VOLUME 83, NUMBER 1 P H Y S I C A L R E V I E W L E T T E R S 5 JULY 1999

Dynamics and Similarity Laws for Adhering Vesicles in Haptotaxis

Isabelle Cantat and Chaouqi MisbahLaboratoire de Spectrométrie Physique, Université Joseph Fourier (CNRS), Grenoble I,

B.P. 87, Saint-Martin d’Hères, 38402 Cedex, France(Received 18 September 1998)

We study vesicle dynamics induced by an adhesion gradient. This kind of migration is namedhaptotaxis in biology. The problem is fully solved as a free boundary one, including hydrodynamicsflows. First, we analyze adhesion at equilibrium. We then determine the propulsion velocity asa function of various parameters. We find a persistent mixture of rolling and sliding. Similaritylaws are extracted analytically both for the adhesion area and propulsion velocity by means ofdimensional and scaling arguments. Our results markedly differ from classical results of nondeformablemigrating entities.

PACS numbers: 87.16.–b, 47.55.Dz, 87.19.– j

Vesicles are closed membranes suspended in aqueoussolution. Extensive investigations have been devoted totheir equilibrium properties to date [1]. There are severalcircumstances in living systems where nonequilibrium fea-tures play a decisive role, however. Of particular inter-est is vesicle movement under external forces. A majorleitmotiv in biological and medical science is the under-standing of how and by which mechanisms cells of theimmune system (e.g., granulocytes) move in response totissue injury. This has led to the development of a numberof experiments in vitro where cells are subject to externalperturbations (e.g., shear flow). This step is essential inidentifying the energetic, and possibly the kinetic, proper-ties that are involved in displacing cells upon applicationof external forces when they are initially under adhesionon a substrate.

In this Letter, we present analytical and numerical re-sults on vesicle dynamics in an adhesion gradient (hapto-taxis). Our results can also be applied to situations wheremovement is due to the presence of a chemoattractant(chemotaxis).

Vesicle mobility induces a coupling of the flow withinthe membrane to the bulk fluid. With the help of Green’stensor techniques, we solve this free-boundary-like prob-lem numerically and determine both the shape and veloc-ity in haptotaxis. A close inspection of the underlyingphysical mechanisms allows us to extract analyticallysimilarity laws for the adhesion area and vesicle veloc-ity as functions of relevant parameters. These laws arein good agreement with the full numerical results. Theyshould constitute an important basis for experimentalstudies. The numerical work is performed in two dimen-sions. The analytical scaling laws will be written, how-ever, for both 2D and 3D.

The various sources of dissipations are the follow-ing. First, the internal dissipation due to phospholipidicmolecules rotation characterized by the Leslie coefficientoccurs on a very small time scale (typically 1025 s),and is then negligible. In turn hydrodynamics relax-

0031-90079983(1)235(4)$15.00

ation occurs on the scale of 1021 s (this time is obtainedwith the curvature rigidity, the water viscosity, and thevesicle size), and constitutes the first noticeable slowprocess for vesicle dynamics. Other sources of dissipa-tions such as bond breaking and restoring associated withadhesion may become important and are currently underinvestigation.

Because the Reynolds number is typically small (Re 1024), the Stokes approximation is justified. The hydro-dynamics equations thus take the form

hDv =p; = ? v 0 . (1)

We require continuity of v through the membrane anda vanishing velocity on the substratum; both demandsexpress the no-slip condition and impermeability. Thethin fluid film between the vesicle and the substratum istreated as in the bulk.

Thanks to linearity of the Stokes equations we can makeuse of the Green’s function method. When viscositiesinside and outside the vesicle are identical, the velocityfield takes the form

vr, t Z

membT ? f0memb ds0

1Z

subsT ?

√2h

≠v 0

≠y01 p0y

!dx0. (2)

Here T is the Oseen tensor [2] and y is a unit vector.For brevity we set v 0 vr0, t and T T r, r0. Thefirst term in Eq. (2) represents the membrane contribution,with fmemb the membrane forces, to be specified below.The second term accounts for the presence of the adhesionwall parametrized by x0. It is worth mention that usecould be made of the special Green’s tensor taking intoaccount the presence of the substratum [3]. In that casethe second term in Eq. (2) disappears at the expense of amore complicated tensor.

© 1999 The American Physical Society 235

VOLUME 83, NUMBER 1 P H Y S I C A L R E V I E W L E T T E R S 5 JULY 1999

The total force on the membrane is given by

fmemb 2dFdr

2d

dr

Zds

"k

2c2 1 z 1 wrs

#

"√k

≠2c≠s2 1

c3

2

!2 cz 2

≠w≠n

r 2 cwr

#n 1

≠z

≠st .

(3)

Here n and t are the outward normal and the tangentto the vesicle; z is a time and space dependent Lagrangemultiplier enforcing a constant local length [whereas areaconservation in implied by Eq. (1)]; c denotes the mem-brane curvature, and k the rigidity. Finally, wr is theadhesion potential. For definiteness, if x, y are the coor-dinates along and perpendicular to the wall we take

wr w0 1 x 2 xm=w d0y2d0y2 2 2 .(4)

The potential is repulsive as 1y4 for y ø d0 and at-tractive as 21y2 at long distance. The optimal distanced0 50 nm between membrane and wall is constant. Theprefactor depends linearly on x with a constant gradient=w. The mean value of the potential under the vesicleis also a constant w0, and xm is the middle point of theadhesion area and is thus moving in the laboratory frame.The vesicle is moving, thanks to an adhesion gradient. Ifthe adhesion potential were fixed, the vesicle would ex-perience an increasing adhesion during its movement. Toavoid this we consider that wr is invariant in the vesicleframe represented by xm.

The numerical strategy is as follows. We first take thelimit on the external variable rsubs and set the left-handside of Eq. (2) to zero. The force on substrate is obtainedby inverting the integral equation. This determines v ev-erywhere, and especially on the membrane. Dynamics aretreated by forward iteration in time. These steps involveseveral technical points which will not be exposed here.

Thus, after transient have decayed, the vesicle acquiresa permanent regime with a constant drift velocity, a situ-ation on which we focus now. First, our numerical resultspredict velocities which lie in the range 0.1 10 mmsfor adhesion differences on both sides of the vesicleof about dww0 10% [R 1 10 mm (vesicle size),k 20 40 kT, w 102 104 kTmm]. It is a bit sur-prising that this value falls in the biological range. Forexample, granulocytes move at about 50 mms in vivo[4]. Another important feature is the persistent mixturebetween a rolling and a sliding motion (that is, the con-tact area has a small relative motion with respect to thesubstrate) (Fig. 1). The vesicle deformability and its ad-hesion length are crucial to capture this feature, because arigid cylinder pulled parallel to a wall feels no torque, andthus only slides [5].

The type of motion (sliding and rolling) found here hasbeen observed by Bongrand et al. for cells under shearflow [6]. Our results are in a marked contrast with theoriesimposing nonrealistic rigid bodies for cells [7]. The abilityfor the vesicle (or cell) to deform implies that the driftvelocity obeys a nontrivial scaling law R22 (in 3D) and

236

R245 (in 2D) with the scale of the vesicle (see belowand Fig. 4). The rigid body constraint leads to a quitedifferent finding, namely the velocity scales as R21 for asphere (up to logarithmic corrections) [7], and as R212 fora cylinder [5].

Because of the complexity of the present problem it ishighly desirable to have at our disposal analytical results.This is very important in order to get insight towards theunderstanding of the underlying physical ingredients whichare responsible for vesicle dynamics. This is what wewould like to sketch now.

A relevant parameter for the vesicle mobility on a sub-strate is the contact area. Let us first consider the 2D caseas in the numerical work. The adhesion length (see Fig. 1for definition) is well defined as long as d0 ø R and is de-noted by Ladh. Let us evaluate this quantity. Ladh dependsonly on the three characteristic lengths: R L

2p , Rs pAp corresponding to the vesicle perimeter L and area

A, respectively, and Rc p

kw0 which is related to thecontact curvature [8]. We assume that Ladh has a weakdependence on velocity, an assumption which turns out tobe quite legitimate in haptotaxis according to full numeri-cal results. Typically three situations may occur, and weconsider them separately.

(A) R ¿ Rc: This situation is relevant in the caseof a strong adhesion, and/or small rigidity, and/or giantvesicles. In reality, only the combination of these threefacts matters. In that case the part of vesicle which is notin contact with the substrate is a truncated quasicircularshape with angular contact point. Indeed if the rigidity isnegligible in comparison to the adhesion, we recover thefluidlike droplet model [1]. Ladh can thus be expressedanalytically in the limit RsR ! 1 (a situation often en-countered in real situations) as

Ladh R1 2 RsR13. (5)

We have explored numerically this regime. Figure 2shows the analytical part compared to the numerical one.The agreement is quite satisfactory. The other limitRsR ! 0 (small swelling) yields obviously Ladh L2.

Ladh

FIG. 1. Vesicle shape at three different times. The arrowindicates the trajectory of a material point on the vesicleillustrating the rolling and sliding motion.

VOLUME 83, NUMBER 1 P H Y S I C A L R E V I E W L E T T E R S 5 JULY 1999

-2.5 -2.0 -1.5 -1.0 -0.5log( 1-Rs/Rl )

-0.20

-0.10

0.00

0.10

0.20

0.30

log

( La

dh/R

l )

numerical resultlinear fit : slope 0.31

FIG. 2. Variation of adhesion length with the swelling. (Nu-merical parameters: R 1.1 and Rc 0.6, with an arbitraryunit of length.)

(B) Rc R: This situation corresponds to the case ofintermediate rigidity, or more precisely when all effectscompete. This case is more subtle. At equilibrium, theincrease in curvature energy due to adhesion must be offsetby the decrease in the adhesion energy. The evaluation ofboth terms will provide us with a scaling law for Ladh.For that purpose, let dcs denote the curvature variationbetween the free and adhering vesicle, which is supposedswelled (Rs R) and impermeable. The conservation ofvesicle perimeter under such a variation entails a vanishingmean value for dcs. Therefore the global curvatureenergy variation is proportional to

Rkdc2 ds. We evalu-

ate dc by means of the volume conservation condition.Under adhesion the volume deficit (the area in 2D) which isbeneath, dy L3

adhR, must be redistributed elsewhere,so as to keep the total volume constant. The estimatefor dy is performed on a portion of a disc with a radiusR and a base Ladh in the limit of small enough Ladhas compared to the total length; see Fig. 4. That is, weconsider the equivalent disc of the real shape. Because adisc has a larger volume, it must be truncated by a quantitydy. This amounts to the sought after curvature increase,since the truncated object has a smaller effective radius ofcurvature. This leads to a radius increase dR dyR R2dc. This implies that the increase in curvature energy isDEc kL6

adhR7. The variation of the adhesion energyis simply given by DEw w0Ladh. A balance betweenDEc and DEw yields

Ladh

√w0

k

!15

R75. (6)

The same arguments can be extended to 3D providingus with

Ladh

√w0

k

!16

R43. (7)

The full numerical analysis provides a good agreementwith the scaling laws (Fig. 3). Higher values of R willcause a crossover to the behavior in Fig. 2, while smaller

0.60 0.80 1.00log ( vesicle size )

-0.6

-0.4

-0.2

0.0

0.2

0.4

log

( ad

hesi

on le

ngth

)

numerical result linear fit slope = 1.5

FIG. 3. Variation of adhesion length with the vesicle size.[Numerical parameters: RsR 0.9 and Rc 1.2 (arbitraryunit)].

values lead to vesicle unbinding (see below), so that thevalidity of the scaling law is limited by these two physicalfacts.

(C) Rc . R: This is the weak adhesion limit close tothe unbinding transition already analyzed in [8]. Since wedo not address here the problem of unbinding this limitwill not be considered later.

Having determined the adhesion length, we are in a po-sition to study dynamics. In this brief exposition we shalllimit ourselves to the situation of intermediate adhesion.Energy is injected by adhesion gradients, and is dissipatedin hydrodynamics flows. Let us first evaluate the viscousdissipation.

The largest velocity gradients occur in the fluid influ-enced by both the vesicle and the substrate. Note that thedissipation in the thin fluid film between the vesicle and thesubstrate is limited by the great rolling ratio. For simplicitywe assume that the vesicle is in direct contact with the sub-strate (no fluid film beneath). The volume of dissipationis that corresponding to the influence region, and scales asL2

adh (as is the case with Laplacian fields—we have =2p 0— the perturbation penetrates over a scale of order Ladh inthe two directions). In these regions (see Fig. 4) the meanlocal velocity is estimated at a distance Ladh from the con-tact point (in the fluid bulk) as Vloc VLadhR using arolling without sliding motion. This relation is geometri-cally obvious from Fig. 4 for a small adhesion length. If

θ

dissipationZone of

Ladh

dv

θ

V

Ω

R

Vloc ~ V sin ~ V Ladh/R

~V/R

FIG. 4. Definition and schematic views of several quantitiesused in the analytical part.

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VOLUME 83, NUMBER 1 P H Y S I C A L R E V I E W L E T T E R S 5 JULY 1999

this is not the case, it is clear on the general ground thatsince the vesicle must adhere on a length Ladh the typicalvelocity gradient must be set on that range, and is equalto VlocLadh. On the global scale of the vesicle the gra-dient is VR. The demand that a permanent regime is es-tablished requires the two gradients to scale in the samemanner, namely VlocLadh VR (these correspond tothe inverse of time scales for dissipation), which is a moreprofound physical view of the relation introduced above.Note also that the greater the adhesion length, the larger isVloc for a given R and V . This is obvious since the vesiclemust evacuate the fluid on a larger length, and in order tomove at the same global velocity V , Vloc must be larger.

Using the dependence R75 for Ladh [Eq. (6)] we deducethe functional dependence of the dissipation (in 2D):

D hL2adh

√Vloc

Ladh

!2

hV 2

√RRc

!45

. (8)

Equating dissipation, D, and source, FdV , with Fd thedriving force, we arrive at

V Fd

h

√Rc

R

!45

2D, V Fd

h

Rc

R2 3D .

(9)

For 3D, dissipation occurs in a volume L3adh, and Ladh

obeys different scaling [Eq. (7)]. This implies a differentdependence on R.

This law works for any driving force Fd . In 2D the forceis defined per unit length. In haptotaxis Fd Ladh=win 2D and Fd L2

adh=w in 3D. Comparison with fullnumerics is presented on Fig. 5. The best fit provides(over about a decade) 0.75 6 0.05, while full analyticalresults predict 45 0.8. It is very important to notethat theories with rigid bodies [7] lead to V R21 in 3Dand V R212 in 2D. Allowing for vesicle deformabilityleads to the nontrivial law R22 and R245 in 3D and 2D,respectively.

Further increase of R leads (as for statics discussed be-fore) to a crossover to a strongly adhering regime discussed

0.45 0.65 0.85 1.05 1.25 1.45log(R)

-2.10

-2.00

-1.90

-1.80

-1.70

-1.60

log(

V/F

)

numerical resultsfit : slope = -0.75

FIG. 5. Variation of the ratio (velocitydriving force) with thescale at constant swelling. [RsR 0.95, Rc 0.5, =ww0 0.2, and d0 0.06 (arbitrary units)].

238

below. In the extreme limit of tensely adhering vesicles,and if the swelling is large, Ladh is given by Eq. (5). Wecan derive the drift velocity in the same manner. BecauseLadh is proportional to the vesicle scale R, and given theform of dissipation and source powers discussed above, weobtain for the drift velocity

V Fdh 2D, V FdhR 3D . (10)We obtain here for 2D and 3D similar results as for acylinder and a sphere (Stokes law) immersed in an infinitemedium (no substrate). We hope to report on extensivenumerical results in the future.

The next step will be to extend the numerical treatmentto 3D. In addition, the relevance of several physical pointsshould be clarified in the future. (i) We have taken into ac-count the first natural dissipation associated with hydrody-namical flow inside and outside the vesicle. This assumesthat the two monolayers form an entity, in that they do notslide with respect to each other. It is likely [9] that this isnot always true, and the effect of the induced dissipationshould be investigated in the future. (ii) Furthermore, per-haps the most serious point that must be emphasized withregard to dissipation in the realm of biology concerns thekinetics of bond breaking and restoring with the substrate.If the time scale involved in this process is larger thanthat associated with hydrodynamics, dissipation should bedominated by chemical kinetics. The time scale dependson the energetics involved in the adhesion, and it seemsat first sight that in the biological world the bond breakingdynamics should be of great importance [10]. (iii) Finally,in order to be closer to realistic situations we should relaxthe assumption of equal viscosities inside and outside thevesicle. In reality, the cytoskeleton exhibits viscoelasticproperties that are necessary to incorporate in future mod-els which are aiming at a more elaborate theory of cellmotility.

We are very grateful to R. Bruinsma, F. Jülicher,and U. Seifert for several enlightening and stimulatingdiscussions.

[1] Structure and Dynamics of Membranes, Handbook of Bio-logical Physics, edited by R. Lipowsky and E. Sackmann(Elsevier, Amsterdam, 1995).

[2] O. A. Ladyzhenskaya, The Mathematical Theory of Vis-cous Incompresible Flow (Gordon and Breach, New York,1969), 2nd ed., Chap. 3.

[3] J. R. Blake, Proc. Cambridge Philos. Soc. 70, 303 (1971).[4] M. B. Lawrence and T. A. Springer, Cell 65, 859 (1991).[5] D. J. Jeffrey and Y. Onishi, Q. J. Mech. Appl. Math. 34,

129 (1981).[6] O. Tissot, A. Pierres, C. Foa, M. Delaage, and P. Bon-

grand, Biophys. J. 61, 204 (1992).[7] A. J. Goldman, R. G. Cox, and H. Brenner, Chem. Eng.

Sci. 22, 637 (1967).[8] U. Seifert, Phys. Rev. A 43, 6803 (1991).[9] U. Seifert and S. A. Langer, Europhys. Lett. 23, 71 (1993).

[10] R. Bruinsma, in Physics of Biomaterials, Fluctuations, SelfAssembly and Evolution, edited by T. Risk and D. Sher-rington, NATO ASI Ser. 332 (Kluwer, Dordrecht, 1996).