BIOEN 6003 Cellular Electrophysiology and Biophysics Modeling of Force Development in Myocytes II Frank B. Sachse, University of Utah
Overview Experimental Studies Sliding Filament Theory Group work Excitation-Contraction Coupling Modeling of Cellular Force Development Group work Integration with Electrophysiological Models Group work Computational Modeling of the Cardiovascular System - Page 2
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 System - Page 3
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 System - Page 4
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 System - Page 5
Sarcomere Length Measurement via Laser Diffraction (Figures from Lecarpentier et al., Real-Time Kinetics of Sarcomere Relaxation by Laser Diffraction, AJP, 1985) Further information: http://muscle.ucsd.edu/musintro/diffraction.shtml Computational Modeling of the Cardiovascular System - Page 6
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 System - Page 7
Force Development: Sliding Filament Theory http://www.sci.sdsu.edu/movies/actin_myosin.html Computational Modeling of the Cardiovascular System - Page 8
Actin-Myosin Interaction A Actin M Myosin ADP Adenosine diphosphate ATP Adenosine triphosphate P i Phosphate (modified from Bers, Excitation-Contraction Coupling and Cardiac Contractile Force, 1991) Computational Modeling of the Cardiovascular System - Page 9
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 System - Page 10
Phases of Force Development A: Actin M: Myosin Z: Z-Disc Begin of contraction Ca release Triggering Relaxation Ca uptake Filling SL: Sarcolemma TTS: Transverse tubule SPR: Sarcoplasmic Reticulum Ca 2+ Release Ca 2+ Uptake Computational Modeling of the Cardiovascular System - Page 11
Calcium Handling and EC-Coupling Transverse Tubule Cell Membrane Sarcoplasmic Reticulum Troponin NCX: Sodium-calcium exchanger ATP: Ionic pump driven by ATP-hydrolysis RyR: Calcium channel (ryanodine receptor) PLB: Phospholamban (modified from Bers, Nature Insight Review Articles, 2002) Computational Modeling of the Cardiovascular System - Page 12
Group Work Which states are important for biophysically detailed modeling of force in myocytes? Which states can be neglected for an efficient model? Computational Modeling of the Cardiovascular System - Page 13
Models of Force Development 1938 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 today Glänzel, Sachse, Seemann Ventricular myocytes Human Computational Modeling of the Cardiovascular System - Page 14
Mathematical Modeling of Myofilament Sliding (Huxley) A + M f g A M A + M: Anzahl Number von of myosins bzgl. 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 System - Page 15
Modeling of Cellular Force Development [Ca] i /µm F/F maxi t [s] Intracellular concentration of calcium Calculated with electrophysiological cell models System of coupled ordinary differential equations t [s] Processes in myofilaments State of troponins, tropomyosins and cross bridges System of coupled ordinary differential equations t [s] Force Development Computational Modeling of the Cardiovascular System - Page 16
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 Some rate coefficients are a function of intracellular calcium [Ca 2+ ] i (Model 1 of Rice 1999 et al./ Landesberg et al.1994) Computational Modeling of the Cardiovascular System - Page 17
4-State Model of Force Development Several states contribute to force development: F = α F : T1+ N1 F max Normalized force [N/N] F max : Maximal simulated force α : Myofilament overlap function Force is dependent on overlap of actin and myosin filaments. Frank-Starling Mechanism Force ~ initial length Diastolic volume ~ cardiac output α Computational Modeling of the Cardiovascular System - Page 18
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 System - Page 19
Group Work Why is modeling of cellular contraction and modeling of ion channels conceptually similar? What are the differences? Computational Modeling of the Cardiovascular System - Page 20
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 System - Page 21
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 Model 3 from Rice et al., 1999 Computational Modeling of the Cardiovascular System - Page 22
Force Development for Different Static Sarcomere Lengths (SL) Force normalized with maximal force [N/N] time [s] Computational Modeling of the Cardiovascular System - Page 23
Model of Glänzel et al. 2002: Activation [Ca] I, λ, XB T TCa TCa, TM on TM off TM on T TCa Troponin Troponin with bound calcium TM off Tropomyosin in nonpermissive state TM on Tropomyosin in permissive state λ 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 et al, IJBC, 2003, Sachse et al, LNCS 2674, 2003) Cooperativity mechanisms XB-TN TM-TM Computational Modeling of the Cardiovascular System - Page 24
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 λ 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 A M ADP P i XB-XB ΤΜ ον, λ, XB Computational Modeling of the Cardiovascular System - Page 25
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 System - Page 26
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 System - Page 27
Coupling of Force with Electrophysiological Models: Basics Noble-Varghese-Kohl-Noble model Rice-Winslow-Hunter model 1st Stimulus at t=35 ms Sarcomere stretch λ Computational Modeling of the Cardiovascular System - Page 28
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 λ Species don t fit Computational Modeling of the Cardiovascular System - Page 29
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 λ Differences in force development due to differences of Ca transient Computational Modeling of the Cardiovascular System - Page 30
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 λ Validation with measured data necessary! Computational Modeling of the Cardiovascular System - Page 31
Group Work Discuss the integration of electrophysiological models with models of cellular force development. When are two models compatible? What other models will be required to reconstruct cellular contraction? Computational Modeling of the Cardiovascular System - Page 32
Summary Experimental Studies Sliding Filament Theory Excitation-Contraction Coupling Modeling of Cellular Force Development Integration with Electrophysiological Models Computational Modeling of the Cardiovascular System - Page 33