1. BENG 230C Cardiovascular Physiology Ionic Models of Excitable Cells
BENG 230C April 10, Cardiovascular Physiology The Cardiac Action Potential Phone: 1

2. Resting Membrane Potential
An imbalance of total ionic charge leads to a potential difference across a cell membrane:
Resting Membrane Potential
Vm = Vo - Vi
Only a slight imbalance is needed to result in a potential difference
If 1/100,000th of available cytosolic K+ ions crossed the membrane of a sherical cell 10µm DØ, the membrane potential changes by 100 mV
Intracellular Ion
Concentrations
Extracellular Ion
Na+
145 mM K+ 5
Mg2+
1-2
Ca2+ H+
4x10-5
Cl-
110
5 - 15 mM
140
0.5
10-4
7x10-5

3. Diffusion Through a Membrane
Fick’s Law:
This is a chemical analog of Ohm’s law for a membrane of thickness L

4. Electrodiffusion

5. Electrochemical Equilibrium Potential
Nernst Equation:
Vm Potential
Co Concentration of ion outside cell
Ci Concentration of ion inside cell
R Gas Constant = NAkB = 8.314 J.K−1mol−1
z Valence
F Faraday’s constant = ×104 C.mol−1
At 37°C, RT/F = 8.314×310/96485 = 26.7 mV
ln(x) = ln10×log10x = ×log10x
 Vm = -61.5log10([Ci]/[Co])/z

6. Goldman-Hodgkin-Katz Current Equation
Since both the voltage and the concentration gradients influence the movement of ions, this is a simplified version of electrodiffusion obtained by solving the Nernst-Planck equation assuming:
The membrane is homogeneous
The electrical field is constant so that Vm varies linearly across membrane
Ions access the membrane instantaneously from intra- & extracellular solutions
The permeant ions do not interact
ΦS is the ionic flux across the membrane carried by ion S (Amp.cm-2)
PS is the permeability of ion S (m3·s-1)
zS is the valence of ion S
Vm is the transmembrane potential (Volt)
[S]i is the intracellular concentration of S, (mmol·l-1)
[S]o is the extracellular concentration of S, (mmol·l-1)
The GHK current equation describes the current density (A/cm2) carried by an ionic species with concentrations c across a cell membrane as a function of the transmembrane potential Vm.

7. Goldman-Hodgkin-Katz Potential
The GHK potential equation can be derived from the GHK current equation by solving for Vm that makes the total current density summed over the ionic species equal zero. For monovalent anions and cations (z = ±1):
Em Membrane Potential at equilibrium
Pion Permeability of membrane to particular ion
[C] Concentration of a monovalent anion or cation
For example:

8. Equivalent circuit model of the cell membrane
Note that the GHK model does not give a linear I-V relation like the Ohmic model above. While both have the same reversal potential, when we consider two or more ions they no longer agree because the individual currents are no longer zero even though the net current is. In practice, different channels have different IV relations. 8

9. Poisson-Nernst-Planck Equations
GHK assumes Nernst-Planck and a constant electric field but ions moving through the channel affect the local electric field and therefore affaect the ion fluxes. In the case of two species of ions S1 and S2 (e.g. with valences +1 and -1), the potential in the channel must satisfy Poisson’s equation:
q unit electric charge
e dielectric constant of channel medium
Two different limits of the PNP equations lead to either a GHK I-V curve or an Ohmic I-V curve

10. Currents that Cross the Cell Membrane
Current density (µA/cm2) = change of charge density over time
Capacitive Current
Changes in capacitive charge of the membrane over time
Charge is related to potential by the definition of capacitance: q = CmVm
q Charge on one side of the capacitor (coulombs/cm2)
Cm Membrane capacitance per area ( farad/cm2 [farad = coulomb/volt])
Vm Membrane potential (mV)
Conductive
Cytoplasm
Conductive Extracellular Space
Non-conductive, dielectric membrane
Ionic currents
Charges on ions crossing the membrane through ion channels
Calculated by channel gating models

11. Hodgkin-Huxley Nerve Cell Analog
Gating equations:
Nernst potentials:
Hodgkin, A., and Huxley, A. (1952): A quantitative description of membrane current and its application to conduction and excitation in nerve. J. Physiol. 117:500–544
(The fifth in a series of papers)

12. Potassium Conductance

13. Potassium Gating (1)
Potassium conductance was found empirically to have behavior:
Corresponds to four protein subunits with equal open probability.
Exponent represents # of gates per channel
n is the gating variable for the K+ current
n = #open/(#open+#closed)
= fraction of open channels
= probability a channel is open
αn = rate of channel opening
βn = rate of channel closing
αn and βn are f(Vm) found by fitting

14. Potassium Gating (2) (1) Solution of (1) neglecting αn and βn is:
where
no is an initial condition,
and n∞ comes from steady state:
So (1) can be written:

15. Fitting Opening/Closing Rates

16. Sodium Conductance

17. Hodgkin-Huxley Nerve Cell Model
Conductances = 1/resistivity
n, m, and h are gating variables
4 ODEs: membrane potential and 3 gating variables
αi and βi are functions of Vm determined by experimental curve fit

18. Hodgkin-Huxley Model Equations and Parameters

19. Hodgkin-Huxley Model Solutions

20. Noble (1962) Purkinje Fiber Model
Noble D (1962) A modification of the Hodgkin-Huxley equations applicable to Purkinje fiber action and pacemaker potential. J Physiol 160:
From Keener and Sneyd (1998) Mathematical Physiology, Springer

21. FitzHugh-Nagumo 2-Variable Model
FitzHugh (1960) Thresholds and plateaus in the Hodgkin-Huxley nerve equations. J Gen Physiol 43:867:896
From Keener and Sneyd (1998) Mathematical Physiology, Springer

22. Beeler-Reuter Ventricular Action Potential Model (1)
Fast inward Na+ current
Slow inward Ca2+ current
Time & voltage dep. outward K+ current
Time indep. outward K+ current
Early application of Hudgkin-Huxley theory to cardiac cells
Beeler, G.W. and H. Reuter, Reconstruction of the action potential of ventricular myocardial fibres. J Physiol, (1): p
4 ionic membrane currents plus a stimulus current are included
Currents are functions of the independent variables of the ODE set:
6 gating variables
Calcium concentration
Membrane potential
Iion = f (Vm, [Ca]i, x1, m, h, j, d, f)

23. Beeler-Reuter Ventricular Action Potential Model (2)
Action potentials are modeled as a system of time dependent ODE’s
Beeler-Reuter model uses 8 ODE’s found by curve fitting experimental patch-clamp data
Included an additional reactivation variable j in INa
6 ODE’s describe the state of gated ion channels
(y represents 6 gating conductance variables x1, m, h, j, d, and f)
the gating parameters αy and βy are calculated with various sets of constants ci from curve fits

24. Beeler-Reuter Ventricular Action Potential Model (3)
1 ODE describes intracellular Ca2+ concentration
1 ODE describes membrane voltage – Statement of Charge Conservation
The ODE set is integrated over time
Given an initial condition, each time step calculated from previous solution
Rapid rise of AP with fast Na+ kinetics => this is a stiff system
Need numerical solvers that can handle this situation
Common choice is implicit Runge-Kutta with adaptive time stepping
Currents Iion are calculated at each time step from the solution of the independent variables at the current time

25. Beeler-Reuter Ventricular Action Potential Model (4)
From Keener and Sneyd (1998) Mathematical Physiology, Springer

26. Luo-Rudy (1991) Guinea Pig Ventricular Cell Model26

27. The AP, the corresponding intracellular calcium transient, and selected ionic currents that generate the AP and determine its morphology and duration are shown in Figure 2. Once activation threshold is reached, the fast inward sodium current (INa) depolarizes the membrane at a very fast rate (maximum dVm/dt = 393 V/s) and generates the fast AP upstroke. INa reaches a very large peak magnitude of –391 µA/µF in 1 ms and quickly inactivates. When the Vm upstroke reaches about –25 mV, the inward L-type calcium current [ICa(L)] activates and provides a depolarizing current that supports the AP plateau against the repolarizing action of the outward delayed potassium currents IKr (r = rapid) and IKs (s = slow). Note that ICa(L) exhibits a "spike and dome" morphology during the AP (71, 200). Its early peak of –4.92 µA/µF is reached in 2.74 ms and contributes very little to the rising phase of the ventricular AP, which is dominated by INa under normal conditions. It plays an important role in triggering Ca2+ release from the sarcoplasmic reticulum (SR) through the calcium-induced calcium release (CICR) mechanism (89) to generate the calcium transient and initiate contraction. The dome of ICa(L) maintains the plateau; it slowly declines as L-type calcium channels inactivate. The two repolarizing potassium currents, IKr and IKs, gradually increase during the plateau, shifting the balance of currents in the outward direction to repolarize the membrane towards its resting potential. The sodium/calcium exchanger (INaCa) is an electrogenic process with a 3 Na+:1 Ca2+ stochiometry (81, 267). Early during the AP it operates in its "reverse mode" to extrude Na+ from the cell, generating a small outward current (20, 31, 59, 162, 230). It then reverses direction and operates in its "direct mode" to extrude Ca2+, becoming a significant inward current that acts to slow repolarization during the late plateau and prolongs the AP duration (17, 71, 77, 80). Finally, there is a large increase (late peak) of the outward (inward rectifier) potassium current IK1, that dominates the late repolarization phase and the return of the membrane to its resting level. Note that the simulation of Figure 2 does not include the transiet outward current Ito. This current is not expressed in guinea pig ventricle or in endocardial cells of other species. In its presence, a "notch" is created in the AP following its peak upstroke, a process termed "phase 1 repolarization" (see Fig. 9 in Ref. 285). 27

28. Michailova-McCulloch Myocyte Ionic Model
Ica,b
Ica,K
INa,Ca
Ip(Ca)
Ca2+
ATP
ADP
Mg2+
JMgxfer
JMgADPxfer
JMgATPxfer
JCaADPxfer
JCaATPxfer
Na+
Ca2+
Ca2+
T- tubule
Ca2+
Ca2+
Jxfer K+
Ito1
calmodulin
calseq
subspace
ICa K+
IKr
Ca2+ K+
IKs K+
IK1
Ca2+
Jrel K+
IKp
Ca2+
JSR
Ca2+
TRPN
NSR
Jtr
Jup
Ca2+
myofilaments
Sarcoplasmic reticulum
Na+
Na+
Na+ K+
INa
INa,b
INa,K
Luo and Rudy, Circ Res, 1994
Winslow R, Rice J, Jafri S, Marban E, O’Rourke B. Circ Res, 1999
Michailova and McCulloch, Biophys J, August, 2001

29. Ion Channels Na+ Closed Open Inactivated
From: Rudy Y, Silva J (2006) Computational biology in the study of cardiac ion channels and cell electrophysiology. Quarterly Rev Biophys Feb;39(1): Epub 2006 Jul 19.

30. States in the H-H Na Channel Model
Sij denotes state of channel with I open m gates and j open h gates
xij is the fraction of channels in state Sij
From Keener and Sneyd (1998) Mathematical Physiology, Springer

31. Markov State Models
Markov and equivalent Hodgkin–Huxley (HH) models
2-state closed (C) – open (O) model with α and β as transition rates. Equivalent HH-type model has a single gating variable.
(b) 4-state model with two independent transitions. C, Closed; O, open; IC, closed-inactivated IO, open-inactivated. The transition rates α, β between IC and IO and between C and O are identical, as are transition rates γ, δ between C and IC and between O and IO. Thus activation and inactivation transitions are independent and readily modeled using the HH formulation. The probability for current activation is m and the probability that it is not inactivated is h; the open probability is m · h.
Three-state model with dependent transitions from C to O and O to I. There is no HH equivalent because of dependent transitions.
K+ channels have four identical subunits, suggesting four independent identical transitions to the activated state. When all subunits are activated, the channel is open. Assigning an activation gate n, the probability of all four subunits being in the activated position, and thus the probability of the channel being open, is n4.
Biophysical analysis has shown that each of the four voltage sensors in certain K+ channels undergoes two transitions before channel opening. A model describing this gating property is shown; R1 is the rest state, R2 is an intermediate state, and A is the activated state. The channel is open when all four sensors are in the activated (A) position. Because transitions from R2 to A depend on transitions from R1 to R2, a HH analog does not exist.
From: Rudy Y, Silva J. Quarterly Rev Biophys Feb;39(1):

32. Two-State Markov Model
From: Rudy Y, Silva J. Quarterly Rev Biophys Feb;39(1):

33. Markov State Models
From: Rudy Y, Silva J. Quarterly Rev Biophys Feb;39(1):

34. Computing Currents
where for a channel x, gsc,x  is the single channel conductance, n is the number of channels per unit membrane area, O is the probability that a channel occupies the open state, and (Vm − VX) is the driving force. -
From: Rudy Y, Silva J. Quarterly Rev Biophys Feb;39(1):

35. Computing Currents
where for a channel x, gsc,x  is the single channel conductance, n is the number of channels per unit membrane area, O is the probability that a channel occupies the open state, and (Vm − VX) is the driving force. -
From: Rudy Y, Silva J. Quarterly Rev Biophys Feb;39(1):