This homework addresses some of the physical properties of long chain biomolecules (proteins and nucleic acids). It is due in my mailbox (in mailroom, 2nd floor of NPB) before 5:00 pm, Tuesday March 19, 2002. You may work together on the homework, but you should hand in your own solution, of course.

1. Prof. Rinzler asked you to examine the paper by Rief et al. [Science 1997], in which an atomic force microscope was employed to mechanically unfold the muscle protein titin. Titin is composed of a number of smaller, individually foldable subunits which are called the immunoglobulin (Ig) domains. By pulling on the two ends of the titin molecule, the researchers observed the sequential unfolding of these individual subunits.

• Notice the sawtooth pattern of bumps in Figure 3, caused by sequential unfolding of individual subunits (domains) of the protein. Why do these bumps vary in overall height or amplitude?

• Can you suggest any benefit that this might have to the organism from which this protein is derived? Feel free to speculate.

• The paper tells you the contour length L of one Ig domain, and it tells you the persistence length p of the molecule (where p = (Effective Segment Length)/2). From those numbers, estimate the energy (Joules) that should be required to stretch a Gaussian chain having this L, p by an amount comparable to its contour length.

• From the data of Figure 1, estimate how much work (Joules) was actually done by the atomic force microscope in one cycle of the sawtooth. How do you account for the discrepancy between this number and your Gaussian chain estimate? What does this tell you about the degree of "stretch" in the rest of the chain when each new domain unfolds?

• We learned in class that proteins are only marginally stable at room temperature, so that there can be a finite (i.e. nonzero) probability of finding one protein molecule unfolded, even though the solution conditions favor the folded state. In fact, the titin molecule has a nonzero rate of spontaneous unfolding under the experimental conditions studied in this paper. Explain why it was necessary in this case to apply force in order to unfold a molecule that already unfolds spontaneously. Shouldn't the molecule unfold without an applied force?

• Look at figure 5 and notice that the data will eventually intercept the x-axis (corresponding to zero force). At what value of pulling speed does that occur? What does this number tell you about the system -- that is, what properties of the protein and/or AFM do you think are revealed by this parameter?

2. We have discussed two "sophisticated" models for the elasticity of a long chain molecule: the freely jointed chain (FJC) and the wormlike chain (WLC). Consider an FJC where each segment has length a, and the total contour length is L. Consider also a WLC of the same contour length, and with a persistence length p = a/2. Let us imagine that we pull on both molecules with a force F, and thereby stretch them by a distance x < L.
   • Use your favorite software to plot the force displacement curve of both molecules in terms of the dimensionless quantities x/L and Fp/kT. It is probably better to plot on a semilog scale, with log(Fp/kT) plotted against x/L. [Recall that we discussed F(r) of the FJC in class. You can find F(r) for the WLC in the Rief paper.]
   • At what values of x/L does the force required to stretch these two chains differ by 10%? How about 100%?
   • For roughly what value of the applied force Fp/kT is there the greatest difference in the extension of the two types of chain? How does this compare to the force applied to the protein by Rief et al. -- that is, was their experiment sensitive to the difference between the FJC and the WLC models?

3. A sample of plasmids (double-stranded rings of DNA) are closed (sealed into rings) and then placed on the upper left corner of a gel for electrophoresis. Application of an electric field moves the DNA down in the vertical direction and - in the process - separates the sample into several spots. See figure below. Marty (the scientist in the lab) stains the gel with a fluorescent dye. This dye intercalates (i.e. slides into the gap) between the base pairs, and thus stains each DNA molecule. The fluorescence intensity is then measured, which reveals the amount of DNA in each spot, and from that Marty knows the relative amount of DNA present. He records the distance traveled by each spot and the amount of DNA it contains, relative to the first spot:

• Spot A: moved 1 mm, intensity = 100
• Spot B: moved 2 mm, intensity = 155.8
• Spot C: moved 3 mm, intensity = 73.6
• Spot D: moved 4 mm, intensity = 21.1
• Spot E: moved 5 mm, intensity = 3.7


(A) Based on these data, identify the values of W associated with each spot. Be sure to explain why spot B contains more DNA than either spots A or C.

(B) We discussed in class how the formation of supercoils is associated with a spring-like potential energy E(W) = (1/2) k W^2 in the DNA. Based on the fact that the double-stranded rings were formed (i.e. closed) at room temperature (22 degrees C), calculate the spring constant k (in Joules) associated with supercoiling of this DNA.

(C) Spot F, located at 6 mm, was not visible on the gel, but what would be its intensity (on the above scale) if it were visible?

(D) Marty has read about 2-dimensional electrophoresis, so he now applies an electric field to the same gel, but now in the horizontal direction. He finds that the spots now separate as follows (see Figure). Why do the spots separate differently this time? Label each spot on the new gel with its value of W.



(E) Marty knows that each plasmid is a ring of 1100 base pairs, and each was originally a B-form helical DNA with a twist of 10.50 base pairs per turn. What is the new twist in the DNA.

4. Space shuttle astronauts use a very long garden hose, which has diameter d = 1 inch, and length L = 2 km, to reach from the International Space Station to the Hubble Space Telescope. When they are finished hosing down the Telescope, they disconnect the garden hose from its faucet at the ISS, but they forget to coil it up and stash it in the Shuttle Cargo Bay. The freely floating hose therefore eventually wiggles into a random configuration.

(A) Estimate the average end-to-end separation of the two ends of the hose. Keep in mind that an object in near-Earth orbit adopts a temperature close to 300 K.[Recall that a garden hose has a similar ratio of diameter to persistence length as does double-stranded DNA.]

(B) Astronaut Smith is sent out to reel in the hose. He begins by pulling on the closest loose end of the hose. How much force does he have to apply to pull that end a distance of 100 meters towards the shuttle?