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'.

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
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.

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 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.