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## Statistical Mechanics of Ion Channels: No Life Without Entropy

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**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