Accurate Reaction Energies in Proteins Obtained by Combining QM/MM and Large QM Calculations

We here suggest and test a new method to obtain stable energies in proteins for charge-neutral reactions by running large quantum mechanical (QM) calculations on structures obtained by combined QM and molecular mechanics (QM/MM) geometry optimization on several snapshots from molecular dynamics simulations. As a test case, we use a proton transfer between a metal-bound cysteine residue and a second-sphere histidine residue in the active site of [Ni,Fe] hydrogenase, which has been shown to be very sensitive to the surroundings. We include in the QM calculations all residues within 4.5 Å of the active site, two capped residues on each side of the active-site residues, and all charged groups that are buried inside the protein, which for this enzyme includes three iron-sulfur clusters, in total, 930 atoms. These calculations are performed at the BP86/def2-SV(P) level, but the energies are then extrapolated to the B3LYP/def2-TZVP level with a smaller QM system, and zero-point energy, entropy, and thermal effects are added. We test three approaches to model the remaining atoms of the protein solvent, viz., by standard QM/MM approaches using either mechanical or electrostatic embedding or by using a continuum solvation model for the large QM systems. Quite encouragingly, the three approaches give the same results within 14 kJ/mol, and variations in the size of the QM system do not change the energies by more than 8 kJ/mol, provided that the QM/MM junctions are not moved closer to the QM system. The statistical precision for the average over 10 snapshots is 1-3 kJ/mol.

Convergence of QM/MM free-energy perturbations based on molecular-mechanics or semiempirical simulations

Lately, there has been great interest in performing free-energy perturbation (FEP) at the combined quantum mechanics and molecular mechanics (QM/MM) level, e.g. for enzyme reactions. Such calculations require extensive sampling of phase space, which typically is prohibitive with density-functional theory or ab initio methods. Therefore, such calculations have mostly been performed with semiempirical QM (SQM) methods, or by using a thermodynamic cycle involving sampling at the MM level and perturbations between the MM and QM/MM levels of theory. However, the latter perturbations typically have convergence problems, unless the QM system is kept fixed during the simulations, because the MM and QM/MM descriptions of the internal degrees of freedom inside the QM system are too dissimilar. We have studied whether the convergence of the MM → QM/MM perturbation can be improved by using a thoroughly parameterised force field or by using SQM/MM methods. As a test case we use the first half-reaction of haloalkane dehalogenase and the QM calculations are performed with the PBE, B3LYP, and TPSSH density-functional methods. We show that the convergence can be improved with a tailored force field, but only locally around the parameterised state. Simulations based on SQM/MM methods using the MNDO, AM1, PM3, RM1, PDDG-MNDO, and PDDG-PM3 Hamiltonians have slightly better convergence properties, but very long simulations are still needed (~10 ns) and convergence is obtained only if electrostatic interactions between the QM system and the surroundings are ignored. This casts some doubts on the common practice to base QM/MM FEPs on semiempirical simulations without any reweighting of the trajectories.

Will molecular dynamics simulations of proteins ever reach equilibrium?

We show that conformational entropies calculated for five proteins and protein-ligand complexes with dihedral-distribution histogramming, the von Mises approach, or quasi-harmonic analysis do not converge to any useful precision even if molecular dynamics (MD) simulations of 380-500 ns length are employed (the uncertainty is 12-89 kJ mol(-1)). To explain this, we suggest a simple protein model involving dihedrals with effective barriers forming a uniform distribution and show that for such a model, the entropy increases logarithmically with time until all significantly populated dihedral states have been sampled, in agreement with the simulations (during the simulations, 52-70% of the available dihedral phase space has been visited). This is also confirmed by the analysis of the trajectories of a 1 ms simulation of bovine pancreatic trypsin inhibitor (31 kJ mol(-1) difference in the entropy between the first and second part of the simulation). Strictly speaking, this means that it is practically impossible to equilibrate MD simulations of proteins. We discuss the implications of such a lack of strict equilibration of protein MD simulations and show that ligand-binding free energies estimated with the MM/GBSA method (molecular mechanics with generalised Born and surface-area solvation) vary by 3-15 kJ mol(-1) during a 500 ns simulation (the higher estimate is caused by rare conformational changes), although they involve a questionable but well-converged normal-mode entropy estimate, whereas free energies estimated by free-energy perturbation vary by less than 0.6 kJ mol(-1) for the same simulation.

Comparison of end-point continuum-solvation methods for the calculation of protein-ligand binding free energies

We have compared the predictions of ligand-binding affinities from several methods based on end-point molecular dynamics simulations and continuum solvation, that is, methods related to MM/PBSA (molecular mechanics combined with Poisson-Boltzmann and surface area solvation). Two continuum-solvation models were considered, viz., the Poisson-Boltzmann (PB) and generalised Born (GB) approaches. The nonelectrostatic energies were also obtained in two different ways, viz., either from the sum of the bonded, van der Waals, nonpolar solvation energies, and entropy terms (as in MM/PBSA), or from the scaled protein-ligand van der Waals interaction energy (as in the linear interaction energy approach, LIE). Three different approaches to calculate electrostatic energies were tested, viz., the sum of electrostatic interaction energies and polar solvation energies, obtained either from a single simulation of the complex or from three independent simulations of the complex, the free protein, and the free ligand, or the linear-response approximation (LRA). Moreover, we investigated the effect of scaling the electrostatic interactions by an effective internal dielectric constant of the protein (ϵ(int) ). All these methods were tested on the binding of seven biotin analogues to avidin and nine 3-amidinobenzyl-1H-indole-2-carboxamide inhibitors to factor Xa. For avidin, the best results were obtained with a combination of the LIE nonelectrostatic energies with the MM+GB electrostatic energies from a single simulation, using ϵ(int) = 4. For fXa, standard MM/GBSA, based on one simulation and using ϵ(int) = 4-10 gave the best result. The optimum internal dielectric constant seems to be slightly higher with PB than with GB solvation.

Comparison of chemical properties of iron, cobalt, and nickel porphyrins, corrins, and hydrocorphins

Density functional calculations have been used to compare the geometric, electronic, and functional properties of the three important tetrapyrrole systems in biology, heme, coenzyme B 12, and coenzyme F430, formed from iron porphyrin ( Por ), cobalt corrin ( Cor ), and nickel hydrocorphin ( Hcor ). The results show that the flexibility of the ring systems follows the trend Hcor > Cor > Por and that the size of the central cavity follows the trend Cor < Por < Hcor . Therefore, low-spin Co I, Co II, and Co III fit well into the Cor ring, whereas Por seems to be more ideal for the higher spin states of iron, and the cavity in Hcor is tailored for the larger Ni ion, especially in the high-spin Ni II state. This is confirmed by the thermodynamic stabilities of the various combinations of metals and ring systems. Reduction potentials indicate that the +I and +III states are less stable for Ni than for the other metal ions. Moreover, Ni – C bonds are appreciably less stable than Co - C bonds. However, it is still possible that a Ni – CH 3 bond is formed in F 430 by a heterolytic methyl transfer reaction, provided that the donor is appropriate, e.g. if coenzyme M is protonated. This can be facilitated by the adjacent SO 3− group in this coenzyme and by the axial glutamine ligand, which stabilizes the Ni III state. Our results also show that a Ni III– CH 3 complex is readily hydrolysed to form a methane molecule and that the Ni III hydrolysis product can oxidize coenzyme B and M to a heterodisulphide in the reaction mechanism of methyl coenzyme M reductase.

Binding affinities in the SAMPL3 trypsin and host-guest blind tests estimated with the MM/PBSA and LIE methods

We have estimated affinities for the binding of 34 ligands to trypsin and nine guest molecules to three different hosts in the SAMPL3 blind challenge, using the MM/PBSA, MM/GBSA, LIE, continuum LIE, and Glide score methods. For the trypsin challenge, none of the methods were able to accurately predict the experimental results. For the MM/GB(PB)SA and LIE methods, the rankings were essentially random and the mean absolute deviations were much worse than a null hypothesis giving the same affinity to all ligand. Glide scoring gave a Kendall's τ index better than random, but the ranking is still only mediocre, τ = 0.2. However, the range of affinities is small and most of the pairs of ligands have an experimental affinity difference that is not statistically significant. Removing those pairs improves the ranking metric to 0.4-1.0 for all methods except CLIE. Half of the trypsin ligands were non-binders according to the binding assay. The LIE methods could not separate the inactive ligands from the active ones better than a random guess, whereas MM/GBSA and MM/PBSA were slightly better than random (area under the receiver-operating-characteristic curve, AUC = 0.65-0.68), and Glide scoring was even better (AUC = 0.79). For the first host, MM/GBSA and MM/PBSA reproduce the experimental ranking fairly good, with τ = 0.6 and 0.5, respectively, whereas the Glide scoring was considerably worse, with a τ = 0.4, highlighting that the success of the methods is system-dependent.

The carbohydrate-binding site in galectin-3 is preorganized to recognize a sugarlike framework of oxygens: ultra-high-resolution structures and water dynamics

The recognition of carbohydrates by proteins is a fundamental aspect of communication within and between living cells. Understanding the molecular basis of carbohydrate–protein interactions is a prerequisite for the rational design of synthetic ligands. Here we report the high- to ultra-high-resolution crystal structures of the carbohydrate recognition domain of galectin-3 (Gal3C) in the ligand-free state (1.08 Å at 100 K, 1.25 Å at 298 K) and in complex with lactose (0.86 Å) or glycerol (0.9 Å). These structures reveal striking similarities in the positions of water and carbohydrate oxygen atoms in all three states, indicating that the binding site of Gal3C is preorganized to coordinate oxygen atoms in an arrangement that is nearly optimal for the recognition of β-galactosides. Deuterium nuclear magnetic resonance (NMR) relaxation dispersion experiments and molecular dynamics simulations demonstrate that all water molecules in the lactose-binding site exchange with bulk water on a time scale of nanoseconds or shorter. Nevertheless, molecular dynamics simulations identify transient water binding at sites that agree well with those observed by crystallography, indicating that the energy landscape of the binding site is maintained in solution. All heavy atoms of glycerol are positioned like the corresponding atoms of lactose in the Gal3C complexes. However, binding of glycerol to Gal3C is insignificant in solution at room temperature, as monitored by NMR spectroscopy or isothermal titration calorimetry under conditions where lactose binding is readily detected. These observations make a case for protein cryo-crystallography as a valuable screening method in fragment-based drug discovery and further suggest that identification of water sites might inform inhibitor design.

Reorganization energy for internal electron transfer in multicopper oxidases

We have calculated the reorganization energy for the intramolecular electron transfer between the reduced type 1 copper site and the peroxy intermediate of the trinuclear cluster in the multicopper oxidase CueO. The calculations are performed at the combined quantum mechanics and molecular mechanics (QM/MM) level, based on molecular dynamics simulations with tailored potentials for the two copper sites. We obtain a reorganization energy of 91-133 kJ/mol, depending on the theoretical treatment. The two Cu sites contribute by 12 and 22 kJ/mol to this energy, whereas the solvent contribution is 34 kJ/mol. The rest comes from the protein, involving small contributions from many residues. We have also estimated the energy difference between the two electron-transfer states and show that the reduction of the peroxy intermediate is exergonic by 43-87 kJ/mol, depending on the theoretical method. Both the solvent and the protein contribute to this energy difference, especially charged residues close to the two Cu sites. We compare these estimates with energies obtained from QM/MM optimizations and QM calculations in a vacuum and discuss differences between the results obtained at various levels of theory.

Reduction potentials and acidity constants of Mn superoxide dismutase calculated by QM/MM free-energy methods

We used two theoretical methods to estimate reduction potentials and acidity constants in Mn superoxide dismutase (MnSOD), namely combined quantum mechanical and molecular mechanics (QM/MM) thermodynamic cycle perturbation (QTCP) and the QM/MM-PBSA approach. In the latter, QM/MM energies are combined with continuum solvation energies calculated by solving the Poisson-Boltzmann equation (PB) or by the generalised Born approach (GB) and non-polar solvation energies calculated from the solvent-exposed surface area. We show that using the QTCP method, we can obtain accurate and precise estimates of the proton-coupled reduction potential for MnSOD, 0.30±0.01 V, which compares favourably with experimental estimates of 0.26-0.40 V. However, the calculated potentials depend strongly on the DFT functional used: The B3LYP functional gives 0.6 V more positive potentials than the PBE functional. The QM/MM-PBSA approach leads to somewhat too high reduction potentials for the coupled reaction and the results depend on the solvation model used. For reactions involving a change in the net charge of the metal site, the corresponding results differ by up to 1.3 V or 24 pK(a) units, rendering the QM/MM-PBSA method useless to determine absolute potentials. However, it may still be useful to estimate relative shifts, although the QTCP method is expected to be more accurate.

A QM/MM study of the binding of RAPTA ligands to cathepsin B

We have carried out quantum mechanical (QM) and QM/MM (combined QM and molecular mechanics) calculations, as well as molecular dynamics (MD) simulations to study the binding of a series of six RAPTA (Ru(II)-arene-1,3,5-triaza-7-phosphatricyclo-[3.3.1.1] decane) complexes with different arene substituents to cathepsin B. The recently developed QM/MM-PBSA approach (QM/MM combined with Poisson-Boltzmann solvent-accessible surface area solvation) has been used to estimate binding affinities. The QM calculations reproduce the antitumour activities of the complexes with a correlation coefficient (r (2)) of 0.35-0.86 after a conformational search. The QM/MM-PBSA method gave a better correlation (r (2) = 0.59) when the protein was fixed to the crystal structure, but more reasonable ligand structures and absolute binding energies were obtained if the protein was allowed to relax, indicating that the ligands are strained when the protein is kept fixed. In addition, the best correlation (r (2) = 0.80) was obtained when only the QM energies were used, which suggests that the MM and continuum solvation energies are not accurate enough to predict the binding of a charged metal complex to a charged protein. Taking into account the protein flexibility by means of MD simulations slightly improves the correlation (r (2) = 0.91), but the absolute energies are still too large and the results are sensitive to the details in the calculations, illustrating that it is hard to obtain stable predictions when full flexible protein is included in the calculations.