(a) Suppose the (unimolecular) rate for U® K is k, and for K® U is k¢. Find the equilibrium ratio of knots to unknots [K] / [U].
(b) Find the free energy difference DG = GK - GU
(c) Given that for the case of 10000 bp circular DNA, [K]/[U] = 0.03, find DG.
Note: This really can be done! Two experiments were done ten years ago that measured this equilibrium, for various size circular DNAs [see S.Y. Shaw and J.C. Wang, Science 260, 533-536 (1993) and A.V. Rybenkov, N.R. Cozzarelli and A.V. Vologodskii, PNAS USA 90, 5307-5311 (1993)]. By the way, most DNA molecules in bacteria are circular. In general DNA ends are bad news: many pathways for degredation of DNA start with reactions at exposed ends.
(a) Is this a diagram of a reaction that occurs in thermal equilibrium? Why or why not?
(b) Find the rate equations for this reaction (i.e. the equations relating between dA/dt, dB/dt and dC/dt).
(c) In the steady state where dA/dt = dB/dt = dC/dt = 0, find the rate of B ® C transitions that occur, normalized to the total number of molecules [A]+[B]+[C].
(d) What kind of processes do you think this kind of reaction could be a model of?
(a) You may recall that the cross-sectional radius of a microtubule is about 12 nm. Find the Young modulus, approximating the microtubule to have a uniform and circular cross-section (i.e. ignore the fact that it is hollow).
(b) Now, suppose one end of the microtubule is clamped down, leaving the other end free to wobble. Estimate the work that must be done to displace the free end of the microtubule by an amount x in a direction perpendicular to the microtubule itself.
Hint: consider the deformation of the microtubule to be a circular bend.
(c) If our microtubule is in thermal equilibrium at room temperature, we can expect it to bend spontaneously, by thermal fluctuation. What is the mean-squared amplitude < x2 > that we should expect?
You will find that B has been measured for microtubules and for actin and for other biofilaments by measurement of spontaneous bending fluctuations, e.g. F. Gitttes et al, J Cell Biol 120, 923-934 (1993).
Also, the bending modulus of carbon nanotubules was also measured a few years ago by this method - after people trying to do electron microscopy noticed that they could not get a free end in focus because it kept wiggling for some darn reason!
x (microns) force (piconewtons = 10-12 newtons)
16.2771
8.28988
15.8929
4.90469
15.2161
2.2769
14.7765
1.06581
13.9411
0.67134
12.9103
0.433313
12.0543
0.251797
8.68331
0.082732
6.49035
0.050713
10.8376
0.17287
10.7692
0.167359
5.16824
0.03441
7.60005
0.070714
9.54139
0.131597
2.17784
0.013115
4.12757
0.026026
determine the persistence length A, by finding a fit of the data to the formula
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These experimental data are from the lab of Vincent Croquette and David Bensimon at the Ecole Normale Superieure in Paris, and were probably actually taken in about 1995 by Terence Strick and Jean-Francois Allemand who were doing their Ph.D.'s there at the time.
That group does incredible experiments on mechanical properties of DNA,
which you can explore at
http://www.lps.ens.fr/~vincent/