Nonlinear Biomechanics of Fibroblast Mechanosensing in Fibrous ECM: Modelling, Analysis and Computation

Transcription

1 Nonlinear Biomechanics of Fibroblast Mechanosensing in Fibrous ECM: Modelling, Analysis and Computation Phoebus Rosakis Department of Mathematics and Applied Mathematics University of Crete & Institute for Computational Mathematics, F.O.R.T.H. 1 / 54

2 Cells Use Stress to Find Each Other in Fibrous Darkness with (IACM Team) Georgios Grekas Charalambos Makridakis and Guruswami Ravichandran Jacob Notbohm Ayelet Lesman David Tirrell 2 / 54

3 Continuum Mechanics 3 / 54

4 Continuum Mechanics mathematical modeling of forces/deformations in deformable solids 3 / 54

5 Cell Biology 4 / 54

6 Fibroblasts cause wrinkling of 2D silicone substrate Harris, Stopak, Wild / 54

7 primarily to fish silicone surface marker particle optimize the me preparation pro and application Arecent dev involves the pre using a curing a non-wrinkling s characteristics. surface is deter dots or lines, ge or gallium arse the surface of th micropattern h and allows the d Moving Fish Epidermal Keratocytes cause wrinkling of 2D silicone substrate Burton, Park, Taylor, 1999 TRENDS in Cell Biology Vol.12 No.2 February 2002 Review (a) (b) rious flexible used to detect orces. (a) Motile ocyte on a silicone m. Arrow direction of. Image kindly by K. Burton. fish keratocyte wrinkling ubstratum. Black ndicate the es of embedded ds; bar, 10 m m. ed, w ith n, from Ref. [14]. nary rat cardiac t causing s on a terned silicone 6 / 54

8 Explants induce densification bands in 3D fibrin extracellular matrix Harris, Stopak, Wild / 54

9 Fibroblasts exert huge forces onto their surroundings 8 / 54

10 Fibroblasts exert huge forces onto their surroundings WHY? 8 / 54

11 mechanosensing cells sense forces/stresses/deformations 9 / 54

12 mechanosensing cells sense forces/stresses/deformations tensotaxis cells protrude/migrate toward regions of higher tension 9 / 54

13 mechanosensing cells sense forces/stresses/deformations tensotaxis cells protrude/migrate toward regions of higher tension durotaxis cells protrude/migrate toward regions of higher stiffness 9 / 54

14 What does the ECM look like? Network of collagen/fibrin fibers 10 / 54

15 The Role of Focal Adhesions 11 / 54

16 The Role of Focal Adhesions Adhesions allow the cell to exert traction on the ECM 11 / 54

17 The Role of Focal Adhesions Adhesions allow the cell to exert traction on the ECM Adhesions act as force/ stress/ deformation detectors. Mechanosensing is active!! 11 / 54

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19 Fibroblasts Use Stress, But WHY? 12 / 54

20 Fibroblasts Use Stress, But WHY? To see 12 / 54

21 Fibroblasts Use Stress, But WHY? To see and be seen 12 / 54

22 Fibroblasts Use Stress, But WHY? To see and be seen and to change things around them 12 / 54

23 Fibroblasts Use Stress, But WHY? To see and be seen and to change things around them i.e. 12 / 54

24 Fibroblasts Use Stress, But WHY? To see and be seen and to change things around them i.e. To detect and approach each other by spreading/protruding 12 / 54

25 Fibroblasts Use Stress, But WHY? To see and be seen and to change things around them i.e. To detect and approach each other by spreading/protruding To remodel the matrix around them (possibly for tissue morphogenesis Stopak-Harris, 1982) 12 / 54

26 Notbohm / 54

27 Notbohm / 54

28 Notbohm / 54

29 Notbohm / 54

30 17 / 54

31 Notbohm-Ravichandran-Lesman-Tirrell hrs 6 hrs 7 hrs 50 µm 18 / 54

32 Notbohm-Ravichandran-Lesman-Tirrell hrs 6 hrs 7 hrs 50 µm Cells contract and tethers form in the ECM joining them. Tethers (white) are regions of high ECM density 18 / 54

33 5 hrs 6 hrs 7 hrs 50 µm 19 / 54

34 5 hrs 6 hrs 7 hrs 50 µm Later, cells grow appendages along the tethers towards each other. Green: cell actin. 19 / 54

35 20 / 54

36 Claim: This behavior relies on a special nonlinearity of the ECM s mechanical behavior 21 / 54

37 Claim: This behavior relies on a special (instability) of the ECM s mechanical behavior this tethering behavior does not occur in homogeneous linear elastic ECM (e.g. hydrogels) 21 / 54

38 Microbuckling Individual fibers will buckle under compression 22 / 54

39 Microbuckling Individual fibers will buckle under compression stiffer in tesnion than in compression (rubber band) 22 / 54

40 A Nonlinear Constitutive Law for Large Deformations 23 / 54

41 A Nonlinear Constitutive Law for Large Deformations Harris-Stopak / 54

42 A Nonlinear Constitutive Law for Large Deformations Harris-Stopak 1981 Macroscopic tether (1.5cm) between contracting (multi-cell) explants (2-3mm) in collagen fibrous ECM 23 / 54

43 A Nonlinear Constitutive Law for Large Deformations Harris-Stopak 1981 Macroscopic tether (1.5cm) between contracting (multi-cell) explants (2-3mm) in collagen fibrous ECM Hypothesis: This can be explained by microbuckling. 23 / 54

44 Elastic Strain Energy Function Start with a single fiber with force stretch relation F (λ) that is weaker in compression than tension and energy W (λ) = λ 1 F (ζ)dζ F (λ) = µ ( λ N 1 ) 24 / 54

45 Elastic Strain Energy Function Start with a single fiber with force stretch relation F (λ) that is weaker in compression than tension and energy W (λ) = λ 1 F (ζ)dζ 24 / 54

46 Elastic Strain Energy Function Start with a single fiber with force stretch relation F (λ) that is weaker in compression than tension and energy W (λ) = λ 1 F (ζ)dζ Suppose ECM (2D) has uniform angular distribution of fibers Ŵ (λ 1, λ 2 ) = 1 2π 2π 0 W ( (λ 1 cosθ) 2 + (λ 2 sinθ) 2 ) dθ 24 / 54

47 Elastic Strain Energy Function 25 / 54

48 Elastic Strain Energy Function Explicitly determined W (F ), F = u Elastic Deformation Energy /unit volume, function of deformation gradient matrix F (strain) 25 / 54

49 Elastic Strain Energy Function Explicitly determined W (F ), F = u Elastic Deformation Energy /unit volume, function of deformation gradient matrix F (strain) 25 / 54

50 Elastic Strain Energy Function Explicitly determined W (F ), F = u Elastic Deformation Energy /unit volume, function of deformation gradient matrix F (strain) Nonlinear Stress-Strain Relation S(F ) = dw (F ) df 25 / 54

51 Elastic Strain Energy Function Explicitly determined W (F ), F = u Elastic Deformation Energy /unit volume, function of deformation gradient matrix F (strain) Nonlinear Stress-Strain Relation S(F ) = dw (F ) df 25 / 54

52 Interesting Properties of W Uniaxial compression 26 / 54

53 Interesting Properties of W Uniaxial compression 26 / 54

54 Interesting Properties of W Uniaxial compression Nonmonotone uniaxial compression: 26 / 54

55 Interesting Properties of W Uniaxial compression Nonmonotone uniaxial compression: densification (compressive) phase transition, 26 / 54

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57 Interesting Properties of W 27 / 54

58 Interesting Properties of W W is a multi-well isotropic strain energy 27 / 54

59 Interesting Properties of W W is a multi-well isotropic strain energy Level curves of Ŵ (λ 1, λ 2 ) 27 / 54

60 Interesting Properties of W W is a multi-well isotropic strain energy Level curves of Ŵ (λ 1, λ 2 ) Two wells : two phases (stable states) because of microbuckling 27 / 54

61 Interesting Properties of W W is a multi-well isotropic strain energy Level curves of Ŵ (λ 1, λ 2 ) Two wells : two phases (stable states) because of microbuckling Instabilities, Discontinuities, Interfaces / 54

62 Elastic Energy of the ECM Model ECM as elastic body with holes (cells/explants) Strain Energy Function W. Elastic Energy of the ECM E[u] = W ( u)dv Ω 28 / 54

63 Elastic Energy of the ECM Model ECM as elastic body with holes (cells/explants) Strain Energy Function W. Elastic Energy of the ECM E[u] = W ( u)dv Minimize Energy Ω 28 / 54

64 Cell Model Cells are the holes. They contract: they apply centripetal forces proportional to distance from their center. Explants: same as cells but at a much bigger scale. 29 / 54

65 Typical Numerical Results (Finite Element Computation) 30 / 54

66 Typical Numerical Results (Finite Element Computation) 30 / 54

67 Typical Numerical Results (Finite Element Computation) Stopak-Harris / 54

68 Typical Numerical Results (Finite Element Computation) Stopak-Harris / 54

69 Mesh Dependence 31 / 54

70 Mesh Dependence 31 / 54

71 Mesh Dependence Discontinuities, Oscillations 31 / 54

72 Mesh Dependence Discontinuities, Oscillations Stopak-Harris / 54

73 Higher Gradients... added to the energy to limit gradient oscillations. Related to discreteness, bending stiffness of the fibers and rotational springs at network nodes. Φ[u] = E[u] + C[u] + ε u 2 2 Ω 32 / 54

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75 34 / 54

76 Length Scale ε << explant size 35 / 54

77 Length Scale ε cell size Compare of simulations with experiments to calibrate ε based on number of protrusions. 36 / 54

78 Conclusions cells use stress to see and be seen by peers 37 / 54

79 Conclusions cells use stress to see and be seen by peers mechanosensing is active (cells exert force, detect resulting stress/deformation). 37 / 54

80 Conclusions cells use stress to see and be seen by peers mechanosensing is active (cells exert force, detect resulting stress/deformation). they exploit a special phase transition (microbuckling of fibrin) to increase detection range by tether formation 37 / 54

81 Conclusions cells use stress to see and be seen by peers mechanosensing is active (cells exert force, detect resulting stress/deformation). they exploit a special phase transition (microbuckling of fibrin) to increase detection range by tether formation (stress decays rapidly in linear solids) 37 / 54

82 Conclusions cells use stress to see and be seen by peers mechanosensing is active (cells exert force, detect resulting stress/deformation). they exploit a special phase transition (microbuckling of fibrin) to increase detection range by tether formation (stress decays rapidly in linear solids) highly nonlinear problem requires specialized modelling/simulation techniques to predict/explain experimental observations by this mechanism 37 / 54

83 Conclusions cells use stress to see and be seen by peers mechanosensing is active (cells exert force, detect resulting stress/deformation). they exploit a special phase transition (microbuckling of fibrin) to increase detection range by tether formation (stress decays rapidly in linear solids) highly nonlinear problem requires specialized modelling/simulation techniques to predict/explain experimental observations by this mechanism yes but why? 37 / 54

84 Cell Networks Lesman-Notbohm-Ravichnadran-Tirrell unpublished experiments 38 / 54

85 Cell Networks Lesman-Notbohm-Ravichnadran-Tirrell unpublished experiments 39 / 54

86 Cell Networks Lesman-Notbohm-Ravichnadran-Tirrell unpublished 40 / 54

87 Cell Networks Lesman-Notbohm-Ravichnadran-Tirrell unpublished 41 / 54

88 Cell Networks Lesman-Notbohm-Ravichnadran-Tirrell unpublished 42 / 54

89 Cell Networks Hypothesis: Matrix Remodeling 43 / 54

90 Cell Networks Hypothesis: Matrix Remodeling Tether network changes ECM mechanical properties (stiffness) 43 / 54

91 Cell Networks 2D Simulation 44 / 54

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