When a resting enzyme is activated, substrate concentration profile evolves in its vicinity, ultimately tending to steady state. We use modem theories for many-body effects on diffusion-influenced reactions to derive approximate analytical expressions for the steady-state profile and the Laplace transform of the transient concentration profiles. These show excellent agreement with accurate many-particle Brownian-dynamics simulations for the Michaelis-Menten kinetics. The steady-state profile has a hyperbolic dependence on the distance of the substrate from the enzyme, albeit with a prefactor containing the complexity of the many-body effects. These are most conspicuous for the substrate concentration at the surface of the enzyme. It shows an interesting transition as a function of the enzyme turnover rate. When it is high, the contact concentration decays monotonically to steady state. However, for slow turnover it is nonmonotonic, showing a minimum due to reversible substrate binding, then a maximum due to diffusion of new substrate toward the enzyme, and finally decay to steady state. Under certain conditions one can obtain a good estimate for the critical value of the turnover rate constant at the transition.