INSTITUTE OF PHYSICS PUBLISHING PHYSICAL BIOLOGY

Phys. Biol. 3 (2006) 209–219 doi:10.1088/1478-3975/3/3/006

Contribution of whole-cell optimizationvia cell body rolling to polarizationof T cellsSergey 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 18 May 2006Accepted for publication 18 September 2006Published 3 October 2006Online at stacks.iop.org/PhysBio/3/209

AbstractDirected secretion of cytotoxins or cytokines by T cells during immune response depends onmigration of the centrosome in the T cell to the interface with the target cell. The mechanismof the centrosome translocation has been elusive. The presented computational analysisdemonstrates that the centrosome should be positioned at the interface if the T cell attemptssimultaneously (a) to minimize its surface area, (b) to maximize the interface area, (c) tomaintain the cell volume and (d) to straighten the microtubules. Live three-dimensionalmicroscopy and measurements show that the optimal position of the centrosome is achieved inlarge part (by about 40%) via rolling of the entire T cell body on the target surface; thismovement appears to entrain the centrosome. The theoretical and experimental results drawattention to the previously unrecognized role of the whole-cell structure and whole-cellmovements in the T cell polarization.

1. Introduction

T-helper and T-killer cells of the immune system performtheir functions via direct interactions with other cells. Uponcontacting a cell that is a target of the immune response, such asa virus-infected cell, a T cell displays a drastic reorganizationof its structure that ensures specificity of the immune response.Most remarkably, an extended contact area between the twocells is developed via deformation of the T cell, and the Tcell centrosome is positioned next to this interface area [1, 2].The centrosome is an intracellular body that was defined byT Boveri as the ‘dynamic center of the cell.’ In T cells,in particular, it determines the location of the vesicles thatstore the effectors of the immune response, for example thecytotoxic compounds that destroy the cells that harbor viruses.Upon binding of the T cell to the target cell, the vesicles fusewith the plasma membrane, releasing the effector compounds[3]. The extended interface area between the T cell and thetarget cell (called immunological synapse) should serve toprevent diffusion of the effectors into the surrounding tissue.At the same time, the positioning of the centrosome next tothe interface with the target cell is thought to ensure thatthe release of the effectors is directed at the target cell and

not at the bystander cells in the tissue [4]. This type ofcentrosome positioning is characteristic not only of killinginteractions of T cells with infected or tumorous cells, but alsoof interactions between T cells with other cell types in theimmune system that can be stimulatory in nature [2], as wellas of the action of natural killer cells [5] that are different fromT cells in other respects. Thus, the centrosome positioning(called polarization) is an intracellular event that is crucial forboth the efficiency and specificity of the immune response.

The mechanism of the centrosome polarization remainsunknown. The aster of the microtubule fibers of the Tcell cytoskeleton, which are attached with one end to thecentrosome, seems to be involved, because if the microtubulesare disassembled in an experiment, the centrosome doesnot polarize [3]. Hypotheses were put forward that relatedthe polarization of the cytoskeleton to either microtubuleshortening or stabilization at the site of contact with thetarget [6, 7], but these hypotheses did not prescribe the actualmechanism for the centrosome reorientation. An observationof microtubule deformation in T cells next to the interfacewith the target cell suggested that molecular motors mightbe involved that could pull on the microtubules, dragging the

1478-3975/06/030209+11$30.00 © 2006 IOP Publishing Ltd Printed in the UK 209

S N Arkhipov and I V Maly

centrosome to the contact site [8]. Direct evidence, however,is lacking.

Given the present difficulties of establishing themechanism of the centrosome polarization in T cellsinteracting with their immunological targets, it is intriguing toobserve that a very similar polarization behavior is exhibitedby cells of various types in a different experimental situation.These experiments have not so far been considered of relevanceto the study of the immunological cell–cell interactions: whenallowed to sediment from suspension onto plain glass, T cellsas well as macrophages, neutrophils, fibroblasts and epithelialcells spread on this inert, artificial substrate and positiontheir centrosomes next to the cell-glass contact area [9–11].Irrespective of its molecular-level cause, the attachment to thesubstrate in these cases appears to be essentially the samecellular activity as the attachment of a T cell to the targetcell. In fact, the T cell polarization has been successfullystudied in experimental models that replace the target cellwith a biomimetic surface. The artificial substrate that mimicsthe target cell surface can be, for example, glass coated withantibodies binding to the T cell receptor [12, 13]. This receptoron the T cell surface normally mediates the interaction withthe target cell. It is striking that the centrosome polarizationto the attachment site is observed in each situation, be it thecell–cell interaction, the interaction mediated by the highlyspecific molecular recognition, the interaction with the plainglass, or even the interaction with the plain glass of a cell verydifferent from a T cell. One may hypothesize that a mechanismshould exist that is at least partly responsible for the emergenceof the functionally important centrosome orientation in the Tcell–target cell interaction, and that should not be identifiedwith any of the pathways or processes specific to T cells. Amechanism of centrosome positioning might exist that couldbe as intrinsic to the cell structure as to be exhibited by ageneric cell that develops a contact with a generic substrateto which it has affinity, the molecular cause of such affinitynotwithstanding.

In this paper, we address the following questions. Whatposition of the centrosome can be expected from the mostgeneric considerations in a cell that is attached to somesurface? In what parsimonious kinematical manner can thisposition be achieved? How does the theoretical expectationas to the kinematics of polarization compare to experimentalmeasurements made on the polarizing T cells?

A fuller description of the geometry of the biomimeticsystem to study T cell polarization should be illuminativeof both the theory and experiments presented in this paper.The T cells in this setup are allowed to settle from thegrowth medium on the optical-glass bottom of the observationchamber, immediately above the objective lens of an invertedmicroscope that is used for observation. These cells arenormally non-adherent and grow in suspension. The chamberbottom, however, is coated with antibodies that bind andstimulate the T cell receptor on the cell surface. This triggersattachment and the centrosome polarization as if the chamberbottom were the target cell. The centrosomes are orientedrandomly in cells that have come in contact with the bottom.In 15 min, however, most attached cells have already spreadslightly and positioned their centrosomes next to the substrate,

commonly in the middle of the attachment site [14, 15].This is known from fixed specimens. Here we employ thisexperimental system for dynamic observations to probe someof the theoretical expectations.

The basic theoretical approach will be structuraloptimality. To introduce this method, a familiar examplewill be useful: the body of a given volume that has theminimum surface area is a ball. A free drop of liquid withsurface tension spontaneously becomes round. Of course,application of this rule to the cell falls short of predictingthe specific shapes of various cell types. Also, the natureof the surface tension and volume maintenance in the caseof the cell is different from that in a liquid drop, being, rather,active contraction of the actomyosin cell cortex and osmoticeffects of ion pumps [16]. Their complex biophysics andlimited applicability notwithstanding, the volume maintenanceand surface minimization well explain the most primitive cellshape, the roughly round one which is acquired by freelysuspended animal cells in culture, so long as the cell body isconcerned but not the thin protrusions. While the geometricalshape which is optimal with respect to these two principles iswell known, the rigorous proof of its being optimal is ratherinvolved. With additional structural elements and optimalityprinciples needed to approximate the polarizing T cell, neitherintuition nor analytical mathematics can be relied on. Theoptimal shape can nonetheless be found numerically.

The centrosome positioning was illuminated from thenumerical optimization standpoint in a study that dealtwith microtubule asters assembled in vitro around artificialcentrosomes in microfabricated cell-size chambers [17].The centrosome position was successfully calculated as theone corresponding to the minimum bending energy of themicrotubules constrained by the chamber walls. Remarkably,when the microtubules were sufficiently long (considerablylonger than the radius of the chamber), their simultaneousunbending positioned the centrosome off center in the roundchamber. This result might not be intuitive, but it demonstratesthat at least the fact that the position of the T cell centrosomes iseccentric can be understood from first principles. Having dealtwith the flat, round chambers, however, this theory cannot beapplied to the problem of why the centrosome in the activatedT cells becomes positioned not only off center, but specificallyin the direction of the target surface.

The liquid-drop model predicts the primitive shape ofa flexible, empty ‘cell,’ whereas the microtubule-unbendingmodel predicts the centrosome position within a given,rigid boundary. Integrating these approaches, we begin bypredicting the centrosome position inside a flexible cell in thegeometrical framework of the experimental model of T cellpolarization. The kinematical implication of the theoreticalalgorithm for the centrosome positioning is then comparedwith live measurements of the T cell polarization.

2. Methods

2.1. Computational details

Microtubules are approximated as chains of straight, freelyjointed segments of constant length and number. (The

210

Geometry of T cell polarization

assumption of the constant length is experimentally validatedand discussed below.) The surface of the free-floating cellis defined as the minimal convex hull enclosing all themicrotubule segments. The structure of the free-floating cellis determined by the set X of the direction angles of all themicrotubule segments—two angles, azimuth and declination,for each segment. The optimal structure is the one thatminimizes the following energy function: Ff(X) = �(V(X) −Vg) − �Vglog(V(X)/Vg) + γ S(X) + κB(X)/2. In thisformula, Vg is the goal cell volume, and V(X), S(X) andB(X) are respectively the actual cell volume, cell surfacearea and microtubule bending as functions of X. The constantcoefficients are � = 25 mm Hg = 3.4 fJ µm–3, the osmoticpressure characteristic of mammalian tissues [18], γ =0.035 dyn cm–1 = 35 aJ µm–2, the cortical (surface) tensionin leukocytes [19], and κ = 88 aJ µm, the flexural rigidityof microtubules. We obtain the latter value by multiplying2.2 × 10−23 N m2 = 22 aJ µm [20], the mid-range valuefor pure microtubules [21], by 4, to reflect the fact thateven though the absolute values for pure microtubules arecontroversial, the four-fold increase was seen consistentlywith the addition to the pure microtubules of microtubule-associated proteins that co-polymerize with them in vivo [22].The terms involving the volume and the osmotic pressurecan be derived from the consideration of work done againstthe osmotic pressure difference across the cell boundary byexpanding or contracting the cell volume, which would beaccompanied by diluting or concentrating the proteinaceousosmogenic solutes within the cell [16]. The bending term B(X)represents the integral, over the length of all the microtubules,of the squared curvature, to which the bending energy isproportional. It is computed by summation, over all thesegment joints, of the inverse square radius of curvature,which the joined segments define. Precisely, if the directionof two joined segments of length s differs by the angle α,then it is assumed that they approximate a radius of curvatures/2∗tan((π − α)/2) over the arc length s. The same applies tothe differences of the directions of the first segments of eachmicrotubule and the equally spaced directions along whichunstrained microtubules should emanate from the centrosome.Optimization was performed using the sequential quadraticprogramming algorithm of the MATLAB software packageby Mathworks, Inc. An aster of straight microtubules wasused as the starting configuration, randomized by adding apseudorandom number between 0 and 0.1 to each segmentdirection angle.

For the attached cell, the microtubule aster is projectedonto the horizontal plane that passes through the lowermostpoint in the aster. The area of the minimal convex polygonenclosing the projection is the area of contact with the target,Sc. The cell surface is calculated as the minimal convex hullthat encloses the projection as well as the microtubule aster.The aster configuration is given by the set of the segment jointangles, which was found optimal in the free-floating cell. Theaster is rotated by two rigid-body rotation angles, θ and φ Theoptimal orientation of the aster is found as the θ , φ pair thatminimizes the energy function Fa(θ , φ) = �(V(θ , φ) − Vg) −�Vglog(V(θ , φ)/Vg) + γ S(θ , φ) − γ adhSc(θ , φ), where γ adh =

2.5 × 10−5 J m–2 = 25 aJ µm–2 is the mid-range estimate of thetypical total (non-specific plus receptor-mediated) adhesionenergy density [23].

2.2. Experimental details

Jurkat cells (a gift of Dr L Kane, University of Pittsburgh) weregrown and prepared for observation essentially as described[24]. The temperature, 37 ◦C, was maintained using a Nevtekair curtain incubator. The cells were incubated with 500 nMOregon Green 488 TubulinTracker (Molecular Probes) in theculture medium for 25 min before being injected into thechambers. For image registration, a Nikon TE 200 invertedmicroscope was equipped with a Hamamatsu ORCA II ERGcooled interline camera, and the Nikon Plan Apo 60x objectivewith numerical aperture 1.4 was driven by a piezo-positionerPIFOC 721 (Physik Instrumente). The camera, the objectivedriver and an illumination shutter (Uniblitz) were coordinatedby the IPLab software. 76 fluorescence images of the opticalsections were taken over 7.5 s beginning at the end of every 5min until 25 min after injection of the cells into the chambers.The formal resolution (voxel size) was 0.22, 0.22, and 0.4 µmin X, Y and Z dimensions, Z being along the optical axis andorthogonal to the glass forming the bottom of the observationchamber. The acquisition of each Z-series of the fluorescenceimages was immediately followed by capturing the image inphase contrast.

The three-dimensional reconstruction from the near-simultaneously taken optical sections was calculated using aray-casting algorithm for fluorescence images and measuredin the same software. The geometrical center (centroid) ofthe cell was calculated as the midpoint between the X, Y andZ bounds of the cell. The position of the centrosome wasdetermined as the point of convergence of the fluorescentmicrotubules, usually corresponding to the point of themaximum brightness due to this convergence. For each cell,only the time points after the cell begins displaying flat, phase-contrast lamellipodial protrusions were used to make certainthat the cell is attached to the substrate and its movementsare active. After the attachment was assured from the phase-contrast images, the vertical coordinates measured in the three-dimensional reconstructions of the fluorescence images wereadjusted to compensate for focus drift between the time pointsby subtracting the current vertical coordinate of the bottomcell boundary from the vertical coordinates of the centrosomeand the top of the cell.

For immunofluorescence, 4% paraformaldehyde (Sigma)was used to fix the cells at room temperature for 30 min. Thecells were then permeabilized with 0.5% Triton (Sigma) for5 min and stained with goat anti-tubulin antibody (Sigma) andTRITC anti-goat antibody (Invitrogen, Carlsbad, CA).

2.3. Prediction of centrosome displacements from cell rolling

For each measured displacement of the cell betweenthe successive observations, the predicted centrosomedisplacement due to cell rolling was calculated as follows. Theangle by which the cell rotates during the time interval is found

211

S N Arkhipov and I V Maly

by dividing the displacement of the centroid by the half-heightof the cell as the idealized rolling radius. The centrosomemoves within the plane which is (a) orthogonal to the planeon which the cell is rolling and (b) parallel to the centroiddisplacement. The initial center of rotation of the centrosomeis the point on its plane of rotation that also lies on the initialaxis of the cell rotation. This axis is horizontal, orthogonalto the centroid displacement, and passes through the initialcentroid position. The distance from the initial position ofthe centrosome to its initial rotation center is the radius of thecentrosome rotation. The initial phase angle of the centrosomeis determined as the angle between (1) the direction fromthe initial centrosome rotation center to the initial centrosomeposition and (2) the direction of the centroid displacement.The final phase is the initial phase plus the angle by whichthe cell rotates. The predicted new position of the centrosomeis obtained by displacing the initial center of the centrosomerotation in parallel with the cell centroid and by adding tothis displacement (1) a displacement in the same direction thatis equal to the centrosome rotation radius multiplied by thecosine of the final phase angle and (2) a vertical displacementthat is proportional to the same radius multiplied by the sine ofthe final phase. The calculation must be done individually foreach cell and each time interval. The calculated displacementsof the centrosome in a given cell can then be summed up for allthe time intervals to get the prediction where the centrosomewould be at the end of the time sequence if it were entrainedin the rolling cell.

3. Results and discussion

3.1. Theoretical optimality of a free T cell

We begin by computing the optimal structure of the free-floating cell with the microtubule aster inside. Thiscorresponds to the T cell structure in the experimental modelbefore the cell comes into contact with the target-like surfaceand undergoes the polarization of the centrosome to thatsurface. Refer to section 2 for the computational details.Microtubules in the theoretical model begin at the commonpoint corresponding to the centrosome. They have constantlength, but can bend arbitrarily. (The assumption of theconstant length is experimentally validated and discussedbelow.) Numerically, the microtubules are represented bychains of straight segments. The cell surface is shrink-wrappedaround the microtubule aster. Mathematically, it is definedas the minimum convex hull enclosing the microtubules.(This surface does not represent the extremely wrinkledplasma membrane, but rather the contractile submembraneactomyosin cortex that actually bounds the leukocyte cellcontents [19].) The numerical optimization begins from arandom configuration (i.e., all the joint angles between themicrotubule segments are randomized). The optimizationalgorithm then attempts to simultaneously minimize the cellsurface and straighten the microtubules while keeping the cellvolume as close as possible to a set value.

One can observe that these are conflicting goals: straightermicrotubules can only be enclosed in a cell surface with a

larger area, and shrinking the surface would also decreasethe volume. We need to determine what configuration of themicrotubule aster will be optimal with respect to all theseobjectives simultaneously. To approach this problem, wecan put in correspondence to each possible configuration anenergy value that would reflect the bending energy of themicrotubules, the energy of the cell surface stretching, and theenergy of deviation of the cell volume from the one dictated bythe osmotic equilibrium with the physiological medium. Thenwe can postulate that the optimal configuration minimizes thisenergy function.

Two distinct cell structures that are optimal in this senseare found depending on the ratio of the microtubule lengthto the goal cell volume. If the microtubules are shortand the cell volume large, the cell acquires a configurationone could expect: an almost round cell with an aster ofmicrotubules, which extend from the centrosome near the cellcenter and bend slightly under the cell surface. Tighteningthe volume constraint on the microtubules, however, causesdislocation of the centrosome in the optimal structure fromthe center. This spontaneous symmetry breaking is alsonot surprising, being very similar to the one seen in theexperiments in vitro and explained by the optimality theoryfor the microtubules constrained in the flat, rigid chambers[17]. Remarkably, we were not able to obtain the asymmetricstructure with the mid-range value of the microtubule flexuralrigidity measured for pure microtubules, 22 aJ µm [20, 21], butonly when we multiplied it by 4; this four-fold increase was themeasured effect of microtubule-associated proteins (MAPs)co-polymerizing with the microtubules in vivo [22]. Evenbeginning with the same initial configuration, separate runsof the optimization algorithm terminate on slightly differentoptimal configurations. At the same time, even starting fromcompletely different initial configurations leads to essentiallythe same final structure. This structure becomes clearlyasymmetric when the microtubules are at least 1.5 times aslong as the radius of a ball that would have the goal cellvolume (figure 1(A)).

This structure (figure 1(A)) shares the following criticalfeatures with the structure of real T cells [8]: the roundedand somewhat onion-shaped cell outline, the microtubulesconverging on the eccentric centrosome, the microtubulesrunning around the cell body as well as through the cellinterior. The most characteristic of the capacity of themodel to capture the nontrivial T cell morphology are thosemicrotubules that are predicted to extend from the centrosomefirst outwards, and then bend where they meet the cellboundary and follow the cell outline toward the ‘pole’ ofthe cell that is opposite the centrosome (figure 1(A)). Thisdistinctive morphological feature has been given attention inthe high-resolution experimental studies of T cells [8].

3.2. Algorithm for optimization upon attachment to the target

The asymmetric microtubule aster configuration predictedfrom the first principles for a suspended cell (figure 1(A))is almost the same as the one seen in polarized T cells incontact with the target, in that the microtubules emanatefrom the centrosome on the side of the cell. The aster of

212

Geometry of T cell polarization

(A) (B)

(C) (D)

Figure 1. Theoretical optimality analysis of the T cell structure and orientation. (A) The optimal structure of a free-floating cell.Microtubules are thick lines converging on the centrosome, the cell surface is triangulated. (B) The same microtubule aster in a cell attachedto the target substrate plane below. (C) The landscape of the energy function of the attached cell. On the axes, the tilt and azimuth ofrotation of the microtubule aster, on the color bar, the values of the energy function. The global minimum is marked with a red cross. (D)The optimal structure of the attached cell, corresponding to the minimum in part C. In this simulation, there are 24 microtubules, each with 62 µm long segments, and the target cell volume is that of a ball with a radius of 8 µm.

microtubules is similarly asymmetric before as well as after thepolarization response to the target binding [3]; the differenceis in the orientation of the asymmetric microtubule system.The essential difference of the microtubule cytoskeleton in asuspended cell from the one seen in T cells polarized to thetarget is that the cytoskeleton orientation in the free cell shouldbe arbitrary. There is no external reference for the free cell inour theoretical as well as experimental model that would makedifferent orientations of the aster distinguishable to the cellor preferred by the optimization algorithm. The orientationselected by the algorithm is essentially random, which canalso be expected of the aster orientation in a cell in suspension.(Strictly speaking, the computed orientation is not random butdepends on the pseudorandom starting configuration, becauseour optimization algorithm is deterministic. However, thedeterministic search passes through a quasi-chaotic regimebefore a near-optimal configuration is found. This effectivelyerases the dependence on the starting configuration.) Thepolarized T cell in a conjugate with the target is most obviouslydifferent from one in suspension in that it is attached tothe target on one side. This could potentially make thedifferent orientations of the asymmetric structure alreadyfound in the free cell non-equivalent, some being more optimalstructurally than others. Next we will explore the possibilityof deriving the polarized structure of the T cell from thefreely floating configuration parsimoniously by optimizing

only the orientation of the microtubule aster while leavingits configuration proper as constant.

To model the T cell attachment to the target surface, whichwould introduce the extracellular asymmetry with which theintracellular one could potentially be aligned, we assumethat the microtubule aster will sit on the target plane and becanopied by the T cell surface. Geometrically, the aster willtouch the plane, and the orthogonal projection of the aster ontothe plane will be enclosed together with the aster in the minimalconvex hull designated as the surface of the cell. The area onthe target plane covered by the microtubule aster will be thearea of contact of the T cell with the target. A structure that isobtained in this manner from the free-floating structure in thesame arbitrary orientation is shown in figure 1(B). The overallshape of the attached model cell can be described very crudelyas a hemisphere on top of a short upright cylinder sitting onthe substrate plane. This geometry also describes reasonablywell the general shape of an attached T cell as seen in three-dimensional experimental images [14, 25]. Evaluating thefidelity of the model, it is important to note that T cells donot flatten upon attachment to the substrate nearly as muchas fibroblasts, epithelioid and other generic cultured cell typesdo. The T cells do send out very flat protrusions over thesubstrate, but most of the cell body retains the dome-on-a-turret shape as seen in confocal reconstructions [14, 25].Most significantly for this study, microtubules in general do

213

S N Arkhipov and I V Maly

not enter the extremely flat lamellipodial protrusions [26], thevolume of the protrusions is insignificant, and the surface areaminimization arises not from the physical surface tension perse but from the active cortex contraction [16] and is thereforerestricted to the compact cell body as modeled, not involvingthe protrusions. In the light of these considerations, thegeometrical construction of the attached cell model appearsadequate.

It can be seen in the theoretical structure (figure 1(B))that the volume and the total surface area of the attached celldepend on the orientation of the microtubule aster, becausethe latter is asymmetric. This dependence alone is sufficient tomake the different orientations of the microtubule cytoskeletoninherited from the free-floating cell non-equivalent. Someorientations would conserve the volume and keep the surfaceminimal after the attachment better than others. These moreoptimal configurations may therefore more readily be adoptedby a T cell attaching to the target. It can also be observed fromfigure 1(B) that the area of contact with the target will dependon the microtubule aster orientation as well. We will requirethat the model T cell maximize the area of contact with thetarget, as this should be favored by the negative surface energydensity associated with receptor-mediated adhesion [23].

Remarkably, the whole-cell optimality associated withthe microtubule aster orientation is a function of only twovariables, the tilt and azimuth angles needed to specify therigid-body rotation of the aster in three dimensions. Thismakes it possible to visualize the relative optimality of allorientations in a single graph. One graphical conventionthat can be exploited is the energy landscape from studiesof molecular conformations, where the energy minimumcorresponding to the most optimal configuration can be foundon plots that are similar to topographic maps. Utilizingthis convention, we can find the optimal orientation of themicrotubule aster in the polarized T cell as the lowest pointon a map obtained by plotting the structural penalty (energyfunction) as altitude with the two rotation angles as thelongitude and latitude. The coordinates of the lowest point onsuch a landscape will be the two angles of rotation necessaryto bring the aster from the unattached cell (whose orientationwas arbitrary) into the optimal (lowest-energy) orientation inthe attached cell. In this operational way, the coordinates ofthe global minimum will specify the optimal orientation.

The energy landscape calculated for the microtubule asterfrom figures 1(A) and (B) is shown in 1(C). It is essentially thesame, although not identical, for the slightly different astersthat, as discussed above, the optimization algorithm can findfor the unattached cell instead of the sample in figure 1.Alongside some irregular features, the structural penaltylandscape displays a creeks-into-the-lake pattern leading to thedeepest point of the global minimum. Taking the aster fromfigure 1(A), rotating it by the angles given by the coordinatesof the global minimum, and canopying it by the cell surfaceas defined above produces the optimal structure of the Tcell attached to the target. It is shown in figure 1(D). Thecentrosome in the optimal structure can be seen over the centerof the interface with the target, quod erat demonstrandum.

3.3. Hypothesis of coupling of optimization to whole-cellmovement

One aspect of the optimization procedure leading to thefunctional position of the centrosome should be easilyobservable because it implies whole-cell movement. Thehypothetical rotation of the extended microtubule aster throughthe dense interior of the cell is unlikely because of theresisting drag force this movement would induce. The rotationprescribed by the model should therefore involve the entirecell. The T cell, however, adheres firmly to the targetsurface. In the experimental model where the surface isthe antibody-coated glass, the adhesion is especially likely topreclude any slippage. The microtubule aster rotation underthese conditions should be manifested as whole-cell rolling.The striking implication of this hypothesis is therefore that thecell should move laterally as the centrosome migrates to thesubstrate.

3.4. Experimental measurement and estimation

To compare the theoretical expectation with the experiment,we followed the course of polarization of a living T cell bytracking the cell outline as well as the centrosome position inthree dimensions and in time. The position of the centrosomewas determined by fluorescently labeling the microtubules thatconverge on it, and recording at successive time points three-dimensional images of the cells.

To validate our live-cell setup against the previous workdone with fixed cells, we fixed and immunostained the cells12 min into one of our live experiments, set up as usualincluding the incubation with the fluorescent probe. 96%,n = 121, cells were polarized in 12 min, compared to 85% in15 min as reported before [15], validating our setup.

In the live experiments, the cells exhibited a firmattachment to the target surface inside the time period usedfor measurements, excluding the possibility that convection,turbulence, etc, add a passive component to the cell movementsthat are registered. The active attachment was distinguishedfrom passive settlement by the presence of lamellipodialprotrusions of the cells on the substrate surface. Theprotrusions were monitored in phase-contrast images acquirednear-simultaneously with each three-dimensional fluorescenceimage in the time sequence. 25 cells were tracked indetail.

Microscopy shows that the cells settle on the biomimeticsubstrate that mimics biochemically the surface of the targetcell and exhibit an active polarization response, as describedbefore [12–15, 25]. The cells attach to the substrate anddemonstrate a downward migration of the centrosome towardsit. After attachment, a cell was observed for 15 ± 1 min onaverage (± SE, n = 25 everywhere). During this time, itdid not display any significant alteration in the outline of itsbody, which is protruding over the substrate and is distinctfrom the thin skirt of lamellipodial protrusions, as previouslydescribed [14, 15, 25]. The diameter of the round cell bodyas viewed from the top comprised initially 14.5 ± 0.5 µm,and the cell height, slightly less than that, 10.6 ± 0.5 µm.At the end of the observation period, the diameter was

214

Geometry of T cell polarization

Figure 2. Real-time observation of the coupled centrosomepolarization and cell movement. All the images show the same Tcell attached to the substrate coated with antibodies to the T cellreceptor. The images in the right column were taken 5 min after theimages in the left column. The images in the top row arephase-contrast images showing a top view of the cell. A skirt of thinlamellipodia is seen around thick, round cell body. The initialposition of the cell centroid is marked with a cross, and the arrow onthe right shows the lateral displacement of the cell centroid fiveminutes later. The images in the bottom row are side views ofthree-dimensional images of fluorescently labeled microtubules.Microtubules diverge like rays from the centrosome, making thelatter the brightest spot. The initial position of the centrosome ismarked with a cross, and the arrow on the right shows the verticaldisplacement of the centrosome towards the transparent underlyingsubstrate 5 min later.

14.4 ± 0.6 µm, and the height, 10.3 ± 0.5 µm. The centrosomemigrated down by 2.4 ± 0.5 µm. Being about 1/4 of thecell height, this displacement is nearly 1/2 of what would beneeded on average for translocating the centrosome from arandom position to the very substrate. The centrosome in Tcells, however, stays away from the very margin of the cell,as discussed above, limiting the possible range of its verticalpositions to less than the vertical dimension of the cell. Thissuggests that our observation window captures most of thecentrosome translocation that takes place.

Most remarkably, the downward migration of thecentrosome is indeed accompanied by lateral migration of thecell on the target substrate. The lateral displacement of the cellcentroid was 2.9 ± 0.4 µm. This is very close in magnitudeto the vertical displacement of the centrosome. At the sametime, being only about 1/5 of the cell diameter, this lateralmigration hardly qualifies as cell locomotion and is easy tooverlook. The quantitative measurements, however, confirmthat the centrosome polarization is accompanied by the lateralmovement of the entire cell. Sample top-view and side-viewsnapshots in figure 2 illustrate this.

Quantitative evaluation of the involvement of the whole-cell movements in the centrosome polarization requires

comparing the actual downward migration of the centrosomewith one that could be expected of a point embedded intothe cell moving as a whole. A geometric formalism will beneeded for deriving the passive component of the centrosomemovements from the movements of the cell. This formalism,effectively a kinematic hypothesis about the T cell polarization,will be used to quantify the theoretically expected contributionof the actual whole-cell movements to the actual centrosomemigration. In the complexity of the simultaneous dynamicsof the cell and its centrosome, we will attempt to discern thecomponent of simple, idealized whole-cell rolling that couldentrain the centrosome.

The geometrical calculation of the trajectory expected ofthe embedded centrosome (see the methods section) is a three-dimensional generalization of the curtate cycloid curve thatis described by a point anywhere along a radius on a rollingwheel. The vertical displacement of such an eccentric pointcan be found given the horizontal displacement of the wheel.The no-slip condition between the wheel and the substratestipulates that the horizontal translocation of the wheel equalsthe radius of the wheel multiplied by its rotation angle. Thus,knowing the horizontal displacement and the rolling radius, therotation angle can be found. Any point on the wheel undergoesrotation around the hub by the same angle. This makes itpossible to find the vertical displacement of any such point. Inthe special case of a point that is initially directly behind theaxis, the vertical displacement will be its distance from the hubtimes the sine of the rotation angle. This calculation strategyis generalizable to the rolling of a three-dimensional ball andthe movement of any off-center point in it (details provided insection 2).

The attached T cell does not have the shape of a ball.The cell therefore cannot roll without internal deformationsthat would complicate the movement of the embedded point,such as the centrosome in our hypothesis. However, if thecomplications are secondary to the main movement which isrolling, one can expect that the movement of the centrosomecan be calculated to a useful precision from the lateralmovement of the cell by following the above strategy. Giventhe deviation of the cell from a ball, it becomes an assumptionthat the rolling radius as used in the calculation of the rotationangle should be the half-height of the cell, and that therotation angle of the cell is also the rotation angle of thecentrosome around the axis passing through the cell centroid.We will calculate the centrosome movement that would beexpected under this idealized cell rolling from the actual lateralmovement of the cell, and compare it quantitatively with theactual centrosome movement.

Applying the formalism individually and then averagingover the cells, we obtain that the centrosome would be 1.0 ±0.3 µm lower than initially due to the cell body rolling. Thecontribution of cell body rolling to the centrosome polarizationcan be calculated as the ratio of the vertical displacementof the centrosome that is expected based on rolling to itsactual vertical displacement (2.4 µm). We obtain that cellbody rolling can be responsible for 43% of the centrosomepolarization in our experiments.

215

S N Arkhipov and I V Maly

3.5. Discussion

In this paper, we have formulated a quantitative theory thatexplains the centrosome polarization in activated T cells. Thesimple geometric principles on the basis of this theory wouldbe applicable to any cell that possesses a microtubule asterand adheres to a surface after being initially suspended freely.The theory thus explains the T cell polarization induced by themolecular recognition of the specific target cell [1] as well asthe orientation of the centrosome to the non-specific substratedisplayed by T cells, macrophages, neutrophils, fibroblastsand epithelial cells that sediment from suspension in culture[9–11]. This broad applicability is seen here as an advantageof the theory and in fact a requirement for any theory of theprocess, but it may raise the question of whether our theoryaccounts for the specificity of the T cell response. The answeris that our theory applies whenever its geometrical conditionsare met by the cell, but the T cells normally meet the conditionof attachment when the target surface induces the specificmolecular recognition, such as between the T cell receptorand the antigen displayed on the target cell. More generally,however, it appears that the polarization response of the Tcells sometimes has to be induced by the specific recognition,as probably is the case in vivo, and sometimes, at least inthe artificial circumstances of an experiment, it need not.The development of the experimental model employed in thisstudy is an illustration. Plain glass induces attachment of Tcells and polarization of the centrosome to the glass substrate[9]. Coating glass with poly-lysine abrogates attachmentof T cells [14], which cannot be rescued by adsorbingvarious kinds of antibodies over the poly-lysine layer, forexample, of antibodies against the LFA-1 integrin, a T cellsurface adhesion molecule [14]. Specific clones of antibodiesto the T cell receptor, however, may be used for coatingover the poly-lysine, and the surface so prepared induces boththe T cell attachment and the centrosome polarization again[14]. Triggering of specific intracellular signaling pathwaysdownstream from the T cell receptors is seen as evidence thatbinding of these specific antibodies mimics, to a degree, thenormal recognition of antigen by the receptors [14]. The realinteraction between a given T cell and a potential target cellis induced by recognition of a specific antigen on the surfaceof the potential target cell; if the antigen is not the one whichthis specific T cell recognizes, the polarization response is notinduced [3]. The polarization in a cell–cell interaction can,however, be induced experimentally by non-specific lectin-mediated binding as well [3]. Comparing the effects of plainglass, poly-lysine, antibodies against the LFA-1 integrins,antibodies against the T cell receptors, and of the specificand non-specific antigens or non-specific lectins on the targetcell, one can conclude that there are biochemically specificsurfaces that induce the attachment and polarization and thosethat do not, as well as that there are biochemically non-specificsurfaces that do and do not. That the polarization is inducedby the highly specific molecular recognition in the naturalsetting does not imply that an explanation of polarization mustaccount for the specificity of response.

In our theory, polarization of the centrosome is linkedgeometrically to spreading of the T cell on the target surface.

The cell spreading and centrosome polarization co-occur inthe variety of the discussed experiments with different celltypes and different substrata. While the spreading of Tcells is known to be driven by actin cytoskeleton dynamics[12, 14], the molecular mechanism of the centrosomepolarization has not been formulated with certainty. Accordingto the present work, a large part of centrosome migrationmay not have a specific molecular mechanism. To beprecise, it may not have a molecular mechanism that wouldbe separate from those that are behind the forces that shapethe geometry of our model, which exhibits the centrosomepolarization. Among them there are the basic elasticityof microtubules, osmotic volume maintenance mechanismsinvolving ion pumps, and actomyosin cortex contractionresponsible for the effective cell surface tension [16]. Thegeometry-shaping mechanism that is the most specific of allto the cell–cell interaction at hand is the actin polymerizationdynamics responsible for spreading of the T cell on the target.Thus, our model, although it does not deal explicitly withmolecular mechanisms, implies that the actin-driven spreadingis part of the molecular mechanism of the polarization of themicrotubule cytoskeleton. The existing evidence for this roleof the actin cytoskeleton comes from the study of signaling inT cells. The signaling cascades that trigger the centrosomepolarization and the T cell spreading, although they havenot been described in their entirety, appear to overlap to asignificant degree. The T cell receptor engagement can induceboth [12, 14, 27, 28], acting through the same, relativelyupstream adaptors and kinases [14, 15, 29]. The secondmessenger Ca++ is also involved in triggering both events[14, 30]. The most suggestive is that the relatively downstreamcytoskeleton regulator Cdc42, whose role in regulating theactin-driven protrusion is comparatively well understood[31], appears to affect both the spreading and centrosomepolarization [6]. Lastly, as T cell precursors develop andexpress more T cell receptors on their surface, their spreadingand centrosome polarization co-vary: improved centrosomepolarization comes with improved spreading [25]. Takentogether with the mentioned co-occurrence of spreading andcentrosome polarization in various cell types on varioussubstrata, this molecular evidence substantiates the implicationof our model that the centrosome polarization is a consequence,among other factors, of the T cell spreading on the targetsurface.

Attachment to the target without the centrosomepolarization was seen in 50% of interactions with a non-specific target cell [3], indicating that the attachment isnot sufficient for polarization. Sufficiency is not arguedhere either, only contribution. However, it is importantfor the present discussion that the natural killer cells inthe cited experiments were attached to the target cells andsimultaneously to the glass substrate. The spreading on theglass clearly dominated. This would explain the absence oforientation of the centrosome to the target cell, precisely ifattachment does direct the orientation, as argued here. Anotherelegant experiment was performed to refute the hypothesisthat the actin-driven spreading is involved in the centrosomepolarization [32]. Engaging different receptors on different

216

Geometry of T cell polarization

sides of the same T cell causes the centrosome to go to the siteof T cell receptor engagement while strong accumulation ofactin is seen at the site of integrin engagement. Is the T cellspreading on the side next to which the centrosome is observedin fixed specimens still the dominant one, quantitatively? Thisis possible especially because the T cell receptors are engagedthere by antibodies coating a rigid bead, while the other targetsurface belongs to a cell expressing the integrin ligands [32].In the absence of the clear predominance of one attachmentsite over the other, the situation in this experiment may besimilar to engagement of several target cells with a singleT cell. Live observations showed that the centrosome thenvisits the separate contact areas stochastically [8]. In theframework of our optimality landscape for the orientation ofthe microtubule aster, the situation in these experiments withmultiple and different targets would lead to appearance ofmultiple attraction basins of different depth and size, betweenwhich the centrosome orientation can jump analogously to amolecule between its alternative conformations. Attemptingto predict which orientation would be preferred would requireprecise, model-guided measurements.

According to our calculations of the actual centrosomedisplacement to the target substrate, only about 40% canbe accounted for by the cell body rolling. The rest ofthe centrosome polarization is not explained by the currentmodel. It should be noticed that our calculations contain asignificant error due to the fact that they assume rolling at astraight line inside the time interval between the consecutiveobservations when the measurements of the cell position aretaken. The so-determined direction of movement, however,changes between the consecutive intervals, indicating thatinside the intervals the cell in fact deviates from the straightline assumed in the calculations. It is possible thereforethat with improved time resolution, the movement of thecentrosome will be more fully explained by cell rolling.The better approximation could also be aided by accounting,in the calculations of the expected centrosome movementfrom the cell movement, for the deformations that shouldaccompany rolling of the cell that is not entirely round. Itis our expectation, however, that the residual unexplainedmovement of the centrosome is not on the whole due tothe imprecision of the measurements and of the theory ofwhole-cell movements. Rather, proper intracellular motilityof the centrosome may exist and contribute significantly to thecentrosome polarization. For example, as proposed earlier,the movement of the microtubule aster could be driven bythe pulling action of molecular motors anchored at the cortex[8]. Discerning the quantitative contribution of such additionalmechanisms in the future would be difficult without firstdetermining the large contribution of whole-cell movementsdriven by whole-cell optimization as described here.

The quantitative test of our theory relied on tracking thecentrosome in three dimensions in living, responding T cells.The observations of the centrosome movements in living Tcells that had been accomplished prior to this work had beenessentially two-dimensional and qualitative. When multipleoptical sections of these cells were acquired, only one per timeframe was nonetheless selected to monitor the centrosome

movement, reducing the latter to two dimensions [15]. In thiscase as well as in the others [8, 33], interaction of T cellswith antigen-presenting cells was followed. The centrosomemovements were accompanied by large displacements of theentire T cell, possibly on the inert substrate as well as onthe target cell, and the target cell itself was not entirelyimmobile. Coupling between the movements of the T celland its centrosome was not analyzed. Here we employed anexperimental model in which the only cellular movements arethose of the T cell on the target surface, and followed themas well as the centrosome movements in three dimensions.This allowed us to discern the previously disregarded lateralmovement of the T cell on the target surface. Earlier,qualitative examination of videomicroscopy data obtained onnatural killer-target cell conjugates did not appear to supportthe similar hypothesis of lateral crawling, which thereforewas rejected [5]. Our measurements not only revealed thisunusual type of movement, but also allowed us to deduce itskinematical connection to the polarization of the centrosomethrough the apparent entrainment of the centrosome by thecell body rolling as a whole. It is an open question whetherthe ‘vectorial translocation’ of the centrosome seen in theprevious two-dimensional data [8] is different from the rollingthat we see in ours, or whether rolling only looked likelinear movement in the two-dimensional projection. Thevectorial translocation was considered consistent with bendingof microtubules as seen in still images next to the interfacewith the target, both observations pointing at the possibilitythat molecular motors are situated at the T cell cortex nearthe interface and pull on the microtubules to translocate thecentrosome [8]. Our computational model, however, explainsboth the centrosome positioning and the very characteristicmicrotubule bending at the interface without invoking anymolecular motors (cf figure 6 of [8] and figure 1(D) here).

For labeling the microtubules in our experiment,we used taxol conjugated with a fluorophore. Atthe submicromolar concentrations used, taxol suppressesmicrotubule dynamics, abrogating normal rearrangements ofthe microtubule cytoskeleton that depend on the microtubuledynamics, arresting cells in mitosis [34]. In our experiments,the centrosome polarization was not inhibited compared toprevious studies not involving taxol. This is consistent withthe fact that other drastic reorientations of the microtubulecytoskeleton in leukocytes, such as its polarization diring theinitiation of migration in neutrophils [35] and retraction intothe uropod in T cells [36], are not affected by abrogationof the microtubule dynamics by taxol. This result validatesnot only our live experimental setup, but also the simplifyingassumption in the model that the microtubules do not changetheir length. In contrast, microtubule dynamics may influencethe centrosome position during other processes in other celltypes, which has been also subject to theoretical investigation[37, 38].

4. Conclusions and outlook

The theoretical and experimental findings in this paper canbe summarized as follows. The eccentric position of the

217

S N Arkhipov and I V Maly

centrosome in T cells can be explained quantitatively asthe one stipulated by whole-cell optimization in the senseof a compromise between microtubule unbending and cellsurface area minimization under the constraint of cell volumemaintenance. The position of the centrosome in activatedT cells next to the interface with the immunological targetcan in theory be obtained parsimoniously by rotating theasymmetric but arbitrarily oriented microtubule aster presentin the T cell before stimulation. The orientation of themicrotubule aster that brings the centrosome to the interfaceis quantitatively optimal in the sense of simultaneous cellsurface minimization, volume maintenance and interface areamaximization. Rotation of the microtubule aster shouldinvolve the entire cell because of the drag resistance, whilethe adherence of the cell to the target should transform therotation into rolling. The lateral movement of the T cell on thetarget surface as expected under rolling is indeed observed inthe experiment simultaneously with the centrosome migrationto the interface. Measurements suggest that rolling accountsfor 43% of the centrosome migration to the target. Anentirely novel picture of T cell polarization emerges fromthese findings. In this picture, the whole-cell structuraloptimization directs the whole-cell movement that entrainsthe centrosome, reorienting it to the target cell as required forthe immunological function of the T cell.

Certain aspects of the polarization mechanism suggestedby our computational and experimental analysis appear tohave deep parallels in seemingly unrelated cell-biologicalphenomena. Cell body rolling with partial slippage on thesubstrate was observed in keratocytes [39], an experimentalmodel of the adherent cell with rapid and highly persistentlocomotion. The centrosome was not monitored in themigrating keratocytes, but the organelles that were, movedaround the cell center similarly to the T cell centrosome in ourexperiments. If the lateral movements of T cells on the targetsurface can be seen as extremely limited locomotion, it seemsnatural that the cell body rolling mechanism would be involvedin this case also. The difference between the keratocytes andT cells is then not in how they move their organelles aroundthe cell center by coupling them passively to the whole-cellmigration through rolling, but in that the keratocytes migrateindefinitely in an apparently random direction and continuecycling their organelles as they move. The T cells, accordingto our analysis, migrate only slightly and on the whole in thedirection dictated by the asymmetry of the microtubule aster,and stop when the centrosome achieves its location next to thesubstrate.

Similarly to the microtubule aster polarization in theT cells, the self-organization of the microtubule aster incytoplasts devoid of centrosomes had also been long believedto arise from intracellular transport of the microtubulesby molecular motors [40]. When the measurements weretaken, they demonstrated immobility of the microtubulesin the cytoplasm [41], also excluding the possibility thatpushing and pulling associated potentially with assembly anddisassembly of the microtubules might be involved. Exclusionof the microtubule movement with respect to the surroundingcytoplasm is the conceptual basis for translating the abstract

rotation of the microtubule aster in our model into rotationof the whole cell body, which in turn is predicted to translateinto rolling because of the perceived no-slip condition on theT cell-target interface. The sequence from the aster rotationto cell body rotation to rolling is purely conceptual; the causalsequence is probably the opposite: T cell rolling on thetarget surface entrains the microtubule aster inside, includingthe eccentric hub of the aster, which is the centrosome.The phenomenon of entrainment of microtubules by a bulkmovement of cytoplasm during cell motility is known fromthe observations in optically favorable regions of motileepithelioid cells, where so-called retrograde flow takes withit microtubules that extend into these lamellar regions [26].The retrograde flow is driven by the actomyosin cytoskeleton,whose organization and the cell shape determine the directionof the flow. During the polarization of the T cell microtubulecytoskeleton, according to our view, the microtubules aresimilarly entrained by the motion of the bulk of the cellbody. The roles, however, are then distributed between themicrotubule and actomyosin cytoskeletons. Whereas themovements are driven by actin, as reviewed above, theirdirection, on the whole, is dictated by the asymmetry of themicrotubule cytoskeleton. Although the geometry of the Tcells conjugated with the target is different from the typicalmotile cell, this microtubule-actin role distribution would bethe same as suggested by experiments where disassembly ofmicrotubules left intact the motility per se, but abrogated itsdirectionality in fibroblasts [42].

Taken together with the possible identical basicmechanism of T and non-T cell polarization response thatmotivated this work, these parallels with cell locomotion seemto point at the fundamental simplicity and universality of cellbehavior. We submit that although the quantitative principlesproposed here are far from an exhaustive explanation of theT cell polarization, the whole-cell movements and whole-cellstructural optimization should play an important role in it,subordinating the underlying molecular mechanisms.

Acknowledgments

The authors acknowledge discussions with Drs C Camacho,M Kim, M Herant, Y-L Wang, M Koonce and E Taylor, aswell as support by NIH grant GM078332 to IVM.

References

[1] Geiger B, Rosen D and Berke G 1982 Spatial relationships ofmicrotubule-organizing centers and the contact area ofcytotoxic T lymphocytes and target cells J. Cell Biol.95 137–43

[2] Kupfer A, Swain S L and Singer S J 1987 The specific directinteraction of helper T cells and antigen-presenting B cells:II. Reorientation of the microtubule organizing center andreorganization of the membrane-associated cytoskeletoninside the bound helper T cells J. Exp. Med. 165 1565–80

[3] Kupfer A and Dennert G 1984 Reorientation of themicrotubule-organizing center and the Golgi apparatus incloned cytotoxic lymphocytes triggered by binding tolysable target cells J. Immunol. 133 2762–66

[4] Kupfer A, Dennert G and Singer S J 1985 The reorientation ofthe Golgi apparatus and the microtubule-organizing center

218

Geometry of T cell polarization

in the cytotoxic effector cell is a prerequisite in the lysis ofbound target cells J. Mol. Cell. Immunol. 2 37–49

[5] Kupfer A, Dennert G and Singer S J 1983 Polarization of theGolgi apparatus and the microtubule-organizing centerwithin cloned natural killer cells bound to their targets Proc.Natl Acad. Sci. USA 80 7224–8

[6] Stowers L, Yelon D, Berg L J and Chant J 1995 Regulation ofthe polarization of T cells toward antigen-presenting cellsby Ras-related GTPase CDC42 Proc. Natl Acad. Sci.USA 92 5027–31

[7] Lowin-Kropf B, Smith Shapiro V and Weiss A 1998Cytoskeletal polarization of T cells is regulated by animmunoreceptor tyrosine-based activation motif-dependentmechanism J. Cell Biol. 140 861–71

[8] Kuhn J R and Poenie M 2002 Dynamic polarization of themicrotubule cytoskeleton during CTL-mediated killingImmunity 16 111–21

[9] Gudima G O, Vorobjev I A and Chentsov Y S 1988 Centriolarlocation during blood cell spreading and motion in vitro: anultrastructural analysis J. Cell Sci. 89 225–41

[10] Albrecht-Buehler G and Bushnell A 1979 The orientation ofcentrioles in migrating 3T3 cells Exp. Cell. Res.120 111–8

[11] Vorobjev I A and Chentsov Y S 1982 Centrioles in the cellcycle. I. Epithelial cells J. Cell Biol. 93 938–49

[12] Parsey M V and Lewis G K 1993 Actin polymerization andpseudopod reorganization accompany anti-CD3 inducedgrowth arrest in Jurkat T cells J. Immunol. 1511881–93

[13] Borroto A, Gil D, Delgado P, Vicente-Manzanares M, AlcoverA, Sanchez-Madrid F and Alarcon B 2000 Rho regulatesTcell receptor ITAM-induced lymphocyte spreading in anintegrin-independent manner Eur. J. Immunol.30 3403–10

[14] Bunnell S C, Kapoor V, Trible R P, Zhang W andSamelson L E 2001 Dynamic actin polymerization drives Tcell receptor-induced spreading: A role for the signaltransduction adaptor LAT Immunity 14 315–29

[15] Kuhne M R, Lin J, Yablonski D, Mollenauer M N,Ehrlich L I R, Huppa J, Davis M M and Weiss A 2003Linker for activation of T cells, ζ -associated protein-70, andSrc homology 2 domain-containing leukocyte protein-76are required for TCR-induced microtubule-organizingcenter polarization J. Immunol. 171 860–6

[16] Bray D 2001 Cell Movements (New York: Garland)[17] Holy T E, Dogterom M, Yurke B and Leibler S 1997 Assembly

and positioning of microtubule asters in microfabricatedchambers Proc. Natl Acad. Sci. USA 94 6228–31

[18] Schmidt R F and Thews G 1989 Human Physiology (NewYork: Springer)

[19] Evans E and Yeung A 1989 Apparent viscosity and corticaltension of blood granulocytes determined by micropipetaspiration Biophys. J. 56 151–60

[20] Gittes F, Mickey B, Nettleton J and Howard J 1993 Flexuralrigidity of microtubules and actin filaments measured fromthermal fluctuations in shape J. Cell Biol. 120 923–34

[21] Kikumoto M, Kurachi M, Tosa V and Tashiro H 2006 Flexuralrigidity of individual microtubules measured by a bucklingforce with optical traps Biophys. J. 90 1687–96

[22] Felgner H, Frank R and Schliwa M 1996 Flexural rigidity ofmicrotubules measured with the use of optical tweezersJ. Cell Sci. 109 509–516

[23] Boal D 2002 Mechanics of the Cell (Cambridge: CambridgeUniversity Press)

[24] Bunnell S C, Barr V A, Fuller C L and Samelson L E 2003High-resolution multicolor imaging of dynamic signaling

complexes in T cells stimulated by planar substrates Sci.STKE 2003 pl8

[25] Hailman E and Allen P M 2005 Inefficient cell spreading andcytoskeletal polarization by CD4+CD8+ thymocytes:regulation by the thymic environment J. Immunol. 1754847–57

[26] Waterman-Storer C M and Salmon E D 1997 Actomyosin-based retrograde flow of microtubules in the lamella ofmigrating epithelial cells influences microtubule dynamicinstability and turnover and is associated with microtubulebreakage and treadmilling J. Cell Biol. 139 417–34

[27] Rubbi C P and Rickwood D 1996 A simple immunomagneticbead-based technique for the detection of surface moleculescapable of inducing T cell functional polarizationJ. Immunol. Methods 192 157–64

[28] Lowin-Kropf B, Smith Shapiro V and Weiss A 1998Cytoskeletal polarization of T cells is regulated by animmunoreceptor tyrosine-based activation motif-dependentmechanism J. Cell Biol. 140 861–71

[29] Bubeck Wardenburg J, Pappu R, Bu J Y, Mayer B, Chernoff J,Straus D and Chan A C 1998 Regulation of PAK activationand the T cell cytoskeleton by the linker protein SLP-76Immunity 9 607–16

[30] Negulescu P A, Krasieva T B, Khan A, Kerschbaum H H andCahalan M D 1996 Polarity of T cell shape, motility, andsensitivity to antigen Immunity 4 421–30

[31] Nobes C D, Hawkins P, Stephens L and Hall A 1995Activation of the small GTP-binding proteins rho and rac bygrowth factor receptors J. Cell Sci. 108 225–33

[32] Sedwick C E, Morgan M M, Jusino L, Cannon J L, Miller Jand Burkhardt J K 1999 TCR, LFA-1, and CD28 playunique and complementary roles in signaling T cellcytoskeletal reorganization J. Immunol. 162 1367–75

[33] Wulfing C, Purtic B, Klem J and Schatzle J D 2003 Stepwisecytoskeletal polarization as a series of checkpoints in innatebut not adaptive cytolytic killing Proc. Natl Acad. Sci.USA 100 7767–72

[34] Jordan M A, Toso R J, Thrower D and Wilson L 1993Mechanism of mitotic block and inhibition of cellproliferation by taxol at low concentrations Proc. NatlAcad. Sci. USA 90 9552–6

[35] Eddy R J, Pierini L M and Maxfield F R 2002 Microtubuleasymmetry during neutrophil polarization and migrationMol. Biol. Cell 13 4470–83

[36] Ratner S, Sherrod W S and Lichlyter D 1997 Microtubuleretraction into the uropod and its role in T cell polarizationand motility J. Immunol. 159 1063–7

[37] Cytrynbaum E N, Scholey J M and Mogilner A 2003 A forcebalance model of early spindle pole separation inDrosophila embryos Biophys. J. 84 757–69

[38] Howard J 2006 Elastic and damping forces generated byconfined arrays of dynamic microtubules Phys.Biol. 3 54–66

[39] Anderson K I, Wang Y-L and Small J V 1996 Coordination ofprotrusion and translocation of the keratocyte involvesrolling of the cell body J. Cell Biol. 134 1209–18

[40] Rodionov V I and Borisy G G 1997 Self-centring activity ofthe cytoplasm Nature 386 170–3

[41] Vorobjev I, Malikov V and Rodionov V 2001Self-organization of a radial microtubule array bydynein-dependent nucleation of microtubules Proc. NatlAcad. Sci. USA 98 10160–5

[42] Vasiliev J M, Gelfand I M, Domnina L V, Ivanova O Y,Komm S G and Olshevskaja L V 1970 Effect of colcemidon the locomotory behaviour of fibroblasts J. Embryol. Exp.Morphol. 24 625–40

219