La molécule d'adn vue comme une poutre élastique : application aux expériences de pinces magnétiques

Transcription

1 La molécule d'adn vue comme une poutre élastique : application aux expériences de pinces magnétiques joint work with: Sébastien Neukirch CNRS & UPMC Univ Paris 6 Institut Jean le Rond d Alembert Michael Thompson (DAMTP, Cambridge, UK) John Maddocks (Ecole Polytechnique Fédérale de Lausanne) Michael Henderson (IBM - T. J. Watson Center, NY, USA) Nicolas Clauvelin (d Alembert - doctorant) Basile Audoly (d Alembert)

2 Elastic filaments climbing plants F a b M M s = 0 theory F cables fibrous proteins applications carbone nanotubes DNA super-coiling

3 Elastic rod in equilibrium p(s) r(s+ds) r(s) F(s) O p(s) = external force gravity, contact, electrostatic,... M(s+ds) M(s) F(s+ds) force balance momentum balance ps sfs sfs0 psf ' s0 Ms sms O Fs s O Fs 0 Ms sms0 rsrs sfs0 M 'sr 'sfs0

4 Cosserat model for 1D elasticity 3 directors d 1,d 2,d 3 on the top of rs e x no shear no extension r 'sd 3 s d 2 d 1 d 2 d 1 d 3 d 1'susd 1 d 2 'susd 2 d 3 'susd 3 evolution in SO(3) usu 1, u 2, u 3 d 1,d 2,d 3 e z us 1, 2, d 1,d 2,d 3 e y u 1, u 2 : curvature and u 3 : twist

5 Linear Constitutive Relations Bending stiffness : K 0 Md 1 K 0 u 1 Md 2 K 0 u 2 Twist stiffness : K 3 Md 3 K 3 u 3 K 0 = E I I : moment of inertia E : Young's modulus filament E Microtubule 1 GPa DNA 1 GPa Actine 2 GPa Collagen 2 GPa Rubber 2 GPa Steel 200 GPa

6 Kirchhoff equations 21 ODEs with variable : s ordinary differential equations d ds Fp d ds MFd 3 d ds r d 3 d ds d i ud i m i K i u i linear elasticity 21 unknowns Fs Ms rs d 3 s d 2 s d 3 s us i1,2,3 A 1 boundary conditions - how the rod is held - few solutions are admissibles D =0 1 D 2 A 1 D 3 d 3 A 1 d 3 A 2 l A 2 A 2 A 2 ra 2 ra 1 kd 3 A 2 DLk A 1

7 Find admissible equilibr ium solutions : shooting method initial conditions r 00,0,0 d 1 01,0,0 d 2 00,1,0 d 3 00,0,1 parameters F0,M0 y x s = 0 z s = L end conditions xl0 yl0 d 3 xl0 d 3 yl0 solution of ODEs u F0,M0 0 u0 L u P this defines a P-L solution manifold

8 1D solution manifold : path following pr edictor-cor r ector scheme 1D solution manifold At each point : 1-(predictor) we take a guess : Z i 2-(corrector) we define a projection : P i u 1,u 2,u 3 0 and we solve : u,u,u u 1,u 2,u 3 0 u 3 predictor P 1 Z 1 corrector Z 2 P 2 A 1 A 2 1 u1,u2,u30 2 u 1,u 2,u 3 0 P i u 1,u 2,u 3 0 A i to obtain u 1 A 0 u 2

9 Find admissible equilibr ium solutions : discr etization methods discretization over N intervals U 0 boundary conditions matching conditions s = 0 U s = L system of nonlinear algebraic equations U 0 U D DU U U U 0 U 0 Newton corrector 1-we take a point U 2-compute Jacobian 3-kernel is tangent plane 4-we project orthogonaly : UU 0

10 2D solution manifold Michael Henderson (IBM)

11 Pulling and twisting DNA N n T S Z/L Z(0) σ n = data from Gilles Charvin (LPS-ENS) n n 0

12 Pulling and twisting DNA Z/L N n T S plectonèmes 0.5 Z(n) 0.4 σ n > 0 data from Gilles Charvin (LPS-ENS) n n 0

13 Results : how a twisted r od coils 1 z L 2 R 170 t TL2 4 2 K n 0 contact(s) Z L 0.95 n8.1 turns

14 Results : how a twisted r od coils 1 z L 2 R 170 t TL2 4 2 K contact(s) n Z L 0.72 n4.3 turns

15 Bifur cation : 0 contact -> 1 contact Z L n Z L n

16 Hard-wall contact, no friction p(s ) 1 force from strand at s 2 acting on strand at s 1 F 1 F 2 r(s ) 1 r(s ) 2 touching conditions : F 1 pf 2 p p rs 1 rs 2 rs 1 rs 2 rs 1 rs 2 thickness rs 1 rs 2 d 3 s 1 rs 1 rs 2 d 3 s 2

17 Results : how a twisted r od coils 1 z L 2 R 170 t TL2 4 2 K contact(s) n Z L 0.62 n3.2 turns

18 Results : how a twisted r od coils 1 z L 2 R 170 t TL2 4 2 K contact(s) n Z L 0.57 n3.7 turns

19 Results : how a twisted r od coils 1 z L 2 R 170 t TL2 4 2 K n 1L1 contact(s) Z L 0.38 n5.7 turns

20 Analytical model for plectonemic DNA T Elastic rod with : M straight tails total length L circular cross-section R 0 bending rigidity K 0 twist rigidity K 3 Z = L - Lp DNA Lp uniform ply electrostatic interaction no endloop N. Clauvelin (PhD work) 3

21 Results ÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁ ÁÁÁÁ Á ÁÁÁ Á Á ÌÌÌÌÌÌÌÌÌÌÌÌÌÌÌÌÌÌÌÌÌÌÌÌÌÌÌÌÌÌÌÌÌÌÌÌÌÌÌÌÌÌÌÌÌÌÌÌÌÌÌÌÌÌÌÌÌÌÌÌÌÌÌÌÌÌÌÌÌ ÌÌÌ Ì ÌÌÌ Ì ÌÌÌ Ì ÌÌ Ì ÌÌ Ì ÌÌ Ì ÌÌ Ì ÌÌÌ Ì Ì data from V. Croquette (LPS-ENS) DNA lambda phage 48kbp phosphate buffer 10 mm

22 M /( q ) Torque prediction N M T S (3/2)πρwlc M = q T (3/2)πρ wlc T

23 M /( q Torque ) prediction N M T S (3/2)πρwlc M = q T (3/2)πρ wlc data from F. Mosconi (LPS-ENS) T

24 Conclusion molécule d ADN tige élastique Perspective simulations numériques avec potentiel électrostatique => système intégro-différentiel