Fluctuations of a membrane interacting with a diffusion field

Transcription

1 EUROPHYSICS LETTERS 1 December 2002 Europhys. Lett., 60 (5), pp (2002) Fluctuations of a membrane interacting with a diffusion field F. Divet 1,T.Biben 1,I.Cantat 2, A. Stephanou 1, B. Fourcade 3 and C. Misbah 1 1 Groupe de Recherche sur les Phénomènes Hors-Equilibre, LSP Université Joseph Fourier et CNRS, Grenoble I BP 87, Saint-Martin d Hères, Cedex, France 2 GMCM, UMR CNRS 6626 Campus de Beaulieu Rennes Cedex, France 3 CEA, DSM/DRFMC - 17 av. des Martyrs, Grenoble Cedex 9, France (received 10 December 2001; accepted in final form 12 September 2002) PACS Kg Molecular interactions; membrane-protein interactions. PACS Ac Theory and modeling; computer simulation. PACS Dg Membranes, bilayers, and vesicles. Abstract. We analyze the dynamics and fluctuations of a phospholipidic membrane containing inclusions which may diffuse along the membrane as well as in the aqueous solution. A kinetic law of exchange between the membrane and the fluid is introduced. It is shown that the membrane may become morphologically unstable. In the stable regime the fluctuations kinetics are analyzed. Diffusion as well as desorption lead to dynamical laws for the fluctuation width which assumes different scaling laws with time (e.g., t 1/2, t ln(t),...) depending on the prevailing dissipation mechanism. The experimental analysis of dynamical fluctuations should give access to the microscopic physical parameters. Membranes and vesicles made of phospholipidic molecules constitute the focus of many researchers in different fields of physics, chemistry, biology and so on. They are regarded as simple systems on which to focus on some basic features which are met in complex entities. On theoretical grounds, the equilibrium shape of a pure phospholipidic vesicle as well as its fluctuation spectrum are fairly understood. Subsequent studies have focused on the interaction between exogenous particles or molecules (e.g., vesicle in contact with nanoparticles and colloids [1], adsorption of polymers [2], interaction between inclusions [3], and fluctuations induced by adsorption of particles for a review see [4]). Little is known, however, in the presence of inclusions that diffuse on the membrane with the ability of exchange with the underlying bulk phases (desorption). Inclusions are known to play several roles, such as inducing a shape transformation, affecting the membrane elasticity [5], or even altering the cell functionality [6]. For example, it is known that clathrin and its adapter can bind to liposomes and cause budding in the absence of ATP or other source of energy [7]. It is the aim of this letter to analyze the membrane fluctuation in the presence of inclusions that are not indefinitely bound to the membrane but may kinetically attach and detach [8]. c EDP Sciences

2 796 EUROPHYSICS LETTERS z Ψ ψ y h(x,y,t) x Fig. 1 A schematic view of the system under consideration. We shall use here indifferently the term inclusions or adsorbed molecules, in both cases we assume an induced curvature. We show that the coupling of the membrane and the diffusion field leads to a rich variety of dynamical behaviours. In order to gain some insight towards the basic phenomena, we restrict our attention to the situation where the system is in global equilibrium and the membrane has an infinite lateral extent. The extension to non-equilibrium situations as well as to vesicles will be reported on in the near future. We consider the situation (fig. 1) where inclusions (polymers, proteins,...) have the ability to diffuse laterally (along the membrane) as well as in the surrounding fluid, and that a permanent exchange between the bulk and the membrane occurs on time scales of interest. Let ψ denote the bulk molecule concentration. It obeys ψ + v ψ = D ψ, (1) t where D is the diffusion constant and v is fluid velocity. The particles may adsorb on the membrane with residence time τ, and diffuse with diffusion constant D s. The concentration field along the membrane obeys the following mass conservation equation: φ t + u φ = j + J n, (2) where u is the tangential component of the velocity of a fluid element moving with the membrane, and stands for the gradient operator along the membrane. J n = n D ψ is the net mass current at the membrane, which expresses the coupling between bulk and the membrane, where n is the normal vector to the membrane. Finally, j is the lateral mass current which is not only driven by the lack of mass homogeneity but is also due to fluctuation of the local membrane profile. This term is related to the membrane chemical potential µ by j = (D s /α) (µ) with µ = δf/δφ, andf is the membrane free energy which we write as (following [9]) F = 1 [ [ ( 2 2 h ) 2 ( )( 2Λ φ φeq 2 h )] + α ( ) 2 ( φ φ eq + β φ ) ] 2 dx dy. (3) The first term is the Helfrich free energy, where h(x, y, t) designates the membrane profile. The second term expresses the coupling between the curvature and the local concentration by taking the reference state as h =0andφ = φ eq. The coupling constant is Λ and has a dimension of a length. The last two terms are the usual Landau-Ginzburg terms. α and β are two constants which are taken positive in order to prevent a spontaneous segregation. Note that membrane (anisotropic) inclusions are known to repel each other via the membrane undulations [3], which is precisely accounted for by assuming α>0 and β > 0 (entropic

3 F. Divet et al.: Fluctuations of a membrane interacting etc. 797 interactions are negligibly small [10]). It will be shown that thanks to the coupling between the membrane and the adsorbed molecules, the adsorbed molecules can undergo a spontaneous segregation, which can be formulated in terms of an effective attraction. Finally, one of the most serious points to be emphasized is to express how the net mass current at the membrane is related to the adsorption and desorption processes. This part is inspired from ref. [11]. For a flat membrane the current would just be given by J n D ξ ψ φ τ ; (4) ξ is the characteristic diffusion length of the molecules in the bulk and τ is the residence time of the molecules on the membrane. From a dimensional analysis D/ξ is a frequency multiplied by an atomic distance. D Ua k ξ νae BT, where a is a molecular length, ν a molecular frequency, and U a is the adsorption barrier. The quantity D s τ is the typical excursion length of an adsorbed molecule on the membrane before it desorbs [11]. At equilibrium the mass current vanishes, entailing that the bulk and membrane concentrations are related by φ eq =(τd/ξ)ψ eq. In reality, the chemical potential of the membrane is not only affected by the concentration field, but also by the membrane configuration. Thus the full mass current must, for a nonplanar membrane, have the following form: J n = D ξ ( ) 1 δf ψ ψeq τα δφ = D ξ ψ 1 ( αφ 2 β φ Λ 2 h ). (5) ατ At equilibrium, ψ = ψ eq and δf/δφ = 0, this dictates the presence of ψ eq as compared to (4). Both expressions (4) and (5) are equivalent as recognized from the second part of eq. (5) for a homogeneous distribution. Any motion of the membrane is accompanied with a fluid flow. In the small Reynolds number the velocity field v obeys the Stokes equation supplemented with the incompressibility condition η 2 v = p, v =0. (6) Finally we impose that the fluid velocity v is equal to the membrane one (this is true as long as the membrane is impermeable to fluid flow) and that the forces balance at the membrane: n j T ij n i = δf δh, t jt ij n i =0, (7) where n i and t i are the i-th components of the normal and the tangent to the membrane, and T ij pδ ij + η( i v j + j v i ) is the liquid stress tensor, see [12]. Since we are interested in small fluctuations, we linearize the set of equations with respect to the membrane amplitude h(r), with h e iq r+ωt and similar expressions for any other quantity. After a lengthy calculation [13] we arrive at [ ωφ q = D s q 2 Qξ + ][[1+ βα ] ]φ τ(1 + Qξ) q2 q + Λq2 α h q, (8) and ωh q = q3 4η h q Λq 4η φ q, (9) where φ q and h q are the Fourier amplitudes of the membrane concentration and its profile. The requirement that the solution be nontrivial leads to the dispersion relation. Note that

4 798 EUROPHYSICS LETTERS Q = q 2 + ω/d and that we impose ψ = ψ eq at z = z, with z q 1. ω 1 and ω 2 will denote the two roots of the dispersion relation. For extreme values of q simple expressions are obtained. For small q we find two modes, ω q 3 and ω q 2. The latter mode is nothing but the bulk diffusion mode which reads ω = Dq 2 and is stable, while the second mode takes more precisely the form ω q3 4η ] [1 Λ2. (10) α For Λ = 0, we recognize the classical bending mode of a free membrane damped by bulk viscosity [14]. That mode expresses the balance between the viscous tension and the Helfrich restoring force. The coupling between the membrane and the concentration field changes the effective rigidity by a factor (1 Λ 2 /α). Thus it is visible that if Λ > Λ c α/ the flat membrane becomes unstable [9]. This instability is easily recognized from the free energy (3). Indeed a minimization with respect to φ yields 2αδφ β 2 (δφ) = Λ 2 h (with δφ φ φ eq) or in Fourier space δφ q = Λq 2 /(α + βq 2 )h q Λq 2 h q. Reporting in the free energy, one gets in Fourier space F eff =(/2)[1 Λ 2 /α]q 4 h q 2, which shows clearly the instability. The physical origin of the instability is that molecules prefer curvature due to their affinity with the membrane (coupling term), that lowers the energy. Local segregation of these molecules enhances this effect. At large enough q, we find two oscillatory damped modes; the real parts of their pulsation are given by ω r = D s (β/α)q 4 and ω r = q 3 /4η. Note that though the bare diffusion and hydrodynamical modes are overdamped, oscillations occur due to coupling (it suffices to consider two modes X and Y described by Ẋ = X + ay and Ẏ = Y + bx and one sees that if ab < 0 one has damped oscillations). The physical reasons are the following: For example if a concentration inhomogeneity takes place on the membrane (just by transporting molecules along the membrane), then there are two channels for relaxation: i) desorption and ii) membrane diffusion. If desorption takes place on a faster time scale then, ahead of the protuberance, the bulk concentration increases, causing thereby a diffusion short circuit through the bulk and a membrane enrichment elsewhere. This induces a slowing-down of the protuberance. If all the effects operate on comparable time scales, the relaxation of the protuberance may operate in an oscillating fashion. The first mode is nothing but membrane lateral diffusion. Indeed, at large q the β term in the free energy dominates. Of course, for intermediate q values both desorption dynamics as well as usual lateral diffusion play a role, as will be discussed below. At long enough time thermodynamic equilibrium is reached. The corresponding time is referred to as the saturation time. For an observation in a time interval which is small in comparison with the saturation time membrane fluctuations should reveal a rich variety of kinetic phenomena. These are related to various dissipation mechanisms. The experimental study of these kinetic phenomena constitute an interesting tool to probe the microscopic degrees of freedom in which energy is dissipated. The general discussion of kinetics may be achieved only with the help of a numerical study. However, in order to gain some insight towards the physical mechanisms we restrict ourselves to the quasi-steady approximation of the bulk diffusion field. This means that diffusion in the bulk is fast in comparison with lateral diffusion and to the time scale on which the membrane configuration changes. This limit amounts formally to setting Q = q in the above dispersion relation. In general, the time and the spatial scales on which different mechanisms operate are quite distinct, and it is under this condition that one can derive scaling laws for the membrane fluctuations. The dispersion relation has two roots that we call ω 1 and ω 2. Four independent

5 F. Divet et al.: Fluctuations of a membrane interacting etc. 799 length scales can be identified which are related to the following crossover wave vectors: q 0 1 ξ, q 1 [ ] 1/3 4η, q 2 1 τ Ds τ, q 3 α β. (11) The first length is the bulk diffusion length, that is close to the membrane; it represents the excursion length before a bulk molecule has a chance to meet the membrane. The last two lengths are obvious, the diffusion length before desorption, and the Landau-Ginzburg wall extent. The second length results from a compromise between desorption time and the time needed for the fluid to flow. At larger lengths hydrodynamics dominates, while at smaller lengths desorption is the limiting dissipation mechanism. Thus, length scales mixing prevents from isolating dissipation mechanisms. Under wide conditions length scales associated with various phenomena are quite distinct [13], and it becomes legitimate to assume lengths separation. Without loss of generality, we assume the following hierarchy: q 0 q 1 q 2 q 3, and any other inequality can be treated along the same lines. Thus, depending on spatial scales, and thus time scales, some phenomena will be prevailing over others. One finds ] [1 Λ2 q 3, q q 1, 4η α 1 τ, q 1 q q 2, ω 1 D s q 2, q 2 q q 3, β D s α q4, q 3 q q3q 2 2/q 2 1, 3 4η q3, q3q 2 2/q q, 1 ω 2 ξ τ q, q q 0, τ, q 0 q q 1, 4η q3, q 1 q q3q 2 2/q 2 1, 3 β D s α q4, q3q 2 2/q q. (12) For example, if one probes the fluctuation frequency w 1 in the spatial interval q 1 q q 2, one should observe a fluctuation which is limited by desorption of molecules, while in the interval q3q 2 2q q that fluctuation would be limited by hydrodynamical flow, and so on. In reality, the dispersion relation contains two modes, w 1 and w 2, and the question arises of whether one is prevailing over the other. A quantity of practical interest which sheds light on this question is the height correlation function h q (t)h q(0), or the mean-square displacement (MSD) ( h(t)) 2 (h(t) h(0)) 2, which are obviously related to each other. Making use of the fluctuation-dissipation theorem, one can write this quantity as A 1 e w1t + A 2 e w2t, where A 1 and A 2 are nothing but the amplitudes of the modes associated with the eigenvalues w 1 and w 2, respectively. The MSD can be decomposed into two contributions, ( h(t)) 2 = ( h 1 (t)) 2 + ( h 2 (t)) 2, each associated to one mode. We introduce two characteristic times, t 0 ( 4η ) 4 ( ) 3 βds = q12 1 α q2 6 τ, t 1 β q6 3 = q2 2 τ. (13) αd s q 2 3

6 800 EUROPHYSICS LETTERS 15 log[2 πq 2 2 <( h(t)) 2 >/k B T] t t 1 t 2/3 τ <( h 1 (t)) 2 > <( h 2 (t)) 2 > log (t/τ ) Fig. 2 Roughness as a function of t. Parameters are chosen such that q i+1/q i =10 2 for i = 0, 1 and 2withq 2 1, and Λ 2 /α =0.1. Simple analytical expressions follow [13]: ( h1 (t) ) 2 k B T 2π 1 Λ 2 3 α D st, t t 0, 1 Λ 2 ( ) 4 α D Ds α st ln β t, t 0 t t 1, 1 Λ 2 ( ) D s t t 2 α 1 Λ 2 /α ln, t 1 t τ, τ 2 ( 3 Γ 2 ) ( ]) 2/3 1 [1 3 1 Λ 2 Λ2 t 2/3, τ t, /α 4η α where Γ(x) is the gamma-function, and 1 ( 2 Γ 1 )( ) 1/2 Ds β t 1/2, t t 0, 2 α ( h 2 (t) ) 2 k B T 2 ( 2π 3 Γ 2 ) ( ) 2/ Λ 2 t 2/3, t 0 t τ, /α 4η ( ) 2/3 τ, τ t. 4η (14) (15) It can be checked that t 0 is so small (molecular time scales) that pure lateral diffusion is ineffective. For large enough time (i.e. t τ), we obtain the classical exponent [15] 2/3 found in the case of a pure membrane where ( h 1 (t)) 2 is the leading amplitude, whereas ( h 2 (t)) 2 exhibits a plateau with time. The exponent 2/3 is due to the elasticity of the membrane and the viscosity of the fluid. Note that for small time (i.e. t t 0 ), we obtain either 1 or 1/2. The exponent 1 is related to a pure diffusive phenomenon of adsorbed molecules. There are two intervals for which the roughness is proportional to t ln t. This is related to membrane/bulk exchange. Such a kind of behaviour occurs also for membranes with active pumps [16]. We observe in fig. 2 a crossover between h 1 and h 2 at time τ. For the chosen parameters the behavior with the exponent 2/3 seems to prevail, but in the range where t

7 F. Divet et al.: Fluctuations of a membrane interacting etc. 801 is close to τ a regime with an amplitude behavior as t ln t may take over, or at least it will be difficult to disregard that contribution in comparison with the traditional one, since both effects are of comparable orders of magnitudes. Thus below but close enough to τ the width should be composed of a bi-exponential, one leading to a width square behaving as t 2/3 (limited basically by hydrodynamics) while the other branch would tend towards a t behaviour (up to a logarithmic correction). The latter branch expresses dissipation due to membrane diffusion. Above but close to t = τ the bi-exponential is related to hydrodynamics for mode 1 and to desorption and hydrodynamics for mode 2. Fits with a bi-exponential would provide a tool to access microscopic quantities such as desorption as well as membrane diffusion. Note that most of parameters (diffusion, desorption, etc.) are activated thermally and a wide range of values for the crossover spatial and temporal scales are possible. Our estimates show a variety of situations where the observations of the crossover are plausible. A full discussion will be given in an extended publication. In summary, we have developed a minimal model for interaction of macro-molecules with a phospholipidic membrane. We have shown that several important pieces of information about kinetic processes can be deduced from analyzing the dynamical fluctuation width. This letter has focused on fluctuations only. When the membrane is morphologically unstable, it is important to settle questions concerning the subsequent nonlinear dynamics. Most of experimental studies on fluctuations are performed on vesicles. It will be of great importance to extend the present study to closed membranes. We have neglected dissipation due a possible sliding of the two monolayers [15]. That dissipation becomes relevant when the fluctuation wavelength is small enough. A systematic comparison of the relative importance of this effect and those studied here will be presented in a forthcoming work. Finally, in order to have access to smaller time scales than those allowed with optical observations, it would be of great importance to study fluctuation of lamellar structures by means of dynamical X-ray scattering. REFERENCES [1] Lipowsky R. and Döbereiner H. G., Europhys. Lett., 43 (1998) 219. [2] Lipowsky R., Colloids Surf. A, 128 (1997) 255. [3] Park J. M. and Lubensky T. C., J. Phys. I, 6 (1996) [4] Manneville J. B., PhD Thesis, Université Paris VII (1999). [5] Safran S. A. (Editor), Statistical Thermodynamics of Surfaces, Interface and Membranes (Addison Wesley, New York) [6] Lodish H. et al. (Editor), Molecular Cell Biology (Scientific American Books, New York) [7] Takei K. et al., Cell, 94 (1998) 131. [8] Vitkova V., PhD thesis, University Rennes I (2002). [9] Leibler S., J. Phys. (Paris), 47 (1986) 507. [10] Marchenko V. and Misbah C., Euro. Phys. J. E, 8 (2002) 477. [11] Valance A. and Misbah C., Phys. Rev. E, 55 (1997) [12] Landau L. D. and Lifchitz E., Mécanique des Fluides (Editions Mir, Moscou) [13] Divet F. et al., Eur. Phys. J. E, 8 (2002) 477. [14] Brochard F. and Lennon J.-F., J. Phys. (Paris), 36 (1975) [15] Seifert U. and Langer S. A., Europhys. Lett., 23 (1993) 71. [16] Granek R. and Pierrat S., Phys. Rev. Lett., 83 (1999) 872.