rf fox fnal 20061 rectified brownian motion in subcellular biology ronald f. fox mee choi william...

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Rectified Brownian Motion in

Subcellular Biology

Ronald F. Fox

Mee ChoiWilliam Mather

School of PhysicsGeorgia Institute of Technology

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2005, Centenary of Einstein’s Annus Mirabilis

Special Relativity

Photoelectric Effect

Brownian Motion

“Joy in looking and comprehending is nature’s most beautiful gift”

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Nanobiology

BiochemistryMolecular biology

Can we learn mechanisms from nanobiologythat are applicable to nanotechnology?

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One such lesson is the constructive use of

thermal energy.

Such a mechanism is called

Rectified Brownian Motion.

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A Few Antecedents

A. F. HuxleyProg. Biophys. Chem. 7, 255 (1957)

M. Meister, S. R. Caplan and H. C. BergBiophys. J. 55, 905 (1989)

R. D. Vale and F. OosawaAdv. Biophys. 26, 97 (1990)

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Rectified Brownian Movement in Molecular and Cell Biology

Phys. Rev. E 57, 2177 (1998)

Rectified Brownian Motion and Kinesin Motion Along Microtubules

Phys. Rev. E 63, 051901 (2001)(with Mee Choi)

Kinesin’s Biased Stepping Mechanism: Amplification of Neck Linker Zippering

Biophysical Journal, 91 2416-2426 (2006)(with William Mather)

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Minnow

m(gm) 134

R(cm) 2

vS (cm/s) 100

vT (cm/s) 3 x 10-8

WS (W) 3.8 x 10-4

WT (W) 3.4 x 10-23

A Minnow

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A Minnow and an E. Coli

Minnow E. Coli

m (gm) 134 2 x 10-12

R(cm) 2 5 x 10-5

vS (cm/s) 100 2 x 10-3

vT (cm/s) 3 x 10-8 0.20

WS (W) 3.8 x 10-4 10-17

WT (W) 3.4 x 10-23 10-13

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A Minnow, an E. Coli and Ubiquinone

Minnow E. Coli Ubiquinone

m (gm) 134 2 x 10-12 1.4 x 10-21

R (cm) 2 5 x 10-5 7.5 x 10-8

vS (cm/s) 100 2 x 10-3 (0.8)

vT (cm/s) 3 x 10-8 0.20 9,300

WS (W) 3.8 x 10-4 10-17 (2.3 x 10-14)

WT (W) 3.4 x 10-23 10-13 3 x 10-6

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Reynolds Number

Minnow E. Coli Ubiquinone

Secular 2 x 104 4 x 10-5 (2.4 x 10-6)

Thermal 4 x 10-6 3 x 10-4 1.6 x 10-3

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Biological Energy Couplings

Photon energy (electron excitation)Electron energy (redox reaction)

Proton energy (pH gradient) Phosphate energy (monomer activation)

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Redox reaction variety

Pure, 1 electron transferIron, copper, zinc…

1 electron and 1 proton transferFADH2, UQH2,..

2 electrons and 1 proton transferNAD+, NADH,..

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),(),( 2

2

txfx

Dtxft

diffusion for reduced ubiquinone

),0(00000 tfx

DYXXdt

d

boundary layer equation

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d

Dr

diffusive rate parameter

d

dd

0

00

reaction rate parameter

r

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rimplications

Weak linear steady state gradients

Negligible energy dissipation associated with the gradients according to

non-equilibrium steady state thermodynamics

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tdt

dm Fvv

~

ttTktt B 2

~~FF

sgmR /105.36 7

scmR

TkD B /1014.1

627

sD

dt D

62

108.22

sR

mR

15109.36

Langevin equation

Einstein’sRelation

1905

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Brownian Work TheoremStR

dt

dm FFvv )(

~6

Secular power from secular force

R

RSS

tF

mt

exp1

1)( 2vF

Stochastic power from Brownian force

Power expended by drag force

R

BTktt

3)()(

~ vF

R

BTkttR

3)()(6 vv

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Rotary Enzymes

Lipoamide1.4 nm long acetyl or succinyl carrier

pyruvate and -ketoglutarate dehydrogenases

Biocytin1.4 nm long CO2 carrier

pyruvate carboxylase and fatty acid synthetase

Phosphopantetheine2.0 nm long thioester carrier

gramicidin and tyrocidine synthetasesfatty acid synthetase, polyketide synthetase

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Kinesin

ProcessivityBias

Coordination

A two “headed” motor proteinthat “walks” on microtubules

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Mechanisms

Direct Chemo-Mechanical Energy Conversion“Power Stroke”

ATP Powered Conformation Change

Rectified Brownian Motion ATPase Switch

Heat Powered Conformation Change

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The trailing head is “thrown forward” in a way that is “akin to a judo

expert throwing an opponent with a rearward-to-forward swing of the arm.”

[Vale and Milligan, Science 288, 88 (2000)]

POWER STROKE

-sheet boundaryhydrogen bonds

ATP powered -sheet closure

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Forces

Covalent C-C, C-N and C-O bonds

4.0 - 4.3 nano-Newtons

Hydrogen bonds

~0.1 - ~50(?) pico-Newtons

Unbound kinesin head neck linker tensions

100 - 200 pico-Newtons

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Measured neck linker free energyof binding to the edge of

the -sheet is only a few kT.

This is enough energy to cause a significantbias for attachment in the forward

direction, or plus end of the microtubule.

Rectified Brownian Motion

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L a n g e v i n e q u a t i o n

)(~ tgcdt

dx

R

tFtgand

R

fc

6

)(~

)(~6

F o k k e r - P l a n c k e q u a t i o n

),(),(),(2

2

txPx

DtxcPx

txPt

B o u n d a r y c o n d i t i o n s

tallfortaPx

DtacP 0),(),(

tallfortbP 0),(

I n i t i a l c o n d i t i o n )()0,( axxP

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B a c k w a r d e q u a t i o n a n d f i r s t p a s s a g e t i m e

12

2

Tx

DTx

c

0)(0)(

aT

xandbT

)(1)(exp

1)( abab

D

c

c

D

caT

sD

abaT c

62

0 1078.12

)(|)(

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Load (pN) MFPT (s)

0 1.7727 x 10-6

1 1.0991 x 10-5

2 1.6465 x 10-4

3 4.0071 x 10-3

4 0.1231

5 4.3005

6 163.1

7 6541

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