Understanding the alpha-helix to coil transition in polypeptides using network rigidity: predicting heat and cold denaturation in mixed solvent conditions

Thermodynamic stability in polypeptides is described using a novel Distance Constraint Model (DCM). Here, microscopic interactions are represented as constraints. A topological arrangement of constraints define a mechanical framework. Each constraint in the framework is associated with an enthalpic and entropic contribution. All accessible topological arrangements of distance constraints form an ensemble of mechanical frameworks, each representing a microstate of the polypeptide. A partition function is calculated exactly using a transfer matrix approach, where in many respects the DCM is similar to the Lifson-Roig model. The crucial difference is that the effect of network rigidity is explicitly calculated for each mechanical framework in the ensemble. Network rigidity is a mechanical interaction that provides a mechanism for long-range molecular cooperativity and enables a proper treatment of the nonadditivity of a microscopic free energy decomposition. Accounting for (1) helix <--> coil conformation changes along the backbone similar to the Lifson-Roig model, (2) i to i + 4 hydrogen-bond formation <--> breaking similar to the Zimm-Bragg model, and (3) structured <--> unstructured solvent interaction (hydration effects), a six-parameter DCM describes normal and inverted helix-coil transitions in polypeptides. Under suitable mixed solvent conditions heat and cold denaturation is predicted. Model parameters are fitted to experimental data showing different degrees of cold denaturation in monomeric polypeptides in aqueous hexafluoroisopropanol (HFIP) solution at various HFIP concentrations. By assuming a linear HFIP concentration dependence (up to 6% by mole fraction) on model parameters, all essential experimentally observed features are captured.     

A stone's throw and its launch window: timing precision and its implications for language and hominid brains

The Chou-Fasman conformational parameters, P, for amino acid residues in proteins are shown to be a linear function of intermolecular force and steric parameters. For α- helix, coil and turn parameters, steric effects are predominant; whereas for β-sheet parameters, intramolecular forces are predominant. Turn and coil parameters show little or no difference in their dependence which is different from that of α-helix and in some ways almost reciprocal. Factors which increase the probability of finding an amino acid residue in an α-helix usually decrease the probability of finding it in coil or turn. Values of P were calculated for several of the less common amino acids.     

Side-chain effect on conformation of ionizable polypeptides in aqueous solution

In order to study the effect of side-chain length on the conformation of polypeptides, conformational changes of various ionic polypeptides with various lengths of side chain, poly-N^ε-glutaryl-L -lysine (PGL), poly-N^δ-glutaryl-L -ornithine (PGO), poly-N^ε-succinyl-L -lysine (PSL), and poly-N^δ-succinyl-L -ornithine (PGO), were investigated by ORD, potentiometric titration, and dilatometric measurements in aqueous solution. The results of optical rotation and potentiometric titration measurements indicate strongly that the α-helix stability increases in the sequence PSO < PSL ～ PGO < PGL, which corresponds to increased side-chain length. The volume change associated with the helix–coil transition also increased in the above sequence. This series of polymers seems to be more hydrophobic compared with poly-L -glutamic acid or poly-L -lysine, as suggested from the values of enthalpy and entropy changes for coil–helix transitions.     

Thermodynamic model of secondary structure for α-helical peptides and proteins

A thermodynamic model describing formation of α-helices by peptides and proteins in the absence of specific tertiary interactions has been developed. The model combines free energy terms defining α-helix stability in aqueous solution and terms describing immersion of every helix or fragment of coil into a micelle or a nonpolar droplet created by the rest of protein to calculate averaged or lowest energy partitioning of the peptide chain into helical and coil fragments. The α-helix energy in water was calculated with parameters derived from peptide substitution and protein engineering data and using estimates of nonpolar contact areas between side chains. The energy of nonspecific hydrophobic interactions was estimated considering each α-helix or fragment of coil as freely floating in the spherical micelle or droplet, and using water/cyclohexane (for micelles) or adjustable (for proteins) side-chain transfer energies. The model was verified for 96 and 36 peptides studied by ¹H-nmr spectroscopy in aqueous solution and in the presence of micelles, respectively ([set I] and [set 2]) and for 30 mostly α-helical globular proteins ([set 3]). For peptides, the experimental helix locations were identified from the published medium-range nuclear Overhauser effects detected by ¹H-nmr spectroscopy. For sets 1, 2, and 3, respectively, 93, 100, and 97% of helices were identified with average errors in calculation of helix boundaries of 1.3, 2.0, and 4.1 residues per helix and an average percentage of correctly calculated helix—coil states of 93, 89, and 81%, respectively. Analysis of adjustable parameters of the model (the entropy and enthalpy of the helix—coil transition, the transfer energy of the helix backbone, and parameters of the bound coil), determined by minimization of the average helix boundary deviation for each set of peptides or proteins, demonstrates that, unlike micelles, the interior of the effective protein droplet has solubility characteristics different from that for cyclohexane, does not bind fragments of coil, and lacks interfacial area. © 1997 John Wiley & Sons, Inc. Biopoly 42: 239–269, 1997     

Carbon-13 and proton NMR studies of helix-coil transition of poly(gamma-benzyl-L-glutamate).

The helix–coil conformational transition undergone by poly(γ-benzyl-L -glutamate) in solutions of trifluoroacetic acid and deuterated chloroform was studied by proton and carbon-13 nmr. The results indicate that in the case of the solvent-induced helix–coil transition, the side chain assumes a helical conformation before the backbone. In the thermally induced helix–coil transition, the results indicate the existence of an intermediate state, which is between the α-helix and random coil and is free from intramolecular hydrogen bonding.     

Elucidating protein thermodynamics from the three-dimensional structure of the native state using network rigidity

Given the three-dimensional structure of a protein, its thermodynamic properties are calculated using a recently introduced distance constraint model (DCM) within a mean-field treatment. The DCM is constructed from a free energy decomposition that partitions microscopic interactions into a variety of constraint types, i.e., covalent bonds, salt-bridges, hydrogen-bonds, and torsional-forces, each associated with an enthalpy and entropy contribution. A Gibbs ensemble of accessible microstates is defined by a set of topologically distinct mechanical frameworks generated by perturbing away from the native constraint topology. The total enthalpy of a given framework is calculated as a linear sum of enthalpy components over all constraints present. Total entropy is generally a nonadditive property of free energy decompositions. Here, we calculate total entropy as a linear sum of entropy components over a set of independent constraints determined by a graph algorithm that builds up a mechanical framework one constraint at a time, placing constraints with lower entropy before those with greater entropy. This procedure provides a natural mechanism for enthalpy-entropy compensation. A minimal DCM with five phenomenological parameters is found to capture the essential physics relating thermodynamic response to network rigidity. Moreover, two parameters are fixed by simultaneously fitting to heat capacity curves for histidine binding protein and ubiquitin at five different pH conditions. The three free parameter DCM provides a quantitative characterization of conformational flexibility consistent with thermodynamic stability. It is found that native hydrogen bond topology provides a key signature in governing molecular cooperativity and the folding-unfolding transition.     

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.     

Correlation between computed conformational properties of cytochrome c peptides and their antigenicity in a T-lymphocyte proliferation assay

By means of conformational energy calculations, we previously showed that the antigenic strength of a series of oligopeptides (derived from the carboxyl terminal sequence of cytochrome c) in a T-lymphocyte proliferation assay depends on their ability to adopt the α-helix conformation. Using experimentally determined statistical weights (within the framework of the Zimm–Bragg theory for the helix–coil transition), here we present a simple free energy analysis of the ability of these peptides to adopt the α-helix conformation in water. The experimental statistical weights have been modified to include the effect of long-range charge–dipole interactions on helix stability. We find that there is a close correlation between the tendency of a peptide to adopt the α-helix conformation and its ability to stimulate antigen-primed T cells. The shortest peptide with a tendency to adopt the α-helix conformation is also the shortest one that exhibits antigenic activity. The rapid and simple method presented here can thus be used to predict relative antigenicities for different peptides derived from cytochrome c.     

Nonadditivity in conformational entropy upon molecular rigidification reveals a universal mechanism affecting folding cooperativity

Previously, we employed a Maxwell counting distance constraint model (McDCM) to describe α-helix formation in polypeptides. Unlike classical helix-coil transition theories, the folding mechanism derives from nonadditivity in conformational entropy caused by rigidification of molecular structure as intramolecular cross-linking interactions form along the backbone. For example, when a hydrogen bond forms within a flexible region, both energy and conformational entropy decrease. However, no conformational entropy is lost when the region is already rigid because atomic motions are not constrained further. Unlike classical zipper models, the same mechanism also describes a coil-to-β-hairpin transition. Special topological features of the helix and hairpin structures allow the McDCM to be solved exactly. Taking full advantage of the fact that Maxwell constraint counting is a mean field approximation applied to the distribution of cross-linking interactions, we present an exact transfer matrix method that does not require any special topological feature. Upon application of the model to proteins, cooperativity within the folding transition is yet again appropriately described. Notwithstanding other contributing factors such as the hydrophobic effect, this simple model identifies a universal mechanism for cooperativity within polypeptide and protein-folding transitions, and it elucidates scaling laws describing hydrogen-bond patterns observed in secondary structure. In particular, the native state should have roughly twice as many constraints as there are degrees of freedom in the coil state to ensure high fidelity in two-state folding cooperativity, which is empirically observed.     

Probing the influence of sequence-dependent interactions upon α-helix stability in alanine-based linear peptides

The observation that short, linear alanine-based polypeptides form stable α-helices in aqueous solution has allowed the development of well-defined experimental systems with which to study the influence of amino acid sequence upon the stability of secondary structure. We have performed detailed conformational searches upon six alanine-based peptides in order to rationalize the observed variation in the α-helical stability in terms of side-chain-backbone and side-chain-side-chain interactions. Although a simple, gas-phase, potential model was used to obtain the conformational energies for these peptides, good agreement was obtained with experiment regarding their relative α-helical stabilities. Our calculations clearly indicate that valine, isoleucine, and phenylalanine residues should destabilize the α-helical conformation when included within alanine-based peptides because of energetically unfavorable side-chain-backbone interactions, which tend to result in the formation of regions of 3₁₀-helix. In the case of valine, the destabilization most probably arises from entropic effects as the isopropyl side chain can assume more orientations in the 3₁₀-helical form of the peptide. A detailed examination of very short-range interactions in these peptides has also indicated that an interaction, involving fewer than five consecutive residues, whose stabilizing effect reinforces that of the (i, i + 4) hydrogen bond may be the basis of the requirement for increased nucleation (σ) and propagation parameters (s) required by Zimm–Bragg theory to predict the α-helical content for compounds in this class of short peptides. Our calculations complement recent work using modified Zimm–Bragg and Lifson–Roig theories of the helix–coil transition, and are consistent with molecular dynamics simulations upon linear peptides in aqueous solution. © 1993 John Wiley & Sons, Inc.     

肽键的统计分析和蛋白质的二级结构

 本文对蛋白质序列的肽键进行了统计分析,计算了二肽构象参数P_α、P_β、P_c和三肽构象参数Q_α、Q_β、Q_c。在此基础上提出了由氨基酸序列预测二级结构的规则。预测的正确率达90％,优于Chou-Fasman方法。这个结果表明二肽(三肽)关联在形成蛋白质二级结构中具有明显的重要性。     

Development of the multiple sequence approximation within the AGADIR model of α-helix formation: Comparison with Zimm-Bragg and Lifson-Roig formalisms

In this work we present the development of the multiple sequence approximation (AGADIRms) and the standard one-sequence approximation (AGADIRIs) within the framework of AGADIR's α-helix formation model. The extensive comparison between these new formulations and the original one [AGADIR; v. Muñoz and L. Serrano (1994),Nat. Struct. Biol., Vol. 1, pp. 399–409] indicates that the standard one-sequence approximation is virtually identical to the multiple sequence approximation, while the previously used residue partition function approximation [Muñoz and Serrano (1994); (1995), J. Mol. Biol., Vol. 245, pp. 275–296] is less precise. The calculations of the average helical content performed with AGADIR are precise for peptides of less than 30 residues and progressively diverge from the multiple sequence formulation for longer peptides. The helicity distribution of heteropolypeptides with less than 50% average helical content is also well described, while those of quasi-homopolymers with high helical content tend to be flattened. These inaccuracies lead to an underestimation of 0.017 kcal/mol for the mean-residue enthalpic contribution in AGADIR, as compared to AGADIRms and AGADIRIs. The other energy contributions to α-helix stability are not affected by the original statistical approximation. We also discuss the particularities of the model for α-helix formation utilized in AGADIR and compare it with the classical Zimm-Bragg and Lifson-Roig theories. Moreover, we develop the mathematical relationships between the basic AGADIR energy contributions and helix nucleation and elongation, which permit the quantitative comparison between formalisms. Remarkably, the comparison between AGADIRms and the Lifson-Roig formalism shows that, despite the differences on treating helix/coil cooperativity, both theories give virtually identical results when an equivalent set of parameters is used. This indicates that the helix/coil transition is a solid theory independent of the particularities of the model for α-helix formation. © 1997 John Wiley & Sons, Inc. Biopoly 41: 495–509, 1997.     

A CD study of the α-helix nucleation hypothesis

One of the dilemmas in predicting the secondary structure of proteins from their amino acid propensity for a given conformation is the presence of all amino acids in all types of secondary structure, regardless of their propensity for that specific structure. One explanation is the nucleation hypothesis that only a few residues with a strong propensity for the secondary structure, such as the α-helix structure, initiates its formation and propagates the structure through indifferent sequences until strong breakers terminate the growth on both ends. Eight 15-mer peptides were studied to examine the α-helix nucleation hypothesis. The nucleation sequence of VAEAK, with high helix propensity, was mixed with an indifferent sequence of TSDSR in all possible permutations. From the percent α-helix structure derived from the CD at 222 nm, it appears that helicity does not propagate through the indifferent sequence. © 1994 John Wiley & Sons, Inc.     

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.     

Stability of polypeptide conformational states as determined by computer simulation of the free energy

A method is developed to extract the entropy of polypeptides and proteins from samples of conformations. It is based on techniques suggested previously by Meirovitch, and has the advantage that it can be applied not only to states in which the molecule undergoes harmonic or quasiharmonic conformational fluctuations, but also to the random coil, as well as to mixtures of these extreme states. In order to confine the search to a region of conformational space corresponding to a stable state, the transition probabilities are determined not by “looking to the future,” as in the previous method [H. Meirovitch and H. A. Scheraga (1986) J. Chem. Phys. 84 , 6369–6375], but by analyzing the previous steps in the generation of the chain. The method is applied to a model of decaglycine with rigid geometry, using the potential energy function ECEPP (Empirical Conformational Energy Program for Peptides). The model is simulated with the Metropolis Monte Carlo method to generate samples of conformations in the α-helical and hairpin regions, respectively, at T = 100 K. For the α-helix, the four dihedral angles of the N- and C-terminal residues are found to undergo full rotational variation. The results show that the α-helix is a more stable structure than the hairpin. Both its Helmholtz free energy F and energy E are lower than those of the hairpin by ΔF ～ 0.4 and ΔE ～ 0.3 kcal/mole/residue, respectively. It should be noted that the contribution of the entropy ΔS to ΔF is significant (TΔS ～ 0.1 kcal/mole/residue). Also, the entropy of the α-helix is found to be larger than that of the hairpin. This is a result of the extra entropy arising from the rotational freedom about the four terminal single bonds of the α-helix.     

Conformational and kinetic studies of synthetic polypeptides by NMR

The α-helix–coil transition of poly-L -leucine, poly-L -alanine and poly-L -methionine in chloroform–trifluoroacetic acid system was studied by nuclear magnetic resonance (NMR) and optical rotatory dispersion (ORD). The kinetics of the hydrogen–deuterium exchange in the peptide was also followed in these polymers by means of NMR. Two types of the NMR spectra and the hydrogen–deuterium exchange reaction were found, corresponding to the high and low molecular weight polypeptides. In high molecular weights, the NH and α-CH resonance lines gave single peaks and the hydrogen–deuterium exchange was expressed as a single first order reaction. In low molecular weights, the NH and α-CH lines were separated into two peaks, corresponding to helical and random-coiled states, respectively, and the exchange react ion was expressed as super-position of a very rapid exchange reaction in the random-coiled part and another slow exchange reaction of the first order in the helical part. These results suggest that the helix–coil interconversion of low molecular weight polypeptides has a longer relaxation time (? 4.5 × 10^?3 sec) than that of high molecular weight polypeptides.     

Infrared spectra and resonance interactions of amide-I and II vibration of alpha-helix.

Amide-I and II vibrations of α-helix have been treated on the basis of the perturbation theory in a dipole–dipole approximation. The infinite helix and its finite fragments have been considered as models. The calculated infrared spectra for the infinite helix are in good agreement with the spectra of synthetic polypeptides. In the case of finite fragments of the α-helix, the characteristic contours of the amide-I and II bands appear when the nuimber of peptide groups is about 10 and 6, respectively.     

Molecular dynamics simulations of a bacterial autotransporter: NalP from Neisseria meningitidis

NalP is an autotransporter secretory protein found in the outer membrane of Neisseria meningitidis. The crystal structure of the NalP translocator domain revealed a transmembrane β-barrel containing a central α-helix. The role of this α-helix, and of the conformational dynamics of the β-barrel pore have been studied via atomistic molecular dynamics simulations. Three simulations, each of 10 ns duration, of NalP embedded within a solvated DMPC bilayer were performed. The helix was removed from the barrel interior in one simulation. The conformational stability of the protein is similar to that of other outer membrane proteins, e.g., OmpA, in comparable simulations. The transmembrane β-barrel is stable even in the absence of the α-helix. Removal of the helix results in an influx of water into the pore region, suggesting the helix acts as a ‘plug’. Water molecules entering the resultant pore form hydrogen bonds with the barrel lining that compensate for the loss of helix-barrel hydrogen bonds. The dimensions of the pore fluctuate over the course of the simulation revealing it to be flexible, but only wide enough to allow transport of the passenger domain in an unfolded or extended conformation. The simulations help us to understand the role of the central helix in plugging the pore and in maintaining the width of the barrel, and show that the NalP monomer is sufficient for the transport of the passenger domain in an unfolded or extended conformation.     

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.