Marangoni transport in lipid nanotubes

EUROPHYSICS LETTERS

15 April 2005

Europhys. Lett., 70 (2), pp. 271–277 (2005) DOI: 10.1209/epl/i2004-10477-9

Marangoni transport in lipid nanotubes P. G. Dommersnes 1 , O. Orwar 1,2 , F. Brochard-Wyart 1 and J. F. Joanny 1 1 Institute Curie, UMR 168 - 26 rue d’Ulm, F-75248, Paris Cedex 05, France 2 Department of Physical Chemistry, and Microtechnology Centre Chalmers University of Technology - SE-412 96 G¨ oteborg, Sweden received 11 August 2004; accepted in final form 23 February 2005 published online 23 March 2005 PACS. 87.16.Dg – Membranes, bilayers, and vesicles. PACS. 47.85.Np – Fluidics. Abstract. – We give a simple picture of transient and stationary transport in lipid nanotubes connecting two vesicles, when a difference of membrane tension is imposed at time t = 0, either by pressing one vesicle with a micro-fiber, or by adding a surplus of membrane lipid. The net result is a transport of membrane from the tense towards the floppy vesicle. In the early stage, the tube remains cylindrical, and the gradient of surface tension gives rise to two opposite flows of the internal liquid: a Marangoni flow towards the direction of high tension, and a Poiseuille flow (induced by Laplace pressures) in the opposite direction. At longer time, the tube reaches a stationary state, where curvature and Laplace pressure are balanced. Marangoni flows dominate for giant vesicles, where Laplace pressure is negligible.

Introduction. – The use of miniaturised automated systems such as “lab on-a-chip” is rapidly expanding and has a strong impact on various areas of research such as genomics, diagnosis in medicine, biochemical reactions, and analysis. Microfluidic motions can be driven by Marangoni effects: the surface tension of liquids depends on temperature, and a temperature gradient creates a surface tension gradient. For a film of thickness e, a shear flow Vx (z) in the direction of the thermal gradient is established Vx (z) = ηz dγ dx . The surface velocity is

1 Vs = ηe dγ dx , and the liquid flux J = 2 eVs . The Marangoni effect allows to direct microscopic flows on patterned surfaces [1]. We focus here on a different form of Marangoni flows [2, 3], used in nanofluidic networks composed of nanotube-conjugated vesicles made from phospholipids. In these networks it is possible to vary the lipid composition, the electrical charge and the tension of the membrane. The connectivity, the container size, the radius and length of the tubes and the angles between the tubes are entirely controlled. The containers within such networks can be chemically different. Materials have been routed between two containers connected by a common nanotube (fig. 1). The principle is to create a membrane tension difference between the two vesicles, e.g., by microinjection of liquid, by squeezing one vesicle with a micro-fiber or by adding an excess of lipid. Our aim here is to discuss the transport induced by a gradient of membrane tension applied along a lipid nanotube connecting a floppy and a tense vesicle. When a tension difference is imposed between two vesicles, a tension gradient is established within a few milliseconds. We call this the first transient state. The connective tube progressively adjusts its shape during the following seconds and reaches a stationary state. We focus first on the fast transient regime: we discuss the Marangoni and the Laplace flows inside the tube for a cylindrical liquid tube of constant radius, submitted to a surface tension gradient. In the last part, we describe the final stationary state, where the shape of the tube has relaxed.

c EDP Sciences


272

EUROPHYSICS LETTERS

σ1

JM

σ2

Fig. 1 – Marangoni transport between vesicles: by pressing a vesicle with a micro-fiber, a flow towards the tense vesicle is created (σ2 > σ1 ).

Structure of tethers. – A point-force acting on the fluid membrane of a vesicle (or a living cell) can extrude a lipid tube [4, 5]. If the tube is extruded by micropipette manipulation [6], the membrane tension σ is well controlled. A steadily growing tube appears beyond a threshold force fc , first calculated in [6]. The free energy of the extruded nanotube is the sum of curvature and surface free energies: 1K F = rL + σrL, 2π 2 r2

(1)

where K is the bending rigidity, r the tube radius and L the tube length. The pressure inside the tube is obtained by the derivative of the free energy, eq. (1), with respect to the volume Ω = πLr2 at constant length: σ 1K . (2) ∆P = − r 2 r3 The pressure in the two compartments at coexistence is equal. In the vesicle, P ≈ where R is the vesicle radius. The equilibrium tube radius is therefore
K rc = . 2σ

2σ R

≈ 0,

(3)

It is of order rc ≈ 100 nm for K = 4 · 10−20 J and σ = 2 · 10−6 Nm−1 . The threshold force fc is derived from the tether energy   ∂F 1 f 3K = + σr. (4) = 2π 2π∂L Ω 4 r 2 √ At the equilibrium radius the threshold force is fc = 2π 2Kσ. This relation has been confirmed experimentally by pulling tubes with optical tweezers, varying the vesicle tension through pipette aspiration in the range from 10−3 to 0.5 mN/m [7], and also by following the retraction of a tube when the force is suddenly suppressed [8]. Vibrational modes of a tube. – The equilibrium nanotube has two types of modes, calculated in the appendix: i) The Lucassen mode associated to fluctuations in lipid density [9]. For a fluctuation of wave vector q, the relaxation rate reads τ1σ = Dσ q 2 , with an effective diffusion constant Dσ = Er η , where E is the elastic modulus associated to the stretching of the fluctuating membrane. For an infinite membrane, E = 8πKσ/kT [10]. For a tube, E is even 3 larger, E ≈ 4σ 8πK kT ≈ 10 σ, as shown in the appendix. ii) The peristaltic mode associated to

P. G. Dommersnes et al.: Marangoni transport in lipid nanotubes

273

fluctuations in the tube radius r. A periodic fluctuation relaxes with a law τ1r = Dr q 2 , with an effective diffusion constant Dr ≈ σr 8η . As σ is much smaller than E, τr can be thousand times larger than τσ . This means that whereas surface tension gradients relax very fast, the equilibration of the tube radius is slow. If q ≈ L1 , τr ≈ ηL σr ≈ few seconds for L = 10 µm, σ = 10−6 Nm−1 , whereas τσ ≈ 1 ms. In the description of flows induced by Marangoni gradients in a tube, the surface tension first relaxes at constant tube radius (transient regime), then the tube radius relaxes to the equilibrium shape (stationary regime). We now consider these two stages separately. Communication between vesicles. – Figure 1 shows a setup to induce tension-driven flow between two nanotube-conjugated vesicles [11]. Initially, the system is at equilibrium, the membrane tension of the vesicles and of the tube are uniform. By pressing one of the vesicles with a micro-fiber, a surface tension difference is created between the two vesicles. The tube membrane is liquid and a flow of lipids is created from the region of low tension to the region of high tension. The fluid and any material trapped inside the tube are dragged along the lipid flow. A tension gradient can also be achieved by micropipette aspiration of one vesicle [11], by addition of excess membrane [12], or by liquid injection [2] to increase the internal pressure. Similar Marangoni flows might be involved in transport and cell/cell communication: the transport between the endoplasmic reticulum and the Golgi apparatus is partly achieved through nanotubes pulled by molecular motors [13]. Tension differences are believed to produce active convective flows rather than diffusive flows between the two compartments. Nanotubes also seem to play a role in secretion. During exocytosis, nanotubes connecting the small vesicle to the cell membrane are involved in the delivery of the container. Looking back at fig. 1, a first guess would be that pressing a vesicle leads to expulsion of the internal liquid! But the Marangoni flow goes in the opposite direction, since the tension in the pressed vesicle increases. Our aim here is to discuss in detail the flow in the tube driven by the gradient of membrane tension. We focus here on long nanotubes. The opposite case of very short junctions between two membranes in close contact with different tensions has been theoretically investigated, and the complex flow patterns analysed numerically [14]. Marangoni flows for a cylindrical tube. – In the initial state, the two vesicles and the tube are at equilibrium. The tension of the vesicles is σ and the tube radius is constant and given by eq. (3). Suddenly, the membrane tension of vesicle 2 is increased by pressing it with a microfiber, whereas vesicle 1 is in its initial state. When the membrane tension increases faster than the equilibration time for the tube radius, a pearling instability can be observed [15]. As 3K shown in [16], a modulation of the tube radius is unstable when σ + ∆σ > 2r 2 = 3σ, showing that bending opposes the pearling instability. We focus here on relative variations of surface tension always smaller than the critical value for pearling ( ∆σ σ < 2), which is the practical case for Marangoni-driven transport. The opposite case of very large tension increase gives rise to phenomena such as pearling front-propagation and moving pearls, as analysed in [17–19]. At times shorter than the tube relaxation time τr ≈ few seconds, the tube is frozen in its initial state with a constant radius rc . On the other hand, the surface tension gradient is established rapidly (τs ≈ ms). It gives rises to two opposite flows pictured in fig. 2 a plug flow dσ and a Poiseuille flow. A tension gradient dσ dx gives a driving force per unit length 2πr dx . This force induces a plug flow inside the tube V = Vs , and also some back-flow in the water outside of the cylinder. The balance between capillary driving force (derived from eq. (4), with K and r constant), and friction force (drag of a cylinder [20]) leads to dσ 4πηVs . = πrc L 1 dx log r − 2

(5)

274

EUROPHYSICS LETTERS

a)

b)

JM

JP

Fig. 2 – Lipid tube submitted to a surface tension gradient: a) Marangoni flow; b) Poiseuille flow.

The surface velocity is comparable to the Marangoni velocity for a film of thickness equal to the −1 Nm−2 , rc = 0.1 µm, η = 10−3 P, radius of the tube. Typical experimental values are dσ dx = 10 which gives Vs ≈ 10 µm/s, in agreement with experimentally measured velocities [11]. The velocity increases logarithmically with tube size L due to hydrodynamic screening. The plug flow inside the tube corresponds to a liquid flux,   πr3 1 dσ L . (6) JM = πrc2 Vs = c log − 4η rc 2 dx For the transient period under consideration, the radius rc is constant, but the Laplace pressure rσc varies along the tube due to the tension gradient, and induces a Poiseuille flow in the 1 opposite direction (shown in fig. 2b). The velocity vx (r) is given by vx (r) = 4η (rc2 − r2 ) r1c dσ dx and the resulting liquid flux JP is JP = −

π 3 dσ r . 8η c dx

(7)

During the early stages (the first second), where the tube remains frozen in its initial configuration, the liquid flux J has therefore two opposite contributions, J = JM + JP . We can check that Marangoni flows overcome Laplace flows, as soon as rLc > 3. The opposite case corresponding to short connections between membranes is studied in [14]. It may be important during exocytosis, where small tense vesicles eject their content. The surfactant flux on the membrane (JS = 2πrc Vs ) can also contribute to the transport of adsorbed solute. Stationary “Marangoni” flows. – After a few seconds, the tube connecting vesicle 1 (radius R1 , membrane tension σ1 ) and vesicle 2 (radius R2 , tension σ2 ) reaches a quasi-static σ1 K 1 trumpet-like shape (profile rc (x)) (fig. 3). For x = 0, the pressure is P1 = 2σ R1 = rc − 2r 3 and for x = L, P2 =

2σ2 R2

=

σ2 rc2

−

K 2rc3

2

1

c1

. The tube profile rc (x), the tension σ(x), the tube

velocity Vs (x) and the liquid flux in the tube are derived from the following equations, with the boundary conditions for rc (x) and p(x) at x = 0 and x = L. – Laplace/curvature pressure p(x): p(x) =

K σ(x) − . rc (x) 2rc3 (x)

(8)

– Conservation of lipid flux Js : Js = rc (x)U (x) = rc1 (x)Vs1 = rc2 Vc2 . 2π

(9)

P. G. Dommersnes et al.: Marangoni transport in lipid nanotubes

σ1

σ2 U(x)

p1

275

r(x)

Js p2

J x f(x)

dx

f(x+dx)

Fig. 3 – Stationary state of a lipid tube connecting tense and floppy vesicles.

– The balance of mechanical and friction forces acting on a tube element dx (fig. 3):   d d 3K 1 f (x) = + σr = ˜2ηVs , dx dx 4 r 2

(10)

where we use the notation ˜ 2 = 2/(log rLc − 12 ). The force contribution from the membrane surface viscosity ηs is negligible, a simple scaling argument gives the ratio between membrane and η 2 −5 , for surface viscosity ηs ∼ 1 µm and typical tube sizes. bulk dissipation ∼ ηηs Lrc2 ( ∆σ σ ) ∼ 10 – Conservation of liquid flux J = JM + JP : πrc2 Vs −

πrc4 dp = J. 8η dx

(11)

1 We now focus on the case σ2σ−σ < 1, where eqs. (9)-(12) can be linearized. The membrane 1 surfactant flux JS deduced from these equations is    rc2 P2 − P1 σ2 − σ1 2π − rc rc JS = . (12) ˜ L 2 L 2η

The internal liquid flow reads     πr3 σ2 − σ1 πr4 1 1 P2 − P1 dσ rc2 dp πr2 πr4 dp − = c − c + . J = c rc − c ˜ ˜2η dx 2 dx 8η dx L 2η 4 2˜ L 2η

(13)

For giant vesicles, “GUV”, the vesicle radius is in the range of a few microns, and the pressure gradient is negligible (the second term in eq. (A.1) is of order r0 /RV smaller than the Marangoni contribution). For small vesicles, “SUV”, where the radius is comparable to r0 , the two terms are important. Conclusion. – Transport in lipid nanotubes is a combination of a membrane flow induced by a gradient of membrane tension and an internal liquid flow which is the sum of a plug flow induced by the membrane flow and a Poiseuille flow induced by a pressure difference between the two vesicles. For large vesicles, the vesicle internal pressures are extremely weak and the dominant effect is the surface tension gradient (Marangoni). We discussed here both the transient regime where the tube is frozen in its initial state and the stationary flow in the long-time limit, where the tube has relaxed to its equilibrium shape. There is reasonable hope for microfluidic setups where the content of one living cell can be modified by exchange

276

EUROPHYSICS LETTERS

with a vesicle reservoir, or with another cell, through nanotubes. If we want to transfer a fraction f of the volume from the emitting vesicle, the time required for transfer, as deduced from eq. (13) is t ≈ ( rR0 )3 σ2Lη −σ1 f . This time is in the range of minutes for GUV vesicles. Taking R = 5 µm, rc = 50 nm, L = 10 µm, f = 0.1 and a rather high difference in tension σ2 −σ1 ≈ 10−5 J/m2 , we obtain t ≈ 100 s. Many practical problems remain due to the physical and chemical vulnerability of the nanotubes and their possible contamination by membrane proteins from one living cell. Our discussion may not be valid for exocytosis, where specific proteins drive the process. ∗∗∗ PGD was supported by a Marie Curie fellowship. OO acknowledges the receipt of a Rotschild-Yvette Mayent-Insitute Curie fellowship. Appendix Fluctuations of lipid tubes. – The two modes are depicted in fig. 4. The peristaltic mode is associated to a modulation of the tube radius. The “Lucassen mode” is associated to the fluctuation of the lipid density. Peristaltic mode: The radius of the tube undulates r = rc + uq eiqx−t/τ . The pressure gradient

dp dx

=

2σ0 dr rc3 dx

rc dp induces a flux J of the internal liquid J = −π 8η dx . Volume conservation

2 implies 2πrc ∂u ∂t +div J = 0. This leads to a diffusion equation for the modulation ∂t u = Dr ∂x u, with a relaxation time 1 rc σ0 . (A.1) = Dr q 2 , Dr = τr 8η

Lucassen mode: A fluctuating membrane is characterised by an elastic modulus E associd ∆A d ∆Γ ated to membrane stretching, defined by E1 = − dσ A = − dσ Γ , where A is the projected area, ∆A the excess area and Γ the lipid density. A fluctuation of Γ (Γ = Γ0 + Γq eiqx−t/τ ) leads to a gradient dσ dx of the membrane tension, which gives rise to a flow (velocity U ) of df ˜ lipids 22πηU = dx = πrc dσ dx . Conservation of lipids implies 2πrc ∂Γ ∂t + div Js = 0, where Js = 2πrΓc U . This results in a diffusion equation for lipid density ∂t Γ = Dσ ∂x2 Γ. The relaxation time τσ is given by 1 = Dσ q 2 , τσ

Dσ =

Erc . 2˜2η

(A.2)

The modulus E has been calculated for a large fluctuating membrane of dimension L × L, from the amplitude of the membrane fluctuations. The excess area for a fluctuation of wave

a)

b) r(x)

Γ (x) rc

x Fig. 4 – a) Peristaltic mode; b) Lucassen mode.

x

P. G. Dommersnes et al.: Marangoni transport in lipid nanotubes

277

∆A

vector q is A q = 12 q 2 u2q , where u2q is derived from the free energy by equipartion of energy,  F = q 21 (Kq 4 + σq 2 )L2 u2q = 12 kT , which leads to an excess area ∆A 1 kT = A 2(2π)2 K



2π a 2π L

   2 2π 2πq kT σ log . dq 2 − log σ ≈ q +K 8Kπ a K

(A.3)

1 ∆A kT The resulting osmotic modulus is E1 = − dσ A = 8πKσ . For usual values of the curvature Dr 8πK kT ≈ 16πK ≈ 104 . modulus (K ≈ 10kT ), one notices that E ≈ kT σ ≈ 103 σ. The ratio is D σ For a lipid tube, E is even larger. We can picture the tube by a ribbon of length L and width 2πr. The wave vector of the eigenmodes, using periodic boundary conditions, can be n2 n2 2 2 written as q 2 = m2 ( 2π L ) + r 2 ≈ qx + r 2 , which leads to an excess area

∆A = A



2π a 2π L

n

max n=1

1 1 kT dqx , 2 2 2 2 K(2π) r qx + (n + 12 ) 2σ K

(A.4)

where nmax = 2πr/a. After integration on qx we get nmax 1 kT r

1 ≈ E 8πKσ a n=1 n2 + 1 + 2

4π 2 rc2 a2

≈

kT 1 . 8πKσ 8

(A.5)

REFERENCES [1] Kataoka D. A. and Troian S. M., Nature, 402 (1999) 794. ¨ mberg A., Ryttsen F. and [2] Karlsson A., Karlsson R., Karlsson M., Cans A. S., Stro Orwar O., Nature, 409 (2001) 150. [3] Karlsson R., Karlsson A. and Orwar O., J. Am. Chem. Soc., 125 (2003) 8442. [4] Waugh R., Biophys. J., 38 (1982) 129. ¨licher F. and Prost J., Phys. Rev. Lett., 88 (2002) 238101. [5] Derenyi I., Ju [6] Evans E. and Yeung A., Chem. Phys. Lipids, 73 (1994) 39. [7] Dai J. and Sheetz M. P., Biophys. J., 77 (1999) 3363. ´nyi I., Buguin A., Nassoy P. and Brochard[8] Rossier O., Cuvelier D., Puech P. H., Dere Wyart F., Langmuir, 19 (2003) 575. [9] Lucassen J., Chem. Eng. Sci., 27 (1972) 1283. [10] Servuss R. M., Harbich W. and Helfrich W., Biochim. Biophys. Acta, 436 (1976) 900. [11] Karlsson R., Karlsson M., Karlsson A., Cans A. S., Bergenholtz J., ˚ Akerman B., Ewing A., Vionova M. and Orwar O., Langmuir, 18 (2002) 4186. [12] Karlsson M. and Orwar O., Langmuir, 17 (2001) 6754. [13] Hirschberg K., Miller C. M., Ellenberg J., Presley J. F., Siggia E. D., Phair R. D. and Lippincott-Schwartz J., J. Cell Biol., 143 (1998) 1485. [14] Chizmadzhev Y. A., Kumenko D. A., Kuzmin P. I., Chernomordik L. V., Zimmerberg J. and Cohen F. S., Biophys. J., 76 (1999) 2951. [15] Bar-Ziv R. and Moses E., Phys. Rev. Lett., 73 (1994) 1392. [16] Bar-Ziv R., Moses E. and Nelson P., Biophys. J., 75 (1998) 294. [17] Goldstein R. E., Nelson P., Powers T. and Seifert U., J. Phys. II, 6 (1996) 767. [18] Goveas J. L., Milner S. T. and Russel W. B., J. Phys. II, 7 (1997) 1185. [19] Olmsted P. and MacIntosh F. C., J. Phys. II, 7 (1997) 139. [20] Happel J. and Brenner H., Low Reynolds Number Hydrodynamics (Martinus Nijhoff) 1983.