Modeling. EC-Coupling and Contraction
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1 Bioeng 6460 Electrophysiology and Bioelectricity Modeling of EC-Coupling and Contraction Frank B. Sachse
2 Overview Recapitulation Group Work Excitation-Contraction Coupling Hill s Muscle Model Sarcomere Modeling Coupling to Electrophysiological Models 3D Simulations Summary Bioeng 6460: Electrophysiology and Bioelectricity - Page 2
3 Group Work: Most Important States of Force Model Work in pairs of 2 Time 5 min Several proteins are involved in force development changing their configuration depending on e.g. Ca concentration and other proteins configurations. Which states would you consider most important for modeling? Bioeng 6460: Electrophysiology and Bioelectricity - Page 3
4 Models of Force Development 1938 today Hill Skeletal muscle Frog Huxley Skeletal muscle - Wong Cardiac muscle - Panerai Papillary muscle Mammalian Peterson, Hunter, Berman Papillary muscle Rabbit Landesberg, Sideman Skinned cardiac - myocytes Hunter, Nash, Sands Cardiac muscle Mammalian Noble, Varghese, Kohl, Noble Ventricular muscle Guinea pig Rice, Winslow, Hunter Papillary muscle Guinea pig Rice, Jafri, Winslow Cardiac muscle Ferret Glänzel, Sachse, Seemann Ventricular myocytes Human Bioeng 6460: Electrophysiology and Bioelectricity - Page 4
5 Hill s Model of Muscle Contraction (1924/1938) Measurement Muscle is fixed at length l 0 Electrical stimulation Isometric, tetanus with max. mechanical tension P 0 Release of fixation Force F is applied with F=mg g: Gravitational constant m: Mass Measurement of time t, when muscle passes mark at length l Calculation of velocity Mark l M u s c l e l 0 Fixation v = l 0! l t m F Bioeng 6460: Electrophysiology and Bioelectricity - Page 5
6 Relationship between Mass and Contraction Velocity Contraction of frog skeletal muscle Contraction velocity v [m/s] measured value Mass [g] Bioeng 6460: Electrophysiology and Bioelectricity - Page 6
7 Modeling of Muscle Contraction ( v + b) ( P + a) = b( P 0 + a) P: Spannung Tension des of muscle Muskels " # $ N m 2 P 0 : Maximale Spannung tension of des muscle Muskels v: Kontraktionsgeschwindigkeit Contraction velocity % &' " m% # $ s &' " # $ N m 2 a,b: Konstanten Constants depending abhängig von l 0 l,, temperature, Temperatur, Ca Concentration -Ionenkonzentration, of Ca,... % & ' m Elastic Element Contractile Element Bioeng 6460: Electrophysiology and Bioelectricity - Page 7
8 Model Extension: Hill s 3-Element Model (1970) Limitations of Hill s Model only force-velocity relationship only tetanized muscle, no information of partial or not stimulated muscle only serial elastic element only quick responses Hills 3-Element Model Inclusion of parallel elastic element Elastic parallel element Elastic serial element Contractile element m Bioeng 6460: Electrophysiology and Bioelectricity - Page 8
9 Modeling of Cellular Force Development [Ca] i /µm F/F maxi t [s] Intracellular concentration of calcium Calculated with electrophysiological cell models t [s] Processes in myofilaments State of troponins, tropomyosins and cross bridges t [s] Force Development System of coupled ordinary differential equations System of coupled ordinary differential equations Bioeng 6460: Electrophysiology and Bioelectricity - Page 9
10 4-State Model of Force Development: State Diagram N0 g N1 Force k l (Ca i ) k -1 (Ca i ) k m (Ca i ) k m (Ca i ) T0 Ca bound f g T1 Ca bound Force Description by set of 1st order ODEs Transfer of states N0, N1, T0, and T1 is dependent of rate coefficients Rate coefficients are partly function of intracellular calcium [Ca 2+ ] i (Model 1 of Rice 1999 et al./ Landesberg et al.1994) Bioeng 6460: Electrophysiology and Bioelectricity - Page 10
![4-State Model of Force Development Several states contribute to force development: F = " F : T1+ N1 F max Normalized force [N/N] F max : Maximal simulated force " : Myofilament overlap function](/docs-images/79/79909577/images/11-5.jpg "Force is dependent on overlap of actin and myosin filaments. Frank-Starling Mechanism Force ~ initial length Diastolic volume ~ cardiac output!")
11 4-State Model of Force Development Several states contribute to force development: F = " F : T1+ N1 F max Normalized force [N/N] F max : Maximal simulated force " : Myofilament overlap function Force is dependent on overlap of actin and myosin filaments. Frank-Starling Mechanism Force ~ initial length Diastolic volume ~ cardiac output! Bioeng 6460: Electrophysiology and Bioelectricity - Page 11
12 4-State Model of Force Development: Matrix Notation # " % % " t % $ N 0 T 0 T 1 N 1 & # ( % ( = % ( % ' $ )K l k l 0 g 0 + g 1 V K l )f ) k l g 0 + g 1 V 0 0 f )g 0 ) g 1 V ) k )l K l 0 0 k )l )g 0 ) g 1 V ) K l &# (% (% (% ' $ N 0 T 0 T 1 N 1 & ( ( ( ' Ca i " T0 T1 Transfer Rate constants: coefficients: K l,k l, f, g 0, g 1,k ) l und V States of Troponin Calcium Cross bridges N 0 unbound weak T 0 bound weak T 1 bound strong N 1 unbound strong t [s] Numerical solution e.g. with Euler- and Runge-Kutta-methods Bioeng 6460: Electrophysiology and Bioelectricity - Page 12
13 3rd Model of Rice 1999 et al.: State Diagram States of Troponin States of Tropomyosin and cross bridges (XB) T unbound N0 0 XB g N1 1 XB k on k off k 1 (TCa,SL) k -1 k -1 k 1 (TCa,SL) TCa bound P0 Permissive 0 XB f g P1 Permissive 1 XB Bioeng 6460: Electrophysiology and Bioelectricity - Page 13
14 3rd Model of Rice 1999 et al.: Matrix Notation y =! T $ # & " TCa% # y = "k on[ca] % $ k on [Ca] k off "k off & ( y ' States of Troponin Transitions x =! # # # " N 0 P 0 P 1 N 1 $ & & & % x = ( ) 0 g0 ( )! f g0 0 ( ) k 1 ( )!g! k 1 "!k 1 k!1 TCa,SL $ k 1 -k!1 TCa,SL $ $ $ # 0 f!g! k!1 TCa,SL 0 0 k!1 TCa,SL % ' ' x ' ' & States of Tropomyosin and cross bridges Transitions Rice et al. Model 3, 1999 Bioeng 6460: Electrophysiology and Bioelectricity - Page 14
15 Force Development for Different Static Sarcomere Lengths (SL) Force normalized with maximal force [N/N] time [s] Bioeng 6460: Electrophysiology and Bioelectricity - Page 15
16 Model of Glänzel et al. 2002: Activation [Ca] I, #, XB T TCa TCa, TM on TM off TM on T Troponin TCa Troponin with bound calcium TM off Tropomyosin in nonpermissive state TM on Tropomyosin in permissive state # Stretch XB Cross-bridges [Ca] I Free calcium in intracellular fluid Ca-binding to Troponin C Shifting of Tropomyosin (Glänzel, Diploma Thesis, IBT, Universität Karlsruhe (TH), 2002, Sachse, Glänzel, and Seemann, IJBC, 2003, Sachse, Glänzel, and Seemann, LNCS 2674, 2003) Cooperativity mechanisms XB-TN TM-TM Bioeng 6460: Electrophysiology and Bioelectricity - Page 16
17 Model of Glänzel et al. 2002: Crossbridge Cycling v M ATP A M ATP v M v A M v A M ADP A Actin M Myosin ATP Adenosine triphospate ADP Adenosine diphosphate P i Phosphate # Stretch v Velocity of filaments XB Cross-bridges weakly coupled strong binding M ADP P i v M ADP v A M ADP Cooperativity mechanism v A M ADP P i TM on, #, XB A M ADP P i XB-XB Bioeng 6460: Electrophysiology and Bioelectricity - Page 17
18 Reconstruction of Static Measurements Study Species: rat Ventricular myocytes Resting length ~ 2µm Protocol Intracellular calcium concentration [Ca 2+ ] i controlled Stretch varied Isometric measurement Normalized Force [N/N] Normalized Force [N/N] Calcium concentration [µm] Measurement ter Keurs et al. 00 Normalized Force [N/N] Normalized Force [N/N] Calcium concentration [µm] Model of Glänzel et al. 02 Calcium concentration [µm] Calcium concentration [µm] Model Peterson et al. 91 3rd model Rice et al. 99 Bioeng 6460: Electrophysiology and Bioelectricity - Page 18
19 Reconstruction of Length Switch Experiments Study Species: rabbit Ventricular myocytes Length Switch Experiment Stretch of 4.6% in 10 ms Return to original configuration in 10 ms Measurement Peterson et al. 91 Normalized Force [N/N] Normalized Force [N/N] Normalized Force [N/N] t [s] Model of Glänzel et al. 02 t [s] t [s] Model Peterson et al. 91 3rd model Rice et al. 99 Bioeng 6460: Electrophysiology and Bioelectricity - Page 19
20 Coupling of Force with Electrophysiological Models: Basics Noble-Varghese-Kohl-Noble model Rice-Winslow-Hunter model 1 Stimulus at t=35 ms Sarcomere stretch # Bioeng 6460: Electrophysiology and Bioelectricity - Page 20
21 Coupling of Force with Electrophysiological Models Luo-Rudy model (Guinea pig ventricular) Rice-Winslow-Hunter model 3 (rabbit ventricular) Stimulus at t=35 ms Sarcomere stretch # Species don t fit Bioeng 6460: Electrophysiology and Bioelectricity - Page 21
22 Coupling of Force with Electrophysiological Models Noble-Varghese-Kohl-Noble model (Guinea pig ventricular) Rice-Winslow-Hunter model 3rd (rabbit ventricular) Stimulus at t=35 ms Sarcomere stretch # Differences in force development due to differences of Ca transient Bioeng 6460: Electrophysiology and Bioelectricity - Page 22
23 Coupling of Force with Electrophysiological Model: Human Priebe-Beuckelmann model (human ventricular) Glänzel et al. model (human ventricular) Stimulus at t=35 ms with frequency Sarcomere stretch # Validation of hybrid models with measured data necessary! Bioeng 6460: Electrophysiology and Bioelectricity - Page 23
24 Modeling Force Development with Cellular Automaton Anatomical Model Physiological Parameters Cellular Automaton Force Transmembrane voltage Calcium concentration F/F maxi t [s] Bioeng 6460: Electrophysiology and Bioelectricity - Page 24
25 Parameters for Simulation: Lookup Tables Known per volume element dv and for time t dv Force (t S ) (instead of transmembrane voltage) Refractory period (t S ) Stimulus Time since activation t S Stimulus frequency f Tissue type Autorhythmicity (t S ) Conduction velocity (f) Excitable neighborhood (constant) Bioeng 6460: Electrophysiology and Bioelectricity - Page 25
26 Modeling of Force Development: Sinus Rhythm Bioeng 6460: Electrophysiology and Bioelectricity - Page 26
27 Modeling of Force Development: Coupling Normalized Force Transmembrane Voltage Bioeng 6460: Electrophysiology and Bioelectricity - Page 27
28 Summary Recapitulation Group Work Excitation-Contraction Coupling Hill Model Sarcomere Modeling Coupling to Electrophysiological Models 3D Simulations Bioeng 6460: Electrophysiology and Bioelectricity - Page 28
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