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• Stress dependent morphogenesis continuum mechanics and system of trusses

  J. Muñoz, V. Conte, M. Miodownik

  (2010). "Continuum mechanics". Nova Publishers, p. 223-243.

  
  

  Abstract

  The mechanical analysis of soft tissues in biomechanics has undergone an increasing progress during the last decade. Part of this success is due to the development and application of some techniques of continuum mechanics, in particular, the decomposition of the deformation gradient, and the introduction of mass, density or volume changes in the reference configuration. Resorting to the common terminology employed in the literature, the changes in biomechanical processes may be classified as growth (change of mass), remodelling (change of density or other material properties such as fibre orientation) or morphogenesis (change of shape). Although the use of those concepts in bone and cardiovascular analysis is well extended, their use in morphogenesis during embryo development has been far less studied. The reasons of this fact may be found in the large shape changes encountered during this process, or the complexity of the material changes involved. In this chapter we develop a general framework for the modelling of morphogenesis by introducing a growth process in the structural elements of the cell, which in turn depends on the stress state of the tissue. Some experimental observations suggest this feedback mechanism during embryo development, and only very recently this behaviour has started to be simulated. We here derive the necessary equilibrium equations of a stress controlled growth mechanism in the context of continuum mechanics. In these derivations we assume a free energy source which is responsible of the active forces during the elongation process, and a passive hyperelastic response of the material. In addition, we write the necessary conditions that the active elongation law must satisfy in order to be thermodynamically consistent. We particularise these equations and conditions for the relevant elements of the cytoskeleton, namely, microfilaments and microtubules. We apply themodel to simulate the shape changes observed during embryomorphogenesis in truss element. As a salient result, themodel reveals that by imposing boundary stress conditions, unbounded elongation would be obtained. Therefore, either prescribed displacements or cross-links between fibres are necessary to reach a homeostatic state.

  Abstract
  The mechanical analysis of soft tissues in biomechanics has undergone an increasing progress during the last decade. Part of this success is due to the development and application [...]

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• Planar differential growth rates initiate precise fold positions in complex epithelia

  M. Tozlouglu, M. Duda, N. Kirkland, R. Barrientos, J. Burden, J. Muñoz, Y. Mao

  Developmental cell (2019). Vol. 51 (3), pp. 299-312

  
  

  Abstract

  Tissue folding is a fundamental process that shapes epithelia into complex 3D organs. The initial positioning of folds is the foundation for the emergence of correct tissue morphology. Mechanisms forming individual folds have been studied, but the precise positioning of folds in complex, multi-folded epithelia is less well-understood. We present a computational model of morphogenesis, encompassing local differential growth and tissue mechanics, to investigate tissue fold positioning. We use the Drosophila wing disc as our model system and show that there is spatial-temporal heterogeneity in its planar growth rates. This differential growth, especially at the early stages of development, is the main driver for fold positioning. Increased apical layer stiffness and confinement by the basement membrane drive fold formation but influence positioning to a lesser degree. The model successfully predicts the in vivo morphology of overgrowth clones and wingless mutants via perturbations solely on planar differential growth in silico.  

  Abstract
  Tissue folding is a fundamental process that shapes epithelia into complex 3D organs. The initial positioning of folds is the foundation for the emergence of correct tissue [...]

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• Computational modeling of epithelial wound healing: Short and long term chemo-mechanical mechanisms

  L. Roldán, J. Muñoz, P. Sáez

  Comput. Methods Appl. Mech. Engrg., (2019). Vol. 350, pp. 28-56

  
  

  Abstract

  During the lifetime of all living multicellular organisms, wounds in their tissues are frequently observed. The capability of closing those gaps is fundamental for a healthy development. If done deficiently, many diseases may occur from simple inflammation to tumor formation. The wound healing process in epithelial tissue occurs in three different stages. The first one is the assembly of a supra-cellular actomyosin cable and its migration towards the wound edge, triggered by biochemical processes in which calcium plays a distinctive role. How this process is orchestrated following damage remains unclear. Later, after its positioning, the cable contracts driving the tissue towards the gap and reducing the wound area. Finally, cell migration towards the interior of the wound ends up sealing the tissue. In this work, we make use of a mechanical continuum model for the first two stages in order to developed and 2D finite element simulations within a monolithically fully implicit implementation. The model for the actomyosin cable formation involves the coupling of transient calcium ions transport, with actin fibers and myosin motors recruitment and non-linear mechanical response of the tissue. The contraction stage, the active deformation of the previously formed actomyosin cable is taken into account. The relative motion of the myosin motors over the actin filaments is modeled so there exists an active tissue contraction in the direction of those fibers. Upon implementation, the model is capable of performing a wide range of biophysical situations reported experimentally, as we demonstrate in our numerical results. We have been able to rationalize through computational mechanics the firing of calcium in the wound right after damage infliction as well as the consequent formation of actin ring, reproducing nicely what has been reported in biological literature. Thereafter, the numerical model of acto-myosin contraction, fully integrated with the non-linear mechanics of the problem, correlates with the mechanics of wound closure at the actin-ring contraction stage. More importantly, the approach is the first of its kind in the modeling of epithelial and embryonic cell layers, where a wide number of complex mechanics has been integrated and solved though computational methods in engineering. We believe that the simulations will help to unravel new insights in open questions of developmental biology.

  Abstract
  During the lifetime of all living multicellular organisms, wounds in their tissues are frequently observed. The capability of closing those gaps is fundamental [...]

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• Stress relaxation in epithelial monolayers is controlled by the actomyosin cortex

  N. Khalilgharibi, J. Fouchard, N. Asadipour, R. Barrientos, M. Duda, A. Bonfanti, A. Yonis, A. Harris, P. Mosaffa, Y. Fujita, A. Kabla, Y. Mao, B. Baum, J. Muñoz, M. Miodownik, G. Charras

  Nature Physics (2019). Vol. 15, pp. 839-847 (Preprint)

  
  

  Abstract

  Epithelial monolayers are one-cell-thick tissue sheets that line most of the body surfaces, separating internal and external environments. As part of their function, they must withstand extrinsic mechanical stresses applied at high strain rates. However, little is known about how monolayers respond to mechanical deformations. Here, by subjecting suspended epithelial monolayers to stretch, we find that they dissipate stresses on a minute timescale and that relaxation can be described by a power law with an exponential cut-off at timescales larger than about 10 s. This process involves an increase in monolayer length, pointing to active remodelling of cellular biopolymers at the molecular scale during relaxation. Strikingly, monolayers consisting of tens of thousands of cells relax stress with similar dynamics to single rounded cells, and both respond similarly to perturbations of the actomyosin cytoskeleton. By contrast, cell–cell junctional complexes and intermediate filaments do not relax tissue stress, but form stable connections between cells, allowing monolayers to behave rheologically as single cells. Taken together, our data show that actomyosin dynamics governs the rheological properties of epithelial monolayers, dissipating applied stresses and enabling changes in monolayer length.

  Abstract
  Epithelial monolayers are one-cell-thick tissue sheets that line most of the body surfaces, separating internal and external environments. As part of their function, they [...]

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• Mechanical oscillations in biological tissues as a result of delayed rest-length changes

  J. Muñoz, M. Dingle, M. Wenzel

  Physical review E: statistical, nonlinear and soft matter physics (2018). Vol. 98 (5), pp. 1-15 (Preprint)

  
  

  Abstract

  Oscillatory behavior in tissue biology is ubiquitous and may be observed in the form of either chemical or mechanical signals. We present here and solve a mechanical cell model that exhibits oscillations emerging from delayed viscoelastic rheological laws. These include a time delay between the mechanical response and the rest-length changes, which evolve proportionally to the delayed cell deformation and use a remodeling rate parameter. We show that different regimes (no oscillatory response, sustained oscillations, and unstable oscillations) are obtained for different values of the delay or the remodeling rate. The results are analytically demonstrated in a one-dimensional problem with one and two cells that are represented by simple line elements. Oscillations of the cell deformations are obtained whenever different delays coexist, or when the delay is size-dependent. We also extend our results to a multicellular two-dimensional vertex model that includes the same rheological law, and which inherits the presence of critical values of the delay or remodeling rate. We numerically show that indeed the size-dependent rest-length changes induce oscillations in the cell shapes and areas.

  Abstract
  Oscillatory behavior in tissue biology is ubiquitous and may be observed in the form of either chemical or mechanical signals. We present here and solve a mechanical cell [...]

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• Computation of forces from deformed visco-elastic biological tissues

  J. Muñoz, D. Amat, V. Conte

  Inverse Problems (2018). Vol. 34 (4), pp. 1-27 (Preprint)

  
  

  Abstract

  We present a least-squares based inverse analysis of visco-elastic biological tissues. The proposed method computes the set of contractile forces (dipoles) at the cell boundaries that induce the observed and quantified deformations. We show that the computation of these forces requires the regularisation of the problem functional for some load configurations that we study here. The functional measures the error of the dynamic problem being discretised in time with a second-order implicit time-stepping and in space with standard finite elements. We analyse the uniqueness of the inverse problem and estimate the regularisation parameter by means of an L-curved criterion. We apply the methodology to a simple toy problem and to an in vivo set of morphogenetic deformations of the Drosophila embryo.

  Abstract
  We present a least-squares based inverse analysis of visco-elastic biological tissues. The proposed method computes the set of contractile forces (dipoles) at the cell boundaries [...]

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• Hybrid cell-centred/vertex model for multicellular systems with equilibrium-preserving remodelling

  P. Mosaffa, A. Rodríguez-Ferran, J. Muñoz

  Int. J. Numer. Meth. Bio. Engng. (2018). Vol. 34 (3), pp. 1-33 (Preprint)

  
  

  Abstract

  We present a hybrid cell‐centred/vertex model for mechanically simulating planar cellular monolayers undergoing cell reorganisation. Cell centres are represented by a triangular nodal network, while the cell boundaries are formed by an associated vertex network. The two networks are coupled through a kinematic constraint which we allow to relax progressively. Special attention is paid to the change of cell‐cell connectivity due to cell reorganisation or remodelling events. We handle these situations by using a variable resting length and applying an Equilibrium‐Preserving Mapping on the new connectivity, which computes a new set of resting lengths that preserve nodal and vertex equilibrium. We illustrate the properties of the model by simulating monolayers subjected to imposed extension and during a wound healing process. The evolution of forces and the Equilibrium‐Preserving Mapping are analysed during the remodelling events. As a by‐product, the proposed technique enables to recover fully vertex or fully cell‐centred models in a seamless manner by modifying a numerical parameter of the model.

  Abstract
  We present a hybrid cell‐centred/vertex model for mechanically simulating planar cellular monolayers undergoing cell reorganisation. Cell centres are represented by a triangular [...]

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• Preface

  J. Font-Llagunes, J. Muñoz, J. Ambrósio

  Multibody system dynamics (2017). Vol. 39 (1), pp. 1-2

  
  

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• Porous-based rheological model for tissue fluidisation

  N. Asadipour, X. Trepat, J. Muñoz

  Journal of the Mechanics and Physics of Solids (2016). Vol. 96, pp. 535-549 (Preprint)

  
  

  Abstract

  It has been experimentally observed that cells exhibit a fluidisation process when subjected to a transient stretch, with an eventual recovery of the mechanical properties upon removal of the applied deformation. This fluidisation process is characterised by a decrease of the storage modulus and an increase of the phase angle. We propose a rheological model which is able to reproduce this combined mechanical response. The model is described in the context of continua and adapted to a cell-centred particle system that simulates cell–cell interactions. Mechanical equilibrium is coupled with two evolution laws: (i) one for the reference configuration, and (ii) another for the porosity or polymer density. The first law depends on the actual strain of the tissue, while the second assumes different remodelling rates during porosity increase and decrease. The theory is implemented on a particle based model and tested on a stretching experiment. The numerical results agree with the experimental measurements for different stretching magnitudes

  Abstract
  It has been experimentally observed that cells exhibit a fluidisation process when subjected to a transient stretch, with an eventual recovery of the mechanical [...]

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• Collective cell durotaxis emerges from long-range intercellular force transmission

  R. Sunyer, V. Conte, J. Escribano, A. Elosegui-Artola, A. Labernadie, L. Valon, D. Navajas, J. García-Aznar, J. Muñoz, P. Roca-Cusachs, X. Trepat

  Science (2016). Vol. 353 (6304), pp. 1157-1161 (Preprint)

  
  

  Abstract

  The ability of cells to follow gradients of extracellular matrix stiffness—durotaxis—has been implicated in development, fibrosis, and cancer. Here, we found multicellular clusters that exhibited durotaxis even if isolated constituent cells did not. This emergent mode of directed collective cell migration applied to a variety of epithelial cell types, required the action of myosin motors, and originated from supracellular transmission of contractile physical forces. To explain the observed phenomenology, we developed a generalized clutch model in which local stick-slip dynamics of cell-matrix adhesions was integrated to the tissue level through cell-cell junctions. Collective durotaxis is far more efficient than single-cell durotaxis; it thus emerges as a robust mechanism to direct cell migration during development, wound healing, and collective cancer cell invasion.

  Abstract
  The ability of cells to follow gradients of extracellular matrix stiffness—durotaxis—has been implicated in development, fibrosis, and cancer. Here, we found multicellular [...]

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