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Coulomb Blockade of Stochastic Permeation in Biological Ion Channels

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Coulomb Blockade of Stochastic Permeation in Biological Ion Channels Outline Coulomb Blockade of Stochastic Permeation in Biological Ion Channels W.A.T.Gibby1 I.Kh. Kaufman1 D.G. Luchinsky1,2 P.V.E. McClintock1 R.S. Eisenberg3 1Department of Physics, Lancaster University, UK 2Mission Critical Technologies Inc., El Segundo, CA USA 3Molecular Biophysics, Rush University, Chicago, USA UPoN Barcelona – 13 July 2015 Gibby, Kaufman, Luchinsky, McClintock, Eisenberg Coulomb Blockade Dynamics Outline Outline 1 The unsolved problem 10 (a) Background 5 ) T 0 B Electrostatic model k ( −5 E Effect of the fixed charge −10 −15 2 Solving the problem? 0 1 20 40 Modelling – Brownian dynamics 2 −40 −20 0 Qf (e) x(A˚) Ionic Coulomb blockade 10 ) (b) T Mechanisms of permeation B 0 k ( 3 Conclusions E −10 −30 −20 −10 0 10 20 30 Prospects x(A)˚ Summary How are ions transported selectively through Ca2+ and Na+ channels? Gibby, Kaufman, Luchinsky, McClintock, Eisenberg Coulomb Blockade Dynamics The unsolved problem Background Solving the problem? Electrostatic model Conclusions Effect of the fixed charge Ion channels Cellular membrane has numerous ion channels (+ pumps and transporters). Ion channel is a natural nanotube through the membrane. Allows ion exchange between inside and outside of cell. Essential to physiology – bacteria to humans. Highly selective for particular ions. Gibby, Kaufman, Luchinsky, McClintock, Eisenberg Coulomb Blockade Dynamics The unsolved problem Background Solving the problem? Electrostatic model Conclusions Effect of the fixed charge Gating Channels spontaneously “gating” open/shut. A stochastic process influenced by e.g. – Voltage Chemicals We are interested only in open channels. Gibby, Kaufman, Luchinsky, McClintock, Eisenberg Coulomb Blockade Dynamics The unsolved problem Background Solving the problem? Electrostatic model Conclusions Effect of the fixed charge Puzzles 1. Selectivity? E.g. Calcium channel favours Ca2+ over Na+ by up to 1000:1, even though they are the same size – example of valence selectivity. Also alike charge selectivity, e.g. potassium channel strongly disfavours sodium, even though Na+ is smaller K+. 2. Fast permeation? Almost at the rate of free diffusion (open hole). 3. AMFE? Na+ goes easily through a calcium channel but is blocked by tiny traces of Ca2+. Gibby, Kaufman, Luchinsky, McClintock, Eisenberg Coulomb Blockade Dynamics The unsolved problem Background Solving the problem? Electrostatic model Conclusions Effect of the fixed charge Puzzles 4. Function of the fixed charge at the SF Ion channels have narrow “selectivity filters” with fixed negative charge... somehow associated with selectivity. What does the charge do, and how does it determine selectivity? 5. Mutations Mutations that alter the fixed charge (alone) can – (a) Destroy the channel (so it no longer conducts), or (b) Change the channel selectivity, e.g. Ca2+ to Na+ or vice versa. 6. Gating Why/how does the channel continually open and close? Gibby, Kaufman, Luchinsky, McClintock, Eisenberg Coulomb Blockade Dynamics The unsolved problem Background Solving the problem? Electrostatic model Conclusions Effect of the fixed charge Atomic structure of KcsA Potassium ion channel So-called “crystal structure” of a bacterial ion channel. Very complicated. Knowledge of the structure does not immediately explain the function – the famous “structure-function problem”. Which features are important? Selectivity filter? Generic features of structure? Need to pick out aspects important for modelling the permeation process. Gibby, Kaufman, Luchinsky, McClintock, Eisenberg Coulomb Blockade Dynamics The unsolved problem Background Solving the problem? Electrostatic model Conclusions Effect of the fixed charge Minimal model of calcium/sodium ion channel Na + Ca 2+ L Qf Ra X R Left bath, Right bath, CL>0 CR=0 Membrane protein Awater-filled, cylindrical hole, radius R = 3Å and length L = 16Å through the protein hub in the cellular membrane. Water and protein described as continuous media with dielectric constants εw = 80 (water) and εp = 2 (protein) The selectivity filter (charged residues) represented by a rigid ring of negative charge Qf = 0 − 6.5e. Gibby, Kaufman, Luchinsky, McClintock, Eisenberg Coulomb Blockade Dynamics The unsolved problem Background Solving the problem? Electrostatic model Conclusions Effect of the fixed charge Calcium and sodium channels We focus on voltage-gated calcium and sodium ion channels – which control muscle contraction, neurotransmitter secretion, transmission of action potentials) – because: They are very similar in structure, but have differing SF loci and hence different Qf . The many mutation experiments to change Qf (destroying the channel or changing its selectivity) are potentially revealing but not yet properly understood. But results and conclusions are more generally applicable. Gibby, Kaufman, Luchinsky, McClintock, Eisenberg Coulomb Blockade Dynamics The unsolved problem Background Solving the problem? Electrostatic model Conclusions Effect of the fixed charge Permeation through an ion channel Typically, the fixed charge Qf provides a strong binding site at the selectivity filter. Dimensions & electrostatics enforce single-file motion. Arriving ions captured at binding site, then fluctuational escape occurs over the potential barrier Eb. Gibby, Kaufman, Luchinsky, McClintock, Eisenberg Coulomb Blockade Dynamics The unsolved problem Modelling – Brownian dynamics Solving the problem? Ionic Coulomb blockade Conclusions Mechanisms of permeation Electrostatics The electrostatic field is derived by self-consistent numerical solution of Poisson’s equation: −∇(εε0∇u)= ezi ni X where ε0 is the dielectric permittivity of vacuum, ε is the dielectric permittivity of the medium (water or protein), u is the electric potential, e is the elementary charge, zi is the valence, and ni is the number density of ions. This equation accounts for both ion-ion interaction and self-action for all ions in their current positions. One result of the calculations is the axial potential energy profile = the Potential of the Mean Force (PMF). Gibby, Kaufman, Luchinsky, McClintock, Eisenberg Coulomb Blockade Dynamics The unsolved problem Modelling – Brownian dynamics Solving the problem? Ionic Coulomb blockade Conclusions Mechanisms of permeation Brownian dynamics The BD simulations use numerical solution of the 1-D overdamped, time-discretized, Langevin equation for the i-th ion: dx ∂u i = −D z + 2D ξ(t) dt i i ∂x i i p where xi stands for the ion’s position, Di is its diffusion coefficient, zi is the valence, u is the self-consistent potential in kBT /e units, and ξ(t) is normalized white noise. Numerical solution is implemented with the Euler forward scheme. We use an ion injection scheme that allows us to avoid simulations in the bulk. The arrival rate jarr is connected to the bulk concentration C through the Smoluchowski diffusion rate: jarr = 2πDRC. Gibby, Kaufman, Luchinsky, McClintock, Eisenberg Coulomb Blockade Dynamics The unsolved problem Modelling – Brownian dynamics Solving the problem? Ionic Coulomb blockade Conclusions Mechanisms of permeation Calcium conduction and occupancy bands Current J and 1 occupation P as a function of charge Qf at J (a.u.) 0 selectivity filter. 50 + 5 6 2 0 0 1 2 3 4 For different Ca [Ca] (mM) Q (e) f concentrations [Ca]. M0 M1 M2 1 We find – 20mM J (a.u.) 40mM 80mM Pattern of narrow 0 3 conduction bands 2 and stop bands. P 20mM 1 40mM Conduction bands 80mM 0 occur at transitions 0 1 2 3 4 5 6 7 Q (e) in channel f occupancy P. Gibby, Kaufman, Luchinsky, McClintock, Eisenberg Coulomb Blockade Dynamics The unsolved problem Modelling – Brownian dynamics Solving the problem? Ionic Coulomb blockade Conclusions Mechanisms of permeation Electrostatic exclusion principle In absence of fixed charge Qf , self-energy barrier Us prevents entry of ion to SF. But Qf compensates Us and allows cation to enter. This effectively restores the impermeable Us for 2nd ion at channel mouth. So for this Qf only one ion can occupy the SF. Implications 1. The SF’s forbidden multi-occupancy is an electrostatic exclusion principle. 2. Like the Pauli exclusion principle in quantum mechanics, it implies a Fermi-Dirac occupancy distribution. 3. For larger Qf similar arguments apply for occupancies of 2,3... Gibby, Kaufman, Luchinsky, McClintock, Eisenberg Coulomb Blockade Dynamics The unsolved problem Modelling – Brownian dynamics Solving the problem? Ionic Coulomb blockade Conclusions Mechanisms of permeation Coulomb blockade: channels vs. quantum dots + Ca2 channel Quantum dot 15 /s) 7 10 (a) M M M 0 1 2 (10 5 Ca J 20mM 0 40mM 4 80mM (b) Z Ca 2 P 2 Z Z 1 3 0 20 (c) U n T) U B M M M G 10 0 1 2 U/(k n=0 n=1 n=2 n=3 0 0 1 2 3 4 5 6 |Q /e| f Ion(s) trapped at SF ⇔ Electron(s) trapped in quantum dot Periodic conduction bands ⇔ Coulomb blockade oscillations Steps in occupation number ⇔ Coulomb staircase Classical mechanics for ion ⇔ Quantum mechanics for electron Stochastic permeation by ion ⇔ Quantum tunnelling by electron Gibby, Kaufman, Luchinsky, McClintock, Eisenberg Coulomb Blockade Dynamics The unsolved problem Modelling – Brownian dynamics Solving the problem? Ionic Coulomb blockade Conclusions Mechanisms of permeation Singular points in ionic Coulomb blockade 15 /s) 7 10 (a) M M M 0 1 2 (10 5 Ca J 20mM 0 40mM 4 80mM (b) Z Ca 2 P 2 Z Z 1 3 0 20 (c) U n T) U B M M M G 10 0 1 2 U/(k n=0 n=1 n=2 n=3 0 0 1 2 3 4 5 6 |Q /e| f Zn = zen ± δZn, Coulomb blockade Mn = ze(n + 1/2) ± δMn Resonant conduction Gibby, Kaufman, Luchinsky, McClintock, Eisenberg Coulomb Blockade Dynamics The unsolved problem Modelling – Brownian
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