On the distance to the transition state of protein folding in optical tweezers experiments

The distance to the transition state ( $x^{‡}$ ) is an important parameter for understanding the energy landscape of chemical reactions. In protein folding, $x^{‡}$ represents the distance to the high energy structure between folded and unfolded states. This correlates with the deformation of the protein as it crosses the energy barrier defining its rigidity. This parameter can be determined by unfolding the protein, analyzing the kinetics of unfolding and refolding, and fitting the data to various models. An approach to determine the $x^{‡}$ is using force as a way to tilt the energy landscape. Force spectroscopy studies, particularly at the single-molecule level, offer a powerful approach for this purpose. One of these techniques is optical tweezers, which allow the application of force by pulling on a bead attached to the protein via spacers, thereby unfolding it. This method provides measurements of force and distance between the folded and unfolded states of the protein. By analyzing force histograms, we can apply different models as the phenomenological Bell-Evans or Kramers theory-based models. Additionally, an alternative direct approach involves summing the distances to the transition state to fit the data of the distance of total protein unfolding. Using this approach, we can plot force versus distance and obtain the $x^{‡}$ and the energy to the transition state from folded to unfolding and vice versa. Furthermore, these results can be correlated with elastic models, such as the worm-like chain model. By integrating these approaches, we can gain deeper insights into protein folding mechanisms.