RF Fox FNAL 2006 1

Rectified Brownian Motion in

Subcellular Biology

Ronald F. Fox

Mee ChoiWilliam Mather

School of PhysicsGeorgia Institute of Technology

RF Fox FNAL 2006 2

2005, Centenary of Einstein’s Annus Mirabilis

Special Relativity

Photoelectric Effect

Brownian Motion

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

RF Fox FNAL 2006 3

Nanobiology

BiochemistryMolecular biology

Can we learn mechanisms from nanobiologythat are applicable to nanotechnology?

RF Fox FNAL 2006 4

One such lesson is the constructive use of

thermal energy.

Such a mechanism is called

Rectified Brownian Motion.

RF Fox FNAL 2006 5

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)

RF Fox FNAL 2006 6

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)

RF Fox FNAL 2006 7

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

RF Fox FNAL 2006 8

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

RF Fox FNAL 2006 9

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

RF Fox FNAL 2006 10

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

RF Fox FNAL 2006 11

Biological Energy Couplings

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

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

RF Fox FNAL 2006 12

Redox reaction variety

Pure, 1 electron transferIron, copper, zinc…

1 electron and 1 proton transferFADH2, UQH2,..

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

RF Fox FNAL 2006 13

RF Fox FNAL 2006 14

RF Fox FNAL 2006 15

),(),( 2

2

txfx

Dtxft

diffusion for reduced ubiquinone

),0(00000 tfx

DYXXdt

d

boundary layer equation

RF Fox FNAL 2006 16

d

Dr

diffusive rate parameter

d

dd

0

00

reaction rate parameter

r

RF Fox FNAL 2006 17

rimplications

Weak linear steady state gradients

Negligible energy dissipation associated with the gradients according to

non-equilibrium steady state thermodynamics

RF Fox FNAL 2006 18

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

RF Fox FNAL 2006 19

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

RF Fox FNAL 2006 20

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

RF Fox FNAL 2006 21

RF Fox FNAL 2006 22

RF Fox FNAL 2006 23

Kinesin

ProcessivityBias

Coordination

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

RF Fox FNAL 2006 24

RF Fox FNAL 2006 25

Mechanisms

Direct Chemo-Mechanical Energy Conversion“Power Stroke”

ATP Powered Conformation Change

Rectified Brownian Motion ATPase Switch

Heat Powered Conformation Change

RF Fox FNAL 2006 26

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

RF Fox FNAL 2006 27

RF Fox FNAL 2006 28

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

RF Fox FNAL 2006 29

RF Fox FNAL 2006 30

RF Fox FNAL 2006 31

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

RF Fox FNAL 2006 32

RF Fox FNAL 2006 33

RF Fox FNAL 2006 34

RF Fox FNAL 2006 35

RF Fox FNAL 2006 36

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

RF Fox FNAL 2006 37

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

)(|)(

RF Fox FNAL 2006 38

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

RF Fox FNAL 2006 39

RF Fox FNAL 2006 40