Transition path from two apposed membranes to a stalk obtained by a combination of particle simulations and string method

Transient pores in hemifusion diaphragms

Exchange of material across two membranes, as in the case of synaptic neurotransmitter release from a vesicle, involves the formation and poration of a hemifusion diaphragm (HD). The nontrivial geometry of the HD leads to environment-dependent control, regarding the stability and dynamics of the pores required for this kind of exocytosis. This work combines particle simulations, field-based calculations, and phenomenological modeling to explore the factors influencing the stability, dynamics, and possible control mechanisms of pores in HDs. We find that pores preferentially form at the HD rim, and that their stability is sensitive to a number of factors, including the three line tensions, membrane tension, HD size, and the ability of lipids to "flip-flop" across leaflets. Along with a detailed analysis of these factors, we discuss ways that vesicles or cells may use them to open and close pores and thereby quickly and efficiently transport material.

Membrane fission via transmembrane contact

Division of intracellular organelles often correlates with additional membrane wrapping, e.g., by the endoplasmic reticulum or the outer mitochondrial membrane. Such wrapping plays a vital role in proteome and lipidome organization. However, how an extra membrane impacts the mechanics of the division has not been investigated. Here we combine fluorescence and cryo-electron microscopy experiments with self-consistent field theory to explore the stress-induced instabilities imposed by membrane wrapping in a simple double-membrane tubular system. We find that, at physiologically relevant conditions, the outer membrane facilitates an alternative pathway for the inner-tube fission through the formation of a transient contact (hemi-fusion) between both membranes. A detailed molecular theory of the fission pathways in the double membrane system reveals the topological complexity of the process, resulting both in leaky and leakless intermediates, with energies and topologies predicting physiological events.

Molecular Insights on the Wetting Behavior of a Surface Corrugated with Nanoscale Domed Pillars

Using all-atom molecular dynamics simulation, we investigated the wettability of a surface texturized with nanoscale pillars of domed, rectangular, or cylindrical shapes. The dewetted and wetted states of the gaps between the pillars were related to the Cassie-Baxter (CB) and Wenzel (WZ) states of a macroscopic water droplet resting on top of the pillars. We uncovered the structures and free energies of the intermediate states existing between the CB and WZ states. The contact line of the liquid-vapor-solid interface could not be depinned for the domed pillars due to their smooth curvatures unlike for the rectangular or cylindrical pillars. The liquid symmetrically penetrated down into the gap between the domed pillars by a liquid-vapor interface shape like a paraboloid, while the penetration for the rectangular or cylindrical pillars was often asymmetrical, giving a half-tubular liquid-vapor interface.

Fast bilayer-micelle fusion mediated by hydrophobic dipeptides

To understand the transition from inanimate matter to life, we studied a process that directly couples simple metabolism to evolution via natural selection, demonstrated experimentally by Adamala and Szostak. In this process, dipeptides synthesized inside precursors of cells promote absorption of fatty acid micelles to vesicles, inducing their preferential growth and division at the expense of other vesicles. The process is explained on the basis of coarse-grained molecular dynamics simulations, each extending for tens of microseconds, carried out to model fusion between a micelle and a membrane, both made of fatty acids in the absence and presence of hydrophobic dipeptides. In all systems with dipeptides, but not in their absence, fusion events were observed. They involve the formation of a stalk made by hydrophobic chains from the micelle and the membrane, similar to that postulated for vesicle-vesicle fusion. The emergence of a stalk is facilitated by transient clusters of dipeptides, side chains of which form hydrophobic patches at the membrane surface. Committor probability calculations indicate that the size of a patch is a suitable reaction coordinate and allows for identifying the transition state for fusion. Free-energy barrier to fusion is greatly reduced in the presence of dipeptides to only 4-5 kcal/mol, depending on the hydrophobicity of side chains. The mechanism of mediated fusion, which is expected to apply to other small peptides and hydrophobic molecules, provides a robust means by which a nascent metabolism can confer evolutionary advantage to precursors of cells.

How does curvature affect the free-energy barrier of stalk formation? Small vesicles vs apposing, planar membranes

Using molecular simulations of POPC lipids in conjunction with the calculation of the Minimum Free-Energy Path (MFEP), we study the effect of strong membrane curvature on the formation of the first fusion intermediate—the stalk between a vesicle and its periodic image. We find that the thermodynamic stability of this hourglass-shaped, hydrophobic connection between two vesicles is largely increased by the strong curvature of small vesicles, whereas the intrinsic barrier to form a stalk, i.e., associated with dimple formation and lipid tails protrusions, is similar to the case of two, apposing, planar membranes. A significant reduction of the barrier of stalk formation, however, stems from the lower dehydration free energy that is required to bring highly curved vesicle into a distance, at which stalk formation may occur, compared to the case of apposing, planar membranes.

How proteins open fusion pores: insights from molecular simulations

Fusion proteins can play a versatile and involved role during all stages of the fusion reaction. Their roles go far beyond forcing the opposing membranes into close proximity to drive stalk formation and fusion. Molecular simulations have played a central role in providing a molecular understanding of how fusion proteins actively overcome the free energy barriers of the fusion reaction up to the expansion of the fusion pore. Unexpectedly, molecular simulations have revealed a preference of the biological fusion reaction to proceed through asymmetric pathways resulting in the formation of, e.g., a stalk-hole complex, rim-pore, or vertex pore. Force-field based molecular simulations are now able to directly resolve the minimum free-energy path in protein-mediated fusion as well as quantifying the free energies of formed reaction intermediates. Ongoing developments in Graphics Processing Units (GPUs), free energy calculations, and coarse-grained force-fields will soon gain additional insights into the diverse roles of fusion proteins.

Density Field Thermodynamic Integration (DFTI): A "Soft" Approach to Calculate the Free Energy of Surfactant Self-Assemblies

Thermodynamic integration is one of the most established methods to quantify excess free energies between different metastable states. Excess intermolecular interactions in surfactant assemblies are on the scale of the energy of thermal fluctuations. Therefore, these materials can be deformed and topologically altered via relatively small mechanical stresses. It is thus intuitive to design reaction paths and associated order parameters that exploit the "soft" nature of these materials to mechanically rather than alchemically morph surfactant assemblies from state to state. Here, we propose a novel method coined "density field thermodynamic integration" (DFTI) that adopts the universality and transferability of alchemical methods while simultaneously exploiting the soft excess interactions between surfactant molecules. DFTI was designed for a rapid quantification of the free energy differences between different metastable structures in soft fluid materials. The DFTI method uses an external field coupled to the local density to mechanically morph the system between metastable states of interest. Here, we explored the capability of the DFTI method to swiftly and accurately calculate free energy differences between states. To this aim, we studied two different coarse-grained lipidic surfactant systems: (i) a fusion stalk and (ii) a worm-like micelle. Our results illustrate that DFTI can provide an efficient, versatile, and rather reliable method to calculate the free energy differences between surfactant assemblies.

Pathways connecting two opposed bilayers with a fusion pore: a molecularly-informed phase field approach

A phase field model with two phase fields, representing the concentration and the head-tail separation of amphiphilic molecules, respectively, has been constructed using an extension of the Ohta-Kawasaki model (Macromolecules, 1986, 19, 2621-2632). It is shown that this molecularly-informed phase field model is capable of producing various self-assembled amphiphilic aggregates, such as bilayers, vesicles and micelles. Furthermore, pathways connecting two opposed bilayers with a fusion pore are obtained by using a combination of the phase field model and the string method. Multiple fusion pathways, including a classical pathway and a leaky pathway, have been obtained depending on the initial separation of the two bilayers. The study shed light on the understanding of the membrane fusion pathways and, more importantly, laid a foundation for further investigation of more complex membrane morphologies and transitions.

Energetics of Flap Opening in HIV-1 Protease: String Method Calculations

HIV-1 protease (PR) is the viral protein responsible for virion maturation, and its mechanisms of action remain incompletely understood. PR is dimeric and contains two flexible, symmetry-related flaps, which act as a gate to inhibit access to the binding pocket and hold the polypeptide substrate in the binding pocket once bound. Wide flap opening, a conformational change assumed to be necessary for substrate binding, is a rare event in the closed and bound form. In this study, we use molecular dynamics (MD) simulations and advanced MD techniques including temperature acceleration and string method in collective variables to study the conformational changes associated with substrate unbinding of both wild-type and F99Y mutant PR. The F99Y mutation is shown via MD to decouple the closing of previously unrecognized distal pockets from substrate unbinding. To determine whether or not the F99Y mutation affects the energetic cost of wide flap opening, we use string method in collective variables to determine the minimum free-energy mechanism for wide flap opening in concert with distal pocket closing. The results indicate that the major energetic cost in flap opening is disengagement of the two flap-tip Ile50 residues from each other and is not affected by the F99Y mutation.

Numerical algorithms for solving self-consistent field theory reversely for block copolymer systems

Besides dictating the equilibrium phase diagram, the rugged free-energy landscape of AB block copolymers gives rise to a multitude of non-equilibrium phenomena. Self-consistent field theory (SCFT) can be employed to calculate the mean-field free energy, F [ ϕ A t a r g e t ] , of a non-equilibrium unstable state that is characterized by a given spatial density distribution, ϕ A t a r g e t , in the incompressible system. Such a free-energy functional is the basis of describing the structure formation by dynamic SCFT techniques or the identification of minimum free-energy paths via the string method. The crucial step consists in computing the external potential fields that generate the given density distribution in the corresponding system of non-interacting copolymers, i.e., the potential-to-density relation employed in equilibrium SCFT calculations has to be inverted (reverse SCFT calculation). We describe, generalize, and evaluate the computational efficiency of two different numerical algorithms for this reverse SCFT calculation-the Debye-function algorithm based on the structure factor and the field-theoretic umbrella-potential (FUP) algorithm. In contrast to the Debye-function algorithm, the FUP algorithm only yields the exact mean-field values of the given target densities in the limit of a strong umbrella potential, and we devise a two-step variant of the FUP algorithm that significantly mitigates this issue. For Gaussian copolymers, the Debye-function algorithm is more efficient for highly unstable states that are far away from the equilibrium, whereas the improved FUP algorithm outperforms the Debye-function algorithm closer to metastable states and is easily transferred to more complex molecular architectures.

Defect Annihilation Pathways in Directed Assembly of Lamellar Block Copolymer Thin Films

Defects in highly ordered self-assembled block copolymers represent an important roadblock toward the adoption of these materials in a wide range of applications. This work examines the pathways for annihilation of defects in symmetric diblock copolymers in the context of directed assembly using patterned substrates. Past theoretical and computational studies of such systems have predicted minimum free energy pathways that are characteristic of an activated process. However, they have been limited to adjacent dislocations with opposite Burgers vectors. By relying on a combination of advanced sampling techniques and particle-based simulations, this work considers the long-range interaction between dislocation pairs, both on homogeneous and nanopatterned substrates. As illustrated here, these interactions are central to understanding the defect structures that are most commonly found in applications and in experimental studies of directed self-assembly. More specifically, it is shown that, for dislocation dipoles separated by several lamellae, multiple consecutive free energy barriers lead to effective kinetic barriers that are an order of magnitude larger than those originally reported in the literature for tightly bound dislocation pairs. It is also shown that annihilation pathways depend strongly on both the separation between dislocations and their relative position with respect to the substrate guiding stripes used to direct the assembly.

Activated Wetting of Nanostructured Surfaces: Reaction Coordinates, Finite Size Effects, and Simulation Pitfalls

A liquid in contact with a textured surface can be found in two states, Wenzel and Cassie. In the Wenzel state the liquid completely wets the corrugations while in the Cassie state the liquid is suspended over the corrugations with air or vapor trapped below. The superhydrophobic properties of the Cassie state are exploited for self-cleaning, drag reduction, drug delivery, etc., while in the Wenzel state most of these properties are lost; it is therefore of great fundamental and technological interest to investigate the kinetics and mechanism of the Cassie-Wenzel transition. Computationally, the Cassie-Wenzel transition is often investigated using enhanced sampling ("rare events") techniques based on the use of collective variables (CVs). The choice of the CVs is a crucial task because it affects the free-energy profile, the estimation of the free-energy barriers, and the evaluation of the mechanism of the process. Here we investigate possible simulation artifacts introduced by common CVs adopted for the study of the Cassie-Wenzel transition: the average particle density in the corrugation of a textured surface and the coarse-grained density field at various levels of coarse graining. We also investigate possible additional artifacts associated with finite size effects. We focus on a pillared surface, a system often used in technological applications. We show that the use of a highly coarse-grained density (a single CV) of the fluid in the interpillar region brings to severe artifacts: errors of hundreds of k_BT in the difference of free energy between the Cassie and Wenzel states, of tens of k_BT in the estimate of the free-energy barriers, and erroneous wetting mechanisms. A proper description of the wetting mechanism and its energetics apparently requires a fine discretization of the density field. Concerning the finite-size effects, we have found that the typical systems employed in simulations of the Cassie-Wenzel transition, containing a single pillar within periodic boundary conditions, prevent the complete break of translational symmetry of the liquid-vapor meniscus during the process. Capturing this break of symmetry is crucial for describing the transition state along the wetting process and the early stage of the opposite process, the Wenzel-Cassie transition.

Generalization of the swelling method to measure the intrinsic curvature of lipids

Via computer simulation of a coarse-grained model of two-component lipid bilayers, we compare two methods of measuring the intrinsic curvatures of the constituting monolayers. The first one is a generalization of the swelling method that, in addition to the assumption that the spontaneous curvature linearly depends on the composition of the lipid mixture, incorporates contributions from its elastic energy. The second method measures the effective curvature-composition coupling between the apposing leaflets of bilayer structures (planar bilayers or cylindrical tethers) to extract the spontaneous curvature. Our findings demonstrate that both methods yield consistent results. However, we highlight that the two-leaflet structure inherent to the latter method has the advantage of allowing measurements for mixed lipid systems up to their critical point of demixing as well as in the regime of high concentration (of either species).

Pulling force and surface tension drive membrane fusion

Despite catalyzed by fusion proteins of quite different molecular architectures, intracellular, viral, and cell-to-cell fusions are found to have the essential common features and the nearly same nature of transition states. The similarity inspires us to find a more general catalysis mechanism for membrane fusion that minimally depends on the specific structures of fusion proteins. In this work, we built a minimal model for membrane fusion, and by using dissipative particle dynamics simulations, we propose a mechanism that the pulling force generated by fusion proteins initiates the fusion process and the membrane tension regulates the subsequent fusion stages. The model shows different features compared to previous computer simulation studies: the pulling force catalyzes membrane fusion through lipid head overcrowding in the contacting region, leading to an increase in the head-head repulsion and/or the unfavorable head-tail contacts from opposing membranes, both of which destabilize the contacting leaflets and thus promote membrane fusion or vesicle rupture. Our simulations produce a variety of shapes and intermediates, closely resembling cases seen experimentally. Our work strongly supports the view that the tight pulling mechanism is a conserved feature of fusion protein-mediated fusion and that the membrane tension plays an essential role in fusion.

Minimum free-energy paths for the self-organization of polymer brushes

A methodology to calculate minimum free-energy paths based on the combination of a molecular theory and the improved string method is introduced and applied to study the self-organization of polymer brushes under poor solvent conditions. Polymer brushes in a poor solvent cannot undergo macroscopic phase separation due to the physical constraint imposed by the grafting points; therefore, they microphase separate forming aggregates. Under some conditions, the theory predicts that the homogeneous brush and the aggregates can exist as two different minima of the free energy. The theoretical methodology introduced in this work allows us to predict the minimum free-energy path connecting these two minima as well as the morphology of the system along the path. It is shown that the transition between the homogeneous brush and the aggregates may involve a free-energy barrier or be barrierless depending on the relative stability of the two morphologies and the chain length and grafting density of the polymer. In the case where a free-energy barrier exists, one of the morphologies is a metastable structure and, therefore, the properties of the brush as the quality of the solvent is cycled are expected to display hysteresis. The theory is also applied to study the adhesion/deadhesion transition between two opposing surfaces modified by identical polymer brushes and it is shown that this process may also require surpassing a free-energy barrier.

Process-Accessible States of Block Copolymers

Process-directed self-assembly of block copolymers refers to thermodynamic processes that reproducibly direct the kinetics of structure formation from a starting, unstable state into a selected, metastable mesostructure. We investigate the kinetics of self-assembly of linear ACB triblock copolymers after a rapid transformation of the middle C block from B to A. This prototypical process (e.g., photochemical transformation) converts the initial, equilibrium mesophase of the ABB copolymer into a well-defined but unstable, starting state of the AAB copolymer. The spontaneous structure formation that ensues from this unstable state becomes trapped in a metastable mesostructure, and we systematically explore which metastable mesostructures can be fabricated by varying the block copolymer composition of the initial and final states. In addition to the equilibrium mesophases of linear AB diblock copolymers, this diagram of process-accessible states includes 7 metastable periodic mesostructures, inter alia, Schoen's F-RD periodic minimal surface. Generally, we observe that the final, metastable mesostructure of the AAB copolymer possesses the same symmetry as the initial, equilibrium mesophase of the ABB copolymer.

Exploiting Lipid Permutation Symmetry to Compute Membrane Remodeling Free Energies

A complete physical description of membrane remodeling processes, such as fusion or fission, requires knowledge of the underlying free energy landscapes, particularly in barrier regions involving collective shape changes, topological transitions, and high curvature, where Canham-Helfrich (CH) continuum descriptions may fail. To calculate these free energies using atomistic simulations, one must address not only the sampling problem due to high free energy barriers, but also an orthogonal sampling problem of combinatorial complexity stemming from the permutation symmetry of identical lipids. Here, we solve the combinatorial problem with a permutation reduction scheme to map a structural ensemble into a compact, nondegenerate subregion of configuration space, thereby permitting straightforward free energy calculations via umbrella sampling. We applied this approach, using a coarse-grained lipid model, to test the CH description of bending and found sharp increases in the bending modulus for curvature radii below 10 nm. These deviations suggest that an anharmonic bending term may be required for CH models to give quantitative energetics of highly curved states.

Calculation of membrane bending rigidity using field-theoretic umbrella sampling

The free-energy change of membrane shape transformations can be small, e.g., as in the case of membrane bending. Therefore, the calculation of the free-energy difference between different membrane morphologies is a challenge. Here, we discuss a computational method - field-theoretic umbrella sampling - to compute the local chemical potential of a non-equilibrium configuration and illustrate how one can apply it to study free-energy changes of membrane transformations using simulations. Specifically, the chemical potential profile of the bent membrane and the bending rigidity of membrane are calculated for a soft, coarse-grained amphiphile model and the MARTINI model of a dioleoylphosphatidylcholine (DOPC) membrane.

Coarse-grained molecular dynamics study of membrane fusion: Curvature effects on free energy barriers along the stalk mechanism

The effects of membrane curvature on the free energy barrier for membrane fusion have been investigated using coarse-grained molecular dynamics (CG-MD) simulations, assuming that fusion takes place through a stalk intermediate. Free energy barriers were estimated for stalk formation as well as for fusion pore formation using the guiding potential method. Specifically, the three different geometries of two apposed membranes were considered: vesicle-vesicle, vesicle-planar, and planar-planar membranes. The free energy barriers for the resulting fusion were found to depend importantly on the fusing membrane geometries; the lowest barrier was obtained for vesicular membranes. Further, lipid sorting was observed in fusion of the mixed membranes of dimyristoyl phosphatidylcholine and dioleoyl phosphatidylethanolamine (DOPE). Specifically, DOPE molecules were found to assemble around the stalk to support the highly negative curved membrane surface. A consistent result for lipid sorting was observed when a simple continuum model (CM) was used, where the Helfrich energy and mixing entropy of the lipids were taken into account. However, the CM predicts a much higher free energy barrier than found using CG-MD. This discrepancy originates from the conformational changes of lipids, which were not considered in the CM. The results of the CG-MD simulations reveal that a large conformational change in the lipid takes place around the stalk region, which results in a reduction of free energy barriers along the stalk mechanism of membrane fusion.

Tuning phase structures of a symmetrical diblock copolymer with a patterned electric field

Electric fields can induce the orientation of the phase interfaces of block copolymers and provide a potential method to tune polymer phase structures for nanomaterial manufacture. In this work, we applied self-consistent field theory to study the self-assembly of a diblock copolymer confined between two parallel neutral substrates on which a set of electrodes was imposed to form a patterned electric field. The results showed that an alternatively distributed electric field can induce the formation of a parallel lamellar phase structure, which exists stably only in the system with selective substrates. The phase structure was proved to be sensitive to the characteristics of the electric field distribution, such as the strength of the electric field, the size and position of the electrodes, and the corresponding phase diagram was calculated in detail. The transition pathway of the phase structure from the perpendicular layered phase to the parallel layered phase was further analysed using the minimum energy path method. It is shown that the path and the active energy barrier of the phase transition depend on the electric field strength. Compound electric field patterns that can be designed to control the formation of novel and complex microphase structures were also examined.

Calculating Transition Energy Barriers and Characterizing Activation States for Steps of Fusion

We use continuum mechanics to calculate an entire least energy pathway of membrane fusion, from stalk formation, to pore creation, and through fusion pore enlargement. The model assumes that each structure in the pathway is axially symmetric. The static continuum stalk structure agrees quantitatively with experimental stalk architecture. Calculations show that in a stalk, the distal monolayer is stretched and the stored stretching energy is significantly less than the tilt energy of an unstretched distal monolayer. The string method is used to determine the energy of the transition barriers that separate intermediate states and the dynamics of two bilayers as they pass through them. Hemifusion requires a small amount of energy independently of lipid composition, while direct transition from a stalk to a fusion pore without a hemifusion intermediate is highly improbable. Hemifusion diaphragm expansion is spontaneous for distal monolayers containing at least two lipid components, given sufficiently negative diaphragm spontaneous curvature. Conversely, diaphragms formed from single-component distal monolayers do not expand without the continual injection of energy. We identify a diaphragm radius, below which central pore expansion is spontaneous. For larger diaphragms, prior studies have shown that pore expansion is not axisymmetric, and here our calculations supply an upper bound for the energy of the barrier against pore formation. The major energy-requiring deformations in the steps of fusion are: widening of a hydrophobic fissure in bilayers for stalk formation, splay within the expanding hemifusion diaphragm, and fissure widening initiating pore formation in a hemifusion diaphragm.

Solvent-exposed lipid tail protrusions depend on lipid membrane composition and curvature

The stochastic protrusion of hydrophobic lipid tails into solution, a subclass of hydrophobic membrane defects, has recently been shown to be a critical step in a number of biological processes like membrane fusion. Understanding the factors that govern the appearance of lipid tail protrusions is critical for identifying membrane features that affect the rate of fusion or other processes that depend on contact with solvent-exposed lipid tails. In this work, we utilize atomistic molecular dynamics simulations to characterize the likelihood of tail protrusions in phosphotidylcholine lipid bilayers of varying composition, curvature, and hydration. We distinguish two protrusion modes corresponding to atoms near the end of the lipid tail or near the glycerol group. Through potential of mean force calculations, we demonstrate that the thermodynamic cost for inducing a protrusion depends on tail saturation but is insensitive to other bilayer structural properties or hydration above a threshold value. Similarly, highly curved vesicles or micelles increase both the overall frequency of lipid tail protrusions as well as the preference for splay protrusions, both of which play an important role in driving membrane fusion. In multi-component bilayers, however, the incidence of protrusion events does not clearly depend on the mismatch between tail length or tail saturation of the constituent lipids. Together, these results provide significant physical insight into how system components might affect the appearance of protrusions in biological membranes, and help explain the roles of composition or curvature-modifying proteins in membrane fusion.

Molecular pathways for defect annihilation in directed self-assembly

Over the last few years, the directed self-assembly of block copolymers by surface patterns has transitioned from academic curiosity to viable contender for commercial fabrication of next-generation nanocircuits by lithography. Recently, it has become apparent that kinetics, and not only thermodynamics, plays a key role for the ability of a polymeric material to self-assemble into a perfect, defect-free ordered state. Perfection, in this context, implies not more than one defect, with characteristic dimensions on the order of 5 nm, over a sample area as large as 100 cm(2). In this work, we identify the key pathways and the corresponding free energy barriers for eliminating defects, and we demonstrate that an extraordinarily large thermodynamic driving force is not necessarily sufficient for their removal. By adopting a concerted computational and experimental approach, we explain the molecular origins of these barriers and how they depend on material characteristics, and we propose strategies designed to overcome them. The validity of our conclusions for industrially relevant patterning processes is established by relying on instruments and assembly lines that are only available at state-of-the-art fabrication facilities, and, through this confluence of fundamental and applied research, we are able to discern the evolution of morphology at the smallest relevant length scales-a handful of nanometers-and present a view of defect annihilation in directed self-assembly at an unprecedented level of detail.

Extended Phase-Space Methods for Enhanced Sampling in Molecular Simulations: A Review

Molecular Dynamics simulations are a powerful approach to study biomolecular conformational changes or protein-ligand, protein-protein, and protein-DNA/RNA interactions. Straightforward applications, however, are often hampered by incomplete sampling, since in a typical simulated trajectory the system will spend most of its time trapped by high energy barriers in restricted regions of the configuration space. Over the years, several techniques have been designed to overcome this problem and enhance space sampling. Here, we review a class of methods that rely on the idea of extending the set of dynamical variables of the system by adding extra ones associated to functions describing the process under study. In particular, we illustrate the Temperature Accelerated Molecular Dynamics (TAMD), Logarithmic Mean Force Dynamics (LogMFD), and Multiscale Enhanced Sampling (MSES) algorithms. We also discuss combinations with techniques for searching reaction paths. We show the advantages presented by this approach and how it allows to quickly sample important regions of the free-energy landscape via automatic exploration.

Free energy analysis along the stalk mechanism of membrane fusion

The free energy profile of the stalk model of membrane fusion has been calculated using coarse-grained molecular dynamics simulations. The proposed method guides the lipid configuration using a guiding wall potential to make the transition from two apposed membranes to a stalk and a fusion pore. The free energy profile is obtained with a thermodynamic integration scheme using the mean force working on the guiding wall as a response of the system. We applied the method to two apposed flat bilayers composed of dioleoyl phosphatidylethanolamine/dioleoyl phosphatidylcholine expanding over the simulation box under the periodic boundary conditions. The two transition states are identified as pre-stalk and pre-pore states. The free energy barrier for the latter is confirmed to be in good agreement with that estimated by the pulling method. The present method provides a practical way to calculate the free energy profile along the stalk mechanism.

Directing the self-assembly of block copolymers into a metastable complex network phase via a deep and rapid quench

The free-energy landscape of self-assembling block copolymer systems is characterized by a multitude of metastable minima. Using particle-based simulations of a soft, coarse-grained model, we explore opportunities to reproducibly direct the spontaneous ordering of these self-assembling systems into a metastable complex network morphology--specifically, Schoen's I-WP periodic minimal surface--starting from a highly unstable state that is generated by a rapid expansion. This process-directed self-assembly provides an alternative to fine-tuning molecular architecture or blending for fabricating complex network structures. Comparing our particle-based simulation results to recently developed free-energy techniques, we critically assess their ability to predict spontaneous formation and highlight the importance of nonequilibrium molecular conformations in the starting state and the local conservation of density.

Budding behavior of multi-component vesicles

For multicomponent vesicles, the line tension of domain boundaries and the component-dependent elastic properties (e.g., spontaneous curvatures) are the two most important factors that mediate the budding behaviors of these vesicles. This paper especially focuses on their effects on the budding types and the budding number of a two-component vesicle. We found that the budding number is mainly determined by the component-dependent elastic properties while the budding types mediated by line tensions. A phase diagram is also obtained showing three different types of phase regions: (i) partially budding, (ii) partially bud-off, and (iii) totally bud-off regions. These inspiring results are, however, derived from a very simple spherical-cap model, and have been tested by computer simulations showing good agreement. We emphasize that, besides testing the spherical-cap model, the computer simulation techniques developed in the current work can be easily extended to other systems involving multicomponent vesicles.