Problems
Problem set due May 1.
Unfolding of amyloid fibers
At the Biophysical Society meeting on March 1 I saw some data for force-induced unfolding (and refolding) of amyloid fibers of yeast prion protein. My recollection was that at about 50 fN of force, a single fiber oscillates back and forth as one of its constituent proteins repeatedly folds and unfolds. At 50 fN the protein spends about half its time in each state; at 120 fN the protein was mostly unfolded (say 90% of the time).
How much does the length of the fiber increase when a single amyloid domain unfolds? This was clearly visible on the force-displacement curve.
How stable (in kBT) is the folded protein compared to the unfolded protein?
Channel charge
Look at Figure 7.2 in PBoC and show that the effective gating charge of a sodium ion channel is 2.5 qe (the charge on the electron) and that Δε0 (the free energy difference between an open and closed channel at 0 V) is about 9 kBT.
If you don't have a favorite nonlinear fitting routine (like SigmaPlot, for instance) to do a general nonlinear regression you will need to construct an analog of the Hill plot and do a linear fit.
Hb binding
Take the fit parameters at the very end of section 7.2 (pp. 277-278) and
- Convert from the Kd's cited in the fitting data to the parameters J, K, L in the Adair model (in units of kBT).
- Pick a standard state (say, 760 mmHg) and compute the value of ε - µ0, also in kBT. Note that you can't tease apart ε (the energy cost to bind one individual O2 to Hb) and µ0 (the free energy cost to remove one individual O2 from the standard state) from this data alone, because one never occurs without the other; you could assign your entire Δε to Hb or to dissolved O2 and it would make no difference in the binding curves.
Calcium gating of a potassium channel
Read the article by Zadek and Nimigean from the course E-reserves and answer these questions. My page numbering may be off since I was reading from a truncated printout and could only see the top 2 or 3 pixels of the page numbers.
- p. 674, eqn (3). I though the Hill equation was (c/Kd)n/(1+(c/Kd)n). Why is theirs different?
- p. 675, eqn (5). What are the "usual meanings" of R, T and F? (The first two I recognize, but I had to reconstruct the last one).
- p. 576, Fig. 1. How are panels A and B related to panel C? That is, how would you construct C from a lot of data like that in A and B?
- p. 577, "Second, ... bilayer". Why exactly does this "lead to uncertainty"?
- p. 577, "If the Hill coefficient ... channel opening". Why do they reject n=20? Do you think this is legitimate?
- p. 577. Fig. 3. Why are the Kd's the same for both Hill fits? Why aren't the errors on the Kd's the same?
- p. 578, "As expected ... more frequent the bursts." How is burst frequency related to burst gap?
- p. 578, "It is noteworthy ... Ca2+ independent." What is the authors' evidence for this?
- p. 579, "The disappearance ... open probability." Explain this statement.
- p. 580, "... (to satisfy microscopic reversibility, f also ... final transition ...)". Explain.
- p. 580, "The cooperativity coefficient (µ) ... higher probability."
- Explain why µ accomplishes this and
- (extra credit, because I don't understand it) why µ enters as successively higher powers as more Ca binds.
- p. 581, Scheme 1. Explain the coefficients 7/2, 2, 5/4, ... for successive Ca binding.
- p. 581, Table 1.
- What is the microscopic Kd for Ca binding?
- Why are there two values for Q? Show that the values agree with the authors' number for the channel gating charge.
- Why is k1=1? What does k1 mean?
- p. 582, "We did not ... above model." If the flicker rate changes so obviously with voltage, how can its effect be "minor"?
- p. 584, "This may occur ... refute this possibility)" Why, "if the voltage also ... in nominally 0 Ca" would this mean that voltage gating operates by a different mechanism from Ca gating?