By developing and making use of the potential theory of the polymer-mediated interaction between spherical colloids, we investigate the many-body effects on the depletion interaction among these colloids in the colloid-polymer mixture. As our main results, we obtain analytic expressions for the polymer end density in the presence of the colloids and the many-body depletion potential acting between these colloids in the "protein" limit. We present a comparison of our theoretical findings with the results of recent computer simulations.