Comment on “Can a Continuum Solvent Model Reproduce the Free Energy Landscape of a β-Hairpin Folding in Water?” The Poisson−Boltzmann Equation

Short solvent model for ion correlations and hydrophobic association

Coulomb interactions play a major role in determining the thermodynamics, structure, and dynamics of condensed-phase systems, but often present significant challenges. Computer simulations usually use periodic boundary conditions to minimize corrections from finite cell boundaries but the long range of the Coulomb interactions generates significant contributions from distant periodic images of the simulation cell, usually calculated by Ewald sum techniques. This can add significant overhead to computer simulations and hampers the development of intuitive local pictures and simple analytic theory. In this paper, we present a general framework based on local molecular field theory to accurately determine the contributions from long-ranged Coulomb interactions to the potential of mean force between ionic or apolar hydrophobic solutes in dilute aqueous solutions described by standard classical point charge water models. The simplest approximation leads to a short solvent (SS) model, with truncated solvent-solvent and solute-solvent Coulomb interactions and long-ranged but screened Coulomb interactions only between charged solutes. The SS model accurately describes the interplay between strong short-ranged solute core interactions, local hydrogen-bond configurations, and long-ranged dielectric screening of distant charges, competing effects that are difficult to capture in standard implicit solvent models.

What makes a good graphene-binding peptide? Adsorption of amino acids and peptides at aqueous graphene interfaces

Molecular dynamics simulations of the aqueous biomolecule–graphene interface have predicted the free energy of adsorption of amino acids and the structure of peptides.

A strategy for reducing gross errors in the generalized Born models of implicit solvation

The "canonical" generalized Born (GB) formula [C. Still, A. Tempczyk, R. C. Hawley, and T. Hendrickson, J. Am. Chem. Soc. 112, 6127 (1990)] is known to provide accurate estimates for total electrostatic solvation energies ΔG(el) of biomolecules if the corresponding effective Born radii are accurate. Here we show that even if the effective Born radii are perfectly accurate, the canonical formula still exhibits significant number of gross errors (errors larger than 2k(B)T relative to numerical Poisson equation reference) in pairwise interactions between individual atomic charges. Analysis of exact analytical solutions of the Poisson equation (PE) for several idealized nonspherical geometries reveals two distinct spatial modes of the PE solution; these modes are also found in realistic biomolecular shapes. The canonical GB Green function misses one of two modes seen in the exact PE solution, which explains the observed gross errors. To address the problem and reduce gross errors of the GB formalism, we have used exact PE solutions for idealized nonspherical geometries to suggest an alternative analytical Green function to replace the canonical GB formula. The proposed functional form is mathematically nearly as simple as the original, but depends not only on the effective Born radii but also on their gradients, which allows for better representation of details of nonspherical molecular shapes. In particular, the proposed functional form captures both modes of the PE solution seen in nonspherical geometries. Tests on realistic biomolecular structures ranging from small peptides to medium size proteins show that the proposed functional form reduces gross pairwise errors in all cases, with the amount of reduction varying from more than an order of magnitude for small structures to a factor of 2 for the largest ones.

Strategies to model the near-solute solvent molecular density/polarization

The solvent molecular distribution significantly affects the behavior of the solute molecules and is thus important in studying many biological phenomena. It can be described by the solvent molecular density distribution, g, and the solvent electric dipole distribution, p. The g and p can be computed directly by counting the number of solvent molecules/dipoles in a microscopic volume centered at r during a simulation or indirectly from the mean force F and electrostatic field E acting on the solvent molecule at r, respectively. However, it is not clear how the g and p derived from simulations depend on the solvent molecular center or the solute charge and if the g(F) and p(E) computed from the mean force and electric field acting on the solvent molecule, respectively, could reproduce the corresponding g and p obtained by direct counting. Hence, we have computed g, p, g(F), and p(E) using different water centers from simulations of a solute atom of varying charge solvated in TIP3P water. The results show that g(F) and p(E) can reproduce the g and p obtained using a given count center. This implies that rather than solving the coordinates of each water molecule by MD simulations, the distribution of water molecules could be indirectly obtained from analytical formulas for the mean force F and electrostatic field E acting on the solvent molecule at r. Furthermore, the dependence of the g and p distributions on the solute charge revealed provides an estimate of the change in g and p surrounding a biomolecule upon a change in its conformation.

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.

Is Poisson-Boltzmann theory insufficient for protein folding simulations?

The Poisson-Boltzmann theory has been widely used in the studies of energetics and conformations of biological macromolecules. Recently, introduction of the efficient generalized Born approximation has greatly extended its applicability to areas such as protein folding simulations where highly efficient computation is crucial. However, limitations have been found in the folding simulations of a well-studied beta hairpin with several generalized Born implementations and different force fields. These studies have raised the question whether the underlining Poisson-Boltzmann theory, on which the generalized Born model is calibrated, is adequate in the treatment of polar interactions for the challenging protein folding simulations. To address the question whether the Poisson-Boltzmann theory in the current formalism might be insufficient, we directly tested our efficient numerical Poisson-Boltzmann implementation in the beta-hairpin folding simulation. Good agreement between simulation and experiment was found for the beta-hairpin equilibrium structures when the numerical Poisson-Boltzmann solvent and a recently improved generalized Born solvent were used. In addition simulated thermodynamic properties also agree well with experiment in both solvents. Finally, an overall agreement on the beta-hairpin folding mechanism was found between the current and previous studies. Thus, our simulations indicate that previously observed limitations are most likely due to imperfect calibration in previous generalized Born models but not due to the limitation of the Poisson-Boltzmann theory.

Elastic bag model for molecular dynamics simulations of solvated systems: application to liquid water and solvated peptides

The fluctuating elastic boundary (FEB) model for molecular dynamics has recently been developed and validated through simulations of liquid argon. In the FEB model, a flexible boundary which consists of particles connected by springs is used to confine the solvated system, thereby eliminating the need for periodic boundary conditions. In this study, we extend this model to the simulation of bulk water and solvated alanine dipeptide. Both the confining potential and boundary particle interaction functions are modified to preserve the structural integrity of the boundary and prevent the leakage of the solute-solvent system through the boundary. A broad spectrum of structural and dynamic properties of liquid water are computed and compared with those obtained from conventional periodic boundary condition simulations. The applicability of the model to biomolecular simulations is investigated through the analysis of conformational population distribution of solvated alanine dipeptide. In most cases we find remarkable agreement between the two simulation approaches.

Direct folding simulation of alpha-helices and beta-hairpins based on a single all-atom force field with an implicit solvation model

Recently, we have shown that a modified energy model based on the param99 force field with the generalized Born (GB) solvation model produces reliable free energy landscapes of mini-proteins with a betabetaalpha motif (BBA5, 1FSD, and 1PSV), with the native structures of the mini-proteins located in their lowest free energy minimum states. One of the main features in the modified energy model is a significant improvement for more balanced treatments of alpha and beta strands in proteins. In this study, using the replica exchange molecular dynamics (REMD) simulation method with this new force field, we have carried out extensive ab initio folding studies of several well-known peptides with alpha or beta strands (C-peptide, EK-peptide, le0q, and gbl). Starting from fully extended conformations as the initial conditions, all of the native-like structures of the target peptides were successfully identified by REMD, with reasonable representations of free energy surfaces. The present simulation results with the modified energy model are consistent with experiments, demonstrating an extended applicability of the energy model to folding studies of a variety of alpha-helices, beta-strands, and alpha/beta proteins.

The transition between the B and Z conformations of DNA investigated by targeted molecular dynamics simulations with explicit solvation

The transition between the B and Z conformations of double-helical deoxyribonucleic acid (DNA) belongs to the most complex and elusive conformational changes occurring in biomolecules. Since the accidental discovery of the left-handed Z-DNA form in the late 1970s, research on this DNA morphology has been engaged in resolving questions relative to its stability, occurrence, and function in biological processes. While the occurrence of Z-DNA in vivo is now widely recognized and the major factors influencing its thermodynamical stability are largely understood, the intricate conformational changes that take place during the B-to-Z transition are still unknown at the atomic level. In this article, we report simulations of this transition for the 3'-(CGCGCG)-5' hexamer duplex using targeted molecular dynamics with the GROMOS96 force field in explicit water under different ionic-strength conditions. The results suggest that for this oligomer length and sequence, the transition mechanism involves: 1), a stretched intermediate conformation, which provides a simple solution to the important sterical constraints involved in this transition; 2), the transient disruption of Watson-Crick hydrogen-bond pairing, partly compensated energetically by an increase in the number of solute-solvent hydrogen bonds; and 3), an asynchronous flipping of the bases compatible with a zipperlike progression mechanism.

Reconciling the "old" and "new" views of protein allostery: a molecular simulation study of chemotaxis Y protein (CheY)

A combination of thirty-two 10-ns-scale molecular dynamics simulations were used to explore the coupling between conformational transition and phosphorylation in the bacteria chemotaxis Y protein (CheY), as a simple but representative example of protein allostery. Results from these simulations support an activation mechanism in which the beta4-alpha4 loop, at least partially, gates the isomerization of Tyr106. The roles of phosphorylation and the conserved Thr87 are deemed indirect in that they stabilize the active configuration of the beta4-alpha4 loop. The indirect role of the activation event (phosphorylation) and/or conserved residues in stabilizing, rather than causing, specific conformational transition is likely a feature in many signaling systems. The current analysis of CheY also helps to make clear that neither the "old" (induced fit) nor the "new" (population shift) views for protein allostery are complete, because they emphasize the kinetic (mechanistic) and thermodynamic aspects of allosteric transitions, respectively. In this regard, an issue that warrants further analysis concerns the interplay of concerted collective motion and sequential local structural changes in modulating cooperativity between distant sites in biomolecules.

A line integral reaction path approximation for large systems via nonlinear constrained optimization: Application to alanine dipeptide and the β hairpin of protein G

A variation of the line integral method of Elber with self-avoiding walk has been implemented using a state of the art nonlinear constrained optimization procedure. The new implementation appears to be robust in finding approximate reaction paths for small and large systems. Exact transition states and intermediates for the resulting paths can easily be pinpointed with subsequent application of the conjugate peak refinement method [S. Fischer and M. Karplus, Chem. Phys. Lett. 194, 252 (1992)] and unconstrained minimization, respectively. Unlike previous implementations utilizing a penalty function approach, the present implementation generates an exact solution of the underlying problem. Most importantly, this formulation does not require an initial guess for the path, which makes it particularly useful for studying complex molecular rearrangements. The method has been applied to conformational rearrangements of the alanine dipeptide in the gas phase and in water, and folding of the beta hairpin of protein G in water. In the latter case a procedure was developed to systematically sample the potential energy surface underlying folding and reconstruct folding pathways within the nearest-neighbor hopping approximation.

Development of a lattice-sum method emulating nonperiodic boundary conditions for the treatment of electrostatic interactions in molecular simulations: A continuum-electrostatics study

Artifacts induced by the application of periodic boundary conditions and lattice-sum methods in explicit-solvent simulations of (bio-)molecular systems are nowadays a major concern in the computer-simulation community. The present article reports a first step toward the design of a modified lattice-sum algorithm emulating nonperiodic boundary conditions, and therefore exempt of such periodicity-induced artifacts. This result is achieved here in the (more simple) context of continuum electrostatics. It is shown that an appropriate modification of the periodic Poisson equation and of its boundary conditions leads to a continuum-electrostatics scheme, which, although applied under periodic boundary conditions, exactly mimics the nonperiodic situation. The possible extension of this scheme to explicit-solvent simulations is outlined and its practical implementation will be described in more details in a forthcoming article.

Free energy surfaces of miniproteins with a ββα motif: Replica exchange molecular dynamics simulation with an implicit solvation model

Designed miniproteins with a betabetaalpha motif, such as BBA5, 1FSD, and 1PSV can serve as a benchmark set to test the validity of all-atom force fields with computer simulation, because they contain all the basic structural elements in protein folding. Unfortunately, it was found that the standard all-atom force fields with the generalized Born (GB) implicit solvation model tend to produce distorted free energy surfaces for the betabetaalpha proteins, not only because energetically those proteins need to be described by more balanced weights of the alpha- and beta-strands, but also because the GB implicit solvation model suffers from overestimated salt bridge effects. In an attempt to resolve these problems, we have modified one of the standard all-atom force fields in conjunction with the GB model, such that each native state of the betabetaalpha proteins is in its free energy minimum state with reasonable energy barriers separating local minima. With this modified energy model, the free energy contour map in each protein was constructed from the replica exchange molecular dynamics REMD simulation. The resulting free energy surfaces are significantly improved in comparison with previous simulation results and consistent with general views on small protein folding behaviors with realistic topology and energetics of all three proteins.

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.

Elastic Bag Model for Molecular Dynamics Simulations of Solvated Systems: Application to Liquid Argon

A new approach is developed to study the dynamics of the localized process in solutions and other condensed phase systems. The approach employs a fluctuating elastic boundary (FEB) model which encloses the simulated system in an elastic bag that mimics the effects of the bulk solvent. This alleviates the need for periodic boundary conditions and allows for a reduction in the number of solvent molecules that need to be included in the simulation. The boundary bag is modeled as a mesh of quasi-particles connected by elastic bonds. The FEB model allows for volume and density fluctuations characteristic of the bulk system, and the shape of the boundary fluctuates during the course of the simulation to adapt to the configuration fluctuations of the explicit solute-solvent system inside. The method is applied to the simulation of a Lennard-Jones model of liquid argon. Various structural and dynamical quantities are computed and compared with those obtained from conventional periodic boundary simulations. The agreement between the two is excellent in most cases, thus validating the viability of the FEB method.