Zimm–Bragg model

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In statistical mechanics, the Zimm–Bragg model is a helix-coil transition model that describes helix-coil transitions of macromolecules, usually polymer chains. Most models provide a reasonable approximation of the fractional helicity of a given polypeptide; the Zimm–Bragg model differs by incorporating the ease of propagation with respect to nucleation.

Helix-coil transition models[edit]

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'.[1]

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,  \theta \ , defined as

 \theta = \frac{\left \langle i \right \rangle}{N}

where

 \left \langle i \right \rangle \ is the average helicity and
 N \ is the number of helix or coil units.

Zimm-Bragg[edit]

Dimer sequence Statistical weight
 ...CC... \  1 \
 ...CH... \  \sigma s \
 ...HC... \  \sigma s \
 ...HH... \  \sigma s^2 \

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  \sigma s \ , where  \sigma \ is the nucleation parameter and

 s = \frac{[H]}{[C]}

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

 \sigma \ll 1 < s \

which makes the propagation of a helix more favorable than nucleation of a helix from coil state.[2]

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

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

where  s \ is the statistical weight and  q \ 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

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

Statistical mechanics[edit]

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[3] 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 Wj 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.

See also[edit]

References[edit]

  1. ^ Samuel Kutter; Eugene M. Terentjev (16 October 2002). "Networks of helix-forming polymers". European Physical Journal E (EDP Sciences) 8 (5): 539–47. arXiv:cond-mat/0207162. Bibcode:2002EPJE....8..539K. doi:10.1140/epje/i2002-10044-x. PMID 15015126. 
  2. ^ Ken A. Dill; Sarina Bromberg (2002). Molecular Driving Forces - Statistical Thermodynamics in Chemistry and Biology. Garland Publishing, Inc. p. 505. 
  3. ^ Zimm, BH; Bragg JK (1959). "Theory of the Phase Transition between Helix and Random Coil in Polypeptide Chains". Journal of Chemical Physics 31: 526–531. Bibcode:1959JChPh..31..526Z. doi:10.1063/1.1730390.