1. Bacterial Foraging. (7 points) Bacteria use swimming to seek out food. Imagine that the bacterium is in a region of low food concentration. For the bacterium to profit from swimming to a region with more food, it has to reach there before diffusion of food molecules makes the concentrations in the two regions the same. Here we find the smallest distance that a bacterium needs to swim so it can outrun diffusion. a) Sketch the distance traveled by a bacterium swimming at a constant velocity v as a function of time t, and the distance over which a food molecule will diffuse in that same time. Indicate on the plot the smallest time and the smallest distance that the bacterium needs to swim to outrun diffusion. b) Make a numerical estimate for these minimum times and distances for an E. coli swimming at a speed of 30m/s. The diffusion constant of a typical food molecule is roughly 500m2/s. c) Estimate the number of ATP molecules the bacterium must consume (hydrolyze) per second in order to travel at this speed, assuming that all of the energy usage goes into overcoming fluid drag. The amount of energy released from one ATP molecule is approximately 20kB T. Disclaimer: the bacterial flagellar motor is actually powered by a proton gradient and this estimate focuses on the ATP equivalents for overcoming fluid drag