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Depolymerization-driven Flow in Nematode Spermatozoa Relates Crawling Speed to Size and Shape Mark Zajac Department of Cell Biology and Center for Cell Analysis and Modeling, University of Connecticut Health Center, Farmington, CT 06030-3505 Brian Dacanay Department of Biomedical Engineering, University of Connecticut, Storrs, CT 06269-2247 William A. Mohler Deptartment of Genetics and Developmental Biology, Center for Cell Analysis and Modeling, University of Connecticut Health Center, Charles W. Wolgemuth 1 Department of Cell Biology and Center for Cell Analysis and Modeling, University of Connecticut Health Center, Farmington, CT 06030-3505
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Depolymerization-driven Flow in Nematode Spermatozoa ...physics.arizona.edu/~wolg/PDF/ZDMW.pdfDepolymerization-driven Flow 3 Cell motility experiments often focus on the cytoskeleton,

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  • Depolymerization-driven Flow in Nematode

    Spermatozoa Relates Crawling Speed to Size

    and Shape

    Mark ZajacDepartment of Cell Biology and

    Center for Cell Analysis and Modeling,University of Connecticut Health Center,

    Farmington, CT 06030-3505

    Brian DacanayDepartment of Biomedical Engineering,

    University of Connecticut, Storrs, CT 06269-2247

    William A. MohlerDeptartment of Genetics and Developmental Biology,

    Center for Cell Analysis and Modeling,University of Connecticut Health Center,

    Charles W. Wolgemuth 1

    Department of Cell Biology andCenter for Cell Analysis and Modeling,

    University of Connecticut Health Center,Farmington, CT 06030-3505

  • Depolymerization-driven Flow 2

    1Corresponding author. Address: Department of Cell Biology, University ofConnecticut Health Center, 263 Farmington Avenue (MC 3505), Farmington,CT 06030-3505, U.S.A., Tel.: (860)679-1655, Fax: (860)679-1269

  • Abstract

    Cell crawling is an inherently physical process that includes protrusion ofthe leading edge, adhesion to the substrate, and advance of the trailing cellbody. Research into advance of the cell body has focused on actomyosin con-traction, with cytoskeletal disassembly regarded as incidental, rather thancausative; however, extracts from nematode spermatozoa, which use MajorSperm Protein rather than actin, provide one example where cytoskeletaldisassembly apparently generates force in the absence of molecular motors.To test whether depolymerization can explain force production during ne-matode sperm crawling, we constructed a mathematical model that simul-taneously describes the dynamics of both the cytoskeleton and the cytosol.We also performed corresponding whole cell experiments using Caenorhab-ditis elegans spermatozoa. Our experiments reveal that crawling speed isan increasing function of both cell size and anteroposterior elongation. Thequantitative, depolymerization-driven model robustly predicts that cell speedshould increase with cell size and yields a cytoskeletal depolymerization ratethat is consistent with previous measurements. Notably, the model requiresanisotropic elasticity, with the cell being stiffer along the direction of motion,to accurately reproduce the dependence of speed on elongation. Our simu-lations also predict that speed should increase with cytoskeletal anisotropyand depolymerization rate.

    Key words: cell crawling; MSP; nematode sperm; cytoskeleton; C. ele-gans; mathematical model

  • Depolymerization-driven Flow 2

    Introduction

    Most papers on amoeboid cell motility start by reciting (16, 25, 31) thecanonical tripartite litany: extend and adhere at the front; advance the cellbody; detach and recede at the rear. Though conceptually distinguished,these are not viewed as separate stages, but rather widely acknowledged (38,42) as tightly integrated physical processes. Even so, individual componentsof the whole mechanism have not received equal consideration. In priorexperiments and mathematical models (13), there has been more focus onleading edge extension (14) rather than advance of the cell body, more focuson forces from cytoskeletal assembly (3, 28, 37) rather than disassembly andmore focus on the solid cytoskeleton rather than the fluid cytosol.

    Research into causes for advance of the cell body (48, 50) favors con-traction of actin bundles by myosin II as most likely. However, observations(22, 23, 56) of Dictyostelium discoideum amoeba show that motor functionof myosin II is not essential for cell crawling. Also, more recent experimentsusing Myosin IIA-deficient fibroblasts showed that the cells without myosinmigrate faster than wildtype cells (12). Therefore, it is possible that thetransolocation of the cell during crawling is driven partially by the dynam-ics of the actin network without the action of molecular motors. Indeed,biomimetic constructs of actin (19, 52) show that polymer network collapsecan generate forces, whether myosin acts as a motor or not. Whatever therole of myosins is in cell motility, constructing quantitative models for cellcrawling will require untangling the physics of the cytoskeleton from theaction of molecular motors.

    Nematode spermatozoa provide an excellent model system for studyingthe basics of cell crawling in the absence of molecular motors. Unlike mostother crawling cells, nematode sperm utilize a cytoskeleton composed of anetwork of Major Sperm Protein (MSP). This protein forms non-polar fila-ments, to which no molecular motors are known to associate (5). However,the motility of these cells still exhibits all three fundamental processes re-quired for standard crawling motility. In these cells, polymerization at theleading edge is believed to drive advance of the front of the cell (40). Themolecular level mechanism for adhesion of these cells to the substrate is stillunknown. Based on in vitro experiments, depolymerization of the cytoskele-tal network has been proposed as the force-producing mechanism for pullingup the rear (30, 55).

  • Depolymerization-driven Flow 3

    Cell motility experiments often focus on the cytoskeleton, although itoccupies less than 5% of typical crawling cells, by volume. Observations ofcell blebbing (7, 9) and Walker carcinoma cells (21) suggest that cytosolicpressure can drive cell extension in regions of cytoskeletal disruption, andthere is evidence (57) that intracellular pressure provides the motive force forAmoeba proteus cells. In this paper, simulations of nematode spermatozooncrawling demonstrate a realistic motility mechanism that relies, in part, oncytosolic forces.

    Spermatozoa from the nematode C. elegans routinely exhibit steady,amoeboid crawling on prepared surfaces. The salient features of a steadilycrawling spermatozoon include an active, laminar pseudopod, at the front,and a passive, domed cell body at the back (Fig. 1 A, B). The pseudopoddevelops transient ruffles and peripheral bulges but otherwise conforms to acharacteristic overall cell geometry, including persistent elongation in the di-rection of motion. Extension of the pseudopod and advance of the cell bodyare not separate stages but rather occur in unison. Variant morphologiesand motilities have been cataloged (33), including periodic velocity cycles(43), but steadily crawling spermatozoa are most amenable to quantitativemeasurement and mathematical modeling.

    In this paper, we construct a mathematical model to describe the crawl-ing motility of nematode spermatozoa (Fig. 2). This model simultaneouslyaccounts for the dynamics of the cytoskeleton and the cytosol, with depoly-merization of the MSP cytoskeleton as the force-producing mechanism foradvance of the cell body (Fig. 1). Previous models (4, 32, 54) have suc-ceeded in capturing certain features of preexisting data but support for ourmodel includes validation against quantitative experiments. To test the re-sults of our model, we measured size, shape and speed of crawling C. eleganssperm. Consistent with a previous experiment (24), we found that largercells crawl faster than smaller cells. Furthermore, cells that are elongated intheir direction of motion crawl more rapidly than rounder cells. We foundthat the model accurately reproduces the dependence of speed on cell sizeand shape, but requires that the cell be stiffer in its direction of motion thanperpendicular to it (Fig. 1 C). We find that cell speed increases with thiselastic anisotropy and also increases with polymer depolymerization rate.

  • Depolymerization-driven Flow 4

    The two-phase depolymerization model

    Anterior extension, during cell crawling, involves cytoskeletal polymerization,with monomers leaving solution, while posterior retraction involves depoly-merization, with monomers returning to the cytosol. This suggests a modelwhich mixes solid and fluid phases. The cytoskeletal volume fraction φ rep-resents the fraction of space filled with solid matter (polymer), for any smallvolume element within a cell. As the cytoskeleton depolymerizes or moves, anequal volume of cytosol must fill the vacated space. Consequently, a fraction(1 − φ) of each volume element is filled with cytosolic fluid.

    In many instances, crawling nematode spermatozoa maintain approxi-mately constant shape; peripheral blebs and ruffles are relatively small, com-pared to average cell diameter. This justifies a model such that local changesin φ are equivalent to rigid body translation, with V0 as the common, steadyvelocity for all points in the cell. The assumption of steady crawling repre-sents a special case. More generally, cytoskeletal flow and depolymerizationchange the volume fraction, over time:

    CytoskeletalDrift

    Equivalent SteadyCrawling Assumed

    ∂φ

    ∂t= −

    ︷ ︸︸ ︷

    ∇ · (φVs) − ksφ︸︷︷︸

    = −︷ ︸︸ ︷

    ∇ · (φV0)

    Polymer Disassembly

    (1)

    where Vs and ks are the velocity and depolymerization rate of the cytoskele-ton, respectively. The assumption of steady crawling amounts to seeking atraveling wave solution for volume fraction kinetics. This yields a steadystate φ profile in a reference frame that moves with the cell.

    In nematode sperm extracts, MSP forms columnar, fibrous comet tailsthat push membrane vesicles (17). Under physiological conditions that pre-vent further growth (30), depolymerization results in comet tail contraction(Fig. 1 C), which can pull a load. The amount of contraction is directlyrelated to the change in MSP polymer density in the comet tail (55). Dur-ing this contraction, the change in length is much faster than the change indiameter (Fig. 1 C) (55), which suggests that the MSP network forms ananisotropic elastic gel (46). Analysis of these cell extract experiments (55)suggests that depolymerization strains the cytoskeleton and justifies a modelin which stress depends on the cytoskeletal volume fraction.

  • Depolymerization-driven Flow 5

    For a crawling cell, on a horizontal plane, consider a Cartesian coordinatesystem, with the x and y axes along the anteroposterior and transverse axes,respectively. Anisotropic shrinkage of cell extracts suggests a model in whichanteroposterior stress exceeds transverse stress. Linear dependence on φ isthe simplest possible assumption:

    σ = −(φ− φ0)

    (σx 00 σy

    )

    with σx > σy (2)

    where σ is the cytoskeletal stress tensor with σx and σy constant and φ0is the cytoskeletal volume fraction for an unstressed volume element. Theoverall negative sign reflects cytoskeletal compression when φ drops belowthe unstressed value. The model does not consider shear stresses.

    Laplace’s Law (51) imposes a boundary condition on stress and pressurein combination. Surface tension in a curved membrane counteracts any im-balance between external hydrostatic pressure and the net outward force oneach unit of cell surface area:

    Effective OutwardPressure

    Bent MembraneCounterbalance

    ︷ ︸︸ ︷

    (p− n̂ · σ · n̂) − p0︸︷︷︸

    =︷︸︸︷

    2κγ

    Ambient Hydrostatic Pressure

    (3)

    where p is the cytosolic pressure, while κ and γ are membrane curvature andsurface tension respectively. By convention, outward pressure and stress atthe boundary carry opposite signs.

    As cell membranes are slightly permeable to water (49), we allow for fluidflow across the cell membrane via a second boundary condition. One unitof membrane area sweeps out volume at a rate of n̂ · V0 where n̂ is thelocal outward unit normal. Following behind the membrane, intracellularflow of solid and fluid matter fills only part of the swept volume, leavingtransmembrane fluid flow to fill whatever space remains:

    n̂ ·V0 =

    Transmembrane Flux

    n̂ · (φVs + (1 − φ)Vf)︸ ︷︷ ︸

    ︷ ︸︸ ︷

    kf (p− p0)

    Intracellular Flux

    (4)

    where Vf is the cytosolic velocity and kf is the membrane permeability towater. Fluid flows across the cell membrane from high pressure to low, witha negative sign denoting fluid ingress.

  • Depolymerization-driven Flow 6

    The cytoskeleton in A. summ spermatozoa (45) is localized near the basalplane. This suggests treating cytosolic and cytoskeletal flow in only two di-mensions, with influx from the third dimension as a source of additionalfluid. At the same time, depolymerization causes monomers to go into solu-tion, thereby joining the fluid phase:

    CytosolicDrift

    TransmembraneFlux

    ∂t(1 − φ) = −

    ︷ ︸︸ ︷

    ∇ · ((1 − φ)Vf) + ksφ︸︷︷︸

    ︷ ︸︸ ︷

    kf(p− p0)

    Monomer Solation

    (5)

    where the fluid influx boundary condition (Eq. 4) ensures that the net trans-membrane flux vanishes over the surface of a cell.

    The total intracellular pressure gradient is split between the cytosol andthe cytoskeleton. A fraction (1 − φ)∇p pushes the fluid phase from regionsof high pressure to regions of low pressure. A fraction φ∇p pushes on thesolid phase, augmented by tractive cytoskeletal forces. For length scales andviscosities pertinent to cells, inertia is negligible. This allows a model whichbalances driving forces against drag forces, over any small region:

    Fluid FractionDriving Force︷ ︸︸ ︷

    − (1 − φ)∇p =

    Intracellular Drag Force︷ ︸︸ ︷

    ζ0(Vf − Vs) (6)

    Solid FractionDriving Force

    ︷ ︸︸ ︷

    −φ∇p+ ∇ · σ︸ ︷︷ ︸

    Cytoskeletal Force

    =

    Extracellular Drag Force

    ζ0(Vs − Vf ) +︷︸︸︷

    ζ1Vs (7)

    where ζ0 and ζ1 are drag coefficients for the motion of the polymer againstthe fluid phase and substrate, respectively. Intracellular drag depends on therelative velocity of cytosolic and cytoskeletal matter. These are absolute ve-locities, measured in a fixed laboratory reference frame. Extracellular dragrepresents surface adhesion, which anchors the cytoskeleton. Experimentssuggest strong anchorage beneath the pseudopod and weak anchorage be-neath the cell body (see Graded Substrate Adhesion in the Results section).Again, by convention, the outward stress and pressure on a volume elementcarry opposite signs.

  • Depolymerization-driven Flow 7

    Ultimately, the model yields a pair of coupled second order, linearizedpartial differential equations, in which φ0, σx, p0, kf , ζ0 and ζ1 are empiricallydetermined (Table 1), while ks and the ratio of σx to σy are free parameters.Combining cytoskeletal drift (Eq. 1) with steady crawling yields the firstequation:

    ∇·(φ0Vs)︸ ︷︷ ︸

    − V0 · ∇φ + ksφ = 0

    φ0Vs = −φ0ζ1

    (

    x̂σx∂φ

    ∂x+ ŷσy

    ∂φ

    ∂y+ ∇p

    )

    (8)

    retaining no term with more than one factor of φ or p (or derivatives thereof),where Vs is obtained from the balance of driving and drag forces (Eq. 6-7).These same equations are easily solved for (Vs −Vf ), which proves useful inthe next step. Combining the dynamic equations for the cytoskeleton (Eq. 1)and the cytosol (Eq. 5) yields the second equation:

    ∇ · (Vs − (1 − φ0)(Vs − Vf︸ ︷︷ ︸

    )) + kf(p− p0) = 0

    (Vs − Vf) =(1 − φ0)

    ζ0∇p

    (9)

    which follows since time derivatives of φ and (1 − φ) must cancel, whensummed. These equations are solved implicitly using empirically determinedcell shapes (Fig. 3 A), with approximately 10,000 interior points, on a regulargrid. Solutions give φ and p directly, from which Vf , Vs and σ are thendetermined (Fig. 2 F-I).

    Observations of steadily crawling A. summ spermatozoa (39, 40) revealthat the MSP cytoskeleton maintains close contact with the cell membrane.For a range of crawling speeds, the cytoskeletal depolymerization rate is com-puted by systematically adjusting ks until n̂ · Vs matches n̂ · V0 at the rearof the cell. At the front of the cell, and other points along the perimeter, thedifference between n̂ · V0 and n̂ · Vs gives the rate at which polymer mustassemble in order to maintain the proscribed crawling velocity, which we willcall the polymer assembly rate (Fig. 2 C). Cytoskeletal growth due to poly-merization must bridge the gap wherever velocity of the existing cytoskeletonfails to keep pace with the cell membrane.

    Actual computation employs dimensionless variables, using L0, p0 andp0/(L0ζ0) to set scales for length, stress and velocity, respectively, where L0is the square root of cell area. In this scheme, changes in non-dimensionaldepolymerization rate are equivalent to changes in L0 at fixed ks. Using

  • Depolymerization-driven Flow 8

    appropriate scale factors to restore units then provides crawling speed as afunction of cell size, once a value for ks is chosen, with p0 and ζ0 determinedempirically (Table 1).

    The model presented here incorporates features which have received priorconsideration on an individual basis. There have been a number of modelsproposed for nematode sperm motility. Of these, many are one-dimensionaland most have not considered depolymerization as a mechanism for contrac-tion (4, 20, 54). Another model in one dimension examined unbundling ofMSP fibers, but did not account for cytosolic fluid flow (32). A detailed modelfor keratocyte crawling (44) considered fluid transport of monomeric actinbut neglected retrograde motion of polymeric actin. A two-phase model hasbeen used to describe cell mechanics during neutrophil aspiration (11, 15).None of the previous models has considered transmembrane fluid flow orcytoskeletal anisotropy.

    Results

    Cell Speed vs. Size and Shape

    We used Differential Interference Contrast (DIC) microscopy to digitallyrecord the crawling of 45 cells at one second intervals, producing 1961 in-dividual frames for analysis. Reliable, automated tracking of motile cellshinged on transforming DIC images into pseudofluorescent images, using anoriginal algorithm (see the Methods section for a more complete description).In some instances, tracked cells appeared to briefly detach from the substrateand then slew sideways or backwards, before regaining traction and then re-suming steady crawling. These events were detected as statistical anomaliesand excluded from trend analysis, leaving 1242 usable frames.

    Our tracking algorithm gave the geometry of the crawling cells in theplane of the substrate. First and second moments of the area distributionthen determined the cell centroid along with maximum and minimum di-ameters. For each image in a sequence, instantaneous velocity was calcu-lated using centroid displacement from the frame behind to the frame aheadof the current frame, divided by twice the time interval between consec-tive frames. Cell elongation along the direction of motion was computed as(a − b)/min(a, b) where a and b are the anteroposterior and transverse celldiameters, respectively. Elongation increases linearly as the ratio of a to b

  • Depolymerization-driven Flow 9

    increases, with a value of zero for a circular cell. Steady crawling often in-cludes slight yet systematic changes in cell speed and geometry, over severalseconds. Therefore, analysis treats the frames from each digital recordingindependently, rather than averaging over each cell.

    We found average values for length, width and speed of typical cellsof 8±1µm, 4.7±0.5µm, and 0.2±0.1µm/s, respectively, which is consistentwith average values reported previously (43). Cell crawling speed was ob-served to increase with increased cell area (Fig. 3 C), consistent with priorreports (24) of larger speeds for cells with greater volume. Crawling speedalso increases with increased anteroposterior elongation (Fig. 3 B).

    Graded Substrate Adhesion

    Within the pseudopod, obtrusive knobs and ridges develop at the leadingedge and then hold station or slowly drift backward, until overrun by thetrailing cell body. These protuberant features within the pseudopod havebeen identified as branching, filamentous, cytoskeletal constructs (1, 40).Spatio-temporal correlation between features from successive frames of a dig-ital recording yields incremental cytoskeletal displacements within the pseu-dopod of a crawling cell (for details, see the Methods section). Dividing eachdisplacement by the time interval between frames then gives cytoskeletal ve-locities throughout the pseudopod. By the same method, static, papillarysurface texture allows tracking of the cell body, as a whole.

    Feature tracking gives drastically different results for the pseudopod andcell body of a steadily crawling spermatozoon, with velocities differing widelyin both direction and magnitude (Fig. 4). Within the pseudopod, minuscule,slightly retrograde cytoskeletal velocities suggest strong adhesion to the sub-strate below. In stark contrast, the rear of the cell moves forward rapidly,suggesting weak adhesion beneath the cell body. The apparent transitionfrom high to low adhesion is quite sharp, occurring at a location just infront of the cell body. These results provide evidence for a widely assumed,yet previously unconfirmed, adhesion gradient beneath crawling C. elegansspermatozoa.

    Simulating Cytoskeletal Disassembly and Stress

    To test our model, we computed the dependence of crawling speed on cellsize and shape, for comparison with our experimental results. Simulations

  • Depolymerization-driven Flow 10

    employed ten empirically determined cell shapes (see the Methods sectionfor a description of how these geometries were determined). The chosenshapes roughly cover the full range of cell elongations, from experiments(Fig. 3 A, B). Working in dimensionless variables allowed scaling of eachshape to cover the full range of measured cell areas. With units then re-stored, simulations gave crawling speed as a function of cell size (as describedpreviously).

    Treating size and shape simultaneously, we fit our simulation results toour experimental data by minimizing the mean squared difference using ksas a free parameter. Results depend on the ratio of σx to σy in the model.A ratio of 5:1 gives the best agreement with our experiments. Using thisanisotropy ratio, simulations then reproduce the observed increase in crawl-ing speed with increased anteroposterior elongation (Fig. 3 B) and simulta-neously reproduce the observed increase in crawling speed for increasinglylarger cells (Fig. 3 C). For a strong adhesion gradient (Fig. 2 D), simula-tions roughly capture the velocity field obtained from cytoskeletal tracking,with small rearward velocities for the pseudopod compared to large forwardvelocities for the cell body (compare Fig. 2 I and Fig. 4).

    Increased speed with increased elongation requires anisotropy. For anisotropic cell, σx = σy, manipulating other parameters of the model gaveeither no increase of or a reduction in speed with elongation. We testedvarious alternatives, such as letting φ0 or ks vary with distance from the backof the cell. Likewise, changes in overall cytoskeletal stiffness were ineffective.

    For optimum anisotropy, a depolymerization rate of 0.05 s−1 gives the bestfit. This is consistent with prior estimates (55) for MSP and is also compa-rable to measured dissociation rates for actin (36). In addition, the optimumks value is also consistent with reports for A. summ sperm, which experiencetotal cytoskeletal collapse in 30-60 seconds under conditions which preventfurther polymerization at the leading edge (18). The polymer assembly rateat the leading edge that we compute (Fig. 2 C) is also consistent with thegrowth rate of MSP fibers in cell extract experiments (17).

    This model also makes two other experimentally testable predictions.First, simulations predict increased crawling speed with increased anisotropy.Crawling speed plummets for anisotropy below optimum and begins to asymp-tote for greater anisotropy (Fig. 5 A). Second, we explored the dependenceof the steady crawling speed on the cytoskeletal depolymerization rate. Wefound that the speed increases roughly linearly with depolymerization for allelongations (Fig. 5 B).

  • Depolymerization-driven Flow 11

    Cytosolic Contributions

    Our mathematical modeling predicts an intracellular pressure gradient (Fig. 2 F)and consequent cytosolic flow. The cytosol is not entrained with the cy-toskeleton (compare Fig. 2 G and Fig. 2 I) which demonstrates the impor-tance of treating solid and fluid phases independently. For any small regionof a cell, the pressure gradient gives the force that propels the cytosol. At thesame point, force derived from cytoskeletal stress can be more than ten timeslarger in magnitude but, summed over the entire cell, cytoskeletal force com-ponents tend to cancel while cytosolic forces accumulate. Simulations givea realistic magnitude of about 5 nN for the net driving force on a typicalC. elegans spermatozoon, with cytosolic forces accounting for roughly 45%of the total. Another interesting feature of our model is that permeabilityof the membrane to fluid produces a small influx of fluid at the leading edge(Fig. 2 E).

    Discussion

    Here we have shown that depolymerization of the MSP network can quantita-tively account for the dependence of C. elegans sperm crawling speed on cellsize and shape. Our model fits the experimental data with a reasonable valuefor the depolymerization rate of the MSP network and requires the cytoskele-ton to be anisotropic. This requirement of anisotropy is well justified by twoobserved features of the MSP network. First, the cytoskeleton of nematodespermatozoa is composed of bundled MSP, where the long axis of each bundlelies roughly parallel to the translational direction (41). Second, contractioninduced in comet tails of MSP formed behind vesicles requires anisotropy toexplain the rate of change of the length of the comet tail with respect to therate of change of the diameter (30, 46, 55). Anisotropic cytoskeletal elastic-ity has yet to receive widespread consideration but our model demonstratesthat anisotropy might be advantageous for any cell that needs to migraterapidly. Indeed, fish keratocytes are known for rapid crawling and providean example where cytoskeletal anisotropy might be important (6, 47).

    The effects of cytoskeletal disassembly and cytosolic forces need not belimited to C. elegans spermatozoa. Even cells that employ molecular motorsmust include cytoskeletal disassembly, as a counterbalance to anterior as-sembly. Therefore, the locomotive efficacy of cytoskeletal disassembly merits

  • Depolymerization-driven Flow 12

    study, as a means of understanding possible contributions to a compoundmechanism. Bovine aortic endothelial cells employ actin for motility yetexhibit increased speed with increased rates of cytoskeletal turnover, consis-tent with simulations based on MSP disassembly (Fig. 5 B). Interestingly,the slope of speed versus turnover rate in these experiments is around 3µm,which is similar to the value of 5µm predicted by our model. There is alsosome evidence (2) that cytosolic pressure contributes to motility of kerato-cytes, which are usually regarded as a model for motility based on actin andmyosin. Keratocytes also exhibit increased speed as width perpendicular tothe crawling direction decreases (10). This has not been directly addressedin prior models but is consistent with possible roles for cytoskeletal depoly-merization and cytosolic flow, as in our model.

    Leading edge fluid influx is a prediction of our model which might haveimplications for anterior cell extension. The proposed “Brownian ratchet”mechanism (35) for protrusion depends on thermal fluctuation opening agap between the membrane and existing cytoskeletal polymer for insertion ofnew monomers into the network. Leading edge fluid influx might advance themembrane, by inflation, and thus reduce the load on protrusive, cytoskeletalpolymer. Consistent with this picture, our model showed a small decrease inpolymer stress as the membrane permeability to fluid was increased (resultsnot shown). Since fluctuations need only make space for a single monomer,even slight effects from membrane permeability might be significant. Local-ization of aquaporins may lead to nonuniform permeability which could givefluid influx a greater influence than predicted by our simulations, where weassumed uniform permeability of membrane to the fluid. In fact, fluorescencedequenching measurements reveal localized water influx at the expanding rimor motile neutrophils (27). Similar experiments could be done on nematodespermatozoa to test the predictions given by the model presented here.

    Methods

    Dissection of Sperm

    Following Royal, et. al. (43), wild-type C. elegans males were isolated and dis-sected in 8µl of sperm medium (50mM HEPES, pH 7.0, 50 mM NaCl, 25mMKCl, 1mM MgSO4, 5mM CaCl2, 8mg/ml Polyvinylpyrrolidone, 0.4mg/mlPronase (53). Liberation of the spermatids and testis was achieved by cut-

  • Depolymerization-driven Flow 13

    ting the worm with a 20 g needle approximately 1/3 the distance from theposterior end of the worm (26).

    Coverslip preparation

    Large and small cover slips (45×50mm lower and 20×20mm upper, bothof thickness #1.5) were washed with 1% Alcanox (detergent), rinsed withdistilled water and left to air dry at room temperature. Cleaned slips werethen coated with polylysine. 20µl of 10µg/ml polylysine was placed on onecover slide while a second cover slide was stacked on top, sandwiching thesolution. These slides were carefully separated and air dried at room temper-ature. Sperm buffer with the activating reagent monensin consisted of theingredients: 50mM HEPES pH7.0, 50mM NaCl, 25mM KCl, 5mM, 1mMMgSO4, and 1mg/ml bovine serum albumin (34). Spermatozoon activationused 1×10−2 M monensin in DMSO (34).

    Imaging of crawling cells

    The sperm, in sperm media, were mounted between a 45×50mm lower cover-slip and a 20×20mm upper coverslip, prepared as described above, supportedby high vacuum grease applied between the coverslips in two parallel strips1.5 cm apart by a 30 g needle. Time series of the sperm were recorded usinga Cooke Sensicam cooled CCD camera coupled to a Nikon TE 300 100×1.4NAPlanApo objective with DIC optics.

    Pseudofluorescent image processing

    DIC microscopy provides high contrast images derived from changes in therefractive index within a sample; the method works best for reasonably trans-parent materials, such as nematode sperm cells. Crudely, a DIC image is cre-ated by breaking the incident light into two paths before sending it throughthe sample. Using a Wollaston prism, light in one of these paths is phase-shifted by an angle 2ψ0 and translated spatially along a shear direction. Oncethrough the sample, the light is recombined. This process highlights gradi-ents in the index of refraction along the shear direction. The intensity, I, ata point in a DIC image can be written as (8):

    I = 2(a2 + a∆a

    )(1 + cos (∆θ + 2ψ0)) (10)

  • Depolymerization-driven Flow 14

    where a is the amplitude, ∆a is the difference in amplitude and ∆θ is thechange in phase angle between the two paths. If θ0 is the phase angle inthe absence of a cell, then θ − θ0 is roughly proportional to the index ofrefraction inside the cell times the cell thickness. Therefore, θ > θ0 insidethe cell and is equal to θ0 outside the cell. We calculate θ from our images byminimizing the difference between the grayscale image intensity in our DICimages and the value expected from Eq. 10 (The full details of this methodwill be published in an upcoming paper). Because the index of refractioninside the cell is larger than outside, the reconstructed θ is large (bright)inside the cell and small (dark) outside the cell. We threshold the cells byweighting the value of θ by the magnitude of the gradient of θ. An initialthresholding determines the region of the cell body. Then, setting the valueof θ inside the cell body equal to the minimum value of θ inside the cell body,we re-threshold the θ intensity, which gives us a binary image of the regionof the whole cell. We use the MATLAB image processing toolbox with thesebinary images to extract the area and the major and minor axes of the cells.

    Cytoskeletal velocity measurement using spatio-temporal

    correlation

    The pseudopods of nematode sperm have sharp, persistent features (MSPbundles) that are visible in DIC images. For a sequence of successive imagestaken at short time intervals, spatial correlation of successive images allowstracking of these cytoskeletal features within a cell as it crawls. We denote theimage intensity at position x and time t as It (x). The intensity at positionx′ and time t + ∆t is It+∆t (x

    ′). We then calculate the normalized cross-correlation coefficient, R, which has been used to measure the deformationof elastic substrata during cell crawling (29),

    R(x,x′) =

    δ

    It (x + δ) It+∆t (x′ + δ)

    √∑

    δ

    I2t (x + δ)√

    δ

    I2t+∆t (x′ + δ)

    (11)

    where the summation over δ visits pixels from a range of around −10 to 10in each direction. For each image, the average intensity is computed andthen subtracted from the image, prior to correlating intensities. We only doour computations for a regularly spaced subset of pixels that lie inside the

  • Depolymerization-driven Flow 15

    thresholded cell region. The value of x′ where R is a maximum defines thevelocity at point x as v = (x′ − x)/∆t.

    Acknowledgements

    We thank T. Roberts for useful discussions and acknowledge support fromgrants NIH RO1HD43156 (WAM) and NSF CTS 0623870 (CW).

    References

    1. Baker, A. M. E., T. M. Roberts, and M. Stewart. 2002. 2.6 angstromresolution crystal structure of helices of the motile major sperm protein(msp) of caenorhabditis elegans. J. Mol. Biol. 319:491–499.

    2. Bereiter-Hahn, J. 2005. Mechanics of crawling cells. Med. Eng. Phys.27:743–753.

    3. Bohnet, S., R. Ananthakrishnan, A. Mogilner, J. J. Meister, and A. B.Verkhovsky. 2006. Weak force stalls protrusion at the leading edge of thelamellipodium. Biophys. J. 90:1810–1820.

    4. Bottino, D. C., A. Mogilner, T. M. Roberts, and G. F. Oster. 2000. Acomputational model of crawling in ascaris suum sperm. Mol. Biol. Cell.11:380A–380A.

    5. Bullock, T. L., A. J. McCoy, H. M. Kent, T. M. Roberts, and M. Stewart.1998. Structural basis for amoeboid motility in nematode sperm. Nat.Struct. Biol. 5:184–189.

    6. Burton, K., and J. Park. 1999. Keratocytes generate traction forces intwo phases. Mol. Biol. Cell. 10:3745–3769.

    7. Charras, G. T., C. K. Hu, M. Coughlin, and T. J. Mitchison. 2006.Reassembly of contractile actin cortex in cell blebs. J. Cell Biol. 175:477–490.

    8. Cogswell, C. J., and C. J. R. Sheppard. 1991. Confocal differential in-terference contrast (DIC) microscopy: including a theoretical analysis ofconventional and confocal DIC imaging. J. Microsc. 165:81–101.

  • Depolymerization-driven Flow 16

    9. Cunningham, C. C. 1995. Actin polymerization and intracellular solventflow in cell-surface blebbing. J. Cell Biol. 129:1589–1599.

    10. Doyle, A. D., and J. Lee. 2005. Cyclic changes in keratocyte speed andtraction stress arise from Ca2+-dependent regulation of cell adhesiveness.J. Cell Sci. 118:369–379.

    11. Drury, J. L., and M. Dembo. 2000. Aspiration of human neutrophils:Analysis of an interpenetrating fluid model. Biophys. J. 78:367A–367A.

    12. Even-Ram, S., A. D. Doyle, M. A. Conti, K. Matsumoto, R. S. Adel-stein, and K. M. Yamada. 2001. Myosin iia regulates cell motility andactomyosin-microtubule crosstalk,. Nat. Cell Biol. 11:63–80.

    13. Gracheva, M. E., and H. G. Othmer. 2004. A continuum model of motil-ity in ameboid cells. Bull. Math. Biol. 66:167–193.

    14. Grimm, H. P., A. B. Verkhovsky, A. Mogilner, and J. J. Meister. 2003.Analysis of actin dynamics at the leading edge of crawling cells: Implica-tions for the shape of keratocyte lamellipodia. Eur. Biophys. J. Biophys.Lett. 32:563–577.

    15. Herant, M., W. A. Marganski, and M. Dembo. 2003. The mechanicsof neutrophils: Synthetic modeling of three experiments. Biophys. J.84:3389–3413.

    16. Huttenlocher, A., R. R. Sandborg, and A. F. Horwitz. 1995. Adhesionin cell-migration. Curr. Opin. Cell Biol. 7:697–706.

    17. Italiano, J., Jr., T. M. Roberts, M. Stewart, and C. A. Fontana. 1996. Re-construction In Vitro of the motile apparatus from the amoeboid spermof Ascaris shows that filament assembly and bundling move membranes.Cell. 84:105–114.

    18. Italiano, J. E., M. Stewart, and T. M. Roberts. 1999. Localized de-polymerization of the major sperm protein cytoskeleton correlates withthe forward movement of the cell body in the amoeboid movement ofnematode sperm. J. Cell Biol. 146:1087–1095.

    19. Janson, L. W., J. Kolega, and D. L. Taylor. 1991. Modulation of con-traction by gelation / solation in a reconstituted motile model. J. CellBiol. 114:1005–1015.

  • Depolymerization-driven Flow 17

    20. Joanny, J.-F., F. Jülicher, and J. Prost. 2003. Motion of an adhesive gelin a swelling gradient: a mechanism for cell locomotion. Phys. Rev. Lett.90:168102.

    21. Keller, H., A. D. Zadeh, and P. Eggli. 2002. Localised depletion of poly-merised actin at the front of walker carcinosarcoma cells increases thespeed of locomotion. Cell Motil. Cytoskeleton. 53:189–202.

    22. Knecht, D. A., and W. F. Loomis. 1987. Antisense RNA inactiva-tion of myosin heavy chain gene expression in Dictyostelium discoideum.Scinece. 237:1081–1085.

    23. Laevsky, G., and D. A. Knecht. 2003. Cross-linking of actin filaments bymyosin ii is a major contributor to cortical integrity and cell motility inrestrictive environments. J. Cell Sci. 116:3761–3770.

    24. Lamunyon, C. W., and S. Ward. 1998. Larger sperm outcompete smallersperm in the nematode caenorhabditis elegans. Proc. R. Soc. Lond. Ser.B-Biol. Sci. 265:1997–2002.

    25. Lauffenburger, D., and A. F. Horwitz. 1996. Cell migration: A physicallyintegrated molecular process. Cell. 84:359–369.

    26. L’Hernault, S. W., and T. Roberts. 1995. Cell biology of nematodesperm. Methods Cell Biol. 48:273–301.

    27. Loitto, V. M., T. Forslund, T. Sundqvist, K. E. Magnusson, andM. Gustafsson. 2002. Neutrophil leukocyte motility requires directedwater influx. J. Leukoc. Biol. 71:212–222.

    28. Marcy, Y., J. Prost, M. F. Carlier, and C. Sykes. 2004. Forces generatedduring actin-based propulsion: A direct measurement by micromanipu-lation. Proc. Natl. Acad. Sci. U. S. A. 101:5992–5997.

    29. Marganski, W. A., M. Dembo, and Y.-L. Wang. 2003. Measurements ofcell-generated deformations on flexible substrata using correlation-basedoptical flow. Meth. Enzymol. 361:197–211.

    30. Miao, L., O. Vanderlinde, M. Stewart, and T. M. Roberts. 2003. Retrac-tion in amoeboid cell motility powered by cytoskeletal dynamics. Science.302:1405–1407.

  • Depolymerization-driven Flow 18

    31. Mitichison, T. J., and L. P. Cramer. 1996. Actin based cell motility andcell locomotion. Cell. 84:371–379.

    32. Mogliner, A., and D. W. Verzi. 2003. A simple 1-D model for the crawlingnematode sperm cell. J. Stat. Phys. 110:1169–1189.

    33. Nelson, G., T. Roberts, and S. Ward. 1982. Caenorhabditis elegansspermatozoan locomotion : Amoeboid movement with almost no actin.J. Cell Biol. 92:121–131.

    34. Nelson, G. A., and S. Ward. 1980. Vesicle fusion, pseudopod exten-sion and amoeboid motility are induced in nematode spermatids by theionophore monensin. Cell. 19:457–464.

    35. Oster, G. 2002. Brownian ratchets: Darwin’s motors. Nature. 417:25–25.

    36. Pollard, T. D., and J. A. Cooper. 1986. Actin and actin-binding proteins.a critical evaluation of mechanisms and functions. Ann. Rev. Biochem.55:987–1035.

    37. Prass, M., K. Jacobson, A. Mogilner, and M. Radmacher. 2006. Directmeasurement of the lamellipodial protrusive force in a migrating cell. J.Cell Biol. 174:767–772.

    38. Rafelski, S. M., and J. A. Theriot. 2004. Crawling toward a unified modelof cell motility: Spatial and temporal regulation of actin dynamics. Annu.Rev. Biochem. 73:209–239.

    39. Roberts, T. M., E. D. Salmon, and M. Stewart. 1998. Hydrostaticpressure shows that lamellipodial motility in ascaris sperm requiresmembrane-associated major sperm protein filament nucleation and elon-gation. J. Cell Biol. 140:367–375.

    40. Roberts, T. M., and M. Stewart. 1997. Nematode sperm: Amoeboidmovement without actin. Trends Cell Biol. 7:368–373.

    41. Roberts, T. M., and M. Stewart. 2000. Acting like actin: The dynamicsof the nematode major sperm protein (msp) cytoskeleton indicate a push-pull mechanism for amoeboid cell motility. J. Cell Biol. 149:7–12.

  • Depolymerization-driven Flow 19

    42. Rodriguez, O. C., A. W. Schaefer, C. A. Mandato, P. Forscher, W. M.Bement, and C. M. Waterman-Storer. 2003. Conserved microtubule-actin interactions in cell movement and morphogenesis. Nat. Cell Biol.5:599–609.

    43. Royal, D. C., M. A. Royal, D. Wessels, S. Lhernault, and D. R. Soll. 1997.Quantitative analysis of caenorhabditis elegans sperm motility and howit is affected by mutants spe11 and unc54. Cell Motil. Cytoskeleton.37:98–110.

    44. Rubinstein, B., K. Jacobson, and A. Mogilner. 2005. Multiscale two-dimensional modeling of a motile simple-shaped cell. Multiscale Model.Simul. 3:413–439.

    45. Sepsenwol, S., H. Ris, and T. M. Roberts. 1989. A unique cytoskeletonassociated with crawling in the amoeboid sperm of the nematode Ascarissuum. J. Cell Biol. 108:55–56.

    46. Shibayama, M., and T. Tanaka. 1993. Volume phase transition andrelated phenomena of polymer gels. Adv. Polymer Sci. 109:1–62.

    47. Small, J. V., and G. P. Resch. 2005. The comings and goings of actin:Coupling protrusion and retraction in cell motility. Curr. Opin. CellBiol. 17:517–523.

    48. Svitkina, T. M., A. B. Verkhovsky, K. M. McQuade, and G. G. Borisy.1997. Analysis of the actin-myosin ii system in fish epidermal keratocytes:Mechanism of cell body translocation. J. Cell Biol. 139:397–415.

    49. Tomita, M., F. Gotoh, M. Tanahashi N .and Kobari, T. Shinohara,Y. Terayama, B. a. O. Yamawaki, T. Mihara, K., and A. Kaneko. 1990.The mechanical filtration coefficient (Lp) of the cell membrane of cul-tured glioma cells (C6). Acta. Neurochir Suppl. (Wien). 51:11–3.

    50. Verkhovsky, A. B., T. M. Svitkina, and G. G. Borisy. 1997. Contractionof actin-myosin ii dynamic network drives cell translocation. Mol. Biol.Cell. 8:974–974.

    51. Vogel, S. 2003. Comparative Biomechanics: Life’s Physical World.Princeton University Press, 41 William Street, Princeton, New Jersey,08540.

  • Depolymerization-driven Flow 20

    52. W., J. L., and D. L. Taylor. 1993. In vitro models of tail contraction andcytoplasmic streaming in amoeboid cells. J. Cell Biol. 123:345–356.

    53. Ward, S., E. Hogan, and G. A. Nelson. 1983. The initiation of spermato-genesis in the nematode caenorhabditis elegans. Dev. Biol. 98:70–79.

    54. Wolgemuth, C., A. Mogilner, and G. Oster. 2004. The hydration dy-namics of polyelectrolyte gels with applications to drug delivery and cellmotility. Eur. Biophys. J. 33:146–158.

    55. Wolgemuth, C. W., L. Miao, O. Vanderlinde, T. Roberts, and G. Oster.2005. MSP dynamics drives nematode sperm locomotion. Biophys. J.88:2462–2471.

    56. Xu, X. X. S., E. Lee, T. L. Chen, E. Kuczmarski, R. L. Chisholm, andD. A. Knecht. 2001. During multicellular migration, myosin ii serves astructural role independent of its motor function. Dev. Biol. 232:255–264.

    57. Yanai, M., C. M. Kenyon, J. P. Butler, P. T. MacKlem, and S. M. Kelly.1996. Intracellular pressure is a motive force for cell motion in amoebaproteus. Cell Motil. Cytoskeleton. 33:22–29.

  • Depolymerization-driven Flow 21

    Table 1: Model Parameters

    Parameter Symbol Value

    Unstressed Polymer Volume Fraction∗ φ0 5 ×10−2

    Membrane Permeability† kf 1.6×10−6 µm3/pN/s

    Atmospheric Pressure p0 1.0×105 pN s/µm2

    Anteroposterior Stress ‡ σx 10 ×p0Transverse Stress σy 2 ×p0Intracellular Drag ζ0 3 ×10

    2 pN s/µm4

    Extracellular Drag § ζ1

    {8

    128×ζ0 (Cell Body)×ζ0 (Pseudopod)

    ∗ Nematode spermatozoa contain roughly 5% MSP by volume.† Consistent with measurement for glioma cells (49) and erythrocytes.‡ Estimate based on the fact that ambient hydrostatic pressure above 50 atmcaused temporary cytoskeletal disruption, while 300 atm causes permanentdisruption (39).§ Consistent with distinct cytoskeletal velocities in the pseudopod and thecell body.

  • Depolymerization-driven Flow 22

    Figure Legends

    Figure 1.

    (A) Side-view schematic of a crawling nematode sperm. Polymerization atthe leading edge pushes the front of the cell forward. Spatially-varying adhe-sion anchors the cell to the substrate and provides traction. Depolymeriza-tion of the cytoskeleton produces contractile force which pulls the cell bodyforward. Coloring denotes volume fraction. (B) Top-view schematic showinganisotropic stress induced by depolymerization, with larger stress induced inthe direction of motion than perpendicular to it. (C) A cultured fibrous col-umn of MSP grows successively shorter over time (right to left). The columnbecomes increasingly faint as depolymerization decreases the optical density.Axial shrinkage exceeds radial shrinkage, suggesting anisotropic cytoskeletalstress. Figure originally published in (30) and reproduced with permission.

    Figure 2.

    A typical C. elegans spermatozoon advances nearly 3µm in 5 seconds (A, B),with little change in shape. The cell has a domed body at the rear (C)and a laminar foot, at the front. Given an empirically determined shape,simulations predict the peripheral polymer assembly rate (C) for a steadilycrawling cell, with a maximum of 0.4µm/s at the leading edge. Simulationsrepresent transmembrane adhesion as external drag (D), with strong adhesionat the front and weak adhesion under the cell body. Relative to the assemblyrate, arrows for fluid flux (E), cytosolic velocity (G) and polymer velocity (I)are scaled by factors of 500, 5 and 1 respectively. Transmembrane fluid flowand cytosolic velocity are plotted in a frame that moves with the cell whilepolymer velocity is plotted in a fixed laboratory reference frame. Simulationsalso yield cytosolic gauge pressure (F) and the magnitude of cytoskeletalstress (H), determined from anteroposterior and transverse components.

    Figure 3.

    Simulations employ real cell shapes with a range of elongations (shapes A,corresponding points B). Working in dimensionless variables allows scalingof each shape to cover the full range of areas. Simultaneous regression showsthat crawling speed depends on both cell elongation and the square root ofcell area (R2 = 0.60). Compared to the best fit (black lines, B and C),

  • Depolymerization-driven Flow 23

    simulation results for optimal anisotropic stress (white lines, R2 = 0.56) fallwithin one standard deviation (gray shading). The fit for simulations withisotropic stress (dashed lines) is not as good (R2 = 0.20). All coefficients ofdetermination are statistically significant (p < 0.001).

    Figure 4.

    Trackable features of a crawling spermatozoon manifest as surface mottlingin a DIC image (above). Feature tracking gives an average speed near0.4µm/s for the cell body (below). Velocities for the anterior cytoskeletonare markedly lower and, from observation, slightly retrograde. Note that nearthe edge of the cell, tracking detects the stationary background, resulting inspuriously low values at some peripheral pixels.

    Figure 5.

    Simulations allow independent manipulation of cytoskeletal anisotropy anddisassembly, with cell size and all other parameters held fixed. With dis-assembly fixed, crawling speed increases rapidly with anisotropy (A), whichis the ratio of anteroposterior stresses to smaller stresses in the transversedirection. With fixed anisotropy, crawling speed increases with increasinglyrapid cytoskeletal disassembly (B).

  • Depolymerization-driven Flow 24

    A B C

    Figure 1:

  • Depolymerization-driven Flow 25

    PSfrag replacements

    A

    B

    2µm

    Body

    FootCPolymerGrowth

    × 1

    DSurfaceAdhesion

    EFluidInflux× 500

    FGaugePressure(kPa)

    GCytosolicVelocity

    × 5

    HPolymerStress(kPa)

    IPolymerVelocity

    × 1

    -0.3

    0.0

    0.2

    0.0

    2.0

    4.5

    PSfrag replacements

    A B

    2µm

    Body

    Foot

    CPolymerGrowth

    × 1

    DSurfaceAdhesion

    EFluidInflux× 500

    FGaugePressure(kPa)

    GCytosolicVelocity

    × 5

    HPolymerStress(kPa)

    IPolymerVelocity

    × 1

    -0.3

    0.0

    0.2

    0.0

    2.0

    4.5

    PSfrag replacements

    A

    B

    2µm

    Body

    FootCPolymerGrowth

    × 1

    DSurfaceAdhesion

    EFluidInflux× 500

    FGaugePressure(kPa)

    GCytosolicVelocity

    × 5

    HPolymerStress(kPa)

    IPolymerVelocity

    × 1

    -0.3 0.0 0.2 0.0 2.0 4.5

    Figure 2:

  • Depolymerization-driven Flow 26

    PSfrag replacements

    A

    B C

    Spee

    d(µ

    m/s

    )Speed

    (µm

    /min

    )

    Elongation (1 − Width / Length)√

    Area (µm)

    1 4 5 6 10

    2 3 7 8 9

    1 2 3 4 5 6 7 8 9 10

    0.2 0.3 0.4 0.5 0.6 5.0 5.5 6.0

    0.4 24

    0.3 18

    0.2 12

    0.1 6Best Fit

    Isotropic Model

    Anisotropic Model

    Figure 3:

  • Depolymerization-driven Flow 27

    PSfrag replacements

    0.1

    0.2

    0.3

    0.4

    Speed (µm/s)

    PSfrag replacements

    0.1

    0.2

    0.3

    0.4

    Spee

    d(µ

    m/s

    )

    Figure 4:

  • Depolymerization-driven Flow 28

    PSfrag replacements

    A B

    0.27

    40.274

    0.39

    4

    0.394

    Elong

    ation

    =0.52

    7

    Elongation = 0.5

    27

    0.12

    0.14

    0.16

    0.18

    0.20

    0.22

    0.24

    0.26

    0.28

    2 4 6 8 10 12 14 16

    0.15

    0.20

    0.25

    0.30

    0.35

    0.03 0.04 0.05 0.06 0.07

    Cra

    wling

    Spee

    d(µ

    m/s

    ) Crawlin

    gSpeed

    (µm

    /s)

    Anisotropy Depolymerization Rate (1/s)

    Figure 5: