Download Biophysik der Zelle -¨Ubung

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