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Introduction & applications Part III

Introduction & applications Part III. magnetic tweezers. “MT”. HW assigned later today (Due next Monday, 3/8/10). March 15 th , 17 th , night of 18 th : Presentations Reports due night of 18 th. Today’s take-home lessons (i.e. what you should be able to answer at end of lecture).

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Introduction & applications Part III

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  1. Introduction & applications Part III magnetic tweezers “MT” • HW assigned later today (Due next Monday, 3/8/10). • March 15th, 17th, night of 18th: Presentations • Reports due night of 18th.

  2. Today’s take-home lessons(i.e. what you should be able to answer at end of lecture) 1. FJC vs. WLC: What’s the difference/ why WLC is better (but more complicated) 2. Twisting vs. Writhe 3. Twisting– anharmonicity of DNA

  3. The Elasticity of a Polymer (DNA) Chain In the presence of a force, F, the segments tend to align in the direction of the force. Opposing the stretching is the tendency of the chain to maximize its entropy. Extension corresponds to the equilibrium. Point between the external force and the entropic elastic force of the chain. Do for naked DNA; then add proteins and figure out how much forces put on DNA.

  4. Two Models of DNA (simple) Freely Jointed Chain (FJC) & (more complicated) Worm-like Chain (WLC) FJC: Head in one direction for length b, then turn in any direction for length b. [b= Kuhn length = ½ P, where P= Persistence Length] Idealized FJC: Realistic Chain: FJC: Completely straight, unstretchable. No thermal fluctuations away from straight line are allowed The polymer can only disorder at the joints between segments FJC: Can think of DNA as a random walk in 3-D. WLC: Have a correlation length

  5. The Freely Jointed Chain Model F b bF Where -Fb cos(q) is the potential energy acquired by a segment aligned along the direction q with an external force F. Other terms are geometric terms. The integral is over all space, where it is much more convenient to do it in spherical coordinates, rather than in Cartesian coordinates. Integration leads to: And for a polymer made up of N statistical segments its average end-to-end distance is: Langevin Function

  6. The Freely Jointed Chain Model At low forces: Thus, at low forces, the chain behaves as a Hookian spring with a spring constant, 3kBT/b = 3kBT/2P, where P is the persistence length of the chain. Lets see its physical meaning….. The spring constant is inversely proportional to the size of the molecule, and to the persistence length. This dependence makes sense. Its easier to align a stiff polymer than a polymer that is very flexible. This is a manifestation of the nature of polymer elasticity: entropic!

  7. Fitting the FJC Model to the Data Small (< 100 fN) and large (>5 pN) forces FJC is good In between have to use WJC. FJC

  8. The Worm-like Chain Model Uses a continuum description of the chain to address these limitations: - The entropic elasticity of the DNA chain involves small deviations of the molecular axis due to thermal fluctuations. - The direction of the chain is correlated over a distance, called the persistence length (= 2x Kuhn length) of the chain. For DNA, at 10 mM NaCl, PDNA = 150 bp or 550 nm. - Bottom line: forces of the order of kBT/P are needed to align and straighten elastic units of these dimensions.

  9. s=L ∫ s=0 Worm-Like Chain Fixed contour length, L. It has a tangent (of fixed length) which varies as you proceed along a contour (s) U = C ds |dt/ds|2 |t(s)=1| If used with Boltzmann weight, gives these decaying correlations—non-trivial If dt/ds large, curves a lot. If C big, persistence length is big. C = Lp·T·kB.  (Large C means large penalty for bending – and this means large Lp.)  Lp=persistence length

  10. The Worm-like Chain Model For forces >kBT/P, the force-extension behavior of the chain can be obtained from the effective energy of a stretched WLC: F z r(s) At high forces, the extension of the WLC approaches its contour length, L, and the tangent vectors fluctuate only slightly around the pulling axis,:

  11. The Worm-like Chain Model “This interpolation formula is an excellent approximation to the exact solution throughout most of the range of forces investigated by the magnetic bead experiment.” Bustamante et al.Science, 265, 1599-1600 (1994)

  12. Fitting the WLC Model to the Data FJC WLC

  13. s Twist Writhe DNA may be Supercoiled:Has both Twist and Writhe Linking Number = Twist + Writhe Lk = Tw + Wr Relaxed DNA: Twist =1 turn/10.4 bp. Writhe = 0. ∆Tw=0 Can’t just twist up DNA and have it all go into twist. Example: Phone cord. Pull on DNA and writhe comes out. Measure relaxed (non super-coiled) DNA and figure out length vs. force.

  14. Supercoiled DNA (s): is the deviation from relaxed linking number. s = (Lk – Lko)/Lko =∆Lk/ Lko Twist (Tw), Writhe (Wr), Linking Number (Lk) Tw: # of times the two strands wrap around each other Wr: # of times C crosses itself. Linking Number = Twist + Writhe Lk = Tw + Wr Tw=2 Tw=0 Wr=0 Wr=2 Charvin, Contemporary Physics,2004 Ex: Hold DNA out straight so that it has no Writhe, add of take out twist, then let fold up (Twist goes into Writhe). Normal DNA is negatively supercoiled, -0.06 = 6 turns for every 100 taken out. Why? Helps unwind DNA– makes it easier to uncoil, separate strands. Enzymes which do this called Topoisomerases. Why? What about archebacteria that lives in hot springs? Positively supercoiled Makes DNA more stable

  15. Movie measureforce.

  16. Some Examples of Tw and WrLinear DNA with Constrained Ends

  17. Application to Eukaryotic Cells In Eukaryotic cells: When a chromosome wants to express a particular gene, it becomes single stranded. Requires a tremendous amount of energy (remember partition function and genes, typically 10-100k a.a., require many H-bonds to be broken: DNA is a very stable structure). Requires enzymes– topoisomerases. In response to supercoiling, they will assume an amount of writhe, just as if their ends were joined.

  18. More Examples of Writhe and Twist DNA’s helical nature (natural twist) not shown. Circular DNA Plectonemes more common: shape bacterial plasmids take. http://commons.wikimedia.org/wiki/Image:Circular_DNA_Supercoiling.png

  19. Low F: symmetric under s - s The shortening corresponds to the formation of plectonemes upon writhing. Torsionally stressed single DNA molecule Playing with phone cord: can you explain graphs? When the force is increased above 0.5 pN, the curve becomes asymmetric: supercoils still form for positive coiling while local denaturation adsorbs the torsional stress for negative s. At forces larger than 3 pN no plectonemes are observed: the torsional stress is adsorbed not by writhe but in local structural changes of the molecule. Extension vs. supercoiling at constant force Three regimes T. Strick et al., J. Stat. Phys., 93, 648-672, 1998

  20. Class evaluation 1. What was the most interesting thing you learned in class today? 2. What are you confused about? 3. Related to today’s subject, what would you like to know more about? 4. Any helpful comments. Answer, and turn in at the end of class.

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