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).

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

αα-Hemolysin:-Hemolysin:

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

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).

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

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.

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 !

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

First results:First results:Transport barrier decreases with salt concentration Transport barrier decreases with salt concentration

STF

LL UUf /)()(0

)8exp( 41

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

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

Does this all explain

experimental data ?

Yes, for wide channels

No, for narrow ones

Channels are ``doped’’Channels are ``doped’’

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

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.

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.

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

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 !

Ca Channels and Ca fractionalizationCa Channels and Ca fractionalization

Almers and McCleskey 1984

λ

λ

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

Interaction potentialInteraction potential

Quantum dot arraysQuantum dot arrays

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

Ion exchange phase Ion exchange phase transitionstransitions

Ion exchange phase Ion exchange phase transitionstransitions

λ

λ

0q

2/1q

0q

1λ

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:

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

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

Doped channel: a simple modelDoped channel: a simple model

)2/1ln(41)(

1/

0

f

xl T

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

'/

Future work: DNA in the Future work: DNA in the channelchannel