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).

New theoretical methodology for elucidating the solution structure of peptides from NMR data. II. Free energy of dominant microstates of Leu-enkephalin and population-weighted average nuclear Overhauser effects intensities.

A small linear peptide in solution may populate several stable states (called here microstates) in thermodynamic equilibrium; elucidating its dynamic three dimensional structure by multi- dimensional nmr is complex since the experimentally measured nuclear Overhauser effect intensities (NOEs) represent averages over the individual contributions. We propose a new methodology based on statistical mechanical considerations for analyzing nmr data of such peptides. In a previous paper (called paper I, H. Meirovitch et al. (1995) Journal of Physical Chemistry, 99, 4847-4854] we have developed theoretical methods for determining the contribution to the partition function Z of the most stable microstates, i.e. those that pertain to a given energy range above the global energy minimum (GEM). This relatively small set of dominant microstates provides the main contribution to medium- and long-range NOE intensities. In this work the individual populations and NOEs of the dominant microstates are determined, and then weighted averages are calculated and compared with experiment. Our methodology is applied to the pentapeptide Leu-enkephalin H-Tyr-Gly-Gly-Phe-Leu-OH, described by the potential energy function ECEPP. Twenty one significantly different energy minimized structures are first identified within the range of 2 kcal/mol above the GEM by an extensive conformational search; this range has been found in paper I to contribute 0.6 of Z. These structures then become "seeds" for Monte Carlo (MC) simulations designed to keep the molecule relatively close to its seed. Indeed, the MC samples (called MC microstates) illustrate what we define as intermediate chain flexibility; some dihedral angles remain in the vicinity of their seed value, while others visit the full range of [-180 degrees, 180 degrees]. The free energies of the MC microstates (which lead to the populations) are calculated by the local states method, which (unlike other techniques) can handle any chain flexibility. The NOE of MC microstate i is calculated as the average <1/r(3)>i(2), and an effective interatomic distance ri(eff) is defined as ri(eff) = i(-1/3), where r is the distance between two protons. Under "initial rate approximation," and neglecting angular modulations, the overall I is the average over ri(eff-6), weighted by the populations of the MC microstates. This treatment is justified under the assumption that the rates at which conformations interconvert within, and among, microstates are faster and slower, respectively, than the rotational reorientation of the molecule. I(-6) leads to the virtual theoretical distances, compared to the corresponding virtual experimental distances, which were obtained previously from a cryoprotective solution of Leu-enkephalin at 280 K. A reasonable fit is found between theory and experiment. Future research directions are outlined.

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.

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.

Introduction of short-range restrictions in a protein-folding algorithm involving a long-range geometrical restriction and short-, medium-, and long-range interactions.

A protein-folding algorithm, based on short-range and geometrical long-range restrictions, is applied to bovine pancreatic trypsin inhibitor (BPTI). These restrictions are used to define a starting conformation, SI, by means of a space-filling model of the protein, whose energy is then minimized. The long-range restriction is the imposition of the native spatial geometric arrangement of the loops (SGAL) formed by the disulfide bonds. The short-range restrictions are applied as follows: the (varphi, psi) map of each residue is divided into six regions (corresponding to the right- and left-handed alpha-helical, extended, right- and left-handed bridge, and coil states) and the individual residues are placed in the states of the native structure [although not in conformations with the correct values of (varphi, psi)]. Minimization of the energy of SI leads to a structure, SF, with a root-mean-square deviation of 4.4 A from NI, a previously energy-optimized version of the x-ray structure. SF is closer to the native structure than is the structure RF, which was obtained previously by imposing only the correct SGAL as a restriction. The energy of SF is much lower than that of RF but still larger than the energy of NF (the energy-refined x-ray structure).

Backbone entropy of loops as a measure of their flexibility: application to a ras protein simulated by molecular dynamics.

The flexibility of surface loops plays an important role in protein-protein and protein-peptide recognition; it is commonly studied by Molecular Dynamics or Monte Carlo stimulations. We propose to measure the relative backbone flexibility of loops by the difference in their backbone conformational entropies, which are calculated here with the local states (LS) method of Meirovitch. Thus, one can compare the entropies of loops of the same protein or, under certain simulation conditions, of different proteins. These loops should be equal in size but can differ in their sequence of amino acids residues. This methodology is applied successfully to three segments of 10 residues of a Ras protein simulated by the stochastic boundary molecular dynamics procedure. For the first time estimates of backbone entropy differences are obtained, and their correlation with B factors is pointed out; for example, the segments which consist of residues 60-65 and 112-117 have average B factors of 67 and 18 A2, respectively, and entropy difference T delta S = 5.4 +/- 0.1 kcal/mol at T = 300 K. In a large number of recent publications the entropy due to the fast motions (on the ps-ns time scale) of N-H and C-H vectors has been obtained from their order parameter, measured in nuclear magnetic resonance spin relaxation experiments. This enables one to estimate differences in the entropy of protein segments due to folding-unfolding transitions, for example. However, the vectors are assumed to be independent, and the effect of the neglected correlations is unknown; our method is expected to become an important tool for assessing this approximation. The present calculations, obtained with the LS method, suggest that the errors involved in experimental entropy differences might not be large; however, this should be verified in each case. Potential applications of entropy calculations to rational drug design are discussed.

Performance of hybrid methods for large-scale unconstrained optimization as applied to models of proteins.

Energy minimization plays an important role in structure determination and analysis of proteins, peptides, and other organic molecules; therefore, development of efficient minimization algorithms is important. Recently, Morales and Nocedal developed hybrid methods for large-scale unconstrained optimization that interlace iterations of the limited-memory BFGS method (L-BFGS) and the Hessian-free Newton method (Computat Opt Appl 2002, 21, 143-154). We test the performance of this approach as compared to those of the L-BFGS algorithm of Liu and Nocedal and the truncated Newton (TN) with automatic preconditioner of Nash, as applied to the protein bovine pancreatic trypsin inhibitor (BPTI) and a loop of the protein ribonuclease A. These systems are described by the all-atom AMBER force field with a dielectric constant epsilon = 1 and a distance-dependent dielectric function epsilon = 2r, where r is the distance between two atoms. It is shown that for the optimal parameters the hybrid approach is typically two times more efficient in terms of CPU time and function/gradient calculations than the two other methods. The advantage of the hybrid approach increases as the electrostatic interactions become stronger, that is, in going from epsilon = 2r to epsilon = 1, which leads to a more rugged and probably more nonlinear potential energy surface. However, no general rule that defines the optimal parameters has been found and their determination requires a relatively large number of trial-and-error calculations for each problem.

Computer simulation of the free energy of peptides with the local states method: analogues of gonadotropin releasing hormone in the random coil and stable states.

The Helmholtz free energy F (rather than the energy) is the correct criterion for stability; therefore, calculation of F is important for peptides and proteins that can populate a large number of metastable states. The local states (LS) method proposed by H. Meirovitch [(1977) Chemical Physics Letters, Vol. 45, p. 389] enables one to obtain upper and lower bounds of the conformational free energy, FB (b,l) and FA (b,l), respectively, from molecular dynamics (MD) or Monte Carlo samples. The correlation parameter b is the number of consecutive dihedral or valence angles along the chain that are taken into account explicitly. The continuum angles are approximated by a discretization parameter l; the larger are b and l, the better the approximations; while FA can be estimated efficiently, it is more difficult to estimate FB. The method is further developed here by applying it to MD trajectories of a relatively large molecule (188 atoms), the potent "Asp4-Dpr10" antagonist [cyclo(4/10)-(Ac-delta 3Pro1-D-pFPhe2-D-Trp3-Asp4-Tyr5-D-Nal6-Leu7-Arg8 -Pro9- Dpr10-NH2)] of gonadotropin releasing hormone (GnRH). The molecule was simulated in vacuo at T = 300 K in two conformational states, previously investigated [J. Rizo et al. Journal of the American Chemical Society, (1992) Vol. 114, p. 2860], which differ by the orientation of the N-terminal tail, above (tail up, TU) and below (tail down, TD) the cyclic heptapeptide ring. As in previous applications of the LS method, we have found the following: (1) While FA is a crude approximation for the correct F, results for the difference, delta FA = FA (TD)-FA (TU) converge rapidly to 5.6 (1) kcal/mole as the approximation is improved (i.e., as b and l are increased), which suggests that this is the correct value for delta F; therefore TD is more stable than TU. (The corresponding difference in entropy, T delta SA = 1.3(2) kcal/mole, is equal to the value obtained by the harmonic approximation.) (2) The lowest approximation, which has the minimal number of local states, i.e., based on b = 0 (no correlations) and l = 1 (the angle values are distributed homogeneously), also leads to the correct value of delta F, within the error bars. This is important since the lowest approximation can be applied even to large proteins. (3) The method enables one to define the entropy of a part of the molecule and thus to measure the flexibility of this part.(ABSTRACT TRUNCATED AT 400 WORDS)