Applications of non-equilibrium models in biological systems

1. Applications of non-equilibriummodels in biological systems Yariv Kafri Technion, Israel

2. General plan • Overview of molecular motors (the biological system we will consider): why study? physical conditions? experimental studies • Theoretical models of single motors:different approaches effects of disorder • Many interacting motors:different kinds of interactions help from driven diffusive systems

3. Is it helpful to use non-equilibrium models to understand such systems? (for example, help understand experiments)

4. D. Nelson D. Lubensky J. Lucks M. Prentiss C. Danilowicz R. Conroy V. Coljee J. Weeks J.-F. Joanny O. Campas K. Zeldovich J. Casademunt,

5. hard diskRAMoutput device Why? The central dogma of biology

6. study in detailthe machines -the dogma in action The central dogma of biology replication DNA transcription RNA translation Protein

7. Molecular Motors: complexes of proteins which use chemical energy to perform mechanical work • Move vesicles • Replicate DNA

8. Produce RNA • Produce proteins • Motion of cells • And much much (much) more MOVIE MOVIE

9. What do motors need to function? (basics for modeling) 1. Fuel (supplies a chemical potential gradient) These vary! (examples before)But for the systems we will discuss typically the following holds (Kinesin) ATP ATP ATP ATP ``discrete’’ fuel ATP ATP

10. How much energy released? created in cell or in experiment for ATP gives about ~ Other sources GTP,UTP,CTP (no TTP) about the same

11. What do motors need to function? 2. Track Again these vary! (examples before) DNA microtubules actin (myosin motors), circular tracks……..one dimensional

12. (started in 1927 drop no. 9) The pitch drop experiment (Ig Noble 2005) R. Edgeworth, B.J. Dalton and T. Parnell Eur. J. Phys (1984) 198-200 swimming in pitch Scales bacteria kinesin ~1 micro-meter (your cells 20 micro-meters) fluid density Reynolds number =inertial forces/viscous forces coefficient of viscosity

13. Another implication of scale local thermal equilibrium motor time scales equilibrium time scale • No inertia (diffusive behavior) • Can assume local thermal equilibrium (namely, transition rates obey a local version of detailed balance – in a few slides) Scale of nm

14. Experimental Technique(s)

15. Single molecule experiments Study behavior of single motor under an external perturbation (force) • deduce characteristics (e.g. force exerted) • understand chemical cycle better tweezers exert forceopposing motion K. Vissher, M. J. Schnitzer, S. M. Block Nature400, 184 (1999) MOVIE

16. velocity force curve Extract velocity for different forces 8nmstep size K. Vissher, M. J. Schnitzer, S. M. Block Nature400, 184 (1999)

17. Velocity-Force Curve K. Vissher, M. J. Schnitzer, S. M. Block Nature400, 184 (1999) stall force The stall force is the force exerted by the motor

18. Kinesin • Utilizes ATP energy • Moves along microtubules, monomer size 8 nm (always in a certain direction) • Processivity about 1 micron (~ 100 steps) • Exerts a force of about 6-7 pN

19. Ingredients for modeling: • No inertia (some sort of biased brownian motion) • Noisy (both temperature and discrete fuel) • Safe to assume local thermal equilibrium Forces ~ pN Distances ~ nM Thermal fluctuations are important!

20. Theory: How do the motors use chemical energy to function? ``two approaches’’ Brownian Ratchets Powerstroke Both rely on the motor havinginternal states Basic idea + + + ATP M ADP P M

21. Powerstroke models (Huxley, 1957) Idea: some internal ``spring’’ is activated using chemical energy description in terms of a biased randomwalker Can complicate by putting in many internal state(Fisher and Kolomeisky on Kinesin)

22. Brownian ratchets (Julicher, Ajdari, Prost,…1994) ``rectify’’ Brownian motion • Two channels for transition, chemical and thermal • If have detailed balance, no motion • Must have asymmetry • Must have rates which depend on the location on the track x

23. Treatment – two coupled Fokker-Plank equations with or Get conditions that under

24. much better ratchet Get conditions that under • asymmetric potentials • no detailed balance effective potential for random walker described by is tilted diffusion with drift

25. Simple lattice version Setup modeled Lattice model

26. Two channels for transition, chemical and thermal • Included external force describe coarse graineddynamics by effectiveenergy landscape

27. force x size of monomer • No chemical potential difference (have detailed balance) • Symmetric potential • Otherwise have an effective tilt diffusion with drift

28. Simple enough that can calculate velocity and diffusion constant diffusion with drift

29. Back to ratchets vs. powerstroke ? Personal opinion: ratchet more generic and can be made to behave as powerstroke

30. Short Summary: • Molecular motors are complexes of proteins which use chemical energy to perform mechanical work. • Single molecule experiments provide data on traces of motors giving information such as:stall force velocity step size ….. ….. • Models including internal states provide a justification for treating the motors as biased random walkers

31. Motors involvedmove alongdisordered substrates (DNA and RNA have given sequences) So far motors which move on a periodic substrate Not always the case!

32. M. Wang et al, Science 282, 902 (1998) Example: RNA polymerase • Utilizes energy from NTPs • Moves along DNA making RNA • very high processivity • Forces • Step size 0.34 nm ~15nm

33. convex M. Wang et al, Science 282, 902 (1998) ~30 bp/s ~15 pN

34. small, simple big, complicated Conventional explanation by model with jumps of varying lengthinto off-pathway state M.E. Fisher PNAS (2001) kinesin – moves along microtubuleswhich is a periodic substrate RNAp – moves along DNAwhich is a disordered substrate

35. Applications of non-equilibriummodels in biological systems Yariv Kafri Technion, Israel

36. Yesterday: Molecular motors on periodic tracks are described by biased random walkers in one hour

37. Motors involvedmove alongdisordered substrates (DNA and RNA have given sequences) Many motors do not move on a periodic substrate

38. M. Wang et al, Science 282, 902 (1998) Example: RNA polymerase • Utilizes energy from NTPs • Moves along DNA making RNA • very high processivity • Forces • Step size 0.34 nm ~15nm

39. convex M. Wang et al, Science 282, 902 (1998) ~30 bp/s ~15 pN

40. small, simple big, complicated Conventional explanation by model with jumps of varying lengthinto off-pathway state M.E. Fisher PNAS (2001) kinesin – moves along microtubuleswhich is a periodic substrate RNAp – moves along DNAwhich is a disordered substrate

41. Recall Randomness??

42. sum over independent random variables fluctuations which grow as Randomness ??? functions of location along track for this setup is not

43. Effective energy landscape is a random forcing energylandscape This results only from the use of chemical energy coupled with the substrate

44. no chemical energy (no ATP) barriers of typical size (diffusion) barriers which grow as (diffusion with drift) effective energy landscape with chemical energy and disorder pauses at specific sites

45. rough energy landscape • anomalous dynamics • shape of velocity-force curve • pauses during motion diffusion with drift no chemical bias (-) heterogeneoustrack with chemical bias periodic track Finite time convex curve

46. Random forcing energy landscapes toy model + assume directed walk among traps (convection by force vs. trapping) with prob prob of a barrier or size rare but dominating events time stuck at trap of this size power law distribution

47. Total time Subballistic consider moves between traps can neglect trapping times larger than

48. Fluctuations in time anomalous diffusion

49. Subballistic exact solution of model with disorder

50. Motor model simple enough to solve exactly