The minimized dead-end elimination criterion and its application to protein redesign in a hybrid scoring and search algorithm for computing partition functions over molecular ensembles

One of the main challenges for protein redesign is the efficient evaluation of a combinatorial number of candidate structures. The modeling of protein flexibility, typically by using a rotamer library of commonly-observed low-energy side-chain conformations, further increases the complexity of the redesign problem. A dominant algorithm for protein redesign is dead-end elimination (DEE), which prunes the majority of candidate conformations by eliminating rigid rotamers that provably are not part of the global minimum energy conformation (GMEC). The identified GMEC consists of rigid rotamers (i.e., rotamers that have not been energy-minimized) and is thus referred to as the rigid-GMEC. As a postprocessing step, the conformations that survive DEE may be energy-minimized. When energy minimization is performed after pruning with DEE, the combined protein design process becomes heuristic, and is no longer provably accurate: a conformation that is pruned using rigid-rotamer energies may subsequently minimize to a lower energy than the rigid-GMEC. That is, the rigid-GMEC and the conformation with the lowest energy among all energy-minimized conformations (the minimized-GMEC) are likely to be different. While the traditional DEE algorithm succeeds in not pruning rotamers that are part of the rigid-GMEC, it makes no guarantees regarding the identification of the minimized-GMEC. In this paper we derive a novel, provable, and efficient DEE-like algorithm, called minimized-DEE (MinDEE), that guarantees that rotamers belonging to the minimized-GMEC will not be pruned, while still pruning a combinatorial number of conformations. We show that MinDEE is useful not only in identifying the minimized-GMEC, but also as a filter in an ensemble-based scoring and search algorithm for protein redesign that exploits energy-minimized conformations. We compare our results both to our previous computational predictions of protein designs and to biological activity assays of predicted protein mutants. Our provable and efficient minimized-DEE algorithm is applicable in protein redesign, protein-ligand binding prediction, and computer-aided drug design.     

Towards the comprehensive, rapid, and accurate prediction of the favorable tautomeric states of drug-like molecules in aqueous solution

Generating the appropriate protonation states of drug-like molecules in solution is important for success in both ligand- and structure-based virtual screening. Screening collections of millions of compounds requires a method for determining tautomers and their energies that is sufficiently rapid, accurate, and comprehensive. To maximise enrichment, the lowest energy tautomers must be determined from heterogeneous input, without over-enumerating unfavourable states. While computationally expensive, the density functional theory (DFT) method M06-2X/aug-cc-pVTZ(-f) [PB-SCRF] provides accurate energies for enumerated model tautomeric systems. The empirical Hammett–Taft methodology can very rapidly extrapolate substituent effects from model systems to drug-like molecules via the relationship between pK_T and pK_a. Combining the two complementary approaches transforms the tautomer problem from a scientific challenge to one of engineering scale-up, and avoids issues that arise due to the very limited number of measured pK_T values, especially for the complicated heterocycles often favoured by medicinal chemists for their novelty and versatility. Several hundreds of pre-calculated tautomer energies and substituent pK_a effects are tabulated in databases for use in structural adjustment by the program Epik, which treats tautomers as a subset of the larger problem of the protonation states in aqueous ensembles and their energy penalties. Accuracy and coverage is continually improved and expanded by parameterizing new systems of interest using DFT and experimental data. Recommendations are made for how to best incorporate tautomers in molecular design and virtual screening workflows.     

Multiple protonation equilibria in electrostatics of protein-protein binding

All proteins contain groups capable of exchanging protons with their environment. We present here an approach, based on a rigorous thermodynamic cycle and the partition functions for energy levels characterizing protonation states of the associating proteins and their complex, to compute the electrostatic pH-dependent contribution to the free energy of protein-protein binding. The computed electrostatic binding free energies include the pH of the solution as the variable of state, mutual "polarization" of associating proteins reflected as changes in the distribution of their protonation states upon binding and fluctuations between available protonation states. The only fixed property of both proteins is the conformation; the structure of the monomers is kept in the same conformation as they have in the complex structure. As a reference, we use the electrostatic binding free energies obtained from the traditional Poisson-Boltzmann model, computed for a single macromolecular conformation fixed in a given protonation state, appropriate for given solution conditions. The new approach was tested for 12 protein-protein complexes. It is shown that explicit inclusion of protonation degrees of freedom might lead to a substantially different estimation of the electrostatic contribution to the binding free energy than that based on the traditional Poisson-Boltzmann model. This has important implications for the balancing of different contributions to the energetics of protein-protein binding and other related problems, for example, the choice of protein models for Brownian dynamics simulations of their association. Our procedure can be generalized to include conformational degrees of freedom by combining it with molecular dynamics simulations at constant pH. Unfortunately, in practice, a prohibitive factor is an enormous requirement for computer time and power. However, there may be some hope for solving this problem by combining existing constant pH molecular dynamics algorithms with so-called accelerated molecular dynamics algorithms.     

Excited states of GFP chromophore and active site studied by the SAC-CI method: effect of protein-environment and mutations

Excited states of fluorescent proteins were studied using symmetry-adapted cluster-configuration interaction (SAC-CI) method. Protein-environmental effect on the excitation and fluorescence energies was investigated. In green fluorescent protein (GFP), the overall protein-environmental effect on the first excitation energy is not significant. However, glutamine (Glu) 94 and arginine (Arg96) have the red-shift contribution as reported in a previous study (Laino et al., Chem Phys 2004, 298, 17). The excited states of GFP active site (GFP-W22-Ser205-Glu222-Ser65) were also calculated. Such large-scale SAC-CI calculations were performed with an improved code containing a new algorithm for the perturbation selection. The SAC-CI results indicate that a charge-transfer state locates at 4.19 eV, which could be related to the channel of the photochemistry as indicated in a previous experimental study. We also studied the excitation and fluorescence energies of blue fluorescent protein, cyan fluorescent protein, and Y66F. The SAC-CI results are very close to the experimental ones. The protonation state of blue fluorescent protein was determined. Conformation of cyan fluorescent protein indicated by the present calculation agrees to the experimentally observed structure.     

Thermodynamic integration to predict host-guest binding affinities

An alchemical free energy method with explicit solvent molecular dynamics simulations was applied as part of the blind prediction contest SAMPL3 to calculate binding free energies for seven guests to an acyclic cucurbit-[n]uril host. The predictions included determination of protonation states for both host and guests, docking pose generation, and binding free energy calculations using thermodynamic integration. We found a root mean square error (RMSE) of 3.6 kcal mol(-1) from the reference experimental results, with an R(2) correlation of 0.51. The agreement with experiment for the largest contributor to this error, guest 6, is improved by 1.7 kcal mol(-1) when a periodicity-induced free energy correction is applied. The corrections for the other ligands were significantly smaller, and altogether the RMSE was reduced by 0.4 kcal mol(-1). We link properties of the host-guest systems during simulation to errors in the computed free energies. Overall, we show that charged host-guest systems studied here, initialized in unconfirmed docking poses, present a challenge to accurate alchemical simulation methods.     

Influence of the membrane potential on the protonation of bacteriorhodopsin: insights from electrostatic calculations into the regulation of proton pumping

Proton binding and release are elementary steps for the transfer of protons within proteins, which is a process that is crucial in biochemical catalysis and biological energy transduction. Local electric fields in proteins affect the proton binding energy compared to aqueous solution. In membrane proteins, also the membrane potential affects the local electrostatics and can thus be crucial for protein function. In this paper, we introduce a procedure to calculate the protonation probability of titratable sites of a membrane protein in the presence of a membrane potential. In the framework of continuum electrostatics, we use a modified Poisson-Boltzmann equation to include the influence of the membrane potential. Our method considers that in a transmembrane protein each titratable site is accessible for protons from only one side of the membrane depending on the hydrogen bond pattern of the protein. We show that the protonation of sites receiving their protons from different sides of the membrane is differently influenced by the membrane potential. In addition, the effect of the membrane potential is combined with the effect of the pH gradient resulting from proton pumping. Our method is applied to bacteriorhodopsin, a light-activated proton pump. We find that the proton pumping of this protein might be regulated by Asp115, a conserved residue for which no function has been identified yet. According to our calculations, the interaction of Asp115 with Asp85 leads to the protonation of the latter if the pH gradient or the membrane potential becomes too large. Since Asp85 is the primary proton acceptor in the photocycle, bacteriorhodopsin molecules in which Asp85 is protonated cannot pump protons. Furthermore, we estimate how the membrane potential affects the energetics of the individual proton-transfer reactions of the photocycle. Most reactions, except the initial proton transfer from the Schiff base to Asp85, are influenced. Our calculations give new insights into the mechanism with which bacteriorhodopsin senses the membrane potential and the pH gradient and how the proton pumping is regulated by these parameters.     

Ab initio structural study of some substituted ibuprofen derivatives as possible anti‐inflammatory agents

The formation energies of a series of substituted derivatives in α‐position of ibuprofen (2‐p‐isobutyl‐phenyl‐propionic acid) are determined, at the ab initio level RHF/6‐311G** with full geometry optimization, in their neutral and anionic forms and in the gas phase and water solution to correlate their physical–chemical properties with their anti‐inflammatory activity. Conformational calculations on the acidic moiety were also performed on five of them. The ab initio methods foresee that all these molecules present two preferred conformations in which the substituting atom in α‐position is lying approximately in the aromatic ring plane, in contrast with the results obtained with semiempirical methods. In this article, the protonation energy in solution, the solvation energy, the HOMO energy of the neutral form, and the lipophilicity will be considered as possible factors of anti‐inflammatory activity. The protonation energy in solution, together with the lipophilicity, are verified to be good activity factors: The smaller the protonation energy and the lipophilicity, the larger the anti‐inflammatory activity. In contrast, the larger the solvation energy, the smaller the activity. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem, 2004     

Dead-end elimination for multistate protein design

Multistate protein design is the task of predicting the amino acid sequence that is best suited to selectively and stably fold to one state out of a set of competing structures. Computationally, it entails solving a challenging optimization problem. Therefore, notwithstanding the increased interest in multistate design, the only implementations reported are based on either genetic algorithms or Monte Carlo methods. The dead-end elimination (DEE) theorem cannot be readily transfered to multistate design problems despite its successful application to single-state protein design. In this article we propose a variant of the standard DEE, called type-dependent DEE. Our method reduces the size of the conformational space of the multistate design problem, while provably preserving the minimal energy conformational assignment for any choice of amino acid sequence. Type-dependent DEE can therefore be used as a preprocessing step in any computational multistate design scheme. We demonstrate the applicability of type-dependent DEE on a set of multistate design problems and discuss its strength and limitations.     

Quantum algorithm for obtaining the energy spectrum of molecular systems

Simulating a quantum system is more efficient on a quantum computer than on a classical computer. The time required for solving the Schr?dinger equation to obtain molecular energies has been demonstrated to scale polynomially with system size on a quantum computer, in contrast to the well-known result of exponential scaling on a classical computer. In this paper, we present a quantum algorithm to obtain the energy spectrum of molecular systems based on the multiconfigurational self-consistent field (MCSCF) wave function. By using a MCSCF wave function as the initial guess, the excited states are accessible. Entire potential energy surfaces of molecules can be studied more efficiently than if the simpler Hartree-Fock guess was employed. We show that a small increase of the MCSCF space can dramatically increase the success probability of the quantum algorithm, even in regions of the potential energy surface that are far from the equilibrium geometry. For the treatment of larger systems, a multi-reference configuration interaction approach is suggested. We demonstrate that such an algorithm can be used to obtain the energy spectrum of the water molecule.     

Effect of methylation on relative energies of tautomers and on the intramolecular proton transfer barriers of protonated nitrosamine: A MR‐CISD study

MR‐CISD, MR‐CISD+Q, and MR‐AQCC calculations have been performed on the minima and transition states (corresponding to intramolecular proton transfer between the protonation sites) of the ground state of protonated nitrosamine and N,N‐dimethylnitrosamine. Our highest level results (MR‐AQCC/cc‐pVTZ) for the smaller system indicate that protonation on the N amino ( 2a ) is practically as favorable as the most favorable protonation on the O atom ( 1a ). They also suggest that protonation on the nitroso N atom ( 2c ) is ～14.5 kcal/mol less favorable than 1a . Results obtained at the MR‐CISD+Q/cc‐pVTZ level indicate that the effect of methylation on the relative energies of the tautomers is, in order of importance, 2a > 2c and increases their energies by ～17.5 and 4.8 kcal/mol, respectively. They also indicate that methylation alters significantly the intramolecular proton transfer barriers. The largest differences between the common geometric parameters of both systems have been found for 2a . © 2015 Wiley Periodicals, Inc.     

Prediction of pKa shifts in proteins using a combination of molecular mechanical and continuum solvent calculations

The prediction of pKa shifts of ionizable groups in proteins is of great relevance for a number of important biological phenomena. We present an implementation of the MM-GBSA approach, which combines molecular mechanical (MM) and generalized Born (GB) continuum solvent energy terms, to the calculation of pKa values of a panel of nine proteins, including 69 individual comparisons with experiment. While applied so far mainly to the calculation of biomolecular binding free energies, we show that this method can also be used for the estimation of protein pKa shifts, with an accuracy around 1 pKa unit, even for strongly shifted residues. Our analysis reveals that the nonelectrostatic terms that are part of the MM-GBSA free energy expression are important contributors to improved prediction accuracy. This suggests that most of the previous approaches that focus only on electrostatic interactions could be improved by adding other nonpolar energy terms to their free energy expression. Interestingly, our method yields best accuracy at protein dielectric constants of epsilonint = 2-4, which is in contrast to previous approaches that peak at higher epsilonint > or = 8. An important component of our procedure is an intermediate minimization step of each protonation state involving different rotamers and tautomers as a way to explicitly model protein relaxation upon (de)protonation.     

Efficient combination of Wang-Landau and transition matrix Monte Carlo methods for protein simulations

An efficient combination of the Wang-Landau and transition matrix Monte Carlo methods for protein and peptide simulations is described. At the initial stage of simulation the algorithm behaves like the Wang-Landau algorithm, allowing to sample the entire interval of energies, and at the later stages, it behaves like transition matrix Monte Carlo method and has significantly lower statistical errors. This combination allows to achieve fast convergence to the correct values of density of states. We propose that the violation of TTT identities may serve as a qualitative criterion to check the convergence of density of states. The simulation process can be parallelized by cutting the entire interval of simulation into subintervals. The violation of ergodicity in this case is discussed. We test the algorithm on a set of peptides of different lengths and observe good statistical convergent properties for the density of states. We believe that the method is of general nature and can be used for simulations of other systems with either discrete or continuous energy spectrum.     

Structures, spectra, and energies of niobium clusters from Nb13 to Nb20

A comprehensive theoretical investigation on structures and properties of niobium clusters in the range from 13 to 20 atoms, in three different charged states, is performed by using the BPW91 and M06 functionals and the cc-pVDZ-PP basis set. These species are predicted to prefer low spin ground state, i.e., singlet (for even electron) and doublet (for odd electron) systems. In terms of growth mechanism, a compact structure with one Nb encapsulated by a cage formed from five and six triangles is found to be favored over an icosahedral evolution. Unlike many 3d metals, whose volumes are much smaller, 13 and 19 Nb atoms clusters do not exist as icosahedra and double-icosahedra. A distinct case is Nb(15) as it bears a slightly distorted bcc structure. For some systems, several lower lying isomers are computed to be so close in energy that DFT computations cannot clearly establish their ground electronic states. The existence of structural isomers with comparable energy content is established for Nb(n) species with n = 13, 18, 19, and 20 in both neutral and charged states. The vibrational (IR) spectra are also calculated. While the spectra of smaller systems are strongly dependent on addition or removal of an electron from the neutral, the spectra of the larger size clusters are mostly independent of the charged state. The neutrals and their corresponding ions usually have a quite similar IR pattern. Electron affinities (EA), ionization energies (IE), average binding energies, dissociation energies, and frontier orbital energy gaps are evaluated. The computed EAs and IEs are generally in fair agreement with experiment. The Nb(15) system is observed to be stable and it can form a highly symmetric structure in all charged states with both open and closed electron shells.     

Rotamer optimization for protein design through MAP estimation and problem‐size reduction

The search for the global minimum energy conformation (GMEC) of protein side chains is an important computational challenge in protein structure prediction and design. Using rotamer models, the problem is formulated as a NP‐hard optimization problem. Dead‐end elimination (DEE) methods combined with systematic A* search (DEE/A*) has proven useful, but may not be strong enough as we attempt to solve protein design problems where a large number of similar rotamers is eligible and the network of interactions between residues is dense. In this work, we present an exact solution method, named BroMAP (branch‐and‐bound rotamer optimization using MAP estimation), for such protein design problems. The design goal of BroMAP is to be able to expand smaller search trees than conventional branch‐and‐bound methods while performing only a moderate amount of computation in each node, thereby reducing the total running time. To achieve that, BroMAP attempts reduction of the problem size within each node through DEE and elimination by lower bounds from approximate maximum‐a‐posteriori (MAP) estimation. The lower bounds are also exploited in branching and subproblem selection for fast discovery of strong upper bounds. Our computational results show that BroMAP tends to be faster than DEE/A* for large protein design cases. BroMAP also solved cases that were not solved by DEE/A* within the maximum allowed time, and did not incur significant disadvantage for cases where DEE/A* performed well. Therefore, BroMAP is particularly applicable to large protein design problems where DEE/A* struggles and can also substitute for DEE/A* in general GMEC search. © 2009 Wiley Periodicals, Inc. J Comput Chem, 2009     

Photo absorption of ‐coumaric acid in aqueous solution: RISM‐SCF‐SEDD theory approach

Photo absorption properties of p‐coumaric acid, the chromophore of photoactive yellow protein, in aqueous solution were investigated by means of reference interaction site model self‐consistent field with spatial electron density distribution (RISM‐SCF‐SEDD) method. RISM‐SCF‐SEDD is a combination methodology of electronic structure theory and statistical mechanics for molecular liquids. Here, time‐dependent density functional theory was coupled with RISM equation to study the electronic structure of p‐coumaric acid in aqueous system. Excitation energies of the chromophore in its neutral, two monoanionic and dianionic forms were computed to elucidate the effect of the deprotonation and solvation on the spectroscopic properties. We found that solvation strongly affects the excitation character of the chromophore, especially for phenolate anion and dianion. The free energy difference among the four protonation states is also discussed. © 2017 Wiley Periodicals, Inc.     

Investigation of the S<Subscript>0</Subscript>S<Subscript>1</Subscript> excitation in bacteriorhodopsin with the ONIOM(MO:MM) hybrid method

 We have investigated the S₀ and S₁ electronic states in bacteriorhodopsin using a variety of QM/MM levels. The decomposition of the calculated excitation energies into electronic and electrostatic components shows that the interaction of the chromophore with the protein electric field increases the excitation energy, while polarization effects are negligible. Therefore, the experimentally observed reduction in excitation energy from solution phase to protein environment (the Opsin shift) does not come from the electrostatic interaction with the protein environment, but from either the interaction ofthe chromophore with the solvent or counter ion, or structural effects. Our high-level ONIOM(TD– B3LYP:Amber) calculation predicts the excitation energy within 8 kcal/mol from experiment, the discrepancy probably being caused by the neglect of polarization of the protein environment. In addition, we have shown that the level of optimization is extremely critical for the calculation of accurate excitation energies in bacteriorhodopsin. Received: 13 October 2001 / Accepted: 6 September 2002 / Published online: 3 February 2003 Contribution to the Proceedings of the Symposium on Combined QM/MM Methods at the 222nd National Meeting of the American Chemical Society, 2001 Correspondence to: K. Morokuma e-mail: morokuma@emory.edu     

On low‐lying excited states of extended nanographenes

Low‐lying excited states of planarly extended nanographenes are investigated using the long‐range corrected (LC) density functional theory (DFT) and the spin‐flip (SF) time‐dependent density functional theory (TDDFT) by exploring the long‐range exchange and double‐excitation correlation effects on the excitation energies, band gaps, and exciton binding energies. Optimizing the geometries of the nanographenes indicates that the long‐range exchange interaction significantly improves the C C bond lengths and amplify their bond length alternations with overall shortening the bond lengths. The calculated TDDFT excitation energies show that long‐range exchange interaction is crucial to provide accurate excitation energies of small nanographenes and dominate the exciton binding energies in the excited states of nanographenes. It is, however, also found that the present long‐range correction may cause the overestimation of the excitation energy for the infinitely wide graphene due to the discrepancy between the calculated band gaps and vertical ionization potential (IP) minus electron affinity (EA) values. Contrasting to the long‐range exchange effects, the SF‐TDDFT calculations show that the double‐excitation correlation effects are negligible in the low‐lying excitations of nanographenes, although this effect is large in the lowest excitation of benzene molecule. It is, therefore, concluded that long‐range exchange interactions should be incorporated in TDDFT calculations to quantitatively investigate the excited states of graphenes, although TDDFT using a present LC functional may provide a considerable excitation energy for the infinitely wide graphene mainly due to the discrepancy between the calculated band gaps and IP–EA values. © 2017 Wiley Periodicals, Inc.     

Charge Movements in the 13-c/s Photocycles of the Bacteriorhodopsin Mutants R82K and R82Q*

We have examined light-induced currents in oriented membranes of the bacteriorhodopsin mutants R82K and R82Q. Our results suggest that two photocurrent components found in R82K, with 30 and 300 us lifetimes, are due to the photocycle of the 13-cis rather than the all-trans form of the pigment. We investigated the pH dependence of these components and their correspondence to absorbance changes at 660 nm characteristic of pho-tointermediates of the 13-cis cycle. The presence of a D₂O effect suggests that the charge motions producing these photocurrents are related to proton or protonated amino acid movement within the molecule. The current amplitudes depend on the protonation states of at least two residues, D85 and (probably) E204. In R82Q, a 10 pis photocurrent is observed that also depends on the protonation state of D85 and is similar to the 30 us current in R82K. We attempt to explain these currents in terms of a model for interacting residues in the extracellular half of the bacteriorhodopsin channel.     

Determination of chromophore charge states in the low pH color transition of the fluorescent protein Rtms5 via time-dependent DFT

The red fluorescent protein Rtms5^H146S displays a transition from blue (absorbance λ_max 590 nm) to yellow (absorbance λ_max 453 nm) upon titration to low pH. The pK_a of the reaction depends on the concentration of halide, offering promise for new expressible halide sensors. The protonation state involved in the low pH form of the chromophore remains, however, ambiguous. We report calculated excitation energies of different protonation states of an RFP chromophore model. These suggest that the relevant titration site is the phenoxy moiety of the chromophore, and the relevant base and conjugate acid are anionic and neutral chromophore species, respectively.