Anatoly B. Kolomeisky Department of Chemistry

1. Anatoly B. KolomeiskyDepartment of Chemistry Growth Dynamics of Cytoskeleton Proteins: Multi-Scale Theoretical Analysis

2. RIGID BIOPOLYMERS actin filaments microtubules intermediate filaments

3. Rigid Biopolymers Many biological functions of rigid biopolymers are determined by their growth dynamics Fundamental problem:To understand the mechanisms of growth and coupling to biological processes

4. Microtubules: • Rigid hollow cylindrical biopolymers • Length-1-10mm, diameter -25nm, thickness of walls 5-6nm • Number of protofilaments – 10-15, the most probable 13

5. Microtubules • 3-start helical structure with a seam • ab-tubulin-GTP subunit • Size of the dimer subunit 8x4x4nm • Polar structure • Plus ends grow faster than minus ends • Polymerization produces forces 1-20pN • Important biological functions: cell division, cell motility and cellular transport

6. Microtubules: Dynamic Instability Microtubules exist in two dynamic phases:growing or shrinking Dynamic instability – non-equilibrium phenomenon Understanding of dynamic instability is not complete

7. Single Microtubules: Experiments • Force generation by single microtubules: • Video microscopy • Optical trap spectrometry Buckling shapes forces Dogterom et al. Appl. Phys. 75, 331 (2002)

8. Actin Filaments: 5.4 nm Two-stranded right handed helix polymer. Protofilaments are half-staggered and wrapped around each other with a 72 nm period.

9. Single Actin Filaments: Experiments Fujiwara et al., Nature Cell Biology, 4, 666-673 (2002) Direct observation of single actin polymerization/depolymerization processes from fluorescently labeled molecules

10. Theoretical Modeling. Multi-Scale Approach: • Macroscopic -phenomenological models • Balance between polymerization and depolymerization processes Structure of the biopolymers, internal interactions, different biochemical transitions and states are neglected

11. 1) Macroscopic Approach: Wrong! For microtubules – phenomenological model V – mean velocity; F- load ; d0=d/n- mean increase in length; d=8.2 nm – dimer size; n=13 – number of protofilaments; Fd1- the most probable work needed to add a single tubulin dimer against the loadF Q+and Q- = 1- Q+ -load-distribution factors

12. Load-Distribution Factors: Effect of an external loadF: load distribution factors activation barrier F >0 F=0 j

13. Microtubules: Phenomenological Model Phenomenological (Thermodynamic) Theory: Dogterom and Yurke (Science,1997) Assumption:d0=d1= d/13=0.63 nm kon= 1791 min-1 koff= -127 min-1 Fit of experimental data Unphysical!Chemical rates are always >0! Phenomenological theory fails! Stalling (V=0) is not an equilibrium

14. Microtubules: Theoretical Description Fisher and Kolomeisky (Biophys. J. 2001) No assumption of thermodynamic equilibrium at stalling (when V=0) d1=ld – complex parameter Stall force kon= 1887 min-1 koff= 0.33 min-1 q+=0.22; l=1-unrealistic Predictions:

15. Fits of Experimental Data: phenomenological theory Fisher, Kolomeisky Stall forceFS

16. Theoretical Modeling. Multi-Scale Approach: 2) Microscopic approach – full atomistic simulations Currently – do not exist! Protein Data Bank: - tubulin subunit More than 10000 atoms!!!

17. Theoretical Modeling. Multi-Scale Approach: 3) Mesoscopic Approach: Takes into account some structural and biochemical properties Polymerization Ratchet Models: Thermal fluctuations create gaps for inserting monomers Mogilner and Oster, Biophys. J. 71, 3030 (1996)

18. Rigid Biopolymers: Theoretical Problems Phenomenological modelsand polymer ratchet modelscannot describe the growth dynamics, especially under external forces and concentration dependence Main problem: Geometrical structure of growing biopolymers, monomer-monomer interactions and biochemical transitions are neglected Our approach:discrete stochastic modelswith lateral interactions, correct geometry of biopolymer’s tips and biochemical transitions

19. Rigid Biopolymers: Theoretical Description To develop the simplest theoretical picture which will take into account the geometry and polymer lattice interactions Our Goal: Problem: Infinite number of polymer configurations!

20. Microtubules: Growth Mechanism Inhomogeneity in growth rates slow ui fast uj Different rates of association and dissociation for different protofilaments

21. Microtubules: Theoretical Description Idea:only few configurations are relevant for growth dynamics Approximate theory: One-Layer Model

22. Microtubules: One-Layer Model Assumption:Only configurations of microtubules with distances from the leading protofilament tip less than d allowed There are N such configurations N-number of protofilaments Explicit expressions for mean growth velocity, V, and for the dispersion, D, for any N and any geometry in terms of {uj,wj}

23. Microtubules: One-Layer Model • How good is the approximation? • Comparison with the full dynamic solution for the specific value of N • Monte Carlo simulations For N=2 the full dynamic solution exists; Relevant for actin filaments

24. Microtubules: Theoretical Description d=a/d-fraction of created or broken lateral bond Compare with N = 2 model Full dynamics : 4 different types of transitions a d 1) u w gv -free energy of creating head-to-tail bond gh – free energy of creating lateral bond gim-free energy of monomer immobilizing

25. Microtubules: Theoretical Description 2) u1-d w1-d 3) u1 w1

26. Microtubules: Theoretical Description 4) u0 w0 Define Assumption: thermodynamics ~ kinetics Growth velocity

27. Microtubules: One-Layer Model N=2 - only 2 configurations ud+w1-d u1-d+wd Compare growth rates: Realistic values: gh3-7 kBT,g  103-107

28. Ratio of Growth Velocities Ratio of exact and approximate velocities for different shifts stronger lateral interactions

29. Comparison with Monte-Carlo Simulations N=13 protofilaments gh-lateral interactions between the monomers in rigid biopolymers ~3-7 kBT Son, Orkoulas and Kolomeisky, J. Chem. Phys. (2005) in press

30. Effect of External Forces Effect of an external loadF: load-distribution factors activation barrier F >0 F=0 j

31. Comparison with Phenomenological Models Concentration dependence – nonlinear! Phenomenological model one-layer modelwithN=13 critical concentration

32. Microtubule Growth: Experiments Biochemistry, 26, 4428-4437 (1997) Non-linear dependence!

33. Description of Experiments on Microtubules Bond energies can be estimated Phenomenol. theory: 2 parameters Our theory: 3 parameters- u0, w0,  force-velocity curve Stall forceFs=5.6pN

34. Theoretical Approach n-layer approximationextension of one-layer approach n=2 full dynamic description n-layer approximation-series expansion around exact result

35. Theoretical Approach Comparison of one-layer and two-layer approximations with exact description for N=2 rigid biopolymers dispersion velocities two-layer one-layer one-layer For realistic lateral interactions (3-7 kBT) two-layer approximation is perfect

36. TheoreticalApproach Kinetic explanations for n-layer approximations Full kinetic scheme for N=2 rigid biopolymers (k,m) – polymer configuration with k monomers in the 1-st protofilament, and m monomers in the 2-nd

37. Theoretical Approach one-layer model two-layer model Kinetic justifications for n-layer approximations

38. Actin Filaments: Fujiwara et al., Nature Cell Biology, 4, 666 (2002) a =2.7 nm Experimental observations:large length fluctuations in actin filaments in stationary phase.D(exp)/D(calc) =35-40!!!

39. Actin Filaments: Hydrolysis is crucial for actin growth dynamics Actin monomers are found in 2 states: ATP or ADP ADP ADP ATP ATP ADP ADP ATP

40. Actin Filaments. Hydrolysis 1) Randommechanism ADP ATP ADP ATP ADP ATP ADP Many interfaces between hydrolyzed and unhydrolyzed segments 1) Sequential (vectorial) ADP ADP ATP ATP ADP ADP ATP One interface between hydrolyzed and unhydrolyzed segments

41. Actin Filaments: Theory wT ADP ADP ATP ADP ATP ADP ATP kTC rh wD ADP ADP ADP ADP ADP kTC-association rate of ATP-actin subunit wT-dissociation rate of ATP-actin subunit wD-dissociation rate of ADP-actin subunit rh-hydrolysis rate C-concentration of free ATP-actin monomers

42. Actin Filaments: Theory wT ADP ADP ATP ADP ATP ADP ATP kTC rh wD ADP ADP ADP ADP ADP IDEA: large fluctuations of length at low concentrations due to dissociation of exposed ADP-actin monomers

43. Actin Filaments: Theory Mean growth velocity Dynamic phase transitions: Above c’ the probability to have a configuration with ADP-actin at the tip of the filament is zero

44. Actin Filaments: Theory Mean dispersion Large length fluctuations at c’ because of ATP-actin dissociation/association and ADP-actin dissociation D(exp)=25-31 sub2s-1 D(theory)=31.6 sub2s-1 c’

45. CONCLUSIONS • Multi-scale analysis of the growth of rigid biopolymers is presented • Mesoscopic models that accounts for geometry, lattice interactions and biochemical transitions are developed • All dynamic properties can be calculated explicitly • n-layer approximations of growth dynamics are presented • Hydrolysis stimulates large length fluctuations in actin filaments at low concentrations

46. Acknowledgements • Dr. E. Stukalin (Rice University) and Prof. M.E. Fisher (U of Maryland) • Financial support: NSF, Welch Foundation, Dreyfus Foundation, Sloan Foundation • Publications: • Kolomeisky and Fisher, Biophys. J., 80, 149 (2001) • Stukalin and Kolomeisky, J. Chem. Phys., 121, 1097 (2004). • Stukalin and Kolomeisky, J. Chem. Phys., 122, 104903 (2005).