A new milestoning procedure using Voronoi tessellations is proposed. In the new procedure, the edges of Voronoi cells are used as milestones, and the necessary kinetic information about the transitions between the milestones is calculated by running molecular dynamics (MD) simulations restricted to these cells. Like the traditional milestoning technique, the new procedure offers a reduced description of the original dynamics and permits to efficiently compute the various quantities necessary in this description. However, unlike traditional milestoning, the new procedure does not require to reinitialize trajectories from the milestones, and thereby it avoids the approximation made in traditional milestoning that the distribution for reinitialization is the equilibrium one. In this paper we concentrate on Markovian milestoning, which we show to be valid under suitable assumptions, and we explain how to estimate the rate matrix of transitions between the milestones from data collected from the MD trajectories in the Voronoi cells. The rate matrix can then be used to compute mean first passage times between milestones and reaction rates. The procedure is first illustrated on test-case examples in two dimensions and then applied to study the kinetics of protein insertion into a lipid bilayer by means of a coarse-grained model.

An improved and simplified version of the finite temperature string (FTS) method [W. E, W. Ren, and E. Vanden-Eijnden, J. Phys. Chem. B 109, 6688 (2005)] is proposed. Like the original approach, the new method is a scheme to calculate the principal curves associated with the Boltzmann-Gibbs probability distribution of the system, i.e., the curves which are such that their intersection with the hyperplanes perpendicular to themselves coincides with the expected position of the system in these planes (where perpendicular is understood with respect to the appropriate metric). Unlike more standard paths such as the minimum energy path or the minimum free energy path, the location of the principal curve depends on global features of the energy or the free energy landscapes and thereby may remain appropriate in situations where the landscape is rough on the thermal energy scale and/or entropic effects related to the width of the reaction channels matter. Instead of using constrained sampling in hyperplanes as in the original FTS, the new method calculates the principal curve via sampling in the Voronoi tessellation whose generating points are the discretization points along this curve. As shown here, this modification results in greater algorithmic simplicity. As a by-product, it also gives the free energy associated with the Voronoi tessellation. The new method can be applied both in the original Cartesian space of the system or in a set of collective variables. We illustrate FTS on test-case examples and apply it to the study of conformational transitions of the nitrogen regulatory protein C receiver domain using an elastic network model and to the isomerization of solvated alanine dipeptide.

Phospholipids are the main components of biological membranes and dissolved in water these molecules self-assemble into closed structures, of which bilayers are the most relevant from a biological point of view. Lipid bilayers are often used, both in experimental and by theoretical investigations, as model systems to understand the fundamental properties of biomembranes. The properties of lipid bilayers can be studied at different time and length scales. For some properties it is sufficient to envision a membrane as an elastic sheet, while for others it is important to take into account the details of the individual atoms. In this review, we focus on an intermediate level, where groups of atoms are lumped into pseudo-particles to arrive at a coarse-grained, or mesoscopic, description of a bilayer, which is subsequently studied using molecular simulation.The aim of this review is to compare various strategies to coarse grain a biological membrane. The conclusion of this comparison is that there can be many valid different strategies, but that the results obtained by the various mesoscopic models are surprisingly consistent. A second objective of this review is to illustrate how mesoscopic models can be used to obtain a better understanding of experimental systems. The advantage of coarse-grained models is that these can be simulated very efficiently, so that phenomena involving large systems, or requiring a large number of simulations, can be studied in detail. This is illustrated with the study of the relation between the phase behavior of a membrane and the structure of the phospholipids, and the membrane structural changes due to molecules (such as alcohols, cholesterol and anesthetics) adsorbed to the membrane. We then discuss the effect of transmembrane peptides on the local structure of a membrane and the mechanism of vesicle fusion and fission.