We present the first time-resolved measurements of the oscillatory velocity
field induced by swimming unicellular microorganisms. Confinement of the green
alga C. reinhardtii in stabilized thin liquid films allows simultaneous
tracking of cells and tracer particles. The measured velocity field reveals
complex time-dependent flow structures, and scales inversely with distance. The
instantaneous mechanical power generated by the cells is measured from the
velocity fields and peaks at 15 fW. The dissipation per cycle is more than four
times what steady swimming would require.Comment: 4 pages, 4 figure

E. H. Harris

J. P. Gollub

Jeffrey S. Guasto

K. Drescher

Karl A. Johnson

S. Childress

English

arXiv

We present the first time-resolved measurements of the oscillatory velocity field induced by swimming unicellular microorganisms. Confinement of the green alga C. reinhardtii in stabilized thin liquid films allows simultaneous tracking of cells and tracer particles. The measured velocity field reveals complex time-dependent flow structures, and scales inversely with distance. The instantaneous mechanical power generated by the cells is measured from the velocity fields and peaks at 15 fW. The dissipation per cycle is more than 4 times what steady swimming would require. A pdf of this published paper has been uploaded elsewhere on this page. Guasto was a post-doctoral student of Jerry Gollub\u27s. --author supplied descriptio

Guasto, J. S.

Johnson, Karl A.

Gollub, Jerry P.

Haverford College: Haverford Scholarship

Haverford College Haverford Scholarship Faculty Publications Biology 2010 Oscillatory Flows Induced by Microorganisms Swimming in Two Dimensions J. S. Guasto Haverford College Karl A. Johnson Haverford College, kjohnson@haverford.edu Jerry P. Gollub Haverford College Follow this and additional works at: https://scholarship.haverford.edu/biology_facpubs Repository Citation Guasto, Jeffrey S., Karl A. Johnson, and Jerry P. Gollub. "Oscillatory flows induced by microorganisms swimming in two dimensions." Physical review letters 105.16 (2010): 168102. This Journal Article is brought to you for free and open access by the Biology at Haverford Scholarship. It has been accepted for inclusion in Faculty Publications by an authorized administrator of Haverford Scholarship. For more information, please contact nmedeiro@haverford.edu. Oscillatory Flows Induced by Microorganisms Swimming in Two DimensionsJeffrey S. Guasto,1 Karl A. Johnson,2 and J. P. Gollub1,31Department of Physics, Haverford College, Haverford, Pennsylvania 19041, USA2Department of Biology, Haverford College, Haverford, Pennsylvania 19041, USA3Department of Physics, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA(Received 26 July 2010; published 11 October 2010; publisher error corrected 14 October 2010)We present the first time-resolved measurements of the oscillatory velocity field induced by swimmingunicellular microorganisms. Confinement of the green alga C. reinhardtii in stabilized thin liquid filmsallows simultaneous tracking of cells and tracer particles. The measured velocity field reveals complextime-dependent flow structures, and scales inversely with distance. The instantaneous mechanical powergenerated by the cells is measured from the velocity fields and peaks at 15 fW. The dissipation per cycle ismore than 4 times what steady swimming would require.DOI: 10.1103/PhysRevLett.105.168102 PACS numbers: 47.63.Gd, 47.15.G, 47.61.k, 87.16.QpMotile single cells use a variety of strategies for loco-motion in low Reynolds number environments. Somemethods produce roughly steady swimming (e.g., rotationof helical flagella in E. coli), while others produce oscil-latory cell motion by beating flagella in a periodic manner[1]. Understanding the hydrodynamics of swimming mi-croorganisms can give insight into complex biologicalprocesses such as predator-prey interactions and flagellarmechanics [2,3]. A variety of mathematical models havebeen proposed to describe interactions between swimmingcells and their environment [1,4,5]. However, experimentalstudies of velocity fields are rare [6], and oscillatory near-field hydrodynamics have not been investigated experi-mentally despite their importance [7,8].The flows induced by swimming organisms are respon-sible for biogenic mixing [9], and microorganisms in par-ticular enhance the transport properties of suspensions[5,10]. Active particles and microorganisms also affectthe rheological properties of suspensions, where the effec-tive viscosity is increased in the case of ‘‘pullers’’ (e.g.,Chlamydomonas reinhardtii) and decreased for ‘‘pushers’’(e.g., E. coli) [11,12]. Hydrodynamic interactions leadingto coherent motion have been studied for concentratedbacterial cells [13], but not for algal cells.In many of these phenomena, the oscillatory swimmingmotion of organisms may be important. Furthermore, theconcentration of swimmers may be sufficiently high thatorganisms are within several radii of one another, closeenough that near-field effects should be considered.Understanding the resulting time-dependent flow fields isrequired for the interpretation of interactions betweenswimming cells and their environments.We present the first time-resolved measurements of theflow field around oscillatory swimming microorganisms(C. reinhardtii). This is accomplished through simulta-neous tracking of swimming cells and passive tracer par-ticles within quasi-two-dimensional (quasi-2D) thin liquidfilms, where direct optical access and long observationtimes are achieved. These measurements reveal complexnear-field flow structures that evolve throughout the beatcycle, and provide a richer picture than time-averaged flowfields and simplified models. The instantaneous mechani-cal power transferred by flagella to the fluid is measuredvia the viscous dissipation. These observations carry im-portant implications for the interpretation and modeling oftransport processes, locomotion, and flagellar mechanics.C. reinhardtii is a single-celled swimming alga with a7–10 m diameter body [14]. It typifies many biflagellatedmicroorganisms with two anterior 10–12 m long flagellathat beat at 50–60 Hz propelling the cell in an oscillatorymanner at a mean speed of 100–200 m=s (see video[15]). The flagella, actuated along their lengths by dyneinmotors, take on asymmetric conformations during thepower and recovery strokes to overcome the time revers-ibility of low Reynolds number flows.Wild-type C. reinhardtii (cc1690) is grown in minimalmedia (M1) on a light cycle (16 hr bright/8 hr dark) toensure uniform size (radius R  3:5 m) and motility[14]. An adjustable wire-frame device [12] is used tostretch a 2 l droplet of the cell suspension into a squarefilm with side length L ¼ 6 mm and measured thicknessh ¼ 15 2 m. A trace amount of surfactant (Tween 20)stabilizes the film without harming the cells. Polystyrenemicrospheres (1 m diam., Thermo Scientific), passi-vated with bovine serum albumin to prevent adhesion toflagella [16], are incorporated with the cells prior tostretching the film.The device is mounted on an upright microscope (400:75 NA objective) where suspensions are observed underbright field illumination with red light (>610 nm) to pre-vent phototaxis [17]. A high-speed digital camera capturesthe motion over 3000 frames (50 or 500 fps). Cells andtracer particles are segmented from one another by size andtracked using a predictive algorithm [18].The thin liquid film ensures that the algae are coincidentwith the focal plane of the objective and prevents distortionPRL 105, 168102 (2010)Selected for a Viewpoint in PhysicsPHY S I CA L R EV I EW LE T T E R Sweek ending15 OCTOBER 20100031-9007=10=105(16)=168102(4) 168102-1  2010 The American Physical Societyfrom out-of-focus particles. This allows detailed observa-tion of the cells (mean speed U0 ¼ 134 m=s) forup to about 8 s at 50 fps. Using high-speed imaging(500 fps), oscillations of the cell body, UðtÞ, becomeevident [Fig. 1 (inset)]. The peak forward velocity is4 times the mean value and can be negative during therecovery stroke. The probability density function (PDF) ofthe beat frequency f for 170 cell tracks is shown in Fig. 1.The oscillations have a fairly narrow distribution with amean frequency f ¼ 53 5 Hz corresponding to a beatcycle period T ¼ 1= f ¼ 18:9 ms.By simultaneously tracking swimming cells and passivetracer particles (see video [15]), we measure the velocityfield induced by the cells by translating and rotating theinstantaneous tracer particle measurements to a commoncoordinate system based on cellular orientation. Swimmertrajectory segments with large curvature or irregular beatfrequencies are excluded [19]. The beat-cycle averagedvelocity field is measured by imaging the cells at a framerate comparable to the cell beat frequency (50 fps), thuscapturing the net motion over a cycle. The velocity fieldresulting from 560 cell tracks is shown in Fig. 2(a), wherethe swimmer motion is to the right and solid lines areinstantaneous streamlines [20].While the general shape of the flow field has somequalitative similarities to a force dipole (‘‘stresslet’’) ap-proximation [1], several unexpected features are apparent[6]. The location of the hyperbolic stagnation point faraway from the anterior of the cell is surprising, since ittypically is thought to be located between the centers ofdrag (body) and thrust (flagella). Two strong vortices arevisible lateral to the organism (associated with the fla-gella), while two weaker vortices appear far from the cellbody beyond the separatrix of the hyperbolic point.The fluid-air interfaces of the liquid film are nearlystress-free, which minimizes velocity gradients transverseto the film. This effect along with comparability of the filmthickness and cell size (h=2R 2) creates a quasi-2Denvironment [21], where we expect longer-range hydro-dynamic disturbances compared to 3D flows. Fluid veloc-ity magnitude normalized by the mean swimmer speed,u=U0, where u ¼ juj, is shown in Fig. 2(b) as a function ofradial distance from the organism, r=R, in various direc-tions. Toward the posterior, fluid speed scales nearly as ur1 up to 20 cell radii away, similar to a force dipole in 2D[22], with slight deviations from u r1 likely due tominor 3D effects. These results are consistent with similarmeasurements in 3D that show the expected u r2 scal-ing [6]. In the lateral direction, the velocity magnitudepasses through a local minimum when traversing the para-bolic stagnation points (vortex), before recovering u r1scaling in the far field. Similarly, in front of the cell, fluidFIG. 1 (color online). Probability density function (PDF) offlagellar beat frequencies, f, measured from unicellular alga(C. reinhardtii) swimming in quasi-2D liquid films ( f ¼ 535 Hz). Inset: velocity oscillations of a single swimming cellmeasured at 500 fps (red circle), where the maximum velocityis 4 times the mean value (solid blue line, 134 m=s).FIG. 2 (color online). (a) The beat-cycle averaged velocityfield around a swimming C. reinhardtii (black disc) in the labframe, where the direction of travel is to the right toward thehyperbolic stagnation point (green diamond). Solid (red) linesare instantaneous streamlines and velocity vectors are shown ona log scale [20]. (b) The fluid velocity magnitude in variousdirections away from the cell demonstrates the predicted ur1 scaling for a force dipole in 2D. Local minima correspond tostagnation points encountered in some directions.PRL 105, 168102 (2010) P HY S I CA L R EV I EW LE T T E R Sweek ending15 OCTOBER 2010168102-2speed decreases rapidly near the hyperbolic stagnationpoint 7–8 radii from the swimmer.The time-averaged velocity field in Fig. 2(a) shows thelimitations of some current swimmer models, as also notedin [6], and raises several important questions. For example,how does the velocity field evolve throughout the oscilla-tory flagellar beat cycle? Using high-speed imaging(500 fps), we measure the instantaneous swimmer phase,and identify tracer particles at corresponding times in thebeat cycle. Velocity fields (resulting from 170 cell tracks)are constructed from tracer velocities at each phase of theoscillation with resolution T=15.A time series of the velocity field during the beat cycle isshown in Fig. 3 (see video [15]). Insets show swimmerspeed and phase (lower left) and approximate flagellarposition (lower right). At the beginning of the power stroke[Fig. 3(a)], the velocity field is neatly divided into foursymmetric quadrants with the hyperbolic stagnation pointlocated slightly forward from the body, in line with con-ventional force dipole swimmer models [1]. As the flagellamove toward the posterior and the power stroke peaks, thevortices lateral to the organism strengthen [Fig. 3(b)], thenshift across the body to the anterior side [Fig. 3(c)–3(e)].After the power stroke ends and the recovery stroke begins,the flagella extend out in front of the organism, and the cellvelocity becomes negative. The flow shown in Fig. 3(f) isqualitatively reversed from Fig. 3(a), changing the sign ofthe dipole. The instantaneous flow field generated by anoscillatory swimmer such as C. reinhardtii is complex andhighly time dependent.In Stokes flows, all of the mechanical energy generatedby a swimmer for locomotion is rapidly dissipated by thefluid. The power transferred to the fluid by the organism iscalculated from the velocity field gradient through theviscous dissipation P ¼ R 2ð:ÞhdA, where  is thefluid viscosity,  ¼ 12 ½ruþ ðruÞT is the rate of straintensor, and dA is a differential area element of the film[15]. The instantaneous mechanical power output, PoscðtÞ,is calculated from the velocity fields [Fig. 3] throughout thebeat cycle and shown in Fig. 4(a). The peak power output(15 fW) occurs during the power stroke and correspondsto the maximum instantaneous speed of the cell body. Asecondary local maximum also occurs at the peak speed ofthe recovery stroke [see Fig. 4(b) (inset)]. Because of theFIG. 3 (color online). Time sequence of the velocity field evolution throughout the beat cycle (period T ¼ 18:9 ms) of C. reinhardtii(oriented to the right) including the hyperbolic stagnation point position (green diamond). Insets show cell speed and beat cycle phase[lower left, see Fig. 4(b) for details], and approximate flagellar shape (lower right, measured separately). (a) Early in the power stroke,the velocity field resembles a (negative) force dipole. (b) At the peak of the power stroke, the vortices lateral to the organism strengthenand sweep toward the posterior. (c)–(e) The vortices then shift to the anterior as the power stroke is completed. (f) At the peak of therecovery stroke, the flow field again takes the shape of a dipole, but with opposite sign. The recovery stroke velocity field (f) is weakerthan the forward stroke, but is enhanced by the log scaling [20].PRL 105, 168102 (2010) P HY S I CA L R EV I EW LE T T E R Sweek ending15 OCTOBER 2010168102-3oscillatory swimming, the average mechanical power out-put over the beat cycle, hPoscðtÞi, is more than 4 times thepower computed from the time-averaged velocity field inFig. 2(a) (i.e., hPoscðtÞi=Pmean flow  4:2).The mechanical power scales with the cell body speed[Fig. 4(b) (inset)] as PU2 with deviations at small Udue to the finite accuracy of measuring tracer velocities[Fig. 4(b)]. This behavior might be expected for drag-basedthrust at low Reynolds number, since the forces scalelinearly with the body speed FU, and the power outputof the organism is P FUU2 [23].This work represents an important step toward under-standing the locomotion of swimming microorganisms andcollective behaviors for which hydrodynamics are respon-sible. Measurements of the time-averaged velocity fieldaround C. reinhardtii in quasi-2D thin liquid films demon-strate the predicted u r1 scaling of the fluid velocitywith detailed resolution of the near-field flow features.Also, high-speed imaging allows for phase-resolved mea-surements of the velocity field throughout the flagellar beatcycle, revealing complex underlying flow structures thatevolve in time. From these measurements, we demonstratethat the mechanical power dissipated by these swimmingmicroorganisms is more than 4 times what the mean ve-locity field predicts. This has important consequences re-garding the input of chemical energy required to poweroscillatory swimming of flagellated cells.We thank R. E. Goldstein, K. Drescher, E. Lauga, andM.D. Graham for helpful discussions, as well as B. Boyesfor technical assistance. This work was supported by NSFGrant No. DMR-0803153.[1] E. Lauga and T. R. Powers, Rep. Prog. Phys. 72, 096601(2009).[2] J. Sheng et al., Proc. Natl. Acad. Sci. U.S.A. 104, 17512(2007).[3] T. J. Mitchison and H.M. Mitchison, Nature (London)463, 308 (2010).[4] G. P. Alexander, C.M. Pooley, and J.M. Yeomans, Phys.Rev. E 78, 045302(R) (2008).[5] P. T. Underhill, J. P. Hernandez-Ortiz, and M.D. Graham,Phys. Rev. Lett. 100, 248101 (2008).[6] K. Drescher et al., preceding Letter, Phys. Rev. Lett. 105,168101 (2010).[7] T. Ishikawa, G. Sekiya, Y. Imai, and T. Yamaguchi,Biophys. J. 93, 2217 (2007).[8] C.M. Pooley, G. P. Alexander, and J.M. Yeomans, Phys.Rev. Lett. 99, 228103 (2007).[9] K. Katija and J. O. Dabiri, Nature (London) 460, 624(2009).[10] K. C. Leptos et al., Phys. Rev. Lett. 103, 198103 (2009).[11] S. Rafaı¨, L. Jibuti, and P. Peyla, Phys. Rev. Lett. 104,098102 (2010).[12] A. Sokolov and I. S. Aranson, Phys. Rev. Lett. 103,148101 (2009).[13] X. L. Wu and A. Libchaber, Phys. Rev. Lett. 84, 3017(2000).[14] E. H. Harris, The Chlamydomonas Sourcebook (AcademicPress, Oxford, 2009), Vol. 1.[15] See supplementary material at http://link.aps.org/supplemental/10.1103/PhysRevLett.105.168102.[16] D. B. Weibel et al., Proc. Natl. Acad. Sci. U.S.A. 102,11 963 (2005).[17] O. A. Sineshchekov, K-H. Jung, and J. L. Spudich, Proc.Natl. Acad. Sci. U.S.A. 99, 8689 (2002).[18] N. T. Ouellette, H. Xu, and E. Bodenschatz, Exp. Fluids40, 301 (2006).[19] M. Polin et al., Science 325, 487 (2009).[20] The velocity field is calculated at 3 m resolution butshown at 6 m resolution using log scaling, where avelocity doubling increases the displayed vector lengthby 0.4% of the panel width.[21] V. Prasad and E. R. Weeks, Phys. Rev. Lett. 102, 178302(2009).[22] A. T. Chwang and T. Y. Wu, J. Fluid Mech. 67, 787(1975).[23] S. Childress, Mechanics of Swimming and Flying(Cambridge University Press, Cambridge, 1981).FIG. 4 (color online). The mechanical power dissipated bythe organism is calculated from the velocity fields. (a) Theinstantaneous dissipation PoscðtÞ is maximal when the swim-mer speed is highest. The dissipated power time-averagedover the beat cycle hPoscðtÞi is more than 4 times the dissipa-tion measured from the time-averaged velocity field (i.e.,hPoscðtÞi=Pmean flow  4:2). (b) The measured power scaleswith the square of the cell body speed, Posc  U2. Inset: meancell body velocity as a function of beat cycle phase.PRL 105, 168102 (2010) P HY S I CA L R EV I EW LE T T E R Sweek ending15 OCTOBER 2010168102-4

Oscillatory Flows Induced by Microorganisms Swimming in Two Dimensions

We present the first time-resolved measurements of the oscillatory velocity field induced by swimming unicellular microorganisms. Confinement of the green alga C. reinhardtii in stabilized thin liquid films allows simultaneous tracking of cells and tracer particles. The measured velocity field reveals complex time-dependent flow structures, and scales inversely with distance. The instantaneous mechanical power generated by the cells is measured from the velocity fields and peaks at 15 fW. The dissipation per cycle is more than 4 times what steady swimming would require

Guasto, Jeffrey S

Johnson, Karl A

Gollub, Jerry P

Kosmopolis

University of Pennsylvania
ScholarlyCommons
Department of Physics Papers Department of Physics
2010
Oscillatory Flows Induced by Microoganisms
Swimming in Two Dimensions
Jeffrey S. Guasto
Haverford College
Karl A. Johnson
Haverford College
Jerry P. Gollub
University of Pennsylvania, jgollub@haverford.edu
Follow this and additional works at: http://repository.upenn.edu/physics_papers
Part of the Physics Commons
Suggested Citation:
Guasto, J., K.A. Johnson, and J.P. Gollub. (2010). Oscillatory Flows Induced by Mircroorganisms Swimming in Two Dimensions. Physical Review
Letters. 105, 168102.
Copyright 2010 American Institute of Physics. This article may be downloaded for personal use only. Any other use requires prior permission of the
author and the American Institute of Physics. The following article appeared in Physical Review Letters and may be found at http://dx.doi.org/10.1103/
PhysRevLett.105.168102.
This paper is posted at ScholarlyCommons. http://repository.upenn.edu/physics_papers/8
For more information, please contact repository@pobox.upenn.edu.
Recommended Citation
Guasto, J. S., Johnson, K. A., & Gollub, J. P. (2010). Oscillatory Flows Induced by Microoganisms Swimming in Two Dimensions.
Retrieved from http://repository.upenn.edu/physics_papers/8
Oscillatory Flows Induced by Microoganisms Swimming in Two
Dimensions
Abstract
We present the first time-resolved measurements of the oscillatory velocity field induced by swimming
unicellular microorganisms. Confinement of the green alga C. reinhardtii in stabilized thin liquid films allows
simultaneous tracking of cells and tracer particles. The measured velocity field reveals complex time-
dependent flow structures, and scales inversely with distance. The instantaneous mechanical power generated
by the cells is measured from the velocity fields and peaks at 15 f W. The dissipation per cycle is more than 4
times what steady swimming would require.
Disciplines
Physical Sciences and Mathematics | Physics
Comments
Suggested Citation:
Guasto, J., K.A. Johnson, and J.P. Gollub. (2010). Oscillatory Flows Induced by Mircroorganisms Swimming
in Two Dimensions. Physical Review Letters. 105, 168102.
Copyright 2010 American Institute of Physics. This article may be downloaded for personal use only. Any
other use requires prior permission of the author and the American Institute of Physics. The following article
appeared in Physical Review Letters and may be found at http://dx.doi.org/10.1103/PhysRevLett.105.168102.
This journal article is available at ScholarlyCommons: http://repository.upenn.edu/physics_papers/8
Oscillatory Flows Induced by Microorganisms Swimming in Two Dimensions
Jeffrey S. Guasto,1 Karl A. Johnson,2 and J. P. Gollub1,3
1Department of Physics, Haverford College, Haverford, Pennsylvania 19041, USA
2Department of Biology, Haverford College, Haverford, Pennsylvania 19041, USA
3Department of Physics, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA
(Received 26 July 2010; published 11 October 2010)
We present the first time-resolved measurements of the oscillatory velocity field induced by swimming
unicellular microorganisms. Confinement of the green alga C. reinhardtii in stabilized thin liquid films
allows simultaneous tracking of cells and tracer particles. The measured velocity field reveals complex
time-dependent flow structures, and scales inversely with distance. The instantaneous mechanical power
generated by the cells is measured from the velocity fields and peaks at 15 fW. The dissipation per cycle is
more than 4 times what steady swimming would require.
DOI: 10.1103/PhysRevLett.105.168102 PACS numbers: 47.63.Gd, 47.15.G, 47.61.k, 87.16.Qp
Motile single cells use a variety of strategies for loco-
motion in low Reynolds number environments. Some
methods produce roughly steady swimming (e.g., rotation
of helical flagella in E. coli), while others produce oscil-
latory cell motion by beating flagella in a periodic manner
[1]. Understanding the hydrodynamics of swimming mi-
croorganisms can give insight into complex biological
processes such as predator-prey interactions and flagellar
mechanics [2,3]. A variety of mathematical models have
been proposed to describe interactions between swimming
cells and their environment [1,4,5]. However, experimental
studies of velocity fields are rare [6], and oscillatory near-
field hydrodynamics have not been investigated experi-
mentally despite their importance [7,8].
The flows induced by swimming organisms are respon-
sible for biogenic mixing [9], and microorganisms in par-
ticular enhance the transport properties of suspensions
[5,10]. Active particles and microorganisms also affect
the rheological properties of suspensions, where the effec-
tive viscosity is increased in the case of ‘‘pullers’’ (e.g.,
Chlamydomonas reinhardtii) and decreased for ‘‘pushers’’
(e.g., E. coli) [11,12]. Hydrodynamic interactions leading
to coherent motion have been studied for concentrated
bacterial cells [13], but not for algal cells.
In many of these phenomena, the oscillatory swimming
motion of organisms may be important. Furthermore, the
concentration of swimmers may be sufficiently high that
organisms are within several radii of one another, close
enough that near-field effects should be considered.
Understanding the resulting time-dependent flow fields is
required for the interpretation of interactions between
swimming cells and their environments.
We present the first time-resolved measurements of the
flow field around oscillatory swimming microorganisms
(C. reinhardtii). This is accomplished through simulta-
neous tracking of swimming cells and passive tracer par-
ticles within quasi-two-dimensional (quasi-2D) thin liquid
films, where direct optical access and long observation
times are achieved. These measurements reveal complex
near-field flow structures that evolve throughout the beat
cycle, and provide a richer picture than time-averaged flow
fields and simplified models. The instantaneous mechani-
cal power transferred by flagella to the fluid is measured
via the viscous dissipation. These observations carry im-
portant implications for the interpretation and modeling of
transport processes, locomotion, and flagellar mechanics.
C. reinhardtii is a single-celled swimming alga with a
7–10 m diameter body [14]. It typifies many biflagellated
microorganisms with two anterior 10–12 m long flagella
that beat at 50–60 Hz propelling the cell in an oscillatory
manner at a mean speed of 100–200 m=s (see video
[15]). The flagella, actuated along their lengths by dynein
motors, take on asymmetric conformations during the
power and recovery strokes to overcome the time revers-
ibility of low Reynolds number flows.
Wild-type C. reinhardtii (cc1690) is grown in minimal
media (M1) on a light cycle (16 hr bright/8 hr dark) to
ensure uniform size (radius R  3:5 m) and motility
[14]. An adjustable wire-frame device [12] is used to
stretch a 2 l droplet of the cell suspension into a square
film with side length L ¼ 6 mm and measured thickness
h ¼ 15 2 m. A trace amount of surfactant (Tween 20)
stabilizes the film without harming the cells. Polystyrene
microspheres (1 m diam., Thermo Scientific), passi-
vated with bovine serum albumin to prevent adhesion to
flagella [16], are incorporated with the cells prior to
stretching the film.
The device is mounted on an upright microscope (40
0:75 NA objective) where suspensions are observed under
bright field illumination with red light (>610 nm) to pre-
vent phototaxis [17]. A high-speed digital camera captures
the motion over 3000 frames (50 or 500 fps). Cells and
tracer particles are segmented from one another by size and
tracked using a predictive algorithm [18].
The thin liquid film ensures that the algae are coincident
with the focal plane of the objective and prevents distortion
PRL 105, 168102 (2010)
Selected for a Viewpoint in Physics
PHY S I CA L R EV I EW LE T T E R S
week ending
15 OCTOBER 2010
0031-9007=10=105(16)=168102(4) 168102-1  2010 The American Physical Society
from out-of-focus particles. This allows detailed observa-
tion of the cells (mean speed U0 ¼ 134 m=s) for
up to about 8 s at 50 fps. Using high-speed imaging
(500 fps), oscillations of the cell body, UðtÞ, become
evident [Fig. 1 (inset)]. The peak forward velocity is
4 times the mean value and can be negative during the
recovery stroke. The probability density function (PDF) of
the beat frequency f for 170 cell tracks is shown in Fig. 1.
The oscillations have a fairly narrow distribution with a
mean frequency f ¼ 53 5 Hz corresponding to a beat
cycle period T ¼ 1= f ¼ 18:9 ms.
By simultaneously tracking swimming cells and passive
tracer particles (see video [15]), we measure the velocity
field induced by the cells by translating and rotating the
instantaneous tracer particle measurements to a common
coordinate system based on cellular orientation. Swimmer
trajectory segments with large curvature or irregular beat
frequencies are excluded [19]. The beat-cycle averaged
velocity field is measured by imaging the cells at a frame
rate comparable to the cell beat frequency (50 fps), thus
capturing the net motion over a cycle. The velocity field
resulting from 560 cell tracks is shown in Fig. 2(a), where
the swimmer motion is to the right and solid lines are
instantaneous streamlines [20].
While the general shape of the flow field has some
qualitative similarities to a force dipole (‘‘stresslet’’) ap-
proximation [1], several unexpected features are apparent
[6]. The location of the hyperbolic stagnation point far
away from the anterior of the cell is surprising, since it
typically is thought to be located between the centers of
drag (body) and thrust (flagella). Two strong vortices are
visible lateral to the organism (associated with the fla-
gella), while two weaker vortices appear far from the cell
body beyond the separatrix of the hyperbolic point.
The fluid-air interfaces of the liquid film are nearly
stress-free, which minimizes velocity gradients transverse
to the film. This effect along with comparability of the film
thickness and cell size (h=2R 2) creates a quasi-2D
environment [21], where we expect longer-range hydro-
dynamic disturbances compared to 3D flows. Fluid veloc-
ity magnitude normalized by the mean swimmer speed,
u=U0, where u ¼ juj, is shown in Fig. 2(b) as a function of
radial distance from the organism, r=R, in various direc-
tions. Toward the posterior, fluid speed scales nearly as u
r1 up to 20 cell radii away, similar to a force dipole in 2D
[22], with slight deviations from u r1 likely due to
minor 3D effects. These results are consistent with similar
measurements in 3D that show the expected u r2 scal-
ing [6]. In the lateral direction, the velocity magnitude
passes through a local minimum when traversing the para-
bolic stagnation points (vortex), before recovering u r1
scaling in the far field. Similarly, in front of the cell, fluid
FIG. 1 (color online). Probability density function (PDF) of
flagellar beat frequencies, f, measured from unicellular alga
(C. reinhardtii) swimming in quasi-2D liquid films ( f ¼ 53
5 Hz). Inset: velocity oscillations of a single swimming cell
measured at 500 fps (red circle), where the maximum velocity
is 4 times the mean value (solid blue line, 134 m=s).
FIG. 2 (color online). (a) The beat-cycle averaged velocity
field around a swimming C. reinhardtii (black disc) in the lab
frame, where the direction of travel is to the right toward the
hyperbolic stagnation point (green diamond). Solid (red) lines
are instantaneous streamlines and velocity vectors are shown on
a log scale [20]. (b) The fluid velocity magnitude in various
directions away from the cell demonstrates the predicted u
r1 scaling for a force dipole in 2D. Local minima correspond to
stagnation points encountered in some directions.
PRL 105, 168102 (2010) P HY S I CA L R EV I EW LE T T E R S
week ending
15 OCTOBER 2010
168102-2
speed decreases rapidly near the hyperbolic stagnation
point 7–8 radii from the swimmer.
The time-averaged velocity field in Fig. 2(a) shows the
limitations of some current swimmer models, as also noted
in [6], and raises several important questions. For example,
how does the velocity field evolve throughout the oscilla-
tory flagellar beat cycle? Using high-speed imaging
(500 fps), we measure the instantaneous swimmer phase,
and identify tracer particles at corresponding times in the
beat cycle. Velocity fields (resulting from 170 cell tracks)
are constructed from tracer velocities at each phase of the
oscillation with resolution T=15.
A time series of the velocity field during the beat cycle is
shown in Fig. 3 (see video [15]). Insets show swimmer
speed and phase (lower left) and approximate flagellar
position (lower right). At the beginning of the power stroke
[Fig. 3(a)], the velocity field is neatly divided into four
symmetric quadrants with the hyperbolic stagnation point
located slightly forward from the body, in line with con-
ventional force dipole swimmer models [1]. As the flagella
move toward the posterior and the power stroke peaks, the
vortices lateral to the organism strengthen [Fig. 3(b)], then
shift across the body to the anterior side [Fig. 3(c)–3(e)].
After the power stroke ends and the recovery stroke begins,
the flagella extend out in front of the organism, and the cell
velocity becomes negative. The flow shown in Fig. 3(f) is
qualitatively reversed from Fig. 3(a), changing the sign of
the dipole. The instantaneous flow field generated by an
oscillatory swimmer such as C. reinhardtii is complex and
highly time dependent.
In Stokes flows, all of the mechanical energy generated
by a swimmer for locomotion is rapidly dissipated by the
fluid. The power transferred to the fluid by the organism is
calculated from the velocity field gradient through the
viscous dissipation P ¼ R 2ð:ÞhdA, where  is the
fluid viscosity, ¼ 1
2 ½ruþ ðruÞT is the rate of strain
tensor, and dA is a differential area element of the film
[15]. The instantaneous mechanical power output, PoscðtÞ,
is calculated from the velocity fields [Fig. 3] throughout the
beat cycle and shown in Fig. 4(a). The peak power output
(15 fW) occurs during the power stroke and corresponds
to the maximum instantaneous speed of the cell body. A
secondary local maximum also occurs at the peak speed of
the recovery stroke [see Fig. 4(b) (inset)]. Because of the
FIG. 3 (color online). Time sequence of the velocity field evolution throughout the beat cycle (period T ¼ 18:9 ms) of C. reinhardtii
(oriented to the right) including the hyperbolic stagnation point position (green diamond). Insets show cell speed and beat cycle phase
[lower left, see Fig. 4(b) for details], and approximate flagellar shape (lower right, measured separately). (a) Early in the power stroke,
the velocity field resembles a (negative) force dipole. (b) At the peak of the power stroke, the vortices lateral to the organism strengthen
and sweep toward the posterior. (c)–(e) The vortices then shift to the anterior as the power stroke is completed. (f) At the peak of the
recovery stroke, the flow field again takes the shape of a dipole, but with opposite sign. The recovery stroke velocity field (f) is weaker
than the forward stroke, but is enhanced by the log scaling [20].
PRL 105, 168102 (2010) P HY S I CA L R EV I EW LE T T E R S
week ending
15 OCTOBER 2010
168102-3
oscillatory swimming, the average mechanical power out-
put over the beat cycle, hPoscðtÞi, is more than 4 times the
power computed from the time-averaged velocity field in
Fig. 2(a) (i.e., hPoscðtÞi=Pmean flow  4:2).
The mechanical power scales with the cell body speed
[Fig. 4(b) (inset)] as PU2 with deviations at small U
due to the finite accuracy of measuring tracer velocities
[Fig. 4(b)]. This behavior might be expected for drag-based
thrust at low Reynolds number, since the forces scale
linearly with the body speed FU, and the power output
of the organism is P FUU2 [23].
This work represents an important step toward under-
standing the locomotion of swimming microorganisms and
collective behaviors for which hydrodynamics are respon-
sible. Measurements of the time-averaged velocity field
around C. reinhardtii in quasi-2D thin liquid films demon-
strate the predicted u r1 scaling of the fluid velocity
with detailed resolution of the near-field flow features.
Also, high-speed imaging allows for phase-resolved mea-
surements of the velocity field throughout the flagellar beat
cycle, revealing complex underlying flow structures that
evolve in time. From these measurements, we demonstrate
that the mechanical power dissipated by these swimming
microorganisms is more than 4 times what the mean ve-
locity field predicts. This has important consequences re-
garding the input of chemical energy required to power
oscillatory swimming of flagellated cells.
We thank R. E. Goldstein, K. Drescher, E. Lauga, and
M.D. Graham for helpful discussions, as well as B. Boyes
for technical assistance. This work was supported by NSF
Grant No. DMR-0803153.
[1] E. Lauga and T. R. Powers, Rep. Prog. Phys. 72, 096601
(2009).
[2] J. Sheng et al., Proc. Natl. Acad. Sci. U.S.A. 104, 17512
(2007).
[3] T. J. Mitchison and H.M. Mitchison, Nature (London)
463, 308 (2010).
[4] G. P. Alexander, C.M. Pooley, and J.M. Yeomans, Phys.
Rev. E 78, 045302(R) (2008).
[5] P. T. Underhill, J. P. Hernandez-Ortiz, and M.D. Graham,
Phys. Rev. Lett. 100, 248101 (2008).
[6] K. Drescher et al., preceding Letter, Phys. Rev. Lett. 105,
168101 (2010).
[7] T. Ishikawa, G. Sekiya, Y. Imai, and T. Yamaguchi,
Biophys. J. 93, 2217 (2007).
[8] C.M. Pooley, G. P. Alexander, and J.M. Yeomans, Phys.
Rev. Lett. 99, 228103 (2007).
[9] K. Katija and J. O. Dabiri, Nature (London) 460, 624
(2009).
[10] K. C. Leptos et al., Phys. Rev. Lett. 103, 198103 (2009).
[11] S. Rafaı̈, L. Jibuti, and P. Peyla, Phys. Rev. Lett. 104,
098102 (2010).
[12] A. Sokolov and I. S. Aranson, Phys. Rev. Lett. 103,
148101 (2009).
[13] X. L. Wu and A. Libchaber, Phys. Rev. Lett. 84, 3017
(2000).
[14] E. H. Harris, The Chlamydomonas Sourcebook (Academic
Press, Oxford, 2009), Vol. 1.
[15] See supplementary material at http://link.aps.org/
supplemental/10.1103/PhysRevLett.105.168102.
[16] D. B. Weibel et al., Proc. Natl. Acad. Sci. U.S.A. 102,
11 963 (2005).
[17] O. A. Sineshchekov, K-H. Jung, and J. L. Spudich, Proc.
Natl. Acad. Sci. U.S.A. 99, 8689 (2002).
[18] N. T. Ouellette, H. Xu, and E. Bodenschatz, Exp. Fluids
40, 301 (2006).
[19] M. Polin et al., Science 325, 487 (2009).
[20] The velocity field is calculated at 3 m resolution but
shown at 6 m resolution using log scaling, where a
velocity doubling increases the displayed vector length
by 0.4% of the panel width.
[21] V. Prasad and E. R. Weeks, Phys. Rev. Lett. 102, 178302
(2009).
[22] A. T. Chwang and T. Y. Wu, J. Fluid Mech. 67, 787
(1975).
[23] S. Childress, Mechanics of Swimming and Flying
(Cambridge University Press, Cambridge, 1981).
FIG. 4 (color online). The mechanical power dissipated by
the organism is calculated from the velocity fields. (a) The
instantaneous dissipation PoscðtÞ is maximal when the swim-
mer speed is highest. The dissipated power time-averaged
over the beat cycle hPoscðtÞi is more than 4 times the dissipa-
tion measured from the time-averaged velocity field (i.e.,
hPoscðtÞi=Pmean flow  4:2). (b) The measured power scales
with the square of the cell body speed, Posc  U2. Inset: mean
cell body velocity as a function of beat cycle phase.
PRL 105, 168102 (2010) P HY S I CA L R EV I EW LE T T E R S
week ending
15 OCTOBER 2010
168102-4


Oscillatory Flows Induced by Microoganisms Swimming in Two Dimensions

Crossref

Physical Review Letters

Oscillatory Flows Induced by Microorganisms Swimming in Two-dimensions

Oscillatory Flows Induced by Microorganisms Swimming in Two-dimensions

Abstract

Similar works

Full text

Available Versions

Haverford College: Haverford Scholarship

Kosmopolis

Crossref