Biological membranes are complex mixtures of lipids, proteins, and other molecules that serve versatile functions within living organisms. The lipid membrane serves as the primary barrier between the cell and the outside world and serves within cells as the boundaries of organelles. Biological membranes play an important role in cellular functions through their many embedded proteins whose collective organization and functions depend importantly on the mechanical interplay with the lipid bilayer. To gain insight into biological processes requires an intrinsic understanding of the basic mechanisms by which inclusions such as proteins interact with the lipid bilayer structures. This has been an active area of research.

We contribute through the development of new models and computational methods at both the level of continuum mechanics and at more mesoscopic scales where essential features of lipids are taken into account at a coarse-grained level to gain efficiencies over fully atomistic descriptions. We use these new approaches to perform a number of investigations of lipid bilayer membranes and protein interactions to gain new insights. First we introduce continuum methods at the level of individual proteins embedded within the bilayer in a manner closely related to the Immersed Boundary Method. We approximate the membrane-protein interaction by introducing a kernel function in the Hamiltonian that modifies the membrane energy in the neighborhood of the protein location. The proteins are naturally represented using a Lagrangian reference frame that tracks location while the membrane is represented using an Eulerian reference.

In the same spirit as the Immersed Boundary Method, the kernel function serves to couple the Eulerian and Lagrangian representations without resorting to explicit boundary conditions. By approximating the interaction in this manner at the level of the Hamiltonian, a set of self-consistent forces are obtained that act to deform the membrane and to perturb the protein. As we shall discuss in more detail, we capture in our model the elastic mechanics of the bilayer, membrane tension, leading-order hydrodynamic interactions, and thermal fluctuations. For proteins that induce curvature, we show how the numerical approach has important consequences for the effective diffusivity of embedded proteins. We also provide a condition which ensures that the system satisfies detailed balance with respect to the Gibbs-Boltzmann distribution, which when analyzing prior work in the literature we find is often overlooked.

This property turns out to be non-trivial in such approximate models since when both the protein and membrane simultaneously fluctuate important stochastic cross-terms are required that are difficult to treat. Some interesting features we can study with our methods include how collective effects arising from different area fractions of the embedded proteins impact the membrane bending mechanics. We find these can have a significant impact on the spectrum of bending fluctuations and on the renormalized flexural rigidity of the membrane bilayer sheet. In recent experiments by the group of Patricia Bassereau (Marie Curie Institute, Paris), protein diffusion within lipid bilayer membranes subjected to various levels of tension have been studied. By using quantuum dots, protein locations have been tracked at the single molecule level providing an ensemble of trajectories for different levels of tension.

Interestingly, whethor tension effects diffusivity depends importantly on the level of apparent curvature a protein induces in the bilayer. We have collaborated with this research group to help understand these findings by performing simulation studies using our membrane-protein model and developing related theory. Overall, our modeling approach and computational methods are quite general and we expect they could be useful in the investigation of a wide variety of phenomena involving membrane-protein interactions. At the molecular level of description we have also investigated coarse-grained models for the lipid bilayer vesicle. Our description of the bilayer is based on the implicit-solvent coarse-grained (IS-CG) lipid model developed by Markus Deserno, Kurt Kremer, and Ira R. Cooke for equilibrium studies.

We extend this work for studies of dynamic processes within bilayers by using the Stochastic Eulerian Lagrangian Method (SELM) to couple the coarse-grained model to fluctuating hydrodynamic fields. This approach provides a thermostat incorporating momentum-transfer from the the hydrodynamics of the surrounding solvent that would otherwise be missing in such implicit-solvent descriptions. In the past, the Deserno-Cooke IS-CG models have primarily been concerned with equilibrium properties of bilayers and used Langevin thermostats which neglect these effects. We have shown that by incorporating the hydrodynamics important correlations are manifested within the bilayer that could play an important role in many processes within the bilayer, such as the transport of proteins or the kinetics of self-assembly. To obtain continuum fields for comparison presents a number of challenges.

We develop a fitting method for lipid bilayer vesicles based on the use of spherical harmonic (SPH) representation of the vesicle shape. We use the SPH representation to perform a spectral analysis of vesicle shape fluctuations.