Problems

Submitted by Nicholas C. Darnton (inactive) on Monday, 2/23/2009, at 11:55 AM

Due Tuesday, March 3.  Since I'm giving you more than a week on this, I will also look for a paper from the literature and append some questions about it to this PS.  Be on the lookout for this addition.

X-ray versus NMR

Look at X-ray and NMR solutions to the human estrogen receptor, PDB files 1hcp and 1hcq. The NMR structures have two regions of extremely high uncertainty / variability.  What sequence (residue number) are these?  Do these correspond to the areas of highest B-factor in the X-ray solution?

Biased random walk. 

You are playing roulette in Atlantic City, betting on odds or evens only.  Starting with a $20 stake, you make 20 consecutive $1 bets and then cash out.

  1. Plot the probability distribution of cashout amounts.  Do not approximate this with a normal distribution; do the problem exactly.
    1. What is the most probable cashout?
    2. More importantly, what is the expected (mean) cashout?  This is why you shouldn't play roulette.
  2. Replot the probability distribution for a $10 initial stake, all other factors as before.  The casino does not offer credit.  You should see a problem with your probability distribution: what is it?  We will see how to treat this issue when we talk about first-passage processes and the martingale.  Misunderstanding of this phenomenon is behind many failed strategies to beat the house.

Random coil end

You're trying to splice a small insert into a 15kb plasmid, but only have one restriction site available.  (Unlikely, but bear with me).   After cutting the plasmid, you want to add a high enough concentration of insert so that most plasmids will incorporate the insert before ligation.

Show that you only need a few nM of insert to successfully compete against self-ligation.

Treat the cut plasmid as a random coil with a persistence length of 150bp.    Ignore the possibility of incorporating several inserts, or of two plasmids ligating together, etc.  As the problem is set up, I think that anything over a few nM insert would result in many fragments inserting back to back; over a few nM plasmid would result in chains of plasmids joining together.  See me for hints if you aren't familiar with restriction enzymes.

More protein force field

Serine chi1 distribution

I finally dug up some data for the distribution of dihedral angles in a sidechain.  For serine, I extracted the data from the figure to a text file.

What value of Vn (in kcal/mol) should you use in the protein force field equation for serine? 

Hints:

  1. Treat this data as if it was taken at room temperature.
  2. The protein force field equation, as written, is a little ambiguous.  Here you should find the values of n, phi0, and Vn that pertain to serine specifically.  (n just a superfluous label attached to V, not an index in a sum, though it kind of looks that way).  Of course here we're talking about the dihedral angle chi1, not phi.
  3. It's an oversimplification to think there's a single distribution for dihedral angles.  Actually the distribution depends on the local secondary structure and dihedrals are not independent of each other (there's a nearest-neighbor anticorrelation due to hindrance effects).

B-S transition in stretched DNA. 

PBoC 8.10 (hard).  There are few enough terms that you can write down the solution without using partition functions, though it's easier to use a partition function if you're familiar with it.  I suggest using Mathematica to plug in numbers and do the actual plotting.  (That's what I do for something with this many terms).

Paper

Read carefully the paper by Find, Oberg and Seshadri, "Discrete intermediates versus molten globule models for protein folding: characterization of partially folded intermediates of apomyoglobin", Folding and Design 3 p. 19-25 (1997), which is available in the course E-reserves.  Answer the following questions.  This is not meant to be an essay assignment – a few sentences explaining the meaning or significance of the relevant passage is enough.  You may well have to hunt around in lecture notes, reading or on the internet to answer some of these question.

  1. p. 20.  "These experiments ... discrete species."  What does being "monodisperse" have to do with the rest of the sentence?
  2. p. 20.  "The relationship ... relatively globular."  Explain the first sentence on general terms.  More specifically, in the second sentence how exactly do the data indicate what the authors claim?
  3. (Optional, unless we get to concentration dependence of equilibria next week, which is doubtful, but you should have seen this in chemistry somewhere)  p. 20.  "For A3 ... to that of A2".  How is the graph "clear[ly] [a] three-state transition"?
  4. p. 21.  "Information concerning ... increasing structure (Table 1)".  What do hydrophobic surfaces have to do with aggregation?  Why can light scattering tell you something about aggregation?
  5. p. 21.  "The presence of the core ... "  What does tryptophan fluorescence have to do with anything?
  6. p. 23.  "The nature of this collapsed state ... large number of substates."  Where do the authors get this idea that the number of substates is large?
  7. p. 23.  "This picture fits ... intermediate cores."  Why do the authors use the word "but" in this sentence?
  8. p. 23.  "Protein concentration ... ε280 of 15,700 M-1 cm-1"  Explain how you would use this value of ε280 to measure protein concentration.
  9. p. 24.  "A flow-cell ... to radiation."  Explain how the flow-cell helps with the radiation damage problem.

 

 

Reading

Submitted by Nicholas C. Darnton (inactive) on Sunday, 2/8/2009, at 9:55 PM

PBoC Chapter 8 sections 8.1 and 8.2 on the random walk / random coil.