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 Department of Physics, U of Minnesota Los Alamos, September 28, 2006 PRL, 95, (2005), Physica A, 359, 129 (2006), PRE 73, (2006). PRL, 95, (2005), Physica A, 359, 129 (2006), PRE 73, (2006).

Ion channels of cell membranes Ion channels of cell membranes (length L>> radius a) water filled nanopores for desalination water filled nanopores in silicon oxide films α -Hemolysin 10 nm

α -Hemolysin: α -Hemolysin:

One ion inside the channel Gauss theorem: Self energy: Parsegian,1969

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).

Transport  How does the barrier depend on the salt concentration ?  H ow does the barrier depend on the salt concentration ?  1d Coulomb potential ! Quarks! M. Akeson, et al Biophys. J. 1999

Two ion barrier  Maximum energy does NOT depend on the number of ions.

Entropy ! Ground state: pair concentration << ion concentration in the bulk L L 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 the optimum value of S happens at and and Barrier in free energy decreases by Result

First results: Transport barrier decreases with salt concentration

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 ! Model

Theory: Partition function: Electric field is conserved modulo ``Quantum number’’ : boundary charge F/TF/T F/TF/T dimensionless salt concentration dimensionless salt concentration

Does this all explain experimental data ? Yes, for wide channels No, for narrow ones

Channels are ``doped’’

Theory of the doped channels: Periodic array: Number of closed pairs inside is determined by the ``doping’’, NOT by the external salt concentration Number of closed pairs inside is determined by the ``doping’’, NOT by the external salt concentration

Theory of the doped channels (cont): ``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 channels? ``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 :  First order phase transitions latent Ca concentration

Ion exchange phase transitions Phase transitions in 1d system !

Ca Channels and Ca fractionalization 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  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 potential

Quantum dot arrays

Phase transitions in divalent salt solutions two competitive groundstates

Ion exchange phase transitions

Energy: Electric field in the channel are discrete values (Gauss 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 state 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 sparse. At high concentration pairs overlap and the concept on pairs is no longer valid. At low concentration pairs are sparse. At high concentration pairs overlap and the concept on pairs is no longer valid.

High concentration: reason of barrier Height of the barrier:

Doped channel: a simple model Doped channel: a simple model

Beyond the simplest model Ratio of dielectrics is not infinity Ratio of dielectrics is not infinity  Electric field lines begin to leave at Channel lengths are finite Channel lengths are finite Ions are not planes Ions are not planes see A. Kamenev, J. Zhang, A. I. Larkin, B. I. Shklovskii cond- mat/

Future work: DNA in the channel