Stability of polypeptide conformational states. II. Folding of a polypeptide chain by the scanning simulation method, and calculation of the free energy of the statistical coil

The scanning simulation method suggested by Meirovitch is extended to a study of the stability of decaglycine at 100 and 300 K. The model is based on the potential energy function ECEPP (Empirical Conformational Energy Program for Peptides) with rigid geometry and without solvent. The free energy of the statistical coil, which is defined over the whole phase space excluding the region of the right-handed α-helix, is calculated. At 100 K, the molecule is found to be unstable in the statistical coil region, and the method generates (i.e., “folds”) conformations that are left-handed or right-handed α-helices with very high preference. Their free energy is found to be comparable with that obtained by another method developed in our previous paper (paper I) [H. Meirovitch, M. Vásquez, and H. A. Scheraga, (1987) Biopolymers 26 , 651–671]. At 300 K the statistical coil becomes the most stable state; sample conformations of the coil are generated efficiently with the scanning method and the free energy is calculated. It appears that both the scanning method and the method of paper I can be used to carry out a complete analysis of the stability of a polypeptide based on free energy considerations.     

Loop formation in polynucleotide chains. II. Flexibility of the anticodon loop of tRNAPhe

The flexibility of hairpin loops containing n bases (residues) has been examined using a theoretical model [N. L. Marky and W. K. Olson (1982), Biopolymers, 21 , 2329–2344] of oligonucleotide loop closure. The study is based on correlated probabilities of chain separation and terminal residue orientation as outlined previously. The probabilities are calculated using standard statistical mechanical methods as functions of local conformational changes of the chain backbone. Our results for an RNA chain of 9 residues suggest that the anticodon loop is a dynamic structure capable of assuming a variety of different spatial conformations. Free energy values related to the various conformations span a narrow range of values (2–4 kcal/mole) and compare well with experimental observations in aqueous solution. Conformational transitions between the loop conformations are within less than 0.5 kcal/mole in free energy. The different spatial loop conformations and the likely pathways between them may have potential relevance to the molecular translation of the genetic code.     

Conformational constraints of amino acid side chains in alpha-helices

The conformational freedom of amino acid side chains is strongly reduced when the side chains occur on an α-helix. A quantitative evaluation of this freedom has been carried out by means of conformational energy computations for all naturally occurring amino acids and for α-aminobutyric acid when they are placed in the middle of a right-handed poly(L-alanine) α-helix. One of the three possible rotameric states for rotation around the C^α ? C^β bond (viz. g⁺) is excluded completely on the helix because of steric hindrance, and the relative populations of the other two rotamers (t and g^?) are altered because of steric interactions and the reduction of hydrogen-bonding possibilities. The computed tendencies of the changes in distributions of rotamers, on going from an ensemble of all backbone conformations to the α-helix, agree with the observed tendencies in proteins. Minimum-energy side-chain conformations in an α-helix have been tabulated for use in conformational energy computations on polypeptides.     

Prediction of the native conformation of a polypeptide by a statistical-mechanical procedure. III. Probable and average conformations of enkephalin

The program SMAPPS (Statistical-Mechanical Algorithm for Predicting Protein Structure) was originally designed to determine the probable and average backbone (?, ψ) conformations of a polypeptide by the application of equilibrium statistical mechanics in conjunction with an adaptive importance sampling Monte Carlo procedure. In the present paper, the algorithm has been extended to include the variation of all side-chain (χ) and peptide-bond (ω) dihedral angles of a polypeptide during the Monte Carlo search of the conformational space. To test the effectiveness of the generalized algorithm, SMAPPS was used to calculate the probable and average conformations of Met-enkephalin for which all dihedral angles of the pentapeptide were allowed to vary. The total conformational energy for each randomly generated structure of Met-enkephalin was obtained by summing over the interaction energies of all pairs of nonbonded atoms of the whole molecule. The interaction energies were computed by the program ECEPP /2 (Empirical Conformational Energy Program for Peptides). Solvent effects were not included in the computation. The results of the Monte Carlo calculation of the structure of Met-enkephalin indicate that the thermodynamically preferred conformation of the pentapeptide contains a γ-turn involving the three residues Gly²-Gly³-Phe⁴. The γ-turn conformation, however, does not correspond to the structure of lowest conformational energy. Rather, the global minimum-energy conformation, recently determined by a new optimization technique developed in this laboratory, contains a type II′ β-bend that is formed by the interaction of the four residues Gly²-Gly³-Phe⁴-Met⁵. A similar minimum-energy conformation is found by the SMAPPS procedure. The thermodynamically preferred γ-turn structure has a conformational energy of 4.93 kcal/mole higher than the β-bend structure of lowest energy but, because of the inclusion of entropy in the SMAPPS procedure, it is estimated to be ～ 9 kcal/mole lower in free energy. The calculation of the average conformation of Met-enkephalin was repeated until a total of ten independent average conformations were established. As far as the phenylalanine residue of the pentapeptide is concerned, the results of the ten independent average conformations were all found to lie in the region of conformational space corresponding to the γ-turn. These results further support the conclusion that the γturn conformation is thermodynamically favored.     

Equilibrium folding and unfolding pathways for a model protein

The folding–unfolding process of reduced bovine pancreatic trypsin inhibitor was investigated with an idealized model employing approximate free energies. The protein is regarded to consist of only C^α and C^β atoms. The backbone dihedral angles are the only conformational variables and are permitted to take discrete values at every 10°. Intraresidue energies consist of two terms: an empirical part taken from the observed frequency distributions of (?,ψ) and an additional favorable energy assigned to the native conformation of each residue. Interresidue interactions are simplified by assuming that there is an attractive energy operative only between residue pairs in close contact in the native structure. A total of 230,000 molecular conformations, with no atomic overlaps, ranging from the native state to the denatured state, are randomly generated by changing the sampling bias. Each conformation is classified according to its conformational energy, F; a conformational entropy, S(F) is estimated for each value of F from the number of samples. The dependence of S(F) on energy reveals that the folding–unfolding transition for this idealized model is an “all-or-none” type; this is attributable to the specific long-range interactions. Interresidue contact probabilities, averaged over samples representing various stages of folding, serve to characterize folding intermediates. Most probable equilibrium pathways for the folding–unfolding transition are constructed by connecting conformationally similar intermediates. The specific details obtained for bovine pancreatic trypsin inhibitor are as follows: (1) Folding begins with the appearance of nativelike medium-range contacts at a β-turn and at the α-helix. (2) These grow to include the native pair of interacting β-strands. This state includes intact regular secondary conformations, as well as the interstrand sheet contacts, and corresponds to an activated state with the highest free energy on the pathway. (3) Additional native long-range contacts are completely formed either toward the amino terminus or toward the carboxyl terminus. (4) In a final step, the missing contacts appear. Although these folding pathways for this model are not consistent with experimental reports, it does indicate multiple folding pathways. The method is general and can be applied to any set of calculated conformational energies and furthermore permits investigation of gross folding features.     

Interaction of calcium ions with α-elastin: An equilibrium dialysis,circular dichroism,and microcalorimetric study

The interaction of calcium ions with α-elastin has been investigated by equilibrium dialysis, CD, and microcalorimetric techniques. Consistent with literature data, it was found that the interaction in water is very poor. In trifluoroethanol (TFE), equilibrium dialysis experiments showed that calcium ions bind to ～-elastin with an association constant of ～250 L × mol^?1. Such a figure is not consistent with highly specific, highly selective binding. It was also found that the CD response is not directly proportional to the amount of bound calcium but depends on the protein concentration. From microcalorimetric experiments it was found that the heat effect relative to the binding process is of the order of 1.9 kcal/g ion. From this figure and from the binding constant, a positive ΔS value of about 17 e.u. was evaluated, leading to the conclusion that the binding process is entropy driven. From microcalorimetric measurements a ΔH of 1.5 kcal/residue was found for the calcium-induced conformational transition of the protein.     

The effect of neutral salts on the conformational transition of poly-alpha-amino acids. I. Potentiometric titration and optical rotatory dispersion studies

The effect of salts on the coil-to-helix transition of poly-α-amino acids was investigated by optical rotatory dispersion and potentiometric titration techniques. Both charge-dependent and charge-independent contributions to the free energy were considered. The free energy of formation ΔF° of the uncharged α-helix from the uncharged random coil for poly-L -glutamic acid (PGA) decreases very rapidly in the limit of zero added salt concentration. This effect probably depends on the uncertainty affecting the choice of the extrapolation of the apparent pK for the random coil at low ionic strength. Above 0.1 M salt, where the free energy determination becomes meaningful, the anions and cations investigated do not affect the value of ΔF°, with the exception of Li⁺. Our data support the point of view that this cation binds to the peptide group. A class of salts produces an increase of the helical content of poly-L -ornithine (PO) both at low and high degree of ionization. This effect appears to be anion dependent. It is shown that (1) no change of ΔF° is involved; (2) recent theories of polyelectrolyte solutions cannot account for our results. We suggest that a true site binding of the anions to the charged amino groups occurs. The role of electrostatic binding in determining the conformational stability of proteins in the presence of some anions is stressed, and a general treatment for the electrostatic binding equilibria is outlined.     

Thermal and charge-induced coil to -helix transition of poly-L-glutamic acid and random L-glutamic acid-L-alanine copolymers

Thermal and charge induced random coil to α-helix transitions of poly-L -glutamic acid (PGA) were measured by optical rotatory dispersion in various solvents. The data of PGA in 0.1M Nacl were analyzed by the Zimm-Rice theory. Enthalpy and entropy changes for the coil-to-helix transition in the unionized state were obtained: ΔH° = ?1020 ± 100 cal/residue mole; ΔH° = ?3.0 ± 0.4 e.u./residue mole. The initiation parameter, σ, of the Zimm-Rice theory was given by a value of 5 ± 1 × 10^-3. Random copolymers of L -glutamic acid and L -alanine containing 10, 30, and 40 molar percents of alanyl residue were synthesized. Stabilities of α-helix of the copolymers were compared to that of PGA. In water and water-ethanol solutions, stabilities of the polymers were almost equal after the simple correction about the ionized charge density of the polymers. In 0.1 M NaCl solution these copolymers showed some deviations from the transition curve of PGA, which would suggest the hydrophobic contribution of the alanyl residues.     

The rate of γ-benzyl-L-glutamate NCA polymerizations

The primary amine initiated homopolymerization of γ-benzyl-L -glutamate NCA in dioxane at 25°C, 35°C, 50°C, and 65°C has been investigated. The reactions were virtually independent of temperature indicating an activation energy of less than 1 kcal/mole. The entropy of activation was estimated to be ?65 entropy units at 300°K. The reaction proceeded in two stages. The first stage was zero-order with respect to monomer, whereas the second was first-order with respect to monomer. Both stages were first-order with respect to initiator. These results were interpreted by assuming that the rate constant for propagation was not independent of the degree of polymerization up to the point where a conformational transition to α-helix occurred.     

Proline-induced constraints in alpha-helices

The disrupting effect of a prolyl residue on an α-helix has been analyzed by means of conformational energy computations. In the preferred, nearly α-helical conformations of Ac-Ala₄-Pro-NHMe and of Ac-Ala₇-Pro-Ala₇-NHMe, only the residue preceding Pro is not α-helical, while all other residues can occur in the α-helical A conformation; i.e., it is sufficient to introduce a conformational change of only one residue in order to accommodate proline in a distorted α-helix. Other low-energy conformations exist in which the conformational state of three residues preceding proline is altered considerably; on the other hand, another conformation in which these three residues retain the near-α-helical A-conformational state (with up to 26° changes of their dihedral angles ? and ψ, and a 48° change in one ω from those of the ideal α-helix) has a considerably higher energy. These conclusions are not altered by the substitution of other residues in the place of the Ala preceding Pro. The conformations of the peptide chain next to prolyl residues in or near an α-helix have been analyzed in 58 proteins of known structure, based on published atomic coordinates. Of 331 α-helices, 61 have a Pro at or next to their N-terminus, 21 have a Pro next to their C-terminus, and 30 contain a Pro inside the helix. Of the latter, 16 correspond to a break in the helix, 9 are located inside distorted first turns of the helix, and 5 are parts of irregular helices. Thus, the reported occurrence of prolyl residues next to or inside observed α-helices in proteins is consistent with the computed steric and energetic requirements of prolyl peptides.     

Examining the dynamic energy landscape of an antiporter upon inhibitor binding

Previously, we applied single-molecule force spectroscopy to detect and locate interactions within the functional Na⁺/H⁺ antiporter NhaA from Escherichia coli. It was observed that the binding of the inhibitor 2-aminoperimidine established interactions different from those introduced by the binding of the native ligand. To understand the inhibitory mechanism of the inhibitor, we applied single-molecule dynamic force spectroscopy to reconstruct the energy landscape of NhaA. Dynamic force spectroscopy revealed that the energy landscape of the antiporter remained mainly unchanged except for the energy barrier of the functionally important transmembrane α-helix IX. Inhibitor binding set this domain into a newly formed deep and narrow energy minimum that kinetically stabilized α-helix IX and reduced its conformational entropy. The entropy reduction of α-helix IX is thought to inhibit its functionally important structural flexibility, while the deeper energy barrier shifted the population of active antiporters towards inhibited antiporters.     

Thermodynamic analysis of purified human placental alpha-L-fucosidase

The temperature dependence for the hydrolysis of both 4-methylumbelliferyl-α-l-fucoside and p-nitrophenyl-α-l-fucoside was determined for purified α-l-fucosidase (EC 3.2.1.51) from human placenta. The inhibition of the enzymatic reaction by l-fucose was also studied using the first of these two substrates at different temperatures. The thermodynamic parameters calculated from the pK_m were for the 4-methylumbelliferyl-conjugate ΔF = ?6.6 kcal/mol, ΔH = ?8.5 kcal/mol, and ΔS = ?6.3 e.u. and for the p-nitrophenylconjugate ΔF = ?5.6 kcal/mol, ΔH = ?12.2 kcal/mol, and ΔS = ?21.1 e.u. The thermodynamic parameters for l-fucose were ΔH = ?12.4 kcal/mol and ΔS = ?20.1 e.u. The lower exothermicity and negative entropy calculated for the 4-methylumbelliferyl substrate compared to the thermodynamic parameters calculated for the p-nitrophenyl substrate and l-fucose suggest the existence of a secondary hydrophobic binding site for the 4-methylumbelliferyl moiety on the enzyme. The difference in the enthalpy for both substrates is also reflected in a difference in activation energy, being 15.8 kcal/mol for the 4-methylumbelliferyl substrate and 20.7 kcal/mol for the p-nitrophenyl substrate. From these results it may be concluded that altered kinetic properties of the enzyme could be the result of the binding of the “aglycone” moiety of the fluorogenic substrate to the enzyme.     

On the multiple-minima problem in the conformational analysis of polypeptides. II. An electrostatically driven Monte Carlo method--tests on poly(L-alanine)

A new approach to the multiple-minima problem in protein folding is presented. It is assumed that the molecule is driven toward the native structure by three types of mechanism. The first one involves an optimization of the electrostatic interactions, whereby the molecule evolves toward conformations in which the charge distribution becomes energetically more favorable. The second mechanism involves a Monte Carlo–energy minimization approach, and the third one is a backtrack mechanism that acts in the opposite direction, increasing the energy—the third type of movement provides a means to perturb the molecule when it is trapped in a stable but energetically unfavorable local energy minimum. This paper describes the implementation of a model based on these mechanisms, and illustrates its effectiveness by computations on different arbitrary starting conformations of a terminally blocked 19-residue chain of poly(L -alanine) for which the global minimum apparently corresponds to the right-handed α-helix. In all cases, the global minimum was attained, even when the starting conformation was a left-handed α-helix. In the latter case, the trajectory of conformations passed through partially melted forms of the left-handed α-helix (because of electrostatic defects at the ends), and then through the formation of structures leading to the more stable right-handed α-helix.     

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 simulations. 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 Ų, respectively, and entropy difference T ΔS = 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. Proteins 29:127–140, 1997. © 1997 Wiley-Liss, Inc.     

The random coil leads to beta transition of copolymers of L-lysine and L-valine: potentiometric titration and circular dichroism studies.

A series of copolymers of L -lysine and L -valine [poly(L -lysine^f L -valine^100-f)] containing 0–13% L -valine have been studied, in 0.10M KF solution, using potentiometric titration and circular dichroism spectroscopy. Incorporation of increasing amounts of valine into the copolymers favors β-sheet formation over α-helix formation at high pH and room temperature. The titrations were analyzed using the method of Zimm and Rice and the partial free energy (ΔG⁰_cβ) for the coil-to-β-sheet transition for valine is estimated at 900 cal/mole at 25°C. From the temperature dependence of the free energy, the partial enthalpy, ΔH⁰_cβ, and entropy, ΔS⁰_cβ, of the transition for valine is estimated to be 854 cal/mole and 6.0 e.u., respectively. The corresponding partial thermodynamic parameters for L -lysine are in agreement with published results. The fraction of β-sheet versus pH has been calculated for poly(L -lysine^86.8 L -valine^13.2) at 25.0°C using the titration data; data obtained from circular dichroism spectroscopy for the same copolymer are in good accord. It is concluded from these results that L -valine is a very strong β-sheet forming amino acid. Furthermore, these results indicate that the Zimm–Rice method is applicable to transitions between the coil and β-sheet states for a polypeptide containing two different residues.     

13C-nmr chemical shift and conformation of L-alanine residues incorporated into poly(β-benzyl L-aspartate) in the solid state

¹³C-nmr spectra of poly(β-benzyl L-aspartate) containing ¹³C-enriched [3-¹³C]L -alanine residues in the solid state were recorded by the cross polarization–magic angle spinning method, in order to elucidate the conformation-dependent ¹³C chemical shifts of L -alanine residues taking various conformations such as the antiparallel β-sheet, the right-handed α-helix, the left-handed α-helix, and the left-handed ω-helix forms obtained by appropriate treatment. The latter two conformations for L -alanine residues are achieved when L -alanine residues are incorporated into poly(β-benzyl L -aspartate). We found that the alanine C_β carbon show significant ¹³C chemical shift displacement depending on conformational change, and gave the ¹³C chemical shift values at about 17 ppm for the left-handed ω-helix, 14 ppm for the left-handed α-helix, 15.5 ppm for the right-handed α-helix, and 21.0 ppm for the antiparallel β-sheet relative to tetramethylsilane.     

Thermodynamics of equilibria of hemoglobins M Milwaukee-I and Saskatoon and isolated chains of hemoglobin a with carbon monoxide

The thermodynamic parameters of the CO-equilibria of isolated chains of hemoglobin A and of two α-chains in hemoglobins M Milwaukee-I and Saskatoon at 25°, pH 7.0 were determined. The parameters for the binding of the first CO molecule to the hemoglobins M were ΔH′=?17 and ?18 kcal/mole heme and ΔS′=?30 and ?29 e.u. for hemoglobins M Milwaukee-I and Saskatoon, respectively. In contrast to this the characteristics of the second step of the binding were ΔH′=+5.9^· and +4.3 kcal/mole and ΔS′=+51 and +49 e.u. These values for the second step were also significantly different from those of the isolated α-chain (ΔH′=?15 kcal/mole and ΔS′=?11 e.u.).     

Calorimetric measurement of enthalpy change in the isothermal helix--coil transition of poly-L-lysine in aqueous solution

The heat ΔH° for converting an uncharged lysine residue from a coil to an α-helical state in poly-L -lysine in 0.1N KCl has been determined calorimetrically to be ?1200 cal/mole at both 15°C and 25°C. Essentially the same value has been obtained for the conversion of an uncharged residue from a coil to a β-pleated sheet state. Titration data provided information about the state of charge of the polymer in the calorimetric experiments, and optical rotatory dispersion data about its conformation. In order to compute ΔH°, the observed Calorimetric heat was corrected for the heat of breaking the sample cell, the heal of dilution of HCl, the heat of neutralization of OH^? ion, and the heat of ionization of the ε-amino group in the random coil. The latter was obtained from similar Calorimetric measurements on poly-D ,L -lysine, which was shown to be a good model for the random coil form of poly-L -lysine. The measured transition heat was ～0.7 cal., which is only 7% of the total heat liberated when a 40 ml solution of 0.25% w/v poly-L -lysine is brought, from pH 11 to pH 7; nevertheless it could be determined with a precision of ±8%. The conformation of poly-L -lysine at pH 11 appears to be completely helical at 15°C, but a mixture of 90% α-helix, 5% β form, and 5% coil at 25°C. Since ΔH° ～ 0 for the α ? β conversion, the polymer behaves like one of 95% α-helix and 5% coil in the calorimeter at 25°C. At neutral pH, poly-L -lysine is an extended coil, like poly-D ,L -lysine.     

Potentiometric titration of poly-L-lysine: the coil-to-beta transition

We have performed potentiometric titrations of poly-L -lysine. From these data we have calculated the free energy and enthalpy changes for the folding of the random coil to the α-helix in 10% ethanol (?120 and ?120 cal/mole) and from the random coil to the β-structure in water (?140 and 870 cal/mole) and in 10% ethanol (?180 and 980 cal mole). Comparison of these values with each other and with values for the coil → α- helix transition in water (?78 and ?880 cal/mole) led to the following conclusions. The stabilization by ethanol of ethanol of the α-helix with respect to the coil is that predicted from the known free energy of transfer of the peptide group from water to 10% ethanol. Similar data to explain the enthalpy difference are not available. The thermodynamic functions for the transition from α-helix to β-structure, obtained by subtracting those for the coil → α-helix and coil → β-structure transitions, are explained from a consideration of the structural differences: non bonded interactions of the polypeptide backbone are less favorable in the β-structure than in the α-helix, causing an increase in the energy, while hydrophobic contacts between side chains raise the entropy of the β-structure as compared with the α-helix, so that the free energy difference between the two structures is small, but enthalpy and entropy differences are large. The observation of only small differences in the free energy and enthalpy changes for the transition from coil β-structure upon going from water to 10% ethanol is expected by considering both the free energy of transfer of the peptide group (as for the α-helix) and the free energy and enthalpy of transfer of the apolar part of the side chain involved in hydrophobic bond formation.     

A quantum mechanical study of the intrinsic helix-forming tendency of α-aminoisobutyric acid and dehydroalanine residues

A quantum mechanical study to compare the ability of α-aminoisobutyric acid (Aib), de-hydroalanine (ΔAla), and alanine (Ala) residues to stabilize helical conformations has been performed. To address the study, the oligopeptides X_n (X = Aib, ΔAla and Ala), where n varies from 1 to 6, were computed with the AM1 semiempirical method. The results show that the residues modified at the C^α carbon atom, Aib and ΔAla, are better helical formers than Ala. Thus, a cooperative energy effect was found for both residues, and especially for ΔAla. These terms permit the understanding the different conformational behaviors between Ala and its C^α-modified residues Aib and ΔAla. This trend is important for de novo protein design, where Aib and ΔAla must be considered useful residues in the design of synthetic helical motifs. © 1994 John Wiley & Sons, Inc.