Hysteresis in Force Probe Measurements: a Dynamical Systems Perspective

Macromolecular binding forces between single protein-ligand pairs have been directly measured with the Atomic Force Microscope (AFM) in several recent experiments. In a typical measurement, the AFM probe, or cantilever, is attached to the ligand and exerts a disruptive force on the bond between the macromolecular pair while the receptor is held fixed; the probe is then moved away from the substrate until the bond is broken. When the bond actually breaks, the tip is observed to slip; in fact, the ligand is jumping to a new equilibrium point determined purely by the cantilever, as if the receptor had been instantaneously moved to infinity. This “jumping-off” or “minimum rupture force” is determined by measuring cantilever deflection. In a similar manner, the two molecules can be brought together and the “jumping-on” force can be determined. These two measurements will result in different estimates of the binding force due to hysteresis. This hysteresis is caused by a cusp catastrophe in the space defined by probe position and cantilever stiffness. The phenomena of “jumping-off” in macromolecular rupture experiments and “jumping-on” when molecules are brought together occur when the system passes through a saddle-node bifurcation as the probe position is varied. Probe approach and withdrawal result in different post-bifurcation equilibria, different energy dissipation, and different force measurements.