  Chemotaxis of enzymes in response to gradients in the concentration of their
substrate has been widely reported in recent experiments, but a basic
understanding of the process is still lacking. Here, we develop a microscopic
theory for chemotaxis, valid for enzymes and other small molecules. Our theory
includes both non-specific interactions between enzyme and substrate, as well
as complex formation through specific binding between the enzyme and the
substrate. We find that two distinct mechanisms contribute to enzyme
chemotaxis: a diffusiophoretic mechanism due to the non-specific interactions,
and a new type of mechanism due to binding-induced changes in the diffusion
coefficient of the enzyme. The latter chemotactic mechanism points towards
lower substrate concentration if the substrate enhances enzyme diffusion, and
towards higher substrate concentration if the substrate inhibits enzyme
diffusion. For a typical enzyme, attractive phoresis and binding-induced
enhanced diffusion will compete against each other. We find that phoresis
dominates above a critical substrate concentration, whereas binding-induced
enhanced diffusion dominates for low substrate concentration. Our results
resolve an apparent contradiction regarding the direction of urease chemotaxis
observed in experiments, and in general clarify the relation between enhanced
diffusion and chemotaxis of enzymes. Finally, we show that the competition
between the two distinct chemotactic mechanisms may be used to engineer
nanomachines that move towards or away from regions with a specific substrate
concentration.
