Zimm–Bragg model
# 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

| 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.

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 $W_j$ 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.

