Zimm–Bragg model

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 (self-replication) with respect to nucleation. It is named for co-discoverers Bruno H. Zimm and J. K. Bragg.

Helix-coil transition models

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

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\$ is the equilibrium constant

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 $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

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 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.

References

^ Samuel Kutter; Eugene M. Terentjev (16 October 2002). "Networks of helix-forming polymers". European Physical Journal E. 8 (5). EDP Sciences: 539–47. arXiv:cond-mat/0207162. Bibcode:2002EPJE....8..539K. doi:10.1140/epje/i2002-10044-x. PMID 15015126. S2CID 39981396.
^ Ken A. Dill; Sarina Bromberg (2002). Molecular Driving Forces – Statistical Thermodynamics in Chemistry and Biology. Garland Publishing, Inc. p. 505.
^ Zimm, BH; Bragg JK (1959). "Theory of the Phase Transition between Helix and Random Coil in Polypeptide Chains". Journal of Chemical Physics. 31 (2): 526–531. Bibcode:1959JChPh..31..526Z. doi:10.1063/1.1730390.

[kutter-1] Samuel Kutter; Eugene M. Terentjev (16 October 2002). "Networks of helix-forming polymers". European Physical Journal E. 8 (5). EDP Sciences: 539–47. arXiv:cond-mat/0207162. Bibcode:2002EPJE....8..539K. doi:10.1140/epje/i2002-10044-x. PMID 15015126. S2CID 39981396.

[dill-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 (2): 526–531. Bibcode:1959JChPh..31..526Z. doi:10.1063/1.1730390.

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