Multiscale Methods for Macromolecular Simulations

Summary

In this article we review the key modelling tools available for simulating biomolecular systems. We consider recent developments and representative applications of mixed quantum mechanics / molecular mechanics (QM/MM), Elastic Network Models (ENM), coarse-grained molecular dynamics and grid based tools for calculating interactions between essentially rigid protein assemblies. We consider how the different length scales can be coupled, both in a sequential fashion (e.g. a coarse-grained or grid model using parameterisation from MD simulations), and via concurrent approaches, where the calculations are performed together and together control the progress of the simulation. We suggest how the concurrent coupling approach familiar in the context of QM/MM calculations can be generalised, and describe how this has been done in the CHARMM macromolecular simulation package.

Introduction

Recent years have seen a significant increase in the ability of computational approaches to help us understand the structure and function of complex biological systems. Alongside the advances in available computer power we have seen steady progress in the development and validation of modelling methods spanning a variety of length and time scales. Explicitly correlated electronic structure methods with linear scaling with system size now offer the prospect of chemical accuracy for the energetics of biological chemical reactions [1]. However, there is a massive disparity in the time and length scales on which the important cellular events occur and those that are accessible using these quantum based tools. Much larger length scales, and longer timescales, can be addressed by atomistic treatments based on much simpler, molecular mechanical expressions for the energy (important examples being MD [2] and the elastic network models[3]), in addition to methods based on larger primitive particles, for example, those in which each bead represents a small number of atoms. At longer length scales still, we will describe approaches which treat whole macromolecules as the building blocks, allowing the study of intermolecular interactions, this being of particular interest in docking and in structural studies of macromolecular assemblies.

To make predictions of macromolecular behaviour from first principles it is clearly necessary to incorporate knowledge from studies at the small length scales into the more coarse-grained approaches. Since every method (except for first principles quantum chemistry) depends on parameters to describe the energy of interactions, one can foresee a process by which we can ascend the length scale ladder, each successive method deployed incorporating parameters taken from the previous one. This approach is already the key to atomistic simulation using MD techniques, as in the widely used parameter sets (force-fields) which incorporate data from QM calculations, and other examples will be given below. This methodology has been referred to in a recent review in this journal [4] as the serial multiscale approach. Others have used the terms sequential [5], implicit, or message-passing methods to describe the approach. The serial approach has the weakness that any fitted parameter set, no matter how carefully constructed, is likely to have limits to its applicability or accuracy. For example, MD simulation using standard force fields will break down when a chemical bond is significantly stretched or an electronically excited state is present. For this reason, it necessary to find other ways to integrate the calculations at the different length scales in a more closely coupled way. Invariably this requires the execution of multiple component calculations as part of a given simulation, leading to their description as parallel multiscale treatments (also known as concurrent [5] or explicit schemes). One obvious approach is to introduce the concept of a spatial decomposition, whereby most of the system treated using the simpler method and the part for which that method fails is treated by a higher level of theory. The most widely used example is the mixed Quantum Mechanics / Molecular Mechanics (QM/MM) approach [6], and one can imagine this description working at other length scales, but with more difficulty, since while QM and MM share the same atomistic character, at the longer length scales mapping between the system descriptions necessarily becomes more complex. In this article we will follow [5] and use the terms sequential and concurrent, reserving the term parallel for use of multiple compute processors.

It is possible to subdivide the sequential and concurrent approaches further, by looking in detail at the nature of the information transfer between the different calculations [4]. This classification of methodologies is perhaps best developed in the field of process engineering [7], where the drive is to integrate detailed materials models into a predictive model of macroscopic behaviour.

QM/MM Methods

As noted already, the close coupling of quantum mechanics and classical molecular mechanics approaches is an important example of a concurrent multiscale scheme. When applied to enzymic catalysis, a quantum mechanical method is used to study the reactive process (active site and some or all of the substrate) and the surrounding, which will include the remainder of the enzyme, solvent, counterions etc, is treated by classical MM models. The QM/MM scheme, especially when used with a cost-effective QM method, such as a semi-empirical theory, allows exploration of the potential energy surface (PES) and molecular dynamics to obtain free energies of reactions. It is now established as the method of choice for the study of enzymatic reactions. It is not necessary to discuss the history of QM/MM models as there are many good reviews of the subject. In particular we refer to the reader to the comprehensive article by Senn and Thiel [6], in which many aspects of the method are described and a summary table of applications provided, and to the forward-looking discussion of Lin and Truhlar [8].

QM/MM Models – the Conventional and Subtractive Approaches

We start with a description of the established QM/MM methodology in which the QM and MM calculations are performed on different parts of the chemical system. The separate regions are coupled, of course, and the coupling terms can take a variety of forms depending on the particular model, usually based on classical electrostatics and parameterized non-bonded interactions. Subsequently Morokuma and his co-workers have introduced a subtractive [6] QM/MM model exemplified by the ONIOM scheme [9]. In this case, the domain chosen for the lower level (or more coarse-grained) calculation includes those parts of the system treated at a higher level. The multiple-counting in tackled by subtracting a third energy, that of the inner domain at the lower level of theory. In this way, the interactions between the domains appear within the lower level calculation and no coupling terms have to be devised. A similar approach has been developed for solid state modelling by Sauer [10], and the technique can simply be applied to other levels of theory. Mixing different QM models, for example DFT + CCSD(T) models, is one important application. At longer length scales, we can now integrate atomistic and coarse-grain treatments, as discussed below.

Energetics of Reaction Pathways

QM/MM calculations have been widely deployed in the study of enzymatic reactions. Many different QM methods have been deployed, ranging from semi-empirical to high level correlated [11] approaches and many of the techniques of small molecule computation chemistry can be used to explore the potential energy surface for macromolecules. While there are serious limitations arising from the lack of sampling inherent in such an approach [12], the gross features of the potential energy surface can provide useful information in the discussion of reaction mechanisms.

Typical QM/MM studies in the past involved a single QM system embedded in a classical MM region. There has been a major effort to streamline QM/MM calculations, so that as many as 100 QM calculations are performed simultaneously in a single computer run. For example, a QM/MM potential energy function has been implemented between the CHARMM [13] and Q-Chem software packages [14]. This interface supports minimization and dynamics via analytical first derivatives at the Hartree-Fock, density functional theory (DFT), and post-HF (RIMP2, MP2, CCSD) levels of theory. In addition, the Replica Path (RPATH) [15], Nudged Elastic Band (NEB) [16], and combination RPATH with restrained distance (RPATH+RESD) [17] reaction pathway methodologies are fully supported. These are powerful techniques for studying reaction pathways in a multi-level parallel (or parallel/parallel) fashion, with each QM/MM pathway point being distributed to a different node of a Beowulf cluster. The software is designed to facilitate the optimization and analysis of a pathway in a single run.

QM/MM approaches to Free Energies

Integrating quantum mechanics into a macromolecular dynamical simulation provides a route to the free energy differences associated with biochemical processes. Direct computation of the free energy of a chemical system from the thermodynamical partition function is rarely practical, and many creative approaches to extracting the relevant free energy changes for processes of biological interest from tractable MD simulations have been devised. The adequate sampling for conformational space remains a challenge, especially when higher-level QM calculations are used. The reader is referred to reference [18] for a review of the methods by which free energies based on ab-initio free energy results may be obtained. In many cases, it is satisfactory to determine the zero-temperature reaction path for the QM system (with the MM surroundings relaxed at each point) and then use MD to sample the entropic contribution of the enzyme or solvent dynamics. Where the motion of the QM atoms needs to be included, a promising approach would appear to be that based on thermodynamic cycles [19], see Scheme 1.

Scheme 1.

The desired transformation (between states A and B at the High level computational technique) is shown by the white arrow. It is not usually directly accessible due to the cost of achieving adequate conformational sampling at the High level of theory. However it can be computed by summing the three gray arrows, whereby the free energy difference between A and B is computed using a cheaper Hamiltonian (either MM or a semi-empirical QM scheme such as EVB [19]. States A and B can be either distinct chemical states in a proposed reaction mechanism, or different chemical systems, in which case the A-B transformations are non physical (known as ‘alchemical’ free energy perturbation). Recently Woods et al [20] have suggested an approach to improving the accuracy the two vertical integrals by using Metropolis-Hastings scheme (performing a Monte Carlo sampling at the MM level but using a QM/MM acceptance test).

In some cases, it is possible to estimate free energy changes without recourse to dynamical sampling. Within a harmonic approximation, the second derivatives of the energy with respect to the atomic coordinates (the Hessian) allows the computational of the vibrational entropy. As a result, there is also a renewed effort to utilize partial or full multiscale Hessians to calculate molecular flexibility and to estimate thermodynamic properties, such as reaction free energies. Expanding on the previous functionality, full QM/MM analytic second derivatives have recently been implemented into Q-Chem and interfaced with CHARMM's VIBRan module [21]. Efficiency in calculating the Hessian can be achieved by limiting the number of environmental (i.e. MM) degrees of freedom and solving couple perturbed calculations on the allowed degrees of freedom. This is most easily done by treating distant molecular groups as rigid objects with only rotational and translational degrees of freedom.

A new vibrational subsystem analysis (VSA) method has been developed and implemented that couples global motion to a local sub-system while including the inertial effects of the environment [21]. The premise of the VSA method is a partitioning of a system into a smaller region of interest and a usually larger part referred to as environment. This method allows for researchers to address a vast array of interesting problems. Examples of these include, but are not limited to: more accurately estimating vibrational free energy contributions for parts of a large system; elimination of the 'tip effect' in Elastic Network Model (ENM) calculations; probing specific degrees of freedom that may contribute to free energy differences; and estimating activation free energies in QM/MM reaction path calculations. The main use of this method will most likely be the estimation of pathway free energies, as opposed to just enthalpies, without the need for extensive dynamic sampling.

Adaptive Schemes

We now consider recent attempts to move beyond a static definition of the QM region. This has been a topic of research in the materials modelling community for some time [22] where it is required for problems such as crack propagation. A recent development of the Learn on the Fly (LOTF) method [23] has lead to its application to biological systems, in which the AMBER QM/MM implementation [24] has been used (Gabor Csanyi, personal communication). For biological systems, the most important application is solvation, where solvent molecules may move into the active site and thus require QM treatment for part of the dynamics trajectory. Heyden et al. have recently proposed the Adaptive Partitioning [25] approach which addresses issues of discontinuities in the energy and gradients characteristic of some of the earlier methods.

Electronic Excitations

To finish this section of the review, it is worth mentioning recent developments in two areas of protein science in which the QM/MM approach can used to explore biologically important electronic transitions. We first consider the application of the QM/MM approach to photochemistry and the study of excited state potential energy surfaces. To date, the most popular QM methods for these studies are those based on multiconfigurational wavefunctions (for example CASSCF) and the availability of 2nd order perturbation theory corrections for dynamical correlation (the CASPT2 scheme) now enables highly accurate energetics to be computed for excited state potential energy surfaces. The CASPT2 scheme has recently been incorporated into an QM/MM approach [26] and used to study the fluorescence of the green fluorescent protein (GFP) used in the design a prototype light-driven molecular switch.

It is also possible to use QM/MM molecular dynamics to explore the influence of enzyme and solvent on redox-active transition metals. One approach, within the framework of Marcus theory, uses classical MD to sample the conformational space, combined with DFT calculations of the vertical ionization process at a number of snapshots. This approach was able to compute a difference in redox potential of 60 mV (between mesophilic and thermophilic forms of Rubridoxin) to within 20 mV [27]. A scheme that incorporates dynamics on both the oxidised and reduced potential energy surface, and switches between them in a grand-canonical MD scheme has been derived and applied to the redox potential of cupper in azurin [28]. Very recently, a new approach to the simulation of redox processes has been suggested. The oxidation state is considered continuously variable, using fractional electron count [29] and can therefore be used as an order parameter in a Thermodynamic Integration free energy scheme.

Elastic Network Models

One of the most important tools for the coarse-grained study of protein conformation changes is the Elastic Network Model (ENM). Given the C_α atomic coordinates for a protein's native structure, we build an elastic network model by using a harmonic potential with a single force constant C to account for pairwise interactions between all C_α atoms that are within a cutoff distance (R_C = 10 Å). The energy in the elastic network representation of a protein is

$$E_{\text{network}}=\frac{1}{2}\sum$$

where d_ij is the distance between the dynamical coordinates of the C_α atoms i and j, and d⁰_ij is the distance between C_α atoms i and j, as given in the reference structure. For recent reviews see [3] and [30]. It is especially useful for multiscale modelling in that it provides a simple coarse grained model that allows protein flexibility while preserving a structure that is reasonably close to an experimentally observed structure. This model is purely harmonic, thus proteins cannot unfold. The low frequency modes from an ENM model present an excellent basis for examining global motion. Limitations of ENM for examining processes with some local character have been observed [31], suggesting that more than a few lowest modes are often needed. However, the robustness of modes to ENM parameter perturbations was found to be useful in identifying functionally important modes [32], which may further reduce the subset of low-frequency modes required for focused conformational sampling.

ENMs can be used to facilitate structure refinement at low resolution [33], and a multi-base generalization of ENM has been the basis for several recently proposed techniques for efficient generation of conformational transition pathways [34][35][36]. ENM-based methods for predicting dynamic coupling and ligand-binding induced conformational changes have been successfully applied to HCV NS3 helicase [37].

Recent work with ENM models demonstrate that ENM modes can provide an efficient sampling of conformational space covered by the large number of available HIV-1 protease structures, in a manner similar to MD trajectory or NMR ensemble analysis [38] and that that existing elastic models (ENM and GNM) can be improved to achieve an optimal description of both protein conformational motions and thermal fluctuations [39].

A more complex elastic network model approach to conformational change is to combine information from multiple conformational states of the protein [40], for example to generate double-well models. This approach has been used to explore e.g. conformational changes in adenylate kinase [36], also see [41], a well studied model system for such transitions [42]. A number of other models for conformational flexibility merit further exploration, including those which combine rigid ‘blocks’ within a protein with flexible connections [43].

Coarse-Grained MD Simulations

Atomistic simulations of complex biomolecular systems provide a detailed picture of e.g. conformational dynamics and protein/ligand interactions. However, even with large scale computational resources and well-scaling simulation codes, it is still challenging to reach long time scales (> 1 μs) for large and complex systems (e.g. [44]). Consequently there has been considerable interest in the application of coarse-grained models to proteins and related biological systems.

A number of different levels of coarse-grained models are possible, varying in the number of atoms which are represented by a single particle. For example, an amino acid residue within a protein could be represented by a single particle or by 3-4 particles where each particle represents 2-5 non-hydrogen atoms. In each case, both bonded and non-bonded interactions have to be parameterized to reproduce either thermodynamic properties of the constituent monomers (e.g. amino acids) or the overall dynamic properties of the resultant protein. Such coarse-graining reduces by an order of magnitude the interactions which have to be evaluated for a given system and also enables the use of longer time steps (e.g. 40 fs for CG rather than 1 fs for atomistic). Together this may provide several orders of magnitude of speedup relative to atomistic simulations, thus enabling longer timescale processes for more complex systems to be explored.

Following the development and initial exploration of coarse-grained models for proteins (previously reviewed by e.g. [3]) and for membranes (previously reviewed by [45]) there has been considerable interest in exploring the potential for this approach and applying it to a number of key systems. In what follows this will be illustrated in terms of: (i) modelling conformational changes of proteins; (ii) applications to membrane proteins; (iii) CG and related multi-scale approaches to more complex assemblies. We will not discuss the history of CG simulations in the context of protein folding as this has recently been reviewed in COSB [46] but will discuss some very recent developments later in this section.

Conformational Changes in Proteins

A number of CG models have been used to explore mechanisms of conformational change in proteins. For all of these a major issue is that of how to parameterise. A promising general approach to this is a force-matching approach in which the CG model is parameterised to match atomistic simulations for the same or a related system. This has been explored for e.g. elastic network models [47] and for two particle per amino acid models [48] of proteins.

Membranes and their Proteins

There has recently been considerable interest in extending CG models previously used for lipid bilayers [49][50][51] and for simple models of membrane proteins [52] [53][54] to more complex membrane proteins and peptides. This has required modification of existing CG models for lipids [55] to also include amino acids and has required issues of parameterization of amino acid/hydrocarbon interactions in CG models to be addressed, both by comparison with experimental thermodynamic (partitioning) data and via comparison with atomistic simulation based free energy profiles for amino acid sidechains vs. location within a lipid bilayer [56].

Initial studies using CG models to simulate membrane/peptide and membrane/protein systems have been very encouraging, reproducing a range of experimental data on the location of model peptides and proteins within membranes [56][57] and reproducing patterns of protein/lipid interactions seen in extended atomistic simulations [58]. It should be noted that in general such models require either dihedral angle or elastic network restraints in order to retain the secondary and tertiary structure of the protein.

CG-MD may be used to self-assemble lipid bilayers around membrane proteins, thus enabling prediction of the localization and orientation of a protein within a lipid bilayer (see Fig. 1). The smaller CPU demands of CG-MD mean that it is feasible to use this method to compare different families of membrane proteins and their interactions with lipids [59]. For example, application to a non-redundant set of ~100 membrane proteins [60] enabled comparative analysis of protein/bilayer interactions, indicating how local bilayer deformations may be related to membrane protein class (e.g. α-helix bundle vs. β-barrel TM domains). CG-MD has used to explore bilayers containing multiple (e.g. 16) rhodopsin molecules [61] to reveal how local bilayer deformation influences protein-protein association within a membrane.

Figure 1.

Schematic representation of a CG-MD simulation of the self-assembly of a lipid bilayer/protein system from randomly positioned lipid (and water – not shown) molecules plus a know membrane protein structure. After ~ 0.25 μs of simulation the protein is stably inserted in a bilayer. The representation of the protein on the left illustrates the use of an elastic network model (in red) to restrain the structure of the protein.

CG-MD simulations may also be used to characterise the interactions of proteins with the surface of a lipid bilayer, as has been shown both for small (~30-residue) toxins that bind to membrane surfaces and inhibit ion channels [62], and for more complex membrane-binding enzymes such as phospholipiase A2 [63].

CG simulations have been used to explore functional aspects of a number of channel and related proteins in membranes. For example, CG simulations of lipid-protein interactions of isolated voltage sensor domains (VSD) from K⁺ channels [64] suggested a degree of bilayer deformation as a result of interactions of lipid headgroups with the charged sidechains of the voltage-sensing S4 helix, as is also seen in (shorter) atomistic simulations [65][66]. CG simulations have been used to study possible mechanisms of gating of a number of channels and related proteins, including the mechanosensitive channel MscL [67], the voltage-gated channel Kv1.2 [68] and the translocon [69].

CG and multi-scale approaches to complex assemblies

One of the key areas for the future which is only just beginning to be explored is the application of CG and multi-scale approaches to more complex systems. A number of groups have successfully applied multi-scale approaches to (reasonably complex) enzymes (e.g. acetylcholine esterase [70] and OmpT [71]) proving the utility of this approach. CG simulations have been used to self-assemble more complex systems such as lipoprotein particles [72]. Initial explorations of combining CG models of protein and of DNA in simulations of the nucleosome [73] are very promising. Highly coarse grained models of lipids combined with radically simplified representations of proteins have been used to simulate protein-induced membrane vesiculation [74]. More recently, a multi-scale approach has been used to explore bending and dynamics of membranes by multiple copies of a BAR domain [75], demonstrating that multi-scale simulations have the potential to link structural and systems descriptions of cell biological processes.

Protein folding in presence of osmolytes - the Molecular Transfer Model

One limitation of many of these coarse-grained models is that the folding/unfolding reaction can only be initiated by a change in temperature, while many in vitro experiments use osmolytes (i.e. small organic molecules such as urea, guanidinium hydrochloride, trimethylamine-N-oxide, and sucrose) to modulate native state stability [76]. Recently, a novel method, referred to as the Molecular Transfer Model (MTM), has been developed to rapidly and accurately predict the effect of osmolytes on the thermodynamic properties of proteins [77]. It does this by combining information from coarse-grained molecular simulations using the BLN (hydrophoBic, hydrophiLic, Neutral) scheme, experimentally measured transfer free energies of individual amino acids, and the Tanford transfer model, all of which estimate the free energy of transferring a given protein conformation from water to aqueous osmolyte solution. With this information the MTM computes the partition function for any osmolyte type and osmolyte concentration of interest, thereby allowing almost any thermodynamic property to be computed. The MTM, being a post-simulation processing technique, is rapid. In a matter of minutes it can predict the effect of a large number of aqueous osmolyte solution conditions on the thermodynamics of protein folding and unfolding. The MTM has been validated against experimentally measured m-values (the rate of change of ΔG as a function of denaturant concentration and a quantitative measure of the breadth of the unfolding transition) and single molecule FRET measurements for protein L and a highly stable Cold Shock Protein [77].

The MTM is a significant breakthrough in modelling because never before have molecular simulations been able to study protein folding/unfolding thermodynamics as a function of osmolyte concentration, it can accurately predict experimentally measurable quantities, and offers a molecular interpretation of experimental results. While, in theory, the MTM can be applied to all-atom simulations in practice it is only currently accurate for coarse-grained models, where adequately converged sampling can be achieved.

Grid-based methods

Molecular modelling has been a powerful approach to provide structural insights into biological procedures at the atomic level. New developments in experimental technologies, such as electron microscopy, provide an approach to obtain low resolution structure information of large molecules and their assemblies. Extracting structure information from these low resolution maps and obtain atomic interpretation of the large biomolecular assemblies has become a powerful tool in modern structural biology. This requires molecular modelling to be conducted in conjunction with these low resolution maps, as well as high resolution atomic structures to maximize the capability in structural biology studies.

Grid based molecular modelling differs from the centre-based models in that molecular properties are assigned to regions of space that are broken up into many smaller volume elements. This is a natural way to interact with electron microscopy data and X-ray data where electron densities are partitioned in the same manner. The main advantage of the grid approach is that operations, such as docking, can be performed rapidly and with a simplified energy function. One major drawback is that it is not easy to incorporate flexibility in a straightforward manner.

As with the development of structural biology, molecular modelling is now applied to larger and larger biomolecular machineries. As the systems become large, the atomic description becomes very inefficient and time consuming and it is more efficient to treat large biomolecules as simplified shape objects, while ignoring their internal structures. Although molecular flexibility plays an important role in biological activity, in many cases molecular geometric shapes together with their surface properties are sufficient to describe many cellular processes such as molecule assembling, protein-protein binding.

As an example of the use of grid based coarse-grained molecular modelling, we summarize the capabilities of CHARMM’s [13] EMAP facility [78] which was originally developed to assist in structure determination with electron microscopy. Later, this facility was expanded to allow protein-protein docking analysis in the absence of experimental data. EMAP uses map objects (Fig. 2) to represent space occupation of molecular structures. Unlike chemical descriptions of molecules that contains atoms and atoms are linked by chemical bonds, a map does not have internal chemical structures. Instead, a map represents a spatial distribution of certain properties, typically, electron density. This distribution generally is described as scalar values at discrete grid points due to the irregularity of the distributions and to limits in storage.

Figure 2.

A map object and its properties. A map object is defined as a grid with a given property mapped on the grid points. This could be either e.g. experimentally determined electron density or properties (e.g. electrostatic charges) generated from reference molecules.

Map comparison provides a metric for fitting one map into another. Four types of cross correlation functions [78] are available for comparison between map objects.

The energetics of molecular systems are the basis of molecular modelling. Calculation of molecular interaction using map objects is the crucial step to a successful modelling or simulation study. For atomic objects interaction calculation is pairwise and is very time-consuming for large molecular assemblies. For map objects, we propose to use field interactions that can be calculated much more efficiently. Four types of energies are defined to describe interaction between map objects: electric field interaction; surface charge-charge interaction; van der Waals interaction; and desolvation interaction. The parameters in these functions are obtained by fitting into energies calculated with using standard force fields, making this another example of a serial multi-scale algorithm.

The grid-threading Monte Carlo (GTMC) search algorithm [78] uses a combination of the grid search and Monte Carlo sampling to provide an efficient way for robustly fitting rigid domains to a target map. The conformational space is split into grid points and short Monte Carlo searches are performed to identify local maxima around the grid points. The global maximum is identified among the local minima.

The derivation of high resolution molecular assembly structures from microscopy maps is a major application of the map approach. This method has been successfully applied into several experimental studies [79]. Fig. 3 illustrates the steps to perform a fitting of high resolution molecular structure into electron microscopy maps.

Figure 3.

Steps to derive molecular assembly structures by fitting molecular structures into electron microscopy maps. This is illustrated using a T-cell receptor (TCR) variable domain (PDB code: 1a7n) as an example complex to illustrate the modelling process with map objects. The TCR variable domain is a complex of two chains. The two chains are first blurred into maps of the same resolution (here 15 Å) as the EM map. Then each map is fitted into the EM map to get complex map. The complex map is projected back to atomic structures, which is the complex structure we are looking for. The root-mean-square (rms) deviations of the fitting result from X-ray complex is 3 Å.

The structure obtained from map fitting generally is not optimized in atomic details. There are often atom overlaps or improper spacing between components. This structural mismatch can be removed by standard approaches such as energy minimization and simulated annealing as long as the fitting result is very close to the right structure. After the minimization, the rms deviation is 0.97Å.

With map energy functions, it is possible to determine complex structures through minimizing the map interaction energy even in cases where the EM complex map is not available. It should be noted that the map object assumes certain rigidity of a molecular object. Certain flexibility of loop region can be accommodated by the low resolution characters, while large flexibilities like domain movement should be dealt with through multiple map objects. Recently, this method was successfully applied to modelling of the peroxiredoxin (Prx) complex [80].

MSCALE – a generalized approach to multiscale simulation

With the exception of QM/MM, all of the multiscale methods reviewed in this article are typically deployed according to the sequential multiscale methodology, data from the high accuracy models being used to parameterise the coarser ones. However it is interesting to consider the potential of a concurrent coupling of these disparate methods, by generalising the ideas underpinning the additive and subtractive QM/MM schemes. Within a single framework, it is possible to support both additive and subtractive approaches (or a combination of these), and to allow a single energy and force calculation to be done with diverse set of software and computer hardware. With such a tool, the multiscale combinations are endless.

CHARMM's recently implemented MSCALE command is such a general tool, supporting multiple independent but connected calculations using defined molecular subsystems. Programs are run in parallel in a client-server mode, with basic communication. The server calculations can employ either CHARMM or several other supported programs (e.g. the quantum chemistry programs NWChem, Molpro, PSI3, and Gaussian 03) with a consistent interface.

This flexible implementation allows any number of subsystems, each running as a separate process, usually on a separate computer (or parallel cluster). The client process maps the forces and Hessian elements from the various atomic or coarse grained centres, either by co-locating them, or connecting them through constraints or restraints. For example the lone-pair facility in CHARMM can be employed to constrain one coarse grained centre to be at the centre of mass of a collection of atoms. In constructing the total energy, a scale factor is applied to each subsystem, additive models will have all positive factors, with negative factors using subtractive models. Scale factors can be tied to parameters (lambda), allowing free energy perturbation simulations with complex multiscale modelling.

With this capability, it is possible to mix models of different resolutions, in a completely general manner. It is also possible to mix models at the same level of resolution, for example combining the two residue-based CG schemes, ENM and BLN, to perform coarse grained protein docking that allows protein flexibility but not protein unfolding. When used with all-atom modelling, it is possible to mix and match force field types in a single calculation. For example in simulating protein absorption on a surface, the protein can be modelled with the CHARMM force field, while the surface could be modelled with the CFF force field.

Conclusions

We have reported encouraging progress in the modelling of biomolecular systems at a range of different length scales. QM/MM techniques are now in routine use and the underlying quantum mechanical methods are becoming steadily more accurate. Significant methological approaches to many chemically significant properties, such as redox potentials, free energy changes and excited state reactivity have been made during the last few years. The coupling of QM and MM Hessian information has led to the development of new vibrational partitioning schemes with application to harmonic free energies analyses for large molecular assemblies. We have reported on recent work with a number of different coarse-grained MD schemes, and while there are still limitations to the types of processes that can be modelled with predictive accuracy, systematic studies of many complex biomolecules and assemblies are made possible by the computational efficiency of the methods. Likewise, elastic network models are specialized for the study of local conformational changes, but recent developments allow the study of a broader range of processes, and they also find application in CG schemes where they can be used as soft restraints on protein structure. At the longest length scales, grid-based methods sacrifice flexibility of the biomolecule to allow rapid computation of interaction energies of multi-million atom systems, and fast interpretation of cryo-EM images.

As the individual methods mature, a key challenge over the next few years will be the closer integration of the methods. The applicability of the coarse-grained schemes will be greatly enhanced if the detailed information required can be generated from accurate studies as part of a closely integrated computational scheme. To date only QM/MM is really used this way, but the generalization of the ideas of additive and subtractive coupling to a wider range of multiscale models offers a route to explore many other concurrent coupling possibilities.

Abbreviations

BLN

HydrophoBic, HydrophiLic Neutral

CCSD(T)

Coupled Cluster Singles and Doubles with perturbative Triples

CG

coarse grained

Cryo-EM

Cryo Electron Micrograph

CPU

Central Processor Unit

DFT

Density Functional Theory

ENM

Elastic Network Model

FRET

Fluorescence Resonance Energy Transfer

GNM

Gaussian Network Model

MD

Molecular Dynamics

MM

Molecular Mechanics

MTM

Molecular Transfer Model

NEB

Nudged Elastic Band

NMR

Nuclear Magnetic Resonance

Omp

Outer Membrane Protein

QM

Quantum Mechanical

RPATH

Replica Path

TM

trans membrane

VSD

Voltage Sensor Domain