Philippe TRACQUI

1. Spatio-temporal Dynamics of Nonlinear Mechano-chemical Processes Driving Anisotropic Rhythmic Contractionof Cardiac Myocytes Philippe TRACQUI in collaboration with Jacques OHAYON CNRS, Laboratoire TIMC-IMAG, Equipe Dynacell Institut de l’Ingénierie et de l’Information de Santé, In3S, Grenoble, France http://www-timc.imag.fr/dynacell

2. Mechanics is a key issue of heart function … … but still remains largely over simplified in analyses and models of cardiac cells and cardiac tissues dynamics • Things are changing with the increasingly recognized importance of the transduction of mechanical signals (mechanotransduction) in cell signaling cascades • There is a real need for the development of the mechanobiology of cardiac cells and tissues, notably through the development of theoretical models as the cell and tissue levels

3. Genes Proteins Biophysical models Constitutive laws Organ model Whole body model Molecular Biology Physiology Bioengineering Clinical medicine Slide from the Physiome project presentation(R. McLeod & P. Hunter) Modeling Hierarchies Databases • Genome • Protein • Physiology • Structural • Bioeng. Materials • Clinical

4. Context:Analysis of excitation-contraction of isolated cardiac myocyte trough a multi-scale integrative approach of the structure and function relationships • Aims: • an analysis of the cardiac performance based on a relevant description of the Ca2+ driven anosotropic and hyperelastic cardiomyocyte contraction • a modelling basis for theoretical and quantitative analysis of the mechano-regulation of cardiomyocyte contraction by mechanotransduction processes Tissue level Cellular level Intracellular level

5. Mean experimental contraction period Mean experimental contraction duration 17,0  5,8 s 1,5  0,4 s Spontaneous contraction of an isolated cardiomyocyte The sarcomere as the contractile unit Mean experimental contraction amplitude ~ 8 m ( 7%) Spontaneous contraction of rat cardiomyocyte size (110 x 20 mm)

6. Associated calcium wave propagation Visualisation of the propagation of an intracellular calcium wave using Ca labelling with the fluorescent Fluo3 probe (Dt = 268ms between two successive images, cell length :110 mm)

7. A theoretical model of the cardiomyocyte self-sustained contraction • expression of Ca2+ oscillations in a domain of the parametric space where travelling waves may exist • introduction of cytosolic Ca2+ variations in the formulation of an active stress tensor, taking into account cell architectural anisotropy • consideration of cardiomyocyte hyperelastic properties with appropriate passive stress-strain relationship • finite element simulation and experimental validation of the dynamical behaviour of the virtual cardiomyocyte in different contexts

8. Modelling calcium waves propagation in cells and tissues Dupont et al. 96 (Means et al., 2006)

9. Autocatalytic process responsible for temporal oscillations: Calcium-Induced-Calcium-Release (CICR) A simplified one calcium -pool model Goldbeter et al. (PNAS, 1990) Z: Cytosolic Ca2+ concentration Y: Ca2+ concentration in the sarcoplasmic reticulum

10. Elastic properties of the cardiomyocyte Passive tension as a function of the sarcomere length (Cazorla et al., 2003) Uniaxial stretching of the cardiomyocyte Tension (kPa) 20 mm Sarcomere length SL (mm)

11. Constitutive stress-strain relationship (1): passive component The cardiomyocyte is considered as an hyperelastic incompressible medium with passive strain energy function • a1 , a2 , cellular material constants • I1 is the first invariant of the right Cauchy- Green strain tensor C (I1=Tr(C))

12. Tmax maximal tension K(SL)=Ca2+50 half-maximal value nH Hill coefficient Constitutive stress-strain relationship (2): active component Active anisotropic Cauchy stress tensor given by: (fs orientation of deformed fibers) with:

13. Interplay of calcium oscillations and cell contraction Model Variables Z(r,t) , Y(r,t) and {u(r,t), v(r,t) } D diagonal diffusion tensor Calcium spatio-temporal dynamics (waves) Active and passive anisotropic mechanical behaviour Integrative mechano-biochemical model of the self-sustained cardiomyocyte contraction

14. Finite element simulation of the cardiomyocyte spontaneous contraction • Geometry extracted from real cell image • Boundary conditions • Stress free boundaries, localized zero displacement in the nucleus area • No calcium fluxes (Neuman conditions) on the cell boundaries • Permeability of the nucleus to cytosolic Ca (Pustoc’h et al., Acta Biotheor. 2005)

15. Simulated self-sustained oscillating contraction of an isolated cardiomyocyte Cardiomyocyte contraction driven by calcium waves originating from cell border (left) or from cell centre (right), as shown by videomicroscopy time-lapse observations Saptio-temporal evolution of cytosolic calcium concentrations (Z(x,y,t)) Calcium spark initiated in the middle of the cell Two solitary waves propagating in opposite directions Cell contracts at both ends simultaneously Triggering calcium spark initiated on the left cell side Soliton propagating from left to right in pace with cell shortening

16. Conclusions and perspectives • A satisfactory and rather simple mechano-biochemical model of the isolated cardiomyocyte oscillatory contraction • Amenable to theoretical analysis (bifurcation analysis of model dynamics) • Exemplified mechanical aspects disregarded by reaction-diffusion models • A quantitative framework for analysing the effect of local mechanotransduction processes (titin, endothelin, ..) • A basis for elaborating of a 2D virtual myocardium in which the global tissue response (arrhythmia, contraction inefficiency, …) to localized perturbations (ischemia, …) can be studied Acknowledgement: This work has been supported by a grant from the CNRS (ACI NIM “MOCEMY”)