The theory of helix-coil transition

2 There is no (0, i//) set associated with the N-terminal a carbon. The (0, VO pair associated with the C-terminal a carbon defines the relative orientation of the C-terminal peptide bond and the terminal COOH group. 

3 The term coil refers to a random unstructured chain in polymer science. The term may be confusing because in common use the word coil does not necessarily invoke a picture of a random structure. 

Computing the partition function for an /V-statc chain requires enumerating all possible states. The clever trick associated with the helix-coil transition theory is to generalize this calculation using the statistical-weight matrix  

For an AFstate system, the first two states can take the form cc, he, ch, or hh, with corresponding Boltzmann weights 1, v, v, and v2. Putting these weights into a vector in the respective order that corresponds to the columns of M, and multiplying by M yields a vector that lists the weights of all possible three-state sequences  

This result is generalized for an iV-state polypeptide (N + 1 residues)  

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In this section we introduce the matrix method to rewrite the GPF of a linear system of m sites in a more convenient form. This is both an elegant and a powerful method for studying such systems. We start by presenting the so-called Ising model for the simplest system. We assume that each urrit can be in one of two occupational states empty or occupied. Also, we assirme only nearest-neighbor (nn) interactions. Both of these assumptions may be removed. In subsequent sectiorrs and in Chapter 8 we shall discuss four and eight states for each subunit. We shall not discuss the extension of the theory with respect to interactions beyond the nn. Such an extension is used, for example, in the theory of helix-coil transition. 

The above-mentioned type of solvent effects has been incorporated into the theory of helix-coil transition by a number of authors (7,24,36 38), with various types of reactions being assumed. These theories explain why AH becomes temperature-dependent in inverse helix-coil transitions and also permit an estimate of AH in pure helicogenic solvents. Their details are surveyed in our companion review article (11). 

S. Lifson and A. Roig. On the theory of helix-coil transition in polypeptides. J. Chem. Phys., 34 1963-1974, 1961. 

Vedenov, A.A., Dykhne, A.M., Frank-Kamenetskii, A.D., Frank-Kamenetskii, M.D., 1967. A contribution to the theory of helix-coil transition in DNA. Molek. Biol. 1 313-319. 

In section 4.7.2, we developed the theory of helix-coil transition in the vacuum. We have seen that the theory is based on the reduction of the entire two-dimensional configurational space of a single amino acid residue, (piy/i into three coarse states helix (H), coiled (C), and impossible (/). These roughly correspond to the a-helix, the /3-strand, and the inaccessible regions in the Ramachandran maps. Since the / states involve large positive energies, their probability of occurrence is very small and therefore can be ignored. Thus, for a linear system of M amino-acid residues, each microstate of the system is translated into a sequence of macrostates, symbolically... 

As is apparent from Figure 10.1, an a-helical structure imposes fairly rigid constraints on the relative positions of successive residues in a peptide chain. Thus there is a loss of entropy that must be overcome energetically in order for an a-helix to form. To explain the underlying biophysics of this system, John Schellman introduced a theory of helix-coil transitions that is motivated by the Ising model for one-dimensional spin system in physics [180, 170],... 

The expressions whose limits give s and <7 are found to converge rapidly and the estimates for s and a so obtained are close to the values obtained experimentally. While improvement of the calculated values of s and <7 will undoubtedly be possible in the future, these results, together with the results of the theory of the one-dimensional Ising lattice, place the description of helix-coil transitions of polyfa-amino acids) on a firm combined molecular and statistical mechanical basis. 

Although the statistical mechanical theories such as those described above yield exact analytic expressions for various quantities characterizing the conformation of an interrupted helix, those expressions are so complicated that it is of both theoretical and practical value to simplify them, with the imposition of suitable restrictions on parameters, to forms that are amenable to straightforward computations and also, hopefully, to direct comparisons with observed data. Various attempts have been made, and they are summarized in Poland-Scheraga s book (10). Though not available at the time this book was published, the approximations worked out by Okita et al. (13) are of great practical use for their wide applicability and simplicity. Their method is described below in some detail, because it has been consistently used in our statistical-thermodynamic analyses of helix-coil transition phenomena. 

Miller and Flory (43) also computed on the idea that, even in helicogenic solvents, actual polypeptides are at certain stages of helix-coil transition and showed that if the values of s and cr are chosen accordingly, chain-length dependence of 1/2 similar to that in Fig. 21 can be reproduced. Recently, Norisuye et al. (49) have confirmed this with the computation of in terms of Nagai s theory (5). 

Water-soluble random copolymers containing L-serine and A/5-(4-hydroxybutyl)-L-glutamine are prepared, and the thermally induced helix-coil transition of these copolymers in water is studied. The Z/mm-Bragg parameters cr and s for the (hypothetical) helix-coil transition in poly(l-serine) in water are deduced. The values of s, computed from both the Lifson and Allegra theories are (.Llfson, o-1 x 10-4 / Allegra, o = 7.5 x 10 5 s - 0.667/0.726 (273 K), 0.757/0.784 (293 K), 0.777/0.792 (313 K), 0.731/0.744 (333 K),... 

Chapter C deals with molecular dimensions of interrupted helices. Typical theories for mean-square radius of gyration and mean-square end-to-end distance are reviewed. Important predictions from theory are compared with the results of recent light-scattering measurements. Complications attendant upon the analysis of light-scattering data for polypeptides in the helix-coil transition region are discussed. 

Chapter E is devoted to the mean-square dipole moment and mean rotational relaxation time derived from dielectric dispersion measurements. Typical data, both in helieogenic solvents and in the helix-coil transition region, are presented and interpreted in terms of existing theories. At thermodynamic equilibrium, helical and randomly coiled sequences in a polypeptide chain are fluctuating from moment to moment about certain averages. These fluctuations involve local interconversions of helix and random-coil residues. Recently, it has been shown that certain mean relaxation times of such local processes can be estimated by dielectric dispersion experiment. Chapter E also discusses the underlying theory of this possibility. 

This chapter summarizes important data for intrinsic viscosity and translational friction coefficient of polypeptides. The first half of the chapter discusses the data obtained in helicogenic solvents and in helix-breaking solvents. It is actually a supplement to the review article by Benoit et al. (61), in which such data published by 1967 were surveyed critically. The second half of the chapter is concerned with the helix-coil transition region. The context here is largely descriptive because of the lack of relevant theory. 

Since the mathematical expression for < u2) is equivalent to that for , measurements of should provide information which can be utilized to check the theory of , e.g. Eq. (C-3), for polypeptides in the helix-coil transition region. This idea, however, cannot be developed in straightforward fashion because there is no available theory to estimate of interrupted helical polypeptides from dielectric dispersion curves. Therefore, we are forced to proceed on some yet unproven assumptions, or even drastic approximations. 

The Ufson-Roig matrix theory of the helix-coil transition In polyglycine is extended to situations where side-chain interactions (hydrophobic bonds) are present both In the helix and in the random coil. It is shown that the conditional probabilities of the occurrence of any number and size of hydrophobic pockets In the random coil can be adequately described by a 2x2 matrix. This is combined with the Ufson-Roig 3x3 matrix to produce a 4 x 4 matrix which represents all possible combinations of any amount and size sequence of a-helix with random coil containing all possible types of hydrophobic pockets In molecules of any given chain length. The total set of rules is 11) a state h preceded and followed by states h contributes a factor wo to the partition function 12) a state h preceded and followed by states c contributes a factor v to the partition function (3) a state h preceded or followed by one state c contributes a factor v to the partition function 14) a state c contributes a factor u to the partition function IS) a state d preceded by a state other than d contributes a factor s to the partition function 16) a state d preceded by a state d contributes a factor r to the partition function. 

Since the Zimm-Bragg parameters o and s of the naturally occurring amino acids (In water) cannot be obtained from studies of the helix-coil transition in homopolymers, because of experimental difficulties, a technique Is developed to circumvent these problems. It involves the study of the thermally induced transition curves for random copolymers of "guest amino acid residues in a water-soluble host" po y(amino acid). The data may be interpreted with the aid of suitable theories for the helix-coil transition in random copolymers to obtain a and s for the "guest" residues. It is shown in this paper that, for the usual ranges of parameters found for polylamino acids), one of the two lowest order approximations (corresponding to earlier treatments by Lifson and Allegra) is completely adequate. In essence, the low-order approximations hoid if o and s for the two constituents of the copolymer do not differ appreciably from each other. 

N 042 "Molecular Theory of the Helix-Coil Transition in Polylamino acids). III. Evaluation and Analysis of s and [Pg.431]

N 106 "Simplified Theory of the Helix-Coil Transition In DNA Based on a Grand Partition Function"... 

N 107 "Theory of the Helix-Coil Transition in DNA Considered as a Copolymer"... 

The statistical mechanical theory of the helix-coil transition of DNA is improved by introducing approximate normalization factors for the unnormalized statistical weights of finding a given molecule of the assembly in a given microscopic state. 

ORDER-DISORDER THEORY AND APPLICATIONS. Phase transitions in binary liquid solutions, gas condensations, order-disorder transitions in alloys, ferromagnetism, antiferromagnetism, ferroelectncity, anti-ferroelectricity, localized absorptions, helix-coil transitions in biological polymers and the one-dimensional growth of linear colloidal aggregates are all examples of transitions between an ordered and a disordered state. 

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