DNA supercoiling: plectonemes or curls?

Transcription

1 DNA supercoiling: plectonemes or curls? Sébastien Neukirch CNRS & UPMC Univ. Paris 6 (France) John F. Marko Northwestern University (IL. USA)

2 Why study DNA mechanical properties? mechanical properties influence biology of the cell 1 meter of DNA in a 10 micron wide nucleus ejection from viral capside transcription (RNApolymerase is torque dependent) protein binding is strain dependent, or induces strain on DNA chromatin compaction/decompaction (cell division) 20kbp dsdna (6800nm) in 40nm wide capside Tang et al (Structure) 2008

3 Pulling and twisting DNA X/ L f 0.9 N n S X(0) n/ n 0 n = Data from G. Charvin (LPS-ENS) 3

4 Pulling and twisting DNA X/ L 0.9 N n f S plectonèmes 0.6 X(n) n/ n 0 n> Data from G. Charvin (LPS-ENS) 4

5 Pulling and twisting DNA X/ L 0.9 f? n N S X(n) curls n/ n 0 n> Data from G. Charvin (LPS-ENS) 5

6 Numerical simulations : twisted rod with self-contact f, R, A given α interpolation fr 2 A 1.7α4 f R n f 6

7 Numerical simulations : twisted rod with self-contact X/L B A C X n = 4πR sin 2α n C B A (based on Swigon+Coleman model for contact in Kirchhoff rods) S. Neukirch, Extracting DNA..., Phys. Rev. Lett. 93 (2004)

8 Limitations of the twisted rod model 1. Electrostatics repulsion : supercoiling radius R is not 1 nm 2. Tails are not straight : «disorded walk» (Worm Like Chain) 8

9 Analytical model with electrostatics repulsion f Elastic rod with : X = L n=lk straight tails DNA uniform ply total length L circular cross-section r 0 bending rigidity A twist rigidity C electrostatic repulsion: 2r U(r, α) r = r 0 N. Clauvelin et al, Biophysical Journal (2009) 9

10 Analytical model with electrostatics repulsion f n=lk V = 1 2 Asin4 α r 2 bending Cτ 2 L twisting X = L straight tails DNA uniform ply +U(R, α) fx with constraint: n = Lk = 1 2π τ L + electro ext. force sin 2α 2r minimize V => (α,r,τ, ) 10

11 Analytical model with electrostatics repulsion : results 2.0 X 6kbp 100mM X n 140 Experimental vs theoretical vs Marko slopes X n dzdn n Data from G. Charvin (LPS-ENS) F pn J. Marko, Torque and dynamics of linking number..., Phys. Rev. E. (2007) 11

12 Statistical mechanics model worm like chain 2.0 X predict transition 1.5 purely plectonemic structure? n 12

13 Abrupt transition 2.2 kbp 150 mm Forth et al Phys. Rev. Let

14 Transition = create first loop WLC single loop Daniels et al (PRE) 2009 Purohit (JMPS) 2008 Brutzer et al (Biophys. J.) 2010 plecto nemes 14

15 Comparing free-energies of straight and supercoiled DNA A Transition Buckling at T=0 K when E(A)=E(B) B unstable stable mechanical model (experiments)

16 Free energy total contour length L = nbp 0.34 nm 16

17 end-loop Free energy Γ n curls m plectonemic plectonemes γ wlc regions curl L s L = nbp 0.34 nm = L s + nγ + m (Γ + ) 17

18 L = L s + nγ + m ( + Γ) Free energy F = F s + nf γ + m (F Γ + F )+F f TS corrections to the work of the external force entropic terms F s = E twist + E bend + E f = 1 2 C sτ 2 s L s g(f)l s F γ = E twist + E bend = 1 2 Cτ 2 p γ +4 Af F = E twist + E bend + E electro = 1 2 Cτ 2 p Asin4 α r 2 + U F Γ = E twist + E bend + E electro = 1 2 Cτ 2 p Γ + q b Af + UΓ F f = 4n Af + q D m Af 18

19 L = L s + nγ + m ( + Γ) Free energy F = F s + nf γ + m (F Γ + F )+F f TS corrections to the work of the external force entropic terms U = U kt kt Ar 2 1/3 confinement in a tube F γ = F γ 1 4π 2 4 Log A(kT) 2 d 4 f 3 L m S(L,, n, m) =(n + m) Log γ loop size fluctuations (d=1nm) n m Log n! Log m! Tonks hard core gas 19

20 Free energy F = F s + nf γ + m (F Γ + F )+F f TS study F under the two constraints: L = L s + nγ + m ( + Γ) n = Lk = τ s 2π L s + τ p 2π 2α (m + nγ + mγ)+sin m + mp + nλ 4πr Twist Writhe we replace τ s,l s to obtain : F nm = F nm (α,r,τ p, ) 20

21 Computation method m = 0 m =0 F nm = F nm (α,r,τ p, ) F n0 = F n0 (α,r,τ p ) 2 L reference state F 00 = gl C s 2πLk L F nm (α,r,τ p, ) =F nm F 00 minimize with regard to α,r,τ p (saddle point approx.) plot as function of 21

22 Experimental data F/kT plecto Results 10 5 m =1 curl n =1 0 5 (0, 0) wlc /L (0, 0) wlc n =1 m = /L

23 Computing mean values At fixed Lk Z = n m e F nm() d Probability of seeing a configuration with n curls and e F nm () d m plectonemic parts: P nm (n = Lk) = e F nm () d n,m Mean extension of the system (considering all possible configurations: <X>(n = Lk) = n,m Xnm ()e F nm() d n,m e F nm () d 23

24 Results P nm wlc plecto curl < X/L > n Lk 0.8 Theo. 2.2 kbp 0.7 Exp. 150 mm 2 pn 0.6 data : Forth et al (Phys. Rev. Let.) n Lk

25 Results kbp 150 mm data : Forth et al (Phys. Rev. Let.) n Lk

26 Results kbp 150 mm data : Forth et al (Phys. Rev. Let.) n n Lk

27 Results n n [ Na + ] = 320 mm F (pn) n bp data : Brutzer et al (Biophys. J.) F=3.5 pn [ Na + ]

28 Mean number of curls Mean number of plectonemes curls <n>= n,m ne F nm () d Z m =1 n =1

29 Mean number of curls Mean number of plectonemes plectonemes <m>= n,m me F nm () d Z m =1 n =1

30 2.2kbp 150mM 0.25 pn blue 0.5 pn green 1 pn cyan 2 pn orange 4 pn red <m> dashed <n> plain

31 10kbp 150mM 0.25 pn blue 0.5 pn green 1 pn cyan 2 pn orange 4 pn red <m> dashed <n> plain

32 10kbp Salt dependence 1 pn => lots of plectonemes and curls at low salt

33 Conclusion WLC single loop plecto nemes 33

34 Thank you Work was done with: N. Clauvelin (Rutgers) B. Audoly (Paris) J. Marko (Northwestern)