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Mechanical properties of DNA under stretching and twisting Why important –

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  1. Mechanical properties of DNA under stretching and twisting Why important – biology: curved/bent DNA important in packing into nuclei and viruses, in regulation of transcription, various enzymes bend/twist DNA during replication, transcription, recombination polymer physics: model for understanding basic force- length relationships for well-defined polymer technology: important for using DNA as tool to pull, twist objects; to study how enzymes that act on DNA work as nano-machines

  2. Entropic effects on linear polymers – Brownian motion randomizes contour; if all contours (micro-states) equally likely, what is most likely end-to-end distance? Freely jointed chain (FJC) model n segments of length b joined at freely rotating joints Brownian motion randomizes fi applied force pulls out chain fixed at one end contour length L = nb (b also called “Kuhn” length) <x>/L = coth(Fb/kBT) – kBT/FB (see Nelson, Biol. Phys. ch 9.2, for derivation, math related to alignment of magnetic domains in applied field H) turns out not to be very good model for DNA F f1 b x

  3. Several groups measured force-extension relation for DNA Bustamante (Science 258:1122 (1992) dimer of dsDNA from l phage, 48kb x2, L ~30mm, one end attached to glass, other to r ~1mm paramagnetic bead observed in microscope attachment point determined by varying flow and magnetic field; for flow v, Fflow= 6phrv(1+9r/16d) Ftotal = Fflow/cos q measure <x> as function of Ftotal <x>

  4. F (pN) b = 50nm 100 200 <x> (mm) Problem – poor fit to FJC model no matter what L or b

  5. Another model, worm-like chain WLC, fits same data much better Randomly oriented smooth chain characterized by: persistence length p = length over which orientational correlation falls to 1/e <x(F)>/L does not have analytic solution, but in high and low force limits, Fp/kBT = ¼ (1-<x>/L)-2 – ¼ + <x>/L DNA ^ t1 q ^ t2 ^ t1 s x 1 <cosq(s)> p s

  6. Bustamante, Science 265:1599 (1994) FJC model WLC model fits force- extension data much better than FJC WLC model

  7. Fp/kBT = ¼ (1-<x>/L)-2 – ¼ + <x>/L analytical approx WLC At low force, <x>/L<<1, Fp/kBT -> 3<x>/2L Hooke’s law F -> (3kBT/2pL) <x> => ksp= 3kBT/2pL p L empirically, dsDNA 50nm 0.34nm/bp ssDNA~1nm ~0.6nm/b ssDNA is “tighter” entropic spring, because more flexible! At high force, Fp/kBT -> ¼ (1-<x>/L)-2 => <x>/L -> 1 – (kBT/4pF)1/2

  8. Corrections to WLC model Slight discrepancy with WLC model at 10-60pN -> estimate of Young’s (elastic) modulus E: F/A=EDx/x ->E = 3x108N/m2 cf~ 109 for nylon 1012 for CNT Dx Note Young’s modulus alone would not fit data; elasticity mainly due to entropic effects (all microstates equally likely)

  9. At F ~ 65pN, DNA suddenly stretches a lot more Further pulling lengthens DNA > L w/ no change in F until new, “fully” stretched state is reached at ~1.7 L Smith et al Science 271:795 (1996)

  10. Stretched “S”-form of DNA probably has base-stacking interactions disrupted -> change in helix pitch >65pN 3.4nm/10bp  5.8nm/10bp Transition from “B” to “S” form DNA equivalent to phase change (e.g. ice->water, adding heat doesn’t change temperature until all ice melted: adding energy via stretching doesn’t change tension until all DNA converted to S-form); can be used to clamp F at 65pN

  11. Stretching experiments often use magnets or laser trapNobel prize to pull on beads Highly focused laser pulls object with higher index of refraction towards brightest part of laser beam (x=0); small displacement x -> restoring force ~ -kx. Given trap stiffness k, observing x, one can infer F Mechanism – light E-field polarizes object with diff. dielectric constant -> attractive dipole force --> -- ++ E

  12. What use are these force-extension relations? Estimate: Average end-end distance of a DNA of given L (at F=0) Average length of ss or ds DNA under given F Study enzymes that pull on DNA (or RNA), or untangle tangled DNA (examples below) ? Use to make sensors that detect single-molecules via their effects on DNA tether formation or length

  13. Study RNA pol In solution RNApol moves along DNA as it makes RNA copy If RNApol immobile, it pulls DNA through it x stuck on glass Measure x by moving trap; infer F from displacement of bead from trap center; infer L from WLC eqn; L shortens over time as RNApol pulls DNA through it; bead rotates as DNA is pulled through RNApol – why? speed dependence on substrate conc. and F provides insignts into how chemo-mechanical motor works

  14. Study DNA pol and helicases infer F from bead displ. WLC predicts fractional ext. of ds and ss regions xds/Lds and xss/Lss as fn(F) xds+ xss = xobs Lds = .3*ndsLss = .5*nss nds + nss= N -> 6eqns, 6unk. x Knowing DNA length N in bp allows one to infer length of ds and ss regions at given F, hence progress of enzyme that converts ss->ds (pol) or ds->ss (helicase); how[ATP], [substrate], load F affect speed -> mech. insights

  15. Study enzymes that change dna topology: topoisomerases If both strands of a dsDNA are bound to a magnetic bead and a glass surface, DNA is torsionally constrained. plectonemes Twisting bead with magnet creates torque. At critical torque, each additional twist -> 1 supercoil (“plectoneme”), which lowers bead detectably. N S F

  16. This system used to study a topoisomerase that nicks ds DNA, allowing it to uncoil if torsionally stressed, and then reseals the nick. What happens to bead height when DNA is nicked? What affects bead’s speed? This expt allowed measurement of friction betw. enz. and substrate DNA at nm length scale – i.e. provides info about interacting surfaces

  17. Mechanical properties of DNA related to twisting Experimental set-up to study twist-torque relations apply torque DNA made by pcr and restr. enz. cutting + ligation; ss nick allows bottom part to swivel observe w ~ t keep const. tension

  18. Example of DNA stretched between bead held on pipet and bead held in laser trap: segment of biotinylated DNA in center binds small SA bead

  19. Torque inferred from rotational velocity t = gw = 14 ph r3w (equiv. to F=gv and g=6phr) 0.7 mm bead 0.9 mm bead 0.9 0.7 0.5

  20. Low torque regime, t = (C/L) (q-q0) (like F = kx) C = 410 +/- 30 pN nm2

  21. Caveat on units torque = r x F, units = [Nm], same as energy But energy = torque x angle, angle “unitless” in radians Less confusing if we refer to energy done by torque as Newton-meter-radians Easy to lose track of radians in equations because units often go unmentioned

  22. Independent measure of C in absence of torque based on equipartition theorem No extra twists, no flow, let bead drift via Brownian motion (C/L) <q2> = kbT, C = 440 +/- 40 pN nm2

  23. C doesn’t change (in linear region) for + or - twist But at tcrit, torque abruptly stops increasing/ decreasing with more twists What’s going on?

  24. Interpretation: At tcrit , DNA undergoes phase change (B->P), more twisting converts more B-form to P-form, t constant until all DNA converted. At tcrit, twisting work like latent heat of melting, converts ice to water without inc. temp.; phase co-existence acts as torque clamp.

  25. Coupling between stretching and twisting Imposing F>65pN converts B DNA to S DNA. S DNA has fewer twists/bp, so DNA unwinds as it extends. On reducing F to 0, DNA collapses to massively under-wound (? partially melted) state, then rewinds to form B DNA. B-S phase equilibrium acts a Force-Torque converter

  26. Hypothesized phase diagram B = standard Watson-Crick DNA S = stretched form (>65pN) P = + twisted form L = - twisted form scP = supercoiled P

  27. Main points For small forces (~pN), DNA acts like entropic spring DNA extension with force best fits WLC model Fp/kBT = ¼ (1-<x>/L)-2 – ¼ + <x>/L (for high and low F) p = 50nm for dsDNA, ~1nm for ssDNA L = 0.34nm/bp for dsDNA, ~0.6nm/b for ssDNA When F<<kBT/2p, F ~ ksp<x> where ksp = 3kBT/2pL When F=0, ksp<x2> = kBT

  28. For larger forces F > 65pN, DNA undergoes structural D B-form DNA (3.4nm/10bp) converts to S-form 2-phase equilibrium (mixture of B and S DNA) can be used to “clamp” force at 65pN For small torques, DNA acts like torsional spring for large torques, DNA undergoes structural changes 2-phase equilibria can be used to make torque clamps Strong forces assoc with structural changes couple elongating and twisting -> force-torque converters

  29. Methods developed in these studies – attaching DNA ends to beads and flat surfaces paramagnetic beads and magnetic tweezers laser trap allow remarkable manipulation of individual macromolecules, observation of their response to pN forces and torques, and their ability to effect pN forces and move sub-mm distances

  30. Applications in nanotechnology Knowledge of DNA elastic properties important for constructing nano objects with DNA ? constant torque wind-up motors ? force-torque converters (Your creative application goes here)

  31. Next week – what can one learn from DNA sequence? Homework – explore www.23andme.com web site What info do they provide about health based on DNA sequence? What technology do they use to determine dna sequence? What are the health implications based on? Do you feel the info is credible, hyped, useful?