# 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 (self-replication) with respect to nucleation.

## 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, ${\displaystyle \theta \ }$, defined as

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

where

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

## Zimm-Bragg

Dimer sequence Statistical weight
${\displaystyle ...CC...\ }$ ${\displaystyle 1\ }$
${\displaystyle ...CH...\ }$ ${\displaystyle \sigma s\ }$
${\displaystyle ...HC...\ }$ ${\displaystyle \sigma s\ }$
${\displaystyle ...HH...\ }$ ${\displaystyle \sigma s^{2}\ }$

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 ${\displaystyle \sigma s\ }$, where ${\displaystyle \sigma \ }$ is the nucleation parameter and

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

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

${\displaystyle \sigma \ll 1

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 ${\displaystyle \theta \ }$. The average helicity ${\displaystyle \left\langle i\right\rangle \ }$ is given by

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

where ${\displaystyle s\ }$ is the statistical weight and ${\displaystyle q\ }$ 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

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

## 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

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

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

${\displaystyle \mathbf {W} _{j}={\begin{bmatrix}s_{j}&1\\\sigma _{j}s_{j}&1\end{bmatrix}}}$

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.