New Theoretical Methodology for Elucidating the Solution Structure of Peptides from NMR Data. 3. Solvation Effects

Exploring Conformational Preferences of Leu-enkephalin Using the Conformational Search and Double-Hybrid DFT Energy Calculations

The conformational preferences of Leu-enkephalin (Leu-Enk) were explored by the conformational search and density functional theory (DFT) calculations. By a combination of low-energy conformers of each residue, the initial structures of the neutral Leu-Enk were generated and optimized using the ECEPP3 force field in the gas phase. These structures were reoptimized at the HF/3-21G(d) and M06-2X levels of theory with 6-31G(d) and 6-31+G(d) basis functions. We finally located the 139 structures with the relative energy <10 kcal mol-1 in the gas phase, from which the structures of the corresponding zwitterionic Leu-Enk were generated and reoptimized at the M06-2X/6-31+G(d) level of theory using the implicit solvation model based on density (SMD) in water. The conformational preferences of Leu-Enk were analyzed using Gibbs free energies corrected by single-point energies calculated at the double-hybrid DSD-PBEP86-D3BJ/def2-TZVP level of theory in the gas phase and in water. The neutral Leu-Enk dominantly adopted a folded structure in the gas phase stabilized by three H-bonds with a βII'-bend-like motif at the Gly3-Phe4 sequence and a close contact between the side chains of Phe4 and Leu5. The zwitterionic Leu-Enk exhibited a folded structure in water stabilized by three H-bonds with double β-bends such as a βII' bend at the Gly2-Gly3 sequence and a βI bend at the Gly3-Phe4 sequence. The calculated ensemble-averaged distance between CGly2 α and CLeu5 α of the zwitterionic Leu-Enk in water is consistent with the value estimated from the simulated annealing using the distance constraints derived from nuclear Overhauser effect spectroscopy (NOESY) spectra in water. Interestingly, the preferred conformations of the neutral and zwitterionic Leu-Enk are new folded structures not predicted by earlier computational studies. According to the refined model of the zwitterionic Leu-Enk bound to δ-opioid receptor (δOR), there were favorable interactions of the terminal charged groups of Leu-Enk with the side chains of charged residues of δOR as well as a favorable CAryl···H interaction of the Phe4 residue of Leu-Enk with Trp284 of δOR. Hence, these favorable interactions would induce the folded structure of the zwitterionic Leu-Enk with double β-bends isolated in water into the "bioactive conformation" like an extended structure when binding to δOR.

Conformations of phenylalanine in the tripeptides AFA and GFG probed by combining MD simulations with NMR, FTIR, polarized Raman, and VCD spectroscopy

Conformational properties of small, flexible peptides are a matter of ongoing interest since they can be considered as models for unfolded proteins. However, the investigation of the conformations of small peptides is challenging as they are ensembles of rapidly interconverting conformers; moreover, the different methods used are prone to different approximations and errors. In order to obtain more reliable results, it is prudent to combine different techniques; here, molecular dynamics (MD) simulations together with nuclear magnetic resonance (NMR), Fourier transform IR (FTIR), polarized Raman, and vibrational circular dichroism (VCD) measurements were used to study the conformational propensity of phenylalanine in the tripeptides AFA and GFG, motivated by the relevance of phenylalanine for the self-aggregation of peptides. The results of this analysis indicate that the F residue predominantly populates the beta-strand (beta) and polyproline II (PPII) conformations in both AFA and GFG. However, while phenylalanine exhibits a propensity for beta-strand conformations in GFG (0.40 < or = beta population < or = 0.69 and 0.29 < or = PPII population < or = 0.42), the substitution of terminal glycines with alanine residues induces a higher population of PPII (0.31 < or = beta population < or = 0.50 and 0.37 < or = PPII population < or = 0.57).

Methods for calculating the absolute entropy and free energy of biological systems based on ideas from polymer physics

The commonly used simulation techniques, Metropolis Monte Carlo (MC) and molecular dynamics (MD) are of a dynamical type which enables one to sample system configurations i correctly with the Boltzmann probability, P(i)(B), while the value of P(i)(B) is not provided directly; therefore, it is difficult to obtain the absolute entropy, S approximately -ln P(i)(B), and the Helmholtz free energy, F. With a different simulation approach developed in polymer physics, a chain is grown step-by-step with transition probabilities (TPs), and thus their product is the value of the construction probability; therefore, the entropy is known. Because all exact simulation methods are equivalent, i.e. they lead to the same averages and fluctuations of physical properties, one can treat an MC or MD sample as if its members have rather been generated step-by-step. Thus, each configuration i of the sample can be reconstructed (from nothing) by calculating the TPs with which it could have been constructed. This idea applies also to bulk systems such as fluids or magnets. This approach has led earlier to the "local states" (LS) and the "hypothetical scanning" (HS) methods, which are approximate in nature. A recent development is the hypothetical scanning Monte Carlo (HSMC) (or molecular dynamics, HSMD) method which is based on stochastic TPs where all interactions are taken into account. In this respect, HSMC(D) can be viewed as exact and the only approximation involved is due to insufficient MC(MD) sampling for calculating the TPs. The validity of HSMC has been established by applying it first to liquid argon, TIP3P water, self-avoiding walks (SAW), and polyglycine models, where the results for F were found to agree with those obtained by other methods. Subsequently, HSMD was applied to mobile loops of the enzymes porcine pancreatic alpha-amylase and acetylcholinesterase in explicit water, where the difference in F between the bound and free states of the loop was calculated. Currently, HSMD is being extended for calculating the absolute and relative free energies of ligand-enzyme binding. We describe the whole approach and discuss future directions.

Conformational heterogeneity of a leucine enkephalin analogue in aqueous solution and sodium dodecyl sulfate micelles: comparison of time-resolved FRET and molecular dynamics simulations

We have undertaken time-resolved Förster resonance energy transfer (FRET) and molecular dynamics simulations to analyze conformations and conformational heterogeneity of an analogue of leucine enkephalin in solution and in the presence of sodium dodecyl sulfate (SDS) micelles. Enkephalins are opioid pentapeptides that interact with opioid receptors in the central nervous system. We used time-correlated single-photon counting to detect energy transfer between the N-terminal tyrosine and a tryptophan residue substituted for phenylalanine at the 4 position. FRET from Tyr to Trp was measured over a temperature range from 5 to 55 degrees C in aqueous solution. By taking into account Tyr rotamer interconversion rates measured previously, we determined average distances between Tyr and Trp for the two populated rotameric conformations of Tyr. Molecular dynamics simulations (100 ns) support this analysis and indicate extensive conformational heterogeneity. The simulations also predict that the FRET orientational factor is correlated with the Tyr-Trp separation. Failure to account for the correlation between orientation and distance results in errors that appear to be largely offset in the leucine enkephalin analogue (YGGWL) by a weighting bias inherent in the R(-6) dependence of the energy-transfer rate. The Tyr lifetimes decrease upon titration of the peptides with SDS, indicating formation of compact conformations of the peptide in the micelle environment. This result is consistent with the conjecture that the lipid environment may induce formation of bioactive conformations of the peptide.

Entropy and free energy of a mobile protein loop in explicit water

Estimation of the energy from a given Boltzmann sample is straightforward since one just has to average the contribution of the individual configurations. On the other hand, calculation of the absolute entropy, S (hence the absolute free energy F) is difficult because it depends on the entire (unknown) ensemble. We have developed a new method called "the hypothetical scanning molecular dynamics" (HSMD) for calculating the absolute S from a given sample (generated by any simulation technique). In other words, S (like the energy) is "written" on the sample configurations, where HSMD provides a prescription of how to "read" it. In practice, each sample conformation, i, is reconstructed with transition probabilities, and their product leads to the probability of i, hence to the entropy. HSMD is an exact method where all interactions are considered, and the only approximation is due to insufficient sampling. In previous studies HSMD (and HS Monte CarloHSMC) has been extended systematically to systems of increasing complexity, where the most recent is the seven-residue mobile loop, 304-310 (Gly-His-Gly-Ala-Gly-Gly-Ser) of the enzyme porcine pancreatic alpha-amylase modeled by the AMBER force field and AMBER with the implicit solvation GB/SA (paper I, Cheluvaraja, S.; Meirovitch, H. J. Chem. Theory Comput. 2008, 4, 192). In the present paper we make a step further and extend HSMD to the same loop capped with TIP3P explicit water at 300 K. As in paper I, we are mainly interested in entropy and free energy differences between the free and bound microstates of the loop, which are obtained from two separate MD samples of these microstates. The contribution of the loop to S and F is calculated by HSMD and that of water by a particular thermodynamic integration procedure. As expected, the free microstate is more stable than the bound microstate by a total free energy difference, Ffree-Fbound=-4.8+/-1, as compared to -25.5 kcal/mol obtained with GB/SA. We find that relatively large systematic errors in the loop entropies, Sfree(loop) and Sbound(loop) are cancelled in their difference which is thus obtained efficiently and with high accuracy, i.e., with a statistical error of 0.1 kcal/mol. This cancellation, which has been observed in previous HSMD studies, is in accord with theoretical arguments given in paper I.

Calculation of the entropy and free energy of peptides by molecular dynamics simulations using the hypothetical scanning molecular dynamics method

Hypothetical scanning (HS) is a method for calculating the absolute entropy S and free energy F from a sample generated by any simulation technique. With this approach each sample configuration is reconstructed with the help of transition probabilities (TPs) and their product leads to the configuration's probability, hence to the entropy. Recently a new way for calculating the TPs by Monte Carlo (MC) simulations has been suggested, where all system interactions are taken into account. Therefore, this method--called HSMC--is in principle exact where the only approximation is due to insufficient sampling. HSMC has been applied very successfully to liquid argon, TIP3P water, self-avoiding walks on a lattice, and peptides. Because molecular dynamics (MD) is considered to be significantly more efficient than MC for a compact polymer chain, in this paper HSMC is extended to MD simulations as applied to peptides. Like before, we study decaglycine in vacuum but for the first time also a peptide with side chains, (Val)(2)(Gly)(6)(Val)(2). The transition from MC to MD requires implementing essential changes in the reconstruction process of HSMD. Results are calculated for three microstates, helix, extended, and hairpin. HSMD leads to very stable differences in entropy TDeltaS between these microstates with small errors of 0.1-0.2 kcal/mol (T=100 K) for a wide range of calculation parameters with extremely high efficiency. Various aspects of HSMD and plans for future work are discussed.

Calculation of the entropy and free energy by the hypothetical scanning Monte Carlo method: application to peptides

A new approach, the hypothetical scanning Monte Carlo (HSMC), for calculating the absolute entropy, S, and free energy, F, has been introduced recently and applied first to fluids (argon and water) and later to peptides. In this paper the method is further developed for peptide chains in vacuum. S is calculated from a given MC sample by reconstructing each sample conformation i step-by-step, i.e., calculating transition probabilities (TPs) for the dihedral and bond angles and fixing the related atoms at their positions. At step k of the process the chain's coordinates that have already been determined are kept fixed (the "frozen past") and TP(k) is obtained from a MC simulation of the "future" part of the chain whose TPs as yet have not been determined; when the process is completed the contribution of conformation i to the entropy is, S(i) approximately -ln Pi(k) TP(k). In a recent paper we studied polyglycine chains, modeled by the AMBER force field with constant bond lengths and bond angles (the rigid model). Decaglycine [(Gly)(10)] was studied in the helical, extended, and hairpin microstates, while (Gly)(16) was treated only in the first two microstates. In this paper the samples are increased and restudied, (Gly)(16) is also investigated in the hairpin microstate, and for (Gly)(10) approximations are tested where only part of the future is considered for calculating the TPs. We calculate upper and lower bounds for F and demonstrate that like for fluids, F can be obtained from multiple reconstructions of a single conformation. We also test a more realistic model of (Gly)(10) where the bond angles are allowed to move (the flexible model). Very accurate results for S and F are obtained which are compared to results obtained by the quasiharmonic approximation and the local states method. Thus, differences in entropy and free energy between the three microstates are obtained within errors of 0.1-0.3 kcal/mol. The HSMC method can be applied to a macromolecule with any degree of flexibility, ranging from local fluctuations to a random coil. The present results demonstrate that the difference in stability, DeltaF(mn)=F(m)-F(n) between significantly different microstates m and n, can be obtained from two simulations only without the need to resort to thermodynamic integration. Our long-term goal is to extend this method to any peptide and apply it to a peptide immersed in a box with explicit water.

Optimization of the GB/SA solvation model for predicting the structure of surface loops in proteins

Implicit solvation models are commonly optimized with respect to experimental data or Poisson-Boltzmann (PB) results obtained for small molecules, where the force field is sometimes not considered. In previous studies, we have developed an optimization procedure for cyclic peptides and surface loops in proteins based on the entire system studied and the specific force field used. Thus, the loop has been modeled by the simplified solvation function E(tot) = E(FF) (epsilon = 2r) + Sigma(i) sigma(i)A(i), where E(FF) (epsilon = nr) is the AMBER force field energy with a distance-dependent dielectric function, epsilon = nr, A(i) is the solvent accessible surface area of atom i, and sigma(i) is its atomic solvation parameter. During the optimization process, the loop is free to move while the protein template is held fixed in its X-ray structure. To improve on the results of this model, in the present work we apply our optimization procedure to the physically more rigorous solvation model, the generalized Born with surface area (GB/SA) (together with the all-atom AMBER force field) as suggested by Still and co-workers (J. Phys. Chem. A 1997, 101, 3005). The six parameters of the GB/SA model, namely, P(1)-P(5) and the surface area parameter, sigma (programmed in the TINKER package) are reoptimized for a "training" group of nine loops, and a best-fit set is defined from the individual sets of optimized parameters. The best-fit set and Still's original set of parameters (where Lys, Arg, His, Glu, and Asp are charged or neutralized) were applied to the training group as well as to a "test" group of seven loops, and the energy gaps and the corresponding RMSD values were calculated. These GB/SA results based on the three sets of parameters have been found to be comparable; surprisingly, however, they are somewhat inferior (e.g, of larger energy gaps) to those obtained previously from the simplified model described above. We discuss recent results for loops obtained by other solvation models and potential directions for future studies.

Simulation method for calculating the entropy and free energy of peptides and proteins

A method called complete hypothetical scanning Monte Carlo has been introduced for calculating the absolute entropy, S, and free energy, F, of fluids. Here, the method is extended to peptide chains in vacuum. Thus, S is calculated from a given sample by reconstructing each conformation step-by-step by using transition probabilities (TPs); at each step, part of the chain coordinates have already been determined (the "frozen past"), and the TP is obtained from a Monte Carlo simulation of the (future) part of the chain whose TPs as yet have not been calculated. Very accurate results for S and F are obtained for the helix, extended, and hairpin microstates of a simplified model of decaglycine (Gly)(10) and (Gly)(16). These results agree well with results obtained by the quasiharmonic approximation and the local states method. The complete HSMC method can be applied to a macromolecule with any degree of flexibility, ranging from local fluctuations to a random coil. Also, the difference in stability, Delta F(mn) = F(m) - F(n) between significantly different microstates m and n can be obtained from two simulations only without the need to resort to thermodynamic integration. Our long-term goal is to extend this method to any peptide and apply it to a peptide immersed in a box with explicit water.

Conformational search of peptides and proteins: Monte Carlo minimization with an adaptive bias method applied to the heptapeptide deltorphin

The energy function of a protein consists of a tremendous number of minima. Locating the global energy minimum (GEM) structure, which corresponds approximately to the native structure, is a severe problem in global optimization. Recently we have proposed a conformational search technique based on the Monte Carlo minimization (MCM) method of Li and Scheraga, where trial dihedral angles are not selected at random within the range [-180 degrees,180 degrees ] (as with MCM) but with biased probabilities depending on the increased structure-energy correlations as the GEM is approached during the search. This method, called the Monte Carlo minimization with an adaptive bias (MCMAB), was applied initially to the pentapeptide Leu-enkephalin. Here we study its properties further by applying it to the larger peptide with bulky side chains, deltorphin (H-Tyr-D-Met-Phe-His-Leu-Met-Asp-NH(2)). We find that on average the number of energy minimizations required by MCMAB to locate the GEM for the first time is smaller by a factor of approximately three than the number required by MCM-in accord with results obtained for Leu-enkephalin.

Efficiency of the multicanonical simulation method as applied to peptides of increasing size: the heptapeptide deltorphin

The advantage of the multicanonical (MUCA) simulation method of Berg and coworkers over the conventional Metropolis method is in its ability to move a system effectively across energy barriers thereby providing results for a wide range of temperatures. However, a MUCA simulation is based on weights (related to the density of states) that should be determined prior to a production run and their calculation is not straightforward. To overcome this difficulty a procedure has been developed by Berg that calculates the MUCA weights automatically. In a previous article (Yaşar et al. J Comput Chem 2000, 14, 1251-1261) we extended this procedure to continuous systems and applied it successfully to the small pentapeptide Leu-enkephalin. To investigate the performance of the automated MUCA procedure for larger peptides, we apply it here to deltorphin, a linear heptapeptide with bulky side chains (H-Tyr(1)-D-Met(2)-Phe(3)-His(4)-Leu(5)-Met(6)-Asp(7)-NH(2)). As for Leu-enkephalin, deltorphin is modeled in vacuum by the potential energy function ECEPP. MUCA is found to perform well. A weak second peak is seen for the specific heat, which is given a special attention. By minimizing the energy of structures along the trajectory it is found that MUCA provides a good conformational coverage of the low energy region of the molecule. These latter results are compared with conformational coverage obtained by the Monte Carlo minimization method of Li and Scheraga.

Optimization of solvation models for predicting the structure of surface loops in proteins

A novel procedure for optimizing the atomic solvation parameters (ASPs) sigma(i) developed recently for cyclic peptides is extended to surface loops in proteins. The loop is free to move, whereas the protein template is held fixed in its X-ray structure. The energy is E(tot) = E(FF)(epsilon = nr) + summation operator sigma(i)A(i), where E(FF)(epsilon = nr) is the force-field energy of the loop-loop and loop-template interactions, epsilon = nr is a distance-dependent dielectric constant, and n is an additional parameter to be optimized. A(i) is the solvent-accessible surface area of atom i. The optimal sigma(i) and n are those for which the loop structure with the global minimum of E(tot)(n, sigma(i)) becomes the experimental X-ray structure. Thus, the ASPs depend on the force field and are optimized in the protein environment, unlike commonly used ASPs such as those of Wesson and Eisenberg (Protein Sci 1992;1:227-235). The latter are based on the free energy of transfer of small molecules from the gas phase to water and have been traditionally combined with various force fields without further calibration. We found that for loops the all-atom AMBER force field performed better than OPLS and CHARMM22. Two sets of ASPs [based on AMBER (n = 2)], optimized independently for loops 64-71 and 89-97 of ribonuclease A, were similar and thus enabled the definition of a best-fit set. All these ASPs were negative (hydrophilic), including those for carbon. Very good (i.e., small) root-mean-square-deviation values from the X-ray loop structure were obtained with the three sets of ASPs, suggesting that the best-fit set would be transferable to loops in other proteins as well. The structure of loop 13-24 is relatively stretched and was insensitive to the effect of the ASPs.

On the transferability of atomic solvation parameters: Ab initio structural prediction of cyclic heptapeptides in DMSO

A statistical mechanics methodology for predicting the solution structures and populations of peptides developed recently is based on a novel method for optimizing implicit solvation models, which was applied initially to a cyclic hexapeptide in DMSO (C. Baysal and H. Meirovitch, Journal of American Chemical Society, 1998, vol. 120, pp. 800-812). Thus, the molecule has been described by the simplified energy function E(tot) = E(GRO) + summation operator(k) sigma(k)A(k), where E(GRO) is the GROMOS force-field energy, sigma(k) and A(k) are the atomic solvation parameter (ASP) and the solvent accessible surface area of atom k, respectively. In a more recent study, these ASPs have been found to be transferable to the cyclic pentapeptide cyclo(D-Pro(1)-Ala(2)-Ala(3)-Ala(4)-Ala(5)) in DMSO (C. Baysal and H. Meirovitch, Biopolymers, 2000, vol. 53, pp. 423-433). In the present paper, our methodology is applied to the cyclic heptapeptides axinastatin 2 [cyclo(Asn(1)-Pro(2)-Phe(3)-Val(4)-Leu(5)-Pro(6)-Val(7))] and axinastatin 3 [cyclo(Asn(1)-Pro(2)-Phe(3)-Ile(4)-Leu(5)-Pro(6)-Val(7))], in DMSO, which were studied by nmr by Mechnich et al. (Helvetica Chimica Acta, 1997, vol. 80, pp. 1338-1354). The calculations for axinastatin 2 show that special ASPs should be optimized for the partially charged side-chain atoms of Asn while the rest of the atoms take their values derived in our previous work; this suggests that similar optimization might be needed for other side chains as well. The solution structures of these peptides are obtained ab initio (i.e., without using experimental restraints) by an extensive conformational search based on E(GRO) alone and E(*)(tot), which consists of the new set of ASPs. For E(*)(tot), the theoretical values of proton-proton distances, (3)J coupling constants, and other properties are found to agree very well with the nmr results, and they are always better than those based on E(GRO).

Ab initio prediction of the solution structures and populations of a cyclic pentapeptide in DMSO based on an implicit solvation model

Using a recently developed statistical mechanics methodology, the solution structures and populations of the cyclic pentapeptide cyclo(D-Pro(1)-Ala(2)-Ala(3)-Ala(4)-Ala(5)) in DMSO are obtained ab initio, i.e., without using experimental restraints. An important ingredient of this methodology is a novel optimization of implicit solvation parameters, which in our previous publication [Baysal, C.; Meirovitch, H. J Am Chem Soc 1998, 120, 800-812] has been applied to a cyclic hexapeptide in DMSO. The molecule has been described by the simplified energy function E(tot) = E(GRO) + summation operator(k) sigma(k)A(k), where E(GRO) is the GROMOS force-field energy, sigma(k) and A(k) are the atomic solvation parameter (ASP) and the solvent accessible surface area of atom k. This methodology, which relies on an extensive conformational search, Monte Carlo simulations, and free energy calculations, is applied here with E(tot) based on the ASPs derived in our previous work, and for comparison also with E(GRO) alone. For both models, entropy effects are found to be significant. For E(tot), the theoretical values of proton-proton distances and (3)J coupling constants agree very well with the NMR results [Mierke, D. F.; Kurz, M.; Kessler, H. J Am Chem Soc 1994, 116, 1042-1049], while the results for E(GRO) are significantly worse. This suggests that our ASPs might be transferrable to other cyclic peptides in DMSO as well, making our methodology a reliable tool for an ab initio structure prediction; obviously, if necessary, parts of this methodology can also be incorporated in a best-fit analysis where experimental restraints are used.

Free energy based populations of interconverting microstates of a cyclic peptide lead to the experimental NMR data

Analysis of nuclear Overhauser enhancement (NOE) intensities data of interconverting microstates of a peptide is a difficult problem in nmr. A new statistical mechanics methodology has been proposed recently, consisting of several steps: (1) potential energy wells on the energy surface of the molecule are identified (the corresponding regions are called wide microstates); (2) each wide microstate is then spanned by a Monte Carlo (MC) or molecular dynamics simulation starting from a representative structure, and the corresponding relative populations are obtained from the free energy calculated with the local states method; and (3) the overall NOEs and 3J coupling constants are obtained as averages over the corresponding contributions of the samples, weighted by the populations. Extending this methodology to cyclic peptides, we are treating here the hexapeptide cyclo(D-Pro1-Phe2-Ala3-Ser4-Phe5-Phe6) in DMSO, which was studied by Kessler et al. using nmr (Journal of the American Chemical Society, 1992, Vol. 114, pp. 4805-4818). They found that at least two structures are required to explain their NOE data, a conclusion also corroborated by our analysis (Journal of the American Chemical Society, 1998, Vol. 120, pp. 800-812) and led to a novel derivation of atomic solvation parameters (ASPs) for DMSO. Thus, the overall interactions within the peptide-solvent system are described approximately by Etot = EGRO + summation operator sigmaiAi, where EGRO is the energy of the GROMOS force field, Ai is the solvent-accessible surface area of atom i, and sigmai is the ASP. In the present paper the validity of these ASPs within the framework of the entire methodology is verified. This requires taking into account 23 microstates. A very good agreement is obtained between experimental and calculated NOEs and 3J coupling constants. The free energy based populations lead to the best results, which means that entropic effects should not be ignored. We have also studied the behavior of the internal angular fluctuations of the proton-proton vectors and discovered that they have a negligible effect on the calculated NOEs; this is due to the relatively concentrated wide microstates spanned by the MC simulations. The applicability of our ASPs to other cyclic peptides in DMSO is being studied in another work and preliminary results are discussed.