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Variability and probability distribution functions
In tissues, cell behaviour differs substantially from that of single cells because of cellcell adhesion: collective processes occur, cells modify their shapes and their physical parameters, such as their stiffness, depending on their neighbours. Moreover, some heterogeneities of physical parameters appear throughout the tissue, in addition to the internal cell variation. This variation is clearly visible if we look at the statistical distribution of these parameters. Among these quantities, the projected cell area appears to have a similar distribution over different cell types that can be fitted with a stochastic model [208] (Fig.I.13a).
Studying the distribution of cell velocity within cellular layers, different groups [36, 166, 197] found an exponential distribution for various cell types with a scale of the order of 100 μm.h−1 (Fig.I.13b) and some modeling with a stochastic equation has been provided in order to explain this behaviour. Similarly, we observe a Laplacian distribution for the traction force (exponential distribution for the absolute value), i.e. with tails larger than for a Gaussian distribution [63, 195]. For MDCK cells, a scale of the order of 10 kPa has been found (Fig.I.13c-d). Using a force-driven heterogeneous elastic membrane sliding over a visI. cous substrate, has allowed [6] to find a similar distribution. But to our knowledge, no study of the relation between the distributions of these different quantities has been performed yet (see Chap.III.3.3).
In Appendix A, we define the univariate probability distribution functions (PDF) used in this work, listing their parameters and the expressions of their mean, mode and variance as a function of the parameters.
Summary of « Tissue fusion over nonadhering surfaces »
The experiment is very similar to the one performed in [32] but the wound domain is treated with PEG (Polyethylene glycol) in order to be nonadhesive for cells. For wounds smaller than a single cell, the tissue simply bridges the gap and does neither sense the nonadhesive substrate nor create an actin ring, confirming that cells have the ability to bridge over small nonadherent defects [159]. On the contrary for large wounds (R > 70 μm in our case), full closure does not occur, even after 7 days (Fig.S8).
In order to study gap closure, we limit our study to radii comprised between 30 and 50 μm. For a given initial radius, the closing time is, as expected, much higher than for the adhesive case (at least ten times higher). Two striking differences appear, (i) the trajectory of the closure is erratic and noisy and (ii) the distribution of the closure time tc at fixed initial radius R is much broader (Fig.2A,C-E). These information led us to write a stochastic force balance equation at the tissue edge. The dynamics of the wound radius r is described by a Langevin equation (I.3.1): dr dt = − ˜ r + p 2D(t) (II.2.1) where ˜ is the reduced line tension and p2D(t) is a noise term of variance 2D. The fraction of closed wounds, f(R, t), obeys a backward Fokker-Planck equation, with appropriate boundary conditions, which we solve numerically. From experimental observations, we choose a reflecting boundary at the initial wound radius r = R, whereas for r = 0 we impose an absorbing boundary condition. A least-squares fit of experimental data yields the following estimates of the parameters (Fig.3): ˜ MDCK = 10 μm2.h−1.
Validation with heterogeneous parameters
In Sec.I.2.6, we have seen that in a tissue, cell areas, velocities, traction forces are distributed along a wide range of values. Furthermore, in [72], a variation of 5% around the mean value of the strain is observed under constant force. These variabilities suggest that cell monolayers contain some heterogeneity and that physical parameters fluctuate around their mean value. In order to test BISM in this case, we simulate a viscous tissue with a non uniform viscous coefficient Eq.(III.2.6). First, we consider a fluctuating viscous coefficient distributed uniformly about its mean value 0 = 103 kPa.μm.s, U(0.75 0, 1.25 0) (Fig.III.3a). We next test a sinusoidal spatial profile of relative amplitude 25% about the same mean value (Fig.III.3b, = 0 + sin(4x/Lx) sin(4y/Ly), = 250 kPa.μm.s, Lx = Ly = 100 μm the system dimensions). These two tests lead to R2 = 0.96, a result similar to the homogeneous case (Fig.III.3d-e).
Since a linear relation between the stiffness and the cytoskeletal stress has been observed in single cells [204, 137], we also consider the case of a linear dependence of the viscous parameter with the pressure: = 0 + 0 2 Tr.
Table of contents :
I Introduction
I.1 Cells and Tissues
I.1.1 Tissues
I.1.2 Cell monolayers
I.1.3 Forces
I.2 Measurement of physical quantities
I.2.1 Single cell measurements
I.2.2 External force measurements: traction force microscopy
I.2.3 Internal force measurements
I.2.4 Velocity measurements
I.2.5 Rheological models
I.2.6 Variability and probability distribution functions
I.3 Theoretical methods
I.3.1 Stochastic calculus
I.3.2 Bayesian inversion
I.3.3 Hyperparameter selection
I.3.4 Likelihood and model selection
I.3.5 Kalman filter
I.4 Outline
II Stochastic wound healing
II.1 Motivation
II.2 Summary of « Tissue fusion over nonadhering surfaces »
II.3 Discussion
II.3.1 Model selection
II.3.2 HaCaT experiments
II.3.3 Perspectives
IIIBayesian Inversion Stress Microscopy (BISM)
III.1 Motivation
III.1.1 Inference from cell geometry
III.1.2 Inference from TFM
III.2 Summary of « Inference of Internal Stress in a Cell Monolayer »
III.3 Discussion
III.3.1 Additional validations
III.3.2 Defects
III.3.3 Distribution of traction forces and internal stresses
III.3.4 Extension to 3D
IV Kalman Inversion Stress Microscopy (KISM)
IV.1 Statistical model
IV.1.1 Observation and evolution models
IV.1.2 Kalman filter
IV.1.3 Hyperparameter values
IV.2 Numerical validation
IV.2.1 Numerical simulation
IV.2.2 Validation
IV.3 Experimental application
IV.3.1 Quasi-1D monolayer in a ring
IV.3.2 2D monolayer in a square domain
IV.4 Summary and perspectives
V Bayesian Inversion Stress Microscopy from substrate displacement (BISMu)
V.1 Traction force measurements on a soft gel
V.1.1 Boundary Element Method
V.1.2 Fourier Transform Traction Cytometry
V.1.3 Other methods
V.2 Statistical models
V.2.1 Real space: BISMu
V.2.2 Fourier space: BISMuf
V.3 Numerical validation
V.3.1 Numerical simulation
V.3.2 Real space
V.3.3 Fourier space
V.3.4 Summary
V.4 Application to experimental data
VI Perspectives
VI.1 Model selection applied to constitutive equations
VI.2 Rotating ring
VI.3 Magnetized aggregate
Résumé en français
A Univariate probability distributions
B Code BISM/ KISM
C Experimental data
C.1 Wound healing on nonadhesive substrate (Guillaume Duclos, Maxime Deforet)
C.2 Rotating ring (Shreyansh Jain)
C.3 Defect (Thuan Beng Saw)
C.4 Square (Grégoire Peyret)
C.5 Magnetized aggregate (François Mazuel)
D Topological defects in epithelia govern the extrusion of dead cells
Bibliography
