Continuum descriptions of membranes and their interaction with proteins: Towards chemically accurate models

Abstract

Biological membranes deform in response to resident proteins leading to a coupling between membrane shape and protein localization. Additionally, the membrane influences the function of membrane proteins. Here we review contributions to this field from continuum elastic membrane models focusing on the class of models that couple the protein to the membrane. While it has been argued that continuum models cannot reproduce the distortions observed in fully-atomistic molecular dynamics simulations, we suggest that this failure can be overcome by using chemically accurate representations of the protein. We outline our recent advances along these lines with our hybrid continuum-atomistic model, and we show the model is in excellent agreement with fully-atomistic simulations of the nhTMEM16 lipid scramblase. We believe that the speed and accuracy of continuum-atomistic methodologies will make it possible to simulate large scale, slow biological processes, such as membrane morphological changes, that are currently beyond the scope of other computational approaches. This article is part of a Special Issue entitled: Membrane Proteins edited by J.C. Gumbart and Sergei Noskov.

Keywords: Bilayer; Biological membrane; Electrostatics; Hydrophobic mismatch; Transmembrane protein.

Figure 1

How can proteins bend membranes? A. Scaffold mechanism. A rigid array of proteins (blue) assemble over the much more compliant membrane deforming the entire system into a new shape. B. Protein crowding mechanism. Thermally driven protein-protein collisions of bound proteins to the membrane surface can create significant lateral pressure and drive bending. C. Spontaneous curvature mechanism. The proteins act locally to distort the surrounding lipid molecules and alter their elastic properties, such as the local spontaneous curvature. These local changes can give rise to new stable morphological structures such as tubules or vesicle budding events. Orange lipids inside the dashed boxes represent the region over which the protein insertion induces local distortions. D. Bilayer-couple mechanism. The asymmetric insertion of many proteins on one side of the membrane generates an area mismatch between the upper and lower leaflets resulting in stress that spreads globally over the entire surface. The generation of curvature relieves the in-plane components of the stress in both leaflets.

Figure 2

Mathematical representation of the membrane. A. A cartoon model representing the upper and lower leaflets and the corresponding lipid molecules. N⃗ is the normal vector of the bilayer midplane (dashed line), n⃗ is the head-to-tail vector of the lipids, and t⃗ is the difference of these two vectors. B. The upper and lower surfaces representing the head-group interfaces with the water from panel A (solid lines) and the bilayer midplane (dashed line). The lipids have been removed in this purely mathematical representation, but the vectorial descriptions N⃗^± and n⃗^± remain.

Figure 3

Examples of curved surfaces. A. cylinder, B. spherical cap, and C. saddle. The geometry of each surface can be defined as a function of the two principal radii of curvature R₁ and R₂. When R₁ and R₂ change in a bilayer there is a curvature energetic penalty in the Helfrich Hamiltonian (Eq. (5)). The mean curvature is equal to the sum of the principal curvatures (inverse of the radius of curvature) $H=\frac{1}{R_1}+\frac{1}{R_2}$ and the Gaussian Curvature is the product $K=\frac{1}{R_1}·\frac{1}{R_2}$.

Figure 4

Lipid tilt degrees of freedom. A. A patch of membrane exhibiting pure twist. In all panels, the top image is a side view of the upper leaflet, and the bottom panel is a top down view of a patch of lipids. Vectors demonstrate the head-to-tail orientation of individual lipids. B. A patch of membrane exhibiting pure splay. C. A patch of membrane exhibiting pure tilt. The lipid density was intentionally decreased for clarity.

Figure 5

Cartoon models of membrane protein interactions. A. The potassium channel KcsA adopts a conical shape in the closed state (left). The hydrophilic residues are blue and the hydrophobic residues are white, and the hydrophobic residues localize to a belt around the protein that creates the energetic ‘seal’ with the membrane. This seal would impose a negative contact angle on the membrane (black lines) potentially causing bending in the simplified geometry on the right. B. The mechanosensitive channel MscL is cylindrical with a more well defined hydrophobic belt (left). This shape would not impose a contact angle on the protein, but if the hydrophobic height of the protein differed from the equilibrium width of the membrane it may impose a hydrophobic mismatch that causes compression or expansion of the adjacent membrane (right).

Figure 6

The geometry of the membrane near an atomistic protein. A. Side view of a membrane protein illustrating the membrane distortions around the protein by h⁺ (upper gray surface) and h⁻ (lower gray surface). B. Close up view of the contact curve, showing the displacement (u⁺) and slope (S⁺) boundary conditions at one point of the upper leaflet contact curve.

Figure 7

Membrane bending around nhTMEM16 determined from fully-atomistic MD and continuum elasticity. A. Membrane distortions caused by nhTMEM16 predicted from continuum elasticity. The protein is represented at the atomistic level, with the upper and lower head group-water interfaces in green and the surfaces delineating the hydrocarbon core gray. All hydrophobic amino acids are white and polar residues are blue. B. Enlarged view from panel A with the hydrophilic groove indicated. C, D. Upper and lower membrane surfaces averaged from fully-atomistic MD simulations. E, F. Upper and lower membrane surfaces determined from continuum elasticity. The protein is gray. White values correspond to the undeformed height of the membrane far from the protein, blue are downward deflections, and red are upward deflections. All color bars are in ångströms. Color scheme is the same throughout. The stars in panel D indicate points of discrepancy between simulations (panel D) and continuum solution (panel F). G, H. Curvatures (G) and membrane heights (H) for upper and lower leaflets along the x equal y axis in panels E and F. The starting point and direction is specified by the dashed arrows in panels E and F.