Exploring Protein Motors Exploring Protein Motors -- from what we can -- from what we can

measure to what we like to knowmeasure to what we like to know

Hongyun WangHongyun Wang

Department of Applied Mathematics and Statistics

University of California, Santa Cruz

Baskin School of Engineering Research Review Day

October 12, 2007

The goal of mathematical studies …The goal of mathematical studies …

Chen & Berg (2000) Yasuda, et al. (1998)

Visscher, Schnitzer & Block (1999)

… is to decipher motor mechanism from measurements.

An example of macroscopic motorAn example of macroscopic motor

Torque of a single cylinder engine:

(1) Intake (2) Compression (3) Expansion (4) Exhaust

How to find the motor forceHow to find the motor forceof a molecular motor?of a molecular motor?

Mathematical framework forMathematical framework for

molecular motorsmolecular motors

Mechanochemical Mechanochemical modelsmodels

Chemical reaction:

€

dSdt

= K x( )⋅S

Mechanical motion and chemical reaction are coupled.

Mechanical motion:

Chen & Berg (2000) Yasuda, et al. (1998)Visscher, Schnitzer & Block (1999)

ζdx

dt= ′φS x( )

Force frompotential

{ + ƒL

Loadforce

{ + 2kBTζdW t( )

dtBrownian

force

1 24 4 34 4

€

SITE[ ] ⇔ SITE∗ATP[ ]

c c

SITE ∗ADP[ ] ⇔ SITE∗ADP ∗Pi[ ]

Characters of molecular motorsCharacters of molecular motorsMolecular motors

Macroscopic motors

• Time scale of inertia << time scale of reaction cycle

• Instantaneous velocity >> average velocity

• Kinetic energy from average velocity << kBT

• Use thermal fluctuations to get over bumps

• Time scale of inertia >> time scale of reaction cycle

• Instantaneous velocity ≈ average velocity

• Kinetic energy from average velocity >> kBT

• Use stored kinetic energy (inertia) to get over bumps

Macroscopic motors use stored kinetic Macroscopic motors use stored kinetic energy to get over bumpsenergy to get over bumps

ΔV

V≈

ΔE

mV 2

ΔE

mV 2≈ 1% ⇒

ΔV

V≈ 1%

This does not work in molecular motors!This does not work in molecular motors!

Molecular motors use thermal Molecular motors use thermal excitations to get over bumpsexcitations to get over bumps

The energy for accelerating a bottle of water to

100 miles/hour

can only heat up the bottle of water by

0.24 degree!

Thermal energy is huge! Thermal energy is huge!

Mathematical Mathematical equationsequations

€

dxdt

= D− ′ φ S x( ) + ƒL

kBT+ 2D

dW t( )dt

( ){

( )1

DiffusionConvection Change ofoccupancy

, 1, 2, ,N

S LS SS S j j

jB

xD k x S N

t x k T x

φρ ρρ ρ

=

′ − ƒ⎛ ⎞∂ ∂∂= + =⎜ ⎟

∂ ∂ ∂⎜ ⎟⎝ ⎠

∑ K1 4 4 2 4 43 1 44 2 4 43

+

( )dx

dt= ⋅

SK S

Langevin formulation(stochastic evolution of an individual motor):

Fokker-Planck formulation(deterministic evolution of probability density):

(mechanical motion)

(chemical reaction)

Efficiencies of a molecular motorEfficiencies of a molecular motor

€

ηTD =f ⋅ v

−ΔG( )⋅ r,

Thermodynamic efficiency of a motor working Thermodynamic efficiency of a motor working

against a against a conservative forceconservative forceMotorsystem

Externalagent

Visscher et al (1999). Nature

Thermodynamic efficiency =Energy outputEnergy input

Viscous drag is not a conservative force:

€

ηStokes =ζ ⋅ v

2

−ΔG( )⋅ rThe Stokes efficiencyThe Stokes efficiency::

Stokes efficiency of a motor working Stokes efficiency of a motor working against a against a viscous dragviscous drag

Hunt et al (1994). Biophys. J.

Energy output = 0

Yasuda, et al (1998). Cell.

Stokes efficiency Stokes efficiency thermodynamic thermodynamic efficiencyefficiency

(experimental observations)(experimental observations)

Visscher et al (1999). NatureHunt et al (1994). Biophys. J.

€

limζ →∞

ζ ⋅ v < fStall

Viscous stall load < Thermodynamic stall load

Which measurement is correct?

Stokes efficiency Stokes efficiency thermodynamic thermodynamic efficiencyefficiency(theory)(theory)

Viscous stall load:

lim vζ

ζ→∞

⋅( ) ( )

1

0

1 1

0 0

Stall

exp

exp exp

B

B B

s dsk T

fsL sL sL

ds s dsk T k T

ψ

φ φ ψ

⎛ ⎞Δ⎜ ⎟⎝ ⎠=

⎧ ⎫+ −⎛ ⎞ ⎛ ⎞Δ⎪ ⎪⋅⎨ ⎬⎜ ⎟ ⎜ ⎟

⎪ ⎪⎝ ⎠ ⎝ ⎠⎩ ⎭

∫

∫ ∫<

% %%

… implies that the motor force is not uniform.

Motor potential profileMotor potential profile

A mean-field potentialA mean-field potentialFokker-Planck formulation

At steady state, summing over S

( ) ƒ0 L

B

xD

x k T x

ψ ρρ

⎡ ⎤′ −∂ ∂⎢ ⎥= +

∂ ∂⎢ ⎥⎣ ⎦

€

ρ x( ) = ρS x( )S =1

N

∑

The motor behaves as if it were driven by a single potential (x)

(x) is called the motor potential profile.

€

′ x( ) =1

ρ x( )φS

′ x( )ρS x( )S=1

N

∑

Probability density of motor at x

Motor force at x (averaged over all chemical states).

The potential profile is The potential profile is measurablemeasurable

Time series of motor positions measured in single

molecule experiments

… does not work well

′ x( ) = x′ t( )x(t)=x

( ) ƒ0 L

B

xD

x k T x

ψ ρρ

⎡ ⎤′ −∂ ∂⎢ ⎥= +

∂ ∂⎢ ⎥⎣ ⎦

( ) L

B

vx fD J

k T x L

ψ ρρ

⎛ ⎞′ − ∂⎜ ⎟+ = =⎜ ⎟∂⎝ ⎠

( ) ( ) ( )0

1logL

B B

xx f x Jx ds

k T k T D s

ψρ

ρ

⋅= − −⎡ ⎤⎣ ⎦ ∫

Therefore, we only need to reconstruct PDF.

A more robust formulation:

Reconstructing potential in a test problemReconstructing potential in a test problem

Extracted motor potential profile

A sequence of 5000 motor positions generated in a

Langevin simulation

SummarySummary

Everything depends on a proper mathematical model

and

analysis of the model!

Time scale of inertiaTime scale of inertia

€

md vdt

= −ζ v − ′ φ S x( ) + ƒL + 2kBTζdW t( )

dt

Time scale of inertia:

Newton’s 2nd law:

€

t0 =mζ

€

m = ρ4π3

r 3 , ζ = 6πη r ⇒ t0 =mζ

= ρ2

9ηr2

For a bead of 1 m diameter:

€

t0 ≈ 56 ×10−9s = 56 ns