Statistical Mechanics of Ion Channels:
No Life Without Entropy
A. Kamenev
J. Zhang
B. I. Shklovskii
A. I. Larkin
Department of Physics, U of Minnesota
Los Alamos, September 28, 2006
PRL, 95, 148101 (2005),
Physica A, 359, 129 (2006),
PRE 73, 051205 (2006).
Ion channels of cell membranes
10nm
nm
10
α-Hemolysin
α-Hemolysin:
One ion inside the channel
L
2a
 water  81
 'lipid  2   water
 water
 'lipid
2a 2 water E0  4e
Gauss theorem:
Self energy:
Parsegian,1969
2e
E0  2
a
E
e
2
UL 
a L 
L
2
8
2a
2
0
2
Barrier:
 There is an energy barrier for the charge transfer
(the same for the positive and negative ions).
Transport
M. Akeson, et al Biophys. J. 1999
 How does the barrier depend on the salt concentration ?
 1d Coulomb potential ! Quarks!
Two ion barrier
 Maximum energy does NOT depend on the number of ions.
Entropy !
L
Ground state: pair concentration << ion concentration in the bulk
Free ions enter the channel, increasing the entropy !
Low salt concentration: collective ion barrier
Ions are free to enter

Entropy at the energy barrier increases
eV / l 0
eV / l 0
S  2( N  N ) ln
 2N ln
N  N
2N
3
3
the optimum value of S happens at
2
2
1
1
1

S

k

N

ck

a
L
N  2 nL  2 ca L
and
B
B
2
Barrier in free energy decreases by
F  TS
Result
f 0 ( )  1  4 ,    ca xT
2
First results:
Transport barrier decreases with salt concentration
F  TS
1 4
exp( 8  )
f0 ( )  U L ( ) / U L
Model
1. 1d Coulomb gas or plasma of charged planes.
2. Finite size of ions.
3. Viscous dynamics of ions.
4. Charges q !
Theory:
Electric field is conserved modulo
2E0
 E 
``Quantum number’’ q  frac 
: boundary charge

 2E0 
Partition function:
dimensionless
salt concentration
 e
Z q  Tr e
 Hˆ q L
 0 ( q ) L
F/T
Does this all explain
experimental data ?
Yes, for wide channels
No, for narrow ones
Channels are ``doped’’
Theory of the doped channels:
Number of closed pairs inside is determined by the
``doping’’ , NOT by the external salt concentration
Periodic array:
Theory of the doped channels (cont):
UL
1  41 ln(  / 2)
``Doping’’ suppresses the
barrier down to about kT
kT
e 11.029 /

 1
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 channels?
 ``Doping’’ plus boundary layers explain
observed large conductances
 They also explain why flux of positive
ions greatly exceeds that of negative ones.
Divalent ions : Ca 2 , Ba 2 , Cd 2
UL
1  103
F
nCa / ndopants
 First order phase transitions
latent Ca
concentration
ln 2
Ion exchange phase transitions
λ
q0
q  1/ 2
q0
Phase transitions in 1d system !
Ca Channels and Ca fractionalization
λ
λ
Almers and McCleskey 1984
Wake up !
 There is a self-energy barrier for an ion in the channel
 Ions in the channel interact as 1d Coulomb gas
 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
Interaction potential
Quantum dot arrays
Phase transitions in divalent salt solutions
Ca
2
Cl 

q0
2
q  0'
q  1/ 2
0'


1
two competitive groundstates
q0
q  1/ 2
1
Ion exchange phase transitions
λ
q0
q0
q  1/ 2
λ 1
Energy:
Electric field in the channel are discrete values
(Gauss law)
 At equilibrium electric fields are integers:
'
 At barrier electric fields are half-integers:

Pairs exist in ground state

A pair in the channel:
Length of a pair:
k BT
a2
xT 

eE0 2l B

Dimensionless concentration of ions:

o
e2
lB 
7A
k BT
  nxT  ca xT
2


Concentration of pairs:
n p  2n2 xT (n p xT  2 2 )
At low concentration  1
pairs are sparse.
At high concentration
pairs overlap
  1
and the concept on pairs is no longer valid.
High concentration: reason of
barrier
E /( 2E0 )  n
E /( 2E0 )  n  12
Height of the barrier:

U L ( )   E 2 P( E )dE

E
E2
   E  cos[
]  exp[ 
]dE
2

E0
2 E 

2
E
E2
   cos[
]  exp[ 
]dE

E0
2  E2 

 exp[ 2 2  E 2  / E0 ]
2

Doped channel: a simple model
  l / xT  1
f 0 (  )  1  4  ln( 1 / 2  )
Beyond the simplest model


•
•
Ratio of dielectrics  /  '
is not infinity
Electric field lines begin to leave at
  a
Channel lengths are finite
Ions are not planes
   eff
see A. Kamenev, J. Zhang, A. I. Larkin, B. I. Shklovskii cond-
mat/0503027
Future work: DNA in the
channel