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 PRL, 95, 148101 (2005), Physica A, 359, 129 (2006), PRE 73, 051205 (2006). Los Alamos, September 28, 2006
(length L>> radius a) • water filled nanopores for desalination • water filled nanopores in silicon oxide films 10 nm 10 nm α-Hemolysin Ion channels of cell membranes
Gauss theorem: Self energy: Parsegian,1969 One ion inside the channel
Barrier: • There is an energy barrier for the charge transfer (the same for the positive and negative ions).
M. Akeson, et al Biophys. J. 1999 • 1d Coulomb potential ! Quarks! Transport • How does the barrier depend on the salt concentration ?
Maximum energy does NOT depend on the number of ions. Two ion barrier
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 the optimum value of S happens at and Barrier in free energy decreases by Result
First results: Transport barrier decreases with salt concentration
Model 1. 1d Coulomb gas or plasma of charged planes. 2. Finite size of ions. 3. Viscous dynamics of ions. 4. Charges q !
Partition function: F/T dimensionless salt concentration Theory: Electric field is conserved modulo ``Quantum number’’ : boundary charge
Does this all explain experimental data ? Yes, for wide channels No, for narrow ones
Periodic array: Theory of the doped channels: Number of closed pairs inside is determined by the ``doping’’ , NOT by the external salt concentration
``Doping’’ suppresses the barrier down to about kT kT Theory of the doped channels (cont): 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.
latentCa concentration Divalent ions : • First order phase transitions
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 • 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
Phase transitions in divalent salt solutions two competitive groundstates
Electric field in the channel are discrete values (Gauss law) At equilibrium electric fields are integers: At barrier electric fields are half-integers: Energy:
Pairs exist in ground state • A pair in the channel: • Length of a pair: • Dimensionless concentration of ions: • Concentration of pairs: • 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:
Beyond the simplest model • Ratio of dielectrics is not infinity • Electric field lines begin to leave at • Channel lengths are finite • Ions are not planes see A. Kamenev, J. Zhang, A. I. Larkin, B. I. Shklovskii cond-mat/0503027