An Assessment of Potential of Mean Force Calculations with Implicit Solvent Models

Solvation Thermodynamics of Solutes in Water and Ionic Liquids Using the Multiscale Solvation-Layer Interface Condition Continuum Model

Molecular assembly processes are generally driven by thermodynamic properties in solutions. Atomistic modeling can be very helpful in designing and understanding complex systems, except that bulk solvent is very inefficient to treat explicitly as discrete molecules. In this work, we develop and assess two multiscale solvation models for computing solvation thermodynamic properties. The new SLIC/CDC model combines continuum solvent electrostatics based on the solvent layer interface condition (SLIC) with new statistical thermodynamic models for hydrogen bonding and nonpolar modes: cavity formation, dispersion interactions, combinatorial mixing (CDC). Given the structures of 500 solutes, the SLIC/CDC model predicts Gibbs energies of solvation in water with an average accuracy better than 1 kcal/mol, when compared to experimental measurements, and better than 0.8 kcal/mol, when compared to explicit-solvent molecular dynamics simulations. The individual SLIC/CDC energy mode values agree quantitatively with those computed from explicit-solvent molecular dynamics. The previously published SLIC/SASA multiscale model combines the SLIC continuum electrostatic model with the solvent-accessible surface area (SASA) nonpolar energy mode. With our new, improved parametrization method, the SLIC/SASA model now predicts Gibbs energies of solvation with better than 1.4 kcal/mol average accuracy in aqueous systems, compared to experimental and explicit-solvent molecular dynamics, and better than 1.6 kcal/mol average accuracy in ionic liquids, compared to explicit-solvent molecular dynamics. Both models predict solvation entropies, and are the first implicit-solvation models capable of predicting solvation heat capacities.

Enhancing backbone sampling in Monte Carlo simulations using internal coordinates normal mode analysis

Normal mode methods are becoming a popular alternative to sample the conformational landscape of proteins. In this study, we describe the implementation of an internal coordinate normal mode analysis method and its application in exploring protein flexibility by using the Monte Carlo method PELE. This new method alternates two different stages, a perturbation of the backbone through the application of torsional normal modes, and a resampling of the side chains. We have evaluated the new approach using two test systems, ubiquitin and c-Src kinase, and the differences to the original ANM method are assessed by comparing both results to reference molecular dynamics simulations. The results suggest that the sampled phase space in the internal coordinate approach is closer to the molecular dynamics phase space than the one coming from a Cartesian coordinate anisotropic network model. In addition, the new method shows a great speedup (∼5-7×), making it a good candidate for future normal mode implementations in Monte Carlo methods.

Structure based descriptors for the estimation of colloidal interactions and protein aggregation propensities

The control of protein aggregation is an important requirement in the development of bio-pharmaceutical formulations. Here a simple protein model is proposed that was used in molecular dynamics simulations to obtain a quantitative assessment of the relative contributions of proteins' net-charges, dipole-moments, and the size of hydrophobic or charged surface patches to their colloidal interactions. The results demonstrate that the strength of these interactions correlate with net-charge and dipole moment. Variation of both these descriptors within ranges typical for globular proteins have a comparable effect. By comparison no clear trends can be observed upon varying the size of hydrophobic or charged patches while keeping the other parameters constant. The results are discussed in the context of experimental literature data on protein aggregation. They provide a clear guide line for the development of improved algorithms for the prediction of aggregation propensities.

CHARMM: the biomolecular simulation program

CHARMM (Chemistry at HARvard Molecular Mechanics) is a highly versatile and widely used molecular simulation program. It has been developed over the last three decades with a primary focus on molecules of biological interest, including proteins, peptides, lipids, nucleic acids, carbohydrates, and small molecule ligands, as they occur in solution, crystals, and membrane environments. For the study of such systems, the program provides a large suite of computational tools that include numerous conformational and path sampling methods, free energy estimators, molecular minimization, dynamics, and analysis techniques, and model-building capabilities. The CHARMM program is applicable to problems involving a much broader class of many-particle systems. Calculations with CHARMM can be performed using a number of different energy functions and models, from mixed quantum mechanical-molecular mechanical force fields, to all-atom classical potential energy functions with explicit solvent and various boundary conditions, to implicit solvent and membrane models. The program has been ported to numerous platforms in both serial and parallel architectures. This article provides an overview of the program as it exists today with an emphasis on developments since the publication of the original CHARMM article in 1983.

The multi-copy simultaneous search methodology: a fundamental tool for structure-based drug design

Fragment-based ligand design approaches, such as the multi-copy simultaneous search (MCSS) methodology, have proven to be useful tools in the search for novel therapeutic compounds that bind pre-specified targets of known structure. MCSS offers a variety of advantages over more traditional high-throughput screening methods, and has been applied successfully to challenging targets. The methodology is quite general and can be used to construct functionality maps for proteins, DNA, and RNA. In this review, we describe the main aspects of the MCSS method and outline the general use of the methodology as a fundamental tool to guide the design of de novo lead compounds. We focus our discussion on the evaluation of MCSS results and the incorporation of protein flexibility into the methodology. In addition, we demonstrate on several specific examples how the information arising from the MCSS functionality maps has been successfully used to predict ligand binding to protein targets and RNA.

Bounding the electrostatic free energies associated with linear continuum models of molecular solvation

The importance of electrostatic interactions in molecular biology has driven extensive research toward the development of accurate and efficient theoretical and computational models. Linear continuum electrostatic theory has been surprisingly successful, but the computational costs associated with solving the associated partial differential equations (PDEs) preclude the theory's use in most dynamical simulations. Modern generalized-Born models for electrostatics can reproduce PDE-based calculations to within a few percent and are extremely computationally efficient but do not always faithfully reproduce interactions between chemical groups. Recent work has shown that a boundary-integral-equation formulation of the PDE problem leads naturally to a new approach called boundary-integral-based electrostatics estimation (BIBEE) to approximate electrostatic interactions. In the present paper, we prove that the BIBEE method can be used to rigorously bound the actual continuum-theory electrostatic free energy. The bounds are validated using a set of more than 600 proteins. Detailed numerical results are presented for structures of the peptide met-enkephalin taken from a molecular-dynamics simulation. These bounds, in combination with our demonstration that the BIBEE methods accurately reproduce pairwise interactions, suggest a new approach toward building a highly accurate yet computationally tractable electrostatic model.

Interpreting the Coulomb-field approximation for generalized-Born electrostatics using boundary-integral equation theory

The importance of molecular electrostatic interactions in aqueous solution has motivated extensive research into physical models and numerical methods for their estimation. The computational costs associated with simulations that include many explicit water molecules have driven the development of implicit-solvent models, with generalized-Born (GB) models among the most popular of these. In this paper, we analyze a boundary-integral equation interpretation for the Coulomb-field approximation (CFA), which plays a central role in most GB models. This interpretation offers new insights into the nature of the CFA, which traditionally has been assessed using only a single point charge in the solute. The boundary-integral interpretation of the CFA allows the use of multiple point charges, or even continuous charge distributions, leading naturally to methods that eliminate the interpolation inaccuracies associated with the Still equation. This approach, which we call boundary-integral-based electrostatic estimation by the CFA (BIBEE/CFA), is most accurate when the molecular charge distribution generates a smooth normal displacement field at the solute-solvent boundary, and CFA-based GB methods perform similarly. Conversely, both methods are least accurate for charge distributions that give rise to rapidly varying or highly localized normal displacement fields. Supporting this analysis are comparisons of the reaction-potential matrices calculated using GB methods and boundary-element-method (BEM) simulations. An approximation similar to BIBEE/CFA exhibits complementary behavior, with superior accuracy for charge distributions that generate rapidly varying normal fields and poorer accuracy for distributions that produce smooth fields. This approximation, BIBEE by preconditioning (BIBEE/P), essentially generates initial guesses for preconditioned Krylov-subspace iterative BEMs. Thus, iterative refinement of the BIBEE/P results recovers the BEM solution; excellent agreement is obtained in only a few iterations. The boundary-integral-equation framework may also provide a means to derive rigorous results explaining how the empirical correction terms in many modern GB models significantly improve accuracy despite their simple analytical forms.

A Kirkwood-Buff derived force field for amides

A force field for the computer simulation of aqueous solutions of amides is presented. The force field is designed to reproduce the experimentally observed density and Kirkwood-Buff integrals for N-methylacetamide (NMA), allowing for an accurate description of the NMA activity. Other properties such as the translational diffusion constant and heat of mixing are also well reproduced. The force field is then extended to include N,N'-dimethylacetamide and acetamide with good success. Analysis of the simulations of low concentrations of NMA in water indicates a high degree of solvation with only 15% of the NMA molecules involved in solute-solute hydrogen bonding. There is only a weak angular dependence of the solute-solute hydrogen bonding interaction with a minimum at an angle of 65 degrees for the N-H and C=O dipole vectors. The models presented here provide a basis for an accurate force field for peptides and proteins.

Secondary structure bias in generalized Born solvent models: comparison of conformational ensembles and free energy of solvent polarization from explicit and implicit solvation

The effects of the use of three generalized Born (GB) implicit solvent models on the thermodynamics of a simple polyalanine peptide are studied via comparing several hundred nanoseconds of well-converged replica exchange molecular dynamics (REMD) simulations using explicit TIP3P solvent to REMD simulations with the GB solvent models. It is found that when compared to REMD simulations using TIP3P the GB REMD simulations contain significant differences in secondary structure populations, most notably an overabundance of alpha-helical secondary structure. This discrepancy is explored via comparison of the differences in the electrostatic component of the free energy of solvation (DeltaDeltaG(pol)) between TIP3P (via thermodynamic Integration calculations), the GB models, and an implicit solvent model based on the Poisson equation (PE). The electrostatic components of the solvation free energies are calculated using each solvent model for four representative conformations of Ala10. Since the PE model is found to have the best performance with respect to reproducing TIP3P DeltaDeltaG(pol) values, effective Born radii from the GB models are compared to effective Born radii calculated with PE (so-called perfect radii), and significant and numerous deviations in GB radii from perfect radii are found in all GB models. The effect of these deviations on the solvation free energy is discussed, and it is shown that even when perfect radii are used the agreement of GB with TIP3P DeltaDeltaG(pol) values does not improve. This suggests a limit to the optimization of the effective Born radius calculation and that future efforts to improve the accuracy of GB models must extend beyond such optimizations.

Conformational sampling with implicit solvent models: application to the PHF6 peptide in tau protein

Implicit solvent models approximate the effects of solvent through a potential of mean force and therefore make solvated simulations computationally efficient. Yet despite their computational efficiency, the inherent approximations made by implicit solvent models can sometimes lead to inaccurate results. To test the accuracy of a number of popular implicit solvent models, we determined whether implicit solvent simulations can reproduce the set of potential energy minima obtained from explicit solvent simulations. For these studies, we focus on a six-residue amino-acid sequence, referred to as the paired helical filament 6 (PHF6), which may play an important role in the formation of intracellular aggregates in patients with Alzheimer's disease. Several implicit solvent models form the basis of this work--two based on the generalized Born formalism, and one based on a Gaussian solvent-exclusion model. All three implicit solvent models generate minima that are in good agreement with minima obtained from simulations with explicit solvent. Moreover, free-energy profiles generated with each implicit solvent model agree with free-energy profiles obtained with explicit solvent. For the Gaussian solvent-exclusion model, we demonstrate that a straightforward ranking of the relative stability of each minimum suggests that the most stable structure is extended, a result in excellent agreement with the free-energy profiles. Overall, our data demonstrate that for some peptides like PHF6, implicit solvent can accurately reproduce the set of local energy minimum arising from quenched dynamics simulations with explicit solvent. More importantly, all solvent models predict that PHF6 forms extended beta-structures in solution, a finding consistent with the notion that PHF6 initiates neurofibrillary tangle formation in patients with Alzheimer's disease.

Implicit versus explicit solvent in free energy calculations of enzyme catalysis: Methyl transfer catalyzed by catechol O-methyltransferase

We compare free energy calculations for the methyl transfer reaction catalyzed by catechol O-methyltransferase using the quantum mechanical/molecular mechanical free energy method with implicit and explicit solvents. An analogous methylation reaction in a solution is also studied. For the explicit solvent model, we use the three-point transferable intermolecular potential model, and for the implicit model, we use the generalized Born molecular volume model as implemented in CHARMM. We find that activation and reaction free energies calculated with the two models are very similar, despite some structural differences that exist. A significant change in the polarization of the environment occurs as the reaction proceeds. This is more pronounced for the reaction in a solution than for the enzymatic reaction. For the enzymatic reaction, most of the changes take place in the protein rather than in the solvent, and, hence, the benefit of having an instantaneous relaxation of the solvent degrees of freedom is less pronounced for the enzymatic reaction than for the reaction in a solution. This is a likely reason why energies of the enzyme reaction are less sensitive to the choice of atomic radii than are energies of the reaction in a solution.

Long dynamics simulations of proteins using atomistic force fields and a continuum representation of solvent effects: calculation of structural and dynamic properties

Long dynamics simulations were carried out on the B1 immunoglobulin-binding domain of streptococcal protein G (ProtG) and bovine pancreatic trypsin inhibitor (BPTI) using atomistic descriptions of the proteins and a continuum representation of solvent effects. To mimic frictional and random collision effects, Langevin dynamics (LD) were used. The main goal of the calculations was to explore the stability of tens-of-nanosecond trajectories as generated by this molecular mechanics approximation and to analyze in detail structural and dynamical properties. Conformational fluctuations, order parameters, cross correlation matrices, residue solvent accessibilities, pKa values of titratable groups, and hydrogen-bonding (HB) patterns were calculated from all of the trajectories and compared with available experimental data. The simulations comprised over 40 ns per trajectory for ProtG and over 30 ns per trajectory for BPTI. For comparison, explicit water molecular dynamics simulations (EW/MD) of 3 ns and 4 ns, respectively, were also carried out. Two continuum simulations were performed on each protein using the CHARMM program, one with the all-atom PAR22 representation of the protein force field (here referred to as PAR22/LD simulations) and the other with the modifications introduced by the recently developed CMAP potential (CMAP/LD simulations). The explicit solvent simulations were performed with PAR22 only. Solvent effects are described by a continuum model based on screened Coulomb potentials (SCP) reported earlier, i.e., the SCP-based implicit solvent model (SCP-ISM). For ProtG, both the PAR22/LD and the CMAP/LD 40-ns trajectories were stable, yielding C(alpha) root mean square deviations (RMSD) of about 1.0 and 0.8 A respectively along the entire simulation time, compared to 0.8 A for the EW/MD simulation. For BPTI, only the CMAP/LD trajectory was stable for the entire 30-ns simulation, with a C(alpha) RMSD of approximately 1.4 A, while the PAR22/LD trajectory became unstable early in the simulation, reaching a C(alpha) RMSD of about 2.7 A and remaining at this value until the end of the simulation; the C(alpha) RMSD of the EW/MD simulation was about 1.5 A. The source of the instabilities of the BPTI trajectories in the PAR22/LD simulations was explored by an analysis of the backbone torsion angles. To further validate the findings from this analysis of BPTI, a 35-ns SCP-ISM simulation of Ubiquitin (Ubq) was carried out. For this protein, the CMAP/LD simulation was stable for the entire simulation time (C(alpha) RMSD of approximately 1.0 A), while the PAR22/LD trajectory showed a trend similar to that in BPTI, reaching a C(alpha) RMSD of approximately 1.5 A at 7 ns. All the calculated properties were found to be in agreement with the corresponding experimental values, although local deviations were also observed. HB patterns were also well reproduced by all the continuum solvent simulations with the exception of solvent-exposed side chain-side chain (sc-sc) HB in ProtG, where several of the HB interactions observed in the crystal structure and in the EW/MD simulation were lost. The overall analysis reported in this work suggests that the combination of an atomistic representation of a protein with a CMAP/CHARMM force field and a continuum representation of solvent effects such as the SCP-ISM provides a good description of structural and dynamic properties obtained from long computer simulations. Although the SCP-ISM simulations (CMAP/LD) reported here were shown to be stable and the properties well reproduced, further refinement is needed to attain a level of accuracy suitable for more challenging biological applications, particularly the study of protein-protein interactions.

Enhancing the accuracy, the efficiency and the scope of free energy simulations

Many different methods exist for computing free energy changes from molecular simulations. Recent advances have led to improvements in the theoretical framework underlying these calculations, as well as in the accuracy and sampling efficiency of the algorithms. Novel methods combining the advantages afforded by various existing approaches offer promising strategies and open up new perspectives to help elucidate the physical basis of important biological processes.