Principles of Protein Structural Organization - G. Schulz 1982

Statistical Mechanics of the Helix-Coil Transition
The Zimm-Bragg Model for the Helix-Coil Transition
Relationship between s and Temperature

Near the transition point, the relative statistical weight is linearly related to Temperature. This relationship is important because helix-coil transitions are frequently observed when the temperature changes. It can be derived by returning to equation (A.4), but reducing it to two states

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Thus, in the transition region (s ≃ 1), the relative change in s as a function of T is approximately constant, i.e., The change in s is approximately proportional to the temperature change (Fig. A.2).

Transition temperature

The transition temperature is approximately proportional to the enthalpy change. The transition occurs at s = 1, that is, at z^a = = z^c or

Assuming that E = E_a = const throughout the a-region of size Ω_a and E = E_c = const throughout the coil region of size Ω_c, we obtain

Transitions occur only when the lower-energy state occupies a smaller conformational space. When Ω_c > Ω_a, the transition temperature becomes negative if the energy of the a-conformation is less favorable than that of the coil conformation, E_a > E_c. Furthermore, transitions take place

Fig. A.2.

Relationship between the relative statistical weight s, temperature T, and enthalpy difference ∆H near the transition point (s = 1, T = θ) for two different polymers.

This is a graphical representation of equations (A.18) and (A.19). Both values ∆H(¹) and ∆H(²) are negative, and ∆H(²)<∆H(¹). The difference in transition temperatures θ(²)—θ(¹) is proportional to the difference ∆H(²) — ∆H(¹). At a given temperature T₀, both polymers have different relative statistical weights s(¹) and s(²). These values of s determine the propensity of the polymers to form a helix at a given temperature.

when the state with lower energy E (stronger bonding) occupies a smaller conformational space 2. In our example, when 2_a (Fig. 2.3), the helical conformation is adopted only if it is energetically favorable. The direction of the transition is such that at temperatures below θ, the lower-energy state prevails, whereas at temperatures above θ, the state with the larger conformational space dominates (Fig. A.2). The higher the transition temperature θ, the smaller the difference in conformational spaces and the greater the number of energetically favorable helical Conformations.