Torque in stretched supercoiled DNA

•Twisting and pulling DNA – “state coexistence” of plectonemic supercoiled DNA and extended DNA

•In mixed state torque depends on force alone, not on linking number

•What is the torque as a function of force?

•Use in studying relaxation of supercoiling by type Itopoisomerases and other supercoil-relaxing enzymes

•Torque-force “phase” diagram (low forces)•What about directly measuring torque?

Magnetic tweezer manipulation of DNA

Rotation of the magnet introduces supercoils, which translates into a reduction of the length of the DNA due to the formation of plectonemic supercoils.

Method originated by Allemand, Strick, Croquette, Bensimon (Science 1995).

Key point is that molecule is held under•constant tension (force) via fixed position of magnet•constant linking number via fixed angle of magnet

If magnet is not moved, then changes of σ (e.g., mediated by topoisomerases) occur at fixed force

Classical model of pure twist for elastic rod (or DNA)

L

θ

nm 1.85 nm 3.4/π2ω

/2πθ TwLk so Wr,no

2Cω kT

Lθ

2C kT

L E

1-0

220

2 ===∆=∆

==2

σ

Twist elasticity described by “twist persistence length”

C = 75 to 115 nm

Why such a big range of values? (95 nm)

σ = − 0.033 σ = 0.000 (relaxed)

σ = − 0.062 (in vivo) σ = − 0.016 Boles, White, Cozzarelli JMB 1991

Plectonemic Supercoiling ( |σ| > 0.01 )

WrTwLk +=

Wr ≈ -1 -1 -1 -1 -1 RH

Separation of helix repeat (3.5 nm) and self-crossing distance (~ A = 50 nm) allows separation of local (twisting) and nonlocal (writhing) contributions to ∆Lk

DNA crossings can soak up ∆Lk, reducing ∆Twand therefore “screening” the twisting energy

Free energy of plectonemic supercoils:

Simulation data:Klenin, Vologodskii,

Cozzarelli, JMB 1991 Vologodskii, Levene, Klenin,

F-K and Cozzarelli, JMB 1992

For experiment data see: Rybenkov, Vologodskii,

Cozzarelli, NAR 1997good survey of Na, Mgconditions

nm 1.85 nm 3.4/π2ω

p ω P kT callC) (P Tw reducesWr

2ω P kT

L F

1-0

20

220

===

<+= mσ

P = 24 nm ok for DNA at physiol salt, range of σ we need

Fully extended DNA under torsional stress

Apply enough force to stretch out DNA even though it is twisted

Chiral fluctuations generate a bit of Wr, slightly screeningtwisting energy (Moroz and P. Nelson, 1997

numerics Bouchiat, 1998pigtails V. Rossetto, 2005).

��

�

AfkT

4AC 1ω C kT (f)c

Af kT f g(f)

x/L (f)g'nm 50 A

2

)f(c )g(f L F

20s

2s

+

−=+−=

==

++−= σ

Intermediate forces – plectoneme/extended mixed state, const f and σ

Determine xs,σs by minimizing total free energy, const f and σ

σσσω

τ

σσσ

σσσ

f, fixedat que torL

)F(1

totalfixed toup addmust but

ed transferrbemay number linking x x

x-1 x

F,FF,in same force )(F x )(F x )F(

o

ppss

sp

pspppsss

∂∂=

=+

=

+=

Minimized F(σσσσ): Common Tangent Constructionphase separation in σ between limiting values of the two “pure states”, σs and σpF(σ) linear in σ between these limits

2

2)( σσ ss cg

LF +−=

2

2)(

σσ p

LFp =

g(f)

σ < σs pure stretched

σ > σp pure plectoneme

σs < σ < σp mixed stateconstant torqueas σ is variedat fixed force

L)F(1

o

σσω

τ∂∂=

Coexistence region properties2

2)( σσ ss cg

LF +−=

2

2)(

σσ p

LFp =

We know g(f) so measuring the coexistence linking numbers (easy) gives us p and cs(f) [can also be done for nonlinear terms]independent of detailed theoretical models

Torque is fully determined and is therefore predicted

L)F(1

o

σσω

τ∂∂=

Torque in Coexistence RegionWe more or less know everything in this formula

P = 24 nm A = 50 nm C = 95 nmg(f) ≈ f – (kT f/A)1/2

Cs ≈ C [ 1 – (C/4A)(kT/A f)1/2 ]

TAf2kτ B=

TAf4kτ B=Love, 1944 linear buckling instability

Strick, Allemand, Croquette, Bensimon, 2001 energy crossing, T=0

Compare to Previous Work

σ

0.06

0.05

0.04

0.03

Comparison of Theory to Data from MC Simulation of Stretched Supercoiled DNA (no fitting)

C=95 nm C=75 nm

Back to Analytical Model: Extension by force

Extension is derivative of free energy so it iscontinuous but not necessarily smooth

L) f,F(

f

Lx σ

∂∂−=

2 pN

1

0.50.25

Extension vs linking, fixed force

Critical forceto start opening plectoneme

Force at which plectonemesare entirely eliminated

This is also the DNA tension inside “free” supercoiled plasmid or domain,

about 0.5 pN for σ = 0.05

less exciting

Comparison with experimental extension versus linking number data(C=95 nm, P=24 nm, A=50 nm)

Super-simple quadratic twist stiffness model describes data well in B-DNA regime…(experimental data of G. Charvin, published in Neukirch PRL 2004)

2.95 pN1.310.740.440.25

For f > 0.5 pN, s < 0 data show a different behavior, due to DNA helix unwinding(C=95 nm, P=24 nm, A=50 nm)

Unwinding of dsDNA breaks L/R symmetry of extension vs σ

2.95 pN1.310.740.440.25

Add denatured DNA state to extended and plectoneme states

Predicted low force vs torque phase diagram – note triple points

2

2)( σσ ss cg

LF +−=2

2)(

σσ p

LFp =

ddd

dd cgL

F εσσσ +−+−= 2)(2

)(

Free energies at 5 pN

Discussion with Vincent CroquetteUntwist DNA with long complementary-palindromic region

(neglecting initiation barrier energy) 2

2)( σσ ss cg

LF +−=

1 ; )(or 0 −== palpalpal /LεL

Fσ

( )00

/211

wrt minimize)1( )()(

ωσ

ωτσ

σσσσ

gccgcg

xFxxFxF

cs

cs

sc

s

sssss

−≈−=−≈−−−=

=−−=

Ingenious techniqueNot thermal equilibriumDNA under >15 pN forceIntrinsically dynamic measurementFinite time window = big errorExperimental uncertainty of roughly

5 pN nm

Direct measurement of torque in DNAusing hydrodynamic drag (Nature 2003)

Direct measurement of torque in DNAusing rotated-polarization light laser tweezer and custom-made

optically anisotropic tethered cylinders (Nature Methods 2007)

Ingenious techniqueStatic measurement possibleDNA under >10 pN forceExperimental uncertainty of

perhaps 3 pN nm

Topo V relaxes scDNA in multiple-turnevents (PNAS 2007)in a fashion similar to type IB topos(Koster et al 2005)

∆∆∆∆Lk of 12.3 ± 1.8

∆∆∆∆Lk of 13.8 ± 1.8

Single molecule experiments for topoisomerase VDNA supercoiling is relaxed in multiturn steps;the step size distribution is exponential

Taneja, Schnurr, Mondragon, Slesarev

Dependence of the relaxation rate on applied force… …and on inferred torque

This indicates that the main source of dissipation in this experiment is rotationaland not translational

Measured rotational friction constant (energy barrier) for Topo V

[ ] [ ] sec nm pN (2)071.0 eeeη

Tk 2rr 2πω τβθτβθβE3B B =+=−=−= −+ −−

−+ ζζτ

m�

π

Following the initial cleavageduring rotation there is some angular range δ under which cleavage and religationoccur at rates k and k’

The probability per turn of religation is

)'(

'/)'(

1 kk

ekkkk

P +

++−

=ωδ

)1/(1Lk 1P−=∆

The distribution of relaxation events follows an exponential distribution

Since both ends of the broken strand have to be close to religate, as the angular velocity increases, the mean step size also increases

kδ=380±130 sec-1 k’δ=33±7 sec-1

Dependence of the mean step size on angular relaxation velocity allows estimate of chemical rates

Summary and Conclusions

We still can’t measure small torques in DNA (< 5 pN nm)

We still don’t have a complete concensus regarding the torsionalstiffness of DNA

We have a pretty good idea of how torques should vary with force inplectoneme-extended DNA coexistence

Many effects can be added to the simple coexistence modelnonlinearities in plectoneme free energysequence effects in plectoneme (and in denaturation)many small plectoneme domains instead of MFT

We really never have observed plectoneme domains coexisting with extended DNA – dynamics, sequence localization effects

NSF PHY-0445565 & DMR-0715099