Mechanical modeling of a developing tissue as both continuous and cellular

Transcription

1 Mechanical modeling of a developing tissue as both continuous and cellular Modélisation mécanique d'un tissu en développement : en quoi un matériau cellulaire diffère d'un matériau continu Cyprien Gay (MSC UMR 7057 Paris-Diderot) Tlili et al. Eur. Phys J. E 38, (2015) ingredients Erratum: 38, 115 (2015) adhesion, extrusion hal growth concentrations, distributions Experiments in vivo in vitro kinematic stress measurements measurements Continuum yet cellular model tests sub-cell based simulation page 1 / 49

2 Living material : more than continuum Guirao et al. elife :e08519 page 2 / 49

3 Living material : more than continuum Guirao et al. elife :e08519 page 3 / 49

4 Living material : more than continuum Rauzi, Nat Cell Bio 2008 page 4 / 49

5 Mechanics = material + environment contact: material: environment: page 5 / 49

6 Mechanics = material + environment contact: material: environment: constitutive equation page 6 / 49

7 Mechanics = material + environment contact: material: environment: constitutive equation page 7 / 49

8 Mechanics = material + environment contact: material: environment: constitutive equation continuum: page 8 / 49

9 Cellularity: a fluid foam is a solid! air + water + interface pressure tension = elastic! V. Leroy page 9 / 49

10 Outline 1. Discrete approach quick (and dirty?) we know the ingredients exactly 2. Continuous approach 3. Technical details and philosophy 4. Topology and elasticity page 10 / 49

11 Cytoskeleton: contractile, elastic, viscous elastic viscous contractile J. Étienne et al. PNAS 2015 page 11 / 49

12 Cell-cell adhesion: dynamical model tensile force bonding at low tension or angle debonding at high tension or angle page 12 / 49

13 Sub-cellular simulation Dynamics Equilibrium Rheology Laplace-Young Plateau Adhesion page 13 / 49

14 Une cellule entre deux plaques A. Asnacios force (t) M. Tortora force (t) page 14 / 49

15 Cytoskeleton: contractile, elastic, viscous M. Tortora tissue G* cortex page 15 / 49

16 Micropipette traction force experiment Vary pipette position Measure: É. Randazavony Infer rheology: page 16 / 49

17 Rheology of cellular aggregate Magnetic cells François Mazuel, Claire Wilhelm page 17 / 49

18 Part 2 2. Continuous approach possibility to simulate large systems challenge: how to incorporate ingredients? Tlili et al. Eur. Phys J. E 38, (2015) Erratum: 38, 115 (2015) hal page 18 / 49

19 From cell to tissue one cell connective tissue no cell-cell detachments to be included: rearrangements shape vs. volume (tensors) growth page 19 / 49

20 Large deformations: cell deformation vs. rearrangements very large deformations and yet cells remain round T1 rearrangements intercalation cell scale: deformation large scale: rearrangements page 20 / 49

21 Growth or apoptosis: cell deformation / change in number growth relieves stretching / apoptosis relieves compression large scale: change in number of cells cell scale: change in volume page 21 / 49

22 Mechanical arrangement: cell / tissue and shape / volume intra inter shape cell shape cell arrangement volume cell volume number of cells page 22 / 49

23 Mechanical arrangement: cell / tissue and shape / volume intra inter rearrangements (visco) elasticity shape contractility fluidification (cytokinesis, apoptosis) (anisotropic) cytokinesis volume swelling +cytokinesis -cytokinesis -apoptosis page 23 / 49

24 Mechanical arrangement: cell / tissue and shape / volume intra inter viscoelasticity rearrangements ck shape fluidization apoptosis - ck volume swelling + ck page 24 / 49

25 Connective tissue: intra + inter = in series intra inter page 25 / 49

26 Mechanical arrangement: cell / tissue and shape / volume intra inter viscoelasticity rearrangements ck shape fluidization apoptosis - ck volume swelling + ck page 26 / 49

27 System of continuum equations incompressible volume conservation mechanical equilibrium constitutive equation page 27 / 49

28 Part 3 3. Technical details and philosophy testing the theory memory: Eulerian/Lagrangian large deformations couplings Tlili et al. Eur. Phys J. E 38, (2015) Erratum: 38, 115 (2015) hal page 28 / 49

29 Measurable detailed kinematics Deformations Graner group, 2003, Several contributions to the deformation rate Guirao et al. elife :e08519 page 29 / 49

30 Measurable detailed kinematics Guirao et al. elife :e08519 page 30 / 49

31 Eulerian description with local deformation memory no memory tissue: huge deformations Tlili, PhD memory no memory page 31 / 49

32 Large deformations: objective derivatives our tensors are built on vectors that are attached to the material page 32 / 49

33 Large deformations: objective derivatives our tensors are built on vectors that are attached to the material => upper-convected derivative (no choice) tensor rotates with material tensor is stretched with material page 33 / 49

34 Large deformations: objective derivatives our tensors are built on vectors that are attached to the material => upper-convected derivative (no choice) tensor rotates with material tensor is stretched with material page 34 / 49

35 Coupling with cell biochemistry growth factors, morphogens + intercalations and cortex anisotropy growth rate pressure Rauzi, Nat Cell Bio 2008 page 35 / 49

36 Dissipation function formalism mechanical fields coupled mechanical and non-mechanical fields page 36 / 49

37 Dissipation function formalism constitutive equation non-mechanical fields page 37 / 49

38 Part 4 4. Elasticity and topology? elasticity relaxation page 38 / 49

39 Part 4 4. Elasticity and topology? elasticity relaxation topology changes break elasticity! page 39 / 49

40 Part 4 4. Elasticity and topology? topology issues in complex materials? elasticity condition common constitutive equations topology changes break elasticity! page 40 / 49

41 Topology issues in complex materials? gel or rubber polymer melt or solution wikipedia wikipedia reptation (de Gennes 1971) liquid foam V. Leroy - MSC topology changes are due to relaxation page 41 / 49

42 Elasticity condition elastic part of constitutive equation should be truly elastic topology changes are due to relaxation page 42 / 49

43 Common constitutive equations elastic part of constitutive equation incompressible Oldroyd B, Giesekus,... Oldroyd A Johnson-Segalman Jaumann Larson page 43 / 49

44 Elasticity condition elastic part of constitutive equation Truesdell, Noll, 1992 page 44 / 49

45 Elastic constitutive equations most general incompressible page 45 / 49

46 Elastic constitutive equations most general incompressible Generalized Oldroyd-B (1st order) Generalized Oldroyd-A (1st order) page 46 / 49

47 Mechanical modeling of a developing tissue as both continuous and cellular Modélisation mécanique d'un tissu en développement : en quoi un matériau cellulaire diffère d'un matériau continu Cyprien Gay Tissus S. Tlili F. Graner Ph. Marcq F. Molino P. Saramito subcellular simulation mechanically active molecules: cortex + adhesion continuum theory tissue-scale mechanics (MSC UMR 7057 Paris-Diderot) M. Tortora V. Boudara É. Randazavony F. Molino G. Frasca V. Du D. Fayol J.C. Bacri F. Gazeau M. Reffay C. Wilhelm Mousses liquides S. Bénito C.-H. Bruneau T. Colin F. Molino cellularity liquid / solid rearrangements / large deformations growth / apoptosis cell number => volume change slow viscous behaviour elasticity / topology page 47 / 49

48 Simulations: why? Vary parameters independently Determine which parameters should be varied Show large scale role of local scale parameters Understand the effect of parameters inaccessible to theory Length scales: cytoskeleton continuum Suggest parameter values page 48 / 49

49 Confined growing aggregate PDMS wall PDMS wall 100 m Desmaison et al, PLoS One 2013 Tlili et al page 49 / 49