= Zimm-Bragg model
{c}
{title2=$f_h=\tfrac12+(s-1)/(2\sqrt{(s-1)^2+4s\sigma})$}
{wiki=Zimm–Bragg_model}
A homogeneous Zimm-Bragg chain assigns propagation weight $s>0$ to each helical unit and initiation factor $\sigma>0$ to each helical run. With states ordered as coil and helix, one possible <transfer matrix for a classical spin chain> is
$$
W=\begin{pmatrix}1&\sigma s\\1&s\end{pmatrix}.
$$
Its largest <eigenvalue> is $\lambda_+=(1+s+\sqrt{(s-1)^2+4s\sigma})/2$. In the long-chain limit the <partition function> grows as $\lambda_+^N$, so logarithmic differentiation gives the helical fraction $f_h=s\,\partial\ln\lambda_+/\partial s$, equal to the displayed expression. If $Q=(s-1)^2+4s\sigma$, then
$$
\frac{df_h}{ds}=\frac{\sigma(s+1)}{Q^{3/2}}>0,\qquad f_h(1)=\frac12,\qquad f_h'(1)=\frac1{4\sqrt\sigma}.
$$
For $\sigma=1$, $f_h=s/(1+s)$ and units are noncooperative. A small initiation factor suppresses boundaries and sharpens the crossover around $s=1$, with width of order $\sqrt\sigma$. For every fixed $\sigma>0$ the long-chain expression is smooth; a discontinuous step appears only in the singular $\sigma\downarrow0$ limit. https://doi.org/10.1063/1.1730390[Theory of the Phase Transition between Helix and Random Coil in Polypeptide Chains, Journal of Chemical Physics 31 (1959), 526–535: propagation and initiation parameters and characteristic-matrix solution]{ref}
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