Monte Carlo studies on potentiometric titration of poly(glutamic acid)

The potentiometric titration of poly(glutamic acid) with special attention to its helix-coil transition is investigated in terms of the previously developed Monte Carlo method. The simulations of the potentiometric titration are carried out for helical and coiled form of the peptide, separately. A cylindrical rod with spherical ionizable groups is adopted as each conformational model of poly(glutamic acid) molecule. A spherical charge with a hard core potential is assumed as a mobile hydrated ion. The helix-coil transition curves are analyzed by the Zimm-Bragg theory. A satisfactory agreement is achieved for the titration curves with the experimental data in most cases. The significance and the limitations of the simulation method are discussed.     

Beyond the nearest-neighbor Zimm-Bragg model for helix-coil transition in peptides

The nearest-neighbor (micro = 1) variant of the Zimm and Bragg (ZB) model has been extensively used to describe the helix-coil transition in biopolymers. In this work, we investigate the helix-coil transition for a 21-residue alanine peptide (AP) with the ZB model up to fourth nearest neighbor (micro = 1, 2, 3, and 4). We use a matrix approach that takes into account combinations of any number of helical stretches of any length and therefore gives the exact statistical weight of the chain within the assumptions of the ZB model. The parameters of the model are determined by fitting the temperature-dependent circular dichroism and Fourier transform infrared experimental spectra of the AP. All variants of the model fit the experimental data, thus giving similar results in terms of the macroscopic observables, such as temperature-dependent fractional helicity. However, the resulting microscopic parameters, such as distributions of the individual residue helical probabilities and free energy surfaces, vary significantly depending on the variant of the model. Overall, the mean residue enthalpy and entropy (in the absolute value) both increase with micro, but combined yield essentially the same "effective" value of the ZB propagation parameters for all micro. Greater helical probabilities for individual residues are predicted for larger micro, in particular, near the center of the sequence. The ZB nucleation parameters increase with increasing micro, which results in a lower free energy barrier to helix nucleation and lower apparent "cooperativity" of the transition. The significance of the long-range interactions for the predictions of ZB model for helix-coil transition, the calculated model parameters and the limitations of the model are discussed.     

Construction of an intermediate-resolution lattice model and re-examination of the helix-coil transition: a dynamic Monte Carlo simulation

In protein modeling, spatial resolution and computational efficiency are always incompatible. As a compromise, an intermediate-resolution lattice model has been constructed in the present work. Each residue is decomposed into four basic units, i.e. the α-carbon group, the carboxyl group, the imino group, and the side-chain group, and each basic coarse-grained unit is represented by a minimum cubic box with eight lattice sites. The spacing of the lattice is about 0.56?Å, holding the highest spatial resolution for the present lattice protein models. As the first report of this new model, the helix-coil transition of a polyalanine chain was examined via dynamic Monte Carlo simulation. The period of formed α-helix was about 3.68 residues, close to that of a natural α-helix. The resultant backbone motion was found to be in the realistic regions of the conformational space in the Ramachandran plot. Helix propagation constant and nucleation constant were further determined through the dynamic hydrogen bonding process and torsional angle variation, and the results were used to make comparison between classical Zimm-Bragg theory and Lifson-Roig theory based on the Qian-Schellman relationship. The simulation results confirmed that our lattice model can reproduce the helix-coil transition of polypeptide and construct a moderately fine α-helix conformation without significantly weakening the priority in efficiency for a lattice model.     

Sequential polypeptides: 5. Statistical mechanical analysis of helix-coil transition in sequential polypeptides

A statistical mechanical theory of the helix-coil transition in sequential polypeptides is developed assuming that the statistical weights of the Zimm-Bragg parameters of a given residue depend on the type of adjacent residues. In the case of a sequential polypeptide consisting of two kinds of residues, the theory describes the helix- coil transition of the polypeptide in terms of the Zimm-Bragg parameters associated with the corresponding residues. The theory is then used to determine this parameter, as a function of temperature, from experimental data for transition temperature as a function of solvent composition, for a series of sequential polypeptides consisting of Glu(OBzl) and Lys(Chz) residues in mixtures of dichloroacetic acid and 1,2-dichlorethane. This parameter is then combined with the Zimm-Bragg parameters for the parent homopolypeptides, and the theory used to predict helix coil transition curves which are in good agreement with the experimental ones for the sequential polypeptides studied.     

Parameters of helix-coil transition theory for alanine-based peptides of varying chain lengths in water.

Thermal unfolding curves have been measured for a series of short alanine-based peptides that contain repeating sequences and varying chain lengths. Standard helix-coil theory successfully fits the observed transition curves, even for these short peptides. The results provide values for sigma, the helix nucleation constant, delta H0, the enthalpy change on helix formation, and for s (0 degree C), the average helix propagation parameter at 0 degree C. The enthalpy change agrees with the value determined calorimetrically. The success of helix-coil theory in describing the unfolding transitions of short peptides in water indicates that helical propensities, or s values, can be determined from substitution experiments in short alanine-based peptides.     

A grammatical theory for the conformational changes of simple helix bundles.

Polymers, including biomolecules such as proteins, have two particularly important types of single-molecule transitions: a helix-coil transition, driven by interactions that are local in the chain, and a collapse transition, driven by nonlocal interactions. A long-standing challenge of polymer statistical mechanics has been to deal with both types of transition in a single theoretical framework. The simplest paradigmatic problem would be a theory of helix-bundle folding. Here, we show how the machinery of formal grammars, originally developed in the context of linguistic analysis and programming-language compilation, provides a simple and general way to combine the Zimm-Bragg model of alpha-helices with the model of Chen and Dill for nonlocal interactions in antiparallel polymeric systems. We use a well-known construction in the theory of formal grammars to give the statistical mechanical partition function for two-helix bundles. Predictions are shown to be quite good in comparison to exact enumerations within a lattice model.     

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.     

Exhaustive Metropolis Monte Carlo sampling and analysis of polyalanine conformations adopted under the influence of hydrogen bonds

We propose a novel Metropolis Monte Carlo procedure for protein modeling and analyze the influence of hydrogen bonding on the distribution of polyalanine conformations. We use an atomistic model of the polyalanine chain with rigid and planar polypeptide bonds, and elastic alpha carbon valence geometry. We adopt a simplified energy function in which only hard-sphere repulsion and hydrogen bonding interactions between the atoms are considered. Our Metropolis Monte Carlo procedure utilizes local crankshaft moves and is combined with parallel tempering to exhaustively sample the conformations of 16-mer polyalanine. We confirm that Flory's isolated-pair hypothesis (the steric independence between the dihedral angles of individual amino acids) does not hold true in long polypeptide chains. In addition to 3(10)- and alpha-helices, we identify a kink stabilized by 2 hydrogen bonds with a shared acceptor as a common structural motif. Varying the strength of hydrogen bonds, we induce the helix-coil transition in the model polypeptide chain. We compare the propensities for various hydrogen bonding patterns and determine the degree of cooperativity of hydrogen bond formation in terms of the Hill coefficient. The observed helix-coil transition is also quantified according to Zimm-Bragg theory.     

Mechanically induced helix-coil transition in biopolymer networks

The quasi-equilibrium evolution of the helical fraction occurring in a biopolymer network (gelatin gel) under an applied stress has been investigated by observing modulation in its optical activity. Its variation with the imposed chain extension is distinctly nonmonotonic and corresponds to the transition of initially coiled strands to induced left-handed helices. The experimental results are in qualitative agreement with theoretical predictions of helices induced on chain extension. This new effect of mechanically stimulated helix-coil transition has been studied further as a function of the elastic properties of the polymer network: crosslink density and network aging.     

Helix-coil stability constants for the naturally occurring amino acids in water. XII. Asparagine parameters from random poly(hydroxybutylglutamine-co-L-asparagine)

The synthesis and characterization of water-soluble random copolymers containing L -asparagine with N⁵-(4-hydroxybutyl)-L -glutamine, and the thermally induced helix-coil transitions of these copolymers in water, are described. The incorporation of L -asparagine was found to decrease the helix content of the polymers in water at all temperatures. The Zimm-Bragg parameters σ and s for the helix-coil transition in poly(L -asparagine) in water were deduced from an analysis of the copolymer melting curves in the manner described in earlier papers. The computed values of s indicate that asparagine destabilizes helical sequences at all temperatures in the range 0–60°C.     

Premelting thermal fluctuational base pair opening probability of poly(dA).poly(dT) as predicted by the modified self-consistent phonon theory.

We employ a mean field, modified, self-consistent phonon theory to evaluate the single base-pair opening rate and the probability of a base pair in the amino proton exchangeable state for the homopolymer poly(dA).poly(dT) at temperatures below the helix-coil transition region. Our calculated premelting single base-pair opening probabilities are in general agreement with several available experimental estimates from imino proton exchange and formaldehyde-induced DNA melting measurements. These calculated opening probabilities, however, are in disagreement with the prediction of the helix-coil transition theory. Possible reasons for the differences are discussed, especially the possible different definition of a meaningful open state in the premelting region. The premelting open state of the modified self-consistent phonon approximation theory seems to be appropriate to describe a solvent-accessible open configuration that is sufficient to facilitate important chemical reactions such as imino proton exchange and formaldehyde reaction with the bases. This can be compared with the completely unstacked open state of the helix-coil transition theory originally defined in the helix-coil transition region. We propose that the amino proton exchangeable state is different from the open state associated with melting and only involves the breaking of the amino interbase H bond. The agreement between the calculated and experimentally estimated probability of a base pair in the amino proton exchangeable state seems to support this hypothesis.     

Mechanism and dynamics of conformational ordering in xanthan polysaccharide

The thermally induced order-disorder transition of xanthan (extracellular bacterial polysaccharide from Xanthomonas campestris) has been investigated by optical rotation, differential scanning calorimetry, stopped-flow reaction kinetics and low-angle laser light scattering, and the results have been analysed in terms of Zimm -Bragg helix-coil transition theory. The reciprocal of the transition midpoint temperature (Tm) varies linearly with the logarithm of cation (K+) the salt dependence of Tm, is in agreement with Manning polyelectrolyte theory the ordered structure. The associated increase in cation binding, calculated from the salt dependence of tm, is in agreement with the Manning polyelectrolyte theory for one of the candidate structures from X-ray diffraction, a 5(1) single helix stabilized by packing of side-chains along the polymer backbone, but not for the alternative double-helix structure that has also been proposed. At each salt concentration, the two fundamental parameters of the Zimm -Bragg theory, s and sigma, were calculated. The equilibrium constant for growth of the ordered structure (s) is derived directly from calorimetric measurement of transition enthalpy (delta Hcal ), and sigma, which quantifies the relative instability of the helix nucleus, is derived from the ratio of delta Hcal to the apparent transition enthalpy (delta Happ ) obtained by van't Hoff analysis of the optical rotation data. The temperature course of conformational ordering calculated theoretically is in good quantitative agreement with experimental results from both optical rotation and scanning calorimetry. The calculated average length of stable, ordered chain-sequences increases with decreasing temperature, but equals or exceeds the total chain length from light scattering only at temperatures more than approximately equal to 70 K below Tm, suggesting that ordered and disordered regions may co-exist within the same xanthan molecule. Consistent with this interpretation, the observed rate of conformational ordering increases sharply under conditions where the starting solution for dynamic measurements is partially ordered, suggesting that ordered sequences within each chain may act as helix nuclei for adjacent disordered regions, so that helix growth, rather than the slower nucleation process, becomes rate limiting.     

Conformational transitions in a model polymer chain with secondary and tertiary structures

The confonnational properties of a simulated polymer chain with secondary and tertiary structures are calculated. The calculation is carried out by a Monte-Carlo procedure for a chain on the cubic lattice consisting of 64 links with Zimm-Bragg parameters s = 0.8–1.3, σ = $\text{1}\text{64}$ and an energy of attraction between links ? = 0–1.0.Different confonnational transitions are investigated: helix-coil, coil-globule, coil-crystal, globule-crystal. It is shown that in all cases formation (or destruction) of a crystal-like structure occurs as a Jump-like transition.The results obtained for a model chain are discussed in relation with confonnational transitions in globular pioteins.     

Predicting helical segments in proteins by a helix-coil transition theory with parameters derived from a structural database of proteins

A novel helix-coil transition theory has been developed. This new theory contains more types of interactions than similar theories developed earlier. The parameters of the models were obtained from a database of 351 nonhomologous proteins. No manual adjustment of the parameters was performed. The interaction parameters obtained in this manner were found to be physically meaningful, consistent with current understanding of helix stabilizing/destabilizing interactions. Novel insights into helix stabilizing/destabilizing interactions have also emerged from this analysis. The theory developed here worked well in sorting out helical residues from amino acid sequences. If the theory was forced to make prediction on every residue of a given amino acid sequence, its performance was the best among ten other secondary structural prediction algorithms in distinguishing helical residues from nonhelical ones. The theory worked even better if one only required it to make prediction on residues that were “predictable” (identifiable by the theory); >90% predictive reliability could be achieved. The helical residues or segments identified by the helix-coil transition theory can be used as secondary structural contraints to speed up the prediction of the three-dimensional structure of a protein by reducing the dimension of a computational protein folding problem. Possible further improvements of this helix-coil transition theory are also discussed. Proteins 28:344–359, 1997. © 1997 Wiley-Liss, Inc.     

The kinetics of DNA helix-coil subtransitions

The kinetic analysis of individual helix-coil subtransitions were performed by comparing melting and renaturation profiles obtained at different temperature change rates. The duration of the three transition stages and its dependence on temperature and ionic strength were determined for a T7 phage DNA fragment. The obtained temperature dependence of the melting time for a stretch flanked by melted regions is in quantitative agreement with that predicted by the theory of slow processes (V.V. Anshelevich, A.V. Vologodskii, A.V. Lukashin, M.D. Frank-Kamenetskii, Biopolymers 23, 39 (1984)). The reasons are discussed for the increasing relaxation time of this stretch in the middle of its transition with decreasing ionic strength. The zipping kinetics of a melted region under essentially nonequilibrium conditions was examined for T7 fragment and pAO3 DNAs. The obtained temperature dependence of the zipping time is in quantitative agreement with calculations based on the theory of slow processes. The renaturation times of stretches flanked by helical regions proved fairly small even at a low ionic strength. These times are several orders of magnitude smaller than the renaturation times of the same stretches with one helical boundary. A formal application of the theory of slow processes failed to account for the small renaturation times of stretches that are zipped from both ends. This is probably due to the non-allowance for the changing entropy of the loop linking two helix-coil boundaries migrating towards each other. Slow processes have been revealed in the intramolecular melting of Col E1 DNA at a high ionic strength. The reason for the long relaxation time of one subtransition is the large size of the loop that separates the melting stretch from the helical part of the molecule. This result can be accounted for by the theory of slow processes.     

Molecular dynamics simulations of synthetic peptide folding

Because the time scale of protein folding is much greater than that of the widely used simulations of native structures, a detailed report of molecular dynamics simulations of folding has not been available. In this study, we Included the average solvent effect in the potential functions to simplify the calculation of the solvent effect and carried out long molecular dynamics simulations of the alanine-based synthetic peptides at 274 K. From either an extended or a randomly generated conformation, the simulations approached a helix-coil equilibrium in about 3 ns. The multiple minima problem did not prevent helix folding. The calculated helical ratio of Ac-AAQAAAAQAAAAQAAY-NH₂ was 47%, in good agreement with the circular dichroism measurement (about 50%). A helical segment with frayed ends was the most stable conformation, but the hydrophobic interaction favored the compact, distorted helix-turn-helix conformations. The transition between the two types of conformations occurred in a much larger time scale than helix propagation. The transient hydrogen bonds between the glutamine side chain and the backbone carbonyl group could reduce the free energy barrier of helix folding and unfolding. The substitution of a single alanine residue in the middle of the peptide with valine or glycine decreased the average helical ratio significantly, in agreement with experimental observations. © 1996 Wiley-Liss, Inc.     

Statistical mechanical studies of polypeptides. 1. Theory of the helix-coil transition including right- and left-handed helical states

The Lifson-Roig and Zimm-Bragg theories of the helix–coil transition in polypeptides are generalized to include both right- and left-handed α-helical states. The partition functions for these more general theories are formulated in terms of the parameters u, v_R, V_L, W_R, and w_L for the generalized Lifson-Roig theory and σ_R, σ_L, s_R, and s_L for the generalized Zimm-Bragg theory. Matrix equations are derived for calculating such average molecular properties as the fraction of the amino acid residues hydrogen bonded into right- and left-handed α-helices, the average number of right- and left-handed helical sequences per molecule, the number-average length (in residues) of the right- and left-handed helical sequences, and the degree of solvent binding to the peptide NH and CO groups. These equations are shown to be conveniently adaptable to machine methods of calculation, thus avoiding the difficulty of solving an eigenvalue problem where the secular equation is of a high order. A discussion is given of the various energetic and entropic effects which determine the screw sense and stability of helices and of the extent to which it is valid to interpret experimental data by adjustment of the parameters of these statistical mechanical theories which include in their formulation only near-neighbor interactions between residues.     

The Kinetics of DNA Helix-Coil Subtransitions

Abstract

The kinetic analysis of individual helix-coil subtransitions was performed by comparing melting and renaturation profiles obtained at different temperature change rates. The duration of the three transition stages and its dependence on temperature and ionic strength were determined for a T7 phage DNA fragment. The obtained temperature dependence of the melting time for a stretch flanked by melted regions is in quantitative agreement with that predicted by the theory of slow processes (V.V. Anshelevich, A.V. Vologodskii, A.V. Lukashin, M.D. Frank-Kamenetskii, Biopolymers 23, 39 (1984)). The reasons are discussed for the increasing relaxation time of this stretch in the middle of its transition with decreasing ionic strength.

The zipping kinetics of a melted region under essentially nonequilibrium conditions was examined for T7 fragment and pAO3 DNAs. The obtained temperature dependence of the zipping time is in quantitative agreement with calculations based on the theory of slow processes.

The renaturation times of stretches flanked by helical regions proved fairly small even at a low ionic strength. These times are several orders of magnitude smaller than the renaturation times of the same stretches with one helical boundary. A formal application of the theory of slow processes failed to account for the small renaturation times of stretches that are zipped from both ends. This is probably due to the non-allowance for the changing entropy of the loop linking two helix-coil boundaries migrating towards each other.

Slow processes have been revealed in the intramolecular melting of Col E1 DNA at a high ionic strength. The reason for the long relaxation time of one subtransition is the large size of the loop that separates the melting stretch from the helical part of the molecule. This result can be accounted for by the theory of slow processes.