Relative Diffusivities of Bound and Unbound Protein Can Control Chemotactic Directionality

Enzyme-based systems have been shown to undergo chemotactic motion in response to their substrate gradient. This phenomenon has been exploited to direct the motion of enzymes and enzyme-attached particles to specific locations in space. Here, we propose a new kinetic model to analyze the directional movement of an ensemble of protein molecules in response to a gradient of the ligand. We also formulate a separate model to probe the motion of enzyme molecules in response to a gradient of the substrate under catalytic conditions. The only input for the new enzymatic model is the Michaelis-Menten constant which is the relevant measurable constant for enzymatic reactions. We show how our model differs from previously proposed models in a significant manner. For both binding and catalytic reactions, a net movement up the ligand/substrate gradient is predicted when the diffusivity of the ligand/substrate-bound protein is lower than that of the unbound protein (positive chemotaxis). Conversely, movement down the ligand/substrate gradient is expected when the diffusivity of the ligand/substrate-bound protein is higher than that of the unbound protein (negative chemotaxis). However, there is no net movement of protein/enzyme when the diffusivities of the bound and free species are equal. The work underscores the critical importance of measuring the diffusivity of the bound protein and comparing it with that of the free protein.