Biological Applications of Electrostatic Calculations and Brownian Dynamics Simulations

Inclusion of Water Multipoles into the Implicit Solvation Framework Leads to Accuracy Gains

The current practical "workhorses" of the atomistic implicit solvation─the Poisson-Boltzmann (PB) and generalized Born (GB) models─face fundamental accuracy limitations. Here, we propose a computationally efficient implicit solvation framework, the Implicit Water Multipole GB (IWM-GB) model, that systematically incorporates the effects of multipole moments of water molecules in the first hydration shell of a solute, beyond the dipole water polarization already present at the PB/GB level. The framework explicitly accounts for coupling between polar and nonpolar contributions to the total solvation energy, which is missing from many implicit solvation models. An implementation of the framework, utilizing the GAFF force field and AM1-BCC atomic partial charges model, is parametrized and tested against the experimental hydration free energies of small molecules from the FreeSolv database. The resulting accuracy on the test set (RMSE ∼ 0.9 kcal/mol) is 12% better than that of the explicit solvation (TIP3P) treatment, which is orders of magnitude slower. We also find that the coupling between polar and nonpolar parts of the solvation free energy is essential to ensuring that several features of the IWM-GB model are physically meaningful, including the sign of the nonpolar contributions.

Enhancing bezlotoxumab binding to C. difficile toxin B2: insights from computational simulations and mutational analyses for antibody design

Clostridioides difficile infection (CDI) is a significant concern caused by widespread antibiotic use, resulting in diarrhea and inflammation from the gram-positive anaerobic bacterium C. difficile. Although bezlotoxumab (Bez), a monoclonal antibody (mAb), was developed to address CDI recurrences, the recurrence rate remains high, partly due to reduced neutralization efficiency against toxin B2. In this study, we aimed to enhance the binding of Bez to C. difficile toxin B2 by combining computational simulations and mutational analyses. We identified specific mutations in Bez, including S28R, S31W/K, Y32R, S56W and G103D/S in the heavy chain (Hc), and S32F/H/R/W/Y in the light chain (Lc), which significantly improved binding to toxin B2 and formed critical protein-protein interactions. Through molecular dynamics simulations, several single mutations, such as HcS28R, LcS32H, LcS32R, LcS32W and LcS32Y, exhibited superior binding affinities to toxin B2 compared to Bez wild-type (WT), primarily attributed to Coulombic interactions. Combining the HcS28R mutation with four different mutations at residue LcS32 led to even greater binding affinities in double mutants (MTs), particularly HcS28R/LcS32H, HcS28R/LcS32R and HcS28R/LcS32Y, reinforcing protein-protein binding. Analysis of per-residue decomposition free energy highlighted key residues contributing significantly to enhanced binding interactions, emphasizing the role of electrostatic interactions. These findings offer insights into rational Bez MT design for improved toxin B2 binding, providing a foundation for developing more effective antibodies to neutralize toxin B2 and combat-related infections.Communicated by Ramaswamy H. Sarma.

Elucidation of bezlotoxumab binding specificity to toxin B in Clostridioides difficile

C. difficile or Clostridioides difficile infection (CDI) is currently one of the major causes of epidemics worldwide. Toxin B from Clostridioides difficile toxin B (TcdB) infection is the main target protein inhibiting CDI recurrence. Clinical research suggested that bezlotoxumab's (Bez) efficiency is significantly reduced in neutralizing the B2 strain compared to the B1 strain. The monoclonal antibody (mAb) functions by binding to the epitope 1 and 2 regions in the combined repetitive oligopeptide (CROP) domain. Some binding residues are distinctively different between B1 and B2 strains. In this work, we aimed to elucidate and compare insights into the interaction of toxins B1 and B2 in complex with Bez by using all-atom molecular dynamics (MD) simulations and binding free energy calculations. The predicted ΔGbinding values suggested that the antibody (Ab) could bind to toxin B1 significantly better than B2, supported by higher salt bridge and hydrogen bonding (H-bonding) interactions, as well as the number of contact residues between the two focused proteins. The toxin B1 residues important for binding with Bez were E1878, T1901, E1902, F1905, N1941, V1946, N2031, T2032, E2033, V2076, V2077, and E2092. The lower susceptibility of Bez towards toxin B2 was primarily due to a change of residue E2033 from glutamate to alanine (A2033) and the loss of E1878 and E1902 contributions, as determined by the intermolecular interaction changes from the dynamic residue interaction network (dRIN) analysis. The obtained data strengthen our understanding of Bez/toxin B binding.

Improving the Accuracy of Physics-Based Hydration-Free Energy Predictions by Machine Learning the Remaining Error Relative to the Experiment

The accuracy of computational models of water is key to atomistic simulations of biomolecules. We propose a computationally efficient way to improve the accuracy of the prediction of hydration-free energies (HFEs) of small molecules: the remaining errors of the physics-based models relative to the experiment are predicted and mitigated by machine learning (ML) as a postprocessing step. Specifically, the trained graph convolutional neural network attempts to identify the "blind spots" in the physics-based model predictions, where the complex physics of aqueous solvation is poorly accounted for, and partially corrects for them. The strategy is explored for five classical solvent models representing various accuracy/speed trade-offs, from the fast analytical generalized Born (GB) to the popular TIP3P explicit solvent model; experimental HFEs of small neutral molecules from the FreeSolv set are used for the training and testing. For all of the models, the ML correction reduces the resulting root-mean-square error relative to the experiment for HFEs of small molecules, without significant overfitting and with negligible computational overhead. For example, on the test set, the relative accuracy improvement is 47% for the fast analytical GB, making it, after the ML correction, almost as accurate as uncorrected TIP3P. For the TIP3P model, the accuracy improvement is about 39%, bringing the ML-corrected model's accuracy below the 1 kcal/mol threshold. In general, the relative benefit of the ML corrections is smaller for more accurate physics-based models, reaching the lower limit of about 20% relative accuracy gain compared with that of the physics-based treatment alone. The proposed strategy of using ML to learn the remaining error of physics-based models offers a distinct advantage over training ML alone directly on reference HFEs: it preserves the correct overall trend, even well outside of the training set.

Explicit ions/implicit water generalized Born model for nucleic acids

The ion atmosphere around highly charged nucleic acid molecules plays a significant role in their dynamics, structure, and interactions. Here we utilized the implicit solvent framework to develop a model for the explicit treatment of ions interacting with nucleic acid molecules. The proposed explicit ions/implicit water model is based on a significantly modified generalized Born (GB) model and utilizes a non-standard approach to define the solute/solvent dielectric boundary. Specifically, the model includes modifications to the GB interaction terms for the case of multiple interacting solutes-disconnected dielectric boundary around the solute-ion or ion-ion pairs. A fully analytical description of all energy components for charge-charge interactions is provided. The effectiveness of the approach is demonstrated by calculating the potential of mean force for Na⁺-Cl^- ion pair and by carrying out a set of Monte Carlo (MC) simulations of mono- and trivalent ions interacting with DNA and RNA duplexes. The monovalent (Na⁺) and trivalent (CoHex³⁺) counterion distributions predicted by the model are in close quantitative agreement with all-atom explicit water molecular dynamics simulations used as reference. Expressed in the units of energy, the maximum deviations of local ion concentrations from the reference are within k _B T. The proposed explicit ions/implicit water GB model is able to resolve subtle features and differences of CoHex distributions around DNA and RNA duplexes. These features include preferential CoHex binding inside the major groove of the RNA duplex, in contrast to CoHex biding at the "external" surface of the sugar-phosphate backbone of the DNA duplex; these differences in the counterion binding patters were earlier shown to be responsible for the observed drastic differences in condensation propensities between short DNA and RNA duplexes. MC simulations of CoHex ions interacting with the homopolymeric poly(dA·dT) DNA duplex with modified (de-methylated) and native thymine bases are used to explore the physics behind CoHex-thymine interactions. The simulations suggest that the ion desolvation penalty due to proximity to the low dielectric volume of the methyl group can contribute significantly to CoHex-thymine interactions. Compared to the steric repulsion between the ion and the methyl group, the desolvation penalty interaction has a longer range and may be important to consider in the context of methylation effects on DNA condensation.

Accuracy Comparison of Generalized Born Models in the Calculation of Electrostatic Binding Free Energies

The need for accurate yet efficient representation of the aqueous environment in biomolecular modeling has led to the development of a variety of generalized Born (GB) implicit solvent models. While many studies have focused on the accuracy of available GB models in predicting solvation free energies, a systematic assessment of the quality of these models in binding free energy calculations, crucial for rational drug design, has not been undertaken. Here, we evaluate the accuracies of eight common GB flavors (GB-HCT, GB-OBC, GB-neck2, GBNSR6, GBSW, GBMV1, GBMV2, and GBMV3), available in major molecular dynamics packages, in predicting the electrostatic binding free energies ( ΔΔ G_el) for a diverse set of 60 biomolecular complexes belonging to four main classes: protein-protein, protein-drug, RNA-peptide, and small complexes. The GB flavors are examined in terms of their ability to reproduce the results from the Poisson-Boltzmann (PB) model, commonly used as accuracy reference in this context. We show that the agreement with the PB of ΔΔ G_el estimates varies widely between different GB models and also across different types of biomolecular complexes, with R² correlations ranging from 0.3772 to 0.9986. A surface-based "R6" GB model recently implemented in AMBER shows the closest overall agreement with reference PB ( R² = 0.9949, RMSD = 8.75 kcal/mol). The RNA-peptide and protein-drug complex sets appear to be most challenging for all but one model, as indicated by the large deviations from the PB in ΔΔ G_el. Small neutral complexes present the least challenge for most of the GB models tested. The quantitative demonstration of the strengths and weaknesses of the GB models across the diverse complex types provided here can be used as a guide for practical computations and future development efforts.

Development of constant-pH simulation methods in implicit solvent and applications in biomolecular systems

pH is a critical parameter for biological and technological systems directly related with electrical charges. It can give rise to peculiar electrostatic phenomena, which also makes them more challenging. Due to the quantum nature of the process, involving the forming and breaking of chemical bonds, quantum methods should ideally by employed. Nevertheless, due to the very large number of ionizable sites, different macromolecular conformations, salt conditions, and all other charged species, the CPU time cost simply becomes prohibitive for computer simulations, making this a quite complex problem. Simplified methods based on Monte Carlo sampling have been devised and will be reviewed here, highlighting the updated state-of-the-art of this field, advantages, and limitations of different theoretical protocols for biomolecular systems (proteins and nucleic acids). Following a historical perspective, the discussion will be associated with the applications to protein interactions with other proteins, polyelectrolytes, and nanoparticles.

Implicit Solvent Model for Million-Atom Atomistic Simulations: Insights into the Organization of 30-nm Chromatin Fiber

Molecular dynamics (MD) simulations based on the implicit solvent generalized Born (GB) models can provide significant computational advantages over the traditional explicit solvent simulations. However, the standard GB becomes prohibitively expensive for all-atom simulations of large structures; the model scales poorly, ∼n2, with the number of solute atoms. Here we combine our recently developed optimal point charge approximation (OPCA) with the hierarchical charge partitioning (HCP) approximation to present an ∼n log n multiscale, yet fully atomistic, GB model (GB-HCPO). The HCP approximation exploits the natural organization of biomolecules (atoms, groups, chains, and complexes) to partition the structure into multiple hierarchical levels of components. OPCA approximates the charge distribution for each of these components by a small number of point charges so that the low order multipole moments of these components are optimally reproduced. The approximate charges are then used for computing electrostatic interactions with distant components, while the full set of atomic charges are used for nearby components. We show that GB-HCPO can deliver up to 2 orders of magnitude speedup compared to the standard GB, with minimal impact on its accuracy. For large structures, GB-HCPO can approach the same nominal speed, as in nanoseconds per day, as the highly optimized explicit-solvent simulation based on particle mesh Ewald (PME). The increase in the nominal simulation speed, relative to the standard GB, coupled with substantially faster sampling of conformational space, relative to the explicit solvent, makes GB-HCPO a suitable candidate for MD simulation of large atomistic systems in implicit solvent. As a practical demonstration, we use GB-HCPO simulation to refine a ∼1.16 million atom structure of 30 nm chromatin fiber (40 nucleosomes). The refined structure suggests important details about spatial organization of the linker DNA and the histone tails in the fiber: (1) the linker DNA fills the core region, allowing the H3 histone tails to interact with the linker DNA, which is consistent with experiment; (2) H3 and H4 tails are found mostly in the core of the structure, closer to the helical axis of the fiber, while H2A and H2B are mostly solvent exposed. Potential functional consequences of these findings are discussed. GB-HCPO is implemented in the open source MD software NAB in Amber 2016.

Protein-Ligand Electrostatic Binding Free Energies from Explicit and Implicit Solvation

Accurate yet efficient computational models of solvent environment are central for most calculations that rely on atomistic modeling, such as prediction of protein-ligand binding affinities. In this study, we evaluate the accuracy of a recently developed generalized Born implicit solvent model, GBNSR6 (Aguilar et al. J. Chem. Theory Comput. 2010, 6, 3613-3639), in estimating the electrostatic solvation free energies (ΔG(pol)) and binding free energies (ΔΔG(pol)) for small protein-ligand complexes. We also compare estimates based on three different explicit solvent models (TIP3P, TIP4PEw, and OPC). The two main findings are as follows. First, the deviation (RMSD = 7.04 kcal/mol) of GBNSR6 binding affinities from commonly used TIP3P reference values is comparable to the deviations between explicit models themselves, e.g. TIP4PEw vs TIP3P (RMSD = 5.30 kcal/mol). A simple uniform adjustment of the atomic radii by a single scaling factor reduces the RMS deviation of GBNSR6 from TIP3P to within the above "error margin" - differences between ΔΔG(pol) estimated by different common explicit solvent models. The simple radii scaling virtually eliminates the systematic deviation (ΔΔG(pol)) between GBNSR6 and two out of the three explicit water models and significantly reduces the deviation from the third explicit model. Second, the differences between electrostatic binding energy estimates from different explicit models is disturbingly large; for example, the deviation between TIP4PEw and TIP3P estimates of ΔΔG(pol) values can be up to ∼50% or ∼9 kcal/mol, which is significantly larger than the "chemical accuracy" goal of ∼1 kcal/mol. The absolute ΔG(pol) calculated with different explicit models could differ by tens of kcal/mol. These discrepancies point to unacceptably high sensitivity of binding affinity estimates to the choice of common explicit water models. The absence of a clear "gold standard" among these models strengthens the case for the use of accurate implicit solvation models for binding energetics, which may be orders of magnitude faster.

Accuracy of continuum electrostatic calculations based on three common dielectric boundary definitions

We investigate the influence of three common definitions of the solute/solvent dielectric boundary (DB) on the accuracy of the electrostatic solvation energy ΔG_el computed within the Poisson Boltzmann and the generalized Born models of implicit solvation. The test structures include small molecules, peptides and small proteins; explicit solvent ΔG_el are used as accuracy reference. For common atomic radii sets BONDI, PARSE (and ZAP9 for small molecules) the use of van der Waals (vdW) DB results, on average, in considerably larger errors in ΔG_el than the molecular surface (MS) DB. The optimal probe radius ρ_w for which the MS DB yields the most accurate ΔG_el varies considerably between structure types. The solvent accessible surface (SAS) DB becomes optimal at ρ_w ~ 0.2 Å (exact value is sensitive to the structure and atomic radii), at which point the average accuracy of ΔG_el is comparable to that of the MS-based boundary. The geometric equivalence of SAS to vdW surface based on the same atomic radii uniformly increased by ρ_w gives the corresponding optimal vdW DB. For small molecules, the optimal vdW DB based on BONDI + 0.2 Å radii can yield ΔG_el estimates at least as accurate as those based on the optimal MS DB. Also, in small molecules, pairwise charge-charge interactions computed with the optimal vdW DB are virtually equal to those computed with the MS DB, suggesting that in this case the two boundaries are practically equivalent by the electrostatic energy criteria. In structures other than small molecules, the optimal vdW and MS dielectric boundaries are not equivalent: the respective pairwise electrostatic interactions in the presence of solvent can differ by up to 5 kcal/mol for individual atomic pairs in small proteins, even when the total ΔG_el are equal. For small proteins, the average decrease in pairwise electrostatic interactions resulting from the switch from optimal MS to optimal vdW DB definition can be mimicked within the MS DB definition by doubling of the solute dielectric constant. However, the use of the higher interior dielectric does not eliminate the large individual deviations between pairwise interactions computed within the two DB definitions. It is argued that while the MS based definition of the dielectric boundary is more physically correct in some types of practical calculations, the choice is not so clear in some other common scenarios.

VERIFICATION OF THE GENERALIZED BORN MODEL AT SHORT DISTANCES

The generalized Born (GB) model, one of the implicit solvent models, is widely applied in molecular dynamics (MD) simulations as a simple description of the solvation effect. In the GB model, an empirical function called the Still's formula, with the algorithmic simplicity, is utilized to calculate the solvation energy due to the polarization, termed as ΔG pol . Applications of the GB model have exhibited reasonable accuracy and high computational efficiency. However, there is still room for improvements. Most of the attempts to improve the GB model focus on optimizing effective Born radii. Contrarily, limited researches have been performed to improve the feasibility of the Still's formula. In this paper, analytical methods was applied to investigate the validity of the Still's formula at short distance. Taking advantage of the toroidal coordinates and Mehler–Fock transform, the analytical solutions of the GB model at short distances was derived explicitly for the first time. Additionally, the solvation energy was numerically computed using proper algorithms based on the analytical solutions and compared with ΔG pol calculated in the GB model. With the analysis on the deficiencies of the Still's formula at short distances, potential methods to improve the validity of the GB model were discussed.

Accelerating electrostatic surface potential calculation with multi-scale approximation on graphics processing units

Tools that compute and visualize biomolecular electrostatic surface potential have been used extensively for studying biomolecular function. However, determining the surface potential for large biomolecules on a typical desktop computer can take days or longer using currently available tools and methods. Two commonly used techniques to speed-up these types of electrostatic computations are approximations based on multi-scale coarse-graining and parallelization across multiple processors. This paper demonstrates that for the computation of electrostatic surface potential, these two techniques can be combined to deliver significantly greater speed-up than either one separately, something that is in general not always possible. Specifically, the electrostatic potential computation, using an analytical linearized Poisson-Boltzmann (ALPB) method, is approximated using the hierarchical charge partitioning (HCP) multi-scale method, and parallelized on an ATI Radeon 4870 graphical processing unit (GPU). The implementation delivers a combined 934-fold speed-up for a 476,040 atom viral capsid, compared to an equivalent non-parallel implementation on an Intel E6550 CPU without the approximation. This speed-up is significantly greater than the 42-fold speed-up for the HCP approximation alone or the 182-fold speed-up for the GPU alone.

An analytical approach to computing biomolecular electrostatic potential. I. Derivation and analysis

Analytical approximations to fundamental equations of continuum electrostatics on simple shapes can lead to computationally inexpensive prescriptions for calculating electrostatic properties of realistic molecules. Here, we derive a closed-form analytical approximation to the Poisson equation for an arbitrary distribution of point charges and a spherical dielectric boundary. The simple, parameter-free formula defines continuous electrostatic potential everywhere in space and is obtained from the exact infinite-series (Kirkwood) solution by an approximate summation method that avoids truncating the infinite series. We show that keeping all the terms proves critical for the accuracy of this approximation, which is fully controllable for the sphere. The accuracy is assessed by comparisons with the exact solution for two unit charges placed inside a spherical boundary separating the solute of dielectric 1 and the solvent of dielectric 80. The largest errors occur when the source charges are closest to the dielectric boundary and the test charge is closest to either of the sources. For the source charges placed within 2 A from the boundary, and the test surface located on the boundary, the root-mean-square error of the approximate potential is less than 0.1 kcal/mol/mid R:emid R: (per unit test charge). The maximum error is 0.4 kcal/mol/mid R:emid R:. These results correspond to the simplest first-order formula. A strategy for adopting the proposed method for realistic biomolecular shapes is detailed. An extensive testing and performance analysis on real molecular structures are described in Part II that immediately follows this work as a separate publication. Part II also contains an application example.

Flexible computational docking studies of new aminoglycosides targeting RNA 16S bacterial ribosome site

Ribonucleic acids (RNAs) have only recently been viewed as a target for small-molecules drug discovery. Aminoglycoside compounds are antibiotics which bind the ribosomal A site (16S fragment) and cause misreading of the bacterial genetic code and inhibit translocation. In this work, a complete molecular modeling study is done for 16 newly derived aminoglycoside compounds where diverse nucleoside fragments are linked. Docking calculations are applied to 16S RNA target and a weak linear correlation, between experimental and calculated data, is obtained. However, one particularity of RNA is its high flexibility. To mimic this behavior, all docking calculations are followed by small molecular dynamic simulations. This last computational step improves significantly the correlation with experimental data and allowed us to establish structure-activity relationships. The overall results showed that the consideration of the RNA dynamic behavior is of great interest.

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.

The electrostatic nature of C3d-complement receptor 2 association

The association of complement component C3d with B or T cell complement receptor 2 (CR2 or CD21) is a link between innate and adaptive immunity. It has been recognized in experimental studies that the C3d-CR2 association is pH- and ionic strength-dependent. This led us to perform electrostatic calculations to obtain a theoretical understanding of the mechanism of C3d-CR2 association. We used the crystallographic structures of human free C3d, free CR2 (short consensus repeat (SCR)1-2), and the C3d-CR2(SCR1-2) complex, and continuum solvent representation, to obtain a detailed atomic-level picture of the components of the two molecules that contribute to association. Based on the calculation of electrostatic potentials for the free and bound species and apparent pK(a) values for each ionizable residue, we show that C3d-CR2(SCR1-2) recognition is electrostatic in nature and involves not only the association interface, but also the whole molecules. Our results are in qualitative agreement with experimental data that measured the ionic strength and pH dependence of C3d-CR2 association. Also, our results for the native molecules and a number of theoretical mutants of C3d explain experimental mutagenesis studies of amino acid replacements away from the association interface that modulate binding of iC3b with full-length CR2. Finally, we discuss the packing of the two SCR domains. Overall, our data provide global and site-specific explanations of the physical causes that underlie the ionic strength dependence of C3d-CR2 association in a unified model that accounts for all experimental data, some of which were previously thought to be contradictory.

A Brownian dynamics study: the effect of a membrane environment on an electron transfer system

During the past few years, three-dimensional crystal structures of many of the important integral membrane proteins responsible for the bioenergetic processes of photosynthesis and respiration have been determined. Moreover, a few crystal structures of protein-protein complexes have become available that characterize the interaction between those membrane proteins and the electron carrier protein cytochrome c. Here, we address the association kinetics for binding of cytochrome c to cytochrome c oxidase (COX) from Paracoccus denitrificans by Brownian dynamics simulations. The effects of ionic strength and protein mutations were studied for two different cytochrome c species: the positively charged, dipolar horse heart cytochrome c and the negatively charged physiological electron transfer partner cytochrome c(552). We studied association toward "naked" COX and toward membrane-embedded COX where the membrane is represented as an uncharged DPPC bilayer modeled in atomistic detail. For the nonnatural association toward "naked" COX, the association rates are >100 times larger for horse heart cytochrome c than for cytochrome c(552). Interestingly, the presence of the lipid bilayer leads to a dramatic decrease of the association rate of horse heart cytochrome c, but slightly enhances association of cytochrome c(552), leading to very similar association rates of both proteins to membrane-embedded COX. This finding from computational modeling studies may reflect the optimization of surface patches and of the total net charge on electron transfer pairs in nature.

The pH dependence of stability of the activation helix and the catalytic site of GART

We have predicted the free energy of unfolding for the pH-dependent helix-coil transition of the activation helix of GART using continuum electrostatic calculations and structural modeling. We have assigned the contributions of each element of secondary structure and of each ionizable residue, within and in the vicinity of the activation helix, to the stability of several fragments of GART that participate in the formation of the catalytic site. We demonstrate that the interaction of His121-His132 contributes 2.2 kcal/mol to the ionization free energy between pH 0 and approximately 6. We also show that the ionization state of a network of five histidines, His108, His119, His121, His132 and His137, and two aspartic acids Asp141 and Asp144, contributes approximately 12 kcal/mol to the stability of the catalytic site of GART, out of a total stability of 16 kcal/mol of the whole enzyme. These interactions are important for the formation of the catalytic site of GART.

Effective Born radii in the generalized Born approximation: the importance of being perfect

Generalized Born (GB) models provide, for many applications, an accurate and computationally facile estimate of the electrostatic contribution to aqueous solvation. The GB models involve two main types of approximations relative to the Poisson equation (PE) theory on which they are based. First, the self-energy contributions of individual atoms are estimated and expressed as "effective Born radii." Next, the atom-pair contributions are estimated by an analytical function f(GB) that depends upon the effective Born radii and interatomic distance of the atom pairs. Here, the relative impacts of these approximations are investigated by calculating "perfect" effective Born radii from PE theory, and enquiring as to how well the atom-pairwise energy terms from a GB model using these perfect radii in the standard f(GB) function duplicate the equivalent terms from PE theory. In tests on several biological macromolecules, the use of these perfect radii greatly increases the accuracy of the atom-pair terms; that is, the standard form of f(GB) performs quite well. The remaining small error has a systematic and a random component. The latter cannot be removed without significantly increasing the complexity of the GB model, but an alternative choice of f(GB) can reduce the systematic part. A molecular dynamics simulation using a perfect-radii GB model compares favorably with simulations using conventional GB, even though the radii remain fixed in the former. These results quantify, for the GB field, the importance of getting the effective Born radii right; indeed, with perfect radii, the GB model gives a very good approximation to the underlying PE theory for a variety of biomacromolecular types and conformations.

Accelerated Poisson-Boltzmann calculations for static and dynamic systems

We report here an efficient implementation of the finite difference Poisson-Boltzmann solvent model based on the Modified Incomplete Cholsky Conjugate Gradient algorithm, which gives rather impressive performance for both static and dynamic systems. This is achieved by implementing the algorithm with Eisenstat's two optimizations, utilizing the electrostatic update in simulations, and applying prudent approximations, including: relaxing the convergence criterion, not updating Poisson-Boltzmann-related forces every step, and using electrostatic focusing. It is also possible to markedly accelerate the supporting routines that are used to set up the calculations and to obtain energies and forces. The resulting finite difference Poisson-Boltzmann method delivers efficiency comparable to the distance-dependent dielectric model for a system tested, HIV Protease, making it a strong candidate for solution-phase molecular dynamics simulations. Further, the finite difference method includes all intrasolute electrostatic interactions, whereas the distance dependent dielectric calculations use a 15-A cutoff. The speed of our numerical finite difference method is comparable to that of the pair-wise Generalized Born approximation to the Poisson-Boltzmann method.

Protein molecular dynamics with electrostatic force entirely determined by a single Poisson-Boltzmann calculation

The electrostatic force including the intramolecular Coulombic interactions and the electrostatic contribution of solvation effect were entirely calculated by using the finite difference Poisson-Boltzmann method (FDPB), which was incorporated into the GROMOS96 force field to complete a new finite difference stochastic dynamics procedure (FDSD). Simulations were performed on an insulin dimer. Different relative dielectric constants were successively assigned to the protein interior; a value of 17 was selected as optimal for our system. The simulation data were analyzed and compared with those obtained from 500-ps molecular dynamics (MD) simulation with explicit water and a 500-ps conventional stochastic dynamics (SD) simulation without the mean solvent force. The results indicate that the FDSD method with GROMOS96 force field is suitable to study the dynamics and structure of proteins in solution if used with the optimal protein dielectric constant.

Molecular mechanics and dynamics of biomolecules using a solvent continuum model

An easy implementation of molecular mechanics and molecular dynamics simulation using a continuum solvent model is presented that is particularly suitable for biomolecular simulations. The computation of solvation forces is made using the linear Poisson-Boltzmann equation (polar contribution) and the solvent-accessible surface area approach (nonpolar contribution). The feasibility of the methodology is demonstrated on a small protein and a small DNA hairpin. Although the parameters employed in this model must be refined to gain reliability, the performance of the method, with a standard choice of parameters, is comparable with results obtained by explicit water simulations. Copyright 2001 John Wiley & Sons, Inc. J Comput Chem 22: 1830-1842, 2001

Proton transfer dynamics of GART: the pH-dependent catalytic mechanism examined by electrostatic calculations

The enzyme glycinamide ribonucleotide transformylase (GART) catalyzes the transfer of a formyl group from formyl tetrahydrofolate (fTHF) to glycinamide ribonucleotide (GAR), a process that is pH-dependent with pK(a) of approximately 8. Experimental studies of pH-rate profiles of wild-type and site-directed mutants of GART have led to the proposal that His108, Asp144, and GAR are involved in catalysis, with His108 being an acid catalyst, while forming a salt bridge with Asp144, and GAR being a nucleophile to attack the formyl group of fTHF. This model implied a protonated histidine with pK(a) of 9.7 and a neutral GAR with pK(a) of 6.8. These proposed unusual pK(a)s have led us to investigate the electrostatic environment of the active site of GART. We have used Poisson-Boltzmann-based electrostatic methods to calculate the pK(a)s of all ionizable groups, using the crystallographic structure of a ternary complex of GART involving the pseudosubstrate 5-deaza-5,6,7,8-THF (5dTHF) and substrate GAR. Theoretical mutation and deletion analogs have been constructed to elucidate pairwise electrostatic interactions between key ionizable sites within the catalytic site. Also, a construct of a more realistic catalytic site including a reconstructed pseudocofactor with an attached formyl group, in an environment with optimal local van der Waals interactions (locally minimized) that imitates closely the catalytic reactants, has been used for pK(a) calculations. Strong electrostatic coupling among catalytic residues His108, Asp144, and substrate GAR was observed, which is extremely sensitive to the initial protonation and imidazole ring flip state of His108 and small structural changes. We show that a proton can be exchanged between GAR and His108, depending on their relative geometry and their distance to Asp144, and when the proton is attached on His108, catalysis could be possible. Using the formylated locally minimized construct of GART, a high pK(a) for His108 was calculated, indicating a protonated histidine, and a low pK(a) for GAR(NH(2)) was calculated, indicating that GAR is in neutral form. Our results are in qualitative agreement with the current mechanistic picture of the catalytic process of GART deduced from the experimental data, but they do not reproduce the absolute magnitude of the pK(a)s extracted from fits of k(cat)-pH profiles, possibly because the static time-averaged crystallographic structure does not describe adequately the dynamic nature of the catalytic site during binding and catalysis. In addition, a strong effect on the pK(a) of GAR(NH(2)) is produced by the theoretical mutations of His108Ala and Asp144Ala, which is not in agreement with the observed insensitivity of the pK(a) of GAR(NH(2)) modeled from the experimental data using similar mutations. Finally, we show that important three-way electrostatic interactions between highly conserved His137, with His108 and Asp144, are responsible for stabilizing the electrostatic microenvironment of the catalytic site. In conclusion, our data suggest that further detailed computational and experimental work is necessary.

Native-state conformational dynamics of GART: a regulatory pH-dependent coil-helix transition examined by electrostatic calculations

Glycinamide ribonucleotide transformylase (GART) undergoes a pH-dependent coil-helix transition with pK(a) approximately 7. An alpha-helix is formed at high pH spanning 8 residues of a 21-residue-long loop, comprising the segment Thr120-His121-Arg122-Gln123-Ala124-Leu125-Glu126-Asn127. To understand the electrostatic nature of this loop-helix, called the activation loop-helix, which leads to the formation and stability of the alpha-helix, pK(a) values of all ionizable residues of GART have been calculated, using Poisson-Boltzmann electrostatic calculations and crystallographic data. Crystallographic structures of high and low pH E70A GART have been used in our analysis. Low pK(a) values of 5.3, 5.3, 3.9, 1.7, and 4.7 have been calculated for five functionally important histidines, His108, His119, His121, His132, and His137, respectively, using the high pH E70A GART structure. Ten theoretical single and double mutants of the high pH E70A structure have been constructed to identify pairwise interactions of ionizable residues, which have aided in elucidating the multiplicity of electrostatic interactions of the activation loop-helix, and the impact of the activation helix on the catalytic site. Based on our pK(a) calculations and structural data, we propose that: (1) His121 forms a molecular switch for the coil-helix transition of the activation helix, depending on its protonation state; (2) a strong electrostatic interaction between His132 and His121 is observed, which can be of stabilizing or destabilizing nature for the activation helix, depending on the relative orientation and protonation states of the rings of His121 and His132; (3) electrostatic interactions involving His119 and Arg122 play a role in the stability of the activation helix; and (4) the activation helix contains the helix-promoting sequence Arg122-Gln123-Ala124-Leu125-Glu126, but its alignment relative to the N and C termini of the helix is not optimal, and is possibly of a destabilizing nature. Finally, we provide electrostatic evidence that the formation and closure of the activation helix create a hydrophobic environment for catalytic-site residue His108, to facilitate catalysis.

Protein-protein association: investigation of factors influencing association rates by brownian dynamics simulations

The rate of protein-protein association limits the response time due to protein-protein interactions. The bimolecular association rate may be diffusion-controlled or influenced, and in such cases, Brownian dynamics simulations of protein-protein diffusional association may be used to compute association rates. Here, we report Brownian dynamics simulations of the diffusional association of five different protein-protein pairs: barnase and barstar, acetylcholinesterase and fasciculin-2, cytochrome c peroxidase and cytochrome c, the HyHEL-5 antibody and hen egg lysozyme (HEL), and the HyHEL-10 antibody and HEL. The same protocol was used to compute the diffusional association rates for all the protein pairs in order to assess, by comparison to experimentally measured rates, whether the association of these proteins can be explained solely on the basis of diffusional encounter. The simulation protocol is similar to those previously derived for simulation of the association of barnase and barstar, and of acetylcholinesterase and fasciculin-2; these produced results in excellent agreement with experimental data for these protein pairs, with changes in association rate due to mutations reproduced within the limits of expected computational and modeling errors. Here, we find that for all protein pairs, the effects of mutations can be well reproduced by the simulations, even though the degree of the electrostatic translational and orientational steering varies widely between the cases. However, the absolute values of association rates for the acetylcholinesterase: fasciculin-2 and HyHEL-10 antibody: HEL pairs are overestimated. Comparison of bound and unbound protein structures shows that this may be due to gating resulting from protein flexibility in some of the proteins. This may lower the association rates compared to their bimolecular diffusional encounter rates.

Generalized born models of macromolecular solvation effects

It would often be useful in computer simulations to use a simple description of solvation effects, instead of explicitly representing the individual solvent molecules. Continuum dielectric models often work well in describing the thermodynamic aspects of aqueous solvation, and approximations to such models that avoid the need to solve the Poisson equation are attractive because of their computational efficiency. Here we give an overview of one such approximation, the generalized Born model, which is simple and fast enough to be used for molecular dynamics simulations of proteins and nucleic acids. We discuss its strengths and weaknesses, both for its fidelity to the underlying continuum model and for its ability to replace explicit consideration of solvent molecules in macromolecular simulations. We focus particularly on versions of the generalized Born model that have a pair-wise analytical form, and therefore fit most naturally into conventional molecular mechanics calculations.

Calculation of weak protein-protein interactions: the pH dependence of the second virial coefficient

Interactions between proteins are often sufficiently weak that their study through the use of conventional structural techniques becomes problematic. Of the few techniques capable of providing experimental measures of weak protein-protein interactions, perhaps the most useful is the second virial coefficient, B(22), which quantifies a protein solution's deviations from ideal behavior. It has long been known that B(22) can in principle be computed, but only very recently has it been demonstrated that such calculations can be performed using protein models of true atomic detail (Biophys. J. 1998, 75:2469-2477). The work reported here extends these previous efforts in an attempt to develop a transferable energetic model capable of reproducing the experimental trends obtained for two different proteins over a range of pH and ionic strengths. We describe protein-protein interaction energies by a combination of three separate terms: (i) an electrostatic interaction term based on the use of effective charges, (ii) a term describing the electrostatic desolvation that occurs when charged groups are buried by an approaching protein partner, and (iii) a solvent-accessible surface area term that is used to describe contributions from van der Waals and hydrophobic interactions. The magnitude of the third term is governed by an adjustable, empirical parameter, gamma, that is altered to optimize agreement between calculated and experimental values of B(22). The model is applied separately to the proteins lysozyme and chymotrypsinogen, yielding optimal values of gamma that are almost identical. There are, however, clear difficulties in reproducing B(22) values at the extremes of pH. Explicit calculation of the protonation states of ionizable amino acids in the 200 most energetically favorable protein-protein structures suggest that these difficulties are due to a neglect of the protonation state changes that can accompany complexation. Proper reproduction of the pH dependence of B(22) will, therefore, almost certainly require that account be taken of these protonation state changes. Despite this problem, the fact that almost identical gamma values are obtained from two different proteins suggests that the basic energetic formulation used here, which can be evaluated very rapidly, might find use in dynamical simulations of weak protein-protein interactions at intermediate pH values.

Kinetics of diffusion-assisted reactions in microheterogeneous systems

This review is focused on the basic theory of diffusion-assisted reactions in microheterogeneous systems, from porous solids to self-organized colloids and biomolecules. Rich kinetic behaviors observed experimentally are explained in a unified fashion using simple concepts of competing distance and time scales of the reaction and the embedding structure. We mainly consider pseudo-first-order reactions, such as luminescence quenching, described by the Smoluchowski type of equation for the reactant pair distribution function with a sink term defined by the reaction mechanism. Microheterogeneity can affect the microscopic rate constant. It also enters the evolution equation through various spatial constraints leading to complicated boundary conditions and, possibly, to the reduction of dimensionality of the diffusion space. The reaction coordinate and diffusive motion along this coordinate are understood in a general way, depending on the problem at hand. Thus, the evolution operator can describe translational and rotational diffusion of molecules in a usual sense, it can be a discrete random walk operator when dealing with hopping of adsorbates in solids, or it can correspond to conformational fluctuations in proteins. Mathematical formulation is universal but physical consequences can be different. Understanding the principal features of reaction kinetics in microheterogeneous systems enables one to extract important structural and dynamical information about the host environments by analyzing suitably designed experiments, it helps building effective strategies for computer simulations, and ultimately opens possibilities for designing systems with controllable reactivity properties.

Simulation of electrostatic effects in Fab-antigen complex formation

A model based on the Poisson-Boltzmann equation has been used to model electrostatics in Anti-p24 (HIV-1) Fab-antigen association. The ionization state at different pH values has been simulated and the results have been used to estimate the stability at different pH values and to generate electrostatic potential maps at physiological ionic strength. The analysis of the electrostatic potential at the solvent-accessible surface shows that residues involved in binding are mostly found in the highest, but also in lowest potential regions. Brownian dynamics simulations have been used to estimate the enhancement of the association rate due to electrostatics which appears limited (approximately 2 at 150 mM ionic strength and approximately 3 at 15 mM ionic strength). A much more pronounced effect is observed upon increase of the charge of the diffusing particle. These results compare well with results obtained previously in similar studies on different systems and may serve to estimate the expected order of magnitude of electrostatic effects on association rates in antibody-antigen systems.

Electrostatic properties of bovine beta-lactoglobulin

Bovine beta-Lactoglobulin (BLG) has been studied for many decades, but only recently structural data have been obtained, making it possible to simulate its molecular properties. In the present study, electrostatic properties of BLG are investigated theoretically using Poisson-Boltzmann calculations and experimentally following pH titration via NMR. Electrostatic properties are determined for several structural models, including an ensemble of NMR structures obtained at low pH. The changes in electrostatic forces upon changes in ionic strength, solvent dielectric constant, and pH are calculated and compared with experiments. pK(a)s are computed for all titratable sites and compared with NMR titration data. The analysis of theoretical and experimental results suggests that (1) there may be more than one binding sites for negatively charged ligands; (2) at low pH the core of the molecule is more compact than observed in the structures obtained via restrained molecular dynamics from NMR data, but loop and terminal regions must be disordered.

A dimeric DNA interface stabilized by stacked A.(G.G.G.G).A hexads and coordinated monovalent cations

We report on the identification of an A.(G.G.G.G).A hexad pairing alignment which involves recognition of the exposed minor groove of opposing guanines within a G.G.G.G tetrad through sheared G.A mismatch formation. This unexpected hexad pairing alignment was identified for the d(G-G-A-G-G-A-G) sequence in 150 mM Na(+) (or K(+)) cation solution where four symmetry-related strands align into a novel dimeric motif. Each symmetric half of the dimeric "hexad" motif is composed of two strands and contains a stacked array of an A.(G.G.G.G).A hexad, a G.G.G.G tetrad, and an A.A mismatch. Each strand in the hexad motif contains two successive turns, that together define an S-shaped double chain reversal fold, which connects the two G-G steps aligned parallel to each other along adjacent edges of the quadruplex. Our studies also establish a novel structural transition for the d(G-G-A-G-G-A-N) sequence, N=T and G, from an "arrowhead" motif stabilized through cross-strand stacking and mismatch formation in 10 mM Na(+) solution (reported previously), to a dimeric hexad motif stabilized by extensive inter-subunit stacking of symmetry-related A.(G.G.G.G).A hexads in 150 mM Na(+) solution. Potential monovalent cation binding sites within the arrowhead and hexad motifs have been probed by a combination of Brownian dynamics and unconstrained molecular dynamics calculations. We could not identify stable monovalent cation-binding sites in the low salt arrowhead motif. By contrast, five electronegative pockets were identified in the moderate salt dimeric hexad motif. Three of these are involved in cation binding sites sandwiched between G.G.G. G tetrad planes and two others, are involved in water-mediated cation binding sites spanning the unoccupied grooves associated with the adjacent stacked A.(G.G.G.G).A hexads. Our demonstration of A.(G. G.G.G).A hexad formation opens opportunities for the design of adenine-rich G-quadruplex-interacting oligomers that could potentially target base edges of stacked G.G.G.G tetrads. Such an approach could complement current efforts to design groove-binding and intercalating ligands that target G-quadruplexes in attempts designed to block the activity of the enzyme telomerase.

Realistic modeling of the denatured states of proteins allows accurate calculations of the pH dependence of protein stability

Computational techniques based on continuum electrostatics treatments have been successful in predicting and interpreting the pKa values of ionizable amino acids in folded proteins. Despite this progress, efforts to reproduce the pH-dependence of protein stability have met with only limited success: agreement with experimental results has been only qualitative. It has been argued previously that the most likely reason for discrepancies is the presence of residual electrostatic interactions in the unfolded state, which cause pKa values to be shifted from their model compound values. Here we show that by constructing atomistic models of the unfolded state with a simple molecular mechanics protocol that uses the native state as a starting point, much improved reproduction of pH effects on protein stability can be obtained. In contrast, when a fully extended model of the unfolded state is used, no such improvement is obtained, a result that suggests that local interactions with residues nearby in the sequence are not sufficient to properly account for the pKa shifts in the unfolded state. In comparison to model compound values, the pKa values of acidic residues in "native-like" unfolded states are typically found to be shifted downwards by approximately 0.3 pH unit, in good agreement with the average downward shift deduced from experimental measurements. Given its success in the present situation, the protocol employed here for developing simple models of the unfolded state may prove useful in other computer simulation applications.

Computer simulation of protein-protein association kinetics: acetylcholinesterase-fasciculin

Computer simulations were performed to investigate the role of electrostatic interactions in promoting fast association of acetylcholinesterase with its peptidic inhibitor, the neurotoxin fasciculin. The encounter of the two macromolecules was simulated with the technique of Brownian dynamics (BD), using atomically detailed structures, and association rate constants were calculated for the wild-type and a number of mutant proteins. In a first set of simulations, the ordering of the experimental rate constants for the mutant proteins was correctly reproduced, although the absolute values of the rate constants were overestimated by a factor of around 30. Rigorous calculations of the full electrostatic interaction energy between the two proteins indicate that this overestimation of association rates results at least in part from approximations made in the description of interaction energetics in the BD simulations. In particular, the initial BD simulations neglect the unfavourable electrostatic desolvation effects that result from the exclusion of high dielectric solvent that accompanies the approach of the two low dielectric proteins. This electrostatic desolvation component is so large that the overall contribution of electrostatics to the binding energy of the complex is unlikely to be strongly favourable. Nevertheless, electrostatic interactions are still responsible for increased association rates, because even if they are unfavourable in the fully formed complex, they are still favourable at intermediate protein-protein separation distances. It therefore appears possible for electrostatic interactions to promote the kinetics of binding even if they do not make a strongly favourable contribution to the thermodynamics of binding. When an approximate description of these electrostatic desolvation effects is included in a second set of BD simulations, the relative ordering of the mutant proteins is again correctly reproduced, but now association rate constants that are much closer in magnitude to the experimental values are obtained. Inclusion of electrostatic desolvation effects also improves reproduction of the experimental ionic strength dependence of the wild-type association rate.

Effect of artificial periodicity in simulations of biomolecules under Ewald boundary conditions: a continuum electrostatics study

Ewald and related methods are nowadays routinely used in explicit-solvent simulations of biomolecules, although they impose an artificial periodicity in systems which are inherently non-periodic. The consequences of this approximation should be assessed, since they may crucially affect the reliability of computer simulations under Ewald boundary conditions. In the present study we use a method based on continuum electrostatics to investigate the nature and magnitude of possible periodicity-induced artifacts on the potentials of mean force for conformational equilibria in biomolecules. Three model systems and pathways are considered: polyalanine oligopeptides (unfolding), a DNA tetranucleotide (separation of the strands), and the protein Sac7d (conformations from a molecular dynamics simulation). Artificial periodicity may significantly affect these conformational equilibria, in each case stabilizing the most compact conformation of the biomolecule. Three factors enhance periodicity-induced artifacts: (i) a solvent of low dielectric permittivity; (ii) a solute size which is non-negligible compared to the size of the unit cell; and (iii) a non-neutral solute. Neither the neutrality of the solute nor the absence of charge pairs at distances exceeding half the edge of the unit cell do guarantee the absence of artifacts.

Electrostatic contributions to the stability of halophilic proteins

Examination of the first crystal structures of proteins from a halophilic organism suggests that an abundance of acidic residues distributed over the protein surface is a key determinant of adaptation to high-salt conditions. Although one extant theory suggests that acidic residues are favored because of their superior water-binding capacity, it is clear that extensive repulsive electrostatic interactions will also be present in such proteins at physiological pH. To investigate the magnitude and importance of such electrostatic interactions, we conducted a theoretical analysis of their contributions to the salt and pH-dependence of stability of two halophilic proteins. Our approach centers on use of the Poisson-Boltzmann equation of classical electrostatics, applied at an atomic level of detail to crystal structures of the proteins. We first show that in using the method, it is important to account for the fact that the dielectric constant of water decreases at high salt concentrations, in order to reproduce experimental changes in pKa values of small acids and bases. We then conduct a comparison of salt and pH effects on the stability of 2Fe-2S ferredoxins from the halophile Haloarcula marismortui and the non-halophile anabaena. In both proteins, substantial upward shifts in pKa accompany protein folding, though shifts are considerably larger, on average, in the halophile. Upward shifts for basic residues occur because of favorable salt-bridge interactions, whilst upward shifts for acidic residues result from unfavorable electrostatic interactions with other acidic groups. Our calculations suggest that at pH 7 the stability of the halophilic protein is decreased by 18.2 kcal/mol on lowering the salt concentration from 5 M to 100 mM, a result that is in line with the fact that halophilic proteins generally unfold at low salt concentrations. For comparison, the non-halophilic ferredoxin is calculated to be destabilized by only 5.1 kcal/mol over the same range. Analysis of the pH stability curve suggests that lowering the pH should increase the intrinsic stability of the halophilic protein at low salt concentrations, although in practice this is not observed because of aggregation effects. We report the results of a similar analysis carried out on the tetrameric malate dehydrogenase from H. marismortui. In this case, we investigated the salt and pH dependence of the various monomer-monomer interactions present in the tetramer. All monomer-monomer interactions are found to make substantial contributions to the salt-dependence of stability of the tetramer. Excellent agreement is obtained between our calculated results for the stability of the tetramer and experimental results. In particular, the finding that at 4 M NaCl, the tetramer is stable only between pH 4.8 and 10 is accurately reproduced. Taken together, our results suggest that repulsive electrostatic interactions between acidic residues are a major factor in the destabilization of halophilic proteins in low-salt conditions, and that these interactions remain destabilizing even at high salt concentrations. As a consequence, the role of acidic residues in halophilic proteins may be more to prevent aggregation than to make a positive contribution to intrinsic protein stability.

Electrostatic and non-electrostatic contributions to the binding free energies of anthracycline antibiotics to DNA

The knowledge about molecular factors driving simple ligand-DNA interactions is still limited. The aim of the present study was to investigate the electrostatic and non-electrostatic contributions to the binding free energies of anthracycline compounds with DNA. Theoretical calculations based on continuum methods (Poisson-Boltzmann and solvent accessible surface area) were performed to estimate the binding free energies of five selected anthracycline ligands (daunomycin, adriamycin, 9-deoxyadriamycin, hydroxyrubicin, and adriamycinone) to DNA. The free energy calculations also took into account the conformational change that DNA undergoes upon ligand binding. This conformational change appeared to be very important for estimating absolute free energies of binding. Our studies revealed that the absolute values of all computed contributions to the binding free energy were quite large compared to the total free energy of binding. However, the sum of these large positive and negative values produced a small negative value of the free energy around -10 kcal/mol. This value is in good agreement with experimental data. Experimental values for relative binding free energies were also reproduced for charged ligands by our calculations. Together, it was found that the driving force for ligand-DNA complex formation is the non-polar interaction between the ligand and DNA even if the ligand is positively charged.

Empirical free energy calculations: a blind test and further improvements to the method

Empirical Gibbs functions estimate free energies of non-covalent reactions (deltaG) from atomic coordinates of reaction products (e.g. antibody-antigen complexes). The function previously developed by us has four terms that quantify the effects of hydrophobic, electrostatic and entropy changes (conformational, association) upon complexation. The function was used to calculate delta deltaG of ten lysozyme mutants affecting the stability of the HyHEL-10 antibody-lysozyme complex. The mutants were computer-modeled from the X-ray structure of the wild-type, and free energy calculations produced a correlation coefficient of 0.5 with the experimental delta deltaG data (average absolute error +/-3 kcal). The following changes were then introduced into the Gibbs function: (1) the hydrophobic force was made proportional to the molecular surface, as calculated by the GEPOL93 algorithm, with the scaling constant of 70 cal/mol/A2; (2) calculation of the electrostatics of binding was carried out by the finite difference Poisson-Boltzmann algorithm, which employed uniform grid charging, dielectric boundary smoothing and charge anti-aliasing; and (3) side-chain conformational entropy was estimated from the CONGEN sampling of torsional degrees of freedom. In the new calculations, correlation with experimental data improved to 0.6 or 0.8 if a single outlying mutant, K96M, was neglected. Analysis of the errors remaining in our calculations indicated that molecular mechanics-based modeling of the mutants, rather than the form of our amended Gibbs function, was the main factor limiting the accuracy of the free energy estimates.

Electrostatic effects in homeodomain-DNA interactions

We report here an investigation of the role of electrostatics in homeodomain-DNA interactions using techniques based around the use of the Poisson-Boltzmann equation. In the present case such a study is of particular interest, since in contrast to other proteins previously studied with this method, the homeodomain is a small, highly charged protein that forms extensive ion pairs upon binding DNA. We have investigated the salt dependence of the binding constant for specific association and for a variety of models for non-specific association. The results indicate that, in line with the models proposed by Manning and Record, the entropy of counterion release accounts for a significant fraction of the salt dependence of the binding free energy, though this is perhaps due to fortuitous cancellation of other contributing terms. The thermodynamic effects of a number of specific homeodomain mutants were also investigated, and partly rationalized in terms of favorable electrostatic interactions in the major goove of DNA. Investigation of the temperature-dependence of the free energy of association indicates that the electrostatic contributions become increasingly favorable as the temperature rises. For this particular system, however, there appears to be no significant electrostatic contribution to the delta(delta C(p)) of association. Finally, an analysis of the free energy of interaction when the homeodomain is moved ca one Debye length from the DNA suggests that pure electrostatic forces are able to steer the homeodomain into a partially correct orientation for binding to the DNA.

Binding of tacrine and 6-chlorotacrine by acetylcholinesterase

Multiconfiguration thermodynamic integration was used to determine the relative binding strength of tacrine and 6-chlorotacrine by Torpedo californica acetylcholinesterase. 6-Chlorotacrine appears to be bound stronger by 0.7+/-0.4 kcal/mol than unsubstituted tacrine when the active site triad residue His-440 is deprotonated. This result is in excellent agreement with experimental inhibition data on electric eel acetylcholinesterase. Electrostatic Poisson-Boltzmann calculations confirm that order of binding strength, resulting in deltaG of binding of -2.9 and -3.3 kcal/mol for tacrine and chlorotacrine, respectively, and suggest inhibitor binding does not occur when His-440 is charged. Our results suggest that electron density redistribution upon tacrine chlorination is mainly responsible for the increased attraction potential between pronated inhibitor molecule and adjacent aromatic groups of Phe-330 and Trp-84.

Molecular modeling methods. Basic techniques and challenging problems

An overview is presented of computer modeling and simulation methods that play an increasing role in drug design: quantum chemical methods, molecular mechanics, molecular dynamics and Brownian dynamics. The application of molecular dynamics for the prediction of thermodynamic properties like free energy differences and binding constants is discussed. The Brownian dynamics method is presented in connection with the calculation of effective electrostatic forces using the Poisson-Boltzmann equation, which allows one to sample ligand-binding geometries and to predict the kinetics of diffusion-limited enzyme reactions. New techniques that have recently been extensively developed, such as the global energy minimization and quantum-classical dynamics methods, are also introduced. The molecular modeling methods are illustrated with selected examples.