Physics 434, 2016: Homework 1

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Back to Physics 434, 2016: Physical Biology.

Please turn on the assignment either as a PDF file to me by email, or as a printout to my mailbox in physics. Detailed derivations (with explanations) and calculations must be present for the problems for full credit.

Undergraduates

1. In class, we briefly discussed E. coli chemotaxis. I told you that once the bacterium reverses the direction of a motor rotation, the flagella bundle falls apart, and the bacterium instantaneously stops and reorients. Nobody objected, even though we all remember that inertia generally does not let a moving particle to stop instantaneously. So let's see if I cheated you or not. Following our "order of magnitude" style of estimates, let's assume that E. coli is a sphere of a radius of 1 micron. The Stokes' law relates the drag force exerted on spherical objects moving in a fluid to the radius of the object and the fluid viscosity, $F_{\rm {drag}}=6\pi \mu Rv$. In this equation, $\mu$ is the viscosity, which for water is 1 mPa*s at 20 degrees Centigrade; $v$ is the velocity of the particle; and $R$ is its radius. Notice the peculiar dependence of the force on the radius -- the linear dependence. This is the property of laminar (or low velocity) flows. Write down the second Newton's law that describes the motion of E. coli once its propulsion mechanism stops. The bacterium is largely made out of water, and so its mass density is about 1 g/cm^3. Solve the equation of motion and find the duration of time it will take for the drag to decrease the bacterial velocity by a factor of two from the initial velocity of 20 micron/s. Then find the total distance the bacterium will move until it stops. How does this distance compare with the size of the bacterium? With a diameter of a water molecule? Was it OK for me to say that E. coli stops instantaneously once its flagella leave the bundle? Consider reading the Physics at low Reynolds number article by Purcell. It has a beautiful pedagogical description of this regime of microscopic fluid flows.
2. This week I would like you to install Matlab or Octave on your computers and make it run. Then solve problem 1.2 from the textbook.
3. Problem 1.4 from the textbook.
4. Problem 1.5 from the textbook.
5. Problem 2.4 from the textbook.
6. Problem 2.6 from the textbook

Graduate students

1. In class, we briefly discussed E. coli chemotaxis. I told you that once the bacterium reverses the direction of a motor rotation, the flagella bundle falls apart, and the bacterium instantaneously stops and reorients. Nobody objected, even though we all remember that inertia generally does not let a moving particle to stop instantaneously. So let's see if I cheated you or not. Following our "order of magnitude" style of estimates, let's assume that E. coli is a sphere of a radius of 1 micron. The Stokes' law relates the drag force exerted on spherical objects moving in a fluid to the radius of the object and the fluid viscosity, $F_{\rm {drag}}=6\pi \mu Rv$. In this equation, $\mu$ is the viscosity, which for water is 1 mPa*s at 20 degrees Centigrade; $v$ is the velocity of the particle; and $R$ is its radius. Notice the peculiar dependence of the force on the radius -- the linear dependence. This is the property of laminar (or low velocity) flows. Write down the second Newton's law that describes the motion of E. coli once its propulsion mechanism stops. The bacterium is largely made out of water, and so its mass density is about 1 g/cm^3. Solve the equation of motion and find the duration of time it will take for the drag to decrease the bacterial velocity by a factor of two from the initial velocity of 20 micron/s. Then find the total distance the bacterium will move until it stops. How does this distance compare with the size of the bacterium? With a diameter of a water molecule? Was it OK for me to say that E. coli stops instantaneously once its flagella leave the bundle? Consider reading the Physics at low Reynolds number article by Purcell. It has a beautiful pedagogical description of this regime of microscopic fluid flows.
2. You should read the Physics at low Reynolds number article by Purcell (find it on the main course page). Find the scallop theorem there, which tells that a microscopic organism cannot move directionally using a one-dimensional reciprocal motion, like scallop opening and closing its halves. (a) What if the motion is two-dimensional? Think of the symmetry properties of the low Reynolds number (overdamped) equations of motion and write down a general form of a displacement as a function of two arbitrary shape coordinates changing in time. Analyze your expression -- does it remind you of any other expressions you've seen in classical or quantum mechanics? Try read the article by Shapere and Wilczek on this topic. (b) Can you think of a way how the scallop theorem can be violated? Hint: what if the water is moving around the scallop while it's going through its open/close cycle? The strategy I am suggesting is called ratcheting.
3. Problem 1.7 from the textbook.
4. Read T2 1.2.6b and set up a simulation that keeps track of both infected and uninflected immune cells. Can you find realistic parameters in this system that would produce a long latency period for HIV?
5. Problem 2.6 from the textbook.
6. Problem 2.7 from the textbook.