Zimm–Bragg model

Helix-coil transition models assume that polypeptides are linear chains composed of interconnected segments. Further, models group these sections into two broad categories: coils, random conglomerations of disparate unbound pieces, are represented by the letter 'C', and helices, ordered states where the chain has assumed a structure stabilized by hydrogen bonding, are represented by the letter 'H'.{{Cite journal|author=Samuel Kutter|author2=Eugene M. Terentjev|author2-link=Eugene M. Terentjev|journal=European Physical Journal E|date=16 October 2002|title=Networks of helix-forming polymers|volume=8|issue=5|pages=539–47|pmid=15015126|publisher=EDP Sciences|doi=10.1140/epje/i2002-10044-x|arxiv = cond-mat/0207162 |bibcode = 2002EPJE....8..539K |s2cid=39981396 }}

Thus, it is possible to loosely represent a macromolecule as a string such as CCCCHCCHCHHHHHCHCCC and so forth. The number of coils and helices factors into the calculation of fractional helicity, $\backslash theta\; \backslash$, defined as

:$\backslash theta\; =\; \backslash frac\{\backslash left\; \backslash langle\; i\; \backslash right\; \backslash rangle\}\{N\}$

where

:$\backslash left\; \backslash langle\; i\; \backslash right\; \backslash rangle\; \backslash$ is the average helicity and

:$N\; \backslash$ is the number of helix or coil units.

The Zimm–Bragg model takes the cooperativity of each segment into consideration when calculating fractional helicity. The probability of any given monomer being a helix or coil is affected by which the previous monomer is; that is, whether the new site is a nucleation or propagation.

By convention, a coil unit ('C') is always of statistical weight 1. Addition of a helix state ('H') to a previously coiled state (nucleation) is assigned a statistical weight $\backslash sigma\; s\; \backslash$, where $\backslash sigma\; \backslash$ is the nucleation parameter and $s\; \backslash$ is the equilibrium constant

:$s\; =\; \backslash frac\{[H]\}\{[C]\}$

Adding a helix state to a site that is already a helix (propagation) has a statistical weight of $s\; \backslash$. For most proteins,

:

which makes the propagation of a helix more favorable than nucleation of a helix from coil state.{{Cite book|author=Ken A. Dill|author2=Sarina Bromberg |title=Molecular Driving Forces – Statistical Thermodynamics in Chemistry and Biology|publisher=Garland Publishing, Inc.|year=2002|page=505}}

From these parameters, it is possible to compute the fractional helicity $\backslash theta\; \backslash$. The average helicity $\backslash left\; \backslash langle\; i\; \backslash right\; \backslash rangle\; \backslash$ is given by

:$\backslash left\; \backslash langle\; i\; \backslash right\; \backslash rangle\; =\; \backslash left(\backslash frac\{s\}\{q\}\backslash right)\backslash frac\{dq\}\{ds\}$

where $q\; \backslash$ is the partition function given by the sum of the probabilities of each site on the polypeptide. The fractional helicity is thus given by the equation

:$\backslash theta\; =\; \backslash frac\{1\}\{N\}\backslash left(\backslash frac\{s\}\{q\}\backslash right)\backslash frac\{dq\}\{ds\}$

The Zimm–Bragg model is equivalent to a one-dimensional Ising model and has no long-range interactions, i.e., interactions between residues well separated along the backbone; therefore, by the famous argument of Rudolf Peierls, it cannot undergo a phase transition.

The statistical mechanics of the Zimm–Bragg model{{Cite journal| last = Zimm | first = BH |author2=Bragg JK | year = 1959 | title = Theory of the Phase Transition between Helix and Random Coil in Polypeptide Chains | journal = Journal of Chemical Physics | volume = 31 | issue = 2 | pages = 526–531 | doi = 10.1063/1.1730390 | author-link = Bruno H. Zimm|bibcode = 1959JChPh..31..526Z }} may be solved exactly using the transfer-matrix method. The two parameters of the Zimm–Bragg model are σ, the statistical weight for nucleating a helix and s, the statistical weight for propagating a helix. These parameters may depend on the residue j; for example, a proline residue may easily nucleate a helix but not propagate one; a leucine residue may nucleate and propagate a helix easily; whereas glycine may disfavor both the nucleation and propagation of a helix. Since only nearest-neighbour interactions are considered in the Zimm–Bragg model, the full partition function for a chain of N residues can be written as follows

:$$

\mathcal{Z} = \left( 0, 1\right) \cdot \left\{ \prod_{j=1}^{N} \mathbf{W}_{j} \right\} \cdot \left( 1 , 1\right)

where the 2x2 transfer matrix W_j of the jth residue equals the matrix of statistical weights for the state transitions

:$$

\mathbf{W}_{j} = \begin{bmatrix}

s_{j} & 1 \\

\sigma_{j} s_{j} & 1

\end{bmatrix}

The row-column entry in the transfer matrix equals the statistical weight for making a transition from state row in residue j − 1 to state column in residue j. The two states here are helix (the first) and coil (the second). Thus, the upper left entry s is the statistical weight for transitioning from helix to helix, whereas the lower left entry σs is that for transitioning from coil to helix.

• Alpha helix
• Lifson–Roig model
• Random coil
• Statistical mechanics

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Category:Polymer physics

Category:Protein structure

Category:Statistical mechanics

Category:Thermodynamic models