Physics 434, 2014: Homework 1
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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.
- In class, we have 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, . In this equation, is the viscosity, which for water is 1 mPa*s at 20 degrees Centigrade; is the velocity of the particle; and 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.
- The first part of the problem is for those of you who had Physics 142 or 152: Let's fill in the blanks in our in-class discussion about photon counting in vision. Let's recall our knowledge of electromagnetism and estimate what would be the size of a diffraction spot created on a retina by a point source far away. We take the size of the pupil ~2mm, and the distance from the pupil to the retina ~2cm. Recall also that humans see colors from violet (about 450 nm) to red (about 700nm); so a number around 500nm is a reasonable number to estimate the wavelength of visible light. How does this size compare to the size of a human photoreceptor? (You may want to search for the photoreceptor size online; Wikipedia is a good starting point). Comment on why do you think this is. If a fly or a bee is sensitive to ultraviolet light, what would you expect the size of the photoreceptor to be? (An insect eye has a very different geometry, but the same laws of physics apply.) And this is for everyone: Let's say that there're about receptors covering 1 steradian (you may want to search Wikipedia for better estimates). Aimed with the typical photon flux number of (which we take from the paper by Stavenga; find it on the class web site), we can calculate the number of photons each photoreceptor gets in ~100 ms of decision-making time. What is this number? Do you expect fluctuations to be important?
- This week I would like you to install Matlab or Octave on your computers and make it run. This is really your main "computational" assignment. The rest of this problem will not be graded. I am providing it only as a task that you can attempt in order to start learning the Matlab language. Don't worry if you do not know it yet -- you will by the end of the semester. Let's start with writing a program that will demonstrate the main topic of our Lecture 1 -- that much of biological information processing is probabilistic. We have discussed this for E. coli in class, and let's now think about a totally different example -- a neuron. Once a human sees a still image, s/he can make a decision if the image contains a certain object, such as an animal, in less than 100 ms. At the same time, an active neuron in the primary visual cortex will fire with a rate of up to 60 spikes/second or so, that is, only about 6 spikes over 100 ms. These come at random times. We will simulate this process in Matlab as follows. Make a vector of 100 numbers, each representing 1 ms of the 100 ms window. These numbers will correspond to whether a spike happened in a particular millisecond. For each of these hundred entrees, draw a random number using the rand function, and set the entree to 1 if the random number is below 0.06 and to 0 otherwise. Using the function sum, find the total number of the spikes that happened in these 100 ms. Repeat the procedure 1000 times using the for loop. Use the mean function to assure yourself that the mean number of spikes in 1000 repetitions of our 100 ms window is, indeed, about 6. Then use the std function to calculate the standard deviation of the number of spikes -- you should get about 2.4. That is, in each of our 1000 repetitions, we may have typically anywhere from 4 to 8 spikes. This spread is very large, and yet a human is able to make a decision about what s/he sees in 100 ms with almost 100 per cent accuracy. Amazing, isn't it? Even though we are so much bigger than E. coli, our information processing is still probabilistic.
- (from Nelson, 2014, book) Use Matlab to plot functions and in the range of . The functions appear qualitatively similar. Now make a semi-logarithmic plots of these two functions. What outstanding feature of the exponential functions jumps out in this representation? Finally, make a log-log graph of both functions and comment.
- For Graduate Students Only: 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.
- For Graduate Students Only: I have also thrown around some numbers describing how long it will take for a molecule of a chemical to leave the area around the bacterium, and for a new molecule to diffuse in. Let's look at the Einstein Relation, which relates diffusivity of a spherical particle in a fluid to the viscosity of the fluid, the particle radius, and the temperature. What will be a diffusion coefficient of a spherical particle of a radius of 1 nm in water at 20 degrees C? Assuming that the mean squared displacement in diffusion goes as (where is the dimensionality of the space), how long will it take for this particle to move 1 micron away from the E. coli? We have discussed that the bacterium can only go straight for about 10 seconds, before its own direction of motion changes due to rotational Brownian motion. How fast should the bacterium's speed be so that in this time it can move away about ten times farther that the diffusing particle would? "Outrunning diffusion" is a crucial criterion that shapes the bacterial run-and-tumble behavior.
- Open ended problem, not graded: In class we briefly mentioned the Berg-Purcell limit on the concentration estimation. What is the similar limit on estimating a concentration gradient?