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IOP PUBLISHING PHYSICAL BIOLOGY

Phys. Biol. 5 (2008) 016006 (14pp) doi:10.1088/1478-3975/5/1/016006

Retractile processes in T lymphocyteorientation on a stimulatory substrate:morphology and dynamicsSergey N Arkhipov and Ivan V Maly

Department of Computational Biology, University of Pittsburgh School of Medicine, 3501 Fifth Avenue,Suite 3064, Pittsburgh, PA 15260, USA

E-mail: maly@ccbb.pitt.edu

Received 9 November 2007Accepted for publication 11 March 2008Published 31 March 2008Online at stacks.iop.org/PhysBio/5/016006

AbstractT cells of the immune system target infected and tumor cells in crowded tissues with highprecision by coming into direct contact with the intended target and orienting the intracellularGolgi apparatus and the associated organelles to the area of the cell–cell contact. Themechanism of this orientation remains largely unknown. To further elucidate it we usedthree-dimensional microscopy of living T cells presented with an artificial substrate mimickingthe target cell surface. The data indicate that long, finger-like processes emanate from the Tcell surface next to the intracellular Golgi apparatus. These processes come in contact with thesubstrate and retract. The retraction accompanies the reorientation of the T cell body whichbrings the Golgi apparatus closer to the stimulatory substrate. Numerical modeling indicatesthat considering the forces involved the retraction of a process attached with one end to the cellbody near the Golgi apparatus and with the other end to the substrate can bring the Golgiapparatus to the substrate by moving the entire cell body. The dynamic scenarios that arepredicted by the quantitative model explain features of the reorientation movements that wemeasured but could not explain previously. We propose that retraction of the surfaceprocesses is a force-generating mechanism contributing to the functional orientation ofT lymphocytes.

1. Introduction

T cells of the immune system come in direct contact withinfected cells and orient their intracellular killing apparatusto the contact site, which ensures that the immune responseis properly directed on the cellular level [1]. The killingapparatus is structurally associated with the Golgi apparatusand with the centrosome, which is the center of convergenceof the microtubule fibers of the cytoskeleton. This complex oforganelles is located eccentrically in the body of a T cell, whilemuch of the cell body volume is occupied by the nucleus. Thecell is nearly spherical when suspended in blood or in cellculture medium, in which most experiments are done, and themain part of the body containing the said organelles remainscompact or nearly spherical even after attachment of the Tcell to the target. The intrinsic eccentricity of the location ofthe Golgi apparatus and of the associated organelles means

that the so-called polarization of this complex to the targetentails not so much, if any, bringing it to the T cell surfaceas positioning it next to the functionally ‘correct’ side of thecell body, rather than next to some other side. We thereforerefer to this positioning as ‘orientation’. Understanding itsmechanism is crucial to understanding the specificity of theimmune response on the cellular level, because this orientationappears responsible for efficient elimination of infected ortumor cells while sparing the healthy bystander cells in thedense tissue environment [1].

The orientation (polarization) of the T cell organellesto the target cell has been known for 30 years [2, 3], yetthe mechanism responsible for bringing about this specificorientation has largely remained elusive. Much effort hasbeen directed at elucidating molecular mechanisms involvedin the T cell polarization. These studies were aided by theintroduction of simplified, and therefore better controlled,

1478-3975/08/016006+14$30.00 1 © 2008 IOP Publishing Ltd Printed in the UK

Phys. Biol. 5 (2008) 016006 S N Arkhipov and I V Maly

experimental systems which replace the target cell by anartificial, biomimetic substrate. T cells exhibit many of theircharacteristic responses to the target cell when presented witha surface coated with a stimulatory clone of antibodies thatbind the T cell receptor molecules on the T cell surface.In particular, preparing in this way the glass bottom of anobservation chamber for live-cell microscopy, and allowingT cells of the Jurkat line to sediment on this surface fromsuspension in the culture medium, creates an experimentalmodel permitting reproducible acquisition of high-qualitythree-dimensional images [4, 5]. This and other experimentalapproaches yielded a substantial body of data on the molecularmechanisms of regulation of the T cell orientation (e.g.,[6, 7]). Yet, in the somewhat paradoxical manner characteristicof today’s cell biology, in comparison to the relative successof the studies of its regulation, progress toward understandingthe very basic mechanism of the orientation itself has beenlimited.

The dominant hypothesis [1] states that the polarization(orientation) of the said intracellular structures in the activatedT cells is brought about by their intracellular movement to the Tcell–target surface contact site from wherever in the T cell theyhappen to be at the moment of contact with the target. It is alsopart of the dominant paradigm that this intracellular movementshould be driven by molecular motors (specifically, by dynein[8, 9]) and (or) by microtubule dynamics (i.e. by assembly ordisassembly of these cytoskeletal fibers [10, 11]). This view isdifferent from the hypothesis put forward by one of the groupsof the discoverers of the T cell orientation [3] that the observedpolarity is a consequence of preferential attachment of the Tcell to the target cell on that side of the T cell, next to which thecentrosome and the associated organelles are already located.The experiments that refuted this preferential attachmenthypothesis, however, were fixed-cell experiments [12, 13].In experiments of this nature, although they yield a measureof dynamic information through observation of different cellpopulations at different times, each individual cell is killed forone-time observation. Therefore, while the interpretation ofthe results obtained as translocation of the organelle complexto the contact site in the individual cell was entirely plausible,these results could not strictly refute that effects on thecell-population level are responsible. The conclusion oftranslocation in individual cells, however, was not deliberatelychallenged in later live-cell microscopy studies. Similarly,the experimental data collected to determine whether thereorientation was intracellular or whether it was a whole-cell movement were two-dimensional videomicroscopy datathat remained unpublished. The conclusion drawn fromthese data that the movement was intracellular [12] wassimilarly not challenged with the advent of higher-resolution,three-dimensional imaging techniques. The supposition thatthe reorientation is driven by dynein and (or) microtubuledynamics is also entirely logical within the framework ofthe intracellular migration paradigm. However, while onlyindirect evidence compatible with these mechanisms waspresented [8–11], involvement of the microtubule dynamicswas directly refuted in experiments with taxol [14] and that ofthe dynein motors had been impossible to demonstrate directly

in experiments with dynactin [15] despite expert efforts. Thus,the interpretations of the early experiments that were the mostfundamental to our understanding of the basic kinematics ofT cell orientation were not re-evaluated with application ofnew and potentially more perceptive microscopy techniques,and the ‘negative’ results of experiments testing the leading(qualitative) hypotheses regarding its driving forces remainedless known.

On the basis of our live, three-dimensional observationsand measurements, we determined that the side of the T cellnext to which the centrosome and the associated organelleswere already located did not exhibit any significant preferentialpropensity for developing the adhesion contact with thebiomimetic substrate. At the same time we did uncover asmall, albeit significant measure of preferential detachment ofT cells developing the contact in the ‘incorrect’ orientation[16]. This is a purely cell-population-level effect, whichresults in enrichment of the population of attached T cellswith those that are functionally oriented. Its contribution,however, was relatively minor (about 15%). Therefore, thereindeed exists a significant measure of actual, mechanicalreorientation of individual cells. Compared to the dominantparadigm, however, the reorientation in our measurementsappeared brought about by rotation of the entire cell body,which entrained the interior structures [17]. We consideredtwo possible, and not necessarily alternative, driving forces forsuch cell body rotation on the stimulatory surface. One wasfree-energy minimization of the overall cell structure [17]. Theother was migration of the cell–substrate contact area aroundthe cell due to polarized recycling of receptors mediatingthe interaction [18]. However this dynamic explanationof the rotation is not as yet satisfactory. Although wehave demonstrated the quantitative plausibility of the free-energy gradient leading to the functional orientation [17],experimental tests have not as yet been conclusive as towhether such an effect plays a role in the actual polarization.At the same time, although the contact migration hypothesiswas kinetically self-consistent [18], our experiments withinhibition of receptor recycling, which was proposed tounderlie this hypothetical mechanism, indicated only a minorquantitative contribution of it to the overall polarization [16].Thus, the driving force of the T lymphocyte orientationremains largely unknown.

To gain further insight into the mechanism of the Tlymphocyte orientation, we made new, higher-resolution(confocal), three-dimensional, live microscopic observations.Their results are reported here, followed by numerical tests oftheir novel implications for the mechanics of this process.

2. Materials and Methods

2.1. Experimental procedures

Jurkat cells were grown and prepared for observationessentially as described before [5, 17, 18]. In brief,cells suspended in RPMI1640 growth medium (Invitrogen,Carlsbad, CA) were injected into the observation chamber(LabTek, Brendale, Austria). The chamber bottom was

2

Phys. Biol. 5 (2008) 016006 S N Arkhipov and I V Maly

glass pre-coated with poly-l-lysine and with anti-TCRantibodies (clone UCHT1, Pharmingen, San Diego, CA).The sedimenting cells were observed on a Nikon TE 200inverted microscope (Nikon, Melville, NY). To visualize thecell surface and the Golgi apparatus, the cell membraneswere labeled with 5 µM BODIPY FL C5-ceramide-BSA(Molecular Probes, Carlsbad, CA) for 45 min at 37 ◦C under5% CO2 before the injection into the observation chambers.The 60× planapochromatic water-immersion objective withnumerical aperture 1.2 (Nikon) was actuated by a PIFOC721 piezo-positioner (Physik Instrumente, Auburn, MA).The camera, the objective driver and a shutter (VincentAssociates, Rochester, NY) were controlled by IPLab software(Scanalytics, Rockville, MD), which was also used for imageanalysis. The temperature (37 ◦C) was maintained usingan ASI 400 air stream incubator (Nevtek, Burnsville, VA).The images were acquired using a CARV II spinning-discconfocal attachment (BD Biosciences, Franklin Lakes, NJ)and an ORCA II ERG cooled interline camera (HamamatsuPhotonics, Bridgewater, NJ). At each time point, stacksof images were taken separately on the wavelength of thefluorescent label and in transmitted incandescent light. Bymoving the objective, three-dimensional images were acquiredat a formal resolution (voxel size) of 0.22, 0.22 and 0.4 µm inthe X, Y and Z dimensions, Z being along the optical axis andorthogonal to the glass forming the bottom of the observationchamber. Each Z-stack was acquired over 7.5 s.

2.2. Mathematical modeling

The mathematical model described in this methods sectionfollows logically the results of the experiments, which aredescribed in the results and discussion section, and to whichthe reader is here referred. To compute the T cell movementscaused by retraction of a process attached on one end to thecell and on the other to the substrate, we follow with necessarymodifications the approach used by Zhao et al [19] in theirstudy of passive leukocyte rolling. To reflect the geometry ofour experimental situation in a somewhat idealized form, weconsider a spherical cell on a substrate plane. For most cellsin the experiments, however, it is a significant idealization ifwe consider only one process that is attached to the substratewith its distal end and that actively retracts, thus setting thespherical cell body in motion. We formulate the model forthis idealized case of one retracting process first, and thendescribe a generalization to two processes. We assume thatthe retracting process is not capable of changing length apartfrom the shortening described by the retraction velocity asdefined below, while it is capable of bending freely to wraparound the cell body, without friction. We introduce thecoordinate system (see figure 10), with the x-axis running fromthe initial cell–substrate contact point to the point of contactbetween the retractile process and the substrate (which pointhas the coordinate xm), and z orthogonally into the medium.The y-axis forms the right-handed system, and the anglesused in the model are counted from the direction of the z-axis counterclockwise around the y-axis. The position andorientation of the model cell are then fully described by the x-coordinate of its body center, xc, and by the angular coordinate

of the point of attachment of the retracting process to the cellbody, θm. Then the kinematic equations of the cell motion are

dxc

dt= vx, (1)

dθm

dt= ωy. (2)

The instantaneous vx and ωy are determined by the balancesof forces and torques exerted on the cell body and by theforce–velocity relationship governing the retraction of the cellprocess. For the latter relationship (which can be regarded asan implicit force-balance equation), we assume the Hill formthat has been used in studies on muscle contraction as wellas on cell locomotion [20, 21] (see the discussion section).The equations from which the instantaneous translational andangular velocities of the cell body are to be determined arethen as follows:

h1vx + h2ωy + F sx + kxF

m = 0 (3)

h2vx + h3ωy − RFsx + RktF

m = 0 (4)

Fm = Fmmax

1 − vm

vmmax

1 + c vm

vmmax

. (5)

Here, h1 = −2.08 pN s µm−1, h2 = 1.24 pN s and h3 =−52.2 pN µm s are hydrodynamic drag coefficients calculatedbelow following Zhao et al [19] from the cell size, the mediumviscosity and the cell surface roughness height. F s

x is thehorizontal component of the substrate reaction force. kx

and kt are factors that determine how the contraction forceFm is projected onto the x-axis and onto the tangent to thecell body, drawn through the point of the process emanationfrom the cell body (figure 10). The expressions for theseand other geometrical factors deriving from the instantaneousconfiguration of the cell and the retracting process are givenbelow. We assume the cell body radius R = 10 µm to reflectthe size of the cells in our experiments. vm is the velocityof the active process retraction. Fm

max = 28 pN, vmmax =

0.26 µm s−1 and c = 5.1 are the parameters of the force–velocity relationship (5), which we determine by a least-squares fit of this function to the data on filopodia retractionin macrophages from the table in [22] (see the discussionsection).

Assuming no slip between the cell body and the substrate,the angular cell body velocity and the velocity of retraction vm

are both proportional to the translational velocity of the cellbody:

ωy = vx

R(6)

vm = kno slipv vx (7)

where kno slipv in the last equation is a kinematic factor, the

expression for which is given below. After substitution of(6) and (7), the instantaneous balance equations (3)–(5) canbe solved for vx , F s

x and Fm. This will involve a quadraticequation for vx , and its root that has the physically correctsign should be taken.

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Phys. Biol. 5 (2008) 016006 S N Arkhipov and I V Maly

Unless we assume that slip is impossible (infinite frictionbetween the cell body and the substrate), it is necessary todetermine whether the substrate reaction force F s

x requiredto maintain the no-slip condition is indeed attainable: F s

x

calculated above needs to be compared with the maximumfriction force, given the friction coefficient η and the normalreaction force

F sx max = −η

(kzF

m + Fgz

), (8)

where kz is the projection factor of the retraction force ontothe z-axis (see below), Fm was solved for above and F

gz is the

projection of the force of gravity on the z-axis,

Fgz = − 4

3πR3�ρg, (9)

with g being the free-fall acceleration and �ρ the densitydifferential between the cell and the medium (assumed tobe 0.08 g ml−1 as was determined for polymorphonuclearleukocytes [23]). If F s

x is not greater than F sx max by magnitude,

the above calculations based on the no-slip assumption will bevalid, and the integration of the kinematic equations (1) and (2)can proceed with the instantaneous vx and ωy obtained above.

Should, however, F sx obtained under the no-slip

assumption exceed F sx max, the no-slip assumptions (6) and

(7) will not be valid. Instead, we will have to assume that thehorizontal substrate reaction force takes the maximum value,given the constant gravity force and the unknown retractionforce:

F sx = −kη

(kzF

m + Fgz

), (10)

where kη has the magnitude of η and the sign of F sx which was

found under the no-slip assumption. Additionally, in the caseof slip,

vm = kslipv vx + kslip

ω ωy, (11)

with the kinematical factors kslipv , k

slipω defined below. After

substitution of (10) and (11), the instantaneous balanceequations (3)–(5) can be solved for vx , ωy and Fm. Thiswill again involve a quadratic equation (for vx or ωy), andits root that has the physically correct sign should be taken.The obtained values of the instantaneous velocities can be usedfor further integrating the motion equations (1) and (2). Weemployed the forward Euler integration method and simulatedthe cell movements in the Matlab programming environment(Mathworks, Inc., Natick, MA).

To calculate the hydrodynamic drag coefficients used in(3) and (4), we employ the approximation derived by Zhaoet al for their study of leukocyte rolling [19]. We assumemedium viscosity of 0.001 pN s µm−2 (1 centipoise), as in thecited work. We also assume roughness height of the cellbody surface of 10 nm, as a presumed minimum realisticvalue, so as to test the ability of the process retraction toreorient the cell under the highest hydrodynamic resistancedue to closest proximity to the substrate. The coefficientsh1–h3 do not depend strongly on the roughness height in thisrange, and the exact choice of it therefore had essentiallyno impact on the model results. Being much smaller thanR, the roughness height does not affect the cell geometryconsiderations and therefore does not appear in the model

apart from the calculation of the hydrodynamic resistancecoefficients.

To derive the geometrical and kinematical factors usedin (3), (4), (7), (8) and (11), let us introduce xm, the x-coordinate of the point where the retracting process is attachedto the substrate, and θ eff

m , which is the angular coordinate(counted the same way as θm, figure 10) of the point wherethe retracting process actually separates spatially from the cellbody. Considering that a part of the process can wrap aroundthe cell body, and assuming the retracting process on the rightside of the cell, as in figure 10 and in the numerical examplesin this paper,

θ effm = max

{θm, π − arctan

xm − xc

R

− arccosR√

(xm − xc)2 + R2

}.

We will also introduce the length of the part of the retractingprocess that is spatially separated from the cell body:

lf =√(

xm − xc − R sin θ effm

)2+ R2

(1 + cos θ eff

m

)2.

Then it can be verified by considering the drawing (figure 10)that

kx = xm − xc − R sin θ effm

lf,

kz = −R(1 + cos θ eff

m

)lf

,

kt = cos(arccos kx − θ eff

m

),

kno slipv =

⎧⎨⎩

(xm − xc)(1 + cos θ eff

m

)lf

, if θ effm = θm

kx + 1, if θ effm �= θm,

kslipv = kx,

kslipω =

⎧⎨⎩

R(xm − xc) cos θ eff

m + R sin θ effm

lf, if θ eff

m = θm

R, if θ effm �= θm.

To generalize to two simultaneously retracting processes,the system of force-balance equations (3)–(5) must beaugmented to incorporate the retractile force of the secondprocess. This force in the general case differs from theretractile force of the first process. This is so even if thegoverning force–velocity relationship is the same for both,due to the kinematic effects: if the attachment points ofthe two processes are not identical, their retraction velocitiescorresponding to the same translation and rotation velocity ofthe cell will be in general different. In addition, the horizontalforce and the torque which the two processes exert on thecell body may be different even with the same retractionforce magnitude in each process. The different kinematicsof the two processes is captured by the geometric factors,which must be calculated separately for each process, usingthe above formulae. In the following equations, the quantitiesreferring to the two retracting processes are distinguished bythe subscripts I and II:

4

Phys. Biol. 5 (2008) 016006 S N Arkhipov and I V Maly

h1vx + h2ωy + F sx + kxIF

mI + kxIIF

mII = 0,

h2vx + h3ωy − RFsx + R

(kt IF

mI + kt IIF

mII

) = 0,

FmI = Fm

max

1 − vmI

vmmax

1 + cvm

Ivm

max

,

FmII = Fm

max

1 − vmII

vmmax

1 + cvm

IIvm

max

.

Solution of this system, in the no-slip case, for example, leadsto a cubic equation for vx , which can be solved numerically,again taking care to select the physically meaningful root.

3. Results and discussion

3.1. Experimental results

In Jurkat cells sedimenting onto the bottom of the observationchamber, we consistently observed plumages of thin processesprojecting into the medium from the side of the sphericalcell body, next to which the Golgi apparatus is located(figures 1(A) and (B)). These processes are dynamic andephemeral. They retract within minutes of sedimentation ofthe cell, often being replaced by a lamellipodial skirt that hasbeen described previously [4, 16]. In comparison to the flatlamellipodia that develop anew in close apposition with thesubstrate and persist for long periods of time, the pre-existingprocesses are usually finger like and can dynamically reachthe substrate from the side of the spherical cell body beforedisappearing (figure 2). We observed that their disappearance(presumed to occur by retraction) consistently accompaniedreorientation of the Golgi apparatus toward the stimulatorysubstrate (figures 2 and 3).

In many cells the processes emanate from a relativelybroad segment of the cell perimeter, often significantly largerthan the Golgi site, and sometimes to the side or even oppositeto the Golgi apparatus. An entirely objective quantificationof the relative position of the Golgi apparatus and the sites ofprocess emanation is difficult due to the fact that the presenceor absence of an individual process at a particular locationcannot always be determined reliably in images of living cellsthat are as thick as they are wide. To test our morphologicalimpression of the predominant emanation of the processesfrom around the Golgi site, we attempted to classify allcells visible in random wide-field images from multipleindependent experiments. An example of the random fieldis shown in figure 1(C). In a fraction of cells, the orientationof the Golgi apparatus cannot be reliably ascertained fromthe fluorescence image. In other cells, the orientation ofthe Golgi does not permit high-throughput assessment of itscolocalization with the processes, because the Golgi locationin them is not peripheral, in the imaging plane. The remainingcells can be classified as (a) unambiguously having processeson the same side as the Golgi (marked ‘+’ in figure 1(D)),(b) unambiguously exhibiting processes not colocalized withthe Golgi (‘−’) or (c) cells displaying processes on all sidesor no processes at all (marked ‘X’). In complete fields from

(A)

(B)

(C)

(D)

(E ) (F ) (G)

Figure 1. Retractile processes emanate from the side of the cell nextto which the intracellular Golgi apparatus is located. (A) two livecells imaged in transmitted light immediately following theirsedimentation onto the bottom of the observation chamber. Scalebar: 20 µm. One of the processes is indicated by the arrow. (B) Aconfocal section of the same field of view imaged on the wavelengthof the fluorescent ceramide-based probe that labels the cellmembranes and reveals the cell outline (plasma membrane) and theintracellular Golgi apparatus (indicated in one cell by the arrow).(C) Vertical projection (top view) of the fluorescence image showingcell outlines and bright Golgi areas in a random field of cells. Cellsdisplaying an unambiguous peripheral position of the Golgiapparatus in this view are those that are classified (marked) in panel(D). (D) Conventional transmitted-light image of the same field.Cells displaying a clear polarization of the surface processes on theside of the cell where the Golgi apparatus is located are marked ‘+’.Cells displaying a clear polarization of the surface processes on theside of the cell opposite to where the Golgi apparatus is located aremarked ‘−’. Cells not displaying a clear polarization of theprocesses with respect to the orientation of the Golgi apparatus aremarked ‘X’. In the unmarked cells, the Golgi orientation itself isunclear or is not peripheral enough in the top view to permitassessment of colocalization with the processes that would bevisible in transmitted light (see panel (C)). ((E)–(G)) A videosequence of retraction of a process. Time indicated is the timeelapsed from the beginning of the experiment. Arrowheads point tothe base and tip of the process, and in the last frame to the formeremanation site on the cell surface. Scale bar: 10 µm.

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Phys. Biol. 5 (2008) 016006 S N Arkhipov and I V Maly

Figure 2. Extension of a process to the substrate precedes, and itsretraction accompanies cell reorientation. The same cell as the oneon the left in figures 1(A) and (B) is shown. Panel columnscorrespond to time points. The time is counted approximately fromthe first contact of the cell with the transparent, non-fluorescentstimulatory substrate. Top row of panels: transmitted-light images,top view. Cell surface processes are indicated by arrows. Scale bar:10 µm. Other panels: fluorescence of the membrane probe showingthe cell outline and the bright Golgi apparatus, in RGB pseudocolor.The brightness scale is consistent for all time points in each view. Inthe row labeled ‘top’, the vertical projection of the three-dimensionalimage is given. The white bars in this row indicate the depth rangesthat were used to make the horizontal projections presented in thenext row, which is labeled ‘side’. The depth for 10 min is the sameas for 5 min. Note the reorientation of the Golgi complex to thesubstrate. In the 2 min side view, a dangling process is indicated byan arrow. The two level marks indicate the position of the confocalsections. Arrows in the sections indicate the process, where seen.

ten experiments done on four separate days, we found 90cells whose Golgi position was clear and permitted to attemptassessment of colocalization. Of these, 42 were found to be ofthe morphological class ‘X’, 31 of the class ‘+’ and 17 of the

Figure 3. Process retraction and body reorientation in the secondcell from figures 1(A) and (B). The row labeled ‘TL top’ consists ofconventional (i.e. top view) transmitted-light images. The rowlabeled ‘FL side’ displays horizontal projections of the three-dimensional images taken on the wavelength of the fluorescentmembrane label at each time point, i.e. the side views. The scale baris 10 µm.

class ‘−’. Overall, the cell classification in random fieldsconfirmed our morphological impression that a significantfraction of cells displays a clear polarity of the processes,and that when the polarity of the processes is clear it tends tofollow the polarity of the Golgi apparatus.

To confirm directly our supposition that the visualdisappearance of the processes is due to their retraction, wetracked individual processes that are visible in transmittedlight as emanating from the cell body in the conventionalimaging plane (above and parallel to the substrate).Higher-speed imaging permitted by this setup showedunequivocally the shortening of the processes, leading to theireventual disappearance (figures 1(E)–(G)). These observationsconfirmed our supposition of retraction.

These processes can reach the substrate even from the topof the cell, if this is where the Golgi apparatus is found uponthe cell sedimentation, by drooping alongside the cell body(figure 4). Correspondingly, the cell can reorient by as muchas 180◦, bringing the Golgi from opposition to apposition withthe substrate, while retracting the processes (figure 5).

In cells that not only retract the initially present thinprocesses, but also develop lamellipodia, the fractions of theGolgi reorientation that occur during the process retractionand during the subsequent lamellipodia development vary. Insome, the reorientation occurs in large part during lamellipodiadevelopment (figure 6), and in others it occurs almost entirelyduring the initial process retraction (figure 7). The verycharacteristic polarized cell morphology with an extendedasymmetric lamellipodium and the Golgi apparatus at itsbase (and therefore next to the substrate) was commonlyseen developing when the initially spherical cell came incontact with the substrate with the Golgi located on theside and with the plumage of thin processes emanatingdirectionally from that area. These processes, however, arenot necessarily transformed into the lamellipodium, but mayretract completely before the onset of the lamellipodiumdevelopment (figure 7).

6

Phys. Biol. 5 (2008) 016006 S N Arkhipov and I V Maly

Figure 4. Filamentous processes long enough to reach to thesubstrate on which the cell is sitting from the Golgi apparatus regionon the top of the cell. In the top, transmitted-light image, the scalebar corresponds to 10 µm. Two processes are visible. In theside-view fluorescence image below it, the process on the left can beseen drooping down from the top of the cell to the substrate. Theprocess on the right appears short from this perspective. The twolevels corresponding to the confocal optical sections shown beloware indicated in the side-view image by the two white bars. Theupper optical section reveals that the right process is in fact long, butgoes mostly horizontally. The left process in this section appearsoptically inseparable from the cell body. The lower section showsits cross-section as clearly separate from the cell body.

As the cells retracted the pre-existing, thin polarizedprocesses, reorientation of the Golgi apparatus occurred inmany of them by apparent rolling of the cell body on thesubstrate without noticeable slippage (figure 8). In others,however, we observed a significant degree of slippage thatallows the cell to reorient without rolling on the substrate, butby spinning essentially in place. Reorientation of the Golgiapparatus to the substrate with the apparent slippage, whichkeeps the cell body in place laterally, can be accompaniedby twisting of the cell body in the plane of the substrate,which by itself does not change the Golgi–substrate distance(figure 9).

3.2. Numerical analysis

Two questions arise from our new observations: consideringthe forces involved (i.e. the dynamics), is it physically feasiblethat the process retraction can drive the cell reorientation?And can the rolling and spinning be both explained by it? Toaddress the theoretical biomechanics of cell movements drivenby the process retraction, we made suitable modifications to

Figure 5. A cell reorienting completely while retracting itsprocesses. Conventions as in the previous figures. Scale bar: 10 µm.

Figure 6. A cell in which approximately one-half of thereorientation occurs during lamellipodia formation that follows theprocess retraction. Conventions as in the previous figures. Scalebar: 10 µm.

the mathematical formalism [19] that had been previouslydeveloped for rolling of leukocytes driven by the bloodstreamand interacting with the blood vessel wall through microvilli.The formalism is based on balancing the driving torque andforce by the torque and force associated with hydrodynamicresistance to the cell movement. Our approach (see thematerials and methods section) was to remove from thisestablished modeling paradigm the driving force of themedium shear gradient and to make the movement driveninstead by microvillus (surface process) contraction. Asthe force–velocity relationship in the process contraction weused the hyperbolic Hill relationship employed in musclecontraction and in cell locomotion research (e.g. [20, 21]),

7

Phys. Biol. 5 (2008) 016006 S N Arkhipov and I V Maly

Figure 7. A reorienting cell in which retraction of a polarizedplumage of processes precedes development of a polarizedlamellipodium. Conventions as in the previous figures, except thatthe transmitted-light images in the top row are not single opticalsections but minimum-intensity projections of the stack of sections,which shows better the relevant structures in the present cell. Scalebar: 10 µm.

Figure 8. A cell exhibiting rolling on a straight line while retractingits processes. Images prepared and labeled the same way as in theprevious figure. In particular, as in all previous figures, all the imageframes refer to exactly the same (three-dimensional) field of view.Reference lines are drawn to better reveal the cell movements. Scalebar: 10 µm.

and fitted it to the data on filopodia retraction in macrophages[22]. It is characterized by the maximum force of 28 pN (atzero velocity) and by the maximum velocity of 0.26 µm s−1

(at zero force). We assume that the cell body is sphericalwith a diameter of 20 µm, as typical in our experiments,that it experiences viscous drag from the surrounding culturemedium, that the shortening process can bend freely to wraparound it and that there is no friction between the process andthe body. The details of the mathematical model are given in

Figure 9. A cell exhibiting twisting and slippage. Images preparedand arranged the same way as in the previous figure, except that thetransmitted-light (‘TL’) images shown here are conventional, singleoptical sections. Reference lines are drawn. Scale bar: 20 µm.

Figure 10. Diagram of the physical model. The circumferenceshows the outline of the spherical cell body. The surface of theplanar substrate on which the cell is sitting coincides from thisperspective (from the side) with the x-coordinate axis shown. Thebold line shows the retracting process attached with one end to thecell body and with the other to the substrate. Also shown are theangle θm characterizing the orientation of the process attachmentpoint to the body, its rate of change ωy, the x-coordinate of the cellbody center xc and its rate of change vx , the substrate reaction forceFs and the process contraction force Fm as exerted on the cell body,the process contraction velocity vm and the constant x-coordinate ofthe process attachment point to the substrate xm. The model detailsare given in the materials and methods section.

the materials and methods section, and figure 10 outlines itdiagrammatically.

The numerical simulation shows that under theassumptions made the process retraction would be able toorient the part of the cell body, to which the retracting processis attached, to the substrate within the time frame of ourexperiments (figure 11).

Notably, however, if no slippage between the cell body andthe substrate is assumed, static equilibrium is reached after thecell body moves laterally by one radius and rotationally byone radian. This does not preclude an essentially completereorientation to the target surface (substrate) of the process–body attachment site (and therefore, presumably, of the Golgi

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Figure 11. Diagrammatic representation of the numerical results ofa simulation of cell movements in the case where the attachmentpoint of the retracting process (bold line) to the body is initially onthe side of the cell and where there is no slip between the sphericalcell body (circumference) and the planar substrate (horizontal line).The side view is shown for each time point in the same referenceframe, which reveals rolling displayed by the model cell body as theprocess attachment point to the body is brought closer to thesubstrate through the process retraction.

apparatus that is typically located in that part of the cell), if,as in figure 11, it is initially on the side of the cell. It mustbe observed that this initial orientation with respect to thesubstrate is the most probable one, if the cell comes in contactwith the substrate in a random orientation. This is the case inour experiments with cells sedimenting from suspension, and,although this is not known, may also be the case in vivo.

The limitation, however, would prevent properreorientation of cells in which the process emanation point(and the Golgi apparatus) is initially on the top. This isillustrated by the simulation shown in figure 13, in which afinite, but very high friction coefficient between the cell bodyand the substrate was assumed (η = 10). The plots in figure 12show that as the lateral and rotational displacements stabilizeat the cell radius and at one radian, respectively, the contractionstalls: its speed vm reaches zero and force Fm the maximum.The reason lies in the kinematics of the cell movements. Thegeometrical factor kt projecting the contraction force onto thetangent to the surface remains constant and equal to 1 dueto wrapping of the retracting process at least partly aroundthe cell body. We assume that the wrapping, as shown inthe model diagrams, will always occur when the processemanation point is initially above the cell’s ‘equator’, whichassumption is as an idealization based on the appearance of

the drooping processes in our microphotographs (figure 4). Incomparison to kt, the factor kx projecting the contraction forceonto the x-axis increases by magnitude from 0 to −1 as thecell body rolls and the process, although it is shortening, endsup wrapped completely around the cell body. At that point thestatic equilibrium is reached, as both the torques and forces ofcontraction and substrate reaction become exactly balanced.The reason this state can be reached is that although the factorkz projecting the contraction force onto the vertical z-axisdecreases by magnitude to essentially zero, even the small,compared to the contraction force, force of gravity acting onthe cell (3.2 pN, see the materials and methods section) issufficient to keep the maximum static friction force F s

x maxabove the tangential component of the substrate reaction forceF s

x at all times with the friction coefficient as high as η = 10(figure 12).

Already in the above situation with the high frictioncoefficient one can observe, however, development ofconditions favoring slip. The eventual decrease of themaximum tangent substrate reaction force F s

x max due to thediscussed kinematic effect favors it, but even the evolutionof the system prior to that does. F s

x max undergoes a periodof increase initially, due to the fact that with the developingstall the retraction force increases according to the force–velocity relationship, and this increase of the retraction forceis rapider than the decrease of the magnitude of the factorprojecting this force onto the normal to the substrate, kz (seethe plots of these quantities versus time in figure 12). Buteven during this initial period of the F s

x max increase, the actualtangent substrate reaction force F s

x rises more steeply yet. Theevolution of F s

x and of its maximum possible value, F sx max, is

therefore at all times toward the former reaching the latter.Thus, the cell rolling due to the process retraction creates, byitself, conditions for overcoming the friction between the cellbody and the substrate. Only reasonable values of the frictioncoefficient are required for this to actually happen, such as η =1 (see the plots in figure 12 and the diagrammatic presentationof the simulation in figure 13).

In the example with η = 1 (figures 12 and 13) the cell bodystarts slipping on the substrate shortly before the complete stallwould be achieved, when the cell body has rolled displacinglaterally by nearly one radius and rotating by one radian.After this happens in about 0.5 min, the cell completes itsreorientation within the next 3 min by rotating essentially inplace. At the very end, the grip is restored and another shortperiod of rolling follows before the equilibrium is reached(the small deviation of F s

x from F sx max during this period is

not apparent in figure 12 due to scale). In the final state,the process has retracted completely and its emanation pointis directed exactly to the substrate. So, presumably, wouldbe oriented also the Golgi apparatus that would be locatedin the cell near the emanation point of the retracting process.With the reasonable friction coefficient, therefore, the processretraction would be capable of reorienting the cell even in themore challenging case of the initial orientation being oppositeto the immunologically functional one.

For completeness, we also consider the case of zerofriction between the cell body and the substrate (η = 0, see

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Figure 12. Results of computer simulation of cell movements under different coefficients of friction (η) between the cell body and thesubstrate. For each of the three values of η, a column of graphs is given, showing how the different variables (see the text and figure 10 forthe nomenclature) of the numerical model change as a function of time.

figures 12 and 13). In this case there is no stage of rolling.The cell initially rotates almost in place, exhibiting only veryslow lateral movement, as can be seen from the plots of thecentroid position xc and the orientation angle θm in figure 12.This leads to complete unwrapping of the retracting processfrom the cell body, at which point the factor kt projecting theretraction force onto the tangent to the cell body deviates from1 and starts a rapid descent to 0 (figure 12). The processbegins to pull the cell laterally toward its attachment pointto the substrate, which is manifested by the transient rise ofthe factor projecting the retraction force onto the horizontalx-axis (kx, figure 12). The lateral movement of the cell bodythen abruptly accelerates while the rotation continues at onlya slightly higher speed than before (see evolution of xc and θm

in figure 12). Thus, the rotation and lateral movements in thezero-friction case are never coordinated as they are in rolling,yet they would still be both exhibited by the cell. Moreover,in this case as well as the other case of complete reorientation(η = 1 above), the cell eventually will have moved laterally byone radius and rotationally by about three radians.

Finally, we consider a generalization of the model to twosimultaneously retracting, arbitrarily anchored processes. Thesimulation shows that even in the absence of slip (figure 14),retraction of a process that is anchored farther away from thecell body (red in figure 14) can lead to the cell rolling over thepoint of attachment of the more proximally anchored process(blue in figure 14). This means that the cell movement cancontinue after it would be expected to stall under the action of

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Figure 13. Diagrammatic representation of the numerical results ofsimulation of cell movements under different values of the frictioncoefficient. The columns of diagrams correspond to the threesimulations for which the numerical details are given in figure 12,and the graphical conventions here are the same as in figure 11.

only one process. The equilibrium is reached only after theretraction of the second process stalls. This happens whenthe point of contact of the cell body with the substrate reachesthe point where the second process is attached to the substrate.The static-equilibrium position is thus predicted to be over themore distant of the two substrate attachment points.

3.3. Discussion

In this work, we have described processes on the surface ofT cells that emanate most commonly from the area next tothe intracellular Golgi apparatus. These processes retractedafter contacting the stimulatory biomimetic substrate. Theretraction accompanied reorientation of the cell body intothe immunologically functional position, with the Golgiapparatus facing the substrate. The new data show thatthe cell reorientation during the process retraction is ingeneral different from, but complementary to the reorientationfollowing the lamellipodium formation, which we describedbefore [17]. Also, the polarized extension of the cell–substrate contact area (i.e. of the lamellipodium), which wepreviously implicated in the reorientation [18], is, accordingto the new data, in general different from the pre-existingpolarized plumage of the finger-like processes, which wedescribe here for the first time. The common direction of thetwo types of protrusions may be indicative of their commongenesis, but this is presently unclear, and one type is not ingeneral seen turning directly into the other. Thus, the T cell

Figure 14. Diagrammatic representation of the numerical results ofsimulation of cell movements with two retractile processes and noslip. The graphical conventions are the same as in figure 11, withthe two processes distinguished by color.

reorientation accompanied by the retraction of pre-existingcell surface processes that is reported here appears entirelynovel. The numerical model also introduced in this paperconfirmed the biophysical plausibility of our hypothesis thatthe process retraction could be the cause of the concomitantcell reorientation. The explanatory power of the new modelwill be discussed below, but first we would like to point outclose similarities in our new observations to those which werepublished previously by others but which have not so far beenconnected to the problem of the mechanism of functionalpolarization in T cells.

The cell processes described here are much longer thanthe microvilli that are commonly considered in models ofleukocyte rolling on the blood vessel wall. These models(e.g. [19, 24]) assume a sub-micron length of the cell surfaceprocesses, based on the measurements made on electronmicrographs [25]. A notable exception is the original workby Tissot et al [26], who measured and modeled microvilli afew microns long. It is necessary to point out that a process

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is likely to appear much shorter in the ultrathin sections forelectron microscopy than it is in reality, because it is likelyto be intersected by the section plane far more proximallyto the cell body than its actual end. In view of this, andawaiting data not biased by this effect, we are not consideringthe processes described here as different in nature from thosein these previous studies: the long processes, which appearedan exception in the electron micrographs and to which Tissotet al called attention, are in fact likely to be common, and to becomparatively easily revealed in favorable projections of thelive three-dimensional confocal images presented here. Themulti-micron length was also recorded in the experiments onfibroblasts [27], which are discussed in more detail below.

Retraction of the cell surface processes following contactwith the substrate stimulating the T cell receptor in ourexperiments is likely to be the same phenomenon as retractionof microvilli in T cells that were stimulated in suspension bysoluble chemokines [28]. At the same time, retraction of cellsurface processes of similar appearance was described longago for fibroblasts sedimenting onto a plain-glass substrate.Retraction in this case similarly followed the processescontacting the substrate and preceded development of a widearea of cell–substrate attachment [27]. It might be indicative ofthe common mechanism of cell orientation that the fibroblastsalso orient their centrosomes (with which the Golgi apparatusis usually associated) to the substrate to which they thusattach [29]. The co-location of the emanation points ofthese processes and the Golgi apparatus or the centrosomewas not addressed in the live-cell fibroblast study discussedabove. It is nonetheless in agreement with the co-location aswell as with the role of the retraction in orientation that thecentrosome was found facing the substrate in fibroblasts veryearly in their multi-stage attachment process, specifically whenthe cell appeared as a sphere with filopodia on the substrate[30] (filopodium is another term for a finger-like cell surfaceprocess; see also the electron micrographs in [31, 32]).

It was proposed already in the early studies of cellattachment that a small radius of curvature is required ofthe cell membrane to overcome electrostatic repulsion andapproach the target substrate closely enough for adhesionmediated by the specific or nonspecific molecular interactions[33]. It was shown that the initial contact indeed occurs at theends of filopodia. This was seen in the fibroblasts adheringto plain glass [31, 32], and in the T cells reacting specificallywith the biomimetic substrate the contact and stimulation alsostarted in small focal points separated from the bulk of the cellsurface [34, 35]. Thus, the small radius of curvature whichmust be exhibited by the cell membrane bounding the thinprocesses apparently makes them or their tips the preferredsites of the contact initiation. In light of our new data, the firmcontact at the process tips that is so initiated may provide thenecessary anchor point, for the subsequent process retractionto actually cause movement of the cell body.

The large moment arm that can be provided by a retractingprocess may be a crucial piece in the T-cell orientationdynamics puzzle. Indeed, the process anchored with its tipon the substrate will connect to the cell body on its surface, i.e.as far away from the cell center as possible, thus creating the

largest possible torque arm to drive the cell reorientation mostefficiently with the given force. The presented theoreticalconsiderations show that, when transmitted to the cell bodyin this manner, the force of retraction in the pico-Newtonrange that has been measured for retracting filopodia [22] canreorient the cell in the time frame of our experiments. Ourmodel predicts complete reorientation in 2–4 min (figures 12and 13). This time scale is compatible with that observedin the experiment, given sufficient time resolution of imageacquisition. As an example, the cell in figure 2 appears toproject one process to the substrate and reorient within thenext 5 min interval.

It is a significant limitation of the model that it does notexplicitly consider the hydrodynamic effects of the relativelylarge surface processes. To estimate its effect on accuracyof the simulations, we compare the coefficient of drag on asphere 6πηR (8πηR3 for rotation) and its approximation fora cylinder 4πηL/ln(L/r) [36]. We roughly estimate that thepresence of about four processes L = 10 µm long and 2r =1 µm thick on the surface of a cell 2R = 20 µm in diametercan double the hydrodynamic resistance to the cell’s rotation aswell as translation. This estimation assumes that the processes’orientation to the direction of motion is especially unfavorable,but neglects the effect of the nearby substrate. On the basisof this estimation, we expect the overall hydrodynamic effectof the large processes to be limited to slowing down the cellmovements at most a few fold, and not affecting the ratio ofthe rotational and translational movement very dramatically.This would not be a cause of any significant discrepancy withthe experiments, because the currently predicted movement, ifanything, may be only somewhat faster than that detected byour imaging, as discussed above.

The form of the force–velocity relationship used in oursimulations was the hyperbolic Hill function (equation (5)in the materials and methods section), which was previouslyapplied to subjects as varied as muscle contraction and actin-driven cell protrusion [20, 21]. We fitted this formula tothe force–velocity datapoints from the table in the study offilopodia retraction in macrophages [22]. The original formulaproposed by the authors of that study, while useful for theirpurpose of elucidating the micro-mechanical mechanism ofretraction, was not suitable for our cell-level simulations dueto the infinite stall force it implied. At the same time, whenfitted by the Hill function, the published force–velocity datayield the maximum retraction rate (0.26 µm s−1) and force(28 pN) in the curve-fitted Hill force–velocity relationshipthat are within the maximum values actually measured inindividual retracting filopodia (0.6 µm s−1 and 45 pN). Sofitted, the data also fix the c parameter in the Hill functionto a value of 5.1, which is very close to the value of about4 for muscle [20]. Both values are vastly different from31, which was obtained in a biochemical reconstitution [21]of the mechanism believed to be responsible for protrusion,rather than retraction, of the cellular processes, and whichhigh value was indicative of the far greater concavity of theforce–velocity relationship in that case. The commonality ofthe c parameter value with that typical of muscle contractionadds confidence in the validity of the approximation we chose.

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At the same time, the phenomenological applicability of theHill function is not by itself an indicator of any specificmolecular mechanism. The cited experiments on macrophagesdemonstrated insensitivity of the filopodia retraction to themyosin inhibitor blebbistatin or to RNAi-mediated disruptionof expression of different myosins [22]. What is of immediateimportance to the present study is that the variety of rathercomplex dynamic scenarios of the reorientation that our modelhas predicted using the Hill force–velocity approximationseem to display a unique capability of explaining ourpreviously made measurements, as discussed below.

New observations reported here reveal substantialvariation between the responses of individual cells. Someapparently have sufficient strength of adhesion to thebiomimetic substrate, while others do not. Sufficient strengthcan be assumed when rotation is coupled to translation in themanner characteristic of rolling (e.g., as in figure 8), whileinsufficient strength can be deduced when the cell appears torotate ‘in place’ (as in figure 9). The numerical model predictsthat with a given, constant friction coefficient the effectivefriction force can be sufficient during part of the cell’s responseand insufficient during another part of it. This predictionstems from the kinematics of the cell response. The evolvingorientation of the retracting process to the cell body changeshow the retraction force is applied to it, and it also changes howthe cell rotation and translation determine the retraction speed,thereby changing the retraction force magnitude as well. Inparticular, this affects the normal force determining the frictionand the tangential force which the friction has to oppose.As a result, the same friction coefficient may be sufficientduring part of the trajectory and insufficient during another.Higher time resolution in microscopic image acquisition willbe needed to validate experimentally the predicted possibilityof alternation of slippage and traction in a given cell. At thesame time, the model can explain the observations of eitherslippage or traction in an individual cell by assuming that theeffective friction coefficient in different cells is different.

Our measurements [17] indicated that of the angularreorientation exhibited by the T cells on the stimulatorysubstrate, 43% could be explained by their simultaneous lateralmovement on the assumption of no slip between the cell bodyand the substrate (i.e. on the assumption of pure rolling). Themechanism that we originally proposed [17] could explainthe end orientation as the one corresponding to the globalminimum of the specially-derived energy function of the cellstructure. This orientation could in principle be attained byrotating the aster of the intracellular microtubule fibers (whichconverge on the centrosome, around which, in turn, the Tcell Golgi apparatus is usually assembled, as described in theintroduction). Rotation of this extended microtubule asterinside the cell, however, is unlikely because of the enormousdrag it would experience in the dense cytoplasm, and there isno direct indication of it (or of the previously reported vectorialtranslocation [8]) in our data. Rotation with the cell, i.e. cellrolling with the microtubule system passively embedded in it,could, however, explain only the 43% of the entire measuredreorientation. The same problem pertains to the othermechanism that we have considered, namely the migration of

the cell–substrate contact area around the cell body to the Golgiapparatus area [18]: it cannot reorient the cell by more than thelateral displacement divided by the cell body radius, whereasour measurements indicate a larger reorientation than that.In contrast, the surface process retraction, according to ournew model, is capable of reorienting the cell body by π whilemoving it as a whole laterally by only one radius. According tothe new model this end result can be achieved through varyingdynamic regimes, but it would nonetheless correspond to afractional apparent contribution of rolling, namely about 30%.Moreover, the new model is capable of predicting both rollingand spinning in place, in varying sequence and proportion,depending on the coefficient of friction between the cell bodyand the substrate. Considering that the friction coefficientcan be highly dependent on the peculiarities of each cell’smicromorphology, the diversity of dynamic regimes permittedby the new theory may be able to explain our new observationsof individual cells that exhibit rolling and spinning in theindividual proportion.

4. Conclusions and outlook

The explanatory power of the new process retractionhypothesis thus appears superior to the other recently proposedmechanisms. There is no indication, however, that orientationof the T cells to their immunological targets, or to thesubstrates that mimic them in the experimental models, shouldbe driven by a singular mechanism. The new data showthat some cells reorient their Golgi apparatus in part ormostly after their surface processes have retracted and thebroad attachment area has developed. The reorientation thencannot be driven by the process retraction, but one shouldexpect the whole-cell structure optimization effects to becomesignificant, which could then drive the reorientation. In thesame or in different cells, a large part or all of the reorientationoccurs concomitantly with the process retraction and beforethe broad attachment and the cell body deformation develop.In these cases, the reorientation, conversely, cannot be drivenby the quasi-elastic whole-cell optimization effects, but theprocess retraction then appears the likely driving force. In acomprehensive framework for T cell polarization, the energyminimization and contact migration that we have analyzedquantitatively before [17, 18], as well as the molecular motor-[8, 9] and microtubule dynamics-based [10, 11] mechanismspostulated by others, and the process retraction reportedhere all need to be carefully considered. Establishing thecommonality and individuality of the sequence, interplayand relative contribution of these mechanisms to the overallpolarization of T cells will require more precise and targetedmeasurements guided by the more comprehensive theoreticalmodels.

Acknowledgments

We thank Dr A Baratt for critically reading the manuscript.This work was supported by grant GM078332 from theNational Institutes of Health.

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