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![Page 1: Statistical Mechanics of Ion Channels: No Life Without Entropy A. Kamenev J. Zhang J. Zhang B. I. Shklovskii A. I. Larkin Department of Physics, U of Minnesota.](https://reader036.fdocuments.net/reader036/viewer/2022081603/56649eca5503460f94bd84cf/html5/thumbnails/1.jpg)
Statistical Mechanics of Ion Statistical Mechanics of Ion Channels: No Life Without Channels: No Life Without
EntropyEntropy
A. KamenevA. Kamenev
J. ZhangJ. Zhang
B. I. ShklovskiiB. I. Shklovskii
A. I. LarkinA. I. Larkin
Department of Physics, U of Department of Physics, U of MinnesotaMinnesota
Los Alamos, September 28, 2006 Los Alamos, September 28, 2006
PRL, 95, 148101 (2005),Physica A, 359, 129 (2006),PRE 73, 051205 (2006).
PRL, 95, 148101 (2005),Physica A, 359, 129 (2006),PRE 73, 051205 (2006).
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Ion channels of cell Ion channels of cell membranesmembranes
(length L>> radius a)•water filled nanopores for desalination•water filled nanopores in silicon oxide films
α-Hemolysin
10 nm10 nm
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αα-Hemolysin:-Hemolysin:
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One ion inside the One ion inside the
channelchannel
waterlipid
water
2'
81
waterlipid
water
2'
81
20
0water2
2
42
a
eE
eEa
20
0water2
2
42
a
eE
eEa
Gauss theorem:Gauss theorem:
lipid'water
L
a2
La
eLa
EUL 2
22
20
28
La
eLa
EUL 2
22
20
28
Self energy: Parsegian,1969Self energy: Parsegian,1969
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Barrier:Barrier:
There is an energy barrier for the charge transfer
(the same for the positive and negative ions).
There is an energy barrier for the charge transfer
(the same for the positive and negative ions).
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TransportTransport
How does the barrier depend on the salt concentration ? How does the barrier depend on the salt concentration ?
1d Coulomb potential ! Quarks! 1d Coulomb potential ! Quarks!
M. Akeson, et al Biophys. J. 1999
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Two ion barrierTwo ion barrier
Maximum energy does NOT depend on the number of ions. Maximum energy does NOT depend on the number of ions.
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Entropy !Entropy !
Ground state: pair concentration << ion concentration in the bulk
Ground state: pair concentration << ion concentration in the bulk
LL
Free ions enter the channel, increasing the entropy ! Free ions enter the channel, increasing the entropy !
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Low salt concentration: collective ion Low salt concentration: collective ion barrierbarrier
Ions are free to enter Ions are free to enter Entropy at the energy barrier increasesEntropy at the energy barrier increases
the optimum value of S happens at the optimum value of S happens at andand
Barrier in free energy decreases byBarrier in free energy decreases by
Result Result
STF
Txacf 20 ,41)(
N
lVeN
NN
leVNNS
2
/ln2
/ln)(2
30
30
LacnLN 221
21 LackNkS BB
221
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First results:First results:Transport barrier decreases with salt concentration Transport barrier decreases with salt concentration
STF
LL UUf /)()(0
)8exp( 41
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1. 1d Coulomb gas or plasma of charged planes.2. Finite size of ions.3. Viscous dynamics of ions.4. Charges q !
1. 1d Coulomb gas or plasma of charged planes.2. Finite size of ions.3. Viscous dynamics of ions.4. Charges q !
ModelModel
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Theory:Theory:
Partition function: Partition function: LHq
qeZˆ
Tr LHq
qeZˆ
Tr Lqe )(0 Lqe )(0
Electric field is conserved modulo Electric field is conserved modulo 02E02E
``Quantum number’’ : ``Quantum number’’ :
02E
Efrac q
02E
Efrac q boundary chargeboundary charge
F/TF/T
dimensionlesssalt concentration
dimensionlesssalt concentration
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Does this all explain
experimental data ?
Yes, for wide channels
No, for narrow ones
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Channels are ``doped’’Channels are ``doped’’
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Theory of the doped channels:Theory of the doped channels:
Periodic array:Periodic array:
Number of closed pairs inside is determined by the``doping’’ , NOT by the external salt concentrationNumber of closed pairs inside is determined by the``doping’’ , NOT by the external salt concentration
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Theory of the doped channels Theory of the doped channels (cont):(cont):
/029.11e
)2/ln(41 1
1
LU
``Doping’’ suppresses the barrier down to about kT
``Doping’’ suppresses the barrier down to about kT kT
Additional role of doping: ``p—n’’ boundary layers create Donnan potential. It leads to cation versus anion selectivity in negatively doped channels.
Additional role of doping: ``p—n’’ boundary layers create Donnan potential. It leads to cation versus anion selectivity in negatively doped channels.
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So ? What did we learn about narrow So ? What did we learn about narrow channels?channels?
``Doping’’ plus boundary layers explain observed large conductances
``Doping’’ plus boundary layers explain observed large conductances
They also explain why flux of positive ions greatly exceeds that of negative ones.
They also explain why flux of positive ions greatly exceeds that of negative ones.
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Divalent ions :Divalent ions :
22 , , CdBaCa2 22 , , CdBaCa2
LU F
2ln
First order phase transitions First order phase transitions
31 10
/ dopantsCa nn
latent Ca concentration
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λ
0q
2/1q0q
Ion exchange phase transitionsIon exchange phase transitions Ion exchange phase transitionsIon exchange phase transitions
λ
Phase transitions in 1d system !Phase transitions in 1d system !
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Ca Channels and Ca fractionalizationCa Channels and Ca fractionalization
Almers and McCleskey 1984
λ
λ
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Wake up !Wake up !
There is a self-energy barrier for an ion in the channel There is a self-energy barrier for an ion in the channel
Ions in the channel interact as 1d Coulomb gas Ions in the channel interact as 1d Coulomb gas
Large concentration of salt in wide channels and ``doping’’ in narrow channels decrease the barrier
Large concentration of salt in wide channels and ``doping’’ in narrow channels decrease the barrier
Divalent ions are fractionalized so that the barrier for
them is small and they compete with Na.
Divalent salts may lead to first order phase transitions
Divalent ions are fractionalized so that the barrier for them is small and they compete with Na.
Divalent salts may lead to first order phase transitions
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Interaction potentialInteraction potential
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Quantum dot arraysQuantum dot arrays
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Phase transitions in divalent salt Phase transitions in divalent salt solutionssolutions
Phase transitions in divalent salt Phase transitions in divalent salt solutionssolutions
0q
2/1q
'0q
2/1q0q
1
'0
1
two competitive groundstates
22 ClCa
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Ion exchange phase Ion exchange phase transitionstransitions
Ion exchange phase Ion exchange phase transitionstransitions
λ
λ
0q
2/1q
0q
1λ
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Energy:Energy:
'
Electric field in the channel are discrete Electric field in the channel are discrete values (values (Gauss lawGauss law))
At equilibrium electric fields are integers:At equilibrium electric fields are integers: At barrier electric fields are half-integers:At barrier electric fields are half-integers:
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Pairs exist in ground Pairs exist in ground statestate
A pair in the channel:A pair in the channel: Length of a pair:Length of a pair:
Dimensionless concentration of ions: Dimensionless concentration of ions:
Concentration of pairs:Concentration of pairs:
At low concentration pairs are At low concentration pairs are sparse. At high concentration sparse. At high concentration pairs overlap and the concept on pairs pairs overlap and the concept on pairs is no longer valid.is no longer valid.
Tp xnn 22
B
BT l
a
eE
Tkx
2
2
0
TT xacnx 2
o
BB A
Tk
el 7
2
11
)2( 2Tpxn
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High concentration: reason of High concentration: reason of barrierbarrier
nEE )2/( 0
21
0 )2/( nEE
Height of the barrier:
]/2exp[
]2
exp[]cos[
]2
exp[]cos[
)()(
20
22
2
2
0
2
2
0
2
2
EE
dEE
E
E
E
dEE
E
E
EE
dEEPEU L
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Doped channel: a simple modelDoped channel: a simple model
)2/1ln(41)(
1/
0
f
xl T
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Beyond the simplest Beyond the simplest modelmodel
Ratio of dielectrics is not infinityRatio of dielectrics is not infinity Electric field lines begin to leave atElectric field lines begin to leave at
• Channel lengths are finite Channel lengths are finite • Ions are not planesIons are not planes
see A. Kamenev, J. Zhang, A. I. Larkin, B. I. Shklovskii see A. Kamenev, J. Zhang, A. I. Larkin, B. I. Shklovskii cond-cond-mat/0503027mat/0503027
a
eff
'/
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Future work: DNA in the Future work: DNA in the channelchannel