On the hydrodynamics of bilayer membranes
Introduction
Bilayer membranes form the outer walls of biological cells and organelles [1]. They are fluid, two-dimensional (2D) structures consisting of two opposite lipid monolayers that may host proteins, carbohydrates and other biological material. Lipids molecules of opposite monolayers contact each other at the level of the extremity of their aliphatic chains while the polar heads of the lipids face the aqueous solvent that surrounds the membrane.
Bilayer membranes are complex systems from the point of view of hydrodynamics: they consist of a pair of contacting 2D fluids with soft out-of-plane elasticity [2] moving in a 3D solvent. The monolayers must be treated as compressible fluids. Indeed, the 2D density is unconstrained as the monolayers can adapt their thickness in order to keep the 3D density constant. Besides, any variation of the lipid density between the two monolayers causes a spontaneous curvature of the bilayer [3], [4]. In the early studies of membrane hydrodynamics the bilayer structure of the membrane was neglected [5], [6]. While this is a good approximations for tensionless membranes at length-scales much larger than microns, experiments and theoretical studies have shown that taking into account the bilayer structure is essential at lower length-scales [7], [8], [9], [10], [11]. This is chiefly due to the importance of the dissipation caused by intermonolayer friction.
The full equations of bilayer hydrodynamics, including curvature and density-difference elasticity, intermonolayer friction, monolayer 2D viscosity and solvent 3D viscosity, were first derived by Seifert and Langer for almost planar membranes [8]. The method employed was a careful balance of in-plane and out-of-plane stresses. Generalization to non-linear membrane deformations, within the approximations of neglecting some dissipation sources, was performed by DeSimone, Arroyo and Rahimi [12], [13], [14] using covariant elasticity and Onsager׳s variational principle [15], [16].
In this paper we apply Onsager׳s variational principle [16] to the detailed derivation of the dynamics of almost planar bilayer membranes, including all types of dissipations (2 Parametrization, 3 Conservation laws and constraints, 4 Dissipation, 5 Elastic energy, 6 Differentiation of the Rayleighian, 7 Dynamical equation, 8 Integration of the bulk dynamical equation, 9 Dynamical equations for the membrane in Fourier space). We recover the linear equations of Ref. [8] with very little modifications (Section 10). We solve in detail these equations for an arbitrary Fourier component and we calculate the associated hydrodynamic 3D flow around the membrane (Section 10). We discuss the relaxation rates below and above a threshold tension separating two distinct behaviors and we physically interpret each relaxation rate in all of the various regimes (Section 11). To study in detail the hydrodynamics of the membrane, we calculate at various length scales the relaxation of a sinusoidal membrane shape (Section 12) and a Gaussian bud (Section 13), each time describing the flow pattern. Finally, we discuss depending on the scale, the legitimacy to neglect some of the dissipation sources, proposing an approximation scheme that allows to retain only the intermonolayer friction at small scales (Section 14).
Section snippets
Parametrization
We describe the membrane shape by the height , above a reference plane (x,y), of the surface separating the two monolayers, as shown in Fig. 1. We call a generic point in this reference plane and a point in 3D space. Because the membrane is assumed to undergo only small deformations, at first order in h the hydrodynamical equations can be derived by considering that both monolayers lie in the z=0 plane. The superscript + (resp. −) will be associated with the monolayer in
Conservation laws and constraints
There are several conservations laws and constraints that must be satisfied. Regarding the bulk solvent as an incompressible fluid, we requirewhere . In the membrane, the lipid mass conservation equation is , where and a dot indicates time derivative. It yields at first-order:In addition, we require a no-slip boundary condition at the interface between each monolayer and the solvent:and a no-permeation condition of the solvent
Dissipation
As mentioned in the Introduction, there are three sources of energy dissipation. The associated dissipation functions, defined as one-half of the dissipated power [15], arewhere B+ (resp. B−) is the volume defined by (resp. ), is the rate-of-deformation tensor in the bulk solvent and is the rate-of-deformation tensor in the monolayer fluids. corresponds to the
Elastic energy
In a bilayer membrane, the two monolayers are strongly held in contact by hydrophobic forces (Fig. 1), hence they are strongly coupled from the point of curvature elasticity. However, the level of interdigitation between the lipids being usually very small, the density fields are essentially uncoupled and the excess elastic energy of the membrane can be written asBecause the monolayers are fluid only the membrane curvature c and the excess densities appear in
Differentiation of the Rayleighian
In the Stokes approximation, i.e., neglecting all inertial effects, the dynamical equations describing the motion of the membrane within the bulk solvent can be obtained by extremalizing with respect to all the dynamical variables the total Rayleighian of the system [16], [12], [13]:Taking into account the constraints (5), (6), (7), (8) through the introduction of the Lagrange multiplier fields , , , and , we need to extremalize the functional
Dynamical equation
Eliminating the membrane Lagrange fields, we obtain the dynamical equations for the membrane:Note that these equations are defined in z=0 and ∇ is the gradient operator in the (x,y) space. In addition, we have the Stokes equation for the bulkand the remaining constraints, which we regroup here for the sake of clarityThese
Integration of the bulk dynamical equation
The bulk Stokes equations can be dealt with by Fourier transforming along the (x,y) plane and solving in the z direction. We define the Fourier transform byfor surface and bulk quantities, respectively. Note that we shall use the same letters for the functions and their Fourier transforms in order to avoid heavy notations. In terms of and , the bulk Stokes equations (26), (27) read
Dynamical equations for the membrane in Fourier space
In Fourier space, the viscous bulk stresses in (24) read , while those in (25) read along and in the direction perpendicular to . Hence, Fourier transforming the membrane dynamical equations (24), (25) yieldswhereappears as a renormalized surface viscosity.
Final dynamical equations and methodology
The procedure to solve the dynamics of a bilayer membrane is the following. Consider at t=0 a membrane with shape and lipid excess densities . First compute and , then set at all times, since, as shown above, the average density relaxes to zero in an infinitesimal time. Next, decompose and into Fourier modes and solve for each mode Eqs. (57), (58):
Relaxation rates
Eqs. (62), (63) that give the time evolution of and can be expressed aswith the dynamical matrix:Its two eigenvalues are the two relaxation rates of the bilayer. They exhibit the avoided crossing1 phenomenon (Fig. 2). Let us discuss the time evolution of a Fourier mode q with initial
Relaxation of a Fourier mode and hydrodynamic flow
Let us consider, at t=0, a single Fourier mode for the membrane deformation. We assume that this shape has been fixed for some time, so the lipid density is at equilibrium. Minimizing the elastic energy (15) with respect to at fixed yields , hence the initial density difference is with . Let us determine the time evolution of the membrane and the hydrodynamic flow it generates.
Relaxation of a Gaussian bud
Let us now consider, at t=0, a Gaussian bud of equation where w is the width at of the height. Again, we assume that this deformation has been fixed for some time, so the lipid density is at equilibrium, which implies . The dynamics of the bud can easily be obtained in the following way. In Fourier components, the bud is described byCalling P(q) the matrix of the eigenvectors of M(q) and defining
Legitimacy of neglecting dissipation sources
Let us discuss (within the framework of almost planar membranes) whether or not it is legitimate to neglect the bulk viscosity η, the monolayer viscosity ηs or the intermonolayer friction coefficient b. Such approximations are encountered in the literature.
Summary
We have derived the hydrodynamical equations of bilayer membranes from Onsager׳s variational principle for almost planar membranes. Membrane tension was taken into account together with all the sources of dissipation, i.e., solvent viscosity, monolayer shear and dilatational viscosities, and intermonolayer friction. The dynamical equations we obtain confirm and generalize those of Seifert and Langer [8]. We showed that there are two regimes for the relaxation rates whether the tension σ is
Acknowledgments
I thank Marino Arroyo for valuable discussions, and I acknowledge financial support form the French Agence Nationale de la Recherche (Contract no. ANR-12-BS04-0023-MEMINT).
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