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2012
Helix-coil transition in polypeptides is an example of a spin model with a preferred spin direction, in the sense that a theoretical formulation of this problem requires to assign a preferred value of spin to the helical conformation in order to account for different symmetries of the helical vs. the coil states. This leads to the spin Hamiltonian of the Generalized Model of Polypeptide Chain (GMPC) variety as opposed to the Potts model variety, both with many-body interactions. We compare the explicit solution of the Potts model and the solution of the GMPC within the transfer-matrix formalism. Comparison of both secular equations reveals that the largest eigenvalue of the Potts model with ∆ many-body interactions is identical to the largest eigenvalue of the GMPC model with ∆ − 1 many-body interactions, indicating the equivalence of both free energies. In distinction, the second largest eigenvalues do not coincide, leading to different thermal behavior of the spatial correlation length, related to the helix-coil transition interval. Spin models with built-in spin anisotropy thus engender different physical properties in the thermodynamic limit that we explore in detail.
Helix-coil transition in terms of Potts-like spins
2013
Abstract. In the spin model of a helix-coil transition in polypeptides a preferred value of spin has to be assigned to the helical conformation, in order to account for different symmetries of the helical vs. the coil states, leading thus to the Generalized Model of Polypeptide Chain (GMPC) Hamiltonian as opposed to the Potts model Hamiltonian, both with many-body interactions. Comparison of explicit transfer matrix secular equations of the Potts model and the GMPC model reveals that the largest eigenvalue of the Potts model with ∆ many-body interactions coincides with the largest eigenvalue of the GMPC model with ∆ − 1 many-body interactions, indicating the identity of both free energies. In distinction, the second largest eigenvalues in both models do not coincide, indicating a different behavior for the spatial correlation length that in its turn defines the width of the helix-coil transition interval. We explore in detail the thermodynamic consequences, resulting from spin models with and without the built-in spin anisotropy, that should indicate which model to favour as a more appropriate description of the equilibrium physical properties pertaining to the helix-coil transition.
Water-Polypeptide Interaction in Classical Models of Helix-Coil Transition
2016
Zimm-Bragg model is the simplest to describe the conformational transitions in biopolymers and is regularly used for preocessing the experimental data. We review the model and its Hamiltonian definition with the goal to introduce the interaction with water into the picture. We show how modified ZB model with the account of water-polypeptide interactions allows to describe both cold denaturation and helix-coil transition and derive such the formula explicitly. The obtained theoretical expression for the helicity degree contains two independent parameters that can be fitted with the experimental data to determine the parameters of cold denaturation and helix-coil transition from a single fit and for a single set of experimental data.
Exactly solvable model for helix-coil-sheet transitions in protein systems
Physical Review E, 2010
In view of the important role helix-sheet transitions play in protein aggregation, we introduce a simple model to study secondary structural transitions of helix-coil-sheet systems using a Potts model starting with an effective Hamiltonian. This energy function depends on four parameters that approximately describe entropic and enthalpic contributions to the stability of a polypeptide in helical and sheet conformations. The sheet structures involve long-range interactions between residues which are far in sequence, but are in contact in real space. Such contacts are included in the Hamiltonian. Using standard statistical mechanical techniques, the partition function is solved exactly using transfer matrices. Based on this model, we study thermodynamic properties of polypeptides, including phase transitions between helix, sheet, and coil structures.
The Journal of Chemical Physics, 2009
The generalized model of polypeptide chains is extended to describe the helix-coil transition in a system comprised of two chains interacting side-by-side. The Hamiltonian of the model takes into account four possible types of interactions between repeated units of the two chains, i.e., helix-helix, helix-coil, coil-helix, and coil-coil. Analysis reveals when the energy I hh + I cc of ͑h-h, c-c͒ interactions overwhelms the energy I hc + I ch of mixed ͑h-c, c-h͒ interactions, the correlation length rises substantially, resulting in narrowing of the transition interval. In the opposite case, when I hh + I cc Ͻ I hc + I ch , nontrivial behavior of the system is predicted where an intermediate plateau appears on the denaturation curve. For the latter case, calculations of the number of junctions and the average length of helical segments indicate rearrangement of helical segments at the transition point. Conceptual links are established with experimentally oriented theories of Ghosh and Dill ͓J. Am. Chem. Soc. 131, 2306 ͑2009͔͒ and Skolnick and Holtzer ͓Biochemistry 25, 6192 ͑1986͔͒, providing a potential explanation for both favorable helix formation and disfavored intersegment interactions from the same theoretical perspective.
The generalized model of polypeptide chain describing the helix-coil transition in biopolymers
Arxiv preprint cond-mat/ …, 2005
In this paper we summarize some results of our theoretical investigations of helix-coil transition both in single-strand (polypeptides) and two-strand (polynucleotides) macromolecules. The Hamiltonian of the Generalized Model of Polypeptide Chain (GMPC) is introduced to describe the system in which the conformations are correlated over some dimensional range ∆ (it equals 3 for polypeptide, because one H-bond fixes three pairs of rotation, for double strand DNA it equals to one chain rigidity because of impossibility of loop formation on the scale less than ∆ ). The Hamiltonian does not contain any parameter designed especially for helix-coil transition and uses pure molecular microscopic parameters (the energy of hydrogen bond formation, reduced partition function of repeated unit, the number of repeated units fixed by one hydrogen bond, the energies of interaction between the repeated units and the solvent molecules). To calculate averages we evaluate the partition function using transfer-matrix approach. The GMPC allowed to describe the influence of a number of factors, affecting the transition, basing on a unified microscopic approach. Thus we obtained, that solvents change transition temperature and interval in different ways, depending on type of solvent and on energy of solvent-macromolecule interaction; stacking on the background of Hbonding increases stability and decreases cooperativity of melting. For heterogeneous DNA we could analytically derive well known formulae for transition temperature and interval. In the framework of GMPC we calculate and show the difference of two order parameters of helixcoil transition -the helicity degree, and the average fraction of repeated units in helical conformation. Given article has the aim to review the results obtained during twenty years in the context of GMPC.
Helix-Coil transition in polypeptides: A microscopical approach
Biopolymers, 1990
ABSTRACT In the framework of an earlier constructed model [N.S. Ananikyan et al. (1990) Biopolymers, Vol. 30, pp. 357-367], some analytical estimates for the correlation length and degree of helicity near the transition point were obtained in the case of an arbitrary topology of hydrogen bond closing (delta). It was shown that the Zimm-Bragg cooperativity parameter sigma is determined by the set of (delta-1) amino acid residues and so is nonlocal. An analytic expression for cooperativity parameters in a heteropolypeptide chain was obtained and numerical calculations showed that in case of heteropolypeptide with random primary structure the nonlocality of cooperativity parameter influenced the temperature dependence of helicity degree.
The helix-coil transition revisited
Proteins, 2007
In this article, we perform a dynamic Monte Carlo simulation study of the helix–coil transition by using a bond-fluctuation lattice model. The results of the simulations are compared with those predicted by the Zimm–Bragg statistical thermodynamic theory with propagation and nucleation parameters determined from simulation data. The Zimm–Bragg theory provides a satisfactory description of the helix–coil transition of a homopolypeptide chain of 32 residues (N = 32). For such a medium-length chain, however, the analytical equation based on a widely-used large-N approximation to the Zimm–Bragg theory is not suitable to predict the average length of helical blocks at low temperatures when helicity is high. We propose an analytical large-eigenvalue (λ) approximation. The new equation yields a significantly improved agreement on the average helix-block length with the original Zimm–Bragg theory for both medium and long chain lengths in the entire temperature range. Nevertheless, even the original Zimm–Bragg theory does not provide an accurate description of helix–coil transition for longer chains. We assume that the single-residue nucleation of helix formation as suggested in the original Zimm–Bragg model might be responsible for this deviation. A mechanism of nucleation by a short helical block is proposed by us and provides a significantly improved agreement with our simulation data. Proteins 2007. © 2007 Wiley-Liss, Inc.
Unified description of solvent effects in the helix-coil transition
Physical Review E, 2014
We analyze the problem of the helix-coil transition in explicit solvents analytically by using spin-based models incorporating two different mechanisms of solvent action: explicit solvent action through the formation of solvent-polymer hydrogen bonds that can compete with the intrinsic intra-polymer hydrogen bonded configurations (competing interactions) and implicit solvent action, where the solvent-polymer interactions tune biopolymer configurations by changing the activity of the solvent (non-competing interactions). The overall spin Hamiltonian is comprised of three terms: the background in vacuo Hamiltonian of the "Generalized Model of Polypeptide Chain" type and two additive terms that account for the two above mechanisms of solvent action. We show that on this level the solvent degrees of freedom can be explicitly and exactly traced over, the ensuing effective partition function combining all the solvent effects in a unified framework. In this way we are able to address helix-coil transitions for polypeptides, proteins, and DNA, with different buffers and different external constraints. Our spin-based effective Hamiltonian is applicable for treatment of such diverse phenomena as cold denaturation, effects of osmotic pressure on the cold and warm denaturation, complicated temperature dependence of the hydrophobic effect as well as providing a conceptual base for understanding the behavior of Intrinsically Disordered Proteins and their analogues.
Biopolymers, 1988
A method is developed to extract the entropy of polypeptides and proteins from samples of conformations. It is based on techniques suggested previously by Mehovitch, 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, 636943751, 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 Car10 method to generate samples of conformations in the a-helical and hairpin regions, respectively, at T = 100 K. For the a-helix, the four dihedral angles of the N-and C-terminal residues are found to undergo full rotational variation. The results show that the a-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 A F -0.4 and A E -0.3 kcal/mole/residue, respectively. It should be noted that the contribution of the entropy A S to A F is significant (TAS -0.1 kcal/mole/residue). Also, the entropy of the a-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 a-helix.