Kalman Filter Estimates of the Contour Length of an Unfolding Protein in Single-Molecule Force Spectroscopy Experiments

Abstract

Force spectroscopy measurements of single molecules using AFM have enabled the study of a range of molecular properties not accessible with bulk methods. These properties of interest must typically be inferred by manually fitting models to selected portions of measured data. As manual intervention in the fitting process easily introduces a bias in the analysis, there is a need for more sophisticated analysis methods capable of interpreting data in an unbiased and repeatable “hands-off” manner. Here we apply an extended Kalman filter to the estimation of protein contour length (Lc) during mechanical unfolding, based on force and extension data from an AFM experiment. This filter provides an online and fully automated estimate of Lc based on a system model, the experimental measurements, and noise statistics. The system model comprises a physical model of the cantilever and a nonlinear WLC approximation of the extended protein. When manually fitting the WLC model to force-extension data from ubiquitin proteins, the estimate of the change in contour length during unfolding is distributed normally as N (22.7 nm, 6.59 nm2). Testing the Kalman filter on the same protein yields ΔLc ∼ N (24.54 nm, 0.24 nm2). As the variance limits resolution in estimating the number of amino acids released by unfolding, it is clear that the Kalman filter presents a substantial improvement over the conventional method. We thereby demonstrate that the Kalman filter provides a powerful unbiased approach to interpreting force spectroscopy data, capable of increasing resolution beyond the traditional experimental limit. Due to the flexibility of this approach, it can be extended to monitoring other state variables of molecular systems observed by various forms of force spectroscopy, including optical and magnetic tweezers.