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Copyright © 2007 The Biophysical Society. All rights reserved.
Biophysical Journal, Volume 93, Issue 11, L49-L51, 1 December 2007

doi:10.1529/biophysj.107.119222

Biophysical Letters

Corrections to the Saffman-Delbrück Mobility for Membrane Bound Proteins

Ali Naji^*, Alex J. Levine^†, ,   and P.A. Pincus^‡

^* Department of Physics and Department of Chemistry & Biochemistry, University of California, Santa Barbara, California
^† Department of Chemistry & Biochemistry and California Nanosystems Institute, University of California, Los Angeles, California
^‡ Materials Research Laboratory, and Departments of Physics and Materials, Biomolecular Science & Engineering, University of California, Santa Barbara, California

Address reprint requests and inquiries to Alex J. Levine.

The diffusivity of transmembrane proteins is a fundamental biophysical parameter controlling the dynamics of protein-protein interactions in the cell membrane. These dynamics underlie such processes as endocytosis and signal transduction 1. Understanding the size dependence of the diffusivity of membrane bound proteins is rather subtle. Saffman and Delbrück (SD) 2 originally demonstrated the significant differences between lateral diffusion in membranes and the better understood problem of diffusion in a bulk solvent. In the membrane, the diffusion constants are only weakly dependent on the size of the diffusing particle, while in bulk solvent the diffusion constant depends inversely on particle size. Although some data appear to support the Saffman-Delbrück theory 3,4,5, more recent experiments exploring the diffusivity of transmembrane proteins over a larger size range 6 using in vitro lipid bilayers show a much stronger protein-size dependence than is consistent with our current understanding of membrane hydrodynamics 2,7,8. These data suggest that the diffusivities of the proteins depend inversely on their size for a variety of proteins and protein aggregates covering about one decade of inclusion radius, and are clearly inconsistent with the SD result.

In this Letter, we address this puzzling discrepancy between theory and experiment by proposing that accounting for local membrane deformations caused by embedded proteins can resolve this conflict. We reexamine the mobility μ of a protein in the lipid bilayer. The mobility defines the linear relationship between a particle’s velocity and the force applied to it via the relation

It is well known that the mobility of a rigid, spherical particle of radius a in a three-dimensional solvent having viscosity η is given by the Stokes result, μ=1/(6πηa), which has an inverse dependence on the particle radius. The mobility of the same particle when embedded in a fluid membrane, however, is more complex. There the particle moves through an effectively two-dimensional liquid that is coupled to the surrounding three-dimensional solvent by the requirement that there be no slip at the interfaces between the lipid membrane and the aqueous solvent. The hydrodynamic coupling between flows in the effectively two-dimensional fluid and the surrounding solvent introduces an inherent length scale into membrane hydrodynamics—the SD 2 length which is set by the ratio of the two-dimensional membrane viscosity η_m to that of the surrounding bulk solvent η. In contrast, the usual low-Reynolds-number hydrodynamics in a bulk liquid is a scale-free theory.

The introduction of this extra length scale profoundly modifies the mobility of particles embedded in the membrane. Because of this, the mobility of a particle of radius a in the membrane is given by

Recent experiments on the diffusion of lipid domains on giant unilamellar vesicles quantitatively support this form of the lateral mobility of embedded objects in membranes 5, but analogous experiments on membrane-bound protein mobilities do not 6. Taken together, these data suggest a resolution of this conflict. The mobilities computed above depend on only a few simple assumptions regarding mass and total momentum conservation: thus they appear unassailable. It is well known, however, that membrane bound proteins typically perturb the membrane structure locally 9,10,11,12. This local perturbation may take many forms including local changes in membrane height or thickness involving local oligomeric chain stretching, local membrane curvature, tilt of the lipids, or changes in local lipid composition relative to that the far field (for mixed lipid systems). Below we show that these membrane perturbations that must be transported along with the proteins can shift the mobility of these composite objects from the SD form to one consistent with D∼1/a scaling. This effect arises from either the enhanced dissipation associated with modifications of the flows in the bulk solvent caused by the protein-induced height or bending deformations, or the enhanced dissipation occurring within the membrane itself in cases where the protein generates local changes in composition, chain stretch, or tilt order. These two scenarios are not mutually exclusive and both give the same D∼1/a scaling, but, as they rely on somewhat different reasoning, we present the arguments independently.

If the protein has a hydrophobic mismatch with the membrane thickness, it will generate a bump on the surface having a lateral dimension on the order of the radius of the protein or the membrane thickness h10,11,12. We now estimate the effective mobility μ_eff of the protein and associated membrane deformation (bump) by considering the power dissipated, P=Fv, when this complex is moved at constant speed v in the membrane in response to an applied force F. Using the definition of mobility, the power input required to move the protein is

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Figure 1

(a) Side-view of the membrane with an embedded protein (shaded), which creates a bump generating additional flows in the solvent (solid arrows) leading to extra dissipation. (b) Top-view of the membrane showing the protein (shaded) and the perturbed membrane surrounding it (hatched).

It is also possible that the principal additional dissipative stresses are associated with degrees of freedom internal to the membrane. Such dissipation may be related to chain stretching or tilt near the protein, or to local demixing of the constituent lipid species resulting from differential affinities between the protein and the various lipid species.

We now estimate the power dissipated in the membrane. As shown in Figure 1b, we posit that the disruption of membrane structure occurs within a distance ξ of the protein. Working in the reference frame of the protein, lipids flow into this modified zone and undergo some entropy-generating (i.e., dissipative) conformational change in some boundary layer around the zone of width δξ, where the power dissipated per lipid is p_lipid=fv=ɛ″ v². Since the dissipative forces f must be odd under time reversal they must be linear in the rate of lipid deformation, which is linear in velocity. Using the area density of the lipids ρ and the area of affected lipids, 2π(a+ξ)δξ, to determine the number of lipids involved in the extra power dissipation we find

We have shown that one may account for the experimentally observed failure of the Saffman-Delbrück diffusivity of membrane bound proteins by positing that the protein carries with it a locally deformed patch of membrane. This local deformation will generate extra flows in the bulk solvent if the protein creates a bump or depression in the membrane; if the protein modifies the internal structure of the lipids in its immediate vicinity, then there is enhanced dissipation in the membrane as the deformation is dragged by the protein. As long as the power dissipated in the membrane or in the surrounding solvent arising from this membrane deformation is at least comparable to the dissipation in the usual flows of the unperturbed membrane, one will observe an inverse radius dependence of the protein diffusivity.

Recent simulations of inclusion mobility 14 have also found deviations from the SD mobility of inclusions. There it was found that μ∼1/a² because of the dissipation enhancement coming from internal soft modes of the inclusion. That work shows yet another way in which extra internal degrees of freedom shift the mobility of the object. These inclusions did not deform the membrane in ways that we suggest here; new simulations having these effects are clearly desirable. Experiments on lipid domain mobilities find agreement with the Saffman-Delbrück expression 5. There one should not expect large perturbations of the surrounding membrane by these domains. Transmembrane proteins, on the other hand, are known to generate static deformations of the surrounding membrane. Little work has been done on examining the dynamic effects of such membrane perturbations. Our simple heuristic analysis suggests that both further theoretical work and more local examinations of protein dynamics in membrane are required to better understand protein transport properties in lipid bilayers.