We will consider a "race" between a listeria bacteria (about 1 μm across) that swims along a straight track inside a (much larger ~15 μm) mammalian cell that it has infected. The listeria emits a pulse of signaling molecules. The listeria itself moves in the cell at a speed of about 5 μm/s. The question we want to tackle is whether molecules emitted by the listeria cell could move ahead of the cell as they diffuse.
A. Assume that a Listeria cell, when at one end of the mammalian cell, emits a pulse of molecules with diffusion constant of D=200 μm2/s. The Listeria cell then moves to the other side of the mammalian cell, reaching the other side of the cell in 3 seconds. How much time does it take the diffusing molecules to reach the other end of the cell?