Facilitated diffusion of DNA-binding proteins

Transcription

1 Facilitated diffusion of DNA-binding proteins

2 Konstantin Klenin, Joerg Langowski, DKFZ Heidelberg, Germany Holger Merlitz, Chen-Xu Wu, Xiamen University

3 Content: What is facilitated diffusion? Two competing modes of transport Analytical estimate of reaction time Numerical simulations Method of excess collisions (MEC) Outlook: Extrapolation to large systems Summary

4 Facilitated diffusion to speed up biochemical reactions? Observation: Proteins find their specific DNA binding site faster than the diffusion limit (A.D. Riggs et al., 970). Suggestion: One dimensional sliding along the DNA? (P.H. Richter, M. Eigen, Biophysical Chemistry 2, p )

5 Model system Spherical cell of radius R Worm-like chain of length L Active binding site (radius = a) in the center of cell and chain Protein (random walker) starts at t=0 at the periphery of the cell r = reaction coordinate, distance of the protein from the center of the cell. If r < a, reaction takes place The sliding-length is a function of walker-chain affinity salt concentration

6 Random walk inside empty cell R Exact solution is available, Szabo et al., J. Chem Phys. 72, p (980): a τ 2 2 R R 3 a + 3D a 2 2R = 2 τ: reaction time, D: Diff. coeff. The leading order term, τ R 3 /(3aD), was already used by Smoluchowski, Z. Phys. Chem. 92, p. 29 (97)

7 R Transport efficiency The time needed to propagate from r to r-dr (A. Szabo 980): dτ = τ ( r dr, r) Z( r) = ρ( r') dr' σ L + c V = r Z( r) = dr D ρ( r) Equilibrium distribution ρ(r) along the reaction coordinate and D the diffusion coefficient. (Normalization, r «R) C = free protein concentration, σ = linear density of proteins on chain dr υ = = dτ Dρ( r) Z( r) Transport efficiency ( velocity )

8 Two modes of transport: υ = 2 3d ( r) 4π r cd3 d υ d = 2σ D d Free diffusion Assumption: Close to binding site, the chain is a stick (long persistence length of DNA) Assumption: Non-specific binding is in equilibrium for all r τ a dr υ 3 + υ d d Where τis the time needed to reach distance a, starting at R» a

9 Analytical estimate of the reaction time + ξ π ξ π ξ ξ τ a D L D V d d arctan ) ( 3 d d D K D 3 2π ξ = Where K = σ/c is the equilibrium constant of nonspecific binding, accessible in experiment! K. Klenin, H. Merlitz, J. Langowski, C.X. Wu, PRL 96, 0804 (2006) d D d L D V 3 ) ( ξ ξ ξ τ + This result is similar to Halford et al., Nucleic Acids Research (2004) ξ is interpreted as average sliding length of the protein on DNA

10 Chain model: Worm-like pipe with attractive step potential r c Attractive protein-chain interaction: Walker is allowed to enter the chain volume. Upon leaving, it is allowed to pass with probability: p = e E pis the exit probability. p = implies free diffusion of the walker 0 / k B T

11 Equilibrium constant and sliding length + = = ) ( exp 2 3 p L r V T K r U r d V c V B eff π = eff V c Effective volume of the cell (r c = chain radius): ( ) 2 = p V r eff c π σ ( ) 2 = = p r c c K π σ = p D D r d d c ξ Analytical expression for sliding length available

12 Simulation parameters Persistence length of chain: (unit length) Chain radius: r c = 0.06 Cell radius: 4.8, Chain length: 346 Step-size of walker: d r = 0.02 D d = D 3d = dr 2 /6 Radius of target: a = 0.06 Protein starts at t = 0 at the periphery r = R Exit probability p = exp(-e/kt) is varied from p= (no -dim diffusion) to p i = 2 -i, i =,, Each data point averaged over 2000 independent cycles

13 Simulation results: Dots: Direct simulation of τ Curve: Analytical estimate ξ min V D d 2π L D3d 0.45 Experiment: ξ < 50 bp (< /3 pers. len.), D.M. Gowers, G.G. Wilson, S.E. Halford, PNAS 02, p (2005)

14 Problem: Binding site is small! Active center (radius a) Path of walker Boundary (radius R) Since a «R, the walker spends much time for search

15 Shall we simulate A B instead of B A? State A: Surface of binding site State B: Periphery of cell Collision: Walker hits A τ BA : Reaction time B A τ AB : Back-reaction time A B N: Average number of collisions per cycle A B A τ R : Recurrence time = Average time between two collisions Starting at A, a walker will experience (in average) N- collisions, then reaching B after τ AB, and returning back. τ AB + τ BA = N τ R

16 How to deal with the recurrence time? τ BA = Nτ R - τ AB When simulating the process A B, τ AB is found, also N- V /V 2 = gray/white (effective) volume τ R ( in) V = τ R = V ~ 2τ R τ R V2 The specific recurrence time: ~ τ = τ R R ( in) V does not depend on the choice of R and can be obtained inside a very small test volume H. Merlitz et al., J. Chem. Phys. 24, (2006)

17 Simulation of the specific recurrence time The specific binding site and a piece of chain are placed inside a box with reflective walls. The walker is allowed to leave the binding site, but the clock is running only inside the site.

18 Simulation results:

19 Results MEC approach: Total speed up of the MEC approach, over all data points: Factor 33

20 Outlook: MEC approach with extrapolation τ BA ( r) N( r) ~ τ V τ ( r) R eff AB N(r) and τ AB (r) are measured inside a small test cell and extrapolated to large cells. τ ( r) N( r) ~ τ V τ ( r) BA R eff AB N( r) τ AB N ( r) ( r r r 6D α a eff eff ) H. Merlitz et al., JCP 24 (2006, in print)

21 Summary: Facilitated diffusion: Sliding ( 50 bp) + hopping of proteins to speed up reaction rate. We have developed simple numerical models Analytical estimate for the reaction times The MEC approach allows to simulate the back reaction A B instead of B A, with a speed up of 0 to 00 times, depending on protein-chain affinity MEC-E approach to extrapolate reaction times for systems of realistic dimensions