-
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: