The evolution of a simple enzyme reaction being convected by Poiseuille flow in a semi-infinite tube is considered. When the effects of diffusion are ignored, the solutions for the concentrations of enzyme and substrate are analogues of the spatially independent case. When small but nonzero diffusion coefficients are admitted the solutions are modified by the smoothing out of any discontinuities present in the conditions at the inlet to the tube. Explicit solutions in each case are given when the nondimensional concentration of the substrate at the inlet, $f(t)$, is the Heaviside step function \[ f(t) = \left\{ {\begin{array}{*{20}c} {1,\quad t > 0,} \\ {0,\quad t < 0.} \\ \end{array} } \right. \]