DNA Bubble Dynamics. Hans Fogedby Aarhus University and Niels Bohr Institute Denmark

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1 DNA Bubble Dynamics Hans Fogedby Aarhus University and Niels Bohr Institute Denmark

2 Principles of life Biology takes place in wet and warm environments Open driven systems - entropy decreases Self organization - pattern formation Thermal fluctuations important Experimental tools: Many, e.g. single molecule experiments Theoretical tools: Several, e.g. statistical physics Bonn, Feb 23, 2007 DNA bubble dynamics 2

3 OUTLINE DNA double helix DNA breathing Experiments Poland-Scheraga model Bubble dynamics Coulomb analogue Results Summary and Conclusion Bonn, Feb 23, 2007 DNA bubble dynamics 3

4 DNA DOUBLE HELIX J. Watson F. Crick Bonn, Feb 23, 2007 DNA bubble dynamics 4

5 Single base Bonn, Feb 23, 2007 DNA bubble dynamics 5

6 Base stacking (T) Thymine (A) Adenine (G) Guanine (C) Cytosine Bonn, Feb 23, 2007 DNA bubble dynamics 6

7 DNA BREATHING DNA stable (self-assembly) Base pair interaction ~2kT Thermal bubble formation DNA breathing affects: DNA replication DNA transcription DNA denaturation Protein binding Quasi 1D structure Temperature Bonn, Feb 23, 2007 DNA bubble dynamics 7

8 Replication and Transcription Bonn, Feb 23, 2007 DNA bubble dynamics 8

9 Denaturation (melting) Denaturation curve of homopolymer DNA (G-C base pairs) Bonn, Feb 23, 2007 DNA bubble dynamics 9

10 EXPERIMENTS Fluorescence correlation spectroscopy Fluorescence quenching Synthetic DNA molecules Tagged with fluorophore and quencher Relaxation dynamics monitored Time scale ~ 50 µsec Ref.: Altan-Bonnet, Libschaber, Krichevsky, PRL 90, (2003) Bonn, Feb 23, 2007 DNA bubble dynamics 10

11 DNA constructs Fluorophore Quencher Bonn, Feb 23, 2007 DNA bubble dynamics 11

12 Melting curve Bonn, Feb 23, 2007 DNA bubble dynamics 12

13 DNA molecular beacon Bonn, Feb 23, 2007 DNA bubble dynamics 13

14 Experimental setup Correlator Detector Detector Sample Lens Bonn, Feb 23, 2007 DNA bubble dynamics 14

15 Correlation function Flourescense intensity: I(t) Correlation function: C(t) = Time average: <A> = lim T 1 T <I(t)I(0)> - <I(0)> 2 <I(0)> T/2 -T/2 A(t)dt 2 Bonn, Feb 23, 2007 DNA bubble dynamics 15

16 Data Relaxation times 30µs µs Multi-exponential behavior Correlation function C(t) Lag time (ms) Bonn, Feb 23, 2007 DNA bubble dynamics 16

17 POLAND-SCHERAGA MODEL F = γ +γx+clog(x), γ = γ (T -T) 0 1 M free energy barrier dissociation entropic melting temperature T>T M T<T M Bonn, Feb 23, 2007 DNA bubble dynamics 17

18 Parameters Bubble size x Barrier γ 0 ~10 kt R Dissociation γ 1 ~4 kt R Entropy coefficient c~2kt R T R = 37 0 C Bonn, Feb 23, 2007 DNA bubble dynamics 18

19 BUBBLE DYNAMICS DNA double helix breathes Thermal excitation of bubbles At low T - few bubbles At high T - many bubbles denaturation Helix-coil phase transition at T M Bonn, Feb 23, 2007 DNA bubble dynamics 19

20 Langevin equation F = γ 0+γ(T)x+clog(x), γ(t) = γ 1(TM-T) dx dt df = -D + ξ(t) dx kinetic noise coefficient dx c = - D γ(t) + + ξ(t) dt x <ξ(t)ξ(0)> = 2DkTδ(t) Langevin equation for bubble dynamics x bubble size (stochastic) ξ thermal noise D kinetic coefficient Bonn, Feb 23, 2007 DNA bubble dynamics 20

21 Fokker-Planck equation 2 P 1 P μ 2 t 2 x x x = + - ε(t) P diffusive term drift term reduced parameters: μ = c/2kt R 1, entropy parameter ε(t) = (γ /2kT )(T/T -1) (T-T ) 1 R M M ε>0 above T, ε<0 below T M kinetic coefficient D absorbed measure time in units of μsec M Fokker-Planck equation for distribution P(x,t) Fokker-Planck equation equivalent to Langevin equation Bonn, Feb 23, 2007 DNA bubble dynamics 21

22 COULOMB ANALOGUE Transform Fokker-Planck equation to Schrödinger equation in imaginary time μ Ψ=x P, μ 1, ε(t) (T-T M ) 2 Ψ 1 Ψ - = - + V(x)Ψ 2 t 2 x μ(μ+1) με(t) ε(t) V(x) = x x 2 2 centrifugal barrier Coulomb potential Bonn, Feb 23, 2007 DNA bubble dynamics 22

23 Coulomb potential a) Potential below transition temperature - continuum states - b) Potential above transition temperature - bound state - Bonn, Feb 23, 2007 DNA bubble dynamics 23

24 General solution Transition probability P(x,x 0,t) from size x 0 to x in time t Ψ n (x) and E n state and energy of stationary Schrödinger equation HΨ n = E n Ψ n, H = -(1/2)d 2 /dx 2 + V(x) ε(t)(x-x 0 ) -Ent P(x,x,t) = e e Ψ (x)ψ (x ) -μ x 0 n n 0 x0 n ε(t) T-T M energy eigenvalue eigenfunction Bonn, Feb 23, 2007 DNA bubble dynamics 24

25 RESULTS Long time solution below T M Transition probability P(x,x 0,t): 2 ( ε x x0 ) t Pxx (,, t) xx e e t 1+ 2 μ ε /2 3/2 μ 0 0 Bubble life time distribution W(t): 2 ε x0 t W() t (1+ 2 μ) x e e t 1+ 2 μ ε /2 3/2 μ 0 exponential power law Bonn, Feb 23, 2007 DNA bubble dynamics 25

26 Bubble life time distribution Exact solution at T M 1+2μ 2x W(t) = e (2t) Γ(1/2+μ) μ = c/2kt R -x /2t -3/2-μ W(t) T M =2T R T M =10T R Power law tail time Bonn, Feb 23, 2007 DNA bubble dynamics 26

27 Correlation function Fluorescence correlations of tagged base pair Measures bubble dynamics on single molecular scale Correlation function C(t) prop. to integrated survival probability Below the transition temperature at long t: Crossover from power law to exponential behavior At the transition temperature (exact): Power law behavior Bonn, Feb 23, 2007 DNA bubble dynamics 27

28 Comparison with data Flourescence correlation spectroscopy (FCS) data from experiment Γ model from analysis Bonn, Feb 23, 2007 DNA bubble dynamics 28

29 SUMMARY AND CONCLUSION Bubble dynamics in double-stranded DNA Single molecule experiments - flourescence techniques - intensity correlations Quasi 1D statistical mechanical model Langevin/Fokker-Planck description Mapping to Coulomb problem Exact solutions - long time behavior Comparison with experiments Conclusion - lesson to be learned Bonn, Feb 23, 2007 DNA bubble dynamics 29