Protein Structure & Folding

Step 1: Define the nucleation and propagation parameters

Each residue is either in coil (c) or helix (h) state. Define the propagation parameter $s$ (equilibrium constant for adding a helical residue to an existing helix) and the nucleation parameter $\sigma$ (penalty for initiating a new helical segment, typically $\sigma \sim 10^{-3} - 10^{-4}$):

$$s = \exp\left(-\frac{\Delta G_{\text{prop}}}{k_BT}\right), \quad \sigma = \exp\left(-\frac{\Delta G_{\text{nuc}}}{k_BT}\right)$$

Step 2: Construct the transfer matrix

The statistical weight of each residue depends on its state and that of its predecessor. The 2x2 transfer matrix $\mathbf{M}$ relates the partition function contributions:

$$\mathbf{M} = \begin{pmatrix} 1 & \sigma s \\ 1 & s \end{pmatrix}$$

where rows represent the current state (c, h) and columns the next state (c, h).

Step 3: Write the partition function

For a chain of N residues, the partition function is obtained by matrix multiplication:

$$Z_N = (1, 0) \cdot \mathbf{M}^N \cdot \begin{pmatrix} 1 \\ 1 \end{pmatrix}$$

Step 4: Solve via eigenvalues

For large N, $Z_N$ is dominated by the largest eigenvalue $\lambda_+$ of $\mathbf{M}$:

$$\lambda_{\pm} = \frac{(1+s) \pm \sqrt{(1-s)^2 + 4\sigma s}}{2}$$

Step 5: Calculate the helical fraction

The fraction of residues in helix state is obtained from the derivative of the partition function:

$$\theta = \frac{\partial \ln Z_N}{\partial \ln s} \approx \frac{1}{2} + \frac{s - 1}{2\sqrt{(1-s)^2 + 4\sigma s}}$$

The transition is centered at $s = 1$ and its sharpness is controlled by $\sigma$: smaller $\sigma$ gives a more cooperative (sharper) transition, consistent with the observation that helix nucleation is the rate-limiting step.

Step 1: The ideal (Gaussian) chain — random walk

Model the unfolded protein as N freely jointed segments of length b. Each step is independent and random. The mean-square end-to-end distance is the sum of uncorrelated steps:

$$\langle R^2 \rangle = \sum_{i=1}^{N}\sum_{j=1}^{N}\langle \mathbf{b}_i \cdot \mathbf{b}_j \rangle = Nb^2 \quad (\text{since } \langle \mathbf{b}_i \cdot \mathbf{b}_j \rangle = b^2\delta_{ij})$$

Step 2: Extract the scaling exponent for the ideal chain

The root-mean-square end-to-end distance scales as:

$$R = \sqrt{\langle R^2 \rangle} = bN^{1/2} \implies R \sim N^{\nu} \text{ with } \nu = \frac{1}{2}$$

This applies when the chain has no excluded volume — for example, in a theta solvent where polymer-solvent and polymer-polymer interactions exactly cancel.

Step 3: Self-avoiding walk — the Flory argument

Real chains cannot overlap. Flory balanced the entropic elasticity (which favors $R \sim N^{1/2}$) against the excluded volume repulsion energy. The free energy as a function of R:

$$F(R) \sim k_BT\left(\frac{R^2}{Nb^2} + \frac{vN^2}{R^3}\right)$$

where $v$ is the excluded volume per monomer. The first term is elastic free energy; the second is the two-body repulsion energy density.

Step 4: Minimize to find the Flory exponent

Setting $\partial F/\partial R = 0$:

$$\frac{2R}{Nb^2} - \frac{3vN^2}{R^4} = 0 \implies R^5 \sim vN^3b^2$$

$$R \sim N^{3/5} \implies \nu = \frac{3}{5} = 0.6$$

Step 5: Biological significance

The exact value from renormalization group theory is $\nu \approx 0.588$ in 3D, close to Flory's estimate. For denatured proteins in good solvent: $R_g \approx 2.0 \times N^{0.59}$ A. In poor solvent (collapse): $\nu = 1/3$ (compact globule). The folding funnel describes the transition from $\nu \approx 0.6$ (unfolded) through $\nu \approx 1/3$ (molten globule) to the native state.