Pradeep Kumar Bal submitted his PhD thesis Mathematical and computational modeling of the active mechanics of multicellular systems: from cell–cell adhesion to epithelial reshaping, supervised by Marino Arroyo at the Universitat Politècnica de Catalunya (September 2025). He is currently expanding his doctoral research as a postdoctoral researcher with Prof. Marino Arroyo.
This thesis develops theoretical and computational frameworks to model two fundamental mechanical functions of multicellular tissues: cell–cell adhesion and epithelial reshaping. Although these processes are governed by subcellular dynamics, they manifest at mesoscopic scales, posing a challenge for existing modeling approaches. The overarching goal is to bridge this gap by providing models that capture the essential physical and biological mechanisms across scales. The work is structured in two parts, each addressing a distinct aspect of tissue mechanics, while sharing a common theoretical foundation based on irreversible thermodynamics and active gel theory.
Part I of the thesis focuses on the dynamic formation and organization of cell–cell adhesions, with particular emphasis on cell doublets as the simplest multicellular unit. A mesoscale theoretical framework is developed that couples the mechanics of the cellular surface with the chemical kinetics of adhesion molecules, their lateral diffusion on the membrane, and their feedback with the actomyosin cortex. The formulation is grounded in Onsager’s variational formalism to ensure thermodynamic consistency and is implemented computationally using both axisymmetric and fully three-dimensional finite element methods.
The emergence of multicellularity represents a key evolutionary transition, enabled mechanically by the ability of cells to adhere and form tissues. This adhesion is primarily mediated by specialized transmembrane proteins, notably those of the cadherin family (Figure 14a). Regulation of cell–cell adhesion plays a central role during early mammalian development [1], with the simplest tissue consisting of a pair of adhered cells. Beyond cohesion, differential adhesion and cortical tension along cell–cell interfaces actively control tissue architecture.
To capture this complexity, we propose a theoretical and computational model that integrates the reaction kinetics of adhesion molecules, their lateral mobility on the membrane, and their coupling to the actomyosin cytoskeleton. The framework combines an active gel description of the actin cortex [2] with a chemo-mechanical model for adhesion dynamics [3] (Figure 14b). These components are coupled using Onsager’s variational formalism for irreversible thermodynamics, resulting in a fully nonlinear and thermodynamically consistent formulation. Axisymmetric and fully three-dimensional solution frameworks are developed to study the self-organization of adhesion in matured cell doublets (Figure 14c), and the model is validated against simulations of force-induced decohesion of cortical surfaces (Figure 14d).
Simulations reveal how out-of-equilibrium mechano-chemical couplings, including force-activated bonds, reduced cortical contractility within adhesion zones, and the immobilization of activated bonds, drive the formation of large, mature adhesion patches enriched in trans bonds (Figure 14c). These results reproduce key experimental observations of adhesion maturation and provide a mechanistic framework for understanding how microscopic activity gives rise to emergent adhesion structures.
Part II focuses on the development of continuum models to study epithelial reshaping, a central driver of morphogenesis. At larger length scales, epithelial sheets represent a prototypical class of cellular tissues in animals, composed of one or a few layers of tightly connected cells forming cohesive barriers that often separate a cavity or free surface from underlying stromal tissue [4]. During development, these tissues envelop embryos and undergo extensive reshaping while performing essential physiological functions such as secretion, absorption, filtration, and protection of underlying tissues.
The functional role of epithelial tissues in development and homeostasis depends critically on their three-dimensional geometry. Epithelial folding underlies many morphogenetic processes, including Drosophila gastrulation and wing development [5], as well as placode invagination in organs such as salivary glands and hair follicles. Out-of-plane deformations arise from apicobasal asymmetries in contractility or from mechanical instabilities such as buckling induced by lateral compression [6].
At cellular and subcellular scales, epithelial reshaping emerges from the interplay of multiple mechanisms, including cell division, apoptosis, and neighbor rearrangements, as well as actively generated forces that drive bending through contractility asymmetries or promote folding via compressive stresses [7]. Morphogenesis often results from the coordinated action of these mechanisms rather than a single dominant process [6].
Existing models, however, fail to connect subcellular cytoskeletal dynamics with the large-scale reshaping of epithelial monolayers, which behave as active continuum shells. The first aim of Part II is thus to develop a continuum shell theory derived from active gel descriptions of the actin cortex. We present a fully nonlinear, viscoelastic, active theory that homogenizes a 3D vertex-like model of individual cell surfaces (see Figure 15 (I)). Each cell surface is treated as an active gel patch, yielding a continuum surface model that captures the collective behavior of viscoelastic active gels undergoing turnover. The theory accounts for geometric and mechanical anisotropy of lateral junctions, as well as apicobasal asymmetries, by relating the deformation of apical, basal, and lateral surfaces to the mid-surface of the tissue. Two formulations are introduced: a Kirchhoff shell theory, with lateral junctions perpendicular to the mid-surface, and a Cosserat theory, allowing junctional tilt. Using a variational formalism, microscopic Rayleighian functionals for each cellular surface are coarse-grained to yield continuum governing equations, which are implemented using finite element methods. Comparison with 3D vertex simulations demonstrates excellent agreement between the continuum and discrete models, and between the Kirchhoff and Cosserat theories (Figure 15 (II)).
The second aim is to develop a finite element approach to approximate this shell theory. The third aim is to apply these models and methods to understand how apicobasal asymmetries and buckling control the reshaping of epithelial shells. More specifically, we aim to elucidate the mechanisms underlying epithelial wrinkling and pattern formation in experiments involving rapidly deflated epithelial shells (see Figure 15 (III and IV))[8]. The continuum model demonstrates how cortical viscoelasticity, viscous drag from the surrounding medium, and curvature anisotropy determine the morphology and spatial patterning of wrinkles in epithelial shells. We then study wrinkling of compressed and asymmetrical epithelial sheets (see Figure 15 (V)), as investigated by [9] using a 1D beam-like continuum model that accounts for tilt and coarse-grains a 2D lateral vertex model, thereby providing a verification of our Cosserat theory. Finally, we examine the effect of apicobasal asymmetry on free-standing tissues by simulating the experiments of [10], finding that the tilt emerging from the Cosserat theory is essential to achieve stable steady states (see Figure 15 (VI)).
The continuum model offers several advantages over detailed cellular models: it enables efficient simulations, is amenable to mathematical analysis, and identifies essential parameters. While this work focuses on the actomyosin cortex, the coarse-graining approach can incorporate additional elements, such as apical belts, intermediate filaments, or adhesion dynamics from Part I. It can also be coupled to deformable substrates or adapted to plant tissues. Although our theory assumes that tissue architecture is fixed, it provides a natural framework to extend to evolving architectures due to cell division, extrusion, or junctional rearrangements (Lemke and Nelson, 2021). Finally, the approach can be combined with models of mechanosensitive signaling (Bollenbach et al., 2007; Hidalgo et al., 2019) to explore the interplay between mechanics and morphogen-mediated regulation, offering a powerful tool to understand epithelial tissue mechanobiology.
References
[1] J. Maître Pulsatile cell-autonomous contractility drives compaction in the mouse embryo Nature cell biology 7 (2015) 849–855.
[2] J. Prost et al. Active gel physics Nature Physics, 2, 111-117.
[3] D. Kaurin, P.K. Bal and M. Arroyo. Peeling dynamics of fluid membranes bridged by molecular bonds: moving or breaking Journal of The Royal Society Interface191, 20220183, 202.
[4] L. A. Davidson, Epithelial machines that shape the embryo Trends in Cell Biology2, 82–87, 20142.
[5] M. Rauzi et al. Embryo-scale tissue mechanics during Drosophila gastrulation movements Nature communications1 (2015), 8677
[6] S. B. Lemke and C.M. Nelson, Dynamic changes in epithelial cell packing during tissue morphogenesis Current Biology18 (2021), R1098–R1110.
[7] K. D. Sumigray et al. Morphogenesis and compartmentalization of the intestinal crypt Developmental cell 2 (2018), 183–197.
[8] N. Chahare et al. Multiscale wrinkling dynamics in epithelial shells bioRxiv (2025), 2025–06.
[9] U. Andrenšek et al. Wrinkling instability in unsupported epithelial sheets Physical Review Letters19 (2023), 198401.
[10] J. Fouchard et al. Curling of epithelial monolayers reveals coupling between active bending and tissue tension Proceedings of the National Academy of Sciences17 (2020), 9377–9383