Review of class 4 – using DNA to purify CNT species

Review of class 4 – using DNA to purify CNT species 1. View CNTs as segment of graphene cut (so C’s are at edge) rolled up and reattached -> n,m classes 2. ssDNA acts like soap to “solubilize” CNTs with bases stacking (flat) against C rings, P’s interacting w/H20 3. DNA binding has little sequence specificity, yet elution of DNA-CNTs from IEX column -> specific peaks of CNT species w/different DNA oligos. Practically very important, but why does it work? 4. How were different CNT species recognized? semi-conductor CNTs have (n,m)-specific abs and emis peaks due to electron band structure

5. Model of ssDNA bound to graphene: bases on alternate sides of sugar-P backbone; adjacent strands anti-parallel, novel base pairing 6. Imagine cutting this DNA-graphene into strips and rolling into CNTs: for DNA strands to be at edges of cuts and to base pair across seam may require special matching between sequence and n,m type 7. Hypothesis: only such uniformly “tiled” CNTs elute in IEX chromatography as narrow peak; non- uniformly tiled species stick in column or elute over broad range of [salt] 8. Nice idea, but not clear what data support hypoth.

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

What we’ll cover: class 5 stretching at low force, concept of entropic spring freely-jointed chain and worm-like chain models at high force – structural change in double helix B->S form, similarity to phase change methods used to study hydrodynamic drag paramagnetic beads laser traps class 6 twisting at low torque, twisting compliance supercoils, topological changes at high torque – structural changes (P, L forms) coupling between twisting and stretching

Linear polymers and Hooke’s Law Freely jointed chain (FJC) model n segments of length b joined at freely rotating joints Brownian (thermal) 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 in 3-d = tanh(Fb/kBT) in 1-d (see Nelson, Biol. Phys. ch 9.2, for derivation) F b f1 x

Where (in the world) do these formulas come from? In 1-d, x = bSsi where si =+1 p({si})=Z-1exp[-(-FbSsi)/kBT] <x> = Sall states p({si}) x = Ss1=+1…Ssn=+1 Z-1 {exp[-(-FbSsi)/kBT]} bSsi = kBT (d/dF) ln {Ss1=+1…Ssn=+1 exp[-(-FbSsi)/kBT]} = kBT (d/dF) ln [exp(Fb/kBT) + exp(-Fb/kBT)]n = nb tanh(Fb/kBT) = L tanh(Fb/kBT) F b f1 x

<x>/L = tanh(Fb/kBT) tanh(z) = (ez – e-z)/(ez + e-z) -> z for z<<1 -> 1 for z>>1 Low force regime F << kBT/b, F -> <x> (kBT/Lb) => ksp = kBT/Lb the longer L, the more compliant the higher T, the less compliant empirically, b ~ 100nm so “low” F < 0.04pN At F = 0, equipartition theorem => ksp<x2> = kBT <x2>1/2 =xrms = (Lb)1/2 note xrms independent of T at F=0 High force regime: F>>kBT/b, <x> -> L

Several groups tried to measure b by pulling on 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 att. pt. 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>

F (pN) b = 50nm 100 200 <x> (mm) Problem – poor fit to 3-d FJC model no matter what L or b rest of paper = complicated disc. of possible explanations

Next paper – WLC model fits same data beautifully! Worm-like chain model - randomly oriented smooth chain with “stiffness” defined by: Persistence length p = length over which orientational correlation falls to characteristic value <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 x DNA q ^ t2 ^ t1 s 1 <cosq(s)> p s

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

Fp/kBT = ¼ (1-<x>/L)-2 – ¼ + <x>/L At low force, <x>/L<<1, Fp/kBT -> 3<x>/2L F -> (3kBT/2pL) <x> => ksp= 3kBT/2pL p L/unit 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

What use are these formulae? Can estimate: Average end-end distance of a DNA of given L Average length of ss or ds DNA under given F For immobilized enzyme pulling on DNA (or RNA), the maximum force it can exert, given <x>/L For enzyme unwinding a dsDNA held at given F, the length unwound, given <x>/L

Next paper – what happens at higher force? 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

At F ~ 65pN, DNA suddenly begins to stretch Further pulling lengthens DNA >L w/ little increase F until new, fully stretched state is reached (~1.7 L) Smith et al Science 271:795 (1996)

Stretched “S”-form of DNA probably has base-stacking interactions disrupted -> change in helix pitch >65pN 3.4nm/10bp  5.8nm/10bp “Cooperativity” of transition (occurrence over just 2-3 pN) suggests S-form segment spreads along DNA; similar to phase change (e.g. ice->water, adding heat energy 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

Nicks in single DNA strands may -> partially unraveling on further stretching -> mix of ss and ds regions with intermediate behavior ds mix F (pN) ss <x> mm

Stretching experiments used laser trapNobel prize to generate high force (only get few pN w/magnets) Highly focused laser pulls object with higher index of refraction towards brightest part of laser beam (x=0); small displacement x -> restoring force ~ -kspx. Given trap strength ksp, observing x, one can infer F and vice versa Mechanism – light E-field polarizes object with diff. dielectric constant -> attractive dipole force --> -- ++ E

Alternative explanation – photons carry momentum; bending ray changes photon momentum; momentum conserva- tion => object feels opposing force; if beam asymmetric, force from brightest region dominates Newman and Block, Rev Sci Instr 75:2787 (2004)

Their set up: apposed lasers with | polarization trap bead; pol. beam splitter (PBS) sends light from left laser to right position detector; bead displacement alters position of laser light on detector; flow and Fdrag = 6phrv can be used to calibrate position detector signal to force PBS DNA attached to streptavidin beads via biotin tags at DNA ends

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 (from next week’s paper)

This kind of set-up can also be used to investigate effect of twisting forces on DNA – next week’s topic apply torque DNA made by pcr and restr. enz. cutting + ligation; ss nick allows bottom part to swivel observe w ~ t keep const. tension

Summary 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/p, F ~ ksp<x> where ksp = 3kBT/2pL When F=0, ksp<x2> = kBT When F > 65pN, B-form DNA (3.4nm/10bp) converts to S-form (1.7x longer); might be able to use mixed B-S form DNA as force clamp

Review of class 4 – using DNA to purify CNT species