Chain Folding Conformation

Let $\mathbf{r} = (\mathbf{r}_1, \mathbf{r}_2, \dots, \mathbf{r}_N)$ be the set of spatial coordinates defining the conformation of a polymer chain of $N$ monomers. The equilibrium conformation minimizes the total free energy functional $F(\mathbf{r})$: \begin{equation*} F(\mathbf{r}) = E_{internal}(\mathbf{r}) + E_{interaction}(\mathbf{r}) - T S(\mathbf{r}) \end{equation*} The internal energy $E_{internal}$ includes bond stretching and angle bending potentials, while $E_{interaction}$ accounts for non-bonded forces (e.g., Lennard-Jones potential $V_{LJ}$): \begin{equation*} E_{interaction}(\mathbf{r}) = \sum_{i$\mathbf{r}_0$ is the state that minimizes $F$: $\mathbf{r}_0 = \arg\min_{\mathbf{r}} F(\mathbf{r})$. The folding tendency is quantified by the free energy difference $\Delta F = F(\mathbf{r}_0) - F(\mathbf{r}_{random})$.