Multiscale modeling of lipid bilayers under equilibrium and non-equilibrium conditions
- Mikko Haataja (Princeton University, USA)
- Julian Shillcock (EPFL, Blue Brain Project, Switzerland)
- Markus Deserno (Carnegie Mellon University, Pittsburgh, PA, USA)
The plasma membrane enveloping mammalian cells is a two-dimensional fluid bilayer consisting primarily of thousands of different types of lipids and proteins. Far from being featureless, it is now well-established that the membrane is “patchy” with spatially organized regions of structure and function, both in terms of lipids and proteins; for an over-view, see Refs. [1-3]. The spatially-extended nature of the membrane “patchiness” together with dynamic membrane processes, due to both thermodynamic fluctuations and non-equilibrium cellular events (such as endo- and exocytosis), provide challenges for theorists and computational scientists alike to develop and simulate quantitative models that seamlessly “funnel” information via coarse-graining from the molecular length and time scales up to the mesoscale.
A. Coarse-graining the static properties of lipid membranes
The challenges at the molecular scale involve developing sufficiently accurate classical force-fields that are capable of describing both the thermodynamic phase behavior of the lipid system with multiple components as well as diffusive lipid transport processes. At the molecular scale, the state of the art is set by atomistic Molecular Dynamics, which is able to probe the interactions between individual lipid molecules and capture the effects of small changes in their chemical structure on the membrane properties, such as oxidized lipids exhibiting a splayed conformation . Such simulations are very costly, and are suitable only for small patches of membrane containing perhaps up to a thousand or so lipid molecules. Recently, however, coarse-grained MD simulations, in which the effective interactions between different molecular groups are calibrated using relevant experimental data  have been able to push this limit up to several thousands of lipids, and include simplified models of proteins. At even larger scales, where tens of thousands of lipids are simulated, even coarser techniques such as Dissipative Particle Dynamics can be used to follow long-time processes, such as membrane remodeling during vesicle fusion . The parameters in a DPD model are typically calibrated using data obtained from more accurate MD simulations, or are determined on an ad hoc basis for the system of interest by comparing its properties with known experimental data and adjusting the parameters until agreement is obtained. Parallel DPD is capable of simulating patches of membrane in excess of 500 nm in linear dimension, and can be directly compared to experiments using light microscopy of large unilamellar vesicles and continuum membrane models based on the Helfrich Hamiltonian. The ability to directly compare a particle-based simulated membrane with continuum models provides an important link between particle-based simulations and theoretical models.
At larger length and time scales, it is very appealing to describe the lipid bilayer as a continuous “sheet” which can deform and whose internal state is determined by the local lipid compositions and membrane protein concentration(s). Such models, while providing useful abstractions of the system, are often phenomenological and not necessarily “microscopically-informed”. The challenge here is to devise strategies that allow one to connect such continuum models based on more microscopic descriptions. An appealing possibility is to employ particle-based simulations on smaller scales to calculate values for the phenomenological parameters that enter into the continuum models, such as the bending rigidity of a lipid bilayer. The membrane then becomes a substrate on which chemical and biophysical processes take place. Sets of differential equations, such as reaction-diffusion models, can then be embedded into the membrane and their behavior studied on the fluctuating surface.
B. Coarse-graining the dynamic properties of lipid membranes in and out of thermal equilibrium
One often overlooked complication in developing physically-based models for membrane domain kinetics resides in the treatment of the surrounding fluid. In particular, fluctuations in the compositional domains will induce flow fields in the surrounding fluid, which will subsequently back-react with the membrane. Particle-based models for membranes, which appropriately account for hydrodynamic effects arising from the surrounding fluid and which provide a detailed microscopic view of collective phenomena associated with lateral lipid diffusion [7-9], have difficulties in probing large-scale compositional domain dynamics due to computational restrictions, although this constraint is gradually being overcome by a combination of hardware development and parallel simulation codes. While rather simple spatially-extended continuum models have been proposed recently, which address domain formation processes in synthetic bilayers across mesoscopic and macroscopic scales [10,11] and elucidate the potential role of active cellular processes on raft-like domain formation dynamics , these models do not explicitly incorporate the role of hydrodynamics on compositional domain formation and dynamics, with the notable exception of Ref.  wherein hydrodynamics within the membrane was included while the coupling between the membrane and the exterior fluid was neglected. In order to develop a better understanding of compositional raft domain dynamics in lipid membranes, both in vivo and in vitro, it is thus necessary to quantitatively assess when hydrodynamics may or may not be neglected, or when it perhaps completely dominates the physics, such as in the vicinity of phase transition points in multicomponent lipid membranes .
Other phenomena that are affected by hydrodynamics include the creation of pores in a membrane during fusion or endocytosis, and the transport of material or nanoparticles through a crowded environment. Forming a pore in a membrane requires pushing the membrane outwards against the viscous drag of the surrounding fluid and this influences the pore’s growth rate, and the force needed to create it. Also, the removal of defects that appear during lipid phase transitions often require the concerted movement of large segments of one phase through another. For example, when an isotropic lipid phase converts to a lamellar phase, pieces of lamellae orient in different directions and lipids have to move through these regions in order to produce the final, aligned, lamellar phase. The appearance of a lamellar phase on a micron length-scale is far beyond what can be simulated with atomistic Molecular Dynamics, so coarse-grained models are required that capture the essential lipid molecular properties but also the fluid behavior on almost-macroscopic dimensions.
Understanding the interactions of nanoparticles with lipid membranes is seen as crucial to predicting their toxicity and suitability for deployment in commonly-used materials, packaging, and medicine. Studies on the penetration of membranes by Fullerenes using Molecular Dynamics simulations have been performed , but are limited to particles with less than around 20 nm diameter. It has been found recently that the size  and shape  of nanoparticles dramatically affect their entry into cells and subsequent toxicity. But as the size of these particles range from 20 nm to 140 nm computer simulations of their interactions with membranes are not feasible without novel extensions of current mesoscopic modelling techniques. However, MD simulations of small nanoparticles can be used to select parameter values for coarse-grained simulations of many nanoparticles interacting with a membrane to construct a system that is capable of predicting the aggregate effects of concentrations of particles on cell membranes. Whether nanoparticles aggregate outside a cell, or penetrate its plasma membrane individually, or act in a concentration-dependent manner are questions that can only be answered using a combination of simulation techniques.
C. Coarse-graining the description of protein-lipid interactions and protein dynamics
Membrane proteins participate in many important cellular functions, including cell-cell signaling, endocytosis, and transport of small ions and molecules through the membrane; they are also often employed as “reporters” of the underlying correlations in the lipid composition in experiments. For instance, Cholera Toxin subunit B (CTB) is frequently used as a marker which binds to ganglioside GM1 (a lipid receptor), which itself is known to associate with rafts. However, proteins can be affected by physical membrane properties in ways that lipids are not. Tian and Baumgart have recently shown  that CTB is efficiently sorted in membrane curvature gradients, while the same is not true for lipids, as was previously predicted in coarse-grained simulations . The larger size of proteins tends to make them more susceptible to their membrane environment, implying that they could sense curvature, composition, hydrodynamic forces, or fluctuations better than individual lipids. Hence, coarse graining techniques are ideally suited to understand the physical basis for the many obvious biological scenarios this implies. Indeed, sophisticated multiscaling approaches have begun to illuminate how for instance BAR (Bin/amphiphysin/Rvs) protein domains individually bend and cooperatively tubulate membranes [20-23], even though the underlying physics of membrane curvature mediated protein aggregation is far from satisfactorily understood.
Models with varying levels of resolution have been employed to study the effects of antimicrobial peptides on bilayers, which range from highly specific questions of individual optimal insertion up to the large-scale fate of cooperatively forming pores [24-27]. Correlated diffusion of proteins, which are coupled by hydrodynamic flow fields in the membrane and through the solvent, have recently been studied by Oppenheimer and Diamant , while efficient simulational approaches of the same problem have been proposed by Naji, Atzberger, and Brown . Correlations between diffusing lipids on a fully atomistic level have also been quantified in simulations by Falck et al.  and Apajalahti et al. . All of these processes happen on mesoscopic length scales and are not exclusively dependent on atomistic details, but they require a thorough understanding of the key interactions and the large scale static and dynamic properties of membranes; hence we can expect coarse-grained simulations to further advance our current understanding.
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