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Cell Membrane Potential-CN ! Colin Nichols Membrane Potential and Ion Channels Background readings: Lodish et al., Molecular Cell Biology, 4th ed, chapter 15 (p. 633-665) and chapter 21 (p. 925-965). Alberts et al., Molecular Biology of the Cell, 4th ed, chapter 11 (p. 615-650) and p. 779-780. Hille Ionic channels of excitable membranes 3rd ed. LIFE – A complex interplay of chemistry and physics The action potential – Physics in cell biology 1 2 2 1 3 4 3 4 Na+ K+ Excitation-contraction coupling The islet of Langerhans Glucose-dependent hormone secretion The Lipid Bilayer is a Selective Barrier inside outside hydrophobic molecules (anesthetics) gases (O2, CO2) small uncharged polar molecules large uncharged polar molecules Ions charged polar molecules (amino acids) water Ion Channel Diversity Cation channel Subunit Topology Cation channel Subunit Topology The ‘inner core’ of cation channel superfamily members Structure of a Potassium Channel Doyle et al., 1998 Selectivity Filter and Space-Filling Model Doyle et al., 1998 Ion selectivity filter in KcsA Doyle et al., 1998, Zhou et al., 2002 Selectivity filter diversity Ion selectivity filter in Na channels Nature Communications , Oct 2, 2012. Structure of a bacterial voltage-gated sodium channel NavMs, NavAb, Kv1.2 pore reveals mechanisms of opening and closing Emily C McCusker,1, 3, Claire Bagnéris,1 Claire E Naylor,1, Ambrose R Cole,1 Nazzareno D'Avanzo,2, 4 , Colin G Nichols2, & B A Wallace1 Views of the Channel Open and Shut Jiang et al., 2002 Views of the Channel Open and Shut Nature Communications , Oct 2, 2012. Structure of a bacterial voltage-gated sodium channel pore reveals mechanisms of opening and closing Emily C McCusker,1, 3, Claire Bagnéris,1 Claire E Naylor,1, Ambrose R Cole,1 Nazzareno D'Avanzo,2, 4 , Colin G Nichols2, & B A Wallace1 Ion gradients and membrane potential Ion gradients and membrane potential Ion gradients and membrane potential Ion gradients and membrane potential The inside of the cell is at ~ -0.1V with respect to the outside -89 mV solution How does this membrane potential come about? Ion gradients and membrane potential K Na Cl K -89 mV How does this membrane potential come about? Ion gradients and membrane potential Na 117 Na 30 K 3 K 90 Cl 120 Cl 4 Anions 0 Anions 116 Total 240 Total 240 [+ charge] = [- charge] [+ charge] = [- charge] 0 mV -89 mV How does this membrane potential come about? Movement of Individual K+ ions + Na 117 Na 30 - K 3 K 90 + Cl 120 Cl 4 - Anions 0 Anions 116 Total 240 Total 240 [+ charge] = [- charge] [+ charge] = [- charge] 0 mV 0 mV Movement of Individual Cl- ions + Na 117 - Na 30 K 3 K 90 Cl 120 Cl 4 Anions 0 + - Anions 116 Total 240 Total 240 [+ charge] = [- charge] [+ charge] = [- charge] 0 mV -89 0 mV Setting a membrane potential – questions! ! How many ions must cross the membrane to set up this membrane potential? ! What makes it stop at this potential? Capacitance Capacitance C Coulombs / Volt Farads F C = Q / V Q = C * V I = dQ / dt = C * dV / dt C µ area I { + + + + + + + + + V C C µ 1 / thickness - - - - - - - - - For biological membranes: Specific Capacitance = 1 µF / cm2 How many ions? ! How many ions must cross the membrane of a spherical cell 50 µm in diameter (r = 25 µm) to create a membrane potential of –89 mV? Q (Coulombs) = C (Farads) * V (Volts) Specific Capacitance = 1.0 µFarad / cm2 Surface area = 4 π r2 Faraday’s Constant = 9.648 x 104 Coulombs / mole Avogadro’s # = 6.022 x 1023 ions / mole Calculations Area = 4 π r2 1 Q = C * V 2 = 4 p (25 x 10-4 cm)2 = 78.5 x 10-12 Farads * 0.089 Volts = 7.85 x 10-5 cm2 = 7 x 10-12 Coulombs = 78.5 x 10-6 µFarads = 78.5 x 10-12 Farads ~ 7 x 10-12 Coulombs / 9.65 x 104 Coulombs per mole 3 = 7.3 x 10-17 moles of ions must cross the membrane = 0.073 femptomoles or ~ 44 x 106 ions ~ 1.1 µM change in [ ] Why does it stop? - The Nernst Equation Calculates the membrane potential at which an ion will be in electrochemical equilibrium. At this potential: total energy inside = total energy outside Electrical Energy Term: zFV Chemical Energy Term: RT.ln[Ion] Z is the charge, 1 for Na+ and K+, 2 for Ca2+ and Mg2+, -1 for Cl- F is Faraday’s Constant = 9.648 x 104 Coulombs / mole R is the gas constant = 8.315 Joules / °Kelvin * mole T is the temperature in °Kelvin Nernst Equation Derivation + + zF.Vin + RT.ln [K ]in = zF.Vout + RT.ln [K ]out + + zF (Vin – Vout) = RT (ln [K ]out – ln [K ]in) EK = Vin – Vout + + = (RT/zF).ln([K ]out/[K ]in) + + EK = 2.303 (RT / F) * log10([K ]out/[K ]in) In General: Eion ~ 60 * log ([ion]out / [ion]in) for monovalent cation at 30° Nernst Potential Calculations ! First K and Cl EK = 60 mV log (3 / 90) = 60 * -1.477 = -89 mV ECl = (60 mV / -1) log (120 / 4) = -60 * 1.477 = -89 mV Both Cl and K are at electrochemical equilibrium at -89 mV ! Now for Sodium ENa = 60 mV log (117 / 30) = 60 * 0.591 = +36 mV When Vm = -89 mV, both the concentration gradient and electrical gradient for Na are from outside to inside At Electrochemical Equilibrium: ! The concentration gradient for the ion is exactly balanced by the electrical gradient ! There is no net flux of the ion ! There is no requirement for any energy- driven pump to maintain the concentration gradient Ion Concentrations Na 117 Na 30 K 3 K 90 Cl 120 Cl 4 Anions 0 Anions 116 Total 240 Total 240 [+ charge] = [- charge] [+ charge] = [- charge] 0 mV -89 mV Ion Concentrations Add water – 50% dilution Na 117 Na 30 K 3 K 90 Cl 120 Cl 4 Anions 0 Anions 116 Total 240 Total 240 [+ charge] = [- charge] [+ charge] = [- charge] 0 mV -89 mV Environmental Changes: Dilution Add water – 50% dilution H2O Na 58.5 Na 30 K 1.5 K 90 Cl 60 Cl 4 Anions 0 Anions 116 Total 120 Total 240 [+ charge] = [- charge] [+ charge] = [- charge] -89 mV Environmental Changes: Dilution Add water – 50% dilution Na 58.5 Na 30 K 1.5 K 90 Cl 60 Cl 4 Anions 0 Anions 116 Total 120 Total 240 [+ charge] = [- charge] [+ charge] = [- charge] EK = -107 mV -89 mV ECl = -71 mV Environmental Changes: Dilution Add water – 50% dilution H2O Na 58.5 Na 30 K 1.5 K <90 Cl 60 Cl <4 Anions 0 Anions 116 Total 120 Total <240 [+ charge] = [- charge] [+ charge] = [- charge] EK = -107 mV -89 mV ECl = -71 mV Deviation from the Nernst Equation 0 Resting membrane potentials Squid Axon in real cells deviate from the Curtis and Cole, 1942 Nernst equation, particularly -25 at low external potassium concentrations. PK : PNa : PCl -50 1 : 0.04 : 0.05 The Goldman, Hodgkin, Katz equation provides a better description of membrane -75 potential as a function of Resting Potential (mV) EK (Nernst) potassium concentration in cells. -100 1 3 10 30 100 300 External [K] (mM) The Goldman Hodgkin Katz Equation ⎛ P ∗[K] + P ∗[Na] + P ∗[Cl] ⎞ V 60mV log⎜ K out Na out Cl in ⎟ m = ∗ ⎜ ⎟ ⎝ PK ∗[K]in + PNa ∗[Na]in + PCl ∗[Cl]out ⎠ ! Resting Vm depends on the concentration gradients and on the relative permeabilities to Na, K and Cl. The Nernst Potential for an ion does not depend on membrane permeability to that ion. ! The GHK equation describes a steady-state condition, not electrochemical equilibrium. ! There is net flux of individual ions, but no net charge movement. ! The cell must supply energy to maintain its ionic gradients. GHK Equation: Sample Calculation Suppose PK : PNa : PCl = 1 : 0.1 : 1 ⎛ 3 +11.7 + 4 ⎞ Vm = 60mV ∗ log⎜ ⎟ ⎝ 90 + 3 +120 ⎠ = 60 mV * log (18.7 / 213) = 60 mV * -1.06 = -63 mV Competing Batteries Model Ohm’s Law: V = I * R, so INa = V1 * gNa IK = V2 * gK ICl = V3 * gCl At rest there is no net current INa + IK + ICl = 0 Þ V1 * gNa + V2 * gK + V3 * gCl = 0 Vm = V1 + ENa = V2 + EK = V3 + ECl Þ V1 = Vm – ENa and V2 = Vm – EK and V3 = Vm – ECl ⇒ 0 = (Vm – ENa) * gNa + (Vm – EK) * gK + (Vm – Ecl) * gCl Vm = gNaENa + gKEK + gClECl Iion = gion * (Vm-Eion) gNa + gK + gCl Recording Membrane Potential To here L1 On-Cell Patch Recording (Muscle) 50 µm Forming a Tight Seal Whole-Cell and Outside-Out Patch 10 µm Channels Exist in at least 2 States To here… Current ~3.2 pA = 3.2 x 10-12 Coulombs/sec Charge on 1 ion ~1.6 x 10-19 Culombs Ions per second = 3.2/1.6 x 10^7 ~20 million Exponential Distribution of Lifetimes mean open time = τopen = 1 / a = 1 / closing rate constant Summary: I. Cell membranes form an insulating barrier that acts like a parallel plate capacitor (1 µF /cm2) II. Ion channels allow cells to regulate their volume and to generate membrane potentials III. Only a small number of ions must cross the membrane to create a significant voltage difference ~ bulk neutrality of internal and external solution IV. Permeable ions move toward electrochemical equilibrium • Eion = (60 mV / z) * log ([Ion]out / [Ion]in) @ 30°C • Electrochemical equilibrium does not depend on permeability, only on the concentration gradient Summary (continued): V. The Goldman, Hodgkin, Katz equation gives the steady-state membrane potential when Na, K and Cl are permeable ⎛ P ∗[K] + P ∗[Na] + P ∗[Cl] ⎞ V 60mV log⎜ K out Na out Cl in ⎟ m = ∗ ⎜ ⎟ PK ∗[K]in + PNa ∗[Na]in + PCl ∗[Cl]out ⎝ ⎠ • In this case, Vm does depend on the relative permeability to each ion and there is steady flux of Na and K The cell must supply energy to maintain its ionic gradients .
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