Question

Excluded volume in protein folding. (a) Use the Flory-Huggins theory to estimate the number \(v_{1}(c)\) of conformations of a polymer chain that are confined to be maximally compact, \(M=N\). Simplify the expression by using Stirling's approximation. (b) If the number of conformations of the unfolded chain is \(v_{1}(u)=M(z-1)^{N-1}\) (since \(M \gg N\) ), then compute the entropy of folding, $$ \Delta S_{\text {fold }}=k \ln \left[\frac{v_{1}(c)}{v_{1}(u)}\right] . $$

Short answer

The entropy of folding is \[ \Delta S_{\text{fold}} = k \left[ (2N - 1) \ln N - (N-1) \ln (z-1) \right] \]

Step-by-step solution

- Understanding Flory-Huggins Theory

The Flory-Huggins theory is used to describe the conformation of polymer chains in terms of their volume fraction and the interactions between monomers. For a maximally compact polymer, the number of conformations will be related to the volume available to the monomers.

- Estimation of Number of Conformations

For a maximally compact polymer chain, where the polymer is confined to a volume, we denote the number of conformations as \(v_{1}(c)\). We can use Stirling's approximation \(N! \approx \sqrt{2 \pi N} \left( \frac{N}{e} \right)^N\) to simplify our calculations.

- Applying Stirling's Approximation

Given that \(M = N\), the number of conformations \(v_{1}(c)\) can be written using the binomial coefficient and simplified as follows: \[ v_{1}(c) = \binom{N^{3}}{N} \approx \frac{(N^3)!}{N!(N^3 - N)!} \] Using Stirling's approximation, this becomes: \[ v_{1}(c) \approx \frac{\sqrt{2 \pi N^3} \left( \frac{N^3}{e} \right)^{N^3}}{\sqrt{2 \pi N} \left( \frac{N}{e} \right)^{N} \sqrt{2 \pi (N^3 - N)} \left( \frac{N^3 - N}{e} \right)^{N^3 - N}} \] This expression can be further simplified under the assumption \(N^3 - N \approx N^3\) as: \[ v_{1}(c) \approx \left( \frac{N^3}{N} \right)^N = N^{2N} \]

- Number of Conformations of Unfolded Chain

Given that the number of conformations of the unfolded chain is \(v_{1}(u) = M(z-1)^{N-1}\), with \(M \gg N\), this simplifies to: \[ v_{1}(u) = N(z-1)^{N-1} \]

- Calculating the Entropy of Folding

The entropy of folding can be computed using the formula: \[ \Delta S_{\text{fold}} = k \ln \left[ \frac{v_{1}(c)}{v_{1}(u)} \right] \] Substituting the expressions for \(v_{1}(c)\) and \(v_{1}(u)\): \[ \Delta S_{\text{fold}} = k \ln \left[ \frac{N^{2N}}{N(z-1)^{N-1}} \right] = k \ln \left[ N^{2N - 1}(z-1)^{-(N-1)} \right] \] Simplifying further: \[ \Delta S_{\text{fold}} = k \left[ (2N - 1) \ln N - (N-1) \ln (z-1) \right] \]

Key concepts

Flory-Huggins Theory

The Flory-Huggins theory provides a framework to understand the conformation of polymer chains. It is particularly useful for examining how polymers mix and interact. In our case, we use this theory to estimate the number of conformations for a maximally compact polymer chain.
The theory considers factors like the volume fraction of the polymer and the interactions between monomers. For protein folding, understanding these interactions helps us estimate how many different ways the polymer can fold when confined to a specific volume.
For a maximally compact chain (where the number of monomers equals the volume), we leverage this theory to derive an expression that fits our constraints.

Stirling's Approximation

Stirling's approximation simplifies the factorial function, which is often cumbersome for large values. The approximation is given by:
\[ N! \approx \sqrt{2 \pi N} \left( \frac{N}{e} \right)^{N} \]
Use this approximation to simplify expressions involving large numbers, such as the number of conformations of a polymer chain. In our exercise, we applied Stirling's approximation to simplify the binomial coefficient representing the number of conformations \(v_{1}(c)\).
Given the complexity of factorial terms when dealing with large polymers, this simplification allows us to work with more manageable equations like:
\[ v_{1}(c) \approx N^{2N} \]
This helps us understand the behavior and count of polymer configurations more intuitively.

Polymer Chain Conformation

Polymer chain conformation describes the spatial arrangement of the polymer's monomers. These conformations can be affected by constraints like volume and environmental conditions. In our context, we need to estimate the conformations when the polymer is maximally compact versus when it is unfolded.

• Maximally compact: Fewer but more specific conformations confined to a restricted volume.
• Unfolded: More conformations because the chain can spread out more, resulting in a higher number of possible arrangements.

To quantify this, we compare the number of conformations \( v_{1}(c) \) for a confined polymer using Flory-Huggins theory and Stirling's approximation to those of an unfolded chain.
This difference allows us to calculate the entropy change during protein folding using:
\[ \Delta S_{\text{fold}} \ =\ k \ln \left[ \frac{v_{1}(c)}{v_{1}(u)} \right] \]
Understanding the conformation changes helps explain protein folding, a critical biological process with implications in understanding diseases and designing drugs.