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Biophysik der Zelle - Übung TUM, WS 2013/2014 Marco Grison ([email protected]) Woche 3, Dienstag 29. Oktober 2013 Exercise 1: bacterial foraging (part II) To seek out food, bacteria swim by means of flagella. Let us consider a situation in which a (round-shaped) bacterium does not possess flagella and relied exclusively on diffusive processes for locomotion. [Remind that Einstein relation γD = kB T also holds for rotational brownian motion with the proper drag coefficient.] (a) Convert the energy of 1 kB T at room temperature in pNnm (approximate it at the first digit) and the water viscosity in pNs/µm2 , which are more appropriate for the problem. (b) If the bacterium wants to reach a region of higher food concentration 1 mm away from its initial position, how long would it take him to reach it? (c) In this period how many times in average it completes a complete rotation? (d) Compare the diffusion constant of such a bacterium with the one of a typical food molecule, which is roughly 500 µm2 /s, and discuss the validity of the result obtain in the previous point. Actually, for a bacterium with flagella 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. (e) Make a plot in which you 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. (f) Make a numerical estimate for these minimum time and distance for an E. coli swimming at a speed of 30 µm/s (as in part I, week 2). (g) How many glucose molecules should it find at a distance of 1 mm to take advantage of the travel? Neglect food diffusion, but discuss in which way it would affect the results. Exercise 2: overdamped systems Consider a protein that is initially held in a strained conformation, perhaps due to an internal strut. One can model such a system with a spring in parallel with a dashpot and a latch (see figure below). If the constraint is suddently removed, the motion equation is the one of an harmonic oscillator in an overdamped system, where a step force is applied. We can use such a system to evaluate the time scale of protein conformational changes. 1 (a) Solve the equation of motion for an harmonic oscillator in an overdamped system, in the presence of an external step force F . (b) For a globular protein of r=1 nm in water, evaluate the drag coefficient. (c) If the stiffness of the system is 1 pN/nm, calculate the relaxation time constant. Compare it with the time scale of local chemical changes, such as the breaking of the bonds between two proteins, that is ∼ 1 ps. 2