Anatoly B. Kolomeisky Department of Chemistry Growth Dynamics of Cytoskeleton Proteins: Multi-Scale Theoretical Analysis

RIGID BIOPOLYMERS actin filaments microtubules intermediate filaments

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

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

Microtubules 3-start helical structure with a seam  -tubulin-GTP subunit Size of the dimer subunit 8x4x4 nm Polar structure Plus ends grow faster than minus ends Polymerization produces forces 1-20pN Important biological functions: cell division, cell motility and cellular transport

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

Single Microtubules: Experiments Force generation by single microtubules: 1)Video microscopy 2) Optical trap spectrometry Buckling shapesforces Dogterom et al. Appl. Phys. 75, 331 (2002)

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.

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

Theoretical Modeling. Multi-Scale Approach: 1)Macroscopic - phenomenological models Balance between polymerization and depolymerization processes Structure of the biopolymers, internal interactions, different biochemical transitions and states are neglected

1) Macroscopic Approach: Wrong! V – mean velocity; F- load ; d 0 =d/n- mean increase in length; d=8.2 nm – dimer size; n=13 – number of protofilaments; Fd 1 - the most probable work needed to add a single tubulin dimer against the load F    and      load- distribution factors For microtubules – phenomenological model

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

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

Microtubules: Theoretical Description Fisher and Kolomeisky (Biophys. J. 2001) No assumption of thermodynamic equilibrium at stalling (when V=0) d 1 = d – complex parameter Predictions: Stall force k on = 1887 min -1 k off = 0.33 min -1   =0.22;  unrealistic

Fits of Experimental Data: phenomenological theory Fisher, Kolomeisky Stall force F S

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

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)

Rigid Biopolymers: Theoretical Problems Phenomenological models and polymer ratchet models cannot 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 models with lateral interactions, correct geometry of biopolymer’s tips and biochemical transitions

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

Microtubules: Growth Mechanism fast u j slow u i Inhomogeneity in growth rates Different rates of association and dissociation for different protofilaments

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

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 {u j,w j }

Microtubules: One-Layer Model How good is the approximation? 1)Comparison with the full dynamic solution for the specific value of N 2)Monte Carlo simulations For N=2 the full dynamic solution exists; Relevant for actin filaments

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

Microtubules: Theoretical Description 2) u 1-  w  u1u1 w1w1 3)

Microtubules: Theoretical Description 4) u0u0 w0w0 Define Assumption: thermodynamics ~ kinetics Growth velocity

Microtubules: One-Layer Model N=2 - only 2 configurations u  +w  u  +w  Compare growth rates: Realistic values: g h  k B T,    

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

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

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

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

Microtubule Growth: Experiments Non-linear dependence! Biochemistry, 26, (1997)

Description of Experiments on Microtubules force-velocity curve Stall force F s =5.6pN Bond energies can be estimated Phenomenol. theory: 2 parameters Our theory: 3 parameters- u 0, w 0, 

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

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

Theoretical Approach 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

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

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!!!

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

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

Actin Filaments: Theory kTCkTC wTwT ADPATP ADP ATP rhrh wDwD k T C-association rate of ATP-actin subunit w T -dissociation rate of ATP-actin subunit w D -dissociation rate of ADP-actin subunit r h -hydrolysis rate C-concentration of free ATP-actin monomers

Actin Filaments: Theory kTCkTC wTwT ADPATP ADP ATP rhrh wDwD IDEA: large fluctuations of length at low concentrations due to dissociation of exposed ADP-actin monomers

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

Actin Filaments: Theory Mean dispersion c’ Large length fluctuations at c’ because of ATP-actin dissociation/association and ADP-actin dissociation D(exp)=25-31 sub 2 s -1 D(theory)=31.6 sub 2 s -1

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

Acknowledgements Dr. E. Stukalin (Rice University) and Prof. M.E. Fisher (U of Maryland) Financial support: NSF, Welch Foundation, Dreyfus Foundation, Sloan Foundation Publications: 1)Kolomeisky and Fisher, Biophys. J., 80, 149 (2001) 2)Stukalin and Kolomeisky, J. Chem. Phys., 121, 1097 (2004). 3)Stukalin and Kolomeisky, J. Chem. Phys., 122, (2005).