Principles of Protein Structural Organization - H. Schultz 1982

Statistical Mechanics of the Helix-Coil Transition
The Zimm-Bragg Model for the Helix-Coil Transition
Matrix Representation

Matrix formulation of the Distribution Function. Kramers and Wannier [790] suggested writing equation (A.6) in a compact matrix form that simplifies the calculation of Z.

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Let us verify that this expression is consistent with equation (A.6) for small N:

These relations can also be extended to large values of N.

The simple case of a homopolymer is satisfactorily described by an Nth-order matrix in terms of eigenvalues. Applying equation (A.7) to homopolymers makes all quantities zia, as well as zci, identical. To further simplify the expressions, the relative statistical weights s can be expressed as

We can treat (zсі)N as a constant and omit it, since we are interested only in The change in Z during the helix–coil transition rather than its absolute value.

By diagonalizing the matrix, this can be reduced to the expression:

When

that is, Z can be explicitly expressed as a function of the eigenvalues.

Since λ1 > 1 > λ2 for all s and σ, the partition function for large N can be approximated as



Last update: 06/08/2026

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