Measuring the Free Energy of Self-assembling Systems in Computer Simulation

Abstract

Membrane fusion is involved in a multitude of biological processes like endo- and exocytosis, viral infection or synaptic release. The detailed mechanism, however, is not well understood because the time and length scales microseconds and tens of nanometers, respectively make a direct experimental observation difficult. The study of this collective phenomena, involving many lipid molecules, is also difficult in simulations with atomistic resolution. While the details of the fusion pathways are still under debate, most fusion scenarios start with the formation of a stalk, which is a hour-glass shaped connection between the two apposing membranes that are going to fuse. Understanding the properties of this initial fusion intermediate is a key to controlling fusion of bilayer membranes. We use a coarse-grained model for bilayer membranes. The lipid molecules are described by a simple bead-spring model and the solvent degrees of freedom are integrated out. The effective non-bonded interactions between the beads of the lipid molecules take the form of a virial expansion. Within the mean-field approximation, the coefficients of the expansion are related to the density and compressibility of the hydrophobic interior of the bilayer membrane and the repulsion between the hydrophilic and hydrophobic units. In order to employ such an excess free energy density functional for the non-bonded interactions in a particle-based simulation, the local densities are calculated from the explicit particle coordinates via a collocation lattice. This soft, solvent-free, coarse-grained model for bilayer membranes allows for an efficient simulation of membrane properties. This coarse-grained model has been employed to study the excess free energy of stalks that form between apposing membranes as a function of the molecular asymmetry of the lipid molecules and the membrane tension. To this end, we have devised a general strategy for calculating free energies in self-assembling systems. The method relies on constructing a reversible thermodynamic path that connects the system of two apposed bilayers and the stalk configuration. This path is constructed in an extended state space using an inhomogeneous, external field that is designed to direct the assembly of the system into the two apposed membranes or the stalk structure in the absence of non-bonded interactions. Using expanded ensemble simulations it is demonstrated that the path is reversible and that the Helmholtz free energy can be obtained with high accuracy. Combining this result with grandcanonical simulations, we have determined the excess free energy of a stalk as a function of the membrane tension. In order to compute the dependence of the excess free energy of a stalk on the molecular architecture, we have used a semi-grandcanonical ensemble, where Monte Carlo moves mmutatellipids of one architecture into molecules with another architecture and vice versa. In this ensemble, the composition of the mixed bilayer membranes is controlled by the chemical potential difference between the species and we can compute the free energy change upon exchanging lipids with different architecture with relative ease. With these computational techniques we systematically investigate the stability of the stalk structure. The simulations show that the excess free energy of stalks in on the order of 10 kBT, where kBT denotes the thermal energy unit. Stalks are comprised of a few tens of lipid molecules and the excess free energy increases with membrane tension. The stability of a stalk strongly depends on the molecular architecture. Amphiphiles with a large head groups give rise to highly metastable stalks, whereas very asymmetric amphiphiles can even reduce the excess free energy of the stalk to negative values, which correspond to a thermodynamically stable structure.