Stabilization of short helices by intramolecular cluster formation

The intramolecular formation of multiple clusters of interacting helices has been characterized in a homopolymer. The configuration partition function permits the formation of clusters in which the number of interacting helices may be as large as the greatest integer in n/2, where n denotes the number of amino acid residues in the chain. The theoretical formulation has its origin in a recent [Mattice, W. L. & Scheraga, H. A. (1984) Biopolymers 23 , 1701–1724], tractable matrix expression for the configuration partition function for intramolecular antiparallel β-sheet formation. Reassignment of the expression for one of the n(n+3)/2 elements in the sparse statistical weight matrix, along with a simple change in notation, converts that treatment into a matrix formulation of the configuration partition function for a chain containing multiple clusters of interacting antiparallel helices. The five statistical weights used are δ, f_l, w, and the Zimm-Bragg σ and s. Each tight bend that connects two interacting helices contributes a factor of δ, f_l is used in the weight for larger loops between interacting helices, and w arises from helix–helix interaction. The influence of the helix–helix interaction is well illustrated by two helix–coil transitions in a chain with n = 156 and σ = 0.001. In the absence of helix–helix interaction, the transition occurs by the nucleation and subsequent elongation of a small number of helices. When helix–helix interaction is attractive, the transition can occur by a different mechanism. Formation of a single pair of interacting helices is followed by addition of new helices to the initial cluster. In the latter process, individual helices experience relatively little growth after they are formed.     

Helix–coil stability constants for amino acids in nonaqueous solvents. Studies of random poly(γ-benzyl-L-glutamate-co-L-alanine) and poly(γ-benzyl-L-glutamate-co-L-leucine) in a mixed solvent

The host–guest technique has been applied to the determination of the helix–coil stability constants of two naturally occurring amino acids, L -alanine and L -leucine, in a nonaqueous solvent system. Random copolymers containing L -alanine and L -leucine, respectively, as guest residues and γ-benzyl-L -glutamate as the host residue were synthesized. The polymers were fractionated and characterized for their amino acid content, molecular weight, and helix–coil transition behavior in a dichloroacetic acid (DCA)–1,2-dichloroethane (DCE) mixture. Two types of helix–coil transitions were carried out on the copolymers: solvent-induced transitions in DCA–DCE mixtures at 25°C and thermally induced transitions in a 82:18 (wt %) DCA–DCE mixture. The thermally induced transitions were analyzed by statistical mechanical methods to determine the Zimm-Bragg parameters, σ and s, of the guest residues. The experimental data indicate that, in the nonaqueous solvent, the L -alanine residue stabilizes the α-helical conformation more than the L -leucine residue does. This is in contrast to their behavior in aqueous solution, where the reverse is true. The implications of this finding for the analysis of helical structures in globular proteins are discussed.     

Theory of protein molecule self-organization. III. A calculating method for the probabilities of the secondary structure formation in an unfolded polypeptide chain

A mathematical model is developed adequately describing an unfolded polypeptide chain without long-range interactions in which fluctuating hydrogen-bonded α-helices, β-bends, fragments of helices 3₁₀, and other local structures are formed. The obtained model is a modification of a one-dimensional Ising model for a heteropolymer and allows one to determine the probability of formation of different secondary structures in various parts of a polypeptide chain, provided the whole set of structural thermodynamic parameters exists.     

Suppression of the statistical coil state during the α ↔ β transition in homopolypeptides

A matrix formulation of the conformational partition function has been used to examine helix ? sheet transitions in homopolyamino acids. α-Helices are weighted by Zimm-Bragg parameters σ and s. Antiparallel β-sheets with tight bends are weighted by the parameters t, δ, and τ, where t is the propagation parameter. In addition, each bend contributes a factor δ, and each residue in the sheet that does not have a partner in the preceding strand contributes a factor τ. The helix can be the dominant conformation in a long chain only if two conditions are satisfied simultaneously: (i) s > 1 , and (ii) either s > t, or σ, δ, and τ are assigned values that inflict a greater penalty on antiparallel sheets than on helices. The maximum amount of coil developed during the helix ? sheet transition is strongly influenced by the size of τ, but it is only weakly dependent on the size of δ. Previously reported optical rotatory dispersion, CD, laser Raman, and nmr studies of thermally induced α ? β transitions in homopolyamino acids, notably poly(L -lysine), demonstrate that little random coil is present. If the random coil content is to remain small during the helix ? sheet transition, τ must be significantly less than unity. A small value for τ means that there is a significant penalty assessed to lysyl residues in an antiparallel sheet that do not have a partner in a preceding strand.     

Molecular theory of the helix–coil transition in polyamino acids. V. Explanation of the different conformational behavior of valine,isoleucine, and leucine in aqueous solution

Even though poly(L -valine) and poly(L -isoleucine) both contain residues that are branched at their β-carbon atoms, they exhibit a different behavior of their Zimm-Bragg helix-growth parameter s in aqueous solution. This quantity increases with temperature for poly(L -valine) but decreases for poly(L -isoleucine). The origin of this behavioral difference was examined by computing theoretical values of s versus temperature from interatomic interaction energies, taking solvent (hydrophobic and hydrophilic) effects into account. The calculated s versus temperature curves for both homopolymers are consistent with the observed experimental behavior. The two homopolymers behave differently because of differences in the change in the number of hydration-shell water molecules accompanying their helix–coil transitions. The larger isoleucine side chains are more crowded together in both the α-helical and coil forms than are those of valine. Therefore, there is a smaller change in hydration of the isoleucine side chains compared to that of the valine side chains in the helix–coil transition. By analyzing the effects of hydration on the s versus temperature curves, it is possible to account also for the experimental curve for poly(L -leucine), which exhibits an intermediate behavior between those for poly(L -valine) and poly(L -isoleucine).     

Theory of protein molecule self-organization. I. Thermodynamic parameters of local secondary structures in the unfolded protein chain

This paper presents the results of a stereochemical analysis of local interactions in unfolded protein chains (sterical repulsions, hydrogen, and hydrophobic bonds, etc.) by means of space-filling modeles. On the basis of this analysis, an evaluation is made of thermodynamic parameters controlling the building-in of all the 20 natural amino acid residues in all the physically possible position of local secondary structures (α-helices, including α-helices with short fragments of helices 3₁₀ at the C-terminus; β-bends of different types, helices 3₁₀, and their combinations) as well as thermodynamic parameters of separate hydrogen bonds of polar side groups with the neighbor peptide groups (“local contacts”). The accuracy of the obtained results is discussed.     

Model calculations on the kinetics of oligonucleotide double helix coil transitions. Evidence for a fast chain sliding reaction

Relaxation data obtained previously for the double helix coil transition of oligoriboadenylates and oligoribouridylates are compared to the results of numerical calculations according to various models. In these models the helix coil transition is described by individual rate constants for the first steps of helix formation, whereas the rate constants of the following steps of helix chain growth are assumed to be uniform. The existence of various helix intermediates containing the same number of base pairs is accounted for by statistical factors. First a quasistationary treatment of a zipper model is used for an analysis of the influence of various model parameters. Then relaxation spectra are calculated including helix coil intermediates explicitly without any assumption of quasistationarity. The relaxation spectrum calculated for any chain length N comprises N—1 fast processes with time constants in the range of 0.1 to 0.5 μs and one slow process with a time constant τ depending upon the nucleotide concentration (τ is usually in the ms time range). The fast processes are associated mainly with the unzippering at helix ends and are usually characterized by relatively small amplitudes, whereas the slow process represents the overall helix coil transition usually characterized by a very large amplitude.Consideration of staggered helix series (where the different helix scries are coupled to each other by the single stranded state) leads to a spectrum of slow relaxation processes with one separate relaxation process for each helix series. It is shown that this “non-sliding” staggering zipper model is not consistent with the experimental results. The measured relaxation curves can be represented by single exponentials for nucleotide chain lengths 8 to 11 (within experimental accuracy). This is also true for conditions where several, clearly separated time constants should be expected according to the theoretical model. The experimental data suggest the existence of a direct coupling between different series of staggered helices by a chain sliding mechanism with a time constant < 1ms. Chain sliding may be explained by diffusion of helix defects along the double helix such as diffusion of small loops. A simple model calculation for the diffusion of a bulge loop assuming quasistationarity suggests a sliding time constant around 100 μs for a helix comprising 10 base pairs.Finally some thermodynamic and kinetic parameters are evaluated according to the “sliding” staggering zipper model: The negative activation enthalpy observed for helix recombination can he described using a series of nucleation parameters indicating reduced stability constants for the first three base pairs. Nucleation may usually be achieved with the formation of the third or fourth base pair depending upon the magnitude of the chain growth parameter. The rate constant of helix chain growth is around 10⁶ s^?1 at 0.05 M [Na⁺] and increases to about 4 × 10⁶ s^?1 at 0.17 M [Na⁺].     

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     

Helix-coil dynamics of a Z-helix hairpin

The helix–coil transition of a Z-helix hairpin formed from d(C-G)₅T₄(C-G)₅ has been characterized by equilibrium melting and temperature jump experiments in 5M NaClO₄ and 10 mM Na₂HPO₄, pH 7.0. The melting curve can be represented by a simple all-or-none transition with a midpoint at 81.6 ± 0.4°C and an enthalpy change of 287 ± 15 kJ/mole. The temperature jump relaxation can be described by single exponentials at a reasonable accuracy. Amplitudes measured as a function of temperature provide equilibrium parameters consistent with those derived from equilibrium melting curves. The rate constants of Z-helix formation are found in the range from 1800 s^?1 at 70°C to 800 s^?1 at 90°C and are associated with an activation enthalpy of ?(50 ± 10) kJ/mole, whereas the rate constants of helix dissociation are found in the range from 200 s^?1 at 70°C to 4500 s^?1 at 90°C with an activation enthalpy +235 kJ/mole. These parameters are consistent with a requirement of 3–4 base pairs for helix nucleation. Apparently nucleation occurs in the Z-helix conformation, because a separate slow step corresponding to a B to Z transition has not been observed. In summary, the dynamics of the Z-helix–coil transition is very similar to that of previously investigated right-handed double helices.     

Dielectric dispersion of the -helix at the transition region to random coil

Dielectric studies have been carried out for the helix–coil transition of poly-β-benzyl-L -aspartate with m-cresol as a solvent. The transition of the solute molecules has been sharply reflected as a characteristic change in the dielectric dispersion curves in changing temperature. Two polarizations, one having a low and the other a high critical frequency, have appeared. According to theoretical considerations of a model of a broken helix, the former is found to come from the orientation. of helical sequences and the latter from the chemical relaxation due to the helix–coil transition. It also seems likely that the unfolded chain may have a polarizability which could not be neglected at the high-temperature side of the transition.     

The formation of the triple helix in a mixed solution of (Pro-Pro-Gly)n and (Pro-Pro-Gly)n'. A model of molecular size recognition

A theoretical analysis is given of the triple-helix–random-coil transition in a mixed solution of poly(Pro-Pro-Gly)_n with two different but defined degrees of polymerization n and n′. Because of the highly cooperative nature of this helix–coil transition, each polypeptide chain tends to form a triple helix with other polypeptide chains with the same degree of polymerization (size recognition). Occurrence of triple helices consisting of polypeptide chains with different degrees of polymerization (error in recognition) is studied in detail as a function of the cooperativity, and n and n′. Implication of this analysis for molecular recognition in general is discussed.     

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.     

Helix–coil stability constants for the naturally occurring amino acids in water. XXI. Glutamine parameters from random poly(hydroxypropylglutamine-co-L-glutamine) and poly(hydroxybutylglutamine-co-L-glutamine)

Water-soluble, random copolymers containing L -glutamine and either N⁵-(3-hydroxypropyl)-L -glutamine or N⁵-(4-hydroxybutyl)-L -glutamine were synthesized, fractionated, and characterized. The thermally induced helix–coil transitions of these copolymers were studied in water. A short-range interaction theory was used to deduce the Zimm-Bragg parameters σ and s for the helix–coil transition in poly(L -glutamine) in water from an analysis of the melting curves of the copolymers in the manner described in earlier papers. The computed values of s indicate that L -glutamine is helix-indifferent at low temperature and a helix-destabilizing residue at high temperature in water. At all temperatures in the range of 0–70°C, the glutamine residue promotes helix–coil boundaries since the computed value of σ is large.     

Role of interstrand loops in the formation of intramolecular cross-beta-sheets by homopolyamino acids

A matrix treatment of the formation of intramolecular anti-parallel β-sheets from a statistical coil has been extended to incorporate interstrand loops of arbitrary size. The behavior of the model is compared with a simpler version in which all pairs of contiguous strands were connected by β-bends. When large interstrand loops are allowed, there are many more types of sheets than is the case when all contiguous strands must be connected by tight or β-bends. For this reason, the larger interstrand loops make it easier to introduce the initial sheet into a statistical coil, and the sheet content is enhanced in the early stages of stages of sheet formation (i.e., at small values of the growth parameter t). As the transition continues (i.e., as t increases), a stage will be reached where occupancy of the statistical coil state is negligible because nearly all residues are in sheets or interstrand loops. Now, additional sheet formation can be accomplished only at the expense of residues in the interior of interstrand loops. For this reason, the larger interstrand loops make it more difficult to complete the final stages of sheet formation. These effects are especially dramatic in the formation of cross-β-sheets.     

Simulation of α-helix–coil transitions in simplified polyvaline: Equilibrium properties and brownian dynamics

A quantitative understanding of helix–coil dynamics will help explain their role in protein folding and in folded proteins. As a contribution to the understanding, the equilibrium and dynamical aspects of the helix–coil transition in polyvaline have been studied by computer simulation using a simplified model of the polypeptide chain. Each amino acid residue is treated as a single quasiparticle in an effective potential that approximates the potential of mean force in solution. The equilibrium properties examined include the helix–coil transition and its dependence on chain position and well depth at the coil–helix interface. A stochastic simulation of the Brownian motion of the chain in its solvent surroundings has been used to investigate dynamical properties. Time histories of the dihedral angles have been used to study the behavior of the helical structure. Auto and cross-correlation functions have been calculated from the time histories and from the state (helix or coil) functions of the residues with relaxation times of tens to hundreds of picoseconds. Helix–coil rate constants of tens of ns^?1 were found for both directions of the transition. © 1993 John Wiley & Sons, Inc.     

Alpha helix capping in synthetic model peptides by reciprocal side chain–main chain interactions: Evidence for an N terminal “capping box”

A significant fraction of the amino acids in proteins are alpha helical in conformation. Alpha helices in globular proteins are short, with an average length of about twelve residues, so that residues at the ends of helices make up an important fraction of all helical residues. In the middle of a helix, H-bonds connect the NH and CO groups of each residue to partners four residues along the chain. At the ends of a helix, the H-bond potential of the main chain remains unfulfilled, and helix capping interactions involving bonds from polar side chains to the NH or CO of the backbone have been proposed and detected. In a study of synthetic helical peptides, we have found that the sequence Ser-Glu-Asp-Glu stabilizes the alpha helix in a series of helical peptides with consensus sequences. Following the report by Harper and Rose, which identifies SerXaaXaaGlu as a member of a class of common motifs at the N termini of alpha helices in proteins that they refer to as “capping boxes,” we have reexamined the side chain–main chain interactions in a varient sequence using ¹H NMR, and find that the postulated reciprocal side chain-backbone bonding between the first Ser and last Glu side chains and their peptide NH partners can be resolved: Deletion of two residues N terminal to the Ser-Glu-Asp-Glu sequence in these peptides has no effect on the initiation of helical structure, as defined by two-dimensional (2D) NMR experiments on this variant. Thus the capping box sequence Ser-Glu-Asp-Glu inhibits N terminal fraying of the N terminus of alpha helix in these peptides, and shows the side chain–main chain interactions proposed by Harper and Rose. It thus acts as a helix initiating signal. Since normal a helix cannot propagate beyond the N terminus of this structure, the box acts as a termination signal in this direction as well. © 1994 John Wiley & Sons, Inc.     

Thermodynamic parameters of the helix-coil transition in polypeptide chains. III. Random copolymers of L-leucine and L-glutamic acid.

Conformational transitions induced by pH changes in random copolymers of leucine and glutamic acid have been studied. Significant differences were observed in the potentiometric titration curves of copolymers with small (up to 4%) and large leucine contents. The helical stability of copolymers with small leucine content, determined from titration curves by the Zimm and Rice method, decreases slightly with an increase in the leucine content, whereas the helical stability of copolymers with large leucine content increases sharply with an increase of the leucine content. It is shown that copolymers with large leucine content aggregate in the region of transition into the helical state, but the increase of their helical state stability is not connected with intermolecular aggregation, as it was also observed for a nonaggregating fraction isolated from one of the copolymers by gel chromatography. A conclusion is made that the helix–coil equilibrium constant s for leucine does not itself exceed the s constant for uncharged polyglutamic acid. The stabilization of the helical state in copolymers with large leucine content is due to intramolecular aggregation of helices in these copolymers. The analysis of the leucine residue distribution between helical and nonhelical regions in globular proteins also gives no real arguments to ascribe special helix-forming properties to leucine.     

A nuclear magnetic resonance study of the helix–coil transition of poly(L-glutamic acid)

There have been many reports that the nuclear magnetic resonance (nmr) spectra of a large number of polypeptides exhibit peak doubling of the α-carbon and the α-carbon proton in the helix–coil transition region. One apparent exception to this generalization has been polypeptides with ionizable side chains, where the helix–coil transition is induced by changes in pH in aqueous solution. Because it is important to establish the proper theoretical reason for the peak doubling and its relation to the rate of conformational change of amino acid residues, we have reexamined the proton and carbon-13 nmr spectra, at high field, for two polydisperse samples of poly(L -glutamic acid). Doubling of the α-carbon proton resonance as well as those of the α- and β-carbon, and backbone carbonyl are observed for a low-molecular-weight sample (DP = 54), while a higher molecular weight sample (DP = 309), exhibits only single resonances. Thus, polydispersity by itself is not sufficient to observe peak doubling; low-molecular weight is also required.