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Biological membranes are complex assemblies of many different molecules of which analysis demands a variety of experimental and computational approaches. In this article, we explain challenges and advantages of atomistic Monte Carlo (MC) simulation of lipid membranes. We provide an introduction into the various move sets that are implemented in current MC methods for efficient conformational sampling of lipids and other molecules. In the second part, we demonstrate for a concrete example, how an atomistic local-move set can be implemented for MC simulations of phospholipid monomers and bilayer patches. We use our recently devised chain breakage/closure (CBC) local move set in the bond-/torsion angle space with the constant-bond-length approximation (CBLA) for the phospholipid dipalmitoylphosphatidylcholine (DPPC). We demonstrate rapid conformational equilibration for a single DPPC molecule, as assessed by calculation of molecular energies and entropies. We also show transition from a crystalline-like to a fluid DPPC bilayer by the CBC local-move MC method, as indicated by the electron density profile, head group orientation, area per lipid, and whole-lipid displacements. We discuss the potential of local-move MC methods in combination with molecular dynamics simulations, for example, for studying multi-component lipid membranes containing cholesterol.

Analysis of the molecular mechanisms underlying membrane lipid organization is of crucial importance for understanding the function of membranes in living cells. Recent interest has focused on the biophysical characterization of model membranes to explore the physical principles governing the behavior of biological membranes. For example, lipid phase behavior and separation was investigated in model membranes by fluorescence microscopy, neutron scattering and NMR, while cholesterol dependent lipid order in endoplasmic reticulum membranes was determined by EPR spectroscopy [

Atomistic molecular dynamics (MD) simulation, which solves explicitly Newton’s equations of motion, is now a very advanced and established technique being able to simulate the behavior of phospholipid membranes up to several hundred nanoseconds [

Together, classical atomistic MD simulations provide information about many relevant membrane processes in the time window of several fs to <~0.5 μs for a few hundred lipid molecules. Using improved sampling schemes, such as replica exchange MD or supercomputer architectures optimized for MD simulations, the simulation time can be extended by a factor of ten. However, even with further improvement of computer instrumentation, it can be expected that other highly relevant but slower phenomena like lipid domain formation or phospholipid flip-flop in membranes taking place in the range of milliseconds to seconds and/or involving larger membrane areas cannot be studied solely by the MD technique. This is a disadvantage of the inherent time dependence of the method, which requires time propagation steps of typically 1–2 fs to follow the fastest processes, such as intramolecular vibrations. As a consequence, membrane phenomena occurring on a large time scale in nature will require lengths of MD simulation runs not accessible by today’s conventional computational power. Supercomputers, such as “Anton”, developed by David E. Shaw’s group, might provide a way to extend MD further towards the relevant mesoscopic scheme [

As alternative to MD simulations of atomistic or CG molecular models, one can circumvent the explicit time dependence of the investigated processes and sample molecular conformations along a Markov chain in the configuration space based on their probability in a Boltzmann distribution [^{N} be the configuration vector comprising positional and rotational coordinates of an ^{N} = {_{1}, _{2}, ... _{N}}, average thermodynamic properties, such as the internal energy, U, of the system are expressed in a statistical sense as average according to:

Here, E(^{N}) is the configuration energy, while the probability of finding the system in state ^{N} is given by

This high-dimensional configuration integral and thereby the thermodynamic average properties, such as the internal energy, cannot be evaluated by numerical integration. However, P(^{N}) can be evaluated by generating an irreducible Markov chain, of which the limiting distribution is the Boltzmann distribution:

Here,

Early attempts to improve sampling efficiency modified the underlying transition matrix, thereby biasing the moves in a certain direction. For example, smart MC developed by Ross

While being a problem in simulating particles in dense fluids, the low acceptance rate of standard Markov chain MC methods has been even a larger hurdle in MC simulations of non-rigid molecules and macromolecular assemblies, such as proteins or lipid bilayers. The reason for this is that intramolecular moves often result in steric clashes associated with high energy and thereby rejection of the attempted move. For example, even a modest change of a torsion angle in the middle of a molecular chain can lead to large movement of the atoms at the end of the chain causing, for example, clashes between neighboring fatty acyl chains of phospholipids in the membrane [

However, these approaches suffer from rather slow convergence, as small changes in torsion angles can be associated with large changes in configurational energy due to van der Waals contacts of distant atoms and steric clashes [

Importantly, the torsion angle changes during a CR move are correlated such that the volume element in the Boltzmann factor exp(−_{1} ... d_{6} is not volume preserving. This necessitates inclusion of a Jacobian,

where {_{i}_{i}

Importance sampling, according to the Metropolis method, basically evaluates the integral over configurations of a subset of the configurational space in Cartesian coordinates. Therefore, the Jacobian has to be included in the Metropolis acceptance criterion to account for the coordinate transformation from cartesian to internal coordinates in the closure procedure. The Jacobian enters the acceptance criterion for the trial move

Here,

Like the other CR methods, this approach is limited to moves in the torsion angle space, since bond lengths and angles are kept constant in the simulation. Fixing bond angles, however, is not in accordance with MD simulations and results in a rough, artificial energy landscape. For example fixing bond angles increases the energy barrier for a cis–trans conversion of alkanes significantly making this constraint less suitable for membrane simulations [

We have recently presented a very different local-move set MC method, the chain breakage/closure (CBC) algorithm, which solves closure equations in the bond/torsion angle space using the constant bond length approximation (CBLA) [

For some aspects of the lipid molecule, we used a united atom representation of DPPC recently developed for the AMBER 4.1. force field [_{3} atoms (atoms # 44–46) on the headgroup nitrogen (atom # 1), the oxygen atoms (atoms # 47,48) on the headgroup phosphate (atom # 5), and the carbonyl atoms (atoms # 49,50) as external atoms (light grey in

A DPPC molecule was built using in-house modeling software. Values for bond angles, torsion angles, and bond lengths were taken from their equilibrium values defined in [

where ^{5} steps of MC simulation of a DPPC bilayer consisting of 64 molecules using the implicit solvent model required ten days of simulation time.

All trajectory analysis was performed using in-house developed Fortran77-based software. Systems energy and entropy were calculated from the trajectories of obtained conformations of the DPPC molecule. A histogram of molecule energies was generated for the one-lipid and bilayer simulation and was fitted to a simple Gaussian model using SigmaPlot 4.0 (SPSS Inc., Chicago, IL, USA). To calculate the entropy associated with the conformational ensemble of the single DPPC molecule, the following formula was used [

Here, _{b}_{ij}

The respective mean value in angular brackets is the average of all positions of an atom in Cartesian space calculated for a given number of MC steps. For example, the covariance matrix after 20,000 MC steps provides the variances and covariances of 20,000 conformations around the respective mean value separately calculated for each _{1}, … _{3}_{N}

Here,

We started our simulations with a single DPPC molecule, as it allows us to assess structural equilibration of the basic unit of the membrane and to perform MC sampling of the whole conformational space. This is important, as thermodynamic properties calculated from bilayer simulations always average over individual lipids, which do not necessarily explore their configurational space individually. Thus, in order to assess equilibrium properties of the used molecular DPPC model, we performed first simulations of single DPPC molecules. This allows us to determine the evolution of structural correlations during MC sampling and thereby to assess the effect of the molecular constraints used in the description of DPPC (e.g., the CBLA and the move sets of external and closure atoms; see _{k}_{k}_{b}·_{V} = 9.43 cal/mol·K [_{p}. As we use the CBLA and thereby ignore stretching vibrations of the bonds, these values are presumably smaller than those obtainable by quantum mechanical calculations. However, it serves as starting point for comparison with the bilayer simulation (see below).

It is possible that the energy landscape for the lipid molecule is quite flat, meaning that a large number of different conformations are associated with very similar total energy. To obtain an independent proof for successful sampling of a representative subset of the conformational ensemble by our MC method, the conformational entropy was calculated. According to the Boltzmann theorem, the entropy is a measure of the probability of the sampled conformational subensemble and should, therefore, tend towards a maximum after long simulations leading to a thermodynamic equilibrium [

The Schlitter entropy calculated from the mass-weighted covariances reaches a plateau after about 1,000,000 MC steps; ^{2} = 0.9999) by a tri-exponential function with fractional half-times of 385 MC steps for the first phase, 11.179 MC steps for the second phase, and 266.595 MC steps for the third phase, respectively (not shown). The first and third phase contributed, each, to about 1/3 of the decrease in entropy, while the second phase contributed half of the entropy drop. Thus, the conformational entropy of our DPPC model is equilibrated by the CBC move set to about 83% of the stable plateau value with a half-time of less than 12,000 MC steps. The rather slow equilibration of the remaining one third of the entropy (the third phase of the fit) is not found for the evolution of conformational energy of the molecule, which became stable already after about 750,000 MC steps (compare ^{6} MC steps, indicates that our atomistic MC algorithm efficiently samples an ensemble of lipid conformations, thereby, faithfully, representing the configurational integral [

Next, the distribution of head group torsions was analyzed from a trajectory of MC simulated DPPC. The start value is given as a dotted line similar to a delta function, and only the alpha-torsions following the definition of Vanderkooi are shown (

Phospholipids are naturally associated in membrane assemblies, and it is therefore important to prove that the MC algorithm faithfully simulates lipid membranes as well. Having shown rapid equilibration of an isolated DPPC molecule does not prove that densely packed DPPC molecules in a bilayer arrangement can be efficiently simulated. Our starting structure is a crystalline-like DPPC bilayer with straight fatty acyl chains and identical conformations for the 64 lipid molecules that comprise our system (see Section 2.2 and _{k}_{p} = 821.63 cal/mol·K~0.822 kcal/mol·K. This can be used for a consistency check of one-lipid and bilayer simulation, as follows. Given the proportionality between heat capacity and system size [_{p} = 0.6 kcal/mol·K (_{p} ~ −0.1 kcal/mol·K per fatty acid chain of 16 carbon length [_{p} = 0.4 kcal/mol·K. Experimental values for the heat capacity of DPPC bilayers are in this range (e.g., ~0.15 kcal/mol measured by Heimburg and colleagues and ~0.3 kcal/mol as determined by Blume’s group, both using various forms of calorimetry [

To quantify the extent of equilibration of DPPC headgroups during transition of the crystalline to the fluid bilayer state, we calculated the torsion angle distribution for selected head group torsion angles as performed for the single DPPC molecule (compare

In contrast, the distribution of the

Next, we evaluated the electron density profile parallel to the z-axis in the laboratory frame. In _{HH}, a measure for bilayer thickness [_{HH} = 46 Å in the crystalline-like start structure and shows a characteristic double peak in the headgroup/glycerol region (grey line in _{HH}, is in line with X-ray diffraction data and earlier MD simulation of DPPC gel phase membranes [_{HH} = 36 Å) than what is found in experiments for fluid DPPC bilayers at 323 K, _{HH} ~ 38 Å [

Another criterion for the transition from the crystalline-like start structure of the bilayer to a fluid membrane is the change of the area per lipid during the MC simulation (^{2} in the start structure to 0.53 nm^{2} after 2200 MC steps, the bilayer expanded in the following 40,000 MC steps to reach a plateau value of 0.6 nm^{2} area per lipid after 60,000 MC steps. Most of the initial drop is a consequence of the volume moves displacing whole lipids thereby resulting in tighter packing and a contraction of the simulation box. The subsequent increase in area per lipid is due to incremental mobility of the fatty acyl chains and their “melting” from the all-trans configuration [

an apparent diffusion coefficient of ^{2}/MC step can be calculated. We note that sampling along a Markov chain in configurational space, especially in internal coordinates, does not provide real dynamic information about the investigated molecular system [

The advantage in sampling efficiency of atomistic Monte Carlo simulations of biomolecular structures can only become significant if an implicit solvent description is used. This is because the computational cost of calculating solute-solvent interactions for explicit solvent surmounts the gain in sampling efficiency by MC compared with other methods [^{2} for _{A} [

The orientation of DPPC head groups was also affected by the parameter values chosen for the slope of the dielectric constant. For

In this study, we give a thorough introduction into atomistic MC simulations of lipid membranes. We describe the challenges and advantages of MC compared to other simulation methods and provide a concrete implementation of local move MC for efficient equilibration of lipid bilayer structures. While previous atomistic MC simulations of membranes use CBMC [

Second, refined force field parameters could be used likely giving bilayer structures in better agreement with experiments [

Third, local move MC simulation of lipid membranes might be improvable by appropriate parallelization of the program code. Due to the sequential moving and updating scheme, MC simulations are traditionally very difficult to parallelize. The only routine, which in principle should be easy to parallelize is the energy calculation, for the whole system as well as for the energy difference prior to and after attempted local moves. One way of parallelizing the whole MC run is by implementing the replica-exchange technique, also called parallel tempering, into the MC algorithm. This method is useful for sampling rare events in MD and MC simulations (see an example for membranes in the Introduction, Section 1.2). As long as the energy distributions of the system simulated at various temperatures overlap, and the system size is not too large, replica-exchange MC simulation can be very efficient for overcoming low-energy barriers [

The last two decades have witnessed an enormous progress of biomolecular simulations, in particular in the field of modeling lipid membranes. In this area, simulations can be very fruitful, since in most experimental techniques only bulk bilayer properties are assessed. In contrast, molecular simulations of membranes can provide structural detail of the underlying lipid and protein dynamics. The simulations classically performed by atomistic MD, however, are very time-consuming and cover only membrane patches of a size of a few nanometer. This naturally calls for several developments: first, hybrid simulations using MD and other simulation approaches. We suggest that the local move MC method presented here is specifically useful in combination with MD simulations of lipid membranes. MD simulations provide dynamic structural information being directly comparable to experimental data. Assessing membrane dynamics with confidence requires proper equilibration of the simulated molecular system with its inherently broad range of time scales. Metropolis MC simulations of macromolecular systems, in particular in case of local and connectivity altering move sets, lack temporal information, as sampling occurs along a phase space trajectory [

A second development in the field concerns multi-scale modeling of lipid membranes. Coarse-graining of the molecular structure of various lipid species has been used to simulate membrane phenomena beyond the reach of classical all-atom simulations including lipid assembly and vesicle formation [

DW acknowledges funding from the Villum foundation and from a fellowship of the Max Delbrück Center for Molecular Medicine during the initial part of this work. DW and HS thank Remo Rohs (Molecular and Computational Biology Program, University of Southern California, Los Angeles, CA 90089, USA) for critical reading of the manuscript.

The authors declare no conflict of interest.

^{+}, Na

^{+}, K

^{+}, F

^{−}, and Cl

^{−}

Molecular model of DPPC for atomistic Monte Carlo simulation. A united atom representation of dipalmitoylphosphatidylcholine (DPPC), which neglects hydrogen atoms, is employed reducing the number of atoms per DPPC molecule to 50 [

Snapshot and energy of a single-molecule Monte Carlo simulation. (^{5} and 1·× 10^{6} MC steps. Large conformational moves of the fatty acyl chains and the lipid head group can be observed; (

Covariance, conformational entropy and free energy of a single DPPC molecule. (

Distribution of DPPC head group dihedral angles from a single-molecule simulation. Head group torsions were defined as in Vanderkooi _{1}); (_{2}); (_{3}); (_{4}); (_{5}); and (_{6}), respectively (compare

Simulation snapshot and system energy of a DPPC bilayer. The membrane simulation started from a crystalline bilayer consisting of 32 DPPC molecules with straight fatty acyl chains in each leaflet. Each molecule was rotated by a random rotation angle around the molecular long axis in the start configuration ((

Head group torsions of DPPC in the membrane. Head group dihedral angles were defined as in Vanderkooi _{1}); (_{1}) (_{4}) and (_{4}).

Electron density profile, area per lipid and lateral displacement in the bilayer. (

Impact of electrostatic interactions on membrane structure. (