Influence of mechanical properties in cell movement across ECM channels.

Transcription

1 Influence of mechanical properties in cell movement across ECM channels. Chiara Giverso Politecnico di Torino Supervisor: Luigi Preziosi 1 / 26

2 2 / 26 Presentation Outline 1 Biological Process Cell motion into matrix Bio-mechanical experiments 2 Micropipette aspiration Models Adhesive Forces Micropipette model results Cell deformation Continuum Model Elastic membrane model Elastic nucleus model Results 3 Conclusions & Future steps

3 Biological Process Cell motion into matrix Cell motion inside extracellular matrix Migration of cells across matrix networks: physiological mechanism used by cells to reach distant sites tumor cells implement it to enhance their invasiveness fundamental in tissue engineering Key factors in 3D cell migration: bonds formation between cells and matrix active force generation ECM degradation (MMPs secretion) environment morphology influence intense mechanical deformations K. Wolf, Y.I. Wu, Y. Liu, J. Geiger, E. Tam, C. Overall, M.S. Stack and P. Friedl. Multi-step pericellular proteolysis controls the transition from individual to collective cancer cell invasion. Nat. Cell. Biol. 9: (2007). 3 / 26

4 4 / 26 Biological Process Bio-mechanical experiments From the biological point of view...

5 5 / 26 Biological Process Bio-mechanical experiments From the mathematical point of view... Nowadays, in continuum mechanics... φ c t + (φ cv c ) = Γ c CELL MASS BALANCE φ ECM = Γ ECM ECM MASS BALANCE t k v c = ν(1 φ ECM ) P DARCY S LAW φ α is the volume fraction of the α-constituent;v c is the velocity of cells (seen as a continuum, moving in a porous rigid ECM); ν is the viscosity of cells; P is the interstitial pressure;k is the permeability;γ α is the rate of production/degradation of the α-constituent k is generally a function of φ ECM and the morphology of pores (e.g. Kozeny-Carman)

6 Biological Process Bio-mechanical experiments What is still missing? Idea: Adhesive properties Mechanical properties Study how the interplay between mechanical and adhesive properties can influence the process of cell entering inside ECM channels. Assumptions: Cell composed of two regions: cytoplasm that can freely enter in the channel nucleus, the most rigid part, that needs to deform to enter in the channel Regular structure of ECM channels 6 / 26

7 (typically Φ = 2.1). 7 / 26 Micropipette aspiration Models Micropipette aspiration Models 1 Cells like an elastic membrane (Chien, 1979) PR p = 2 L ( p 1+log 2 L ) p γ R p R p R p is the radius of the pipette;l p is the aspired cell length ; γ is the shear elastic modulus of the membrane. 2 Cells like an elastic solid (Theret, 1988) W.R. Jones et al. Journal of Biomechanics. 32(2): (1999). P E = 2π 3 ΦL p R p E is the Young s modulus for the cell; Φ depends on the ratio between external and internal radius of the pipette

8 Micropipette aspiration Models If we apply these models to cell entrance into an ECM channel... Chien s and Theret s equation describe the deformation of the nucleus (i.e. the "elastic solid" part of the cell) L p is the length of the aspired portion of the nucleus; P is related to adhesive bond forces, F adhesion, P FZ adhesion πrp 2. 8 / 26

9 9 / 26 Adhesive Forces Adhesive Forces F adhesion is the sum, over the region of contact, of all the bond forces, F bond, generated through mechano-transduction, weighted by the density of expressed and activated integrins, ρ b, the surface ratio of substratum ligands (ECM adhesive sites), α ECM = S ECM /S channel ; Assuming that ρ b and α ECM do not depend on time and are homogeneously distributed F adhesion = ρ b α ECM S F bond ds. { } S = (X,Y,Z) : X 2 + Y 2 = R 2 p, L0 n < Z < L0 n + L b ; L b is the length of the the surface for which ECM-bonds are expressed and it remains constant in time during cell deformation, L b = L 0 cell Rp L0 n = L0 cell = = R p [ 4 3 R 3 c R3 n R 3 p ( R 3 L 0 ) ( 2 n R n 1 n) ] p 3 L0 R n R 2 n R2 p.

10 Adhesive Forces Single bond force 1 linear force F Z bond = k b Z F Z adhesion = πr pρ b α ECMkL 2 b 2 constant force F Z bond = F M b F Z adhesion = 2πR pρ b α ECMF M b L b 3 constant force over a bounded regionf Z bond = F M b χ L M b (Z) F Z adhesion = 2πR pρ b α ECMF M b L b ρ b is the density of bonds on cell surface (linked to the capability of cells to express integrins); α ECM = S ECM is the ECM surface channel fraction; S channel } k is bond stiffness; L b is the length of the cell along which bonds are formed ; L b {L = min b,l M b. 10 / 26

11 Micropipette model results Micropipette model results Chien s Model G k γ Linear Force 2 L ( p 1+log 2 L ) p R p R p L 2 b Constant (bounded) Force 2 L ( p 1+log 2 L ) p R p R p G F γ 2 L ( ) b Chien s Model G k γ G F γ G F γ (bound) Characteristic Numbers: ratio between adhesive and mechanical properties. G k γ = ρ bα ECMk b R 2 n γ G F γ = ρ bα ECMF M b R n γ,. 0 R p / 26

12 Micropipette model results Interpretation of the results Bar charts represents the range for which a cell, with a given G k γ or G F γ can enter inside the channel, for the different hypothesis of bond forces Chien s Model G k γ G F γ G F γ (bound) CELL 1 G k γ = ρ bα ECM k b R 2 n γ, R p CELL 2 G F γ = ρ bα ECM F M b Rn γ. CONSTANT LINEAR BOUNDED 12 / 26

13 13 / 26 Micropipette model results Theret s Model Linear Force G k E 2π 3 Φ L p L 2 b Constant (bounded) Force G F E 2π 3 Φ L p 2 L ( ) b Theret s Model G k E G F E G F E (bound) Characteristic Numbers: ratio between adhesive and mechanical properties. G k E = ρ bα ECMk b R n E G F E = ρ bα ECMF M b E., Rp

14 14 / 26 Micropipette model results Results seems promising, showing that cells can pass through channels within a certain range of diameters, depending on the mechanical properties of cell nucleus and adhesive capabilities...but... Chien s equation has been obtained for an infinite 2D membrane; Theret s model has been derived for an infinite 3D half space and empirically simplified to the well-known simple equation. = Both models cannot be applied to describe the cell totally aspired inside the pipette!

15 15 / 26 Cell deformation Continuum Model Cell deformation Continuum Model Assumptions: mechanical representation of the nucleus 1 elastic membrane 2 elastic solid nuclear deformations (a) ellipsoid (b) cigar-shaped

16 16 / 26 Cell deformation Continuum Model 1. Elastic membrane model The energy required to increase the surface area, W S tot, is W S tot = λ( S) 2 S is the increase in the surface area of the cell passing from an initial spherical shape to its final shape. (a) ellipsoidal S ellips = S ellips S sphere = 2πR 2 p with e = 1 R 2 p h 2 e (b) cigar-shaped and h e = R3 n R 2 p S cigar = S cigar S sphere = 4πR 2 p ( 1+ h ) e R p e arcsin(e) 4πRn 2 [ ( Rn R p ) 3 ( Rn R p ) 2 ], whereh = 2 3 Rp R 3 n R3 p R 3 p

17 17 / 26 Cell deformation Continuum Model 2. Elastic nucleus model The energy required to deform the nucleus is obtained, once that a proper constitutive equation is chosen. We assume an incompressible neo-hookean constitutive law for the nucleus of the cell W V = µ 2 [tr(c) 3], C = F T F and µ is the shear modulus of the nucleus. To calculate F, we assume that undeformed parallel planes remain parallel in the deformed configuration (a) Deformation Gradient for an ellipsoidal deformation (cylindrical coordinates) F = diag { Rp, R } p, R2 n R n R n Rp 2 = diag { R p, R p, } 1. R p 2

18 18 / 26 Cell deformation Continuum Model (b) Deformation Gradient for a cigar-shaped deformation F = F c = F N pole = { FN pole for H < Z R n F c for H Z H F S pole for R n Z < H. R p R 2 n Z 2 0 Z R 2 p (z h)2 where z = z(z) and Γ(Z) = ( R 2 n Z 2) 3/2 0 R pzρ (R 2 n Z2 ) 3/2 R p 0 R 2 n Z 2 and R 2 n 0 0 R 2 p R 2 p (z h)2 R 2 n Z 2 0 Γ(Z)ρ 0 R 2 p (z h) 2 R 2 n Z (z h) R 2 n Z2, (R 2 p (z h)2) 3/2 R 2 n Z 2 R 2 p (z h) 2,

19 19 / 26 Cell deformation Continuum Model Energy calculation The elastic energy stored per unit volume for (a) an ellipsoidal deformation W V = µ 2 (b) a cigar-shaped deformation ( ) 2 R2 p R 2 + R4 n n R 4 3 p [ Wc V = µ R 2 p 2 2 R 2 n Z 2 + R2 pz 2 ρ 2 ( R 2 (R 2 n Z 2 ) 3 + n Z 2) ] 2 R 4 3, p [ ] WN pole V = µ 2 R2 p (z h) 2 2 R 2 n Z µ Z R 2 p (z h) 2 2 (z h) R 2 2 (R 2 n Z 2 ) 3/2 + n Z 2 ( R 2 p (z h) 2) ρ 2 + 3/2 + µ 2 [ ( R 2 n Z 2) 2 ( R 2 p (z h) 2) 2 3 ].

20 20 / 26 Cell deformation Continuum Model The total energy required to pass from the initial spherical configuration to the cell totally deformed inside the channel: (a) ellipsoid (b) cigar-shaped W V tot = W V dv = 2 ( Vr 3 µπr3 n 2 R2 p R 2 + R4 n n R 4 3 p ( ) Wtot V = 2 Vr c+ Wc V dv + Vr N pole WN pole V dv = = 2πµR 2 p H + µ ( ) ] 2 πr2 p [R n tanh 1 H + HRn + π µ [R 6n R 4 H R4n H R2n H5 17 ] ( H7 3πµ R 2 n H H3 p 3 Rn πµ [ 2 2 H πµ 1 Rn 2 2 H [ ] R 2 p (z h)2] dz + R 2 p (z h)2 (z h) R 2 n 2 Z2 ( ( R 2 n Z 2) + R 3/2 (R 2 2 p (z h)2) 3/2 n Z2) 2 dz + Z πµ Rn (R 2 n Z2) 3 2 H (R 2 p (z h)2) 2 dz ) ( ) 2 ( R n R 2 n R2 p 2R n + R 2 n R2 p. ) + )

21 21 / 26 Cell deformation Continuum Model The cell can enter inside the channel if W adhesion W tot, where W adhesion is the work done by adhesive forces W adhesion = F Z adhesion L, L = L fin n L 0 n is the total displacement of the cell inside the channel

22 Cell deformation Continuum Model Elastic membrane Shape Linear Force Constant (bounded) Force [ ( ) ] 2 [ ( ) ] 2 1 R 2 p 1 + sin 1 (ẽ) 1 Ellipsoid G k λ 16π 2 R 3 1 R 2 p 1 + sin 1 (ẽ) pẽ G F λ 8π 2 R 3 1 pẽ Cigar G k λ 16π R p L2 b L ellips ( 1 R 2 p ) 2 3 R 1 p R p L2 b L cigar G F λ 8π R p L( ) L ( b ellips 1 R 2 p ) 2 3 R 1 p R p L( ) L b cigar Linear forces, ( G k ) λ ellipsoid cigar shaped Constant forces, ( G F ) λ ellipsoid cigar shaped Bounded adhesive region, ( G F λ 5 ellipsoid cigar shaped ) G k λ = ρ bα ECM k b λ G F λ = ρ bα ECM Fb M R 1 n λ Rp Rp Rp 22 / 26

23 Cell deformation Continuum Model Elastic nucleus Shape Linear Force Constant (bounded) Force 2 R 2 p + 1 R4 p 3 2 R 2 p + 1 R4 p 3 Ellipsoid G k µ 2 3 R p L 2 b L G F µ 2 ellips 3 R p L( ) b L ellips Cigar G k µ 4 I( R p) G 3 R F µ p L2 b L 2 I( R p) cigar 3 R p L( ) b L cigar Linear forces, ( G k ) µ ellipsoid cigar shaped Constant forces, ( G F ) µ ellipsoid cigar shaped Bounded adhesive region, ( G F ) µ 5 ellipsoid 4 3 cigar shaped G k µ = ρ bα ECM k b R n µ G F µ = ρ bα ECM F M b µ Rp Rp Rp 23 / 26

24 24 / 26 Cell deformation Continuum Model Elastic nucleus High G F µ High ρ b,α ECM, F M b or low µ cell can enter inside channels with small radii. Given mechanical and adhesive properties establish the minimum pore size of the network Constant forces, ( G F ) µ ellipsoid cigar shaped Bounded adhesive region, ( G F ) µ 5 ellipsoid 4 3 cigar shaped G k µ = ρ bα ECM k b R n µ 0.4 CELL 1 CELL G F µ = ρ bα ECM F M b µ Rp Rp

25 Conclusions & Future steps Conclusions simple general framework, valid even for more complicated nuclear constitutive equations and bonds forces; qualitative agreement with biological findings: cells are able to enter inside ECM-channels only for pore radii bigger than a critical one; application: evaluation of critical scaffold pore size in tissue engineering. 25 / 26

26 Conclusions & Future steps Future steps need of more experimental works in order to validate the model and obtain quantitative results study the whole 3D cinematic process (necessary but non sufficient condition) integration of the cytoplasmic membrane deformation inclusion of these results in the permeability coefficient 26 / 26