Electrophysiological Modeling of Membranes and Cellsfs/lessons/bioeng6460/lesson1.pdf•Parameterization and optimization of electrical nerve stimulators, ... IbCa ICa,L,Ca,ds INaCa

Bioeng 6460Electrophysiology and Bioelectricity

Electrophysiological Modeling ofMembranes and Cells

Frank B. [email protected]

CVRTI

Bioeng 6460: Electrophysiology and Bioelectricity - Page 2

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Overview

• Motivation and Principles• Electrical Modeling of Membranes

• Resistor-Capacitor Circuit• Nernst Equation• Hodgkin-Huxley Model

• Cardiac Myocyte Models• Beeler-Reuter Model• Luo-Rudy Model• Noble et al. Model

• Summary


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Course Material

In general:

http://www.cvrti.utah.edu/~macleod/bioen/be6460/


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Electrophysiological Models of Cells: Motivation

Description of Insights inPrediction of

Applications• Representation and integration of measurement data• Education and teaching

• Cardiology• Bioengineering• Pharmacology

• Development of diagnostic and therapeutic instrumentation• Parameterization and optimization of electrical nerve stimulators,

defibrillators, and pace maker• electrode material, shape and position • signal

• Evaluation and approval of pharmaceuticals...

electrophysiological phenomena}


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• Hodgkin-Huxley axon membrane giant squid• Noble Purkinje fiber - • Beeler-Reuter ventricular myocyte mammal • DiFrancesco-Noble Purkinje fiber mammal• Earm-Hilgemann-Noble atrial myocyte rabbit• Luo-Rudy ventricular myocyte guinea pig• Demir, Clark, Murphey, Giles sinus node cell mammal• Noble, Varghese, Kohl, Noble ventricular myocyte guinea pig• Priebe, Beuckelmann ventricular myocyte human• Winslow, Rice, Jafri, Marban, O’Rourke ventricular myocyte canine• Seemann, Sachse, Weiss, Dössel ventricular myocyte human…

1952

today

Models describe cells by set of ordinary differential equationsEquations are assigned to a whole cell and/or a small number of its compartments

Models of Cellular Electrophysiology


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Development of Electrophysiological Cell Models

Mathematical model

Measurement data37°

Cell

Space-, voltage- and patch-clampVoltage sensitive dyesChannel blockers,e.g. TTX for Na channels…

Measurement system

Action voltage, membrane currents,conductivities, ion concentration,membrane capacitance,length, volumes…

Commonly, system of 1st orderODEs of Hodgkin-Huxley and Markov type


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Modeling of Membrane: Resistor-Capacitor Circuit

Region i

Membrane

Region e

!

"e

!

"i

!

Cm

!

Rm

!

Cm =Q

Vm

Cm : membrance capacity F[ ]

Q : electrical charge As[ ]

Vm = " i # "e : voltage over membrane V[ ]

d

dtVm =

d

dt

Q

Cm

=Im

Cm

Im : Current through membrane A[ ]

Rm = #Vm

Im

Rm : Resistance of membrane $[ ]


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Modeling of Membrane: Nernst-Equation

Region i

Membrane • permeable for ion type k• homogeneous, planar, infinite

Region e

!

k[ ]i

!

k[ ]e

!

jD,k

!

jE,k

!

"e

!

"i

!

k[ ]i: Concentration of k in region i k[ ]

e: Concentration of k in region e

" i : Potential in region i "e : Potential in region e

jD,k : Ionic current by diffusion jE,k : Ionic current by electrical forces


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Modeling of Membrane: Nernst-Potential

In Equilibrium

Malmivuo, Plonsey3.2.3

!

jE,k + jD,k = 0

Vm,k = " i # "e = #R T

zk Fln

k[ ]i

k[ ]e

k : Ion type

Vm,k : Nernst potential V[ ]

R : Gas constant [J/mol/K]

T : Absolute temperature K[ ]

zk : Valence

F : Faraday$s constant C/mol[ ]

k[ ]i: intracellular concentration of ion type k M[ ]

k[ ]e: extracellular concentration of ion type k M[ ]


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Modeling of Membrane: Nernst Equation - Example

!

Vm,K = "310K

+1

R

FlnK[ ]

i

K[ ]o

= "61mVlogK[ ]

i

K[ ]e

K[ ]i=150M

K[ ]e

= 5.5M

Vm,K = "88mV

Nernst equation explains measured transmembrane voltage of animal and plant cells

For potassium (monovalent cation) at temperatures of 37ºC:

For typical intra- and extracellular concentrations:

Commonly, several types of ions are contributing to transmembrane voltage!


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Hodgkin and Huxley: Measurements

Measurement and mathematical modeling of electrophysiological propertiesof cell membrane (published 1952, Nobel prize 1963)

“Giant” axon from squid with ~0.5 mm diameter

Techniques• Space clamp• Voltage clamp

Simplifications:

Extracellular space:Salt solution

Semi permeable membrane

Intracellular space:Axoplasm

Na K L Na: Sodium ionsK: Potassium ionsL: Other ions (primarily chlorine)


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Hodgkin-Huxley Model: Equivalent Circuit Diagram

IC INa IK IL

Im

Cm GNa GK GL

+ - -- + +

Φi

Φe

Extracellular space

Intracellular space

Vm= Φi - Φe

VNa Vk VL

Membran

GNa, GK, GL Membrane conductivity of Na, K and other ions [S/cm2]

INa, IK, IL Currents of Na, K and other ions [mA/cm2]

VNa, VK, VL Nernst voltages of Na, K and other ions [mV]

Cm , Im , Vm Membrane capacitor [F/cm2], current [mA /cm2] and voltage [mV]

!

Im

= Cm

dVm

dt+ V

m"V

Na( )G

Na+ V

m"V

K( )G

K+ V

m"V

L( )G

L


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Hodgkin-Huxley Model: Principles

!

GNa

=INa

Vm"V

Na

GK

=IK

Vm"V

K

GNa

=IL

Vm"V

L

!

GNa

=INa

VNa

GK

=IK

VK

GNa

=IL

VL

Ohm’s law:

Nernst voltages for correction!


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Hodgkin-Huxley Model: Constants

Voltages are related to resting voltage Vr

Conductivity and capacitance are related to membrane area

Relative Na voltage Vr -VNa -115 mV

Relative K voltage Vr - Vk 12 mV

Relative voltage of Vr - VL -10.6 mVother ions

Membrane capacitance Cm 1 µF/cm2

Maximal conductivity of Na GNa max 120 mS/cm2

Maximal conductivity von K GK max 36 mS/cm2

Conductivity for other ions GL 0.3 mS/cm2

All ion channels open}


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Hodgkin-Huxley Model: ODEs Describe Conductivities

!

GNa

= GNa max

m3

hdm

dt= "

m1#m( ) # $

mm

dh

dt= "

h1#h( ) # $

hh

GK

= GKmaxn4

dn

dt= "

n1#n( ) # $

nn

GL

= const

"m

=0.1 25 #V%( )e0.1 25#V%( )

# 1

1

ms$m

=4

eV%/18

1

ms

"h

=0.07

eV%/ 20

1

ms$h

=1

e0.1 30#V%( ) + 1

1

ms

"n

=0.01 10 #V%( )e0.1 10#V%( ) #1

1

ms$n

=0.125

eV%/ 80

1

ms

Voltage and time-dependent

Sodium current}Potassium current}Current by other ions}


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Hodgkin-Huxley Model: Simulation of Voltage Clamp Measurements

http://www.bbb.caltech.edu/GENESIS


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Hodgkin-Huxley Model: Simulation of Voltage Clamp Measurements

}}

Na

K!

GNa

=GNa,max

m3h

GK

=GK,max

n4


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Hodgin-Huxley Model: Activation by Hyperpolarization


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Hodgkin-Huxley Model: Stimulus After Refractory Period


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Hodgkin-Huxley: Stimulus During Refractory Period


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Schematic Electrophysiology of Cardiac Myocytes

Depolarisation:After reaching of threshold voltage:

Short term increase of gNa+

Plateau phase:Fast increase followed by slow decrease of gCa2+

Fast decrease followed by slow increase of gK+

Repolarisation:Return of gNa+, gK+ and gCa2+ to resting valuesPartly, gK+ increase leads to hyperpolarisation

[Na]Extracellular spaceMembrane

Intracellular space

[K] [Ca]

[Ca][K][Na]

Time and voltagedependent, ion selective ion channels

Res

ting

volta

ge

Actio

n v

olta

ge

Upstroke

Fast Nachannel

Slow KchannelsCa

channels


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Beeler-Reuter Model 1977

Electrophysiological model of mammalian ventricular myocyte membrane

Parameterization by measurement with clamp techniques

OutsideMembraneInside

INaI ICa IK INaO

INa: Inward current of sodium

IS: Inward current (primarily calcium)

IK1: Outward current of potassium

IX1: Outward current (primarily potassium)

Time and voltage dependent}Time independent}

} Time and voltagedependent

Vm

[Ca2+]i


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Beeler-Reuter: Equations for Currents

!

iX1 = X1 0.8e

0.04 Vm +77( ) "1

e0.04 Vm + 35( )

#

$ %

&

' ( iNa = gNa m

3h j + gNaC( ) Vm " ENa( )

iK1 = 0.354e

0.04(Vm + 85) "1

e0.08(Vm + 53)

+ e0.04 Vm + 53( ) +

0.2 Vm + 23( )1" e"0.04 Vm + 23( )

#

$ %

&

' ( is = gs d f Vm " Es( )

Es = "82.3 "13.0287 ln Ca2+[ ]

iENa = 50 mV

iX1,iNa,iK1,is: Stromdichten [µA / cm2]

Vm: Transmembranspannung [mV]

Es,ENa : Nernst - Spannung für an is beteiligte Ionen bzw. Natrium [mV]

gs: Leitfähigkeit für an is beteiligte Ionen [1/ cm2 / k)]

gNa: Leitfähigkeit für Natrium bei vollständig offenen Natriumkanälen [1/ cm2

/ k)]

gNaC : Leitfähigkeit für Natrium bei geschlossenen Natriumkanälen [1 / cm2 / k)]

d,m,X1: Aktivierungsparameter von Ionenkanälen als Funktion von t und Vm

f ,h, j: Inaktivierungsparameter als Funktion von t und Vm

Ca2+[ ]i: Konzentration von Calcium [mMol / cm3]

Current densities [µA/cm2]Transmembrane voltage [mV]

is and sodium Nernst voltages [mV]

Conductivity [mS/cm2]

Conductivity of open Na channels [mS/cm2]

Concentration of intracellular calcium [mmol/cm3]

Conductivity of closed Na channels [mS/cm2]Activation state (described by ODE)

Inactivation state (described by ODE)


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Beeler-Reuter: Equations for Currents and Concentrations

!

dVm

dt= "

1

Cm

iK1 + i

X1 + iNa

+ iCa

+ iexternal

( )

d Ca2+[ ] i

dt= "10"7

is+ 0.07(10"7 " Ca

2+[ ] i)

Cm

= 1µF

cm2 : Membrankapazität pro Fläche

Results of simulations for stimulus frequency of 1 Hz

Membrane capacitanceper area


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Luo-Rudy Models

Electrophysiological model of ventricular myocyte of guinea pig

Parameterization by measurement with clamp techniques • Phase I: 1991• Phase II: 1994 • ...

Motivation

• Improved measurement techniques (e.g. single ion channel measurements)• Deficits of Beeler-Reuter, e.g. Fixed extracellular ion concentrations Neglect of calcium transport and buffering in sarcoplasmic reticulum Neglect of cell geometry …


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Luo-Rudy Model 1994

Myoplasm

Extracellular space

Sarcoplasmic reticulum

ICa,b ICa INaCa Ip(Ca)

Pump

IUp Ileak

Ins(Ca) IKp IK1 IK INaK INa INa,b

Irel

Itr

Geometrycylinder-shapedlength: 100 µmradius: 11 µm


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Noble-Kohl-Varghese-Noble Model 1998

Mathematical description of ionic currents and concentrations, transmembrane voltage, and conductivities of guinea-pig ventricular myocytes

Myoplasma

extracellular space

Sarcoplasmic reticulum

IbCa ICa,L,Ca,ds INaCa INaCa,ds

pump

IUp

Ip,Na Ib,Na ICa,L,Na INa,stretch

INaK

IK1 Ib,K

Irel

Itr

Geometrycylinder-shapedlength: 74 µmradius: 12 µm

Mechano-electricalfeedback bystretch activatedion channels

Neural influence by transmitter activatedion channels etc.

INa

ICa,L,Ca

ICa,L,KIKIK,stretch

ICa,stretch

IK,ACh

TroponinItrop


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Noble-Kohl-Varghese-Noble Model 1998

Cal

cium

con

cent

ratio

n [m

M]

Results of simulations for stimulus frequency of 1 Hz


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Prediction of Mechano-Electrical Feedback

Reduction of actionpotential duration (APD)by strain

Increase of restingvoltage by strain

SL: sarcomere length


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Prediction: Triggering of Action Potential by Strain

t=1 s: Electrical stimulus t=2 s: Strain for 5 ms

Triggering of action potential for SL>2.7 µm

Sarcomerelength [µm]


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Summary

• Motivation and Principles• Electrical Modeling of Membranes

• Resistor-Capacitor Circuit• Nernst Equation• Hodgkin-Huxley Model

• Cardiac Myocyte Models• Beeler-Reuter Model• Luo-Rudy Model• Noble et al. Model

Documents

Electrophysiological Modeling of Membranes and Cellsfs/lessons/bioeng6460/lesson1.pdf•Parameterization and optimization of electrical nerve stimulators, ... IbCa ICa,L,Ca,ds INaCa