Electrophysiological Modeling of Membranes and Cells

Transcription

1 Bioeng 6460 Electrophysiology and Bioelectricity Electrophysiological Modeling of Membranes and Cells Frank B. Sachse

2 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 Bioeng 6460: Electrophysiology and Bioelectricity - Page 2

3 Course Material In general: Bioeng 6460: Electrophysiology and Bioelectricity - Page 3

4 Electrophysiological Models of Cells: Motivation Description of Insights in Prediction of } electrophysiological phenomena 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... Bioeng 6460: Electrophysiology and Bioelectricity - Page 4

5 Models of Cellular Electrophysiology 1952 today 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 Models describe cells by set of ordinary differential equations Equations are assigned to a whole cell and/or a small number of its compartments Bioeng 6460: Electrophysiology and Bioelectricity - Page 5

6 Development of Electrophysiological Cell Models Cell 37 Measurement system Measurement data Space-, voltage- and patch-clamp Voltage sensitive dyes Channel blockers, e.g. TTX for Na channels Action voltage, membrane currents, conductivities, ion concentration, membrane capacitance, length, volumes Mathematical model Commonly, system of 1st order ODEs of Hodgkin-Huxley and Markov type Bioeng 6460: Electrophysiology and Bioelectricity - Page 6

7 Modeling of Membrane: Resistor-Capacitor Circuit C m = Q V m C m : membrance capacity F [ ] " i Region i Q : electrical charge [ As] [ ] V m = " i # " e : voltage over membrane V C m R m Membrane d dt V m = d dt Q C m = I m C m [ ] I m : Current through membrane A " e Region e R m = # V m I m R m : Resistance of membrane $ [ ] Bioeng 6460: Electrophysiology and Bioelectricity - Page 7

8 Modeling of Membrane: Nernst-Equation [ k] i Region i [ k] e " i " e j D,k j E,k Membrane permeable for ion type k homogeneous, planar, infinite Region e [ k] i : Concentration of k in region i k " i : Potential in region i " e : Potential in region e [ ] e : Concentration of k in region e j D,k : Ionic current by diffusion j E,k : Ionic current by electrical forces Bioeng 6460: Electrophysiology and Bioelectricity - Page 8

9 Modeling of Membrane: Nernst-Potential In Equilibrium j E,k + j D,k = 0 Malmivuo, Plonsey [ ] i [ ] e V m,k = " i # " e = # R T z k F ln k k k : Ion type [ ] V m,k : Nernst potential V R : Gas constant [J/mol/K] [ ] T : Absolute temperature K z k : 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] Bioeng 6460: Electrophysiology and Bioelectricity - Page 9

10 Modeling of Membrane: Nernst Equation - Example Nernst equation explains measured transmembrane voltage of animal and plant cells For potassium (monovalent cation) at temperatures of 37ºC: V m,k = " 310K +1 [ K] i =150M [ K] e = 5.5M R F ln K K [ ] i [ = "61mVlog K ] i [ ] o [ K] e For typical intra- and extracellular concentrations: V m,k = "88mV Commonly, several types of ions are contributing to transmembrane voltage! Bioeng 6460: Electrophysiology and Bioelectricity - Page 10

11 Hodgkin and Huxley: Measurements Measurement and mathematical modeling of electrophysiological properties of 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 ions K: Potassium ions L: Other ions (primarily chlorine) Bioeng 6460: Electrophysiology and Bioelectricity - Page 11

12 Hodgkin-Huxley Model: Equivalent Circuit Diagram Φ i V m = Φ i - Φ e Intracellular space I m I C I Na I K I L C m G Na G K G L V Na V k + V L + M e m b r a n G Na, G K, G L Membrane conductivity of Na, K and other ions [S/cm 2 ] I Na, I K, I L Currents of Na, K and other ions [ma/cm 2 ] V Na, V K, V L Nernst voltages of Na, K and other ions [mv] Φ e Extracellular space C m, I m, V m Membrane capacitor [F/cm 2 ], current [ma /cm 2 ] and voltage [mv] I m = C m dv m dt + ( V m " V Na )G Na + ( V m " V K )G K + ( V m " V L )G L Bioeng 6460: Electrophysiology and Bioelectricity - Page 12

13 Hodgkin-Huxley Model: Principles Ohm s law: G Na = I Na V Na G K = I K V K G Na = I L V L Nernst voltages for correction! G Na = I Na V m " V Na G K = I K V m " V K G Na = I L V m " V L Bioeng 6460: Electrophysiology and Bioelectricity - Page 13

14 Hodgkin-Huxley Model: Constants Voltages are related to resting voltage V r Conductivity and capacitance are related to membrane area Relative Na voltage V r -V Na -115 mv Relative K voltage V r - V k 12 mv Relative voltage of V r - V L mv other ions Membrane capacitance C m 1 µf/cm 2 Maximal conductivity of Na G Na max 120 ms/cm 2 Maximal conductivity von K G K max 36 ms/cm 2 } All ion channels open Conductivity for other ions G L 0.3 ms/cm 2 Bioeng 6460: Electrophysiology and Bioelectricity - Page 14

15 Hodgkin-Huxley Model: ODEs Describe Conductivities G Na = G Na max m 3 h dm dt = " m ( 1 # m) # $ m m dh dt = " h 1 # h dn G K = G K max n 4 dt = " n 1 # n G L = const ( ) # $ h h ( ) # $ n n } } } Sodium current Potassium current Current by other ions ( ) # V% 1 " m = #V% e ( ) # 1 ms " h = e V%/ 20 ms 0.01( 10 # V% ) 1 " n = #V% e ( ) #1 ms $ h = $ m = 4 e V%/18 1 ms #V% e ( ) + 1 $ n = e V%/ 80 ms 1 ms Voltage and timedependent Bioeng 6460: Electrophysiology and Bioelectricity - Page 15

16 Hodgkin-Huxley Model: Simulation of Voltage Clamp Measurements Bioeng 6460: Electrophysiology and Bioelectricity - Page 16

17 Hodgkin-Huxley Model: Simulation of Voltage Clamp Measurements G Na = G Na,max m 3 h G K = G K,max n 4 } Na } K Bioeng 6460: Electrophysiology and Bioelectricity - Page 17

18 Hodgin-Huxley Model: Activation by Hyperpolarization Bioeng 6460: Electrophysiology and Bioelectricity - Page 18

19 Hodgkin-Huxley Model: Stimulus After Refractory Period Bioeng 6460: Electrophysiology and Bioelectricity - Page 19

20 Hodgkin-Huxley: Stimulus During Refractory Period Bioeng 6460: Electrophysiology and Bioelectricity - Page 20

21 Time and voltage dependent, ion selective ion channels 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 values Partly, gk+ increase leads to hyperpolarisation Action voltage Extracellular space [Na] [K] [Ca] Membrane Intracellular space [Na] [K] [Ca] Resting voltage Schematic Electrophysiology of Cardiac Myocytes Upstroke Fast Na channel Ca channels Slow K channels Bioeng 6460: Electrophysiology and Bioelectricity - Page 21

22 Beeler-Reuter Model 1977 Electrophysiological model of mammalian ventricular myocyte membrane Parameterization by measurement with clamp techniques V m I NaI I Ca I K I NaO [Ca 2+ ] i Outside Membrane Inside I Na : Inward current of sodium I S : Inward current (primarily calcium) I K1 : Outward current of potassium I X1 : Outward current (primarily potassium) Time and voltage }dependent } } Time Time independent and voltage dependent Bioeng 6460: Electrophysiology and Bioelectricity - Page 22

23 Beeler-Reuter: Equations for Currents ( ) "1 i X1 = X1 0.8 e0.04 V m +77 # & % e 0.04( V m $ + 35) ( ' i Na = g Na m 3 h j + g NaC 4e 0.04( V m # + 85) "1 i K1 = 0.35 e 0.08(V m + 53) + e 0.04 ( V m + 53 ) ( V + 23 m ) & % 1" e "0.04 ( V m + 23 ) ( $ ' i s = g s d f V m " E s ( )( V m " E Na ) ( ) E s = "82.3 " ln [ Ca 2+ ] i E Na = 50 mv i X1,i Na,i K1,i s : Stromdichten Current densities [µa / [µa/cm 2 ] 2 ] V m : Transmembranspannung Transmembrane voltage [mv] E s,e Na : Nernst i s and -sodium Spannung Nernst für an voltages i s beteiligte [mv] Ionen bzw. Natrium [mv] g s : Leitfähigkeit Conductivity für [ms/cm an i s beteiligte 2 ] Ionen [1/ cm 2 / k)] g Na : Leitfähigkeit Conductivity für of open Natrium bei channels vollständig [ms/cm offenen 2 ] Natriumkanälen [1/ cm 2 / k)] g NaC : Leitfähigkeit Conductivity für of Natrium closed Na bei channels geschlossenen [ms/cm Natriumkanälen 2 ] [1 / cm 2 / k)] d,m,x1: Aktivierungsparameter Activation state (described von Ionenkanälen by ODE) als Funktion von t und V m f,h, j: Inaktivierungsparameter Inactivation state (described als Funktion by ODE) von t und V m [ Ca 2+ ] i : Konzentration von Calcium [mmol / cm 3 ] Concentration of intracellular calcium [mmol/cm 3 ] Bioeng 6460: Electrophysiology and Bioelectricity - Page 23

24 Beeler-Reuter: Equations for Currents and Concentrations dv m dt = " 1 ( i C K1 + i X1 + i Na + i Ca + i external ) m d[ Ca 2+ ] i dt = "10 "7 i s (10 "7 " [ Ca 2+ ] i ) C m = 1 µf : Membrankapazität Membrane capacitance pro Fläche cm 2 per area Results of simulations for stimulus frequency of 1 Hz Bioeng 6460: Electrophysiology and Bioelectricity - Page 24

25 Luo-Rudy Models Electrophysiological model of ventricular myocyte of guinea pig Parameterization by measurement with clamp techniques Phase I: 1991 Phase II: 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 Bioeng 6460: Electrophysiology and Bioelectricity - Page 25

26 Luo-Rudy Model 1994 Extracellular space Pump I Ca,b I Ca I NaCa I p(ca) I Up I leak I rel Sarcoplasmic reticulum Geometry cylinder-shaped length: 100 µm radius: 11 µm I tr Myoplasm I ns(ca) I Kp I K1 I K I NaK I Na I Na,b Bioeng 6460: Electrophysiology and Bioelectricity - Page 26

27 Noble-Kohl-Varghese-Noble Model 1998 Mathematical description of ionic currents and concentrations, transmembrane voltage, and conductivities of guinea-pig ventricular myocytes extracellular space pump I Ca,stretch I bca I Ca,L,Ca I Ca,L,Ca,ds I NaCa I NaCa,ds I NaK Geometry cylinder-shaped length: 74 µm radius: 12 µm Myoplasma I Up Sarcoplasmic reticulum I rel Troponin Mechano-electrical feedback by stretch activated ion channels I tr I trop Neural influence by transmitter activated ion channels etc. I Na I p,na I b,na I Ca,L,Na I Na,stretch I K,stretch I K I K1 I Ca,L,K I b,k I K,ACh Bioeng 6460: Electrophysiology and Bioelectricity - Page 27

28 Noble-Kohl-Varghese-Noble Model 1998 Calcium concentration [mm] Results of simulations for stimulus frequency of 1 Hz Bioeng 6460: Electrophysiology and Bioelectricity - Page 28

29 Prediction of Mechano-Electrical Feedback Reduction of action potential duration (APD) by strain Increase of resting voltage by strain SL: sarcomere length Bioeng 6460: Electrophysiology and Bioelectricity - Page 29

30 Prediction: Triggering of Action Potential by Strain Sarcomere length [µm] t=1 s: Electrical stimulus t=2 s: Strain for 5 ms Triggering of action potential for SL>2.7 µm Bioeng 6460: Electrophysiology and Bioelectricity - Page 30

31 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 Bioeng 6460: Electrophysiology and Bioelectricity - Page 31