Extended Kalman filter estimates the contour length of a protein in single molecule atomic force microscopy experiments

Abstract

Atomic force microscopy force spectroscopy has become a powerful biophysical technique for probing the dynamics of proteins at the single molecule level. Extending a polyprotein at constant velocity produces the now familiar sawtooth pattern force-length relationship. Customarily, manual fits of the wormlike chain (WLC) model of polymer elasticity to sawtooth pattern data have been used to measure the contour length L(c) of the protein as it unfolds one module at a time. The change in the value of L(c) measures the number of amino acids released by an unfolding protein and can be used as a precise locator of the unfolding transition state. However, manual WLC fits are slow and introduce inevitable operator-driven errors which reduce the accuracy of the L(c) estimates. Here we demonstrate an extended Kalman filter that provides operator-free real time estimates of L(c) from sawtooth pattern data. The filter design is based on a cantilever-protein arrangement modeled by a simple linear time-invariant cantilever model and by a nonlinear force-length relationship function for the protein. The resulting Kalman filter applied to sawtooth pattern data demonstrates its real time, operator-free ability to accurately measure L(c). These results are a marked improvement over the earlier techniques and the procedure is easily extended or modified to accommodate further quantities of interest in force spectroscopy.