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How Microorganisms Move through Water: The hydrodynamics of ciliary and flagellar propulsion reveal how microorganisms overcome the extreme effect of the viscosity of water Author(s): George T. Yates Reviewed work(s): Source: American Scientist, Vol. 74, No. 4 (July-August 1986), pp. 358-365 Published by: Sigma Xi, The Scientific Research Society Stable URL: http://www.jstor.org/stable/27854249 . Accessed: 18/07/2012 14:21

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http://www.jstor.org George T. Yates How Microorganisms Move through Water

The hydrodynamicsof ciliaryand flagellar propulsion reveal how microorganisms overcome theextreme effect of the viscosityof water

the various The various organisms that propel of and the forces experienced by the fields of engineering, an and themselves through aqueous smallest organisms. physics, applied mathematics, a and environment span over 19 orders of A single drop of water from biology (see for example Gray mass. one or Holwill Wu et al. magnitude in body At pond stream will usually contain Hancock 1955; 1966; extreme are blue whales, the largest hundreds of organisms invisible to 1975; Brennen and Winet 1977). animals ever to inhabit the earth, and the naked eye, a large percentage of are un at the other are microscopic bacteria. which capable of swimming The effects of viscous In terms of absolute the der their own There are more speed, larger power. flows organisms swim faster;when speeds than 50,000 known species of Proto zoa more relative to body size are compared, alone, with new ones being de The larger and familiar free an more as however, microorganisms are the scribed at average rate of swimming animals, such fish and a true champions. Single-celled organ than one day, and most swim at cetaceans, derive their propulsive some water isms, typically between 2 and 1,000 stage of their life cycle (Jahn et thrust by accelerating back (xm long, can sustain swimming al. 1979). Besides being found swim ward, taking advantage of the equal com to speeds up to about 100 body lengths ming freely, microorganisms and opposite inertial reaction force overcome per second. Large pelagic fish like monly occur inside larger organisms, the viscous fluid force or tuna can maintain speeds of only where they exist in symbiotic para which would otherwise slowly arrest are re about 10 body lengths per second, sitic relationships, and some theirmotion. The dominance of iner cases while the world record for human sponsible for disease. tial forces in these is evident maneuvers a swimmers is just over one body Detailed observations of the from the of large ship a an length per second. movements of these tiny organisms approaching harbor. As experi an accom In all cases, organism became possible only after the inven enced mariner will explain, the snip's must or even plishes self-propulsion by putting its tion of the lightmicroscope. The elec power be turned off, or body appendages through period tron microscope has added a new reversed, well before the ship icmovements. After one cycle of the dimension to the study of theirmotil reaches the dock, because the action on motion, the organism and its appen ity, by providing sharp, highly mag of viscosity the hull only gradually mo dages return to their original configu nified fixed images for close examina affects the ship's overwhelming ration, but themean body position is tion of themicrostructures associated mentum. moved forward. The types of appen with locomotion. With the aid of The ratio of inertial to viscous em a dages used, the types of motion these powerful tools, cilia and flagel forces in fluid flows is indicated by ployed, and the very nature of the la have been identified as the cellular nondimensional quantity called the are a fluid forces encountered differ great appendages whose movements Reynolds number. For body of di ly for the various swimmers through most frequently responsible for self mension L moving with velocity U in out a the vast size range. This article propulsion of microorganisms (Fig. viscous fluid, the inertial force is will concentrate on the movements proportional to p(UL)2, the viscous To appreciate fully how propul force varies like (xUL, and the Reyn sion is achieved and olds number is defined as Re = George T. Yates is Senior Scientist in Engineering by ciliary flagel pUL/ lar it is essential to under where is the and the Science at Caltech. He received his B.S. from activity, jx, p density jjl Purdue and hisM.S. and Ph.D. stand some fundamental of University from principles viscosity the surrounding fluid. Caltech. His research on a re or focuses fundamental fluid of hydrodynamics. After brief For fish humans, the Reynolds mechanics, with on we special emphasis view of the relevant principles, number is typically about one mil current biofluiddynamics. His interests include the will look at the structures of cilia and lion, and inertia dominates the fluid mechanics and physiology of swimming and With of both flu motion. As the dimension di extraction winds and flagella. knowledge body flying, optimal energy from id and in hand, we minishes to the scale of currents, and The work dynamics biology microscopic free surface flows. will then be able to discuss the me viscous forces dominate described in this paper has been carried out with organisms, chanics of in detail. This inertial forces, and the support from theNational Science Foundation and propulsion Reynolds a subdivision of number becomes 0.1 or smaller. At theOffice ofNaval Research. Address: subject, biofluiddyn is its nature interdisci the even smaller dimension of the Engineering Science, Caltech, Pasadena, CA amics, by very 91125. plinary; it requires collaboration by individual cilia and flagella, which

358 American Scientist, Volume 74 oscillate typically between 10 and 30 cycles per second, the Reynolds number is usually less than 10~5. Because the inertial effects are of no significance at these small scales of motion, microorganisms are unable to take advantage of the inertial reac tion and must derive their forward thrust from viscous reactions. The typical viscous forces which act on a human swimmer are at least ' 6 orders of magnitude smaller than _ i the inertial forces. To the 1 ji.m appreciate nature of the viscous forces experi enced by microorganisms, we would have to swim in a fluid one million times more viscous than water. Even a in pool filled with honey the rela tive strength of viscous to inertial forces would be two orders ofmagni tude smaller than it is formicroorga nisms swimming in water, and a more appropriate fluid would be mo lasses or even tar. Whenever the Reynolds number is small, the rate of change in mo over mentum time can be neglected relative to the viscous and pressure components, and time becomes sim ply a parameter. This means that the fluid motion generated in response a to microorganism's movement de pends on its instantaneous velocity, and not on its velocity at any previ ous instant or on the rate of change of any other quantity with time. Fur thermore, because the fluid inertia (mass times acceleration) is negligibly small, any external force must be balanced by the viscous stress and the pressure in the fluid. can now We begin to appreciate the difficulties encountered in achiev ing self-propulsion at low Reynolds numbers. For a swimmer?for exam 1. Cilia and are the Figure flagella ple, a human in an ocean of molas appendages that propel microorganisms ses?who moves his arms from a water. are from through They distinguished to a final con each other by their length relative to the cell given starting position as the instantaneous fluid body, well as by their structures and figuration, are characteristic movements. Flagella often force depends linearly on the velocity are attached to one end of the body, and of the motions. When the more than twice the of the body frequently length swimmer not body. The bacterium Salmonella abortus stops moving, only a does his absolute motion but all equi (top) has many flagella, which form stop, bundle that is behind the 10 comes clearly visible jJLlTl the surrounding fluid to rest cell in this electron The bundle and thus body micrograph. intermittently unwraps reforms, immediately, because there are no a sea Ciona is changing the swimming direction. A spermatozoon from squirt (bottom left) inertial effects. Furthermore, if the to 200 flashes shown swimming in sea water in this series of flash photographs (top bottom, swimmer retraces the taken with a The wave is and to exactly paths per second) light microscope. propulsive planar propagates followed his arms to their initial the the the cell moves to the left. Cilia (bottom by right along flagellum. Consequently, right), he will find himself are numerous in rows on the configuration, which are usually much shorter than the body, and arranged the the cell (and the in the same body surface. Gradual variation in the phase of the beat cycle along length of fluid) exactly fromwhich he started. How leads to the coordinated wave pattern seen in this scanning electron micrograph of the position then can creatures microorganism Paramecium. (Top photograph from Routledge 1975; bottom left photograph living accomplish from Omoto and Brokaw 1982; bottom right photograph from Tamm 1972.) self-locomotion under such condi

1986 July-August 359 tions? The answer to this question length of the cilium and are joined by The mechanics of lies in the detailed dependence of the cross arms or bridges which permit viscous forces on the flow velocity them to slide against each other (Satir propulsion themotion of and orientation, which I will discuss 1974). By the bonding and unbond The fluid force resisting an at a after looking at the tools nature has ing of these cross arms, the cilium object moving low Reynolds can a number to the provided to enable microorganisms generate bend anywhere along is linearly proportional and to swim freely in their environment. its length. velocity of the object, depends on the for Eukaryotic flagella actively bend flow orientation. Except to The structures of cilia and along their length in two- or three simple geometries, it is difficult and dimensional waves, and prokaryotic calculate this force the resulting fluid velocities. thin flagella flagella rotate as more or less rigid The long, geom cilia and lends itself to The etymology of the terms cilium helices. Although the internal struc etry of flagella some and flagellum indicates not only the tures and mechanisms of producing simplifying approximations that to appearance of these structures but motion of eukaryotic and prokaryotic have been used develop resis also the type of movement they per flagella are greatly different, we will tive force theory and slender body two have been form. "Cilium" is derived from the consider themechanics of propulsion theory. These theories or in the Latin word meaning eyelid, and of both classes together. The single highly successful explaining brings to mind many closely spaced multiple flagella of these microorga mechanics of ciliary and flagellar pro are one hairs capable of rapid, synchronized nisms relatively long (at least pulsion. are movement. "Flagellum" comes from half and frequently considerably For objects that much longer more than are wide and have circular the Latin word forwhip, suggesting than the length of the body) they cross fluid unit a long, flexible rope which can be and are often attached to one end of sections, the force per into a made to move in a wide variety of the body. The body length of flagel length is conveniently split and a trans ways. A closer examination of both lates rarely exceeds 50 |mm,while longitudinal force (Fs) verse two the structure and the movement of their flagella are often longer than force (Fn). These compo nents can be cilia and flagella reveals some impor 100 jam. of force estimated from the tant differences which give them spe Cilia are relatively short (much formulas or are cial advantages over eyelashes less than the body length), usual = and are in Fs -|xCsUs whips. ly numerous, arranged and rows on In both prokaryotes (bacteria) the body surface. In terms of = - are Fn ^CnUn and eukaryotes (organisms that have overall body size, ciliates general a mem some their cell nuclei enclosed in ly larger than flagellates, with where Us and Un are the tangential brane, such as protozoans and algae) species attaining lengths of 1 to 2 and normal velocities of the particu and in the sperm cells of higher mm. The individual cilia have a fairly lar cross section relative to the fluid. animals, the propulsive organelle has regular beat pattern with two distinct The constants of proportionality Cs a or traditionally been called flagellum, phases. The forward (power effec and Cn are generally different and now a but it is considered form of tive) stroke is rapid, with the cilium depend on the shape of the body, are cilium in eukaryotes (Jahn et al. fully extended. The return (recovery) but independent of the fluid vis as a 1979). The relatively rigid, helical stroke, which appears if bend cosity (jland the velocities Us and Un. are = shaped flagella of bacteria about were propagated from the base to the The ratio of the force coefficients 7 0.02 |xm in diameter and are con tip of the cilium, is slower, with the Cs/Cn takes on the value 1 for a a structed from subunits of protein cilium bent close to the cell surface. sphere and approaches 0.5 in the a core we a called flagellin with about As shall see, this asymmetry in limiting case of slender body that an re 0.003 |mm in diameter. Although it the beat is essential feature for becomes longer and longer while varies greatly from species to species, effective functioning of the cilium. taining the same width. the length of a bacterial flagellum is Because the phase of the beat These two formulas constitute more one almost always than 100 times cycle varies gradually from cili the basis of resistive force theory. um over an its thickness. A rotary joint attaches to another the body, The essential features of this theory wave the flagellum to the cell body and organized is observed. This are illustrated in Figure 2, where it is it to rotate wave same a allows during locomotion may travel in the direc applied to the settling of long, thin as or an an a (Berg 1975;Routledge 1975;Macnab tion the effective stroke in rod inclined at angle from the and Aizawa 1984). opposite direction. The types of both horizontal. The buoyancy force Fb, wave are Cilia, eukaryotic flagella, and the and beat patterns widely the object's mass times the accelera essen numerous flagella of sperm cells all have varied among the species tion of gravity minus the hydrostatic tially the same anatomy, which dif of ciliates. The control of the coordi pressure force, acts downward and, fers greatly from that of bacterial nated motions of the cilia in these in the absence of inertia, must be a flagella. Nine outer pairs of microtu single-celled organisms, which lack balanced by the viscous fluid forces. a nervous un bules surround central pair of mi system, remains largely The surprising result is that the rod a crotubules, all contained within known. The most likely underlying does not fall straight down in the cross roughly circular section about mechanisms include changes in direction of the buoyancy force but 0.2 or concen fxm in diameter. These microtu membrane potentials ion rather has a component of velocity can or bules bend but are nearly inex trations, with without hydrody perpendicular to gravity. This lateral run tensible. They almost the entire namic coupling. migration of the rod occurs only

360 American Scientist, Volume 74 < when y 1 and becomes more pro Stokes equations are linear in the ing combined flow may represent the as a nounced 7 becomes smaller. fluid velocity and therefore allow the swimming of microorganism. use a a The possibility of self-propulsion of general solution technique As basic building block, we a as use is direct result of y being less than known the singularity method. will a solution of Stokes equa one. an a Take the example of ice skat This technique is commonly applied tions which corresponds to placing er a a who is in themiddle of pond and to problems throughout engineering point forcewith given magnitude F = wishes to reach the bank without and science when the governing 8tt|xoi in the fluid. The fundamen are lifting his skates off the ice. If the equations linear. If two different tal singular solution of this problem, were to use a a skater pair of poorly solutions of Stokes equations are called stokeslet (Figs. 3 and 4), has a = a designed skates forwhich Cs Cn (7 known, say Ui and u2, then third velocity field which decays like 1/r, = u r 1), then the resistive force would solution can be found by taking a where is the distance from the = be aligned parallel to the motion and linear combination of these two (u external force to themeasuring point. a + he would be unable to produce net Aui Bu2, where A and B are This extremely slow decrease of ve no forward movement, matter how arbitrary constants). The process can locity with distance from the distur hard he tried. Making application of be repeated again and again, and bance is especially noteworthy: it in a the results shown in Figure 2, he thus few relatively simple funda dicates that the stokeslet will have a use a can con should pair of skates that have mental solutions be used to far-reaching influence on the flow a a very small resistance to movement struct whole family of solutions. In field. parallel to the blades (small Cs) and the present application, various fun Ifwe allow mass to be created at a that have very large resistance to damental solutions can be distributed a point inside the flow field, Stokes or sideways movement (large Cn). By along the cilia flagella, cell body, equations admit another singular so at an a holding his feet outward-point and flow boundaries, and the result lution, source (Fig. 4), which has a and ing angle pushing them apart, radially outward flow velocity that he will begin to move forward (just falls off like 1/r2.A sink is the oppo as a the rod in Figure 2 drifts laterally). site of source, having an inward When he pulls his feet back together, flow velocity. From appropriate com he should point his toes inward to binations of stokeslets, sources, and a produce further forward propul sinks, other higher-order singulari sion. ties which satisfy the governing The major strengths of resistive equations can be obtained (Batchelor force theory are its simplicity and 1970; Blake 1971; Chwang and Wu ease relative of application for obtain 1976). One of these, the potential ing approximate results. Its weak doublet, is the limiting case of a nesses are source a that appropriate values for and sink approaching each are the resistive force coefficients not other. It results in a velocity field always obvious, the local fluid veloci which decays like 1/r3. ties are not predicted, and interac A potential doublet with = tions between the flagellum and the strength ? a3U/2 satisfies the or are cell body flow boundaries diffi boundary condition of no normal ve a cult to account for in systematic locity on a sphere of radius a translat manner. more The rigorous slender ing?that is, moving without rotat overcomes body theory these diffi ing?with constant velocity U. This culties and provides a systematic doublet also satisfies Stokes equa method of approximating the flow tions; however, there are tangential field and on 2. Resistive force on hence the forces long, Figure theory helps explain fluid velocities the sphere surface, thin Slender is the fluid forces acting on an organism which are in viscous objects. body theory whose movement is dominated viscous impossible based on the fundamental by flows. For an at a low concepts effects. The is illustrated here an object moving of mass and momentum conserva theory by the fluid inanimate object, a long, slender rod settling Reynolds number, velocity tion at low which at a at the must Reynolds numbers, under the action of gravity low body surface be equal to result in a number. The rod system of linear partial Reynolds migrates laterally the velocity of the body, a require as it falls. The net force is differential equations, the so-called buoyancy Fb ment called the balanced the fluid resistive force F, no-slip boundary Stokes These by condition. This condition can be sat equations. equations, which has two components, one normal (Fn) with condi isfied for a appropriate boundary and theother tangentialto the rod (FJ. sphere by linearly super can be in a These forces are to the a a = tions, solved variety of linearly proportional posing stokeslet of strength normal and of the = ways; I will outline one ap tangential components 3aU/4 and a doublet of ? only rod's and A strength which has been velocity, U? Us respectively. -a3U/4 The of proach successfully relation between a and 0 results (Fig. 3). importance to the of microor simple the stokeslet is seen when applied swimming from the quotient Fs/Fn, and depends only strength The = we ganisms. advantage of slender on the ratioof the forcecoefficients 7 Q/ consider the resulting fluid force whenever a is 0? or 6 body theory is that solutions of de Cn. By symmetry, 90?, found by integrating the stresses sired can be and is zero, and thus 0 must have a maximum over accuracy obtained, = the surface of the which for some intermediate values of a. For 7 sphere, its is that it can ? the Stokes formula disadvantage require 0.5, this occurs when a 35? and results in yields drag tedious and involved a maximum lateral of the rod with computations. migration ? ? It 0 ? 19?. = = should be emphasized that F $7T\X0L 6Tru,aU

1986 July-August 361 From this we note that the mean interactions strong among adjacent force theory to each differential ele stokeslet is related cilia, the cell and ment strength directly body, neighboring along the length of a flagellum, to the mean external force on cells or and this in turn acting boundaries, Gray and Hancock were able to esti the This association remains causes some in body. difficulty obtaining mate the fluid force and moment per valid for all bodies results or in out undergoing gener analytic carrying unit length. The total force and mo al motions at a low num numerical ment are Reynolds computations. then obtained by summing ber. In approximating the flow field the contribution from each element. More difficult can be far from the we can combine If the problems body, organism is self-propelling, the solved the various like into sin total mean by distributing singularities equivalent thrust of the flagellum within a or on the singularities body gularities positioned at the center of (the total mean fluid force in the surface, and the cell. the detailed flow body by adding together Although direction of motion) must be equal to the fields from all the velocities close to the are velocity singu very body the drag of the head, which is ap larities. The of the lost, the overall fluid flow as seen strengths singular proximated by the drag on a sphere ities are chosen so that the some distance from the can of radius no-slip organism a?the Stokes drag formula condition on the sur be boundary body easily estimated by only one or noted above. face is satisfied. the two The Evaluating equivalent singularities (Fig. 5). rate of work done by the strengths involves a solving compli flagellum per unit length amounts to cated of system integral equations. How and Cn(Un)2 + CS(US)2. When this is inte The slenderness of cilia and fla flagellates over grated the entire flagellum and allows the flow to ciliates swim gella singularities averaged over time, and the result is be distributed an along organelle's Using resistive force coefficients, added to the rate of work done in center line. It further all the Hancock and and Han permits (1953) Gray moving the head through the fluid, stokeslets outside cock made contributions mean singularities except (1955) major the total power consumption a certain "near field" to be to research on region flagellates when they of the can be found. We = = organism neglected, and thus the 0.5 and can then integral proposed using 7 Cs/Cn estimate the efficiency tj of are Ve a formula for which equations greatly simplified. simple Cs de this mode of propulsion as the ratio locities due to distant singularities pends only on X/b, where X is the of the mean rate at which useful with distance and of the wave and decay rapidly length propulsive b work is done to the totalmean power make contribu is the radius of the cross section only higher-order (Fig. consumption. Using this definition, tions. The stokeslet must be retained 6). Further refinements of the coeffi Lighthill (1975) has shown that the the entire because cients have been made for three along body length, maximum efficiency of propulsion its field dimensional with a velocity decays very slowly, shapes, 7 ranging for general planar wave is bounded off, as we have seen, like 1/r. between 0.6 and 0.7 falling (Cox 1970; Tillet by The wide-ranging influence of the 1970;Lighthill 1976). stokeslets that there will be implies By direct application of resistive

stokeslet doublet doublet + stokeslet* 3. Slender resistive force Figure body theory supplements theory, tangential to the surface. To conserve linear momentum, the a method of providing systematic approximating the flow field pressure forces, viscous forces, and external forces must be an always around object the Stokes with the in a by solving equations no-slip equilibrium. For stokeslet, the viscous forces and the pressure condition. The use of boundary flow singularities and the balance forces combine to balance an external force. In the case of a of forces in a viscous fluid at low numbers are illustrated the Reynolds doublet, total normal stress exactly balances the net shear stress, here the distribution of fluid forces on an by imaginary sphere, and an external force cannot be balanced. The combination of a which are attributable to the pressure (black arrows) and to viscous doublet and a stokeslet of appropriate strengths yields the flow forces (color arrows). The viscous forces have two a field components, around the sphere. The stokeslet and doublet have both normal stress perpendicular to the surface and a shear stress magnitude and direction (indicated by a and ?).

362 American Scientist, Volume 74 stokeslet source doublet

Figure 4. The flow fields resulting from three simple singular solutions of Stokes equations?a stokeslet, source, and potential doublet?are shown here. The magnitude of the fluid velocity varies along the streamlines, decreasing in all cases to zero far from the singularity. The magnitude and direction of the stokeslet and doublet are indicated by a and ?. The circles show the location of the sphere used for evalu ating the fluid forces in Figure 3. = Clearly if 7 1, the propulsive effi pattern would be more advanta and Brokaw 1979). zero no ciency is and swimming is geous. Three basic models for the pro possible, which agrees with our earli Using slender body theory to pulsion of ciliates have been devel er an example of ice skater. Propul study flagellate swimming, Higdon oped to adapt resistive force and sive efficiency is generally poor for (1979) and Johnson (1980) have slender body theory to the special organisms that swim at low Reynolds found unfformly valid solutions for characteristics of cilia, their relatively = numbers, because, for 7 0.5, tj is the entire flow field around microor large number, short length, beat pat less than 0.09; itbecomes even small ganisms. Direct comparisons be tern, and coordinated movements: er for larger values of 7. tween the resistive force model and the envelope model, the sublayer a The force distribution along slender body theory indicate that if model, and the traction layer model. are assumes planar beating flagellum will vary the resistive force coefficients in The envelope model from thrusting in the regions where creased by about 35%, with 7 remain that the effects of all the individual can the flagellar segments are undergo ing nearly unchanged, better agree cilia be combined and the instan ing transverse motion to drag in the ment with the more precise slender taneous positions of the cilia tips can a regions where they align nearly par body theory is obtained (Johnson be considered as "body surface," allel with the swirnming direction. This variation causes an increased energy consumption for a given thrust (Lighthill 1976). In contrast, a spiral or helical wave generates thrust uniformly along its length and thus may lead tomore effective pro pulsion. Since they rotate, bacterial fla gella must use helical motions for propulsion. Spiral beat patterns are also commonly observed in eukary otic flagella. For planar beat patterns the linear forces balance; for helical waves, however, the torque must also be balanced, and this require ment cannot be fulfilled without a rotation of thewhole organism. Rota tionmust, in turn, cost the organism Figure 5. The role of the singular solutions of Stokes equations is illustrated by the marked a additional energy. For organisms differences in the flow fields of free-swimming organism (right) and one being drawn the fluid an external force The flow is revealed in these with spherical heads and helical fla through by (left). time-exposure 1 in diameter. The streaks made the gellarbeats, Chwang andWu (1971) photographs by suspended particles |xm by particles found the head size to form streamline patterns which can be compared to those in Figure 4. In both the flow optimum created a dead under toward between 10 and 40 times pattern by paramecium (P. multimicronucleatum) settling gravity range great the bottom of the and the flow around a er than the radius of the picture pattern living paramecium swimming freely flagella. in a horizontal plane toward the lower left, the net stokeslet is proportional to the efficiencies from strength Propulsive ranged external force acting on the body. For a free-swimming organism, where the external forces 0.10 to 0.28. For with are organisms relatively insignificant, the sum of all the viscous forces and thus the resultant stokeslet smaller heads less than about a (a/b 5), strength become insignificant. Hence, far from the body, the velocity field behaves like a they concluded that planar beat potential doublet.

1986 July-August 363 an sheet on which the no all the elemental on an oscillating singularities all ling cell and of object forced condition is the slip boundary applied. cilia and the image singularities. through the fluid are very different model Blake solved Taylor's (1951) original has this integral equation (see Fig. 5), and their interactions been extended by Blake (1971) and for the swimming speed by taking with external boundaries must be Brennen whose studies time thus the different as (1974), pre averages, obtaining well. For free-swimming dicted that maximum ve mean of the fluid propulsive velocity profile ciliates, the flow field decays rapidly and would be at to locity efficiency tangential the wall. The sublayer away from the body, and we can tained when the cilia underwent tips model is used when temporal varia expect the effects of walls tomanifest an and obtained tions in are not elliptic trajectory velocity needed, and themselves only when the organism reasonable estimates of overall swim when the cilia are and is a widely spaced very close to boundary (Winet to observed the interactions between them are a ming speeds compared 1973). The effects of boundary may swimming speeds. The envelope weak. be more evident when the thrust model is when the concen In an effort to account for more justified the force is widely separated from tration of cilia is and when their over high variations in velocity time, the drag forces, as in the case of wave move in the same di which can patterns be quite large, Keller and flagellates. rection as the effective stroke. Itmust his a colleagues (1975) introducedthe The head of flagellate can be be abandoned whenever the flow traction which as an layer model, smooths considered inert object moving inside the is to ciliary layer be consid out the ciliary forces to form a contin under an external force, and there is ered or the of an uous no assumption imper body force inside the ciliary question that, for the same ap meable surface is as be and retains the ve violated, may layer oscillatory plied force, the velocity will decrease the case forwave in patterns moving locities. The resulting iterative solu when the head moves close to a solid a direction to the effective tion shows that the in opposite oscillations boundary. At low Reynolds numbers stroke and for cilia. are of the same widely separated velocity order of this boundary effect is very strong, The first as the mean flow can sublayer model, pro magnitude compo and the speed easily be reduced Blake addresses nents posed by (1972), the within the ciliary layer and by more than 5% even when the flow within the where ciliary layer, decay exponentially outside the lay object is ten head radii from the wall on the force each cilium, and hence er. (Lee and Leal 1980). the stokeslet is found a strength, using For slender body moving near slender Since the cilia a body theory. The effects ofwalls wall, both the normal and tangen operate close to a wall (i.e., the cell tial resistive force coefficients in the and In are crease. body), singularity types nature, motile cells found both It is important to note that Cn must be modified. For a at some distance in strengths from and the increases proportionally more than stokeslet above a a = single wall, sys immediate neighborhood of walls. Cs, so that 7 Cs/Cn decreases and tem of which in Almost all observations of even image singularities, swimming may fall below 0.5 (Yang and cludes another stokeslet of are made in the vi Leal equal microorganisms 1983). Recalling that propulsion but strength opposite sign and two cinity of solid boundaries in the form becomes more efficient as 7 is re other is higher-order singularities, of glass microscope slides. Both in duced, we see that there are two needed below the wall to the nature and in the the a satisfy laboratory, competing effects as flagellate ap condition on the of walls a no-slip boundary presence may considerably proaches solid boundary: one to wall The flow field is an (Blake 1971). modify organism's motions. enhance the performance (increase then as an over a represented integral TTie flow fields of self-propel T|) and the other to retard the motion (increase the drag). The current data are insufficient to tell which effect predominates. Although data on the swimming speed of flagellates in relation to the a are distance from wall obscured by variations in the speed of individuals and possible changes in the rate of work, several other observations clearly show the important influence of the walls. Flagellates swim in curved near a paths solid boundary. For eukaryotic flagellates the typical ly three-dimensional beat pattern is 6. The kinematics of a are an Figure swimming flagellate illustrated by organism with a flattened and the head of radius a and a of diameter moves approaches limiting spherical single flagellum 2b. As the organism to case of a two-dimensional the at a constant mean a planar right speed V, propulsive wave propagates backward the along beat. In addition, the beat at c. In this the remains in the of the frequency flagellum speed simplified example, flagellum plane is and both the wave h the wave X are lowered (Katz et al. page, amplitude and length constant. The highlighted noticeably of the is v segment flagellum moving downward with velocity in addition tomoving 1981). forward with V. For a velocity The segment's total velocity U can be resolved into normal and flagellate swimming paral The F can lel to thewall and a tangential components. viscous force per unit length be estimated by resistive using helical beat force or theory by slender body theory. It has a component in the direction of motion (thrust) pattern, the segments of the flagel which, for a is balanced the on the head. lum near free-swimming organism, by drag the wall experience larger

364 American Scientist, Volume 74 forces than those parts away from Brennen, C, and H. Winet. 1977. Fluid me Lee, S. H., and L. G. Leal. 1980. Motion of a chanics of a the wall, and the forces and mo propulsion by cilia and flagella. sphere in the presence of plane interface. Ann. Rev. Fluid co ments on the head are in turn Mech. 9:339-98. Part 2; An exact solution in bipolar acting A. on ordinates. Fluid Mech. 98:193-224. Besides the for Chwang, T., and T. Y. Wu. 1971. A note /. changed. altering the helical movement of M. 1975. Mathematical ward and the rate of rotation microorganisms. Lighthill, J. Biofluiddyna speed Proc. Royal Soc. London B178:327-46. mics. SIAM. around the direction of -. 1976. -. swimming, Hydromechanics of low-Reyn 1976. Flagellar hydrodynamics: The this asymmetrical distribution of olds-number flow. Part 4: Translation of John von Neumann lecture, 1975. SIAM Rev. forces creates a torque which rotates spheroids. /. Fluid Mech. 75:677-89. 18:161-230. R. the organism in the plane of thewall. Cox, G. 1970. The motion of long slender Macnab, R. M., and S.-I. Aizawa. 1984. Bacte bodies in a 1: Thus, cells move in viscous fluid. Part General rial motility and the bacterial flagellar motor. free-swimming Fluid Mech. 44:791-810. Ann. Rev. 13:51-83. curved near a wall. theory. /. Biophys. Bioeng. paths and G. Hancock. 1955. The C. and C Brokaw. 1982. Struc A of the effects Gray, J., J. propul Omoto, K., J. rigorous analysis sion of sea-urchin spermatozoa. /. Exp. Biol. ture and behaviour of the sperm terminal ofwalls can be solv accomplished by 32:802-14. filament. /. Cell Sei. 58:385-409. the Stokes distri G. ing equations using Hancock, J. 1953. The self-propulsion of Routledge, L. M. 1975. Bacterial flagella: Struc butions of singularities. Blake (1971) microscopic organisms through liquids. ture and function. In Comparative Physiolo an Proc. Soc. London A217:96-121. Structural Materials, constructed image system for an Royal gy?Functional Aspects of ed. L. H. P. and K. a J. J. L. 1979. The of Bolis, Maddrell, isolated stokeslet in the vicinity of Higdon, hydrodynamics Helical waves. Fluid Schmidt-Nielsen, pp. 61-73. North-Holland. wall, and others have flagellar propulsion: /. rigid expanded Mech. 94:331-51. Satir, P. 1974. The status of the on this present sliding concept (Yang and Leal 1983). micro-tubule model of motion. In Holwill, M. E. J. 1966. Physical aspects of ciliary between observations movement. Cilia and ed. M. A. 131? Comparisons flagellar Physiol. Rev. 46:696 Flagella, Sleigh, pp. and these theories should lead to 785. 42. Academic. some S. L. 1972. motion in Parame interesting work in the future. Jahn, T. L., E. C. Bovee, and F. F. Jahn. 1979. Tamm, Ciliary How to Know the Protozoa. 2nd ed. Wm. C. cium. J. Cell Bio. 55:250-55. Both resistive force theory and Brown. G. I. 1951. of the slender body theory shed consider Taylor, Analysis swimming R. of Proc. Soc. able on the basic features of Johnson, E. 1980. An improved slender microscopic organisms. Royal light for Stokes flow. Fluid Mech. London A209:447-61. fluid flows at low body theory /. very Reynolds 99:411-31. Tillett, J. P. K. 1970. Axial and transverse numbers. The influence Stokes flow slender bod far-reaching Johnson, R. E., and C. J. Brokaw. 1979. Flagel past axisymmetric of the stokeslet is notewor lar ies. Fluid Mech. 44:407-17. especially hydrodynamics. A comparison between /. as resistive-force and the H. on thy, is its association with the total theory slender-body Winet, 1973. Wall drag free-moving on a ]. 25:113-27. ciliated Biol. external force body. Microorga ory. Biophys. micro-organisms. /. Exp. 59:753-66. nisms as Katz, D. F., T. D. Bloom, and R. H. BonDur accomplish self-propulsion ant. a direct result of the fluid force's 1981. Movement of bull spermatozoa in Wu, T. Y., C. J. Brokaw, and C. Brennen, eds. cervical mucus. Bio. 25:931-37. 1975. and in vols. 1 on Reproduction Swimming Flying Nature, the flow orientation and 2. dependence Keller, S. R., T. Y. Wu, and C. Brennen. 1975. Plenum. and the movement of the A traction asymmetric layer model for ciliary propulsion. Yang, S. M., and L. G. Leal. 1983. Particle cilia or In and in ed. T. Y. near a organisms' flagella along peri Swimming Flying Nature, motion in Stokes flow plane fluid C. and a odic paths. Our knowledge remains Wu, J. Brokaw, C. Brennen, vol. 1, fluid interface. Part 1: Slender body in pp. 253-72. Plenum. fluid. Fluid Mech. 136:393-421. incomplete, especially regarding the quiescent J. effects of boundaries, interactions among cilia, three-dimensionality of the flows, and swimming in nonlin ear fluids such as mucus. There is no doubt that our understanding will continue to be expanded by collabo ration among the various branches of biology and engineering and by close interaction between observation and School of Olo?Y analysis. AhlTUROP.>\' arcvaa?.n.o ,pACTER?.toy ei. References Q4TOIA.217 etym. 221 Batchelor, G. K. 1970. Slender body theory for cross ?e. -m particles of arbitrary section in Stokes ? flow. /. Fluid Mech. 44:419-40. PAl?OHT.113 - - 31' Berg, H. C. 1975. Bacterial movement. In PUYSt Swimming and Flying inNature, ed. T. Y. Wu, .209 C. J. Brokaw, and C Brennen, vol. 1, pp. 1 7 11. Plenum. Toxic.So R. 1971. A note on Blake, J. the image system a a for stokeslet in no-slip boundary. Proc. Cambridge Phil Soc. 70:303-10. -. 1972. A model for the micro-structure V in ciliated organisms. /. Fluid Mech. 55:1-23. C. 1974. An Brennen, oscillating-boundary layer theory for ciliary propulsion. /. Fluid Mech. 65:799-824.

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