Mechanical properties of DNA under stretching

Why important – biology: curved/bent DNA important

in packing into nuclei, into viruses,in regulation of transcription, variousenzymes bend/twist DNA during replication, transcription, recombination

technology: important for using DNA as toolto pull, twist objects; to study howvarious enzymes that act on DNA work;to build nanoscale objects using DNA

What we’ll cover:stretching – low force

concept of entropic springfreely-jointed chain and

worm-like chain modelshigh force –> structural change in double helix

B->S form, similarity to phase change

methods used to studyhydrodynamic dragparamagnetic beadslaser traps

example of how used to study mechanism of enzyme that works on DNA

Linear polymers and Hooke’s Law

Freely jointed chain (FJC) model n segments length b joined

at freely rotating joints Brownian (thermal)

motion randomizes fi

applied force pulls out chain fixed at end contour length L = nb

<x>/L = tanh(Fb/kBT) for 1-d model (see Nelson, ch 9.2)

tanh(z) = (ez – e-z)/(ez + e-z) -> z for z<<1-> 1 for z>>1

f1

Fb

x

<x>/L = tanh(Fb/kBT)

Low force regime F << kBT/b, tanh(z)-> z for z<<1F -> k<x> where k = kBT/Lb

the longer L, the more compliantthe higher T, the less compliant

equipartition theorem: k<x2> = kBT

<x2>1/2 = xrms = (Lb)1/2 = n1/2b n=L/bxrms independent of T, F at low forcethermal energy randomizes fi

High force regime: F>>kBT/b, <x> -> L

Several groups tried to measure b by pulling on DNA

Bustamante (Science 258:1122 (1992)

phage DNA of known L ~30mm attached at 1 end to glassother end to r ~ 1mm para-magnetic bead

att. pt. determined by varyingflow and magnetic field

knowing flow v, Fflow = 6phrvmeasuring q, Ftotal = Fflow/cos qmeasure <x>

<x>

<x> (mm)

F (pN)

Problem – poor fit to 3-d FJC model no matter what L or b

b = 50nm 100 200

Worm-like chain model randomly oriented chain with “stiffness” defined by:

Persistence length p = length over which orientational correlation falls exponentially 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 at low F, <x>/L << 1, F = ksp<x> where ksp = (3/2)kBT/pL

st1

t2^^

<cosq(s)>

q

sp

1

DNA

x

FJC model

WLC model

WLC modelfits force-extension data much better than FJC

Bustamante, Science 265:1599 (1994)

WLC model also fits ssDNA if you change p

pds @ 50nm (~150 base pairs)pss @ 1nm which is “wiggilier”?

--------- relative

ksp = (3/2)kBT/pL pds = 50nm pss = 1nm

Is the spring stiffer for ss or ds DNA?

Why does a more flexible DNA chain (ss) act like a stiffer spring?

Contour length L (= length of fully pulled out chain)Lss @ .5nm/b * # bLds @ .3nm/bp * # bp

Why is the contour length of ds DNA shorter/bp?(think base stacking in helix…)

Fp/kBT = ¼ (1-<x>/L)-2 – ¼ + <x>/L L=n*l/bp or l/b

Could you estimate length of ss or ds DNA of length n in bases (or base pairs) at given F?

Why is n-base ssDNA longer at large F but shorter at low F than n-bp dsDNA?

--------- relative

At F ~ 65pN, dsDNA suddenly begins to stretch

Further pullinglengthens DNA >Lw/ little increaseF until new, fullystretched 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

3.4nm/10bp 5.8nm/10bp

“Cooperativity” of transition suggest S-form segmentspreads along DNA (takes less energy to expand anS-form region than to initiate one); similar to phase change ice->water, adding heat doesn’t change temperature until all ice melted, more pulling work doesn’t change tension until all DNA converted to S-form.

Stretching experiments used laser trapNobel prize

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 strength k, observing x, one can infer F

Mechanism: light E-field polarizes object with diff.dielectric constant -> attractive dipole force

--> -- ++ in gradient E, polarized object feels net force

E

Newman and Block, Rev Sci Instr 75:2787 (2004)

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

Moving laser trap stretches DNATrap position reports DNA end-to-end length

Quadrant photo-detector reports beaddisplacement Dx fromtrap center, i.e. reportsF given trap stiffness ksp since F = ksp

Numerator = observed D length (compared to all ds)Denominator = max D length if all ss compared to all ds

Ratio = fractional D in length ~Nss/Ntot

What is length of mixed ds-ssDNA?

What enzyme did Bustamante et al add tothis system?

Enzyme + dNTP added to ds/ss tetherData collected every 0.125s; how fast does enz. move?Bottom curve = slope averaged over sliding 3s windows How might you interpret the “bumps”?

Watching DNA polymerase act in real time

Where on velocity trace is enz. active?Why doesn’t velocity -> 0 between bumps? Why is “off time” (1/koff) the aver. time enz. is on?Can you estimate koff, kon from this data?

Complicated scheme of E + D <->ED where E can bind as polymerase (p), then bind dNTP, add base (n->n+1) or as exonuclease (x) then remove a base (n->n-1), or convert between p and x configurations

Rate, binding constantsfrom literature, “bulk” expts.

You could compare your single-molecule kon, koff

to data from bulk expts; this might strengthenyour interpretation but does not advance the field

What is biological role of exonuclease function?

What happens to misincorporation rate if you mutate (eliminate) exo function?

Effect of tension (F) on enzyme velocity

Why are error bars bigger ~6pN?Why might velocity decrease as tension increases?

Does data strongly support n = 1, 2, or 3?

Complicated model for enzyme pulling a few (n) bases of template ss into configuration of ds; this requires work W(n) against tension; velocity ~e-W(n)/kT; how do models of different n’s fit the data?

n = 1n = 2

n = 3

Above stall force ~40pN, only exo activity (this is how they converted ds tethers to partially ss!)

What does inset show?Is conversion reversible?

How would you interpret “bumps” in exo velocity?

Unfortunately, obs. koff, kon’s suggest bumps can’t be enz.falling off, rebinding, but involve pol <-> exo conversions

What can single-molecule expts. show that would be very hard to learn from bulk expts.?

Are enzyme molecules heterogeneous or all the same?

Is enzyme rate sequence-dependent?

Is enzyme rate slowed by tension? This could informdetailed models of how enzyme works

What makes enzyme interconvert between pol and exo conformations?

Summary

laser traps/magnets/tethered bead expt’l. system: allow application of pN forcesmeasurement of pN forces and DNA/RNA lengths with near nm precision

WLC model predicts DNA mechanical properties accurately(extension as function of force, twist and

buckling as function of torque)

Clever experimental systems -> real-time observationof single enzymes/assemblies at work, potentiallyelucidating mechanistic details

Lots of other examples of single-molecule studies:

RNA polymerases that partially melt dsDNAand make RNA copies

Motors that pack DNA into virus particles

Helicases that unwind ds DNA/RNA

Topoisomerases that nick, religate DNA, relievingtorsional strain and topological entanglement

Ribosomes that copy RNA into protein

These studies combine nano-scale biology andengineering -> new discipline

For now, mostly research applications…

Understanding nanoscale biosystems provideinsight, tools potentially applicable tonon-biological nanosystems

Example – experimental test of basic physics prediction of relation between work W done on non-equilibrium system and free energy change DG at equil.

W > DG (due to dissipation) classical eqn<e-W/kT> = eDGJarzynski prediction 1999

slow

fast

nfold

efold

W = area betwcurves

Science vol 296p1832, 2002

Next week – DNA sequencingwhy the interestfirst “next generation” method

Homework problems on DNA mechanics

Midterm due by end of weekend