the spatial gradient of the collision rate. Comparison of the numerical results with a simple continuum model and a Monte Carlo simulation shows that those conventional models can capture the basic features of the present results. These new findings pave the way towards a mathematical description of the dispersion of microorganisms in various environments.

Flutter in a quasi-one-dimensional model of a collapsible channel.

The effects of wall inertia on instabilities in a collapsible channel with a long finite-length flexible wall containing a high Reynolds number flow of incompressible fluid are studied. Using the ideas of interactive boundary layer theory, the system is described by a one-dimensional model that couples inviscid flow outside the boundary layers formed on the channel walls with the deformation of the flexible wall. The observed instability is a form of flutter, which is superposed on the behaviour of the system when the wall mass is neglected. We show that the flutter has a positive growth rate because the fluid loading acts as a negative damping in the system. We discuss these findings in relation to other work on self-excited oscillations in collapsible channels.

Fluid particle diffusion in a semidilute suspension of model micro-organisms.

We calculate non-Brownian fluid particle diffusion in a semidilute suspension of swimming micro-organisms. Each micro-organism is modeled as a spherical squirmer, and their motions in an infinite suspension otherwise at rest are computed by the Stokesian-dynamics method. In calculating the fluid particle motions, we propose a numerical method based on a combination of the boundary element technique and Stokesian dynamics. We present details of the numerical method and examine its accuracy. The limitation of semidiluteness is required to ensure accuracy of the fluid particle velocity calculation. In the case of a suspension of non-bottom-heavy squirmers the spreading of fluid particles becomes diffusive in a shorter time than that of the squirmers, and the diffusivity of fluid particles is smaller than that of squirmers. It is confirmed that the probability density distribution of fluid particles also shows diffusive properties. The effect of tracer particle size is investigated by inserting some inert spheres of the same radius as the squirmers, instead of fluid particles, into the suspension. The diffusivity for inert spheres is not less than one tenth of that for fluid particles, even though the particle size is totally different. Scaling analysis indicates that the diffusivity of fluid particles and inert spheres becomes proportional to the volume fraction of squirmers in the semidilute regime provided that there is no more than a small recirculation region around a squirmer, which is confirmed numerically. In the case of a suspension of bottom-heavy squirmers, horizontal diffusivity decreases considerably even with small values of the bottom heaviness, which indicates the importance of bottom heaviness in the diffusion phenomena. We believe that these fundamental findings will enhance our understanding of the basic mechanics of a suspension of swimming micro-organisms.

Run and tumble chemotaxis in a shear flow: the effect of temporal comparisons, persistence, rotational diffusion, and cell shape.

Escherichia coli is a motile bacterium that moves up a chemoattractant gradient by performing a biased random walk composed of alternating runs and tumbles. This paper presents calculations of the chemotactic drift velocity v (d) (the mean velocity up the chemoattractant gradient) of an E. coli cell performing chemotaxis in a uniform, steady shear flow, with a weak chemoattractant gradient at right angles to the flow. Extending earlier models, a combined analytic and numerical approach is used to assess the effect of several complications, namely (i) a cell cannot detect a chemoattractant gradient directly but rather makes temporal comparisons of chemoattractant concentration, (ii) the tumbles exhibit persistence of direction, meaning that the swimming directions before and after a tumble are correlated, (iii) the cell suffers random re-orientations due to rotational Brownian motion, and (iv) the non-spherical shape of the cell affects the way that it is rotated by the shear flow. These complications influence the dependence of v(d) on the shear rate gamma. When they are all included, it is found that (a) shear disrupts chemotaxis and shear rates beyond gamma approximately 2 s(-1) render chemotaxis ineffective, (b) in terms of maximizing drift velocity, persistence of direction is advantageous in a quiescent fluid but disadvantageous in a shear flow, and (c) a more elongated body shape is advantageous in performing chemotaxis in a shear flow.

Bacterial tracking of motile algae assisted by algal cell's vorticity field.

Previously published experimental work by other authors has shown that certain motile marine bacteria are able to track free-swimming algae by executing a zigzag path and steering toward the algae at each turn. Here, we propose that the apparent steering behaviour could be a hydrodynamic effect, whereby an algal cell's vorticity and strain-rate fields rotate a pursuing bacterial cell in the appropriate direction. Using simplified models for the bacterial and algal cells, we numerically compute the trajectory of a bacterial cell and demonstrate the plausibility of this hypothesis.

Coherent structures in monolayers of swimming particles.

We report various types of coherent structures in suspensions of spherical particles swimming in a monolayer. We solve the fluid dynamics precisely from far-field hydrodynamic interactions to lubrication between two near-contact surfaces. The simulation results clearly illustrate that coherent structures, such as aggregation, mesoscale spatiotemporal motion, and band formation, can be generated by purely hydrodynamic interactions.

Orientational relaxation time of bottom-heavy squirmers in a semi-dilute suspension.

One of the important quantities to characterize unsteady behaviour of a cell suspension is the orientational relaxation time, which is the time scale for a micro-organism to re-orientate to its preferred direction from disorientated conditions. In this paper, a swimming micro-organism is modelled as a squirming sphere with prescribed tangential surface velocity, in which the centre of mass of the sphere is displaced from the geometric centre (bottom-heaviness). The orientational relaxation time of bottom-heavy squirmers in a suspension is investigated both analytically and numerically. The three-dimensional movement of 64 identical squirmers in a fluid otherwise at rest, contained in a cube with periodic boundary conditions, is dynamically computed, for random initial positions and orientations. The effects of volume fraction of squirmers, the bottom-heaviness and the squirming mode on the relaxation time are discussed. The results for a semi-dilute suspension show that both the mean stresslet strength and the orientational relaxation time decrease from those for a dilute suspension. We also observe a stress overshoot in some cases. The mechanism for this is different from that for a visco-elastic fluid, and is explained by the change with time of the orientation of squirmers.

Measurement of cell velocity distributions in populations of motile algae.

The self-propulsion of unicellular algae in still ambient fluid is studied using a previously reported laser-based tracking method, supplemented by new tracking software. A few hundred swimming cells are observed simultaneously and the average parameters of the cells' motility are calculated. The time-dependent, two-dimensional distribution of swimming velocities is measured and the three-dimensional distribution is recovered by assuming horizontal isotropy. The mean and variance of the cell turning angle are quantified, to estimate the reorientation time and rotational diffusivity of the bottom-heavy cell. The cells' phototactic and photokinetic responses to the laser light are evaluated. The results are generally consistent both with earlier assumptions about the nature of cell swimming and quantitative measurements, appropriately adjusted. The laser-based tracking method, which makes it possible to average over a large number of motile objects, is shown to be a powerful tool for the study of microorganism motility.

A model for time-dependent flow in (giraffe jugular) veins: uniform tube properties.

Computations are reported for a one-dimensional model of time-dependent flow in collapsible tubes representing long mammalian veins. The tubes are taken to have uniform intrinsic properties and we concentrate on the effect of longitudinal gravity. The main application is to the jugular vein of the upright giraffe, with given inflow rate from the head, a given pressure, slightly above the external, atmospheric pressure, at the downstream (vena caval) end, and a variety of initial conditions. We show that: (i) previously calculated steady flows are the long time limits of unsteady computations, although only after a considerable time in which slowly-decaying waves and elastic jumps propagate up and down, (ii) steady flows are indeed not found when the steady-flow analysis shown them not to exist, although the consequent unsteadiness is of such small amplitude as to be practically unimportant, (iii) the time taken for the flow to become steady when the neck is raised from the horizontal or the head-down position can be several seconds longer than the neck-raising time itself (3-7s). We also find that roll-waves do not develop despite having been previously predicted for long collapsible tubes. Further application is made to the effect of postural changes on human neck and leg veins.

The influence of turbulence on plankton predation strategies.

The importance of predation in regulating the size of competing plankton and larval fish populations has long been appreciated. However, it has only recently been recognized that turbulence must have a significant influence on predator-prey interactions because most rival species of microorganisms co-exist in oceanic or fast moving fresh water flows. Turbulence is likely to influence predation strategies in two ways. The extra energy imparted to a micro-organism from the flow field will enhance the number of encounters or "contacts" between predators and prey. At the same time, because the velocity of a predator relative to its potential prey will be increased, the time-scale over which a capture must be completed is reduced. Balancing the benefits of extra encounters with the drawbacks of more difficult captures, will dictate an optimal predation strategy, either foraging behaviour or ambush feeding, on the predator. This will depend on its own and the prey's swimming capabilities, as well as the characteristics of the turbulent environment. In this paper some previous work, examining the increased encounter rate in turbulence, will be extended to look at the capture problem. The main proposal is that the capture event should be encapsulated in a capture probability function, from which the optimal predation strategy can be derived. As an illustration, plausible capture probability functions will be postulated and the resulting predictions tested against numerical simulations carried out in a turbulent-like flow field. Good agreement between the predictions and the simulations is demonstrated.

Modelling run-and-tumble chemotaxis in a shear flow.

The biased random walk undergone by chemotactic bacteria such as Escherichia coli will be influenced at the microscopic level by flow in the ambient medium. In this paper, we model swimming bacteria being advected and rotated by a simple shear flow. Under certain scaling assumptions, we obtain an advection-diffusion equation for cell density, when the chemotactic response is small, which shows a coupling between the rotation and chemotaxis. We also present an alternative method for calculating the chemotactic flux in an unbounded region which is valid for more general chemotactic responses.

Planktonic contact rates in homogeneous isotropic turbulence: theoretical predictions and kinematic simulations.

The key role played by turbulence in the environment of plankton and larval fish populations has become appreciated in recent years. In particular, the turbulent enhancement of encounter rates between different species of microorganisms, either swimming or passively advected by the flow, is well established. However, most of the current modelling approaches are rather ad hoc, giving rise to ambiguities in the specification of certain key parameters. In this paper, the encounter problem in a turbulent flow of large Reynolds number is re-examined from first principles and a number of new formulae will be established for different swimming strategies. The key innovation is the proposal of a model form for the conditional joint probability density function of predator and prey velocities when the organisms are separated by their given contact radius, R. Particular attention will be paid to the case when a microorganism follows a random trajectory, due to a combination of its own swimming and the action of the flow. The theoretical predictions are subsequently tested against corresponding quantities derived from a series of kinematic simulations of a turbulent-like flow field. Good agreement is demonstrated between the predictions and simulations.

Vascular endothelial cells minimize the total force on their nuclei.

The vascular endothelium is a cellular monolayer that lines the arterial walls. It plays a vital role in the initiation and development of atherosclerosis, an occlusive arterial disease responsible for 50% of deaths in the Western world. The focal nature of the disease suggests that hemodynamic forces are an important factor in its pathogenesis. This has led to the investigation of the effects of mechanical forces on the endothelial cells themselves. It has been found that endothelial cells do respond to stresses induced by the flowing blood; in particular, they elongate and align with an imposed flow direction. In this paper, we calculate the distribution of force exerted on a three-dimensional hump, representing the raised cell nucleus, by a uniform shear flow. It is found that, for a nonaxisymmetric ellipsoidal hump, the least total force is experienced when the hump is aligned with the flow. Furthermore, for a hump of fixed volume, there is a specific aspect ratio combination that results in the least total force upon the hump, (0.38:2.2:1.0; height:length:width). This is approximately the same as the average aspect ratio taken up by the cell nuclei in vivo (0.27:2.23:1.0). It is possible, therefore, that the cells respond to the flow in such a way as to minimize the total force on their nuclei.

Large-amplitude undulatory fish swimming: fluid mechanics coupled to internal mechanics.

The load against which the swimming muscles contract, during the undulatory swimming of a fish, is composed principally of hydrodynamic pressure forces and body inertia. In the past this has been analysed, through an equation for bending moments, for small-amplitude swimming, using Lighthill's elongated-body theory and a 'vortex-ring panel method', respectively, to compute the hydrodynamic forces. Those models are outlined in this review, and a summary is given of recent work on large-amplitude swimming that has (a) extended the bending moment equation to large amplitude, which involves the introduction of a new (though probably usually small) term, and (b) developed a large-amplitude vortex-ring panel method. The latter requires computation of the wake, which rolls up into concentrated vortex rings and filaments, and has a significant effect on the pressure on the body. Application is principally made to the saithe (Pollachius virens). The calculations confirm that the wave of muscle activation travels down the fish much more rapidly than the wave of bending.

Alteration of mean wall shear stress near an oscillating stagnation point.

The site opposite an end-to-side anastomosis, resulting from femoral bypass surgery, and the carotid sinus are two regions well known to be prone to fibrous intimal hyperplasia or atherogenesis, respectively. The blood flow at these two sites features a stagnation point, which oscillates in strength and position. Mathematical models are used to determine some of the features of such a flow; in particular, the mean wall shear stress is calculated. The positional oscillations cause a significant change in the distribution and magnitude of the mean wall shear stress from that of the well-studied case of a stagnation point that oscillates only in strength. It is therefore proposed that the recorded effect of time dependence in the flow upon atherogenesis could still be a result of the distribution of the mean and not the time-varying components of the wall shear stress.

An active membrane model for peristaltic pumping: Part I--Periodic activation waves in an infinite tube.

A model for the coupled problem of wall deformation and fluid flow, based on thin-shell and lubrication theories, and driven by a propagating wave of smooth muscle activation, is proposed for peristaltic pumping in the ureter. The model makes use of the available experimental data on the mechanical properties of smooth muscle and accounts for the soft material between the muscle layer and the vessel lumen. The main input is the activation wave of muscular contraction. Equations for the time-dependent problem in tubes of arbitrary length are derived and applied to the particular case of periodic activation waves in an infinite tube. Mathematical (small amplitude) and numerical analyses of this case are presented. Predictions on phase-lag in wall constriction with respect to peak activation wave, lumen occlusion due to thickening lumen material with contracting smooth muscle, and the general bolus shape are in qualitative agreement with observation. Some modifications to the mechanical, elastic, and hydrodynamic properties of the ureter that will make peristalsis less efficient, due for example to disease, are identified. In particular, the flow rate-pressure rise relationship in linear for weak to moderate activation waves, but as the lumen is squeezed shut, it is seen to be nonlinear in a way that increases pumping efficiency. In every case a ureter whose lumen can theoretically be squeezed shut is the one for which pumping is most efficient.