Coupling between Lipid Shape and Membrane Curvature

INTRODUCTION

There are hundreds of different naturally occurring lipids that make up the internal and enclosing membranes of the cell. They are not just homogeneously distributed but are found in different concentrations in different organelles. Remarkably, this specificity occurs in a highly dynamic environment in which lipids are constantly recycled throughout the cellular machinent, yet lipids themselves lack lock-and-key type binding sites found on proteins and other function specific biomolecules. Examples of lipid sorting are numerous and include the differentiation between apical and basal regions of Golgi and endoplasmic reticulum ( 1 ). the function of sorting endosomes (2). and lipid selection during viral budding (3-5). Just what drives this sorting is currently one of the key mysteries of cell biology (6).

Consider the example of sorting that occurs between the endoplasmic reticulum and the plasma membrane. These regions differ markedly in composition although they are connected via vesicular endocytic and exocytic pathways. In fact, it is these pathways that are thought to he responsible for the sorting, and an indicator toward the mechanism by which they differentiate between lipids is the presence of highly curved tubular or vesicular regions that form and bud off from the parent membrane ( 1,2,6). The basic principle is that highly curved regions may selectively include or exclude certain lipids on the basis of their shape or stiffness. Neither the influence of shape or membrane stiffness is yet adequately understood, but recent key experiments involving simple model membranes have demonstrated that lipids with unsaturated tails (that typically form less stiff bilayers) become concentrated in tubular regions pulled from a vesicle (7), and in vivo experiments have demonstrated that lipids with greater tail unsaturation are sorted into pathways involving highly curved tubular intermediates (8). While these experiments have begun to address the issue of lipid fluidity/ stiffness in partitioning different lipids, the influence of lipid shape has so far received little attention (2.8) and to verify theoretical predictions (9) it is essential that its coupling with membrane curvature he quantified.

It is important to remember that lipids can be sorted in different ways; either between leaflets of the bilayer (interleaflet) or between highly curved regions and Hat regions of the membrane (interregion). Interregion and interleaflet sorting involve processes on different timescales (diffusion versus flip-flop) and depend differently on the makeup of the lipid mixture that comprises the membrane. To see why this is the case, consider the fact that there are always two leaflets to a hilayer and that wherever one leaflet has a curvature K, the other will have curvature [asymptotically =]-K. This means that in a binary lipid system the curvature coupling will be approximately equal but opposite in each bilayer sheet. Such a situation is conducive to strong interleaflet sorting when there are two lipid species with opposing preferred curvature. On the other hand, interregion sorting is likely to require a third species with intermediate curvature that will partition into Hat regions of the membrane. Other possible mechanisms for interregion sorting can be envisaged in biological systems where flip-flop is very slow, but our purpose here is not to examine such mechanisms. Instead we will focus on the fundamental effect that is common to both interleaflet and interregion sorting, that is. the coupling between lipid shape and membrane curvature.

As a final point of interest it is worthwhile considering the role of lipid shape when combined with nonequilibrium processes such as active concentration of lipids via llippases (10). This is essentially the reverse situation to passive sorting and represents a mechanism by which the shape of cellular components may be altered. Such active sorting is now thought to be commonplace and represents an important component in vesicle budding and movement processes (11).

Regardless of whether we are concerned with active or passive lipid sorting, one of the key physical processes underlying biological function is the coupling between lipid shape and membrane curvature. The aim of this study is to systematically examine the physics of this coupling. Firstly we would like to determine whether lipids with different shapes will preferentially distribute into monolayers with a different curvature. More importantly, we aim to quantify this effect, both as a function of the curvature and of the deviation of the lipid from a cylindrical shape.

MODEL SURFACTANTS AND SIMULATION METHODOLOGY

The goals of this study represent a particularly difficult simulation task. On the one hand, very large systems (10,000 lipids) and very long timescales are required. Yet at the same lime, our goal of observing the effects of molecular shape necessitates the use of a particle-based model to represent individual lipids. This requirement clearly excludes non-particlc-based techniques used for large-scale systems such as dynamic triangulation, and alternatively, our need for large systems rules out the use of fully atomistic simulations. This leaves us with coarse-grained particle-based lipid models, which cover a wide range of complexity from single beads per lipid (12) up to models that capture some chemical detail (13,14), and including many that lie in between (15-17). Most of these coarse-grained models employ explicit solvent particles and for certain two-dimensional geometries such as a flat bilayer this does not pose a serious performance drawback. For three-dimensional geometries such as vesicles and budding bilayers, however, the solvent typically accounts for >95% of computation time and limits the extent to which large length and timescales may be studied. Of course there remain situations where the effects of the solvent itself are of interest, or where hydrodynamics are important. In such cases the solvent particles are treated explicitly (15,18). An alternative to the explicit approach is to model the solvent implicitly via attractive interactions between tails, thereby achieving more than an order-of-magnitudc gain in efficiency. This aim has been pursued for more than a decade (12) but only recent advances have led to models with well understood and tunable properties suitable fora wide variety of applications (see the review by Brannigan et al. (17) and references therein). In this study, we shall employ a model developed by us that is based on the use of broad tail attractions (19). Since this has previously been described in detail (20) we provide only a brief description here.

Lipids

The three beads are linked by two almost inextensible bonds and the lipid is made stiff through the use of a harmonic spring with rest length 4s between head bead and second tail bead. Further details may be found in the reference (20).

This describes an attractive potential with depth e that for r = r^sub c^ smoothly tapers to zero. By tuning the range w^sub c^ it is possible to alter the stiffness and fluidity of the resulting bilayer, which effectively mimics real lipid characteristics such as tail length and saturation.

Comparison with real membranes

Given the strongly coarse-grained nature of our lipids it is remarkable just how well the physical properties of the resulting simulated membranes correspond with those of real systems. Parameters such as stiffness, area stretching modulus, and rupture tension all lie within their experimentally observable ranges (20). Nevertheless, there is one important parameter that is different between simulated and real lipid systems. That parameter is the flip-Hop rate, which is much faster in the simulation (20). and would be of concern if our study were to focus on the dynamics of exchange between leaflets. Since this study is not concerned with dynamics, a fast flip-Hop rale is actually very useful since it allows us to reach equilibrium quickly. Interestingly, in biological (as opposed to synthetic experimental) systems, llippases catalyze the flip-Hop process (21). Although some types of tlippases are unidirectional-that is. they catalyze flip into one leaflet only-our simulations are also relevant for the passive case where catalysis is bidirectional.

Colloidal particles

In the section "Lipid Shape Coupling During Budding", we simulate the partial wrapping of a colloidal particle by a section of bilayer membrane. This situation rather closely models the acquisition of membrane coats via budding as seen in many common animal viruses (22). In such cases the viral particles are wrapped due to their adhesion with the parent membrane. Nevertheless, our purpose was much more general. We simply wanted to obtain a membrane geometry close to that which would be found generally for all types of budding events, i.e.. a spherical region, a neck, and a Hat parent membrane.

For various technical reasons it was not convenient to model the colloid as a single large particle with an offset repulsive potential. Instead, it was constructed by creating a spherical mesh of small hard spheres enclosing a collection of soft filler particles. The mesh itself was generated by placing a single bead of size s at each point in an optimal spherical covering (23). FENE bonds were then imposed between each bead and its near neighbors. To make the colloid rigid it was inflated w ith soft spheres with radius 2.5s. These spheres interacted with each other via a simple harmonic repulsive potential U(r) - (k^sub f^/ 2)(r^sub cut^ - r)2 where r is the interparticle distance. k^sub f^ = 2Oe is the strength of interaction, and r^sub cut^ = 2.5s is the cutoff distance. In each of the wrapping simulations in this work a single colloid of radius R^sub colloid^ = 10s consisting of 1002 cage atoms and 1000 filler spheres was used. Using a particle-based colloid model such as this entails several advantages. The first is that there are no particularly large particles in the system, which allows optimizations such as cell and particle lists in the integration to retain their efficiency gains. The second is that we now have additional control over the particle shape and interaction. In particular, we have constructed colloids in which one part of the cage surface adheres to the membrane and the other part does not. Such attractions, when used, were always imposed via a simple Lennard-Jones attractive potential with well-depth e and equilibrium-distance s.

Simulation details and units