A Smoluchowski Equation for Force-Modulated Chemistry in Single Molecule Pulling Experiments

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Thioredoxins (Trx) are a class of enzymes, which catalyze the reduction of disulphide bonds between two cysteine residues, commonly found in proteins. An experimental investigation into the reaction mechanisms employed by various species of Trx was carried out by Perez-Jimenez et al. using single molecule force-clamp spectroscopy. The experiment involved applying a pulling force along the disulfide bond of the protein substrate, and measuring the rate of the Trx-catalyzed reduction as a function of the pulling force. One interesting finding of the experiment was that some forms of thioredoxin exhibit a biphasic relationship for reduction rate as a function of force magnitude. For this project, a mathematical model of this system was created, which employs a Smoluchowski formalism in the vein of Agmon-Hopfied or Sumi-Markus models. The model describes the time evolution of the probability distribution function of the protein's configuration within a space defined as the internal protein coordinate, as it diffuses over a potential which distorts under the applied force, while losing probability density to the Bell model type “sink” term, which is also a function of the applied force, representing reactants going to products (disulphide bond cleavage). By numerically solving the Smoluchowski equation and integrating the resulting surface over both time and the protein coordinate to calculate lifetime for increasing values of applied force, the model successfully reproduced the experimentally observed values for disulphide bond reduction rate as a function of applied force. Parameterizing the Smoluchowski equation to fit the experimentally measured data points provided a means of drawing insights into a physical interpretation of the model including the relationship between degree of biphasic behavior and the distance along the reaction coordinate from the bottom of the reactant well to the top of the transition state.