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 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
\theta=
| \left\langlei\right\rangle | |
| N |
\left\langlei\right\rangle
N
| Dimer sequence | Statistical weight | |
|---|---|---|
...CC... | 1 | |
...CH... | \sigmas | |
...HC... | \sigmas | |
...HH... | \sigmas2 |
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
\sigmas
\sigma
s
s=
| [H] | |
| [C] |
s
\sigma\ll1<s
From these parameters, it is possible to compute the fractional helicity
\theta
\left\langlei\right\rangle
\left\langlei\right\rangle=\left(
| s | \right) | |
| q |
| dq | |
| ds |
q
\theta=
| 1 | \left( | |
| N |
| s | \right) | |
| q |
| dq | |
| ds |
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
l{Z}=\left(0,1\right) ⋅ \left\{
| N | |
| \prod | |
| j=1 |
Wj\right\} ⋅ \left(1,1\right)
where the 2x2 transfer matrix Wj of the jth residue equals the matrix of statistical weights for the state transitions
Wj=\begin{bmatrix} sj&1\\ \sigmajsj&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.