Reconstructing Free-Energy Landscapes For Cyclic Motor Proteins

Abstract

Cyclic motor proteins are responsible for energy conversion processes inside the cell. Their dynamical behaviour can be described theoretically in terms of over-damped Brownian motion on a tilted periodic free-energy potential landscape. The periodic potential reflects the cyclic behaviour of the proteins and the tilt an external force driving the system out of thermal equilibrium. This theoretical description requires a priori knowledge of the free-energy landscape. However, the free-energy landscape of a particular protein is often not known. In this thesis we develop a novel way to take advantage of the increasing availability of data from single molecule experiments to construct free-energy landscapes. The method uses the individual stochastic trajectories measured in single molecule experiments to compute the steady-state probability distribution. Then by exploiting the periodicity, we can express the Smoluchowski equation (describing over-damped Brownian motion) in the Fourier space (k-space) and invert it to reconstruct the free-energy landscape. We prove the validity of the reconstruction method by presenting numerical examples for a range of one-dimensional model potentials. The method is shown to be robust for common experimental uncertainty and partial knowledge of parameters. The key advantages of the method over previous similar approaches are its ease of implementation and accuracy for both deep and shallow well regimes. Our method is also easily extended to higher dimensional systems as it does not rely on a closed solution of the Smoluchowski equation.

This is important as energy conversion in motor proteins requires the consideration of at least two degrees of freedom (usually one chemical and one mechanical). We show that if both degrees of freedom of the stochastic trajectories are measured then our method provides a reliable method of construct 2D landscapes. In single molecule experiments it is usually only the mechanical degree of freedom that is measured in single molecule experiments. We therefore investigate under what circumstances a two-dimensional landscape can be inferred from measurements in only one dimension. We also determine a method for identifying the presence of energy coupling from information in only one degree of freedom.