Voltage Gated Ionic Current

Transcription

1 Eryk Wolski (Free University of Berlin) Supervisor: Wilhelm Huisinga

2 Contents The Membrane Model Current through Resistance Models for Voltage Dependent Gating Basics of Ionic Battery Capacitive current The Volt Clamp

3 The Membrane Model C - Capacitance; R - resistance of ion channel; V r - Nernst Potential; Image Source: Christof Koch Sep-21-97: koch/biophysics-book/

4 Current through Resistance Current caused by Ion i: I ioni = V i V R i = g i (V V i ) sum of the individual ionic currents: I ion = i g i(v V i ) V i - reversal potential at which no current are flowing through channel i. g i - conductance. g i = ḡ i f 0 ḡ i are the conductance if all channels are open. f 0 - fraction of open channels

5 Models for Voltage Dependent Gating Ion Channels Channels are selective for ions of a particular type. Individual channels switch among states very rapidly, often more rapidly than detectable by commonly used electrophysiological methods. Switching probabilities of many channels depend upon membrane potential (voltage dependence) or upon the binding of neurotransmitters to the membrane (ligand dependence)

6 Models for Voltage Dependent Gating A simple molecular Channel Switching between an open O and closed C State Law of mass action. C k+ k O This transition is called a uni-molecular process because involve only one channel molecule. It is Reversible what is indicated by the arrows

7 Models for Voltage Dependent Gating A simple molecular Channel Law of Mass Action The rate of a process is proportional to the product of the concentrations of the molecular species involved in the process [O] = f o = N o /N f o fraction of the open channels. N total number of channels N o number of open channels (similarly, f c and N c refer to the closed state). k + - rate constant. units s

8 Models for Voltage Dependent Gating A simple molecular Channel flux O C = j = k f 0 flux C O = j + = k + (1 f 0 ) Description of the transition from closed to open channels. df 0 dt = j + j = k f 0 + k + (1 f 0 ) = (k + k + )(f 0 f 0 fraction of open channels. k + (k + k + ) )

9 Models for Voltage Dependent Gating A simple molecular Channel Lets define: = 0) at the membra- Fraction of channels open at the equilibrium ( df dt ne potential V. f = k+ k +k + Factor determining the half time of reaction. τ = 1/(k + + k ) The fraction of open channels satisfies the differential equation. df 0 dt = (f 0 f ) τ

10 Models for Voltage Dependent Gating The Voltage dependent channel Show image of Voltage dependent potassium channel. Voltage dependent Ionic channels are composed of proteins with charged amino acid side chains The potential difference across the membrane can influence the rate at which the transition form the open to closed state occur

11 Models for Voltage Dependent Gating Arrhenius Equations Ilia has already introduced the Arrhenius Equation! The membrane potential V contributes to the energy barrier for the transitions O C. k + = k + 0 e αv k = k 0 e βv Where k + 0 and k 0 are independent of V

12 Models for Voltage Dependent Gating The Voltage dependent channel f = k+ k + k + τ = 1 k + + k Substitute: k + = k + 0 e αv into the expression of f and τ. S 0 = 1 α β Determines the steepness of the dependency of f on V k = k 0 e βv V 0 = ln(k 0 /k+ 0 ) β α Determines the voltage at which half of the channels are open

13 Models for Voltage Dependent Gating The Voltage dependent channel f = 1 1+e (V V 0 )/S 0 it is a sigmoidal function τ = e(αv ) 1 1+e (V V 0 )/S 0 k0 Sigmoidal function combined with exponential function. f_inf T(ms) V(mV) V(mV)

14 Models for Voltage Dependent Gating Excitability and Action Potential A system is excitable when small perturbations return to steady state, but larger (i.e, above a threshold V 0 ) perturbations cause large transient deviations away from the steady state. For the voltage dependent ion channel excitability means that the channel are switching from the open to the closed state or vice versa only if the voltage transcends V 0. How fast this switching occurs depends on S

15 Basis of the Ionic Battery Equilibrium Potential - where the electrical and osmotic forces are balanced is given by the Nernst Equation. It is derived from the change in Gibbs Free Energy G. At equilibrium G is zero. Rearranging gives the Nernst Potential. V = V Nernst = RT zf ln [ion] out [ion] in Nernst Potential is determined by the concentration gradient across the membrane. R Gas constant; F Faraday constant; T temperature in kalvin ;z valence of the ion

16 Capacitive Current Given C-capacitance, V m - membrane Potential, Q-Charge. Q = V m C Current capacitive current I = dq dt I c = C dv m(t) dt

17 The Membrane Model Kirchhoffs Law The sum of all currents flowing into or out of any electrical node must be zero. 0 = I cap + I ion + I app Iapp - any current that might be applied. C dv dt = g i (V V i ) + I app

18 Voltage Clamp Image Source: Christof Koch Sep-21-97: koch/biophysics-book/

19 Voltage Clamp Equations: C dv dt = f o ḡ(v V rev ) + I app Differential equitation of membrane potential. We keep the Potential constant by applying I app I app = ḡf o (V V rev ) df 0 dt = (f 0 f ) τ Gating Equation with f = 1 1+e (V V 0 )/S 0 and τ = e V (α+β)/2 2 k 0 + k 0 cosh((v V 0)/2S 0 )

20 Voltage Clamp df 0 dt = (f 0 f ) τ With ḡ = 2,Vo = 25,So = 5;A = (e (V (α+β)/2) )/(2 k 0 + k 0 ) = 10, V = 40, 30, 20, 10, 10 respectively

21 Voltage Clamp I app = ḡf o (V V rev ) With ḡ = 2; Vrev = 65mV and V = 40, 30, 20, 10, 10 and inserting fo we can compute Iapp

22 Voltage Clamp A different channel with S 0 = 1 and V 0 = 15 will react in a different way to the same experimental conditions. I app = ḡf o (V V rev ) With ḡ = 2; Vrev = 65mV and V = 40, 30, 20, 10, 10 and inserting fo we can compute Iapp

23 Thank you