One can change parameters to describe the set of cases
given by Hille (2001)"Ionic Channels of Excitable Membranes" page 19, Figure 1.6
There are two potassium currents, one a leak, the other a gated channel.
A. Start with only the leak current. Set conductance gKstep = 0.
a. Set up plot:
Set X axis to Em with range from -150 to +150 mV
Select Y variable = Jelect. Set Yaxis scaling to autoscale.
b. Run solution:
Question 1. Is Ohms law obeyed exactly over the range of Em?
Show by calculating three points.
B. Set up gKleak using the loops. Put the name gKleak as the first
parameter under Inner loop Configurator. The value of gkLeak
automatically appears next to the right. Under other values type:
@*3
and under #times type:
3
This will give you 3 solutions, multiplying the original value
of gKleak by 3 and then 9 for the second and third loops.
Question 2. What is the slope of the solution for gKleak = 9.
Is is pretty exacly three time that for gKleak=3?
Question 3. What is the Em when the current = zero? This is
the potassium reversal potential in this system, EK.
Compare this to the Nernst potential for K+.
C. Change the intracellular K+ concn to 50 mM. Rerun.
Question 4. What is the reversal potential now?
Question 5. By how many mV did the Ek change for a 10-fold
change of concentration?
D. Reset gKstep (the conductance for the voltage gated channel) back to 0.003.
Run a solution.
Question 6. What is the membrane conductance around -100 mV? Is this
the leak or the channel conductance?
Question 7: What is the conductance when Em = +50 mV? How do you
account for this conductance?
E. Set Zg = 100, giving the channel a valence of 100 charges. Run the solution.
Question 8. There is now a sharp change in the current at a particular
voltage. What voltage? To what is the sharp change due?
F. Reverse the concentrations: Set Ko = 150 mM and Ki = 5 mM. Run a solution.
Question 9: At what voltage is the current reversal? Is this the Nernst?
check the value for VK in the list of output variables.
G. Obtain a set of parameters to produce solutions matching Part D of Hille's
Figure. What is the minimum number of parameters you can change to
reproduce the the 4 curves in the figure. (Ans.Just 2.) Give your values
for the parameters of the 4 curves.
H. Change valence and its sign. Explain results.
I. A time lag, tau represents the (fixed time dconstant for conformational state change. Adjust tau to fir yur expectation for H-H kinetics, using an average value.
How many time constants are required for the lagged conductance to get to within 1% of the max conductance.
How long for the lagged current to reach within 1% of the instantaneous current? Does it ever reach it?
J. EXERCISE: Define an equation for a voltage dependent time constant, for example by makng one in the fashion used by Hodgkin and Huxley (1952d). Insert it in this code, taking care to indicate the time dependencies.
JSim v1.1
/*
MODEL NUMBER: 0106
MODEL NAME: VboltzmannLagged
SHORT DESCRIPTION: Model of Boltzmann gated channel conductance vs. transmembrane
voltage for any ion. Conductance change has fixed timelag, tau.
*/
import nsrunit; unit conversion on;
math VboltzmannLagged {
realDomain t msec; t.min = 0; t.max = 1000; t.delta =0.2;
//PARAMETERS are defined here, with units. // 2 slashes start comments:
real Em(t) mV, // Em , membrane potential
gKleak = 0.0003 siemens/cm^2, // Permeant bkgd leak channel
gKstep = 0.0030 siemens/cm^2, // Max Perm of gated channel; Siemen = 1/ohm
Am = 1 cm^2, // Surface area of membrane
Ko = 5 mM, Ki = 150 mM, // K concns outside and in
R = 8.31441 J*mol^(-1)*K^(-1), // gas constant
Temp = 310.16 K, // 37C
RT = R*Temp, // 19.340*10^6 mmHg*cm^3*mol^(-1)at 37C
Farad = 96485 coulomb*mol^(-1), // Faraday
qe = 1.6022e-19 coulomb, // elementary charge
//Nav = 6.0221e23, // Avagadro's #, no.molecules/mole
kB = 1.3807e-23 volt*coulomb*K^(-1), // Boltzmann's const., J*deg^-1
RToF = (1000 mV/volt)*RT/Farad, // mV, RToF=26.73 mV at 37C,
zg = 10, // # gating charges per channel, various
boltzg = (0.001 volt/mV)*zg*qe/(kB*Temp), // 1/volt, RToF = kB*Temp/qe
valence= 1, // boltcheck = qe/(kB*Temp),
RTozF = RToF/valence, // RTozF = RT/(Farad*z), mV;
loge10 = ln(10), // natural log of 10
VK = loge10*RTozF*log(Ko/Ki), // mV, ENernst for K, mV
EKchan = -40 mV; // mean Em for chan opening, mV
// VARIABLES: with units:
real Jelect(t) amp, // Jelect is electrical current as if gK change were instanteous
Jelecttau(t) amp, // Jelecttau is current when conductance change is slowed
zero(t) = 0, // zero(Em) is for plotting the zero current level
tau = 10 ms, // time constant for conformational change
gK(t) siemens/cm^2, // instantaneous conductance change (leak + channel)
gKlag(t) siemens/cm^2, // lagged conductance change
gKchannel(t) siemens/cm^2,// conductance of Boltzmann channel only, not leak
gchannellag(t) siemens/cm^2, // lagged conductance of Boltzmann channel only, not leak
rate = 1 mV/ms; // rate of change of Em
// INITIAL CONDITIONS:
when (t= t.min) {gchannellag = gKstep/(1+exp(-boltzg*(Em-EKchan))); Em = -150;}
// ODEs:
Em:t = rate;
gKchannel = gKstep/(1+exp(-boltzg*(Em-EKchan))) ; //instantaneous change
gK = gKleak + gKchannel; // instantaneous, as in VBoltzmann
Jelect = Am*(Em-VK)*gK; // instantaneous current, without lag
gchannellag:t= (1/tau)*(gKchannel-gchannellag); //the lag equation
gKlag = gKleak + gchannellag; // gchannel is lagged, not the leak
Jelecttau = Am*(Em-VK)*gKlag; // current with lagged gK change
} // END
/*
DETAILED DESCRIPTION: Provides current voltage relationships for potassium currents
through a passive leak conductance and through a voltage-gated channel
in a membrane. Gives Nernst potentials. Allows changing of gate charges
and mean Em for the gate, EKchan, as well as concentrations and temperature
in order to explore the Boltzmann relationships.
Model is essentially that of Vboltzmann but to demosntrate a time lag in the
conformational change and the conductance, the transmembrame voltage Em is
changed to be a function ot time, so the independent variable is "t". The
lag is not in the Hille description on p19, but later in the book.
Model for monovalent cation in water driven by electrochem gradient
Membrane separates 2 regions of fixed concentrations, Ko and Ki
Solute activities are unity. Currents do not reach S.S. instantaneously
but are delayed by a single exponential lag.
Single channel conductance about 300 pSiemens; 10^6 chan -> 300 uS.
See Notes, section J, within this .proj file for a suggestion to create a tau(t, Em).
SHORTCOMINGS/GENERAL COMMENTS:
KEY WORDS:
Channel, Nernst potential, ionic current, gating current, conformational state,
Tutorial, membrane potential, Boltzmann, electrophysiology
REFERENCES:
Hille B. Ion Channels of Excitable Membranes, Third Edition.
Sunderland, Massachusetts: Sinauer Associates, 2001, 814 pp. page 19, Figure 1.6
REVISION HISTORY:
Written by JBB 5mar11
Revised by BEJ 8mar11 to update comments' format.
COPYRIGHT AND REQUEST FOR ACKNOWLEDGMENT OF USE:
Copyright (C) 1999-2011 University of Washington. From the National Simulation Resource,
Director J. B. Bassingthwaighte, Department of Bioengineering, University of Washington, Seattle WA 98195-5061.
Academic use is unrestricted. Software may be copied so long as this copyright notice is included.
This software was developed with support from NIH grant HL073598.
Please cite this grant in any publication for which this software is used and send an email
with the citation and, if possible, a PDF file of the paper to: staff@physiome.org.
*/