1. CALCIUM CHANNELS The Nernst-Planck equation is used to describe ion flow: (1) where J i, D i,  i, and μ i are the local flux density, diffusion coefficient, density, and electrochemical potential, respectively, of ion species i. By averaging these equations over equi-concentration/potential surfaces with area A, this can be reduced to a one- dimensional approximation: (2) J i is now the flux of species i, not the flux density, and is a constant. This equation can be integrated from bath to bath across the channel, which for symmetric baths gives: (3) If V is small and IV curves linear, the channel conductance g is given by the electrical resistances R X and R Ca of monovalent X + and Ca 2+ throughout pore: (4) (5) These resistances are shown in this equivalent-circuit diagram: SYNTHETIC NANOPORES THEORY: RESISTORS-IN-SERIES A common mechanism for the anomalous mole fraction effect in biological calcium channels and in wide synthetic nanopores A theory of the anomalous mole fraction effect (AMFE) in biological calcium channels is presented that does not assume single-filing of multiple ions in the pore. In a mole fraction experiment, a mixture of two ion species is at a fixed total concentration and the channel conductance is measured while the relative concentrations are changed. In some cases, the conductance of the mixture is less than that of the conductances either species individually (an AMFE). The textbook explanation of the AMFE is the coordinated movement of multiple ions moving through a single-file pore. The theory that is presented takes a different approach and builds on the ideas of Nonner et al. that depletion zones of ions in the pore correspond to high resistance elements in series. Experimental verification that the AMFE can exist in wide pores without correlated ion motion is shown in a micrometer-long pore in plastic with a negative surface charge. The same theory with a reduced model of this abiotic pore reproduces the experimental currents. Dirk Gillespie 1, Dezső Boda 1,2, Yan He 3, and Zuzanna Siwy 3 1 Rush University Medical Center, Chicago, IL; 2 University of Pannonia, Veszprém, Hungary; 3 University of California, Irvine ABSTRACT Wide nanopores have AMFEs in mixtures of Ca 2+ and Li +, Na +, K +, Cs + [Ca 2+ ] + [X + ] = 100 mM [Ca 2+ ] + [X + ] = 20 mM block of K + current by Ca 2+ [K + ] = 100 mM Ca 2+ is added A single double-conical (hour-glass) nanopore is made in a 12 μm-thick polyethylene terephthalate (PET) by irradiating it with a single, accelerated, heavy ion (UNILAC, GSI, Darmstadt, Germany) and then chemically etching the track left by the ion with 9 M NaOH. Etching from both sides produces the double-conical shape shown in this electron micrograph by Dr. P. Apel (JINR, Dubna, Russia). Each pore we studied has a 50 Å diameter at its narrowest point (no single filing) a uniform negative surface charge of COOH - (-1e/nm 2 ) very high cation selectivity (P K /P Cl > 50; P Ca /P Cl > 100) We use Monte Carlo simulations to compute the concentration profiles. In the Monte Carlo simulations, Ca 2+ is added symmetrically. In the Nernst-Planck equation, we only consider the selectivity filter where the diffusion coefficients D i and the area A are constant. Then the resistances and normalized conductance are Theory vs. Experiments The selectivity filter of the L-type Ca 2+ channel is modeled as having radius 3.5 Å length 10 Å protein dielectric 10 “Upswing” branch exaggerated in experiments because Ca 2+ was added asymmetically. This increases the driving force of Ca 2+ as [Ca 2+ ] > 0.1 mM Half-block of Na + current by 1 μM Ca 2+ does not depend on Ca 2+ diffusion coefficient. Ca 2+ displaces Na + and binds in the pore, but Ca 2+ does not conduct CONCLUSIONS The AMFE occurs because the resistances R i in different parts of the pore vary differently with mole fraction the resistances are in series. The general idea is that the resistance of the selectivity filter does not change linearly with mole fraction because one ion species is preferentially selected over the other the resistance of the uncharged regions varies like the mole fraction because the occupancy of the uncharged regions reflects the bath concentrations. The AMFE does not require single filing of ions indicate a “multi-ion” pore (rather, occupancy is mainly determined by electrostatics) 1.AMFE: Nonner, W., D. P. Chen, and B. Eisenberg. 1998. Biophys. J. 74:2327-2334. 2.model of L-type Ca 2+ channel: Boda, D., M. Valiskó, B. Eisenberg, W. Nonner, D. J. Henderson, and D. Gillespie. 2007. Phys. Rev. Lett. 98:168102. REFERENCES Ca 2+ binds only in the pore center no Ca 2+ in outer regions = high resistance = no Ca 2+ current at all left outer region right outer region pore center The AMFE is caused by Ca 2+ being preferentially bound, displacing Na + from the pore, causing Na + current to decrease but Ca 2+ only binds in the center of the pore leading to a low resistance for Ca 2+ in the center of the pore (even at low [Ca 2+ ]) that saturates quickly because Ca 2+ occupancy saturates and a very high resistance for Ca 2+ in the outer regions of the selectivity filter so that there is no appreciable Ca 2+ current to compensate the decrease in Na + current Theory Each nanopore is a double cone with an angle of only ~2 . Therefore, in a short region of length L, the resistance is inversely proportional to the local concentration: For simplicity, we used the radially-averaged concentrations from MC simulations in the Nernst-Planck equation and one effective diffusion coefficient. Resistances of the pore center and mouths change differently with mole fraction because Ca 2+ is preferentially bound [Ca 2+ ] + [K + ] = 100 mM The local resistances within the pore is inversely proportional to the local ion concentrations. Therefore, the resistances near the pore mouths vary linearly with mole fraction but, the resistances in the pore center vary nonlinearly with mole fraction because the pore binds Ca 2+ much better than K +, producing the AMFE. 0 Ca 2+ 5% Ca 2+ lines = theory symbols = experiment X + = Li +, Na +, K +, and Cs + Ca 2+ bath [Na + ] = 30 mM lines = modelsymbols = data from Almers and McCleskey