Problems PHY315 Due Tuesday April 20.
1. Coulomb’s law states that the force between two charges q1 and q2 separated by a
distance r is 12
2
0
1
4
qq
rπε ε
=F , where ,0 is the permattivity constant, equal to
8.854Η10-12 C2/NΑm2 and , is equal to 1 for a vacuum, about 3 for oils and 80 for water.
Calculate the force between two electronic charges separated by 1 nm in pure water. Note
that the force would be smaller in a salt solution due to screening of the charge by salt
ions, but it will be larger in the interior of proteins where the dielectric constant is closer
to that of oil varying between 5 and 30.
2. Bacterial swimming.
a. A bacterium with a drag coefficient of 20 nNΑs/m can swim with a speed of 25
:m/s what is its power output? (Recall this is the Low Reynold’s number regime.)
b. Chemical energy for hydrolysis of ATP is 10-19 Joules. If a bacterium were driven
by ATP (it is not) how many hydrolysis per second would minimally be needed to
power the movement?
3. You have seen the model of the rotating ATP synthase, both the F0 part that traverses the
membrane and the F1 part on the inside of the membrane. The rotation of the F1
component has been directly visualized by attaching a fluorescent-labeled actin filament
to the ( subunit. It is found that the speed depends on the third power of the length of the
filament, and that the rotational speed is 0.85 revolutions per second when a 2-µm long
actin filament is attached at its end.
a. Assuming that the average height of the filament is about 100 nm above the
surface, that the diameter of the filament is 3 nm, and that the viscosity is 0.89
Pa/s, calculate the torque that the motor mus generate against the viscous drag.
Note 33
12
4
() (
3ln(2 / )
torque L L
hr
πη ω≅)
+T where Τ is the angular velocity
(radians/sec), 0 is the viscosity, h is the height above a nearby plane, r is the
radius of the cylinder that is rotated, L1 and L2 are the lengths from the rotation
point to the ends.
b. Given that three ATP molecules are hydrolyzed per revolution, calculate the work
done per ATP hydrolyzed.