A physical model of cell crawling motion

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A physical model of cell crawling motion

Didier Sornette

To cite this version:

Didier Sornette. A physical model of cell crawling motion. Journal de Physique, 1989, 50 (13),

pp.1759-1770. �10.1051/jphys:0198900500130175900�. �jpa-00211029�

A physical model of cell crawling ^motion

Didier Sornette

Laboratoire de Physique de la Matière Condensée (*), Faculté des Sciences, ^Parc Valrose, ⁰⁶⁰³⁴

Nice Cedex, ^France

(Reçu ^{le 30} septembre ^1988, révisé le 14 mars 1989, accepté ^{le 16 mars} 1989)

Résumé.

On propose ^un modèle simple des mécanismes du mouvement de rampement ^de cellules sans cils ni flagelles. L’idée essentielle repose ^sur « l’exocytose dirigée

^découverte

récemment dans de nombreuses cellules possédant ^une capacité ^{motrice. On} suggère que

l’exocytose dirigée ^{est créée par une}

pompe

osmotique fonctionnant à l’intérieur de la cellule.

Cette hypothèse permet ^{d’obtenir une} description quantitative précise de l’écoulement lipidique

membranaire d’avant en arrière de la cellule. On propose enfin que ^cet écoulement membranaire induit un écoulement du fluide environnant la cellule conduisant à une poussée ^{d’arrière en avant}

exercée par le fluide ^sur la cellule. Les valeurs typiques ^{obtenues avec ce modèle sont} comparées

aux données expérimentales existentes. L’accord quantitatif suggère que le ^mouvement de rampement de cellules repose ^non seulement sur une interaction cellule-substrat mais aussi sur une interaction cellule-fluide environnant.

Abstract.

A simple

prototype

model of the motion mechanisms of cells without flagella ^or

cilia is proposed. It relies essentially ^{on the}

^directed exocytosis

discovered recently in many motile cells. An osmotic

pump » working inside the cell is suggested ^{as a} plausible ^{motor for the}

establishment of the directed exocytosis. ^The quantitative features of the cell lipidic ^membrane

backflow are obtained explicitly. This membrane flow induces a fluid motion outside the cell, leading ^{to a} thrust for forward motion exerted by ^{the fluid on the cell.} Typical values compare well with existing data : this suggests that cell crawling ^motion might ^{involve not} only ^cell-

substrate but also cell-fluid interactions.

Classification

Physics ^Abstracts

87.25

87.45F

87.45D

1. Introduction.

Movement is a fundamental property ^of living organisms intricately coupled ^to their intemal

machinery. ^{In this letter, a} simple

prototype

^{model of} the motion mechanisms of cells without flagella ^{or cilia is} proposed. ^This form of motion is involved for example ^{in the}

movement of fibroplasts (which ^must migrate ^{into a wound to take} part ^{in the} healing process), ^{of white} blood cells (which ^move through ^the body ^to fight pathogens), ^{of certain} spermatozoa ^{and some} unicellular organisms [1]. ^{It is believed} generally ^{that movement.}

occurs as in amoeba and depends ^{on the} production of transient cytoplasmic extensions of the cell surface [2,3]. ^{The view defended here is that} lipid ^membrane flow, which is known to

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0198900500130175900

occur in these cells, ^{cannot be} neglected and should provide ^a stimulus for re-evaluating ^the

models of movement.

In general, ^the understanding of individual cell motion involves three major problems : 1) ^{what is the}

^motor

of the cell ? This problem touches upon the complex ^{structure and}

function of the cell interior and is an active open field of research in biology ^and biochemistry [4]. ^Different types of cells have different motors. Schematically, ^the

^{motor »} operates ^often by conformational changes ^{in a} protein (actin, myosin...) ^{which can be utilized to slide}

filaments past ^{one another as} in muscles [4, 5]. ^{One can think of other} types ^{of motors such as} the « pumps

which create gradients in various species concentrations within the cell ;

2) what is the mechanism of movement ? Here, ^the activity ^{of the motor is taken for} granted, ^the question of interest is : to what use is the motor put ? ^Nature has invented many types ^of locomotion modes, amongst ^them propulsion by ^different types ^of flagella [6, 7], ciliary ^motion [8] ^and crawling motion in cells without flagella ^{or cilia} [1-3] ;

3) ^{what are} the factors controlling the cell movement ? Several generic phenomena ^have

been proposed ^{such as contact} inhibition, chemotaxis, haptotaxis, galvanotaxis, phototaxis...

and the role of some of them has been well established [2]. ^The triggering factor is distinct for different cells or may involve ^a conjonction of several ingredients within ^a single ^cell.

However, the detailed biochemical mechanisms involved in these taxis phenomena ^are largely

unknown and remain an important field of research.

Due to the complexity ^{and often extreme} specificity ^{of cell} behaviour, ^a physical ^{model is}

bound to be too naive and general ^for really describing biological ^{motions. However, one} may

hope ^{that the} simplistic ^model presented here may become ^a starting point for refined

analysis. The three main ideas which are presented ^{below are the} following :

i) ^{an osmotic} gradient within the cell drives a net flux of intracellular vesicles in ^a direction which determines the direction of motion of the cell ;

ii) the internal vesicle flux polarises ^the endocytosis/exocytosis cycle [9-11] thereby inducing ^a net return flow of lipid in the cell surface. This allows one to obtain the quantitative

features of the cell lipidic membrane backflow by the condition of conservation of lipid material ;

iii) ^the lipid surface flow is viscously coupled ^{to the} surrounding ^fluid medium, leading ^{to a}

net thrust for forward motion exerted by ^{the fluid on the cell,} which leads to locomotion. Its interest may rely ^{on the fact} that, starting ^{from a} specific hypothesis ^{for the motor} operation (the « pump motor »), everything ^{can be} determined ! In particular, the mechanism of

motion, the direction of migration, ^the velocity ^and efficiency ^of propulsion ^{can be} simply estimated, as we see below.

2. Ingrédients of the model.

It is known that a living ^cell continuously recycles part of its external membrane in its interior, using endocytosis-exocytosis cycles [1]. ^{In an} elementary endocytosis, ^a portion ^{of the}

membrane invaginates ^{and detaches from the membrane, thus} forming ^a closed vesicle in the interior. In an exocytosis ^{event, a small vesicle,} initially inside the cell, ^{comes in contact with}

the outer membrane and fuses with it. The physical integrity of the cell membrane is a

dynamical process resulting ^{from an} equilibrium ^between exocytosis ^and endocytosis ^{on each}

point of the membrane. As a consequence, ^a fraction of the membrane (for example ^{in a}

fibroplast cell, ^two percent ^{of the} membrane) ^is constantly found in the interior of the cell

under the form of small closed vesicles which were once part ^{of the outer membrane.}

Now, ^we suppose that the « pump ^{motor »} of the cell consists in creating ^{a net flux of}

vesicles in the direction, say, Ox. This vesicle flux induces ^a bias in the endocytosis-exocytosis equilibrium. ^More vesicles suffer

collisions » with the front of the cell membrane resulting ⁱⁿ

an increased exocytosis ^{in this} part of the membrane. Conversely, there appears ^a depletion

of vesicles on the opposite side of the cell (the back) leading ^{to a} decrease of exocytosis ^{on the}

rear end.

Such an intracellular vesicle transport may take place ^via components ^{of the} cytoskeleton,

the microtubules or the microfilaments and are ATP-dependent [5, 12]... ^Another possibility,

which is explored ^{here, concerns} the role of ^a chemical gradient along the Ox axis which is created by the cell in its interior. If the corresponding ^{molecules cannot} permeate freely through the vesicle membrane, ^{a net osmotic force} appears ^on each vesicle pushing ^{it towards}

the low concentration region [13]. This defines the front of the cell as the place ^{where more}

vesicles collide.

In short, ^more material is put ^{in the} membrane-front than is taken away resulting ^{in a net}

surface increase in the front. Less material is put in the cell membrane at the rear end than is taken away, resulting ^{in a net membrane} depletion in the back. This

dipole

source-sink structure creates a hydrodynamic flow from the front to the back within the cell membrane.

This last picture ^was proposed ⁱⁿ [9-11].

Now, ⁱⁿ the presence of this membrane circulation, several mechanisms may be involved in the cell motion. The simplest ^{is to} recognize that the membrane flow couples with the fluid outside the cell and induces by continuity ^{a backward outer} fluid flow as seen in the cell frame

(see Fig. 1). Cell motion follows along ^{Ox with a} velocity roughly of the order of the backward lipid velocity in the cell membrane.

Fig. ^1. - Simple picture of the external fluid flow created by the cell membrane as seen in the cell frame.

This model is based on recent results obtained by ^Bretscher [1, 9-11] ^on lipid circulation in the cell membrane. The backward lipid ^flow hypothesis within the membrane is in fact observed via the backward motion of small test particles ^{attached to the} membrane _[1],

showing the existence of a continuous lipid flow from the leading edge ^{to the} trailing edge.

The hypothesis that this flow is triggered by ^the exocytosis-endocytosis processes is evidenced

by (see [1] ^and references therein for a general discussion) i) the efficient recycling ^{of about}

roughly ^two per ^cent per minute of the cell membrane in the cytoplasm in the form of

vesicles), ii) ^the experimental observation that ferritin receptors proteins (which ^{are carried} by ^the vesicles) ^{are found in} majority in the cell front (their recirculation with the membrane backflow dilutes their concentration), ^a fact consistent with the transport of vesicles towards the front, iii) the observation that proteins, ^{which are} crosslinked such that they ^diffuse only slowly (D - ^{10- 10} cm2/sec), ^{and which are not} transported by the vesicles and therefore

always remain in the membrane, ^{are found at the rear of the} cell, ⁱⁿ agreement with what is

expected in presence of ^a backward membrane lipid ^flow carrying ^imbedded proteins. ^In fact, they ^are deplected from the front end of the cell. Of course, proteins which have normal diffusion coefficients (D ~ ^{10- 8 - 10- 9} cm2/sec) ^{do not} form detectable gradients since, ⁱⁿ

this _case, the diffusion is more rapid than membrane flow.

Among ^the many open questions ^{on cell} motion, ^the speculative ^answers proposed ^within

the physical model of this letter are the following : i) ^what distinguishes ^the leading edge ^{of a}

motile cell may be ^a gradient of chemical species built inside the cell ; ii) ^this gradient (or

whatever the motor is) ^{works as a} gear, converting the power used in establishing ^{it into}

viscous vesicle motion, which itself is converted in viscous lipid membrane motion via

exocytosis ; iii) this backward membrane flow creates a viscous backward external fluid flow

(seen in the cell frame) ^which implies ^a forward thrust exerted by the external fluid on the cell.

1 shall thus present ^the quantitative arguments for the model which subsequently ^{will be} compared ^more carefully ^with biological ^facts.

3. Movement of a semipermeable ^vesicle through ^{an osmotic} gradient [13].

Suppose ^a gradient of solute concentration gradient

is established by ^{the cell} along the Ox direction. Consider small vesicles with a semi-

permeable membrane which lets the solvent permeate freely, whereas the solute cannot cross

the membrane. The nature of the relevant solute molecules is not addressed here : many molecules may qualify ^{in a} biological ^{cell as solute} species (such ^as ions and many types ^of molecules and proteins...) ^{which cannot} permeate passively through vesicle membranes [14].

In the presence of ^a solute concentration gradient, ^{the osmotic} pressure given by ^{van’t Hoff} expression Ap = R T (C ’ - C - ) ^{across a} single flat membrane induces ^{a net} fluid flow

through ^it

where Lp ^{is a hydraulic} coefficient characteristic of the membrane permeability property ^with respect ^{to the} solvent, ^{R is} the ideal gas constant, T the temperature ^and C + (- ) is the value of the solute concentration just ^{to the} right (left) of the membrane along ^the Ox axis. Note that the solvent flows towards large ^solute concentration regions. ^{Now take a} single vesicle of radius « a

at rest with an inner homogeneous solute concentration C ln = Co. Suppose ^for example ^{that C -} C;n ^over the forward _cap (in ^{the direction} Ox) ^and C + > Cln ^{over the rear}

cap of the vesicle. According ^to equation (2), ^{solvent must be} pulled into the vesicle over the forward _cap and expelled from the vesicle over the rear cap (see Fig. 2). Therefore, ^{if the}

external fluid is steady, the vesicle cannot remain at rest but must move in the direction of lower concentration (Ox ). ^{In fact,} it is easy ^to convince oneself that the movement of the vesicle through ^the gradient will go ^{on even} if the inequality ^{C -} C in ^{C + is not} fulfilled. In

fact, ^the only necessary condition is C - C + .

Fig. ^2.

^Schematic representation ^{of a} vesicle in an osmotic gradient.

This effect has been studied in [13] ⁱⁿ great detail. We can recover the main results by ^the following simple analysis. ^The stationary velocity ^{of a} single vesicle is simply given by ^{the net}

fluid flux through its surface divided by ^{its surface area 4} ira 2

^{and reads}

valid in the low solute concentration limit. Equation (3) ^states that the modulus of the vesicle

velocity ^is essentially that of the solvent across a semipermeable membrane with a difference of concentration C ’ - C - = a a. The limit of high solute concentration can also be evaluated

[13] ^{but is not relevant to the case} discussed here. Taking typically a = 10- 2 moles/cm4,

a -- r 102 _nm, Lu .-- 10- 7 cm2 sec/gm (Lp ^{may show} large variations depending ^{on the nature of}

the different impurity molecules imbedded in the vesicle membrane and can range typically

from 10- 6 to 10- 11 cm2 sec/gm, the last value corresponding ^to pure bilayers [14]), ^{at room}

temperature, ^{one obtains} vves = ^2.5 um/sec = ¹²⁵ um/min.

4. Lipid backward flow in the cell membrane.

If N denotes the number concentration of vesicles affected by this osmotic motion within the

cell, ^{the flux} of vesicles incident upon ^{an area} dS of the surface of the cell membrane front is

Nv,e, . n ^{dS where n} is the normal to the surface dS. The

surface flux

is then given by

since each vesicle brings ^{in a surface 4} 7Ta2. In the reasonable limit where the lipid ^membrane

fluid is considered to be incompressible, ^this incoming

surface » flux triggers ^a gradient ^of

cell membrane lipid velocity such that the lipid incompressibility constraint is satisfied. As an

illustration, ^{let us take the} simple ^{case of a} spherical ^{cell of radius A} (extension ^to arbitrary shapes ^involves straightforward algebra). ^{In the} spherical ^case, the membrane lipid ^motion

goes from the front pole ^{to the rear} pole along great circles, ^{with a} rotational symmetry around the Ox axis, ^as depicted ⁱⁿ figure 3. With the coordinates defined in figure 3, 0 being

the

latitude », the condition of lipid ^fluid incompressibility in the presence of the incoming

flux of vesicles fusing with the membrane leads to a fluid lipid velocity Vlip ( (0) ^{in the}

membrane given by

Fig. ^3.

a) ^Schematic representation ^{of a}

spherical

cell. The vesicle flux which is incident on each surface element of the cell membrane triggers ^a lipidic membrane flow shown in b) such that the condition of conservation of lipids ^{is fulfilled.}

where df

A sin 0 d Q , ^dS

A 2 sin 0 d 0 dç ^and vves - ni 1 Vves 1

^{cos 0.} The solution of

equation (5) ^is

where

Note that vo ^is proportional ^to Nv,e, a2 since this corresponds ^to the flux of lipid ^surface brought by the vesicles. The proportionality ^to the cell radius A comes from a cancellation between the cell internal surface cross-section A 2 offered to the vesicle flux and the line cross-

section A for the two-dimensional backward membrane lipid ^flow.

If a fraction X --= 2 % of the membrane is in the form of vesicles, ^their concentration, supposed ^to be uniform inside the cell, ^is

Inserting (8) ⁱⁿ (7) gives vo

3 Xv ves = ^{6 x} 10- 2 _vves. Note that the dependence ^of

uo with the cell enters only through ^the parameter X ! ^{With our estimate} vves = ¹²⁵ um/min,

this leads to vo ~, ¹⁰ um/min.

The existence of such a flow even in absence of a substrate is suggested ^{in some} cells such as

neutrophils ^{which can} show surface capping ^{when in} suspension.

5. External fluid flow and cell motion.

The hydrodynamic lipid membrane backward flow interacts with the outer medium. In

general, ^{cells are in contact with a substrate or with an} extracellular matrix. It is a common

hypothesis ^{to assume that} receptors ^{in the membrane which are carried} along ^the lipid ^flow

may bind ^to the substrate or to the matrix and therefore form a

foot

which provides ^the

thrust for forward motion [15, 2] ; the backward motion of the membrane with respect ^{to the} cell becomes equivalent ^{to a} forward motion of the cell with respect ^{to the substrate.}

In the present model, ¹ want to stress that there is no real need of substrate for the cell to move. Indeed, by continuity of the fluid velocity ^{due to} viscosity leading ^{to a « stick »} boundary condition, the membrane lipid ^{flow must induce a} fluid flow in the surrounding

medium represented ⁱⁿ figure ^{1 as seen in} the cell frame. For a spherical cell of radius A and a

surface lipid ^flow given by equation (6), the external fluid flow can be calculated exactly following [16]. If V is the cell velocity (to ^be determined) along ^{Ox in a fluid at rest at} infinity,

the fluid velocity field reads [16]

where ⁿ

r/r, ^t is the unit vector tangent ^to the membrane along the direction of the lipid

backward flow and a and a2 ^{are two constants to} be determined from the following boundary

conditions :

Equations (10) ^and (11) ^{lead to}

The total force exerted by ^{the fluid on} the cell is then given by Stokes formula [16]

were 17 ^{is the} dynamical viscosity (n = 10 - 3 kg/s . m ^for water). ^{The minus} sign ⁱⁿ expression (13) ^{allows to recover} the usual result that a sphere moving ^with velocity

+ V in a fluid suffers a resistance to its motion, i.e. the fluid exerts a force on the sphere ^{in the} opposite ^{direction to the} velocity. However, if al becomes negative, ^{the reverse effect will}

occur as we now see.

The equilibrium ^state of the cell motion is determined from the condition that the cell is an

isolated system ^{and thus no extemal forces are} acting ^on it : this reads F

0 which implies

al

0. This determines V

(2/3) vo ^{as the cell} velocity. With the numerical values given above, ^a typical ^cell velocity ^{is V = 6} um/min which compares well with the values observed in vivo and in vitro [1-3]. This result suggests ^{that so-called} crawling ^{cell motions} usually

considered to occur via a specific cell-substrate interaction may well involve ^a non-negligible

interaction between the cell and the outer fluid. In other words, the thrust needed for forward motion can be taken from the fluid as well as from a solid substrate.

Let us give ^more precisions ^{on the} force distribution. First, ^it is easy ^{to see} that the fluid

pressure is uniform around the cell due ^to the fact that al

0 (see ^for example ^Ref. [16]). ^The dynamic force distribution exerted the fluid on the cell is given by

and [16]

With a1 = 0 and a2 = - A 3/2, ^{we have the following fluid velocity field using the} cylindrical symmetry ^{around the} direction of cell motion :

This yields finally ^the contribution of the quadrupolar velocity ^{field on} the force in the direction of cell motion :

The monopolar contribution is zero due to the condition al

0. We verify ^{that a} complete integration of the force field given by equation (17) ^over the cell surface gives ^a total force F

equal ^{to zero, as} already stated above.

In order to understand more precisely ^the origin of the thrust given by ^{the fluid on the} cell,

suppose that ^at time _zero, the cell, initially ^{at rest,} suddenly established its membrane lipid

flow. At first, ^its velocity V is small but steadily ^{the cell} accelerates until it acquires ^its equilibrium velocity ^V

(2/3) vo. This scenario is obtained directly ^on examination of

equations (12a) ^and (13). ^As long ^{as the cell} velocity V is smaller then (2/3) vo, ^the coefficient al is negative and therefore the total force exerted by ^{the fluid on} the cell is positive ^{in the}

direction of the cell motion. This means that the cell receives a positive accelerating ^thrust

from the fluid which stops ^{as the} equilibrium velocity ^V

(2/3 ) vo is reached. This is indeed a true equilibrium ^{since a} larger ^{V would} change ^the sign of al ^and produce ^{a force} opposing ^the

cell motion which would bring ^{it back to its} equilibrium velocity. ^{In the case where} V # (2/3 ) vo, ^the complete spatially dependent force field can be obtained from a derivation

parallel ^{to that} leading ^to equation (17). ^{Let us focus on the} monopolar component ^{which is} the only ^{one which} gives ^a non-vanishing total force [16] :

To the dynamic ^force distribution exerted by ^{the fluid on the cell} given by equation (14), ^one

must add the static pressure contribution which ^{no more} vanishes for al = 0. The result for the force along the direction of cell motion is :

With equation (12a), ^this yields

The fluid thrust on the cell is thus proportional ^to the factor [ (2/3 ) vo - V ]. ^The positive sign

of f,, ^{means that when} (2/3 ) vo

V, ^the fluid indeed accelerates the cell. The force field (20)

is represented ⁱⁿ figure ^4.

Fig. 4. - Case when V (2/3 ) vo. ^The monopolar force field is represented ^{as a} function of the lateral

angle ^{8. The other} quadrupolar contribution averages ^{to zero over} the whole cell surface and thus does not contribute to the thrust provided by ^{the fluid on} the cell. The positive sign ^of f x ^{means that when}

(2/3 ) Vo > V, the fluid indeed accelerates the cell.

In fluid motion at such low Reynolds ^{number as Re}

AV / v ~, ^{10- 5} (where v = q /p ^is

kinematic viscosity ^{and p} is the fluid density), inertia is totally irrelevant [6]. ^{This means that,}

if the cell motor stops suddently, ^{the cell} stops almost instantly (within = ^0.3 >sec) ^after coasting ^{over a distance no} larger ^{than 0.1} À ! This implies ^{that the} direction of the cell motion is completely determined by the solute gradient and the response of the cell motion ^to any change in its internal gradient is immediate. Of course, the cell may ^not be able to change

its gradient ^so rapidly but this is a different story (see ^Sect. 7). ^{In sum,} crawling ^{over a solid or}

«

swimming

^{in a fluid at} very low Reynolds number is very similar with respect ^{to the thrust} needed for forward motion.

The lipid membrane flow should also induce a circulating fluid flow within the cell. If the fluid inside the cell had the viscosity ^{of water, its} typical ^{order of} magnitude ^{would be} again given by Vo = 2 w 10- 2 Vves which is much smaller than Vves: ^{vesicles would not feel the} presence of this flow. However, this order of magnitude is very much overestimated since the

cytoplasm ^{is a} complex viscoelastic fluid [17] ^which impedes notably internal fluid motions.

6. Power efticiency.

In order to sustain the solute concentration gradient against spontaneous ^diffusion j

^D VC, where D is the solute diffusion coefficient, ^{the cell must} continuously spend

some power Posm. ^The ideal gas law shows that the energy spent ^for bringing n ^solute

molecules from a pressure po ^{to a} pressure p is

With p

RT (Co ⁺ aA ), po

RTC o from van’t Hoff equation and with n = 7T A 2 j solute

molecules transported per unit time, the power spent by ^{the cell to} maintain the solute

concentration is

in the limit of small aA/Co- Only a fraction of this power is used ^to propel the vesicle. The power spent per vesicle is given by Stokes formula 8 7TTJaÍ (vves)2, ^where a1 is the coefficient of the r-1 term in the fluid velocity ^{field around a vesicle} given by ^an equation of the form of

equation (9). ^It is determined from the boundary conditions [13] :

wich give ai = 6 a/5. ^{For a} total of 4 TTA 3 N /3 ^vesicles, the power efficiency ^{of the} (osmotic pressure) -> (vesicle flux) transduction is e

{4 TT A 3 N /3} ⁴⁸ TT ’YJa (vves)2 /5 Posm ^which

reads

where we have used 4 TT A 3 N /3 = X (A / a )2. Note that e is independent of the solute concentration gradient ^a. Taking ^D

10- 8 cm2/sec, ^X

² %, ’n = 10- 3 gls.m, ^{we obtain}

e ~, 2 x 10- 4 which is small by industrial standards ! In fact, ^{it turns out that} efficiency ^is really

not the primary problem of cell motion. For cells with a specific motility apparatus ^{such as} flagella, e ^is of the order of 10- 3 [6]. This is very low but ^not unphysical for cells. Crawling efficiency appears ^to be only slightly ^{worse than} specific motility efficiency [18] !

The power carried by the vesicles is in turn transformed into viscous dissipation ^{in the}

backward lipid membrane flow and in fluid flow in the outer medium.

7. Concluding ^remarks.

A physical ^{model of a certain} type ^of crawling motion of cells without flagella ^or cilia has been

proposed. ^Some concluding ^{remarks are of order.}

1) ^The mechanism for the vesicle flux (discussed ^{in Sect.} 3) ^is independent ^{from the} argument for the establishment of the lipid membrane flow by exocytosis (Sect. 4) ^{and the} corresponding cell motion (Sect. 5) : section 4 and section 5 remain valid (with appropriate

numerical modifications) ^{if the « motor » does not function as a}

pump » but rather ^{as a}

«

rolling foot-path » [5] for the vesicles involving ^the cytoskeleton.

2) If the « pump » hypothesis ^{is correct,} what is the mechanism by which the solute

gradient concentration is established ? It is well known that many functions in biological ^cells operate by setting up concentration ^or ionic gradients. ^These gradients ^almost always ^involve

membrane (external ^and internal) components of the cell. It has been suggested [14] ^{that the} endoplasmic reticuluum might qualify ^{has a} relevant « pump ^». In fact, the notion that osmotic driving ^{forces are} involved in extensions of cell bodies is not new and physical analyses of such mechanisms already ^exist (see ^for example ^Ref. [20]).

3) ^{A number of} observations, collectively ^termed persistence phenomena, suggest ^{that cells}

can retain a memory of migration for many minutes. Persistence phenomena ^{are difficult to}

account for if the cell migration is actin-mediated [2]. On the other hand, ^the

pump

^motor model proposed here is in agreement with the existence of a memory effect. It corresponds ^to

the time ^T necessary for the solute gradient (1) ^{to relax to a} uniform concentration profile :

is the time taken by ^a solute molecule to cover a typical distance A. With D = 10- 8 cm2/sec

and A = 10 _um, ^{T .~} 1 min whereas for A = 100 _um,

3 h !

4) Equation (8) together with the result of section 5 (V

2/3 vo) predict that the cell

velocity ^{V is} independent of the cell type ^as long ^as the fraction X of the cell membrane which is in the form of vesicles is independent of the cell. This is in agreement ^{with the} observation that velocities of many different types ^{of cells are} quantitatively similar : 9 um/min ^{on Ascaris} extract, ^6-11 um/Bmin for nematode sperm cells, ^15-25 ktm/min ^for leukocytes, ^6-12 um/min

for Fundulus « deep » cells, ^6-14 um/min ^for Dictyostelium mucoroides... (see [19] ^and

references therein).

5) Crawling ^{in a} fluid via the backward membrane flow mechanism is reasonable if Brownian motion can be neglected. The dimensionless number Dcell/V A, ^where Dceii

kT/6 ^7T7JA is the diffusion coefficient of a spherical cell of radius A, weights ^the importance

of cell Brownian diffusion with respect ^to the directed cell motility. With V .- 10 um/min, we

find Dceu/V A = ^{10- 3 for A}

^{10 um and 10-1 for A = 1 iim} which shows that Brownian motion can indeed be neglected.

6) ^The discussion has been restricted to the case of a very simple spherical ^cellular shape.

Also, ^an implicit hypothesis ^{is that} exocytosis-endocytosis processes do ^not change ^the shape

of the membrane. This is valid in the limit of small vesicles fusing slowly, ^with rapid lipid ajustment. ^In reality, ^the finite size of the vesicles and the finite lipid flow lead to the appearance of ^a ruffling edge ^at the front of the cell, signaling that this is the place ^{where the}

«

action » in the cell motion takes place. ^This limitation in our model should not change ^the

main points of this paper. In any case, this « growth » process involving ^{« surfaces} patchworks » ^{seems a} promising field for future researches.

7) The « pump » mechanism proposed in section 3 does not involve any kind of chemotaxis. Indeed, ^{in a neutral} medium, the cell may choose ^{to move} by setting up ^an internal solute concentration gradient which is the source for vesicle motion. This is different from an osmotic force acting directly upon the cell when embedded in ^a solute concentration

gradient.

8) The naive views which have been presented here may ^not be relevant for most biological

cells. However, Nature shows an incredible variety of cell locomotion mechanisms : it would be surprising indeed if this one has nothing ^to do with that chosen by ^some organism. ^{Let us} hope ^{that the} present ideas will stimulate a reassessement of models of motion without cilia or

flagella.

Acknowledgments

It is a pleasure ^to acknowledge stimulating discussions with J. Febvre, ^J. Pouyssegur, ^G.

Romey ^and C. Sardet and to thank J. P. Boon and N. Ostrowsky ^{for a critical} reading ^{of the} manuscript. ¹ dedicate this work to Paul-Emmanuel Sornette whose birth coincided with that of the ideas presented ^here.

References

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