By using two different computer simulation methods, of which one produces exact results while the other is based on the Wilemski-Fixman closure approximation, we evaluate the utility of closure approximation in calculating the rates of diffusion-controlled reactions involving a reactant with multiple reaction sites. We find that errors in the estimates of steady-state rate constants due to closure approximation are not so large. We thus propose an approximate analytic expression for the rate constant based on the closure approximation.