This exercise will help you get a feel for the timing of enzyme-catalyzed reactions. Consider the standard Michaelis-Menten mechanism. Imagine we start with a subtrate concentration of cS0 at time t=0. Initially the conversion of substrate to product will proceed at its maximal rate. a) Let =1cS/cS0 be the fractional conversion of substrate to product. Derive an expression for the amount of time t it takes to achieve . This expression should be in terms of the Michaelis constant KM,kcat (which is equal to k2 in the standard Michaelis-Menten mechanism as presented in class), cS0, and cE0. b) The maximal rate of reaction occurs when at time t=0 when cS=cS0. Over time the rate of the reaction will slow down. Derive an expression for the time t at which the reaction rate is f times the maximal rate. That is, f is the fractional reaction rate as substrate gets consumed. c) Plot the functions you derived in parts (a) and (b). You should make plots for various values of cS0. When making the plots, it is a good idea to plot against dimensionless time, as defined in the previous problem. The plots may be a bit complicated to sketch, so you may want to plot it with plotting software. Comment on what you see