BIOEN 6900 Computational Modeling of the Cardiovascular and Neuronal System Modeling of Force Development in Myocytes Overview Recapitulation Modeling of Conduction Modeling of Force in Skeletal Muscle Microscopic Anatomy Experimental Studies Modeling of Cellular Force Development Application Summary Computational Modeling of the Cardiovascular and Neuronal System - Page 2
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 v = l 0! l t Mark l Computational Modeling of the Cardiovascular and Neuronal System - Page 3 M u s c l e m F l 0 Fixation Relationship between Mass and Contraction Velocity Contraction velocity v [m/s] Contraction of skeletal muscle from frog measured value Mass [g] Computational Modeling of the Cardiovascular and Neuronal System - Page 4
Modeling of Muscle Contraction ( v + b) ( P + a) = b( P 0 + a) P: Spannung Tension des of muscle Muskels " N % # $ &' m 2 " N % P 0 : Maximale Spannung tension of des muscle Muskels # $ m 2 & ' " m% v: Kontraktionsgeschwindigkeit Contraction velocity # $ s &' a,b: Konstanten Constants depending abhängig von l 0 l,, temperature, Temperatur, Ca Concentration -Ionenkonzentration, of Ca,... m Elastic Element Contractile Element Computational Modeling of the Cardiovascular and Neuronal System - Page 5 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 relaxed 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 Further extensions necessary for realistic modeling of cardiac muscles! m Computational Modeling of the Cardiovascular and Neuronal System - Page 6
Sarcomeres in Cardiac Muscle (Fawcett & McNutt 1969) Sarcoplasmic reticulum Terminal cisternae Mitochondrion Transverse tubule Z-Disc Sarcoplasmic reticulum Dyad Computational Modeling of the Cardiovascular and Neuronal System - Page 7 Sarcomeres in Skeletal Muscle (Fawcett & McNutt 1969) Sarcolemma Transverse tubule Sarcoplasmic reticulum Terminal cisternae I-Band A-Band Z-Disc Triad Computational Modeling of the Cardiovascular and Neuronal System - Page 8
Sarcotubular System T system (transverse tubule) Sarcoplasmic reticulum (longitudinal tubule) Sarcolemma (cell membrane) Triad (skeletal muscle) Dyad (cardiac muscle) Mitochondrion (modified from Porter and Franzini- Armstrong) Computational Modeling of the Cardiovascular and Neuronal System - Page 9 Electron Micrograph of Sarcomere in Striated Muscle A - anisotrop I - isotrop (Modified from Lodish et al., Molecular Cell Biology, 2004) Computational Modeling of the Cardiovascular and Neuronal System - Page 10
Myofilaments (Modified from Lodish et al., Molecular Cell Biology, 2004) Computational Modeling of the Cardiovascular and Neuronal System - Page 11 Proteins of Sarcomere 2!m Computational Modeling of the Cardiovascular and Neuronal System - Page 12
Involved Proteins and Regulation of Force 2 µm Actin Myosin Actin TnC TnI TnT Actin Head-to-tail overlap Tm Tn: Troponin Tm: Tropomyosin A: Actin Z: Z-Disk (adapted from Gordon et al. 2001) Computational Modeling of the Cardiovascular and Neuronal System - Page 13 Regulation of Force Tn: Troponin Tm: Tropomyosin (Modified from Lodish et al., Molecular Cell Biology, 2004) Computational Modeling of the Cardiovascular and Neuronal System - Page 14
Contraction of Myocyte by Electrical Stimulation Microscopic imaging of isolated ventricular cell from guinea pig http://www-ang.kfunigraz.ac.at/ schaffer Computational Modeling of the Cardiovascular and Neuronal System - Page 15 Measurement of Force Development in Single Cell Myocyte clued between glass plates Force transmission to measurement system Stimulator Fluid at body temperature Mechanical Fixation (variable strain) Glass plates Measured force per myocyte: 0.15-6.0 µn Computational Modeling of the Cardiovascular and Neuronal System - Page 16
Measurement Techniques Permeabilization of sarcolemma/skinning of myocytes by saponin or Triton X-100 Direct control of intracellular concentrations of ions, drugs etc. [ Ca 2+ ] = Ca 2+ i [ ] o Transillumination of myocyte or muscle strands with laser light Diffraction pattern ~ sarcomere length Computational Modeling of the Cardiovascular and Neuronal System - Page 17 Sarcomere Length Measurement via Laser Diffraction (Figures from Lecarpentier et al., Real-Time Kinetics of Sarcomere Relaxation by Laser Diffraction, AJP, 1985) More information: http://muscle.ucsd.edu/musintro/diffraction.shtml Computational Modeling of the Cardiovascular and Neuronal System - Page 18
Force Development: Sliding Filament Theory Cellular force development by sliding myofilaments (Huxley 1957), i.e. actin and myosin, located in sarcomere Attachment of myosin heads to actin Filament sliding Detachment Spanning of myosin heads Computational Modeling of the Cardiovascular and Neuronal System - Page 19 Force Development: Sliding Filament Theory http://www.sci.sdsu.edu/movies/actin_myosin.html Computational Modeling of the Cardiovascular and Neuronal System - Page 20
Actin-Myosin Interaction A Actin M Myosin ADP Adenosine diphosphate ATP Adenosine triphosphate P i Phosphate (Bers, Excitation-Contraction Coupling and Cardiac Contractile Force, 1991, modified) Computational Modeling of the Cardiovascular and Neuronal System - Page 21 Coupling of Electrophysiology and Force Development Rest small Ca 2+ Stimulus Contraction Ca 2+ release from SR high Ca 2+ High concentration of intracellular Ca 2+ Force development Contraction of sarcomere/cell Computational Modeling of the Cardiovascular and Neuronal System - Page 22
Phases of Force Development Begin of contraction Ca release Relaxation Ca uptake Triggering Filling A: Actin M: Myosin Z: Z-Disc SL: Sarcolemma TTS: Transverse tubule SPR: Sarcoplasmic Reticulum Ca 2+ Release Ca 2+ Uptake Computational Modeling of the Cardiovascular and Neuronal System - Page 23 Calcium Handling and EC-Coupling Transverse Tubule Cell Membrane Sarcoplasmic Reticulum NCX: ATP: RyR: PLB: Troponin Mitochondrion Sodium-calcium exchanger Ionic pump driven by ATP-hydrolysis Calcium channel (ryanodine receptor) Phospholamban (Bers, Nature Insight Review Articles, 2002, modified) Computational Modeling of the Cardiovascular and Neuronal System - Page 24
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 Computational Modeling of the Cardiovascular and Neuronal System - Page 25 Mathematical Modeling of Myofilament Sliding (Huxley) A + M f " g A # M A + M: Anzahl Number von of bzgl. myosins Actin not ungekoppeltem bound to actin Myosin A # M: Anzahl Number Querbrücken of myosins bound (Actin/ to Myosin actin gekoppelt) f,g: Ratenkonstanten constants for für binding Bindung and bzw. unbinding Ablösung d [ A # M] ( [ ] # [ A # M] ) # g[ A # M] = f Max A # M dt Max[ A # M]: Maximale number Anzahl of an myosin Querbrücken proteins bound to actin filament Computational Modeling of the Cardiovascular and Neuronal System - Page 26
Group Work Which states are important for a detailed modeling of force in myocytes? Which states can be neglected for an efficient model? Computational Modeling of the Cardiovascular and Neuronal System - Page 27 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 Computational Modeling of the Cardiovascular and Neuronal System - Page 28
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) Computational Modeling of the Cardiovascular and Neuronal System - Page 29 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! Computational Modeling of the Cardiovascular and Neuronal System - Page 30
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 Computational Modeling of the Cardiovascular and Neuronal System - Page 31 State Diagram of 3rd Model of Rice 1999 et al. States of Troponin States of Tropomyosin and cross bridges 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 Computational Modeling of the Cardiovascular and Neuronal System - Page 32
Matrix representation of State Diagram 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 g ( ) # f g 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 Computational Modeling of the Cardiovascular and Neuronal System - Page 33 Force Development for Different Static Sarcomere Lengths (SL) Force normalized with maximal force [N/N] time [s] Computational Modeling of the Cardiovascular and Neuronal System - Page 34
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 Computational Modeling of the Cardiovascular and Neuronal System - Page 35 Model of Glänzel et al. 2002: Crossbridge Cycling v M ATP A M ATP v M A M v 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 Computational Modeling of the Cardiovascular and Neuronal System - Page 36
Reconstruction of Static Measurements Study Species: rat Ventricular myocytes Resting length ~ 2µm Protocol Calcium concentration constant Stretch constant 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 Computational Modeling of the Cardiovascular and Neuronal System - Page 37 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 Computational Modeling of the Cardiovascular and Neuronal System - Page 38
Coupling of Force with Electrophysiological Models: Basics Noble-Varghese-Kohl-Noble model Rice-Winslow-Hunter model 1st Stimulus at t=35 ms Sarcomere stretch # Computational Modeling of the Cardiovascular and Neuronal System - Page 39 Coupling of Force with Electrophysiological Models Luo-Rudy model (Guinea pig ventricular) Rice-Winslow-Hunter model 3rd (rabbit ventricular) Stimulus at t=35 ms Sarcomere stretch # Species don t fit Computational Modeling of the Cardiovascular and Neuronal System - Page 40
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 Computational Modeling of the Cardiovascular and Neuronal System - Page 41 Coupling of Force with Electrophysiological Model: Human Priebe-Beuckelmann model (human ventricular) Glänzel-Sachse-Seemann model (human ventricular) Stimulus at t=35 ms with frequency Sarcomere stretch # Validation with measured data necessary! Computational Modeling of the Cardiovascular and Neuronal System - Page 42
Modeling Force Development with Cellular Automaton Anatomical Model Physiological Parameters Cellular Automaton Force Transmembrane voltage Calcium concentration F/F maxi t [s] Computational Modeling of the Cardiovascular and Neuronal System - Page 43 Parameters for Simulation: Lookup Tables Known per volume element dv and for time t dv Stimulus Time since activation t S Stimulus frequency f Tissue type Force (t S ) (instead of transmembrane voltage) Refractory period (t S ) Autorhythmicity (t S ) Conduction velocity (f) Excitable neighborhood (constant) Computational Modeling of the Cardiovascular and Neuronal System - Page 44
Modeling of Force Development: Sinus Rhythm Computational Modeling of the Cardiovascular and Neuronal System - Page 45 Modeling of Force Development: Coupling Normalized Force Transmembrane Voltage Computational Modeling of the Cardiovascular and Neuronal System - Page 46
Summary Recapitulation Modeling of Conduction Modeling of Force in Skeletal Muscle Microscopic Anatomy Experimental Studies Modeling of Cellular Force Development Application Computational Modeling of the Cardiovascular and Neuronal System - Page 47