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 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'.
, defined as where The Zimm–Bragg model takes the cooperativity of each segment into consideration when calculating fractional helicity.
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 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.
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 where the 2x2 transfer matrix Wj of the jth residue equals the matrix of statistical weights for the state transitions 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.