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

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 InstabilityMicrotubules 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 shapes forces

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, 666-673 (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

offon kckV

Structure of the biopolymers, internal interactions, different biochemical transitions and states are neglected

1) Macroscopic Approach: Wrong!

)]/exp(

)/exp([)(

1

10

TkFdk

TkFdkdFV

Boff

Bon

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 load F

and load-distribution factors

For microtubules – phenomenological model

Load-Distribution Factors:Effect of an external load F:

TkFdoffoffoff

TkFdononon ekFkkekFkk B1B1 //

)0()(,)0()(

and load distribution factors 1

activation barrier aE

1Fd

1Fd

F=0F >0

j

TkEon

aek B/

Microtubules: Phenomenological Model

])/exp([)( 10 offBon kTkFdkdFV

Phenomenological (Thermodynamic) Theory: Dogterom and Yurke (Science,1997)

Assumption: d0=d1= d/13=0.63 nmkon= 1791 min-1

koff= -127 min-1Fit 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)

)]/exp()/exp([)( 110 TkFdkTkFdkdFV BoffBon

No assumption of thermodynamic equilibrium at stalling (when V=0)

d1=d – complex parameter

Predictions:

off

onBS k

k

d

TkF ln

1

Stall force

kon= 1887 min-1

koff= 0.33 min-1

=0.22; unrealistic

Fits of Experimental Data:

phenomenological theory

Fisher, Kolomeisky

Stall force FS


2) Microscopic approach – full atomistic simulations

Currently – do not exist!

Protein Data Bank:

- tubulin subunit

More than 10000 atoms!!!


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 uj

slow ui

Inhomogeneity in growth rates

Different rates of association and dissociation for different protofilaments


Approximate theory:

One-Layer Model

Idea: only few configurations are relevant for growth dynamics

Microtubules: One-Layer ModelAssumption: 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}

Microtubules: One-Layer ModelHow 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


Compare with N = 2 model

Full dynamics : 4 different types of transitions

u

w

da

]2

exp[Tk

ggg

w

u

B

imvh

gv -free energy of creating head-to-tail bond

gh – free energy of creating lateral bond

1)

gim- free energy of monomer immobilizing

=a/d-fraction of created or broken lateral bond


2) u1-

w

])1(2

exp[1

1

Tk

ggg

w

u

B

imvh

u1

w1

3)

]2

exp[1

1

Tk

ggg

w

u

B

imvh


4)

u0

w0

]exp[0

0

Tk

gg

w

u

B

imv Define

]exp[Tk

g

B

h

2

0

0

1

1

)1(2

0

0

1

1

2

0

0

w

u

w

u

w

u

w

u

w

u

w

u

Assumption: thermodynamics ~ kinetics

)1

11)((

2)(

/)2(

/0

0 dda

dawu

dexactV

Growth velocity

1,,1,0 where

, ,

i

wwuu iii

Microtubules: One-Layer Model

N=2 - only 2 configurations

u+w

u+w

)1

11)((

2)(

1)()-(

)12(0

0

)12(0

0

w

ud

exactV

wudlayeroneVCompare

growth rates:

)()( For layeroneVexactV

Realistic values: gh kBT,

]exp[Tk

g

B

h

Ratio of Growth Velocities

for different shifts

stronger lateral interactions

Ratio of exact and approximate velocities

Comparison with Monte-Carlo Simulations

]exp[Tk

g

B

h

gh-lateral interactions between the monomers in rigid biopolymers

~3-7 kBT

Son, Orkoulas and Kolomeisky, J. Chem. Phys. (2005) in press

N=13 protofilaments

Effect of External Forces

Effect of an external load F:TkadF

jjjTkadF

jjjjjjj ewFwweuFuu BB /)(/)( )0()(,)0()(

jj and load-distribution factors 1

jj

activation barrier aE

1Fd

1Fd

F=0F >0

j

TkEj

aeu B/

Comparison with Phenomenological Models

Concentration dependence – nonlinear!

critical concentration

one-layer model with N=13

Phenomenological model

offon kckV

Microtubule Growth: Experiments

Non-linear dependence!

Biochemistry, 26, 4428-4437 (1997)

Description of Experiments on Microtubules

force-velocity curve

Stall force Fs=5.6pN

Bond energies can be estimated

Phenomenol. theory:

2 parameters

Our theory:

3 parameters-

u0, w0,

Theoretical Approachn-layer approximation extension of one-layer approach

n=2

n

full dynamic description

n-layer approximation- series expansion around exact result

Theoretical ApproachComparison of one-layer and two-layer approximations with exact description for N=2 rigid biopolymers

velocities dispersion

For realistic lateral interactions (3-7 kBT) two-layer approximation is perfect

one-layer one-layer

two-layer

Theoretical ApproachKinetic 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)

)(2

1 2offon kckaD 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 ADP

ADP ADP

ATP ATP

ATP

Actin monomers are found in 2 states: ATP or ADP

Actin Filaments. Hydrolysis1) Random mechanism

1) Sequential (vectorial)

ADP

ADP

ADP

ADP

ADP

ADP

ADP ADP

ATP ATP

ATP

ATP ATP

ATP

One interface between hydrolyzed and unhydrolyzed segments

Many interfaces between hydrolyzed and unhydrolyzed segments

Actin Filaments: Theory

kTC

wT

ADP ATP

ADP

ADP

ADP ADP ADP

ADP

ADP

ADP

ATP

ATP

rh

wD

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


kTC

wT

ADP ATP

ADP

ADP

ADP ADP ADP

ADP

ADP

ADP

ATP

ATP

rh

wD

IDEA: large fluctuations of length at low concentrations due to dissociation of exposed ADP-actin monomers


Mean growth velocity Dynamic phase transitions:

T

HT

k

rwc

'

Above c’ the probability to have a configuration with ADP-actin at the tip of the filament is zero

Actin Filaments: TheoryMean dispersion

c’

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

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,

104903 (2005).

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