pore formation.

Chemistry and Physics of Lipids 185 (2015) 109–128

Contents lists available at ScienceDirect

Chemistry and Physics of Lipids journal homepage: www.elsevier.com/locate/chemphyslip

Mechanics of membrane fusion/pore formation Marc Fuhrmans ∗ , Giovanni Marelli, Yuliya G. Smirnova, Marcus Müller Georg-August-Universität Göttingen, Institut für theoretische Physik, Friedrich-Hund-Platz 1, D-37077 Göttingen, Germany

a r t i c l e

i n f o

Article history: Available online 1 August 2014 Keywords: Fusion Pore formation Simulation Coarse-grained models

a b s t r a c t Lipid bilayers play a fundamental role in many biological processes, and a considerable effort has been invested in understanding their behavior and the mechanism of topological changes like fusion and pore formation. Due to the time- and length-scale on which these processes occur, computational methods have proven to be an especially useful tool in their study. With their help, a number of interesting findings about the shape of fusion intermediates could be obtained, and novel hypotheses about the mechanism of topological changes and the involvement of peptides therein were suggested. In this work, we try to present a summary of these developments together with some hitherto unpublished results, featuring, among others, the shape of stalks and fusion pores, possible modes of action of the influenza HA fusion peptide and the SNARE protein complex, the mechanism of supported lipid bilayer formation by vesicle spreading, and the free energy and transition pathway of the fusion process. © 2014 Elsevier Ireland Ltd. All rights reserved.

1. Introduction Lipid bilayers are aggregates of large numbers of individual lipid molecules. Simply put, these layers are held together by the lipids’ amphiphilic nature in an attempt to isolate their hydrophobic tails from the aqueous environment. Lipid bilayers combine properties of solids and liquids in that they maintain their structure at the macroscopic level, but at the same time allow the lipids to diffuse and deform, changing both their location and conformation within the boundaries dictated by the topology of the aggregate (Singer and Nicolson, 1972; Vereb et al., 2003). This dual nature is a requirement for their biological function as cell membrane. These membranes are a crucial part of the organization of living organisms and need to reliably maintain the division into cellular and subcellular compartments. However, at the same time they need to be flexible enough to allow (regulated) transport of molecules ranging in size from relatively small water molecules or ions on the one hand to complete strands of mRNA on the other. Also, the cell membrane has to accommodate a large amount of different proteins serving various functions including enzymatic activity, transport, signal transduction, and cell–cell recognition. In fact, with the respiratory chain and photosynthesis, the very heart of the cells’ energy metabolism is not only located right inside the cell membrane, but directly requires the membrane for its function (Alberts et al., 2002).

∗ Corresponding author. Tel.: +49 551399566. E-mail address: [email protected] (M. Fuhrmans). http://dx.doi.org/10.1016/j.chemphyslip.2014.07.010 0009-3084/© 2014 Elsevier Ireland Ltd. All rights reserved.

The actual topology of the compartmentalization is not constant, but subject to frequent, though carefully regulated, changes. These changes can be divided into three groups: poration-, fission- and fusion-events. The most straightforward of these changes is the poration, in which a hole in the lipid bilayer is introduced. In fission events, an initially connected region of space surrounded by a continuous lipid bilayer is divided into two disconnected regions of space, each surrounded by its own closed bilayer shell. The reversion of this process is fusion events, in which two initially separate bilayer shells are combined into a single closed lipid bilayer surrounding a continuous volume. Since the majority of this review will be discussing findings related to fusion, we will describe the fusion process in a bit more detail. Fig. 1 schematically illustrates the terms used in this review and gives a summary of the suggested pathways published in literature. As can be seen, except for the net effect described above, what is going on at the molecular level is not known with certainty at present. However, it is a logical requirement, and therefore generally agreed, that a connection between the two separate bilayers needs to be introduced, and at least one pore has to be opened. The first step of the connection introduced between the two-bilayer shells is usually referred to as stalk. The stalk is understood to be a metastable bridge connecting the hydrophobic tails of the outer (cis) monolayers of the two membranes about to be fused, whereas the inner (trans) monolayers remain at approximately the same distance as before the connection. The second stage of the fusion process that is a common feature of all suggested pathways is the last step that completes the topological changes involved in the fusion process. This state is conventionally called a fusion pore, and describes an

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M. Fuhrmans et al. / Chemistry and Physics of Lipids 185 (2015) 109–128

Fig. 1. Schematic illustration of the fusion process, trying to give an overview of the suggested pathways connecting the fusion intermediates stalk (A), hemifusion diaphragm (B), fusion pore (C), circular stalk (D), stalk-pore complex (E), and -shaped hemifusion diaphragm (F). Black arrows represent radial stalk expansion, red arrows indicate linear stalk elongation, and blue arrows show pore formation. A detailed description is given in the article text. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of the article.)

hourglass-shaped section of bilayer with a lumen connecting the interiors of the fused bilayer shells. It is therefore not a pore in the sense of a perforation of the bilayer, and the bilayer of the fusion pore is in fact already continuous and can be formed into a closed sphere without necessity of further topological changes. Apart from these two elements, however, the exact pathway and sequence of events are not known with certainty, and it is not unlikely that the fusion process follows different pathways under different conditions. A commonly accepted option is the stalkhemifusion route (Chernomordik and Kozlov, 2005), in which the stalk radially expands in area until the trans monolayers meet and form a bilayer that separates the two volumes as a so-called hemifusion diaphragm. Subsequent rupture of this diaphragm forms the fusion pore and thereby completes the fusion process. This corresponds to the sequence ABC in Fig. 1. An alternative to this route has been reported as the linear stalk elongation (Müller et al., 2002; Marrink et al., 2008; Risselada et al., 2012) in which the stalk does not perform a radial expansion, but elongates linearly along a circular path. In this way the trans monolayers do not make direct contact, but the creation of a closed, circular stalk forms a small third compartment in-between the bilayers. In this scenario, two pores need to open to connect all compartments: one to form a topology similar to the hemifused state (except that the diaphragm originates from different membrane leaflets), and a second to form the final fusion pore. This is illustrated by the sequence ADFC in Fig. 1. However, a predicted mutual facilitation between stalks and pores supports such mechanism, and a sequence in which the first pore opens before the stalk elongation has been argued to be an energetically feasible, leaky fusion pathway (Katsov et al., 2006), corresponding to the sequence EFC in Fig. 1. A third option completely foregoes the hemifusion diaphragm as a metastable state and finds rapid fusion directly after stalk formation without further interruption (Kasson et al., 2006), which is shown in Fig. 1 as the sequence AC. Due to the microscopic level on which these events occur (nm) and the very short time scale (␮s) over which they persist, it is near impossible to gain direct experimental insight into the mechanism of the pathway and the nature of the intermediate states. However, spectroscopic methods have been employed to study the interesting phase behavior of lipid mixtures as reviewed by Seddon (1990), Seddon and Templer (1995). Since some of these lyotropic phases share structural key elements with fusion intermediates, it is possible to make predictions on the fusion mechanism based on changes in the phase diagram. An example for this would be the induction of inverted phases by certain fusion peptides (Yeagle

et al., 1991; Epand and Epand, 1994; Epand et al., 1994; Davies et al., 1998; Peisajovich et al., 2000; Aranda et al., 2003), which has been interpreted as an indication of the fusogenicity of these peptides. Another experimental method is to use fluorescent labels on the different lipid vesicles and their cargo, which makes it possible to trace events like stalk formation via lipid mixing, and pore formation or completion of the fusion process via content leakage or mixing, respectively. A good overview of these findings is given in a recent review by Jahn et al. (2003). If the goal is to gather information on the exact shape of a fusion intermediate or on the mechanism of a certain step of the process, it is therefore necessary to employ theoretical calculations and/or simulations. A common strategy for calculations is to consider the lipid bilayer as an elastic sheet and minimize the free energy to find the optimal shape for a given topology (Markin and Albanesi, 2002; Kozlovsky and Kozlov, 2002). This approach tries to relate the membrane behavior to experimentally known physical properties of lipid bilayers like, e.g., bending stiffness and compressibility. In addition, such calculations require comparatively little computational resources and are therefore widely available. In many cases, however, a particle based description of the lipid bilayer is necessary to capture the complex interactions of the individual lipids, especially if additional, non-lipid, agents like, e.g., peptides are also involved. On top of that, only in simulations it is possible to directly observe topological changes as they happen, and to follow them at molecular resolution. Thanks to the universality of the behavior of lipid bilayers, a wide range of models using different levels of detail is able to correctly reproduce the experimentally known phenomena (Müller et al., 2006), and atomistic (Berger et al., 1997), as well as a large number of coarse-grained and even solvent-free models are available, as reviewed by Venturoli et al. (2006). The choice of model should reflect the properties believed to play a role in the process under observation. It should be noted, however, that due to the collective nature of lipid aggregates, it is often necessary to simulate large numbers of molecules for a considerable time to realistically represent most processes. As every particle and interaction included in the simulation increases the computational costs, coarse-graining is a necessity, but has been shown to be successful in studying many different aspects of lipid behavior, as recently reviewed in Müller et al. (2006), Venturoli et al. (2006), Marrink et al. (2009). The aim of this review is to give a perspective onto recent findings in the field of membrane fusion and poration, with a focus on the different pathways along which these topological changes occur. Due to this focus, the presented findings are based on theoretical calculations and simulation which, at the moment,


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provide an unparalleled possibility to discern the exact morphology of intermediate structures at molecular resolution. However, due to the inherent model character of all results obtained with these methods, the results have to be validated and checked for agreement with experimental observations, and we will take care to relate the presented findings to available experimental data. We will subdivide the review into reports concentrating on the shape of fusion intermediates (Section 3), reports focusing on the mechanism of topology changes involved in fusion and vesicle spreading (Section 4), and reports combining the previous two aspects by studying the optimal transition pathway according to its free energy (Section 5). In addition, we will start the article with a short overview of the theoretical and simulation methods used, aiming to give an impression of their different scope and limitations to the unfamiliar reader (Section 2).

(Hamm and Kozlov, 2000). In addition, not all molecular degrees of freedom are captured by the description, and it is assumed that the lipids’ hydrophobic core is incompressible and no lateral stretching of the bilayer occurs, which may not be a good approximation in the interstices at the three-monolayer junctions encountered in fusion intermediates or in the highly curved stalk structure. Continuum models have predominantly been used to obtain the shape and free energy of lipid aggregates, and are well established as tools for that purpose. However, they may have additional use in studying the dynamics of such systems, or be used to embed a particle-based description in a hybrid simulation scheme, as reviewed by Brown (2008).

2. Overview of methods

While elastic theory represents a top-down approach in that it uses phenomenological properties like the energy of bending or the energy of splay and tilt as input parameters for its models, molecular dynamics (MD) or Monte Carlo (MC) simulations are more of a bottom-up approach as they start with a particle-based interaction scheme. Nevertheless, the interaction parameters are still chosen so that a selection of important phenomenological properties is matched. If an atomistic representation is desired, the so-called “Berger lipids” (Berger et al., 1997) are a popular choice. However, even with access to large distributed computing networks (Kasson et al., 2010), the limitations with respect to system size and simulation time are an important factor. In addition, since coarse-grained models are very powerful at capturing the behavior of lipid mixtures (Müller et al., 2003), it is likely that in the foreseeable future coarsegrained models will precede atomistic representations in testing new hypotheses of membrane fusion, especially when changes of topology in which large numbers of molecules are involved have to be simulated. However, since it is possible to create atomistic coordinate sets from coarse-grained simulations (Rzepiela et al., 2010), atomistic representations can be used to study crucial stages of coarse-grained trajectories in higher resolution. The time and length scales obtainable by coarse-grained models, ms and ␮m, are very attractive for the simulation of topological changes in lipid bilayers in which even the smallest topological elements like stalks or pores require the collective interplay of a large number of lipids. The current models are able to capture the structural elements and enough of their environment to study the phenomenon and its dependence on the immediate surroundings like presence of additional molecules, local curvature or tension. However, if simulation on a significantly larger, e.g., cellular scale, is desired, an accurate description of biological systems needs to take into account additional factors like the rather complex environment as well as the cytoskeleton or the glycocalyx, which go beyond the scope of simple lipid models reviewed in this article, but present an interesting challenge for future simulations. In this article, we will mainly discuss findings using the popular MARTINI model (Marrink et al., 2007; Monticelli et al., 2008), and an even coarser, solvent-free lipid model (Hömberg and Müller, 2010). As coarse-grained models, both face the general dilemma of having to compensate a loss of degrees of freedom with effective, coarsegrained interactions. Nevertheless, compared to calculations based on elastic theory they are much less limited in the bilayer configurations they can present, and can even reproduce topological changes. However, it is not necessary that a parameter set that has been obtained to match one aspect of lipid behavior is equally suited to reproduce other aspects with equal precision, and one should always seek experimental validation for hypotheses based on simulations.

2.1. Elastic theory If the general shape of a fusion intermediate is available from intuition or experimental evidence, it is possible to refine that shape by formulating a free energy density (i.e., free energy per unit area) and optimizing the surface so that the integral over the free energy density is minimized. Such a free energy density should be based on known physical properties of lipid bilayers. A popular approach is based on the Helfrich energy of bending g=

m 2 (h − h0 ) + g k, 2

(1)

where h and k are the mean and Gaussian curvature, respectively, and h0 is the spontaneous curvature (Helfrich, 1973). The bending rigidity m is readily available from experimental measurements (Niggemann et al., 1995). The Gaussian curvature modulus g is much harder to obtain (Siegel and Kozlov, 2004), but is usually not required, since the integral of the Gaussian curvature over the surface of the stalk is a constant that does not depend on the stalk’s exact shape, and the Gaussian part of the energy contribution can safely be neglected in the calculation of the optimal stalk geometry, which has become customary since the first application to the shape of a stalk (Kozlov and Markin, 1983). One should, however, note that Eq. 1 is only valid for small curvature, and some care needs to be taken when assuming that m is constant even in highly curved fusion intermediates. Since each monolayer has its own spontaneous curvature, it is best to treat the bilayer as two surfaces corresponding to the monolayers. This requires to make assumptions on how the monolayers are coupled, which is especially difficult for multi-bilayer junctions like the stalk or hemifusion diaphragm. This problem can be somewhat lessened by a more detailed model that allows the lipids to adapt their orientation to improve the packing efficiency. Such a model has been formulated by Kozlovsky and Kozlov (2002). It expresses the energy in terms of lipid splay J and tilt t. The lipid splay J = divn describes the mean curvature for pure bending, but also allows for cases in which the orientation of the lipids n does not remain parallel to the surface normal N of the monolayer. The tilt t = n/(n · N) − N is a measure for this deviation between the orientation of the lipids and the monolayer. The total energy per unit area in this model is defined as s t (J − J0 )2 + t2 , f = (2) 2 2 where the spontaneous splay J0 is equal to h0 and the splay modulus s is equal to h (Hamm and Kozlov, 2000), and t is the tilt modulus. This model provides a more complete description than Eq. 1, but it still represents an approximation that is primarily valid for radii of curvature that are large compared to the monolayer width

2.2. Coarse-grained models

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2.3. Self-consistent-field theory Like MD simulations, self-consistent-field theory (SCFT) calculations (Matsen, 2006) are a bottom-up approach and start with a particle-based description of the interactions in the lipid bilayer. Unlike molecular-dynamics simulations, however, these interactions are not computed pair-wise, but are instead reformulated via the (exact) Hubbard–Stratonovich transformation as a system of independent particles interacting with an external fluctuating field. This reformulation lends itself to simplification with a mean-field approximation, which ignores the fluctuations and allows to self-consistently calculate the mean field. In SCFT, the lipids are typically represented by polymer-like Gaussian chains, because efficient numerical procedures have been developed for this standard model. In principle, however, other molecular architectures can be considered using a partial enumeration scheme (Ben-Shaul et al., 1985; Szleifer and Carignano, 1996; Müller and Schick, 1998). One important advantage of SCFT is the ease with which the free energy of self-assembled structures can be calculated. This allows to systematically explore the free energy of putative transition states of membrane poration, stalk formation, and bilayer fusion (e.g., Katsov et al., 2004, 2006; Müller and Schick, 2011). It is possible to constrain the system to a certain value of a chosen reaction coordinate, e.g., the diameter of a stalk, and calculate the free energy of the constrained system. For metastable states (local minima) and transition states (saddle points), the constraining force vanishes, and the obtained value is independent on the choice of the reaction coordinate. However, the details of the transformation path between these states depend on the reaction coordinate. Moreover, the choice of a suitable reaction coordinate requires physical insights, and a wrong reaction coordinate may lead a topological change along a biased path that does not correspond to what one would observe in molecular simulations.

The choice of the degrees of freedom that describe the structure is crucial. Since lipids in the fluid phase of a membrane laterally diffuse in a monolayer and change between monolayers on large time scales, the configuration of a specific lipid (e.g., center-of-mass position of the 1024th lipid) is not a suitable variable. For these collective transformations of the membrane topology, the structure can be characterized by the spatially varying distribution function P({r}1 ) of single-lipid conformations, where {r}1 = r1 , . . ., rN denotes the N coordinates of the interaction centers that define the conformation of a single lipid molecule. Such a single-lipid conformation quantifies the position and orientation of the lipid molecule as well as its internal conformational degrees of freedom. The free energy is a functional of this single-lipid distribution function (Müller, 2006). In practical calculations, however, the single-lipid distribution function contains too much information and, in previous applications (Cheng et al., 2010; Ting et al., 2011; Müller et al., 2012), a simple scalar collective variable field, m(r), that quantifies the local density difference between hydrophilic and hydrophobic interaction centers has been employed to approximately characterize the structure. This collective variable can be computed from the microscopic coordinates of all lipids according to

ˆ m(r|{r}) ∝

˛,hydrophilic

ı(r − rˇ )

(3)

ˇ,hydrophobic

where the sums run over all hydrophilic and hydrophobic interaction centers, respectively. Via the summation over all particles in this definition of the collective variable one automatically accounts for the permutation symmetry of identical lipids. Characterizing the structure only by this scalar order parameter field one can define the free energy functional, F[m], by the constrained partition function of all system configurations in which the microscopic collective ˆ variable, m(r|{r}), is compatible with the order parameter m(r):

F[m] ≡ −kB T ln

D[{r}]e−

H({r}) kB T

ˆ ı(m(r) − m(r)).

(4)

r

2.4. String method A way to overcome the difficulties in calculating the transition path of topological changes described in the previous section is the (improved) string method. This method also works with MD simulations and alleviates the problem of having to choose a potentially biased reaction coordinate. Often changes of the membrane topology can be regarded as thermally activated processes. Such transformation processes are characterized by one or multiple transition states. The rate, r, with which the transformation proceeds, is related to the excess freeenergy, F, of the transition state via r ∼ exp(− F/kB T). Such a description will be appropriate if the activation barrier, F, significantly exceeds the thermal energy kB T (rare event) and, more precisely, if different trajectories of systems that change the membrane topology will pass through a similar sequence of states during the transformation process. Then, the transition state is defined as the state with maximal free energy along the transformation path (Ren and Vanden-Eijnden, 2002; Vanden-Eijnden, 2006, 2010; Dellago et al., 1998; Bolhuis et al., 2002; Dellago and Bolhuis, 2009). Under these conditions one can use a free-energy functional, F, which assigns an excess free energy to each structure, to capture the transformation path. A transformation path connects two (meta)stable states via a sequence of intermediate structures. This sequence of structures is denoted as the “string”. The contour variable, 0 ≤ s ≤ 1, along this string serves as a reaction coordinate that quantifies how far the transformation has proceeded. The most likely transformation path consists of a sequence of structures that minimize the free energy in all directions perpendicular to the string – the minimum free energy path (MFEP) (Ren and VandenEijnden, 2007).

ı(r − r˛ ) −

Unlike SCFT calculations, in computer simulation of particle systems the free energy, F[m], of a structure characterized by the order parameter m is not directly accessible because they cannot be expressed as thermal averages of an observable that is a function of the particle coordinates. However, the chemical potential, (r) ≡ (ıF[m]/ım(r)), can be obtained by field-theoretic umbrella sampling (Müller, 2009, 2011; Müller and de Pablo, 2013). To this end, one augments the interactions of the particle model by an additional Hamiltonian of the form Hfup ({r}) kB T

=

dr

2 ˆ [m(r) − m(r|{r})] 2

(5)

The Boltzmann factor of this field-theoretic umbrella potential converges to the ı-function constraint in Eq. (4) in the limit → ∞ (Müller, 2009; Maragliano and Vanden-Eijnden, 2006)

H →∞ fup

exp −

kB T

→

ˆ ı[m(r) − m(r|{r})]

(6)

r

where we have ignored proportionality constants. In this limit the free energy of the restrained system, F [m], with the field-theoretic umbrella potential converges toward the free-energy functional, F[m], and the chemical potential can be calculated according to

(r|m) ≡

ıHfup ıF →∞ ıF ˆ =

= kB T m(r) − m(r)

→ ım(r) ım(r) ım(r) = (r|m)

(7)

where the average · · · is performed in the restrained system. Independent from , this average has to be sampled for about


one molecular relaxation time for accurate calculation of the local chemical potential (Müller, 2009). To ensure that the limit is a valid assumption, one has to chose sufficiently large so that the energetic contributions from the umbrella potential are higher than those of any other interactions in the system and the chemical potential no longer depends on . To implement the above equations in numerical calculations, the spatial variation of the order-parameter field, m(r), is approximated by its values, mc = m(rc ), on a collocation lattice, where rc denotes the location of a lattice site and the discrete index c runs over all NV sites of the collocation lattice. Thus the free-energy functional is approximated by a free-energy function, F({mc }), of the values of the order parameter on the site of the collocation lattice. An efficient and straightforward algorithm for obtaining the MFEP in the free-energy landscape, F({mc }), which can be combined with field-theoretic umbrella sampling (“on-the-fly” string method (Maragliano et al., 2006; Maragliano and Vanden-Eijnden, 2007)), is the improved string method (Ren and Vanden-Eijnden, 2007). In this method, one discretizes the MFEP mc (s) in a finite set of configurations, ms,c where the contour parameters s adopts Ns discrete values between 0 and 1. The numerical procedure is then comprised of two steps: 1. For each configuration at position s along the string reduce the free energy: The aim of this step is to improve the order parameter field that is used to constrain the system to the configurations of the string. In the simplest case, one can do this by using a down-hill descent. In this approach, one updates the order parameter at each point of the collocation lattice according to ms,c → ms,c − s,c where s,c ≡ (∂F({ms,c })/∂ms,c ) denotes the local chemical potential of the structure ms,c and is a small parameter. This procedure is sufficient to locate the saddle points of the MFEP but the MFEP does not locally conserve the density and therefore need not describe a physical path (Müller and Sun, 2013). Alternative techniques that account for local conservation laws are available (Maragliano et al., 2006; Zhang et al., 2012). 2. Uniformly redistribute the configurations along the string: During the first step all configurations move down-hill on the free-energy surface away from the saddle-points and toward the (meta)stable free-energy minima, which means that the spacing of the configurations along the string is no longer even. One therefore has to restore the uniform distribution along the path, which can be done by interpolating the value of the order parameter at each site c of the collocation lattice by a spline in the contour variable s, i.e., ms,c → mc (s). Using this continuous inter polation in s, one defines the new string by ms,c = mc (s) with s = 0, 1/(Ns − 1), . . ., 1. This re-parameterization acts like a thermodynamic force along the string and compensates the downhill motion of the first step in the direction along the MFEP. If this procedure converges the free energy is minimal in all directions except for the direction along the string, which exactly is the defining condition of the MFEP. The main benefit of this technique is its ability to explore the free-energy landscape and capture the most probable path of collective transformations including the rate-determining saddlepoints. Importantly, the improved string method does not a priori require the definition of a reaction coordinate. The contour variable s provides such a parameter, but it need not be obviously related to a simple change of a structural characteristic. 2.5. Markovian state models If the activation barriers are not much larger than the thermal energy scale, one can also use a large number of “ordinary” MD

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simulations to construct a kinetic model of transition pathways. For these Markovian state models (Singhal et al., 2004), data from simulations of (usually irreversible) topological changes are used to obtain a series of snapshots at high frequency which are used as starting point for a large number of short MD simulations. By clustering and comparing the initial and final state of these simulations, a set of transition probabilities between macrostates can be build, which, together with the number of snapshots in each state, allow calculation of free energies for the fusion intermediates corresponding to the identified clusters. Due to the large number of simulations that need to be analyzed, it becomes necessary to automate the morphological description of the lipid aggregate. A relatively straightforward reaction coordinate to monitor fusion is to use lipid mixing between the different leaflets involved in the fusion process (Kasson et al., 2006). However, the morphological information that can be extracted with this approach is limited, and a more direct descriptor of the morphology of the system like persistent voids (Kasson et al., 2007) can improve the model. This approach is not linked to a specific model and can therefore include different levels of detail, as long as one has the resources available to perform the large number of simulations needed. A central assumption is that the states are indeed Markovian, i.e., not influenced by their history, which can be expected to be true for topological changes like fusion, but will not hold when hysteresis effects are important like for the liquid-to-gel phase transition of lipid bilayers. 3. Shape of fusion intermediates When talking about the shape of lipid aggregates, it is worthwhile to note that “shape” is a somewhat misleading term, as it is normally used as a property of ordinary, “every-day” objects. When we are talking about the shape of a lipid aggregate, however, the dimensions of the object at hand are on the same length scale as the molecules that the object is composed of themselves, and the concept of shape can only be taken as a crude approximation. In fact, with the relatively small numbers of lipid molecules comprising the rim of a pore or a stalk, it is clear that the lipids’ order at any given moment will not be a tidy arrangement distributed nicely across the structure like some conceptual sketches suggest, but rather a wildly fluctuating “conglomerate” that only reveals a shape when averaged over time. We will therefore compare different methods to obtain the shape of lipid aggregates, examining how important the lipids’ molecular aspect is in the description. The focus will be primarily on the shape of stalks, with a short discussion of the fusion pore in comparison. 3.1. Stalk shape Under the right conditions, e.g., low hydration, negative spontaneous curvature, asymmetric membranes or tension, stalks are stable structures that can even be the equilibrium state in certain areas of the phase diagram, giving rise to the rhombohedral stalk phase (Yang and Huang, 2002, 2003). In addition, due to their high metastability, it is also possible to form and simulate stalks in systems for which it does not constitute a thermodynamically stable equilibrium, and a local energy minimum is sufficient to simulate them long enough to study their shape. For the following discussion, we will distinguish between the internal, molecular organization or ‘structure’ of the stalk, and its overall, external shape, the ‘morphology’ or ‘geometry’ of the stalk. Recent X-ray experiments (Aeffner et al., 2012) and molecular simulation studies (Daoulas and Müller, 2013) predict that the

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Fig. 2. Lipid representation in the different models. The hydrophobic tail beads are shown in green, and the hydrophilic head beads in blue. (a) Polymer model: 11 hydrophilic and 21 hydrophobic beads. (b) Solvent-free model: a head of 4 hydrophilic beads is connected to a hydrophobic bead bearing two 4 bead long hydrophopbic tails. (c) MARTINI model (POPC): the lipids consist of several different bead types (left) that can be mapped onto hydrophobic tails and hydrophilic head as shown on the right. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of the article.)

geometric shape of the stalk is an universal property that is surprisingly independent on the molecular architecture (e.g., spontaneous curvature of the lipids) and external factors like membrane tension. Despite such a conserved stalk shape in parameter space, however, the concomitant excess free-energy of the stalk substantially depends on both molecular architecture and tension (Daoulas and Müller, 2013; Norizoe et al., 2010). Aside from the free energy cost due to the commonly discussed bending stresses and tilting of the amphiphiles (Kozlovsky and Kozlov, 2002), a considerable part of the stalk’s excess free energy has been attributed to the presence of hydrophobic interstices, which are characterized in both X-ray studies and simulations by two less dense areas in the hydrophobic region above and below the center of the stalk. These regions can only be filled by bending the trans monolayers and/or stretching of the lipid tails. Both of these options represent energetically unfavorable conformational frustrations (Kozlovsky and Kozlov, 2002), that are avoided by the reduced density at the interstices. Experiments have shown that the presence of small hydrophobic molecules, such as e.g., hexadecane, enhances the rate of lipid mixing and decrease the phase-transition temperature toward inverted lipid phases independent of their molecular shape (Pincet et al., 2005), suggesting that small oil-like molecules may favorably partition in the sparse hydrophobic regions of the stalk, thus reducing the present frustration and excess free energy of the stalk. Such a partitioning, however, has never been directly observed nor is it known how the stalk’s structure is affected by their presence. In order to test how well the apparent universality of the stalk shape extends to its representation in different coarse-grained models in MD simulation, we performed a series of simulatations of stalks using the MARTINI model (Marrink et al., 2007) and a solvent-free model (Hömberg and Müller, 2010). In addition, we used available data from MC simulations based on a polymer-like lipid model (Katsov et al., 2004; Müller et al., 2012) as comparison. The solvent-free model includes only two bead species, one for the hydrophobic tail beads and one for the hydrophilic head beads. As described in Hömberg and Müller (2010), the beads comprising the lipids are chained together by application of harmonic bond potentials and a simple angle potential, while a weighted-density functional of the bead densities defines the non-bonded interactions. The specific interaction parameters were chosen so that the lipids spontaneously assemble into bilayers and can be found in Fuhrmans and Müller (2013). The polymer-like model (Müller et al., 2012) also includes only two bead species and represents the lipids as linear diblock copolymers. In contrast to the solvent-free model, however, it also includes solvent as linear homopolymers of the

same length as the lipid molecules. In the MARTINI model, the lipids are composed of a larger number of different particle types, parameterized to reproduce the respective hydration free energies of the moiety they represent. Water is included as a single bead representing 4 atomistic water molecules, corresponding to the general coarse-graining level of roughly 4:1. A graphical comparison of the lipid representation in the different models is shown in Fig. 2. For the MD simulations with MARTINI and the solvent-free model, we simulated a system consisting of a single stalk between two closely apposed lipid bilayers. The stalks were artificially introduced with the help of an additional external potential. After stalk formation, the extra potential was removed and the systems were equilibrated before the actual analysis. The simulations were performed in the NPT ensemble, arranging the pressure coupling so that the bilayers were at zero tension. In addition, we examined the effects of the hydration level for MARTINI and the solventfree model. In the MARTINI model, this could be accomplished in a straightforward fashion by varying the number of solvent beads in the inter-membrane space to represent between 5 and 12 water molecules per lipid. For the solvent-free model, the ‘implicit’ solvent is not present as particles in the simulations, but is indirectly taken into account by the strength of the repulsion between the headgroup beads governed by the vBB coefficient, which we varied between −0.1 (weakly dehydrated) and −1.0 (strongly dehydrated). An overview of the simulated systems is given in Table 1. For each system, at least 500 statistically independent snapshots were used for the analysis. To account for the different length scales inherent to the simulation models, we used the bilayer thickness as a natural unit to allow an unbiased comparison between the models. Fig. 3 gives an overview of the different morphological descriptors used in the analysis: dt is the distance of the trans monolayers, dw and db are the water layer and bilayer thickness at the system’s periphery, respectively, and ds is the stalk neck diameter. Table 1 Overview of the simulated systems. Model

Hydration level

Hydrophilic interactions

Number of water molecules per lipid

vBB

solvent-free

–

MARTINI polymer

5, 6, 7, 8, 9, 10 7

−1, −0.5, −0.1 – –


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Table 2 Ratios of the lengths defining the stalk geometry for the different models and X-ray experiments. Nw /Nl is the number of water molecules between the two bilayers divided by the number of lipids in the proximal monolayers, RH is the relative humidity. dw (nm) 3.33 3.88 2.04 2.11 2.24 2.12 2.75 4.08 2.02 Fig. 3. Morphological descriptors used for the analysis of the stalk: dt is the trans monolayer distance, dw and db are the water layer and bilayer thickness at the periphery, respectively, and ds is the stalk neck diameter.

In experiments (Aeffner et al., 2012) these descriptors are evaluated using electron density profiles. The maximum of the electron density profile corresponds to the lipid headgroup region, and is associated with the phosphate group of a lipid molecule. In our coarse-grained simulations we can define a similar surface by calculating number density profiles, and identifying the position of the equimolar dividing surface, i.e., the region where the hydrophobic and hydrophilic densities are equal, with the phosphate groups. To investigate whether the model-specific properties give rise to differences in the stalk structure, we compared stalks obtained using the three models described above. Fig. 4 shows axial–radial number density plots of the stalks obtained with the three models. The normalized descriptors of the stalk geometry for each model are summarized in Table 2, using different levels of hydration for the MARTINI and the solvent-free model. From this quantitative comparison of stalk morphologies observed in different simulation models and experiments, we conclude that the structure is rather insensitive to the details of the interactions. Most notably among these, the description of water in the computational models vastly differs from spherical Lennard–Jones particles representing four water molecules in the MARTINI model over polymer-like solvent-chains in the

X-ray, RH = 78% (DPhPC) X-ray, RH = 34% (DOPC) MARTINI, 5Nw /Nl MARTINI, 6Nw /Nl MARTINI, 7Nw /Nl solvent-free, vBB = −1.0 solvent-free, vBB = −0.5 solvent-free, vBB = −0.1 polymeric

dt /db

ds /db

dw /db

2.3 2.1 2.1 2.2 2.3 2.13 2.33 2.16 2.6

0.9 0.91 0.95 0.78 0.74 1.13 1.08 0.97 0.8

0.68 0.73 0.67 0.70 0.73 0.60 0.78 1.16 1.1

The experimental data were adopted from Aeffner et al. (2012).

polymer-like model to the complete absence of a particle-based representation in the solvent-free model, and it appears that, while the free-energy of the entire system is affected (Norizoe et al., 2010; Daoulas and Müller, 2013), the actual stalk geometry remains largely unaffected by the structure of the water layer and its influence on the lipid interactions within and across the bilayers. The water between the bilayers next to the stalk should therefore be considered directly hydrating the lipid headgroups with virtually no bulk water remaining. Remarkable is also that in the X-ray diffraction measurements (Aeffner et al., 2012), stalks were found to form when the bilayer separation went below a critical distance that did not depend on the lipid type, and lipids with different headgroups showed the same behavior. Since the bilayer thickness in the experimentally studied systems was approximately constant, these findings agree nicely when compared to the (normalized) results of our simulations, and it appears that stalks become stable at the same volume ratio of hydrophobic to hydrophilic components. Indeed in recent self-consistent field calculations (Daoulas and Müller, 2013), the stalk geometry was demonstrated to be largely independent on the external conditions such as membrane tension, and to be defined by the architectural asymmetry, i.e., the relative length of headgroup and tails. In addition, the morphological descriptors corresponding

Fig. 4. Plot of the number density of the hydrophobic units, normalized by the bulk density and truncated at 0.71. In addition to the color-coded density, the equimolar dividing surface identifying the hydrophobic/hydrophilic interface is shown as a purple line. We used a discretization of 100 bins per dimension, and the length along the axial and radial direction is normalized by the hydrophobic thickness of the equilibrated membrane db (a) MARTINI model with 5 water molecules per lipid between the bilayers; (b) polymer-like model; (c) solvent-free model with headgroup interaction parameter vBB = −0.1. The system size is 512 lipids per bilayer for the MARTINI model, 2150 for the solvent-free model, and 4193 for the polymer model. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of the article.)

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Fig. 5. (left) Plot of the electron density obtained from X-ray diffraction measurements for a stalk between DPhPC bilayers at 78% relative humidity, adapted from Aeffner et al. (2012). (Middle) Plot of the number density of the hydrophobic units, normalized by the bulk density for a stalk in the MARTINI model at a hydration level of 5 water molecules per lipid. The system is identical to that shown in Fig. 4, but the density range plotted has been chosen to show the differences between the interstices and the stalk itself to the hydrophobic bulk. (Right) One octant of a stalk shown as isosurface of the hydrophobic density obtained from SCFT calculations. The lower isolevel, i.e., the outer of the two surface-sets, corresponds to the hydrophobic/hydrophilic interface, whereas the higher isolevel, i.e., the inner surface-set, has been chosen to visualize the lower density directly below the stalk. In addition, for two headgroup positions marked as black and red dot, the tail distribution is shown as a cloud with red indicating high and green indicating low probability. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of the article.) Reprinted from Daoulas and Müller (2013) with permission from The Royal Society of Chemistry.

to these calculations are dt /ds = 2.3 and ds /db = 0.89, and thus agree well with our results. On the other hand the stalk structure predicted by an elastic Helfrich-like model in Kozlovsky and Kozlov (2002) deviates from our results. Although in this elastic model the molecular degrees of freedom are taken into account by making the energy a function of the tilt and splay of lipid chains, it relies on the assumptions that the splay is small compared to the monolayer thickness, and that the lipids are incompressible and no lateral bilayer stretching occurs. All of this is not the case in the available X-ray data (Aeffner et al., 2012) and the results from our simulations. Regions of reduced hydrophobic density are found directly above and below the stalk, and the stalk itself also shows a lowered density, as shown in Fig. 5. Similar results are found in the SCFT calculations (Daoulas and Müller, 2013), for which also the configurational space of lipids in the stalk could be analyzed. This analysis, shown in Fig. 5, indicates an increased number of available orientations, including extreme tilt and stretching, for lipids close to the stalk, and illustrates the disordered lipid structure within the stalk. In addition, the elastic calculations assumed a contact point in the center of the stalk in which lipids from all four monolayers meet to avoid empty interstices. Such a point is not found in experiments and any of our

simulations, as reflected by the ratio of the stalk diameter and the minimum trans monolayer distance ds /dt , which is close to 2 for all of the three models as well as the experimental measurements. Considering the stability of the simulated stalks, we note a dependence on the bilayer distance. While stable stalks occur in a small region of the phase diagram of certain lipid at low hydration (Yang and Huang, 2003), for most systems stalks are only metastable structures and occur during phase transitions or other topological changes. When reducing the hydration level in the MARTINI model or the headgroup repulsion in the solvent-free model, the simulated stalks did not retain their spherical (lateral) cross-section but underwent a linear elongation as shown in Fig. 6. Such an elongation is associated with the lamellar-to-invertedhexagonal phase transition (Marrink and Mark, 2004), and indicates the lipids’ preference to form inverted phases at low hydration. On the other hand, when increasing the hydration level or headgroup repulsion, artificially formed stalks spontaneously disappeared. For the MARTINI and the solvent-free model, we also performed simulations in which oil molecules, represented by chains of four hydrophobic beads, were present in the lipid bilayers at a 1:10 ratio. Fig. 7 shows the partitioning of these molecules within the stalk system, finding an enrichment of oil molecules in the interstices

Fig. 6. Snapshots illustrating the stalk behavior at different hydration level/headgroup repulsion. Panels (A) and (C) are snapshots of the MARTINI model with 5 and 7 water molecules per lipid in the intermembrane space, respectively. Panels (B) and (D) are snapshots of the solvent-free model with headgroup interaction parameters vBB = −1.0 and -0.1, respectively. Each snapshot is shown as cross section through the stalk in side (left) and top view (right). Hydrophobic beads are rendered in green, and hydrophilic beads in blue. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of the article.)


Fig. 7. (a) Axial–radial plot of the oil number density, normalized by its maximum value and truncated at 0.5: (a) MARTINI model with seven water molecules per lipid between the bilayers, (b) solvent-free model with vBB = −0.1. The number density is expressed in units of nm−3 . The purple lines indicate the hydrophobic/hydrophilic interface.

directly above and below the stalk. Interestingly, the stalk geometry itself did not show a significant change in the presence of oil. 3.2. Fusion pore shape A recent study (Yoo et al., 2013) investigated the shape of the fusion pore using coarse-grained MD simulations with the MARTINI model and compared their findings to calculations using an elastic model based on the Helfrich energy of bending, Eq. (1). In the MD simulations, a manually created fusion pore was equilibrated, while artificially held open pores allowed for an easy exchange of lipids between the monolayers as well as solvent repartitioning. The simulations found a previously unknown “bowing”

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Fig. 8. Plot of the phosphate bead density in coarse-grained simulations of a fusion pore. The dashed horizontal lines illustrate the bowing feature by indicating the maximum monolayer distance and comparing it to the distance at the system edge (vertical arrows). Reprinted from Yoo et al. (2013). Copyright 2013 Elsevier.

feature of the fusion pore: the inter-bilayer separation displays a maximum at close distance to the center of the fusion pore (Fig. 8). This bowing feature was found not to be an artifact of a restraint of the inter-bilayer distance at the system boundaries, but spontaneously formed in systems with unrestrained inter-bilayer distance. If, however, the inter-bilayer separation at the periphery was restrained to values larger than the equilibrium distance, the bowing feature was found to disappear (Fig. 9). In addition, a thinning of the lipid bilayer in the curved center of the fusion pore was found. The study also included calculations of the optimal shape with a simple continuum model that were compared to the simulations with restrained inter-bilayer distances. While the continuum description did not allow for changes in bilayer thickness, the

Fig. 9. Neutral surface of a fusion pore for systems of DPPC (A) and DOPC (B) at different restrained peripheral intermembrane distances Rb . Shown are fits to data obtained from simulations with the MARTINI model (CG) and optimizations of these fits based on elastic theory (continuum). Reprinted from Yoo et al. (2013). Copyright 2013 Elsevier.

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Fig. 10. Phase diagram for pure DOPE, a 1:1 mixture of DOPE and DOPC, and pure DOPC constructed with the help of a spontaneous aggregation approach. Shown are the effects of the wildtype (WT) of the influenza HA fusion peptide at both fusogenic (pH 5.0) and non-fusogenic pH (pH 7.4), as well as the non-fusion G1V and W14A mutants. A, B, and C indicate hydration levels of 8.0–8.9, 9.7–10.4, and 11.2–12.0 water molecules per lipid, respectively. Each block in the bar graph represents a single simulation, with the color indicating the phase adopted at the end of the 12 ␮s simulation. Reprinted with permission from Fuhrmans and Marrink (2012). Copyright 2012 American Chemical Society.

calculated bilayer shape showed a remarkable similarity to the average shape obtained from MD simulation (Fig. 9) for intermediate bilayer distances. For small (restrained) inter-bilayer distances, the continuum model overestimated the bowing feature, showing the limitation of the elastic model when the radii of curvature approach the monolayer thickness. In general, however, the qualitative agreement between simulation and elastic model are still significantly better in the case of the fusion pore than in the description of the stalk shape (cf. Section 3.1). This is likely due to the fact that the general bilayer structure, compared to a planar membrane, is relatively well preserved in the fusion pore which still is a curved but otherwise intact bilayer, whereas a stalk locally disrupts the bilayer structure, making it more important to include packing effects and density variations in its description. 4. Topological changes So far, the focus has been on the shape of lipid aggregates. Even more interesting, as well as more challenging, is to study the way in which transitions between these shapes occur. Also here, computational methods are a valuable tool to test the feasibility of putative pathways as well as generate data to suggest new ones. A topic that has received a large amount of attention in the last years is the involvement of peptides in the fusion and poration of lipid bilayers. We will present findings on the fusion peptide of influenza HA as a representative of a group of small peptides that can act by insertion into the lipid bilayer alone, and findings on the SNARE complex as an example of a protein whose mode of operation requires to exert actual mechanical work on the fusing bilayers. In addition, we present recent findings on the creation of supported lipid bilayers by spreading vesicles on attractive substrates, as an illustration of other external factors that can trigger topological changes in lipid bilayers. A common feature of the selected examples is that they illustrate how different sequences of events can lead to topologically similar outcomes, and how intricate the balance between these paths is. 4.1. Phase behavior Due to the high metastability of lipid aggregates, it is difficult to observe topological changes within the short time frame available to computer simulation. It is therefore often found that a chosen starting configuration will persist even under conditions that would favor a different configuration in equilibrium. A seldom used but surprisingly powerful method to overcome this problem is to

make use of the relatively fast self assembly of coarse-grained models (Marrink et al., 2001), and start from a randomly distributed mixture of lipids and solvent. Such mixtures are relatively easy to generate, since the absence of hard interactions in coarse-grained models makes the setup much less sensitive to unfavorable contacts which would give rise to problematically high forces in an atomistic representation. Marrink et al. demonstrated that such a spontaneous aggregation approach successfully reproduces the phase diagram of DOPE, DOPC and different mixtures of these lipids (Marrink and Mark, 2004). This approach can be taken further by including other agents that are known to affect fusion into the setup, and has been used to study the effects of the influenza HA fusion peptide in simulations based on the MARTINI model (Fuhrmans and Marrink, 2012; Fuhrmans et al., 2009). These 20 amino-acid long peptides are located at the N-terminus of the HA2 unit of the influenza HA protein, and have been shown to be required to fuse the influenza virus with its intended host cell (Steinhauer et al., 1995). In addition, in vitro experiments revealed that even the fusion peptides alone can induce lysis or vesicle fusion (Han and Tamm, 2000), lower the transition temperature for cubic phases (Epand and Epand, 1994), and affect the lamellar-to-inverted-hexagonal phase transition in a concentration-dependent manner (Siegel and Epand, 2000). A recent review has been published by Cross et al. (2009). As shown in Fig. 10, presence of the peptides has a clear effect on the adopted phase in spontaneous aggregation simulations. Most noticeably a change from the inverted hexagonal to the stalk phase is found for pure DOPE, and a change from the stalk phase to the lamellar phase for pure DOPC and an 1:1 mixture of DOPC and DOPE, both of which represent a shift toward positive mean curvature. In addition, the fusion peptides were found to stabilize cubic phases, which indicates a reduced cost of negative Gaussian “saddle-splay” curvature. While experimental findings showed similar results for the effects on the phase diagram (Davies et al., 1998; Epand and Epand, 1994), the simulations provided additional information by revealing the peptides’ preferred location in the lipid–water interface as regions of high negative Gaussian curvature that where formed by pores adjacent to stalks. This induction of positive mean curvature is at odds with earlier hypotheses linking the fusion peptides’ function to a stabilization of stalks as the initial step in the fusion process (Epand, 2003), since stalks are characterized by an overall negative mean curvature. However, judging from the peptides’ position in the lipid monolayer that locates their main volume in the headgroup region (Li et al., 2005; Lagüe et al., 2005; Fuhrmans and Marrink, 2012), it


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appears hard to envision how the peptides could stabilize negative mean curvature, whereas a stabilization of positive mean curvature is easily understood. In addition, direct attempts to study fusion between dehydrated bilayers in the presence of the peptides showed an inhibition of stalk formation (Fuhrmans and Marrink, 2012), suggesting a different role in the fusion process. A possible candidate for that role was suggested to be the stabilization of pores as structures with a positive mean and a negative Gaussian curvature, combining both of the peptides’ curvature preferences found in the phase diagram. In addition, the peptides’ location in the adopted phases and especially the induction of the cubic single diamond phase (Fuhrmans et al., 2009) were seen as an indication that stalk–pore complexes may also be linked to the fusion peptide’s function, as will be discussed in more detail in the next section. In addition, with the spontaneous aggregation approach not only the finally adopted phase contains information. The aggregation process does usually not directly go from the random mixture to the final result, but the lipids rather form smaller aggregates first that only gradually merge into a stable phase. During this process, typical structural elements like pores and stalks that would also play a role in actual phase transitions will be formed or eliminated, and one could gain additional insight into the mechanics of the process by noting how these elements interact, and if the average life time of certain structures is significantly enhanced or shortened under some conditions. An example for this approach is the self-assembly of large vesicles using SCFT calculations performed by Sevink and Zvelindovsky (2005). 4.2. Influenza-HA-induced fusion The mode of action of the Influenza HA fusion peptide discussed in the previous section was also studied in a more direct manner using coarse-grained simulations based on the MARTINI model (Risselada et al., 2012). These simulations were started from a setup of a preformed stalk–pore complex between two lipid bilayers of either DOPE or DOPC at a distance of approximately 2 nm, and, optionally, up to eight fusion peptides in one of the facing monolayers. The outcome of the simulations can be summarized by four main results: (i) Even for pure DOPC bilayers, the stalk elongated around the pore and formed a circular three bilayer junction along the upper rim of the pore, whereas in the absence of a pore the stalk did not elongate at all. The resulting bilayer configuration was termed “-hemifusion-diaphragm” (Fig. 11), and agrees with earlier SCFT calculations corresponding to the heterogeneous nucleation of a stalk next to a pore or vice versa (Katsov et al., 2006). (ii) Fusion peptides were able to stabilize otherwise unstable preformed pores by forming a bundle of at least four peptides within the pore. The lipid rim of this pore was of a hydrophobic nature, with the hydrophilic sides of the fusion peptides lining the axis of the pore. (iii) A pore stabilized by peptides also resulted in the formation of a -hemifusion-diaphragm in DOPC bilayers (in these simulations, the stalk was created after the peptide bundle stabilizing the pore had formed). During this process, the hydrophobic rim of the peptide-stabilized pore is replaced with a hydrophilic rim lined with lipid headgroups, and the peptide bundle is partially expelled from the pore to allow for this replacement, as shown in Fig. 12A. (iv) Models of non-fusogenic mutants were unable to support the formation of a -hemifusion-diaphragm. The G1V and W14A mutants (Qiao et al., 1999; Lai et al., 2006) were not able to form a bundle stable enough to stabilize the pore, whereas the G1S mutant (Qiao et al., 1999) formed a pore-stabilizing bundle that turned out to be too stable to be replaced by the elongating stalk. These effects correlate well with the experimental findings – G1V and W14A inhibit lipid mixing, and G1S displays lipid mixing, but does not complete the fusion process (Fig. 12B).

Fig. 11. Snapshots illustrating the development of a preformed stalk–pore complex into a -hemifusion-diaphragm. The stalk elongates along the pore rim until a closed circular stalk is formed. This process illustrates how the presence of an energetically unfavorable pore rim can drive the elongation (and formation) of a stalk by replacing the pore rim with a slightly more favorable three-bilayer junction. In contrast to the “traditional” hemifusion diaphragm which is composed of the trans monolayers of the two apposed bilayers, the -hemifusion-diaphragm originates entirely from one bilayer. Reprinted from Risselada et al. (2012).

Together with estimates of the line tension of a pore rim compared to that of the elongated three-bilayer junction encountered in the -hemifusion-diaphragm, these findings were interpreted as a line-tension driven step of the fusion process. Starting with the formation of a pore, the unfavorable energy of the pore edge is used to drive the fusion process by reducing the (relative) energy cost of stalk formation and elongation. In this interpretation, the fusion peptides’ role is to sufficiently stabilize a pore to initiate the whole process. However, the exact amount of stabilization is a delicate issue, and a too stable peptide-lined pore (e.g., in the case of the G1S mutant) will not have enough line tension to gain anything by replacing part of its rim with a stalk and, eventually, the elongated three-bilayer junction of the -hemifusion-diaphragm. Such an intimate connection between pores and stalks not only explains experimental observations of leaky fusion induced by fusion peptides (Frolov et al., 2003), but also the striking

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Fig. 12. Snapshots of systems including pores stabilized by bundles of the wildtype HA fusion peptide (A) and the non-fusogenic G1S mutant (B). The pore stabilized by the wildtype peptides allows stalk elongation along its rim while the peptides are partially expelled from the pore, whereas the G1S mutant stabilizes the stalk–pore complex but inhibits stalk elongation. Reprinted from Risselada et al. (2012).

resemblance of the effects of fusion peptides (Epand, 2003; Tamm et al., 2002) on the one hand, and the effects of antimicrobial (Brogden, 2005) and cell-penetrating peptides (Järver and Langel, 2006) on the other. All of these peptide families, whose functions in vivo are related only to fusion or only to poration, can induce both fusion and lysis or content-leakage under the right conditions in in vitro experiments (Lai and Tamm, 2010; Yang et al., 2010; Nicolas, 2009; Almeida and Pokorny, 2009; Hale and Hancock, 2007), and have been shown to stabilize non-lamellar, inverted cubic phases (Epand and Epand, 1994; Haney et al., 2010; Mishra et al., 2008). This similarity has been rationalized as an effect of their main common feature: they all partition to the lipid/water interface and thereby exert considerable curvature stress on the affected monolayer (Bechinger, 2009). By pore formation this stress can be relieved, and, if a second membrane is in close vicinity, formation of a stalk–pore complex can further reduce the stress if the associated reduction in free energy of the pore rim is competitive with that of the peptides. In vivo, viral fusion peptides do not appear isolated but as part of a larger membrane protein complex which is embedded in the viral envelop. If the fusion peptides are lodged into the target membrane, the virus and host-cell are therefore automatically tethered via the fusion protein which, potentially, may bring the bilayers into the close contact required for stalk formation. Another role proposed for the fusion peptides based on both atomistic (Kasson et al., 2010; Li et al., 2010) and coarse-grained simulations (Vaidya et al., 2010) is to create membrane-stress that facilitates tail protrusions that, in turn, allow stalk formation by creating hydrophobic contacts between the apposed lipid bilayers. Such lipids that have their tails splayed between apposed bilayers

have been suggested to be not only precursors of stalk formation, but to rather constitute the real energetic barrier in this process (Risselada and Grubmüller, 2012), with the actual stalk representing a metastable local minimum in the energy landscape that is reached within a few ns after occurrence of a splayed lipid (Stevens et al., 2003; Smirnova et al., 2010; Mirjanian et al., 2010; Kasson et al., 2010; Smeijers et al., 2006). 4.3. SNARE-induced fusion A protein for which its role in the fusion process, including stalk formation, could be studied directly in computer simulation is the SNARE complex governing synaptic fusion. The three proteins belonging to this complex form a tight coiled-coil bundle, in which two ˛-helical peptides are anchored to the membrane of the synaptic vesicle (synaptobrevin) and the presynaptic membrane (syntaxin) via C-terminal transmembrane domains. It is believed that the formation of the bundle exerts a force on the membrane anchors that pulls the membranes together and triggers membrane fusion, as reviewed by Jahn and Scheller (2006). Unlike the fusion peptides discussed above, the SNARE complex is considered to perform actual mechanical work during the fusion process, and the feature could convincingly be captured in the MD simulations. The simulations, performed by Risselada et al. (2011), used the available X-ray structure (Stein et al., 2009) to build a representation of the peptide bundle in the coarse-grained MARTINI model. The bundle was partially “unbundled” by pulling on the two transmembrane anchors to obtain a structure mimicking an earlier stage of the complex assembly. The membrane anchors of this structure


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Fig. 13. Snapshots of coarse-grained simulations of SNARE-mediated vesicle fusion, showing the process as cross sections in side (A) and top view (B). The unbending/zipping of the SNARE bundles brings the vesicles close and a stalk forms without further unbending of the peptides (I). The stalk elongates following a circular path until an invertedmicelle-like compartment is created between the vesicles (II). Further unbending/zipping of the SNARE bundles then pulls the C-termini of the transmembrane anchors toward each other in a closing-scissors motion while simultaneously pores form in the bilayers separating the vesicle interiors (III, IV), suggesting that the peptides provide mechanical energy used in the pore formation. Reprinted from Risselada et al. (2011). Copyright 2011 John Wiley and Sons.

were embedded in two vesicles, resulting in a starting configuration in which the stiff linker region between the transmembrane domains and the remainder of the linear peptides was bent in both of the anchoring peptides (cf. Fig. 13A). From this setup, the simulations proceeded to complete fusion of the vesicles following a pathway illustrated in Fig. 13. By monitoring the peptides state of (re)assembly during this process as well as artificially switching off the rigidity of the linker regions at different stages, several roles for the SNARE proteins were postulated (Risselada and Grubmüller, 2012). The SNARE proteins provide energy in the form of bending stress stored in their conformation to the vesicles. This energy is used for two steps: pulling the bilayers toward each other, and actively participating in the creation of the fusion pore by performing a “closing-scissors” movement of the transmembrane domains which exerts a pull on the trans monolayers of the fusing membranes. In addition, the membrane anchors (but no further mechanical unbending work) appear to be needed to facilitate stalk formation. In their presence, the stalks appeared after occurrence of splayed lipids, whereas no stalks formed between bilayers that were artificially brought into similar proximity but contained no transmembrane domains. The actual pathway reported in Risselada et al. (2011) was found to follow the circular elongation route described above (Section 4.2). However, theoretical considerations aided with additional simulations have been used to discuss the possible scenarios of fusion behavior in more detail Risselada and Grubmüller (2012), narrowing the field to two sets of contrasting options along with the factors favoring them: radial stalk expansion versus linear stalk elongation, and leaky versus non-leaky fusion. The linear stalk elongation was associated with a regime corresponding to conditions favoring the inverted hexagonal (HII ) phase, as the lamellar-to-inverted-hexagonal phase transition has been shown to proceed via such an elongation (Marrink and Mark, 2004; Knecht et al., 2006). The radial expansion, on the other hand, has been predicted to face a considerable energy barrier (Katsov et al., 2004). While this barrier could be lowered by lipids with a negative spontaneous curvature in the cis monolayers (Lee and Schick, 2007), such a spontaneous curvature would at the same time drive the system toward the regime in which linear stalk elongation is favored, making the balance between the two outcomes hard to

predict. However, even under conditions where linear stalk elongation constitutes the favorable stalk behavior, it could be prevented from happening by spatial restraints like a too small contact area locked in-between a cluster of multiple SNARE complexes. In such a scenario, the combined unbending of the SNARE bundles can be easily envisioned to create the hemifusion diaphragm by dragging the trans monolayers toward each other, or, directly form a fusion pore without an expanded hemifusion diaphragm as intermediate state. If, however, the actual fusion pathway mediated by SNARE proteins was a linear stalk elongation, the question arises whether the process can occur without significant content leakage. As was discussed for the influenza HA fusion peptides (cf. Section 4.2), it is often favorable to form a pore adjacent to a stalk, especially, when the stalk can elongate along the pore rim. However, spontaneous formation of such a pore would allow leakage, which in vivo would be detrimental to the synaptic transmission and is therefore considered not to constitute a viable option. In vitro assays of SNARE induced fusion, on the other hand, frequently report content-leakage (Chen et al., 2006; Dennison et al., 2006; Wang et al., 2009; Südhof, 2007), which are tempting to explain via linear stalk elongation together with spontaneous pore formation. One therefore has to ask if there are factors that could prevent pore formation during the stalk elongation pathway in vivo, and the high sterol content of the synaptic membrane which is known to stabilize lipid bilayers (Baoukina and Tieleman, 2011) was suggested to constitute a possible candidate (Risselada and Grubmüller, 2012). In addition, one should take into consideration that the transient pores occurring in association with the elongating stalk are very small and, unlike the fusion pore, do not spontaneously expand. It is therefore possible that, even if a pore forms, content-leakage would be limited to small molecules and the larger neurotransmitters in the synaptic vesicle would still be retained until completion of the fusion process. 4.4. Vesicle spreading Another process for which a potential pathway has been suggested from computer simulation is the creation of supported lipid bilayers by vesicle spreading. In this technique, a solution of

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Fig. 14. Snapshots illustrating the deciding role of interaction range in the determination of the pathway chosen and the corresponding orientation of the resulting SLB. Shown are rupture at the side of the vesicle’s top at short potential range (top, receding-top mechanism) and poration in the vesicles bottom at long potential range (bottom, parachute mechanism). The respective ranges of the potentials used are indicated by the dashed black lines. Gray circles represent lipid head groups, and red and green lines represent lipid tails from the inner and outer monolayer, respectively. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of the article.) Reprinted with permission from Fuhrmans and Müller (2013). Copyright 2006 American Chemical Society.

vesicles is brought into contact with an attractive solid support, and the vesicles spontaneously assemble into a continuous lipid bilayer covering the substrate (Brian and McConnell, 1984). While it is obvious that poration of the vesicles is required in this process, the exact location of the pore has not been resolved experimentally, and some experiments reported an inside-up orientation of the supported bilayer (Reimhult et al., 2009), whereas others reported an outside-up orientation (Contino et al., 1994; Salafsky et al., 1996). In addition, atomic force microscopy (AFM) showed configurations that were interpreted as a sliding mechanism and predicted a mixed orientation of the supported bilayer (Jass et al., 2000). The process was simulated using a solvent-free coarse-grained model (Hömberg and Müller, 2010) and representing the substrate with a simple 9–3 Lennard–Jones potential. Depending on the conditions, three different mechanisms were observed (Fuhrmans and Müller, 2013): a tension driven burst mechanism, and two curvature driven mechanisms. The tension driven mechanism occurred when the vesicle was filled with cargo molecules that imposed a volume constraint and tension on the membrane. The deformation caused by the vesicle’s adsorption to the substrate increased the tension further, and led to an unspecific rupture location somewhere in the vesicle’s unadsorbed top, and, in consequence, an inside-up orientation of the created supported bilayer. In simulations without the volume constraint, corresponding to hyperosmotic conditions or (slow) spreading accompanied by content permeation, a curvature-driven mechanism was observed, as indicated by a specific position of pore formation located directly next to the highly bend outer edge of the flattened adsorbed vesicle (Fig. 14). Whether the pore opened in the vesicle’s bottom or its unadsorbed side was found to depend on the range of the interactions with the substrate. With short-ranged interactions, the pore always nucleated in the side of the vesicle, and rapid expansion of the pore led to an inside-up orientation of the created supported bilayer. With long-ranged interactions, however, multiple small pores opened in the peripheral region of the vesicle’s adsorbed bottom. Through these pores, lipids from the inner monolayer came into contact with the substrate and allowed the vesicle to increase the adsorbed area by creating a ring-shaped bulge that

grew outward from the vesicle with an outside-up orientation. In this pathway, the pores did not enlarge, and only the lipids from the outer monolayer that had originally participated in the adsorption ended up facing the substrate in the final bilayer configuration. Correlated with the two curvature-driven pathways, referred to as the “receding-top” and the “parachute” mechanism, a pronounced thinning of the membrane at the rupture location was found (Fig. 15). Remarkably, the respective regions themselves had virtually no curvature, and the thinning was interpreted as a nonlocal effect of the strong curvature at the outer vesicle edge. Such nonlocal effects of curvature and packing frustration are known to also appear adjacent to positive hydrophobic mismatch (Sintes and Baumgärtner, 1997; Venturoli et al., 2005; West et al., 2009; Brannigan and Brown, 2006) and kink grain boundaries in block copolymer bilayers (Matsen, 1997). The sensitivity to the range of the substrate interactions was identified as a competition between curvature-induced thinning and an increased density in the adsorbed bottom. Only if the attractive range was long enough to reach the lipids of the inner monolayer, thinning and subsequent poration in the bottom turned out to be energetically feasible. It therefore appears, that both an inside- and an outside-up orientation of supported lipid bilayers created by vesicle spreading are possible, depending on the range of the substrate interactions. No evidence of a sliding mechanism in which the vesicle’s top becomes disconnected from the adsorbed bottom was found in the simulations. However, when using short-ranged interactions of sufficient strength, the simulations showed simultaneous formation of multiple pores following the receding-top mechanism, which gave rise to a sequence of configurations showing a remarkable resemblance to those interpreted as sliding in the AFM study (Jass et al., 2000). Since the membrane connectivity cannot be deduced from the AFM data, a reinterpretation of the reported data as a display of the receding-top mechanism is a possibility. 5. Free energies and transition pathways In this section, we will review a selection of recent findings on the free energy and transition pathway of the formation of pores


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Fig. 15. Shape of the outer edge of absorbed vesicles prior to rupture. Shown are a vesicle spreading via the receding-top mechanism at short-ranged interactions with the substrate (left) and a vesicle spreading via the parachute mechanism at long-ranged interactions (right). The points indicate the location of the midplane of the bilayer. In addition, vertical bars indicate the thickness of the hydrophobic core, and horizontal bars indicate the curvature found at the respective points. Reprinted with permission from Fuhrmans and Müller (2013). Copyright 2006 American Chemical Society.

and stalks. In addition, we include hitherto unpublished data on the mean free energy path of vesicle fusion. 5.1. Pore formation A basic change of the membrane topology is the formation of a pore. Pore formation may occur if the membrane is subjected to lateral stress, , e.g., by osmotic swelling or by a strong electric field pulse (electroporation). In analogy to classical nucleation theory for first-order transitions, an idealized description uses the radius r of the pore as a reaction coordinate and quantifies the excess free energy F(r) of a structure with pore radius r by the free energy costs of the line tension, , of the rim of the pore and the reduction of free energy by the elimination of energetically costly tense membrane area (Taupin et al., 1975), F(r) = 2 r − r2 . The free energy as a function of this reaction coordinate, r, has a maximum at r* = / and F* = 2 /. These saddle-point values characterize the transition state, and this simple approach rationalizes, for instance, that a reduction of the line tension of a pore will dramatically increase the nucleation rate of pores. In general, such a description only becomes accurate if the membrane tension becomes vanishingly small. In this limit, however, the critical pore size r* diverges, the free-energy barrier F* exceeds the thermal energy by far, and nucleation events are extremely rare. Another problem is, that for values of r that are not large compared to the bilayer thickness, the line tension will depend on r because of the strong bending of the pore’s rim and interactions across the pore. In addition, the line tension defined by the size dependence of F(r) will itself depend on the tension. Intuitively one also expects that the structure of “pores” with a radius that is comparable or smaller than the bilayer thickness markedly differs from a scaled-down version of a macroscopic pore because the molecular nature of its components is no longer compatible with a “smooth”, homogeneous description. Thus pronounced deviations form the simplified description of classical nucleation theory are to be expected. Ting et al. (2011) used the string method (cf. Section 2.4) in conjunction with SCFT theory to investigate pore formation in lipid bilayers under tension. These calculations provide insights into the MFEP without any a priori assumptions about the structures along the transformation path. For vanishingly small tension they confirmed the predictions of classical nucleation theory. Interestingly, the very initial stages of pore formation or critical “pores” at very

large tension rather correspond to a local thinning of the membrane without substantial molecular reorganization. Thus, very small “pores” resemble equilibrium fluctuations of the membrane thickness. Along the pore-formation path such a thinned membrane patch first converts into a channel of hydrophilic lipid head groups that bridge the membrane. In these channels, headgroups from all directions overlap, and they do not constitute an actual unobstructed path across the bilayer. Only at larger pore sizes the pore rim resembles an edge of a bilayer. These calculations are in qualitative agreement with particle simulations (Müller and Schick, 1996). 5.2. Stalk formation The MFEP of the shape transformation from two apposed bilayers to a stalk structure has been studied by computer simulations of the polymer-like membrane model introduced in Section 3.1. The sequence of structures and the excess free energy along the transformation path are presented in Fig. 16 (Müller et al., 2012). The structures depict the density difference between hydrophilic and hydrophobic segments in a plane that is perpendicular to the two parallel apposed lipid bilayers and runs through the center of the stalk. In the early stages of the transformation the bilayer adopts a dimple-like shape where the two cis-monolayers approach each other but the outer trans-monolayers largely remain unperturbed. For this ascending part of the MFEP the minimal distance between the apposed cis-monolayers may be a suitable reaction coordinate. The transition state corresponds to a structure where the two dimples from the apposing monolayers join and form a continuous hydrophobic bridge or “pre-stalk”. A snapshot of a molecular configuration with an order parameter that corresponds to this saddle point is depicted in the inset of the right panel of Fig. 16. For the rather low molecular density in a lipid bilayer as compared to the polymer-like model, this transition state involves only a very small number of lipid molecules. Beyond the saddle point the diameter of this hydrophobic bridge increases. The qualitative character of the reaction coordinate therefore differs on the two sides of the saddle point, and with “conventional” reaction coordinates, two different coordinates, e.g., cis-monolayer distance on one side and stalk diameter on the other, would have been necessary to constrain the system. In addition, the density in the coarse-grained particle model slightly decreases at the monolayer junctions above

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Fig. 16. Minimum free-energy path (MFEP) of stalk formation between two apposed lipid bilayers. (left) Contour plot of the order parameter, ms (r), in the midplane of the system; every second configuration along the string is depicted. Hydrophobic regions are colored red, hydrophilic regions are shown in blue. The contour parameter, s, increases from left to right and top to bottom in uniform increments. (Right) Free energy along the MFEP (black, left axis) and the probability that configurations at position s along the MFEP decay into two unconnected bilayers within a molecular relaxation time after the removal of the field-theoretic umbrella potential, Eq. (5) (blue, right axis). Results have been obtained for 256 independent configurations at each value of s. The thermal energy scale, kB T, is indicated for our coarse-grained membrane model (amphiphiles) and biological lipids. The inset presents a configuration snapshot at the saddle point, s* = 0.532. Hydrophilic beads are colored yellow, hydrophobic beads are shown in red, solvent particles are not shown. Only every 10th linear amphiphile is depicted. When we identify one double-tailed lipid with two linear amphiphiles, the molecular density in the snapshot matches that of a lipid bilayer. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of the article.) Reprinted from Müller et al. (2012).

and below the stalk structure in agreement with the packing frustration discussed in Section 3.1, and it is also in accord with the changes of the molecular conformations observed in SCFT calculations (Daoulas and Müller, 2013). The free-energy difference is measured in units of int d02 , where int denotes the tension of the oil–water interface and d0 the bilayer thickness. The free-energy difference between the two apposed bilayers and the stalk nicely agrees with predictions of SCFT calculations (Katsov et al., 2004). The right panel additionally shows the probability that a structure along the path will evolve backwards into the starting structure of two apposed bilayers without stalk. This probability sharply varies in the vicinity of the saddle point. On the ascending part of the MFEP, structures predominantly evolve to the starting structure, whereas structures that are beyond the transition state proceed to form a metastable stalk. In the vicinity of the transition state this probability crosses the 50–50 threshold. This information corroborates the identification of the transition state and its significance for dynamic simulations.

5.3. Vesicle fusion Using a similar setup as that used for stalk formation (cf. Section 5.2), we have also studied the MFEP between a stalk and a fusion pore. In these simulations we have imposed symmetry with respect to the center of the simulation box at which the stalk is located in the initial structure. This allows us to increase the accuracy of the measured order parameter field by averaging over the “octants”, but limits the MFEP to radially symmetric paths. Our results correspond to the stalk-hemifusion pathway and we observe a radial expansion of the stalk to first a hemifusion diaphragm and subsequently a fusion pore. In Fig. 17 we depict the sequence of structures from the metastable stalk to the fusion pore and the concomitant free-energy profile. First the diameter of the stalk increases which raises the free energy of the system. The MFEP passes through a first transition state between the metastable stalk and a metastable hemifusion diaphragm. This hemifusion diaphragm, however, does not expand significantly in its diameter

Fig. 17. Axially symmetric pathway of membrane fusion between two apposed lipid bilayers. (Left) Snapshots of system configurations along the minimum free-energy path as obtained by the improved string method. The system of two apposed bilayers is cut in the midplane between the bilayers and the lower bilayer is viewed from the top. Hydrophobic segments are colored red and the hydrophobic–hydrophilic interface is shown in blue. The “red spot” shows the hydrophobic core of a stalk that connects the two apposed bilayers. In the course of the fusion pathway it radially expands to a hemifusion diaphragm and, eventually, a fusion pore is nucleated at its center. (Right) Preliminary results on the free-energy along the symmetric fusion pathway indicating two barriers associated with the radial expansion of the stalk and the rupture of the hemifusion diaphragm. The free-energy is measured in units of the bilayer thickness do and the interfacial tension o between the oil and water. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of the article.)


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Fig. 18. Markovian state model for fusion between vesicles linked by a crosslinker molecule. (a) Suggested mechanism with reaction rates calculated from the transition matrix. (b) Temporal development of fusion intermediates. (c) Free energy values of the fusion intermediates. Note that the fused and hemifused states are further subdivided according to the k-means clustering. Reprinted from Kasson et al. (2006). Copyright 2006 National Academy of Sciences, USA.

but instead the MFEP passes through a second saddle point where a fusion pore is formed at the center of the diaphragm and the final topology is established. Beyond the second transition state, the radius of the fusion pore expands until, in the canonical ensemble, a metastable fusion pore of finite radius is obtained. Using a different, complementary numerical technique, the Markovian state model (cf. Section 2.5), Kasson et al. (2006) studied the kinetics, free energy and transition pathway of fusion between lipid vesicles based on MD simulations with the MARTINI model. To perform the large number of simulations required to evaluate the transition probabilities, a distributed computing network (Shirts and Pande, 2000) was used, and a crosslinker molecule connecting the vesicles was included in the setup to overcome the activation barrier and enable spontaneous fusion in the (unguided) MD simulations. The crosslinker brought the vesicles in close proximity and potentially also participated in facilitating the occurrence of splayed lipid tails that are believed to be a mandatory precursor of the stalk (cf. Section 4.2). Using lipid and content mixing either alone (Kasson et al., 2006) or aided by an analysis based on persistent voids (Kasson et al.,

2007) to identify the different stages of the fusion process, transition likelihoods between the unfused vesicles, stalk-like states, metastable hemifusion states and fused vesicles were obtained. Interestingly, the transition probabilities indicated a branched pathway for vesicle fusion, and in addition to the “classical” stalkhemifusion pathway, a direct path connecting stalk and fused state was observed that comprised approximately 20% of the performed trajectories. In addition, the free energy differences between separated vesicle and stalk were calculated as G = −5.9 kB T, highlighting the role of the crosslinker in enabling the fusion process. The hemifused state further reduced the free energy by 5.1 kB T, with a further reduction of 4.1 kB T upon completion of the fusion process. 6. Conclusions It has long been known that the “repertoire” of topologies available to lipid aggregates in aqueous solvent is limited. An example of this fact is that the same lyotropic phases are observed for different lipids, albeit in very different regions of the phase diagrams. With

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the presented simulations of stalks and fusion pores, is appears that the universality of these shapes extends well to the very different ways lipid/water systems can be represented in computer simulations, and is, in fact, a very fundamental property of lipid-like amphiphiles that directly follows as long as they have the ability to form lamellar bilayers. Especially remarkable is that not only the general topology (Fig. 18), but the actual geometrical shape appears to be conserved over different lipids in experiment as well as over different models in simulations, which may well suggest that for the most, it is the hydrophilic to hydrophobic volume ratio which, together with the hydrophobic core thickness, limits the available topologies and at the same time determines the geometry, with only the free energy depending on the details of the lipids. In agreement with this interpretation is the relative success of simple elastic continuum models in calculating the geometry of fusion intermediates and lyotropic phases. However, as has become apparent in the comparison to data from computer simulations, for small radii of curvature a particle-based description becomes necessary as the assumptions at the basis of the continuum description no longer hold. Somewhat more delicate is the study of the mechanisms of topological changes. The presented examples show that different sequences of events can lead to the same outcome, and that the relative free energies of the alternative transition states are subject to multiple factors that may tip the favor in the question which branch of a pathway is taken. For the lamellar-to-inverted-hexagonal phase transition, it has been accepted that the topological change occurs via the formation of stalks that subsequently elongate in a linear fashion to create the lipid cylinders forming the backbone of the HII phase (Marrink and Mark, 2004). In addition, it has been argued that the required topological plasticity of biological membranes stems, at least in part, from a lipid composition that shifts the equilibrium phase part of the way toward inverted phases (Yorek, 1993). It is therefore well within the realms of possibility, that a local apposition and dehydration of lipid bilayers or an accumulation of (fusion) peptides may regionally shift the phase equilibrium to the HII phase and cause linear stalk elongation. Whether this actually constitutes the mode of operation of fusion peptides or SNARE proteins cannot be decided with certainty at present, but the intimate relation of such a pathway to pores is a tempting option to rationalize observations of content-leakage in in vitro fusion assays or the similar behavior of such different peptide families as viral fusion peptides on the one hand, and pore-forming antimicrobial and cell-penetrating peptides on the other. The presented examples of topological changes also illustrate the manifold ways in which energy can be provided to overcome the often substantial barrier to trigger a transition, even though the resulting configuration is usually at a lower free energy than the starting point. It can be as straightforward as performing direct mechanical work like in the case of the bundling of the SNARE proteins that transfers its motion to the bilayers in which the different peptides are anchored via rigid linkers. However, it can also be more indirect as in the example of vesicles spreading on attractive substrates, where adsorption energy is primarily converted to bending, and only indirect (and even nonlocal) effects of the bending cause a thinning of the bilayer that leads to pore formation. Even more indirect, as well as somewhat counterintuitive, are the findings for the influenza HA fusion peptides, which could not be linked to a direct stabilization of stalks, but were found to instead stabilize pores. While these pores are not a necessary part of the topological changes required for membrane fusion, they can still provide energy in the form of the line tension of their rim, that can indirectly drive stalk formation and elongation up to hemifusion and subsequent fusion pore formation. It remains to be seen whether such a, potentially leaky, fusion mechanism can be reconciled with the physiological requirements of fusion in vivo. It should be noted,

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