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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