The hydrodynamics of filter feeding in choanoflagellatesklinck/Reprints/PDF/pettittEJProt2002.pdfThe hydrodynamics of filter feeding in choanoflagellates Michala E. Pettitt 1,*, Belinda

The hydrodynamics of filter feeding in choanoflagellates

Michala E. Pettitt1,*, Belinda A. A. Orme2, John R. Blake2 and Barry S. C. Leadbeater3

1Department of Earth Sciences, Wills Memorial Building, University of Bristol, Queen’s Road, Clifton, Bristol, BS81RJ, UK; Fax: 0117 925 3385, E-mail: [email protected].

2School of Mathematics and Statistics, University of Birmingham, Edgbaston, Birmingham, B15 2TT, UK3School of Biosciences, University of Birmingham, Edgbaston, Birmingham, B15 2TT, UK

Received: 8 October 2001; 8 July 2002. Accepted: 16 July 2002

Choanoflagellates are filter feeding heterotrophic nanoflagellates. Fluid flow generated by the motionof the flagellum delivers suspended prey particles to the surface of a crown-like filtering apparatus. Hy-drodynamic characteristics are consequently of great importance to understanding prey capture in theseorganisms.This paper reports on the use of video-microscopy to investigate the hydrodynamics of filter feeding inchoanoflagellates. The cell and collar morphology, and the flagellar and fluid motion of three species ofchoanoflagellates, one from each family of the phylum, are described. Inter-species differences in theseparameters are compared to one another and to optima theoretically determined by Higdon (1979b).The motion of a flagellum close to an infinite boundary is modelled using a line of stokeslets. This modelpredicts the occurrence of viscous eddies in the fluid streamlines. Viscous eddies are observed in the farfield fluid flow of two species of sessile choanoflagellates.Balancing forces over a control volume and using slender body theory allows estimates of pressure dropacross the collar of the three species of choanoflagellate to be made. The magnitude of pressure drop(∆p) was found to be consistent with previous publications, suggesting that ∆p may be a constraining pa-rameter for filter feeders.

Key words: Choanoflagellates; Fluid Flow; Pressure Drop; Slender Body Theory; Viscous Eddies; Food par-ticle filtration.

they may be responsible for the majority of thegrazing pressure exerted on the bacterial standingstock (Fenchel 1982).

Mechanism of filter feeding

The basic mechanism of filter feeding, commonto all choanoflagellates, is illustrated in Figure 1.The round or ovoid protoplast (cell), typically10–20 µm in length, is modified at the distal endinto a crown, or collar like structure. The radiallysymmetrical collar is formed from a palisade ofidentical microvilli, the number and length of

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Introduction

The Choanoflagellida, or choanoflagellates, aresingle celled heterotrophic nanoflagellates of thephylum Choanozoa (Cavalier-Smith 1993). Theseorganisms are widely distributed geographicallyand representatives of the group can be found inmost aquatic habitats. Some species may be entire-ly pelagic but most are particularly associated withsuspended debris or solid surfaces (Carrias et al.1998). Recognised as primarily filter feeding bacte-riovores, the choanoflagellates occupy a vital nichein the microbial loop. Under some circumstances

*corresponding author

Europ. J. Protistol. 38, 313–332 (2002)© Urban & Fischer Verlaghttp://www.urbanfischer.de/journals/ejp

which vary in a species-specific manner. Thelength of microvilli can also vary between individ-uals of the same species, being influenced by boththe size and the physiological state of the cell.Functionally, the collar acts as a filter, trappingbacteria on its outward-facing surface, beforetransporting them towards the cell where they arephagocytosed. Bacteria encounter the filter bybeing entrained in the fluid flow generated by asingle smooth flagellum that extends from the cellapex at the centre of the collar. The flagellum prop-agates an approximately planar wave, which travelsfrom the base to the tip, drawing fluid through thecollar and driving it distally away from the proto-plast.

The phylum comprises three families, distin-guished mainly by differences in their extracellularstructures (Thomsen and Buck 1991). In this paper

314 M. E. Pettitt et al.

Fig. 1. Cross sectional diagram of filter feeding inchoanoflagellates, showing some of the measures usedin this paper. H is the mid-point of the cell above aboundary, A is the radius of the cell protoplast and L thelength of the flagellum. λ is the wavelength and β theamplitude of the wave. Grey lines show direction offluid flow, grey spheres indicate the manner and sites ofbacterial capture and ingestion.

the effect these extracellular structures may haveon fluid flow is assessed by describing the flow inone species from each family. All species were se-lected on the basis of cell size and reliability offlagellar motion during microscopical examina-tion.

i. The Codosigidae are naked apart from a veryfine investment. Many species are sessile, being at-tached to solid surfaces via a fine, flexible stalk,termed a pedicel. The length of the pedicel is high-ly variable, and is dependent on species and life-cycle stage. In this paper Codosiga gracilis (Fig.2a), a marine species, is used to assess how varyingthe length of the pedicel affects fluid flow. Al-though one pedicel may support between five andsix cells in this species, this study limits itself toconsideration of the fluid flow around a singlestalked individual.

ii. The Salpingoecidae are identified by the pos-session of a firm ‘cup-like’ theca. The theca, whichsurrounds the protoplast, attaches cells directly tosolid surfaces. Salpingoeca amphoridium (Fig. 2b),a relatively large freshwater species, was selectedfor this study.

iii. The Acanthoecidae are exclusively marineand often pelagic. They are surrounded by a ‘bas-ket-like’ lorica composed of siliceous costal strips.The morphology of the lorica is highly variable be-tween species. The lorica of Stephanoeca diplo-costata (Fig. 2c), comprises between 150 and 185costal strips in total, and is divided by a waist intoanterior and posterior chambers (Leadbeater1979). Although a siliceous pedicel, by which thecell may attach to surfaces, does occur in some in-dividuals of this species, all Stephanoeca diplo-costata cells described here were pelagic.

Hydrodynamic environment

The hydrodynamic environment of size andflow speed within which choanoflagellates propeland filter water is dominated by viscous forces.Such environments are described by their lowReynolds number. The Reynolds number is theratio of inertial to viscous forces; the two forcesconcerned in the motion of a fluid. Inertial force isassociated with the motion of a mass of fluid. Vis-cous force is associated with the sliding of layers offluid over one another (Vogel 1994). Reynoldsnumber (Re) is defined as:

Re = ρνl/µ

where ρ and µ are the density and viscosity of thefluid respectively, l is a characteristic length of asolid surface around which the fluid is moving, andν is a characteristic velocity of the fluid (Vogel1994). The micrometre scale of choanoflagellatesand their prey determine that fluid flow is domi-nated by viscous forces. Inertial forces will have noappreciable effect on flow. This circumstance is de-scribed by a Reynolds number of much less thanunity. Typical Re values for flagellate cells rangefrom 10–2–10–4 (Higdon 1979a; Sleigh 1991). Insuch hydrodynamic environments flow is alwaysstrictly laminar, and small particles entrained in thefluid flow have negligible momentum. In flow en-vironments dominated by viscous forces, it is pos-sible to develop mathematical models of the hy-drodynamics that can lead to an enhanced under-standing of the observed phenomenon.

Mathematical modelling of the flow

Many papers (Lighthill 1976; Higdon 1979a, b, c;Liron and Blake 1981), utilise the concept pioneeredby Hancock (1953) of using slender body theory toreplace the flagellum by a distribution of ‘stokeslets’and ‘potential dipoles’ along the centre line.

Higdon (1979b) analysed the generation of feed-ing currents by a sessile choanoflagellate from theCodosigidae (Codosiga) in this way. Higdon de-scribes the choanoflagellate by a limited number ofparameters. The spherical cell body radius (A), andthe radius (a) and length (L) of the smooth cylin-drical flagellum are used to geometrically modelthe flagellate. The cell body is situated at a height(H) above the surface to which the cell is attached.The flagellar motion is defined as a sinusoidal wavein terms of the amplitude (β), the wavelength (λ)and the total number of waves (N) occurring overthe length of the flagellum (see Fig. 1). Using theseparameters, Higdon determined the velocity fieldand power consumption for a cell with varyingL/A, a/A and H/A ratios. Varying these ratios al-lowed Higdon to determine the theoretical opti-mal configuration of cellular dimensions and flag-ellar waveform. The present paper attempts to ver-ify, or refute, the optima identified by Higdon(1979b), by comparing them to measurements onthe cellular parameters and flagellar waveforms ofthree species of choanoflagellates.

Liron and Blake (1981) also employed ‘stokes-lets’ to determine the occurrence, number andshape of viscous eddies generated by point forces

Hydrodynamics of filter feeding in choanoflagellates 315

Fig. 2. Light micrographs of a) Codosiga gracilis, b) Salpingoeca amphoridium and c) Stephanoeca diplocostata.Scale bars indicate 5 µm.

near solid boundaries. The theoretical presence ofthese eddies is pertinent to choanoflagellates asmany species attach directly, or via pedicels tosolid surfaces. The present paper expands on theapproach of Liron and Blake (1981) by using a lineof stokeslets normal to a planar boundary to ap-proximate the flow fields generated by the flagel-lum. The length of the line of stokeslets, and theheight at which they are situated above the surface,is varied to simulate the cellular characteristics ofthe two sessile species of choanoflagellate used inthis study, Salpingoeca amphoridium and Codosigagracilis. This paper compares the occurrence andshape of the streamlines predicted by our streamfunction formulation to the pattern of fluid flowgenerated by these two species of choanoflagellate.

Influence of the collar

The presence of the collar as a structure throughwhich fluid must pass has been largely discountedwhen modelling fluid flow around flagellates. Hig-don (1979b) assumes that the collar tentacles aretoo fine to affect the dynamics of flow. We believethis assumption to be unfounded, and that thestructure of the collar may significantly affect thehydrodynamics of filter feeding in choanoflagel-lates. The present paper aims to build upon Hig-don’s work by describing the effect of the collar onfluid flow in terms of the relationship between fluidvelocity, pressure drop, and the variable collar ge-ometry of the three species of choanoflagellates.

Pressure drop can be ascribed to the resistancefluid encounters as it passes through a filter, in thisinstance the palisade of microvilli that comprisethe collar. The pressure drop across a filter is deter-mined by the head-on velocity of the fluid, fluidviscosity, and the relative spacing and radius of thecollar tentacles (the filter elements). The propor-tion of filter elements to filter pores (γ) will be acomponent in determining the resistance to flow.If fluid is forced through this filter at a given veloc-ity, pressure drop will increase with increasing val-ues of γ. If, however, pressure drop is taken as aconstant, head-on fluid velocity will fall as γ in-creases. In practice in choanoflagellates the flagel-lar beating drives water away from the cell, reduc-ing the pressure within the collar. This creates apressure drop across the collar drawing water be-tween the tentacles. An organism can change theflow behaviour by altering the flagellar beat (fre-quency, wavelength, amplitude) or by changing the

collar characteristics (number, length and spacingof the tentacles and their angle to the cell body).

Fenchel (1986) estimated the pressure dropacross the filters of a number of protozoan filterfeeders, using the model of Tamada and Fujikawa(1957). He concluded that pressure drop across afilter was relatively constant between species, and,therefore, may represent a physical constraint onfilter feeders. The direct inference of this conclu-sion is that filter feeders with a high proportion offibres to pores will necessarily generate lower ve-locities than filter feeders where values of γ arelower. The model of Tamada and Fujikawa (1957),used by Fenchel (1986), represents a filter as a sin-gle layer of parallel, identical filter elements. Thegap between filter elements, h, is therefore con-stant. This is not an accurate approximation for thefiltering apparatus of choanoflagellates, where hincreases as you move distally along the collar. Thepresent study employs slender body theory and acontrol volume approach to estimate pressuredrop across the collar. This model accomodates theincreasing values of h, and the decreasing values ofγ, encountered with distal movement along thecollar. Details of the model are presented in Ap-pendix 1.

Materials and methods

Late exponential stage cultures of the three species ofchoanoflagellates were prepared for video recordingwork. Cultures of Salpingoeca amphoridium weregrown at 25 °C in Pratt’s medium (Karpov and Lead-beater 1997) with the addition of the bacterium Kleb-siella pneumoniae as a nutritive source, and 1cm lengthsof human hair as an attachment substrate. Cultures ofCodosiga gracilis and Stephanoeca diplocostata weregrown at 14 °C in seawater/distilled water medium. Thismedium was prepared as a 99:1 ratio of 0.2 µm-filteredseawater and distilled water. ‘C2’, a co-isolated bacteri-um, was added as a nutritive source to cultures of Co-dosiga gracilis, along with 1 cm lengths of human hair asan attachment substrate. Three ml of a nitrate/phos-phate additive (5 g NaNO3 and 0.75 g Na2HPO4 dis-solved in 100 ml distilled water) was added to 1 L of sea-water/distilled water medium for culturing of Stepha-noeca diplocostata. This additive stimulated the growthof co-cultured bacteria. The attachment substrate wasomitted from cultures of Stephanoeca diplocostata.

Sixteen hours prior to use the hair substrate with itsattached cells was removed from cultures of Codosigagracilis and Salpingoeca amphoridium and transferredto fresh medium. This ‘starvation’ step reduced ambientbacterial concentration, and increased the percentage of


the cell population showing flagellar activity and ingest-ing latex microspheres during observation.

Fluid motion was observed using a Leitz ortholux IImicroscope, fitted with phase contrast objectives.Recordings were made using a Sony hyper-HAD CCDcamera and a Panasonic S-VHS recorder. The recorderwas fitted with a Jog/Shuttle control to allow frame byframe replay. Playback was displayed on a 14′′ Toshibamonitor giving a final magnification of ×3,660 for ×40objective and ×4,800 for x100 oil immersion objective.

Microscope slides were specially adapted for record-ings of fluid flow by creating a well 20 mm square and0.5 mm deep. Such precautions are necessary to reducethe hydrodynamic distortions that close slide-to-cover-slip spacing may impose on the flow field generated by acell.

Latex microspheres 0.5 µm in diameter (Poly-sciences) were used throughout for tracking flows andestimating flow speeds. Microspheres were suspendedin Pratt’s medium to a concentration of 5 × 107 micro-spheres ml–1 for use with Salpingoeca amphoridium.Adsorption of bovine serum albumin (BSA), to the sur-face of the microspheres considerably reduced the ag-gregation of the microspheres when suspended in sea-water (Polysciences, 1991). BSA-coated microsphereswere suspended to a concentration of 5 × 107 micro-spheres ml–1 in seawater/distilled water medium for usewith Codosiga gracilis and Stephanoeca diplocostata.

One hundred microlitres of the appropriate micro-sphere suspension was added to the well of an adaptedslide, along with a 1cm length of hair bearing thechoanoflagellate cells. Cells were initially examined at×40 for feeding behaviour. Once a possible subject hadbeen identified, recordings of the motion of the micro-spheres around the cell were made, either at ×40 or ×100oil immersion. All recordings were made at room tem-perature, between 19 °C and 22 °C.

Video recordings were made onto Fuji Pro S-VHStape at a frequency of 50 Hz, giving a temporal separa-tion between adjacent frames of 0.02 s. Frame by frameplayback allowed the position of individual micro-spheres entrained in the fluid flow around a cell, to berecorded directly from the screen onto a transparentoverlay sheet. This was done either every 0.02 s or every0.2 s depending on the velocity of the microsphere.Flagellar waveforms were recorded in a similar manner;the outline of a flagellum was traced directly onto atransparent overlay sheet. A separate tracing was madefor each successive frame of the recording.

Collar parameters, tentacle number and tentacle ra-dius, were determined for Salpingoeca amphoridiumusing transmission electron microscopy (TEM). Lateexponential phase cultures were pelleted by centrifuga-tion, dehydrated through an ethanol series and fixedand embedded according to Karpov and Leadbeater(1998). Dehydration of samples for TEM can be associ-ated with a level of shrinkage in the cell tissue. The mea-sured values of tentacle radius presented here should

therefore be considered as a minimum. The same collarparameters for Codosiga gracilis were taken from Lead-beater and Morton (1974), and for Stephanoeca diplo-costata from Leadbeater (1994).

Results

Cell characteristics and flagellar motion

Table 1 displays average cell characteristics forthe three species of choanoflagellate studied.Height above the substratum (H) refers to theheight of the mid point of the cell body above theattachment surface, consistent with Higdon(1979b). No values of H are given for Stephanoecadiplocostata, as all cells examined were freely sus-pended. Values of H for Codosiga gracilis are ex-pressed as the observed range, because cells wereselected to reflect the range of pedicel lengths ob-served in this species. The calculated values L/Aand H/A respectively non-dimensionalise the flag-ellar length (L) and the height of the midpointabove the substratum (H), by dividing by a mea-sure of the cell size, the cell radius (A).

It can be seen in Table 1 that cells of Codosigagracilis and Stephanoeca diplocostata have similardimensions. Salpingoeca amphoridium has a slight-ly larger cell radius and much longer flagellum re-sulting in a larger value for L/A, the non-dimen-sionalised flagellar length.

Table 2 displays measurements that describe theflagellar waveform and motion for the threespecies of choanoflagellate studied. All measure-ments were determined using traces of flagellataken direct from the video recording. The flagellarwaveform in both Codosiga gracilis and Stepha-noeca diplocostata appears to contain less than onefull wavelength. Wavelength in these species wasdetermined by extrapolating the closest observedapproximations to a sinusoidal waveform. Thismust be regarded as an approximation as bothspecies appear to have a large three dimensionalcomponent to their waveform. The value N is theaverage number of linear wavelengths on a com-plete flagellum.

Wavespeed (Vw), was calculated from the dis-tance travelled by the maximum amplitude (β) ofthe waveform in consecutive frames. αk is a mea-sure used in Higdon (1979b), as an indicator of theforward thrust generated by the flagellum. It is cal-culated as the maximum slope of the flagellum per-pendicular to the cell body.



Table 2. Flagellar motion and waveform measurements for Salpingoeca amphoridium, Codosiga gracilis andStephanoeca diplocostata. n is the number of cells in the data set. N is the number of complete waves in a flagellum.αk is the maximum slope of the flagellum perpendicular to the cell body, a measure of forward thrust as defined inHigdon (1979b).

Species n Minimum Beat Wavelength N Wavespeed (Vw). αkFrequency (Hz) µm (λ) µm sec–1 (arb)

Salpingoeca 10 17 ± 0.31 17.92 ± 1.15 1.2 304.64 ± 1.58 1.7 ± 0.012amphoridiumCodosiga 9 10 ± 0.33 10.27 ± 0.51 0.81 45.6 ± 9.38 2.0 ± 0.034gracilisStephanoeca 4 10 ± 0 8.63 ± 0.47 0.96 41.3 ± 8.17 2.0 ± 0.021diplocostata

Table 3. Average collar characteristics and velocity measurements for Salpingoeca amphoridium, Codosiga gracilisand Stephanoeca diplocostata, expressed as the mean and the standard deviation for the data set. n is the number ofcells in the data set. CNo is the number of tentacles comprising the collar, CD the tentacle diameter. Collar Angledescribes the angle at which collar tentacles protrude from the protoplast, subtended to the vertical. Collar length ismeasured from the surface of the protoplast. hmin and hmax are calculated as the narrowest and widest gaps betweenadjacent collar tentacles. VHO is the head-on fluid velocity; the distance a particle travels in the 0.1 s before intersect-ing with the collar. Vmax is the maximum particle velocity observed.

Species n CNo CD Collar Collar hmin hmax VHO Vmax(µm) Angle (°) Length (µm) (µm) (µm) (µm sec–1) (µm sec–1)

Salpingoeca 10 27 0.1 19.9 8.05 0.1 0.70 26.0 204.0amphoridium ± 0.64 ± 0.01 ± 5.12 ± 2.50 ± 0.012 ± 0.081 ± 2.36 ± 13.21

Codosiga 9 30 0.15 17.2 4.5 0.11 0.39 20.5 109.0gracilis ± 1.67 ± 1.49 ± 0.009 ± 0.078 ± 2.11 ± 19.72

Stephanoeca 4 41 0.15 15.5 4.9 0.14 0.32 14.0 90.0diplocostata ± 1.26 4± 1.12 ± 0.011 ± 0.098 ± 1.28 ± 11.86

Table 1. Cell characteristics for Salpingoeca amphoridium, Codosiga gracilis and Stephanoeca diplocostata. Meanvalues, ± the standard deviation, are shown for all species and parameters with the exception of the height of Co-dosiga gracilis above the substratum, which is expressed as a range. n is the number of cells in the data set. FlagellarLength (L) was measured in non-feeding cells as the flagellum is characteristically held rigid and straight when notin motion.

Species n Flagellar Length (L) Cell Radius (A) Height Above L/A H/Aµm µm Substratum (H)

µm

Salpingoeca 10 20.65 ± 2.92 2.31 ± 0.68 6.20 ± 0.48 9.03 ± 0.31 2.75 ± 0.29amphoridiumCodosiga 9 8.30 ± 1.83 1.84 ± 0.16 12.9–23.5 4.68 ± 0.12 5.96–13.07gracilisStephanoeca 4 8.58 ± 1.95 1.81 ± 0.63 N/A 4.81 ± 0.32 N/Adiplocostata

This study did not use stroboscopic illumina-tion to determine the frequency of the flagellarbeat in these choanoflagellates. We are therefore,unable to state categorically that the flagellar beatfrequencies are, as given in Table 2, 17 and 10 Hzrespectively; they may be a multiple of these num-bers, but it is unlikely that any small multiplecould give the series of flagellar profiles shown inFig. 3.

Given that, in general terms, the motion of aflagellum can be described as:

Frequency × Wavelength = Wavespeed,

consideration of Table 2 shows that the observedwavespeeds for Codosiga gracilis and for Stepha-noeca diplocostata are approximately half thatwhich would be predicted from the above equa-tion. The likely cause for this disparity lies in thepossible three-dimensional nature of the wave-forms in these two species. Observations on theflagellar movement in Codosiga gracilis and

Stephanoeca diplocostata suggest that the wave-forms may have a helical component, a motion thatwould result in significant movement of the flagel-lum in the ‘z’ plane of focus. This unrecorded mo-tion may account for the deviation of the observedwavespeeds from those predicted. From the aboveequation the predicted wavespeeds for Codosigagracilis and Stephanoeca diplocostata are 102.7 µmsec-1 and 86.3 µm sec-1 respectively. These are com-parable with the observed maximum flow rates inthese species (Vmax in Table 3). This consistencylends credence to the hypothesis that there is acomponent to the flagellar waveform in Codosigagracilis and Stephanoeca diplocostata that couldnot be recorded using this method.It can be seen in Table 2 that Codosiga gracilis andStephanoeca diplocostata have similar flagellarwaveform and motion. The flagellar motion ofSalpingoeca amphoridium is distinct from theother species, having a higher beat frequency andthus generating a higher wavespeed. None of the


Fig. 3. Traces of the flagellar waveform in consecutive still frames, with a temporal separation of 0.02 sec. A singleflagellar beat cycle for, Codosiga gracilis 1A–F, Stephanoeca diplocostata 2A–F, and Salpingoeca amphoridium3A–D.


species considered exhibited a strictly sinusoidalwaveform. The waveform in all species showed in-creasing amplitude as the wave moved distallyfrom the cell. This is illustrated in Figure 3, whichshows one cycle of the flagellar beat, for each ofthe three species considered.

Fluid motion

Figures 4a–f, illustrate typical patterns of fluidflow generated by the two sessile species ofchoanoflagellate studied; Salpingoeca amphoridi-um and Codosiga gracilis. Two examples of fluidflow in Codosiga gracilis are presented, showingflow around a cell with a long (Fig. 4b and 4e) or ashort (Fig. 4c and 4f) pedicel, respectively. Figures4a–4c show the direction of particle movement inthe fluid flow induced by the flagellar beating. Fig-ures 4d–4f show details of some particle move-ments in close proximity to the collar. In Figures4d–4f solid circles represent the position of a parti-cle at 0.02s intervals. The dashed lines between thesolid circles represent the path the particle hastaken in the intervening period, and is a direct in-terpretation of the ‘smearing’ of particles seen onthe video recording

Figures 4g–4i show the streamlines generated bya line of stokeslets which, situated above an infiniteplane, act together to create a force akin to that cre-ated by a single flagellum. The length of the line ofstokeslets, and the distance above the infinite planeat which it is positioned, is varied in the formulationto approximate the parameters of flagellar length(L), and cell height (H), representative of Salpingoe-ca amphoridium and Codosiga gracilis. The streamfunction formulation is detailed in Appendix 1. Fig-ure 4g shows the streamlines generated when valuesof H and L similar to those for Salpingoeca am-phoridium are used, and is therefore directly com-parable to Figure 4a. Similarly Figure 4h and 4ishow streamlines generated when values of H and Lsimilar to Codosiga gracilis with a long and shortpedicel respectively, are used. These predictedstreamlines can be compared to the actual flow lines

generated by Codosiga gracilis with varying pedicellengths (Fig. 4b and 4c respectively).

Far field fluid flowFigures 4a–4c confirm the occurrence of re-cir-

culating viscous eddies in the outer flow fields ofSalpingoeca amphoridium and Codosiga gracilis.

The far field fluid flow generated by Salpingoecaamphoridium (Fig. 4a) contains a toroidal-shapedviscous eddy, the cross-sectional ‘centre’ of whichis approximately the same distance from the at-tachment substrate as the end of the flagellum. The‘centre’ of the eddy in the fluid flow generated byCodosiga gracilis (Fig. 4b) is also approximatelylevel with the end of the flagellum, but the opticalcross-section represented by the recorded stream-lines differs in shape from the eddy cross-sectionseen in Salpingoeca amphoridium. The cross-sec-tional shape of the eddy observed around Salpin-goeca amphoridium is roughly circular with itsmajor axis at 65° angle to the attachment substrate(see Fig. 4a). In Codosiga gracilis, the optical cross-section of the eddy is more elliptical in shape and isorientated so that the major axis is approximatelyparallel with the attachment surface.

The cross-sectional shape of the viscous eddy inCodosiga gracilis (Fig. 4b–4c) appears to be influ-enced by the parameter H, the height of the midpoint of the cell above the attachment substrate.The positioning of the ‘centre’ of the eddy, relativeto the end of the flagellum and the attachment sur-face remains the same. However, the shape of theeddy cross-section is not a symmetrical circle or el-lipse, it is flattened on the side adjacent to the at-tachment surface giving the cross-section a moresemicircular appearance.

The pattern of fluid flow generated by thestream function formulation (Fig. 4g–4i) also pre-dicts the occurrence of an eddy. The shape, and theposition of that eddy above the infinite plane altersas the input values of L, the length of the line ofstokelets, and H, the height of that line above theinfinite plane, are varied. The velocity field in thefar field decays as the square of distance.


Fig. 4. Fluid flow induced by flagellar beating in Salpingoeca amphoridium (4a/4d), Codosiga gracilis H = 23.5 µm(4b/4e), H = 12.9 µm (4c/4f). Streamlines generated by stokeslets force models with varying values of H and L(4g–i). Scale bars in Figures 4a–4c represent 5 µm. Scale bars in Figures 4d–4f represent 2 µm. Black dots in Figures4a–4c represent the extent (length) of the flagellum.

where A is the radius of the cell body, R is the dis-tance along the collar, taking the surface of the cellbody as zero, θ is the collar angle, CD is the collartentacle diameter and CN° is the number of collartentacles that comprise the collar, as illustrated inFigure 6.

The value VHO is the head-on fluid velocity, andis calculated from the distance a particle moves inthe 0.1 s prior to its intersection with the collar. AsVHO varies along the length of the collar, an averagevalue is derived from the intersection of (n × 10)particles with the collar halfway along its length.This was the best estimate of average VHO possible.Determining the full range of VHO from the proxi-mal to the distal extremes of the collar, in order tocalculate the average, was not feasible. At the distalend of the collar it was difficult to judge whether a

Near field fluid flowFigures 4a–4c illustrate that there is a strong di-

rectional streaming of fluid along the flagellar axis.This streaming is due to the net force that the flag-ellum is generating in that direction. Examinationof the flow in close proximity to the flagellum, Fig-ures 4d–4f, shows that there is an increase in thevelocity of flow close to the flagellum; the en-trained microsphere travels greater distances be-tween successive video frames. The velocity fieldin this region decays logarithmically with distance,a slow decay rate away from the flagellum.

In addition to an increase in the velocity of flowclose to the flagellum, there is an alteration in thenature of flow. Far field fluid flow is characterisedby the relatively smooth linear progression of themicrosphere. In close proximity to the flagellumhowever, lateral movements of the particle becomepronounced; flow becomes more oscillatory in na-ture (Fig. 4d–4f). The increase in lateral movementof the suspended microspheres may indicate thatfluid motion within the collar is closely coupled tothe lateral movements of the flagellum.

Figure 5 illustrates the details of fluid flowaround the lorica of the pelagic Stephanoeca diplo-costata.

Examination of Figure 5 reveals that fluid flowaround Stephanoeca diplocostata displays similari-ties to and differences from the flow around thetwo sessile species of choanoflagellate studied. Re-

circulation of fluid in Stephanoeca diplocostata wasnot observed. The viscous eddy associated withflow in Salpingoeca amphoridium and Codosigagracilis was not seen in this species. However, thestrong directional streaming of fluid flow along theaxis of the flagellum, a feature present in both Salp-ingoeca amphoridium and Codosiga gracilis, alsooccurs in Stephanoeca diplocostata. There is noperceptible increase in the velocity of flow as itpasses through the lorica; the distance a suspendedparticle travels between successive video framesdoes not increase until it passes through the collar.

Pressure drop and the influence of the collar

Table 3 summarises the collar characteristics ofthe three species of choanoflagellate studied, alongwith average and maximal flow velocities generatedby the flagellum. The collar angle describes theangle at which collar tentacles protrude from theprotoplast, subtended to the vertical, when the ver-tical plane is parallel to the axis of the flagellum.Since the collar tentacles, the filter elements in theseorganisms, emerge from the protoplast at an angle,they do not lie parallel to one another. Conse-quently, the gap between tentacles, h, increaseswith distance along the collar. Two values of h aregiven for each species. The minimum value of h,hmin, occurs when the filter elements are most tight-ly packed, i.e. as the collar emerges from the proto-plast. The maximum value of h, hmax, occurs at themost distal end of the collar. h is calculated as:

2π(A + R) sinθ – (CN° × CD) (1.1)CN°


Fig. 5. Fluid flow induced by flagellar beating inStephanoeca diplocostata. The scale bar represents 5 µm.The black dot represents the extent of the flagellum.

ticle is in close proximity to the flagellum (see Fig.4d–4g). This approach to studying fluid flow in ra-dially symmetrical organisms produces a two di-mensional representation of what is a three dimen-sional process. Consequently, it is impossible torecord particle movement in the third dimension(depth), and thus the velocities stated here are ‘bestestimates’

Table 3 documents a number of inter–speciesdifferences in collar morphology.

The collar of Salpingoeca amphoridium has thesmallest number of tentacles, which protrude fromthe protoplast at the largest collar angle. These twofeatures combine to give the collar of Salpingoecaamphoridium the highest values of hmax; it is a rela-tively porous structure. The collars of Codosigagracilis and Stephanoeca diplocostata are less porousthan that of Salpingoeca amphoridium. Althoughthe three species have similar values of hmin, the in-creased number of collar tentacles in the collar ofCodosiga gracilis, and more particularly Stephanoe-ca diplocostata, combine with a smaller collar anglein both species, to reduce the magnitude of hmax.Measures of velocity, VHO and Vmax are greater in


Fig. 6. Illustration detailing the collar parameters usedto calculate values of h, the distance between adjacentcollar tentacles.

Fig. 7. Pressure drop (atmospheres) across the collars of Salpingoeca amphoridium (SA), Codosiga gracilis (CG)and Stephanoeca diplocostata (SD) as a function of the distance along the collar ( µm), where zero is the midpoint ofthe cell.

particle actually intersected with the collar or wassimply swept around the end, and in the proximalzone particle velocities were too slow to be accu-rately determined. Vmax is the maximum measuredparticle velocity, which usually occurs when a par-


Salpingoeca amphoridium than in either Codosigagracilis or Stephanoeca diplocostata. This may resultfrom the longer flagellum in Salpingoeca amphorid-ium which may beat at a higher frequency.

The information summarised in Table 3 wasused in conjunction with slender body theory in acontrol volume to estimate and provide insight onthe pressure drop (∆p) across the collars of thethree species studied. In our calculations, fluid ve-locity and viscosity were regarded as being con-stant along the collar, and the pressure drop (∆p)was allowed to co-vary with increasing values of h(It would have been equally valid to maintain ∆p asthe fixed variable, and allow velocity to vary as afunction of h) The relationship between the pres-sure drop (∆p) and head-on velocity is linear, so aproportionate decrease in ∆p would occur if VHO

changed. Appendix 2 summarises the calculations,and results are presented in Figure 7.

Figure 7 shows that for all three species ofchoanoflagellate studied the pressure drop de-creases towards the distal end of the collar, as val-ues of h increase.

The magnitude of the pressure drop over thecollar is not the same in the three species studied.The collar of Stephanoeca diplocostata experiencesthe largest difference in pressure drop between thebase and the distal end. The pressure drop is rela-tively high at the base of the collar, decreasing to0.4 × 10–6 atm at the distal end. The averaged pres-sure drop over the entire collar length is 1.6 × 10–6

atm. The collar of Salpingoeca amphoridium hasthe lowest pressure drop of the three collars mod-elled. Pressure drop in this species varies from 6 ×10–7 atm close to the cell protoplast, to 5 × 10–8 atmat the far distal end of the structure (not shown),with an average pressure drop over the wholelength of 1.3 × 10–7 atm. The collar of Codosigagracilis experiences an intermediate pressure drop,ranging from 2.5 × 10–6 atm at the base, to 2 x 10–7

atm at the distal end of the collar, with an averageover the collar length of 5.0 × 10–7 atm.

Discussion

Comparing cell and flagellar morphologies

Higdon (1979b), derived optimal values for anumber of cell and flagellar parameters that max-imised flow rate (U/V), and/or minimised powerrequirement (η–1). U is the flow rate through a cir-cular disk of radius 2A averaged over one flagellarbeat cycle and V is the average linear wave speed.Higdon used this as an estimate of the potentialflow through the collar. The minimum power re-quirement is estimated as:

η–1 = P/6πµAU2

where P is the power consumption of the flagel-lum, µ is the viscosity of the surrounding medium,A the cell radius and U the flow rate definedabove.

The optima for some of these parameters, suchas L/A, the non-dimensionalised flagellar length,and H/A, the non-dimensionalised height, are in-terdependent. Under such circumstances, Higdon(1979b) calculated the optima for the dependentvariable for one or more values of the fixed vari-able. In the evaluation of this data, we derive theoptima for the independent variable from the clos-est approximation available of the fixed variable.Thus, in assessing the optimal value for H/A, wewould use the line of L/A = 5 for Codosiga gracilisand Stephanoeca diplocostata, but the line of L/A =10 for predicting the optimal height for Salpingoe-ca amphoridium (see Table 1). Higdon’s study wasbased on planar sinusoidal waves whereas there issome evidence of three dimensional flagellar mo-tion in our observations. This will introduce amore three dimensional component to our flowpatterns and will result in particles moving in andout of the focal plane. In a more elaborate study ofthe hydrodynamics generated by a helical beat,Orme, Blake and Otto (2002) predict a more heli-coidal averaged particle motion which retains thesmall scale oscillations seen in Figures 4d–f. Fig-


Figs 8–11. Redrawn with permission from Higdon (1979b). Figure 8, the relationship of U/V (a) and η–1 (b), to L/A.Figure 9, the relationship of U/V (a) and η–1 (b), to H/A. Figure 10, the relationship of U/V (a) and η–1 (b), to N, thenumber of wavelengths in a flagellum. Figure 11, the relationship of U/V (a) and η–1 (b), to αk, the maximum slopeof the flagellum.

ures 8–11 are adapted for the purpose of this studyfrom Higdon (1979b) and are reproduced withpermission.

Cell characteristicsThe non-dimensionalised average flow rate,

U/V, shows a hyperbolic relationship with L/A,the non-dimensionalised flagellar length (Fig. 8a).The average L/A ratio of 9.03 for Salpingoeca am-phoridium lies near the top of the linear part of thegraph, suggesting that the length of the flagella inSalpingoeca amphoridium may be optimised tomaximise flow rate. The L/A ratio in this species isslightly larger than the optimum for minimisingpower consumption, η–1 (Fig. 8b). The relationshipdescribed in Figure 8b shows a defined minimumat L/A = 7, with η–1 increasing linearly with Lthereafter. Codosiga gracilis and Stephanoecadiplocostata are cells of similar dimensions andproduced L/A ratios of 4.68 and 4.81 respectively.These are smaller than would be predicted to min-imise η–1, and produce values of U/V of just overhalf that of Salpingoeca amphoridium.

The sensitivity of U/V to H/A, the non-dimen-sionalised height, is dependent on the value of L/A,the non-dimensionalised flagellar length (Fig. 9a).When values of L/A are smaller, L/A = 5 for exam-ple, and all portions of the flagella lie near to the cellbody, the value of H/A has relatively little effect onflow rate. This results from short flagella being rela-tively ineffectual at generating flow close to a solidsurface. As the flagellum lengthens and becomesmore effective, and L/A increases, the effect of H/Aon flow rate becomes more pronounced. Despitethis, the values of U/V and η–1 are relatively insensi-tive to H/A as long as the height of the cell body ex-ceeds the length of the flagellum (H>L). Therefore,Higdon concluded that the height of the midpointof the cell being equal to the length of the flagellum(H = L) may be the lower limit for an efficient or-ganism. Salpingoeca amphoridium does not complywith this prediction. The average flagellar length (L)of Salpingoeca amphoridium is far in excess of theheight of the mid-point of the cell above the sub-stratum (H). Consequently, flagellar motion inSalpingoeca amphoridium is likely to be relativelyinefficient in terms of power consumption. Thegraph in Figure 9b predicts that η–1 will be over2000 for the case L/A = 10 (H/A for this species is2.75). This compares to values of η–1 of 1200–1400(L/A = 5) for Codosiga gracilis. Power requirementis low in Codosiga gracilis as values of H/A vary be-

tween 5.96–13.07. In all but one of the Codosigagracilis cells from which these calculations are de-rived, the height of the midpoint of the cell abovethe attachment substrate exceeded the flagellarlength, (H > L).

Flagellar characteristicsThe number of wavelengths in a flagellum (N)

has a defined optimum dependent on flagellarlength. For L/A = 10, the optimum N for bothU/V and η–1 is between 1.1 and 1.3, Figures 10aand 10b respectively. Salpingoeca amphoridiumconforms to this prediction with a value of N = 1.2.When L/A = 5, the optimal number of wavelengthsdrops below 1, for both U/V and η–1. Table 2 re-veals that Codosiga gracilis and Stephanoeca diplo-costata both have values of N of less than 1.

The parameter αk, which is a measure of themaximum slope of a flagellum, gives an indicationof the forward thrust the flagellum will generate.For all values of L/A, U/V increases approximatelylinearly with increasing values of αk (Fig. 11a).Consequently, no optimal value can be deter-mined. The relationship of η–1 to αk, does yieldL/A dependent optima (Fig. 11b). At L/A = 10, theoptimum is around 2.5, dropping to 2 when L/A =5. For all values of L/A, as αk falls below 1.5 thevalue of η–1 rises dramatically. Examination ofTable 2 shows that both Codosiga gracilis andStephanoeca diplocostata have values of αk close tothe predicted optimum of 2 (L/A = 5). In Salpin-goeca amphoridium the value of αk is 1.7, belowthe optimal value of 2.5 (L/A = 10). The shape ofthe relationship of η–1 to αk between the values 1.7and 2.5, determines that power consumption is rel-atively insensitive to changes in αk over this range.

In conclusion we can state that the flagellarwaveforms of the three species of choanoflagellateexamined conform to the optima predicted byHigdon (1979b) in terms of N, and for Codosigagracilis and Stephanoeca diplocostata also in termsof αk. The value of αk for Salpingoeca amphoridi-um is less than the predicted optimum, but theconsequential increase in η–1 will be small with lit-tle reduction in the efficiency of the organism.

Cell characteristics show less agreement with thepredicted optima. The flagellar length in both Co-dosiga gracilis and Stephanoeca diplocostata is short-er than would yield an optimal value of L/A = 7. Co-dosiga gracilis complies with Higdon’s assertion thatH = L may be the minimum ratio of cell height toflagellar length for an efficient organism. Salpingoe-


ca amphoridium does not, and consequently mayhave a higher power requirement to drive a flagellarmotion adapted to maximising flow rate.

Occurrence and description of viscous eddies

Validity of viscous eddiesThe occurrence of far field viscous eddies has

been described previously, in the flow field gener-ated by other protozoan species (Sleigh and Bar-low 1976). Descriptions of the position and shapeof these eddies has been criticised as being an arte-fact. It has been suggested (Liron and Blake, 1981;Fenchel, 1986), that the occurrence of these vis-cous eddies is due mainly to the influence of themicroscope slide and coverslip on fluid flow. Twoof the three species of choanoflagellate examinedin this study generated a flow field that containedan eddy structure. We do not believe these eddiesto be due to the external constraints of the appara-tus for the following reason. The maximum di-mension of any cell observed was slightly in excessof 25 µm, the minimum depth of the well in whichthe cells were contained was 500 µm. Therefore, incomparison with previous studies, which haveused larger protozoa and unmodified microscopeslides and coverslips, the choanoflagellate cellswere further from the influence of these solidboundaries

Similarities between theoretical and experimental fluid flows

It can be seen by comparing Figures 4a and 4gthat the eddy in the far field fluid flow of Salpin-goeca amphoridium resembles the eddy predictedby the stream function formula, for the same val-ues of H and L, in a number of respects. Themajor axes of the cross-sections of both actualand predicted eddies are orientated at an angle tothe attachment substrate/infinite boundary. Ob-servations suggest that this angle is greater for theeddy produced by the flagellar motion of Salpin-goeca amphoridium, than for the eddy predictedby the model. The model predicts that the eddywill be fairly ‘tight’, that is, that there will be rapidrecirculation of fluid. The fast velocities of fluidflow generated by the flagellar motion in Salpin-goeca amphoridium (Table 3), suggest that thisrapid recirculation may also occur. The positionof the eddy relative to the Salpingoeca amphoridi-um cell, agrees with the predictions of Liron and

Blake (1981), and Higdon (1979b), that the centreof the eddy will be parallel with the end of theflagellum.

The cross-sectional shape of the eddy predictedby the stream function formula when values of Hare large, and values of L are small, is a good matchfor the shape of the fluid flow generated by Co-dosiga gracilis with a long pedicel. In both actualand predicted fluid flows, the major axis of theeddy cross-section is orientated parallel to the at-tachment substrate. The centre of the eddy gener-ated by Codosiga gracilis occurs at a greater dis-tance from the cell than the eddy generated bySalpingoeca amphoridium. The eddy is also morediffuse in the former species, suggesting slower re-circulation of fluid. This prediction is in concor-dance with the documented velocity of fluid flowin Codosiga gracilis (Table 3).

Dissimilarities between theoretical and experimental fluid flows

The streamlines we use to model the flow fieldare represented by constant forces. In Stokes flow,this creates constant streamlines, ‘tori’, alongwhich particles will travel. A particle entrained in acircular torus will follow the same smooth pathand cycle indefinitely.

Experimental results do not support the exis-tence of smooth, constant streamlines. Examina-tion of Figures 4a–4f and Figure 5, reveals that theflow paths followed by particles are far fromsmooth. In far field flow (Fig. 4a–4c, Fig. 5), parti-cles exhibit minor lateral movements, rather than acontinuous smooth forward motion. In near fieldflow (Fig. 4d–4f), microspheres have a large oscil-latory component to their motion. We suggest thatthis discrepancy between theoretical and experi-mental descriptions of flow results, in part, froman incomplete modelling of the reality of flagellarmotion by the stream function formulation wehave used.

Such discrepancies may result from taking theforce created by the flagellum as being evenly dis-tributed, in one net direction (away from theboundary/substrate interface). This is not an accu-rate description of the motion the flagellum actual-ly undertakes. In practice, force is applied to thefluid by the whole surface of the flagellum, the ori-entation of which is changing constantly over thecourse of the flagellar beat cycle.

As the direction and magnitude of the force ex-erted by the flagellum on the surrounding fluid is


constantly changing, so are the flow streamlines.This constant re-orientation will be most pro-nounced in the fluid immediately adjacent to theflagellum, and may help to explain the oscillatorycomponent of microsphere movement in the nearfield flow. In addition, the constant re-orientationof the force applied by the flagellum may alter thetrajectory of microspheres, allowing them to crossfrom one streamline to another. This leads to amuch more complicated pattern of particle flow,reminiscent of the chaotic flow pattern describedby Orme, Otto and Blake (2001a, b).

Brownian motion may also have a rôle in mov-ing small particles from one streamline to another.Over one period of a flagellar beat it can be esti-mated that the root mean displacement for a0.5 µm microsphere attributable to Brownian mo-tion will be around 0.25 µm to 0.5 µm (i.e. 0.5–1particle radius). The influence of Brownian motionwill be more observable in the far field, because ofthe weaker influence of the deterministic flow fieldassociated with the flagellar beat pattern, but it canlead to streamline hopping by microspheresthroughout the flow field, resulting in a more ran-dom behaviour.

The presence of the collar may also destabilisethe streamlines. Fluid driven by flagellar motionwill diverge to pass around the collar tentacles. Al-though the collar is referred to as a static object,most collars exhibit 1–2 µm of lateral movement,relative to the point of anchorage, in response tothe lateral forces exerted by the flagellum. Conse-quently, streamlines are diverging around a non-stationary object. This may impose an additionallevel of randomisation on the particle flow pat-tern.

The influence of the boundaryThe shape of the eddy in the flow field of Co-

dosiga gracilis alters in response to varying valuesof H (Fig. 4b–4c). In Figure 4b, where the pedicellength is 23.5 µm, the cross-sectional shape of theeddy is approximately symmetrical about a majoraxis in the horizontal plane. This axis runs throughthe centre of the cross-section. In Figure 4c, wherethe pedicel length is 13.1 µm, this is not the case,and a flattening of the underside of the eddy is ap-parent. It may be hypothesised that this is the in-fluence of the boundary on flow close to its sur-face. Irrespective of the cause, it is worth notingthat none of the particles from which traces weretaken in Figure 4c, actually intersected with the

collar. This change in eddy shape is not accuratelypredicted by changing the value of the heightabove the infinite plane, H, in the stream functionformula. Reducing H, rather than producing thesemi-circular eddy cross section observed in cellsof Codosiga gracilis with a short pedicel, producesan eddy similar in cross-section to that producedby Salpingoeca amphoridium. The predicted eddyis orientated with the major axis at an angle to theattachment substrate. A reduction in the value ofH, also predicts a smaller eddy, closer to the appli-cation of the point force. This is not observed incells of Codosiga gracilis with short pedicels.

It is noteworthy that although the height of themid-point of the cell (H) in Salpingoeca amphorid-ium is considerably less than for Codosiga gracilis(Table 1) the influence of the boundary is not de-tected as a change in the cross-sectional shape ofthe eddy in the former species. This is consistentwith the observation that the mid-point of thecross-section of the eddy is the same distance fromthe attachment substrate as the tip of the flagellum.The greater flagellar length (L) and, to a lesser ex-tent, cell radius (A) in Salpingoeca amphoridiumcombine to ensure that the positioning of the eddyis outside the influence of the attachment sub-strate.

Effect of the loricaExamination of Figure 5, and an inter species

comparison of the values of VHO (Table 3), offersno evidence to support the notion that the silicalorica of Stephanoeca diplocostata acts as a ‘flume’increasing the velocity of fluid flow around thecollar, as proposed by Andersen (1989). Codosigagracilis and Stephanoeca diplocostata are specieswith similar flagellar waveform and dynamics, andsimilar collar characteristics. We may therefore,postulate that values of VHO in these species maybe similar. However, a Students ‘T’ test shows thatthe head-on fluid velocity (VHO), is significantly (p < 0.05) slower in Stephanoeca diplocostata thanin Codosiga gracilis. Although this is an interestingqualitative comparison, a quantitative analysis intothe effect of the lorica on the velocity of fluid flowwould require the use of a loricate species with andwithout its lorica. In practical terms, this is almostimpossible to achieve. Culturing loricate species inlow silica media will reduce the breadth of thecostal strips, but not remove them entirely (Lead-beater 1994). It may also interfere with other as-pects of their physiology.


∆P as a constraining feature on collar morphology and flagellar dynamics

Figure 7 illustrates pressure drop across the col-lar as a function of the distance from the protoplastand hence, increasing space between the tentacles(h). All the species of choanoflagellate studiedshow a similar response; pressure drop across thecollar decreases as the collar itself becomes moreporous and presents less resistance to flow. Ananalogous argument can be used to explain inter-species differences. The collar of Stephanoecadiplocostata has the highest ratio of filter elementsto filter pores (γ). This results from a higher num-ber of filter elements, 41, and a smaller collar anglesubtended to the vertical, 15.5°. The collar ofStephanoeca diplocostata consequently presents ahigher resistance to flow. This circumstance willresult in either a larger pressure drop, if velocity isthe fixed parameter, or a slower flow if pressuredrop is the reference constant. By contrast, the col-lar of Salpingoeca amphoridium has relatively fewfilter elements, just 27, and a larger angle subtend-ed to the vertical, 19.9°. Consequently, less resis-tance is offered to flow and pressure drop is small-er. For a given pressure drop over the collars ofStephanoeca diplocostata and Salpingoeca am-phoridium, we would expect to observe a fasterfluid flow for the latter, than for the former. In-spection of Table 3, reveals that that is the case.

Fenchel (1986), determined the pressure dropacross the filter feeding apparatus of a range offlagellates and ciliates, using the model of Tamadaand Fujikawa (1957). He concluded that pressuredrop is a conserved feature between species andmay therefore describe a physical limitation on fil-ter feeding. The results presented in this paper givetacit support to this assertion.

Fenchel (1986) expressed pressure drop in theCGS unit of dyne cm–2, and reported an estimatedrange of pressure drops from 9 to 16 dyne cm–2,with an average of around 10 dyne cm–2. One at-mosphere is equivalent to 1.01325 × 106 dyne cm–2,thus Fenchel (1986) predicted pressure drop tooccur within the range of 8.8 x 10–6–1.5 × 10–5 atm.Two choanoflagellates are reviewed by Fenchel,Diaphanoeca sp. and Monosiga sp. These specieswere calculated as having a pressure drop of 1.2 ×10–5 atm and 1.4 × 10–5 atm, respectively. Inspectionof Figure 6 reveals that the estimates of pressuredrop presented here are lower than those ofFenchel (1986). Although the lowest pressure drop

calculated, that associated with the distal collar ofSalpingoeca amphoridium, was around two ordersof magnitude less than the estimates of Fenchel(1986), there was some overlap in the two sets ofpredictions. Averaging pressure drop over thelength of the collar, results in estimates of pressuredrop closer to those proposed by Fenchel (1986).Average pressure drops range from 1.3 × 10–7 atmsfor the porous collar of Salpingoeca amphoridium,to 1.6 × 10–6 atms for the dense collar of Stephanoe-ca diplocostata. Considering the differences in themodels used to determine pressure drop in thesetwo studies, the consistency of the data does indi-cate that pressure drop is relatively constant in arange of filter feeding organisms.

A number of physical constraints could con-tribute to the narrow range of pressure drops ob-served in filter feeders. Filter feeders must generatesufficient flow to deliver prey to the filter surface,and have filters with adequate surface area and con-figuration to intercept those prey particles. At itsmost basic, this configuration of flow and filter ar-chitecture will determine the lower limit to pressuredrop. The upper limit of pressure drop may also bedetermined by considerations governing the deliv-ery of prey to the filter surface. The majority of bio-logical filters, so far described, are unbounded; thatis the fluid is free to diverge and pass around, as wellas through the filter. Increases in pressure drop,whether it be effected by an increase in flow rate orby an increase in the proportion of filter elements(γ), will encourage fluid to flow around rather thanthrough the filter. This would result in a decrease inthe rate of prey delivery to the filter and may influ-ence the upper limit of tenable pressure drop.

In conclusion this study has revealed that filterfeeding in choanoflagellates is not a standard pro-cess. There is wide variation in the feeding strate-gies adopted by different species.

Salpingoeca amphoridium maximises the rate offlow through and around its porous collar by rapidbeating of its long flagellum. Maximising the rate offlow rather than the efficiency of its cellular mor-phology and flagellar motion may ultimately be thestrategy that optimises phagocytosis and energyutilisation for this species. In spite of the larger gapsbetween adjacent collar tentacles, hmax, Salpingoecaamphoridium is able to maintain the pressure dropacross the collar by producing fast fluid flows.Maximising flow rate is energetically costly. Thismay indicate that Salpingoeca amphoridium isadapted to moderate or high prey environments.


Maximising flow rate has a number of interrelat-ed consequences for the cell and its prey capture.The forces generated by the beating flagellum haveto be balanced in a sessile organism by tensile orcompressive forces at the point of anchorage(Sleigh, 1991). These forces will be substantial inSalpingoeca amphoridium, and may provide an ex-planation of why cells of this species attach directlyto the substrate, rather than producing a pedicel.The compressive forces can be estimated as being ofthe magnitude µρβ2L/λ where µ is the viscosity ofthe surrounding fluid, ρ is the radian frequency, βthe amplitude of the flagellar beat, L is the flagellumlength whilst λ is the corresponding wavelength.

Applying this estimation to Salpingoeca am-phoridium suggests that the average compressiveforces generated by the flagellar motion in thisspecies will be 10–100 nN. A fine pedicel may notbe able to withstand such large compressive forces.As a consequence of this, cells of Salpingoeca am-phoridium will inhabit a region where the sub-strate to which they are attached may exert an in-fluence on the prevailing flow regime. In these re-gions, ambient flow may be reduced, and the highvelocities generated by the cell may be vital tomaintaining bacterial delivery to the collar surface.

Choanoflagellates may take several minutes tophagocytose the bacteria entrapped on the outsideof the collar. Retention of bacteria on the collarwhilst awaiting phagocytosis is therefore an im-portant aspect of feeding in these protozoa. As thevelocity of fluid passing through and around thecollar increases, so does the drag on the entrappedbacteria. Consequently, for a given prey type, asthe head-on fluid velocity (VHO) increases, the re-tention of bacteria drops (Shimeta and Jumars1991; Shimeta 1993). Salpingoeca amphoridium hasvery low levels of particle retention on its porouscollar (Pettitt 2001). Under these circumstances theoccurrence of an eddy in the flow field will be anadvantage to the feeding cell, as it allows recaptureof bacteria that have been lost from the collar. Thisis in contrast to Fenchel’s (1986), conclusions onthe effect of eddies. He states that the occurrenceof eddies would have a deleterious effect on sessilefilter feeders, as it would result in constant re-fil-tration of water already cleared of prey. Publica-tions by Fenchel however, have concentrated onlarger filter feeders, which may have higher reten-tion efficiencies than choanoflagellates. Underthese circumstances, an eddy would be disadvanta-geous, as there would be less recirculating prey,

and less time for new prey to be recruited from thebulk flow into the eddy structure.

Codosiga gracilis and Stephanoeca diplocostataadopt an alternative strategy to Salpingoeca am-phoridium. They optimise their cellular morpholo-gy and flagellar motion for energy efficiency. Inthese two species this may be the strategy that op-timises phagocytosis and energy utilisation, andmay indicate that both species are adapted to preypoor or prey limited environments. The smallergaps between adjacent collar tentacles, hmax, inthese two species impose a slower rate of flow inorder to maintain pressure drop. The compressiveforces, generated by the slower flagellar motion inthese two species, will be lower than those experi-enced by Salpingoeca amphoridium. As a conse-quence, Codosiga gracilis is able to overcome therestrictions in fluid flow associated with its attach-ment substrate, by producing a pedicel.

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Appendix 1: Estimating pressure dropSlender Body Theory (SBT) is one technique that can

be used to estimate the pressure drop across the collar ofthe three species of choanoflagellate studied. Other the-ories that could be used to estimate pressure drop in-clude lubrication theory, which relies on the gap be-tween the filter elements, h, being of much smallerwidth than the radius of the filter element, χ. In the caseof choanoflagellates, variation exists in the width of the

gap between the collar tentacles, h. The gap is smallclose to the cell protoplast, and large at the distal end ofthe collar. By varying the value of h in the formulationfor pressure drop, it will be possible to demonstratewhere pressure drop is greatest, or velocity smallest, asthey are linearly related.

The estimate of pressure drop is derived by examin-ing force balances between pressure drop and the forceneeded to move fluid through the collar. The pressuredrop over the gap between the tentacles is specificallycompared to the force per length needed to move fluidthrough the collar. All values used in the calculations arein dimensional form, and are expressed in units of sec-onds, µm and kg.

The force balance per unit length between two tenta-cles leads to an expression for the pressure change(Batchelor 1971). The force per unit length on each ten-tacle can be calculated approximately as:

4πµVHO (1.2)log (b/χ)

where µ is the dynamic viscosity of the fluid; derivedfrom the kinematic viscosity and density of the fluidand equals 10–1 kg µm s–1. VHO is the velocity of fluidflow in the direction perpendicular to the collar in µms–1. b is the gap between the centre of the tentacles ( µm)as shown in Figure 12, and χ is the tentacle radius ( µm),(Batchelor 1971).

Fig. 12. A horizontal section through two adjacent ten-tacles, showing the location of the parameters used inSBT to determine pressure drop (∆p).


Since the force per unit length must be equal to thepressure drop multiplied by the gap between the tenta-cles, ∆p x b, where ∆p denotes pressure drop, we obtain,

∆p × b = 4πµVHO (1.3)log (b/χ)

Therefore, on rearranging we obtain the relationship:

∆p = 4πµVHO (1.4)b log (b/χ)

The variation in pressure drop can therefore be com-puted over the whole collar length as a function of thegap between adjacent tentacles. To enable us to computethe variation we require that: the radius of the collartentacles is a fixed parameter, and that the velocity offlow approaching perpendicular to the collar is constantat all distances along the collar.

An equally valid approach that could be adopted,would be to designate pressure drop as the constantvalue, and determine the velocity necessary to force fluidthrough the varying gap, h, between the filter elements.Average pressure drop over the whole collar is calculatedusing the approximate discretization formula as:

∆p = 1 n

(1.5)n ∑ ∆ pii=1

Where n is the total number of intervals at whichpressure drop is determined, and ∆pi is the pressuredrop at each interval.

Appendix 2: Stream function formulationIn Liron and Blake (1981) the streamlines generated

by a single point force above an infinite boundary werecalculated. This paper builds on that work by more ac-curately representing the flagellum as a line of pointforces. The length of this line, and the height at which itis positioned above the infinite boundary, are altered toapproximate to the various flagellar lengths and cell

heights from the substrate seen in the two sessile speciesof choanoflagellate studied.

The force generation by the line of point forces thatmodel the flagellum acts, on average, normal to theboundary above which it is situated. An axisymetricflow field will therefore develop. This allows us to in-troduce a Stokes stream function to represent the distri-bution of forces along the flagellum.

If we represent the radial and vertical directions bythe polar co-ordinate system (r, z) then the stream func-tion (Ψ) is defined in Aderogba and Blake (1978) by:

Ψ= F r2 r2 hir2z ]8πµ [(r2+ (z+hi)2)1/2 –(r2+(z+hi)2)1/2

+(r2+(z+hi)2)3/2

(1.6)

where the parameters are defined as follows:µ represents the viscosity of the fluid that surrounds

the cell. hi gives the distance above the boundary that theforce is situated. When the force is spread over the lengthof the flagellum, hi varies for each point in that spread.

F is the strength of the force acting. The force may beacting at one point, or be distributed along a ‘stretchedstraight’ flagellum. We non-dimensionalise the flagellarlength by dividing through by the height above theboundary. Thus the parameters of H and L used in thestream function formulation to model Salpingoeca am-phoridium will be H = 1, L = 3.33 (see Table 1). The forcethat represents the action of the flagellum is then subdi-vided into a number of points spread at even intervalsalong the flagellar length. The longer the flagellum, thegreater the number of points, and the greater the spreadof the force. Thus the force per interval (FINT) would be:

FINT = F/10

when 10 intervals were used.

When representing the streamlines generated in twodimensions, the co-ordinates (r, z) are defined as thehorizontal and vertical axes respectively.

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The hydrodynamics of filter feeding in choanoflagellatesklinck/Reprints/PDF/pettittEJProt2002.pdfThe hydrodynamics of filter feeding in choanoflagellates Michala E. Pettitt 1,*, Belinda