Electrophysiological Modeling of Membranes and Cells

Transcription

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

2 Overview Recapitulation Electrical Modeling of Membranes Cardiac Myocyte Models Modeling of Diseases Numerical Solution Homework Summary Bioeng 6460: Electrophysiology and Bioelectricity - Page 2

3 Electrophysiology of Mammalian Sinoatrial Node Cell Opening of Na channels Depolarization starting at resting voltage (approx. -60 mv) leads to upstroke Autorhythmicity with a frequency of ~3 Hz Bioeng 6460: Electrophysiology and Bioelectricity - Page 3

4 Cellular Electrophysiology: Normal and Failing Simulation of normal and failing human ventricular myocytes with modified Priebe-Beuckelmann model Pathology: Hypertrophy Significant changes of density of proteins relevant for calcium transport: sarcolemmal NaCa-exchanger sarcoplasmic Ca-pump (Sachse, Seemann, Chaisaowong, and Weiss. JCE, 2003) Calcium concentration [µm] Transmembrane voltage [mv] t [ms] Bioeng 6460: Electrophysiology and Bioelectricity - Page 4

5 Markov Modeling of Ion Channels and Mutations Markov models allow reconstruction of single channel behavior to be based upon thermodynamic principals assignment of physical meaning to rate constants Example: State diagram of cardiac sodium channel model O: Open, I: Inactivated, C: Closed (L.A. Irvine, M.S. Jafri, and R.L.Winslow. Biophys J. 1999) Markov models consist of sets of 1st order ODEs Commonly, one channel description of a traditional Hodgkin-Huxley type cell model is substituted by an appropriate Markov model Recently, the inclusion of Markov models in newly developed models increased Bioeng 6460: Electrophysiology and Bioelectricity - Page 5

6 2-State Markov Model do dt = " C #$ O dc dt = $ O # " C O : Probability of channel is in open state C : Probability of channel is in closed state ",$ : Rate coefficients. Function of e.g. V m and ion concentration 1 0 Closed Open α β Bioeng 6460: Electrophysiology and Bioelectricity - Page 6

7 Markov Models for WT and 1795insD Cardiac Na Channels Wild-type Na channel 1795insD Na channel Background mode (normal) Burst mode (pathological) Clancy and Rudy. Circulation 2002;105: Bioeng 6460: Electrophysiology and Bioelectricity - Page 7

8 Timothy Syndrome Bioeng 6460: Electrophysiology and Bioelectricity - Page 8

9 Modeling of Calcium Channel Mutation: Timothy Syndrome Channel Modeling Differences of steady state inactivation between wild type (WT) and mutated channels Integration in Myocyte Model Prediction of course of transmembrane voltage in myocyte Changes dependent on % of mutated channels Significant increase of action potential duration (and intracellular calcium concentrations) Numerical optimization Bioeng 6460: Electrophysiology and Bioelectricity - Page 9

10 Calcium Channel Defect: Timothy Syndrome Significant reduction of voltagedependent inactivation of L-type calcium channels (Ca v 1.2) Characterization with electrophysiological studies in oocytes with normal (WT) and G406R Ca v 1.2 Prolonged QT time (LQT) in patient ECGs Ion channel Prediction of cellular behavior with electrophysiological model of WT and G406R Ca v 1.2 Ventricular myocyte (Splawski, Timothy, Decher, Kumar, Sachse, Pierson, Beggs, Sanguinetti, and Keating, PNAS, 2005) Bioeng 6460: Electrophysiology and Bioelectricity - Page 10 t [s]

11 Timothy Syndrome: Increased Risk Of Arrhythmia Protocol: Stimulus frequency of 3 Hz, pause Transmembranspannung [mv] Calciumkonzentration [nm] Spontaneous opening of sarcoplasmic release channel leads to delayed afterdepolarisation! Bioeng 6460: Electrophysiology and Bioelectricity - Page 11

12 Mutation of Slow Inward Rectifying K-Current I Ks WT KCNQ1 + KCNE1 50 % WT / 50 % mutation KCNQ1 with gain of function mutation + KCNE1 (S140G, V141M) Bioeng 6460: Electrophysiology and Bioelectricity - Page 12

13 Prediction of Ventricular and Atrial Myocyte Behavior Human ventricular myocyte at 1 Hz Modified Iyer-Mazhari-Winslow model APD - short QT syndrome high risk for sudden cardiac death (Hong, Piper, Diaz-Valdecantos, Brugada, Burashnikov, Santos de Soto, Grueso-Montero, Brugada, Sachse, Sanguinetti, and Brugada, Cardiovasc. Res., 2005) Human atrial myocyte at 1, 2, and 3 Hz WT Modified Courtemanche-Ramirez-Nattel model APD - facilitates atrial fibrillation (Seemann, Weiß, Sachse, and Dössel, Proc. CinC, 2004) WT/ S140G Bioeng 6460: Electrophysiology and Bioelectricity - Page 13

14 1st Order ODEs in Electrophysiological Models 1 Closed Open Extracellular ion 1-n α n Membrane V m Intracellular β n n 0 I m = I i + C m d dt V m do dt = " C #$ O dc dt = $ O # " C Bioeng 6460: Electrophysiology and Bioelectricity - Page 14

15 Numerical Solution of ODEs Procedure Discretization: "u "x # u # x Choose appropriate step length x: Distance between x n and x n+1 Step length determines numerical error Numerical Methods Euler Method Runge-Kutta Method 2. Order Runge-Kutta Method 4. Order Richardson-Extrapolation, Bulirsch-Stoer Method Predictor-Corrector Methods Bioeng 6460: Electrophysiology and Bioelectricity - Page 15

16 Euler Method " u " x = f x,u ( ) Finite Difference Approximation u n+1 # u n = f ( x x n+1 # x n,u n ) n hf( x n,u n ) u n+1 u n Rewrite x n x n+1 u n +1 = u n + h f ( x n,u n ) Bioeng 6460: Electrophysiology and Bioelectricity - Page 16 h

17 Euler Method: Example dv m dt = I stim (t) " 1 C m I ion (t,v m ) ion Finite Difference Approximation V m V n +1 " V n t n +1 " t n = I stim (t n ) " 1 C m I ion (t n,v n ) Rewrite $ V n+1 = V n + #t I stim (t n ) " 1 ' & I C ion (t n,v n ) % m ( Bioeng 6460: Electrophysiology and Bioelectricity - Page 17

18 Runge-Kutta Method 2nd Order " u " x = f x,u ( ) Discretization k 1 = hf( x,u n ) k 2 k 1 /2 u n+1 # k 2 = hf x n h,u n k & % 1( $ ' u n x n x n+1 Step h/2 h/2 u n+1 = u n + k 2 Bioeng 6460: Electrophysiology and Bioelectricity - Page 18

19 Runge-Kutta Method 2nd Order: Example dv m dt = I stim (t) " 1 C m I ion (t,v m ) ion Discretization V m $ k 1 = #t I stim (t n ) " 1 ' & I C ion (t n,v n )) % m ( $ k 2 = #t I stim (t n + h 2 ) " 1 I C ion (t n + h m 2,V n + k 1 & % 2 ) ' ) ( Step V n +1 = V n + k 2 Bioeng 6460: Electrophysiology and Bioelectricity - Page 19

20 Runge-Kutta Method 4th Order " u " x = f x,u ( ) Discretization k 1 = h f x,u n # $ # k 3 = hf x n h,u n k & % 2( $ ' k 4 = h f x n + h,u n + k 3 ( ) k 2 = hf% x n h,u + 1 n 2 k 1 ( ) & ( ' Step u n +1 = u n + k k k k 4 6 Bioeng 6460: Electrophysiology and Bioelectricity - Page 20

21 Runge-Kutta Method 4th Order: Example u n+1 k k k k 1 u n x n x n+1 h/2 h/2 Bioeng 6460: Electrophysiology and Bioelectricity - Page 21

22 Homework Analysis of electrophysiological model of a cardiac myocyte Before starting a project, read the following page completely Matlab code for Luo-Rudy 1991 model is available (only code for action potential generation and voltage clamp is necessary) Bioeng 6460: Electrophysiology and Bioelectricity - Page 22

23 Homework Work on your project individually Ask, if you should get stuck or something is unclear via to or Send reports with results via before 29. Sept to Include figures and data tables in text. The material should be sufficient to reconstruct results. Bioeng 6460: Electrophysiology and Bioelectricity - Page 23

24 Homework: Assignment 1 Action potentials only fire if the stimulus rises the transmembrane voltage above a threshold. Determine at least ten combinations of stimulus strength and duration that will just barely illicit a normal action potential and plot these value pairs (strength versus duration) in an X-Y graph. In one case, document in a sequence of overlay plots of the action potentials the progression from subthreshold response to a fully developed action potential as stimulus strength or duration rise. The Map_demo.m file provides a template that can be the starting point for this. What do these results tell you about how the (simulated) cell responds to stimulation? Bioeng 6460: Electrophysiology and Bioelectricity - Page 24

25 Homework: Assignment 2 In order to simulate the experiments performed out by Hodgkin and Huxley on the nerve axon, carry out a set of simulations in which you systematically alter the concentration gradient of sodium (use reasonable values, holding the intracellular concentration constant) and observe the effects on action potential shape and amplitude. Why does the action potential not disappear completely even when the sodium gradient approaches zero? What is providing the inward current in this case? Bioeng 6460: Electrophysiology and Bioelectricity - Page 25

26 Homework: Assignment 3 We discussed briefly in class the need to select a fine enough time resolution for the simulations in order to capture the dynamics of the membrane. With this exercise, we will investigate the effect of the time step on the generation of action potentials with the model. Starting from the existing Map.m file, create a new version that permits you to adjust the time step parameters, starting with the maximum and minimum values. Then manipulate the algorithm used for setting the adjustable time step in whatever ways you can think of and observe the results. Bioeng 6460: Electrophysiology and Bioelectricity - Page 26

27 Summary Recapitulation Electrical Modeling of Membranes Cardiac Myocyte Models Modeling of Diseases Numerical Solution Homework Bioeng 6460: Electrophysiology and Bioelectricity - Page 27