# Conformational Entropy as Collective Variable for Proteins

## TL;DR

This paper proposes using the NMR $S^2$ order parameter as a collective variable for enhanced sampling in protein folding simulations, enabling efficient folding/unfolding studies and free energy calculations.

## Contribution

It introduces the novel application of the $S^2$ order parameter as a collective variable for enhanced sampling in protein simulations, linking entropy and folding dynamics.

## Key findings

- Successfully reversible folding and unfolding of a small protein using $S^2$ with Metadynamics.
- Efficient computation of free energy and unfolding rates at various temperatures.
- Demonstrates the relation between $S^2$, entropy, and protein folding energetics.

## Abstract

Many enhanced sampling methods, such as Umbrella Sampling, Metadynamics or Variationally Enhanced Sampling, rely on the identification of appropriate collective variables. For proteins, even small ones, finding appropriate collective variables has proven challenging. Here we suggest that the NMR $S^2$ order parameter can be used to this effect. We trace the validity of this statement to the suggested relation between $S^2$ and entropy. Using the $S^2$ order parameter and a surrogate for the protein enthalpy in conjunction with Metadynamics or Variationally Enhanced Sampling we are able to reversibly fold and unfold a small protein and draw its free energy at a fraction of the time that is needed in unbiased simulations. From a more conceptual point of view this implies describing folding as a resulting from a trade off between entropy and enthalpy. We also use $S^2$ in combination with the free energy flooding method to compute the unfolding rate of this peptide. We repeat this calculation at different temperatures to obtain the unfolding activation energy.