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Interacting active surfaces: A model for three-dimensional cell aggregates [1]

['Alejandro Torres-Sánchez', 'Theoretical Physics Of Biology Laboratory', 'The Francis Crick Institute', 'London', 'United Kingdom', 'Max Kerr Winter', 'Guillaume Salbreux', 'Department Of Genetics', 'Evolution', 'University Of Geneva']

Date: 2023-01

Abstract We introduce a modelling and simulation framework for cell aggregates in three dimensions based on interacting active surfaces. Cell mechanics is captured by a physical description of the acto-myosin cortex that includes cortical flows, viscous forces, active tensions, and bending moments. Cells interact with each other via short-range forces capturing the effect of adhesion molecules. We discretise the model equations using a finite element method, and provide a parallel implementation in C++. We discuss examples of application of this framework to small and medium-sized aggregates: we consider the shape and dynamics of a cell doublet, a planar cell sheet, and a growing cell aggregate. This framework opens the door to the systematic exploration of the cell to tissue-scale mechanics of cell aggregates, which plays a key role in the morphogenesis of embryos and organoids.

Author summary Understanding how tissue-scale morphogenesis arises from cell mechanics and cell-cell interactions is a fundamental question in developmental biology. Here we propose a mathematical and numerical framework to address this question. In this framework, each cell is described as an active surface representing the cell acto-myosin cortex, subjected to flows and shape changes according to active tensions, and to interaction with neighbouring cells in the tissue. Our method accounts for cellular processes such as cortical flows, cell adhesion, and cell shape changes in a deforming three-dimensional cell aggregate. To solve the equations numerically, we employ a finite element discretisation, which allows us to solve for flows and cell shape changes with arbitrary resolution. We discuss applications of our framework to describe cell-cell adhesion in doublets, three-dimensional cell shape in a simple epithelium, and three-dimensional growth of a cell aggregate.

Citation: Torres-Sánchez A, Kerr Winter M, Salbreux G (2022) Interacting active surfaces: A model for three-dimensional cell aggregates. PLoS Comput Biol 18(12): e1010762. https://doi.org/10.1371/journal.pcbi.1010762 Editor: Saúl Ares, CNB: Centro Nacional de Biotecnologia, SPAIN Received: March 23, 2022; Accepted: November 26, 2022; Published: December 16, 2022 Copyright: © 2022 Torres-Sánchez et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. Data Availability: The source code used to produce the results and analyses presented in this manuscript is available at the Git repository: https://github.com/torressancheza/ias. Funding: ATS, MKW and GS acknowledge support from the Francis Crick Institute, which receives its core funding from Cancer Research UK (FC001317, https://www.cancerresearchuk.org), the UK Medical Research Council (FC001317, https://www.ukri.org/councils/mrc/), and the Wellcome Trust (FC001317, https://wellcome.org). GS and ATS acknowledge support from a grant to GS, Chris Dunsby, Axel Behrens from the Engineering and Physical Sciences Research Council (EP/T003103/1, https://epsrc.ukri.org). The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript. Competing interests: The authors have declared that no competing interests exist.

1 Introduction Tissue morphogenesis relies on the controlled generation of the cellular forces that collectively drive tissue-scale flows and deformation [1, 2]. The interplay between cell-cell adhesion, cellular mechanics and the cytoskeleton plays a key role in determining how biological tissues self-organise [3]. These ingredients are also crucial for the growth of in vitro organoids, organ-like systems derived from stem cells which can self-organise into complex structures reminiscent of actual organs [4–6]. Several classes of models have been proposed to describe the mechanics of multicellular aggregates, such as cellular Potts models [7–13], phase field models [14–18] and vertex models [19]. Vertex models, and the closely related Voronoi vertex models [20, 21], describe cells in a tissue as polyhedra that share faces, edges and vertices forming a three-dimensional junctional network [19, 22–24]. Cell deformations are encoded by the displacement of the vertices X a of the network. These displacements are dictated by the balance of vertex forces F a stemming from cell pressures P c , surface tensions t f , and line tensions Γ e that are coupled to virtual changes in cell volume δV c , face area δA f , and edge length δl e respectively in a work function (1) where to get to the last expression one needs to express δV c , δA f and δl e in terms of a virtual displacement of the vertices δX a . Two and three-dimensional versions of the vertex model have been employed for many applications, for instance to study cell packing [23, 25], cell sorting [26], wound closure [27], cyst formation [28], tumourigenesis in tubular epithelia [29] among many other [19]. However because of their definition, vertex models do not explicitly resolve cortical flows on the cell surface. The effect of cell-cell adhesion is implicitly introduced in the surface tension t f , which mixes together physical processes arising from cell-cell adhesion and surface forces in the cell membrane and in the actomyosin cortex. In vertex models with vertices positions as degrees of freedom, topological transitions leading to cell-neighbour exchange are encoded explicitly by formulating rules to change edges in the network. At the single cell scale, a number of studies have shown the relevance of coarse-grained, continuum models to describe the mechanics of the cell surface. In animal cells, the cell surface is composed of a lipid membrane surrounding the actin cortex, a layer of cross-linked actin filaments undergoing continuous turnover of its constituents. Because of this turnover, the actin cortex behaves as a viscoelastic material with a characteristic remodelling time of tens of seconds [30, 31]. When looking at dynamics at time-scales beyond this relaxation time, the cortex can be seen as a viscous fluid layer. Because of the large cortical surface tension and effective 2D viscosity compared to the cell membrane [30, 32], models of cell surface mechanics have been focusing on the mechanics of the cortex. In this approach, an active fluid theory taking into account cellular cortical flows, gradients of active cytoskeletal tension and their regulation, and orientation and filament alignment in the actin cortex, has proven successful to describe the mechanics of cell polarisation, cell motility or cell division [33–42]. From a computational perspective, there has been a growing attention to the simulation of the dynamics of fluid interfaces both with prescribed [43–46] and with time-evolving shape [41, 42, 47–53]. In addition, recent discrete deformable cell models have described cellular aggregates by representing cells by triangular meshes with viscoelastic edges [54–57]. However, to our knowledge no computational framework has attempted to provide a physical description of three-dimensional cellular aggregates taking into account explicitly the mechanics of a single cell surface described as an active fluid surface, as well as cell-cell adhesions. Here we bridge this gap and introduce a new modelling and simulation framework, and a freely available code [58], for the mechanics of cell aggregates in three dimensions (Fig 1). We describe cells as interacting active surfaces [59]. The governing equations for the cell surface mechanics are discretised using a finite element method. In this method, each cell is represented by a three-dimensional mesh with vertices positions X a . In analogy with Eq (1) in vertex models, we start from the virtual work theorem for interfaces (Eq (2)) and find the net forces at the vertices F a (Eq (11)), which vanish in the absence of inertia. This condition allows us to obtain cortical flows and cell shape changes. In this framework, cell-neighbour exchange appears as a natural output of the remodelling of cell-cell interactions and is not treated explicitely. PPT PowerPoint slide

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TIFF original image Download: Fig 1. Schematic of the interacting active surface framework. A tissue is described by a collection of triangular meshes representing each cell. The dynamics of the tissue is described by the dynamics of the vertices making the cellular meshes, similar to how the movement of the vertices of a vertex model describe the deformation of a tissue. The motion of the cell mesh is obtained by coarse-graining continuum mechanics equations of a theory of active surfaces via the finite element method. In this theory, cortical flows, cortical tensions, intracellular pressures, bending moments and forces arising from cell-cell interactions are taken into account. https://doi.org/10.1371/journal.pcbi.1010762.g001 The main benefits of our method are that it (1) can incorporate complex descriptions of the physics of the cell surface, including sources of tension, in-plane and normal moments [59], (2) accounts for cell-cell adhesion explicitly through constitutive laws that can be adapted to represent different biological scenarios, (3) resolves cell shape and cortical flows with arbitrary resolution given by the mesh size of the finite element discretisation with a well-defined continuum limit. The nonlinear equations of the model are treated computationally with the use of nonlinear solvers. Since we consider only the discretisation of the cell surface, the number of degrees of freedom is considerably smaller than in 3D Cellular Potts or phase-field models, both of which require a 3D discretisation. On the other hand, to capture cell shape, cortical flows and cell-cell interactions accurately, we need to use many more vertices per cell than in a vertex model, which leads to a greater computational cost. To limit computational time, the code is parallelised to allow each cell to be stored on a different partition, each possibly using a pool of cores (hybrid MPI-OpenMP method). As such, it is possible to simulate several tens of cells on a computing cluster. We now turn to the description of our framework. In Section 2 we describe the mathematical formulation and the discretisation of our method. We show some examples of application in Section 3 and end in Section 4 with conclusions, summary and ideas for future work.

4 Discussion The framework of interacting active surfaces introduced here is a novel method to study the mechanics of cell aggregates such as early developing embryos or organoids, and opens the door to their systematic modelling and simulation. We have demonstrated here that it can be used to study in detail the shape of adhering cell doublets, simple epithelia, as well as growing cellular aggregates. Our method is well-suited to capture the mechanics of tissues and organoids connecting it to cell level processes such as cortical flows, cortical tension and cellular adhesion in the organisation of a cellular aggregate. In this study we have restricted ourselves to relatively simple constitutive equations for the tension and bending moment tensors (Eq (3)), and we have considered situations with a uniform and constant surface tension within each cell. Our method is based on using the virtual work principle (Eq (2)), a very general statement of force and torque balance for a surface, to obtain a set of algebraic equations for the cell surface described with finite elements. As such it is versatile and we expect that more complex constitutive equations, corresponding to more detailed physical descriptions of the cell surface, can be easily introduced in our description. We now discuss some of these possible extensions of our framework. We have not included here apico-basal polarity, an axis of cell organisation which results from a spatially segregated protein distribution and inhomogeneous cytoskeletal structures [73]. To take this into account, one could introduce a polarity field in each cell and consider an active tension γ on the cell surface whose value at each point depends on the polarity field orientation. This could be used to introduce, for instance, differences in apical, basal and lateral surface tension which are taken into account in 3D vertex models [19]. It would be natural to introduce a concentration field on the cell surface, describing a regulator of the cortical tension, such as myosin concentration. At the level of a single surface, such coupling between cortical flows and its regulator can give rise to pattern formation, spontaneous symmetry breaking and shape oscillation [40, 74, 75]. The dynamics of the concentration per unit area, c, of such a regulator can be obtained from the balance equation on the surface: (42) where D t c is the material derivative of c (∂ t c in a Lagrangian description), j is the flux of c relative to the centre of mass, and r is a reaction rate. A natural choice for the flux would be j = −D∇c to represent diffusion according to Fick’s law. A natural choice for the reaction rate would be r = k on − k off c for turnover dynamics, with target concentration c 0 = k on /k off and typical turnover time . The discretisation of such fields can be easily introduced in our framework, following the methods detailed in [41, 76]. One could then model the effect of this concentration on the active tension by assuming e.g. γ(c) = γ 0 c/c 0 with γ 0 a reference tension at c = c 0 . Importantly, the model introduced here is based on an interaction potential between cells, which can be motivated microscopically from a description of cell-cell linkers that equilibrate quickly to their Boltzmann distribution, with free linkers on the surface in contact with a reservoir imposing a constant concentration. From a computational perspective, the exact integration of the interaction potential requires the computation of double integrals, which have a large computational cost. Alternatively, one could approximate the double integrals further in the limit by considering the interaction of each point on surface only with its closest point projection on , following classical numerical approaches for the adhesion between interfaces (Ref. [77] and references therein). On the other hand, the interplay of adhesive cell-cell linkers such as E-cadherin with cortical dynamics also plays an important role in orchestrating cell adhesion [78]. Unlike the specific adhesion of solid interfaces, cell-cell adhesion dynamics involves a complex interplay between the diffusion, advection and binding dynamics of linkers [79]. Notably, E-cadherin junctions have been shown to be mechanosensitive [80] and to act to regulate the actomyosin levels at junctions [81]. To take these effects into account, an explicit description of E-cadherin concentration on the cell surface might be required. Thus, an extension of our model could introduce explicitly two-point density fields c IJ (X I , X J ) representing the concentration of bound linkers between cells I and J, as well as a concentration field of free linkers on each cell c I . Alternatively, one could introduce cell-cell adhesion by considering a finite number of explicitely described individual linkers [82]. Our model does not account for the friction generated by relative surface flows between cells that adhere to each other, which is likely to play an important role during cell rearrangements. The effective friction stemming from an ensemble of transiently binding and unbinding linkers can be modelled effectively with a friction coefficient motivated by microscopic models such as a Lacker-Peskin model [83], which lead to predictions of force-velocity relations which depend on whether linkers are force-sensitive, e.g. slip or catch bonds [84, 85]. One could include these terms systematically in our finite element discretisation following the ideas in [86, 87]. In its current version and with these additions, we hope that the interacting active surface framework will be a useful tool to investigate the mechanics and self-organisation of cellular aggregates.

Supporting information S1 Appendix. Details of derivations and computational framework used in the main text. https://doi.org/10.1371/journal.pcbi.1010762.s001 (PDF) S1 Fig. (A) Side view, cut by a plane perpendicular to the adhesion patch of an adhering doublet for larger than the critical value. The system develops a buckling instability that grows with time. The instability eventually leads to self-intersections (right-most image, where the blue cell has collapsed). (B) Coloured lines: pressure for simulations with different values of and . Black dotted line: theoretical approximation valid in the limit of . (C) Convergence of the method evaluated by computing the inner cell pressure P for different average mesh sizes h, and comparing the results with a simulation with h/ℓ ≈ 2 ⋅ 10−2 (finer). For each h, we compute a box plot using different values of and fixed , . (D) Side view, cut by a plane perpendicular to the adhesion patch of an adhering doublet with asymmetric tension for α = 0.7; the cell with lower tension (blue) engulfing the cell with higher tension (red) develops a self-intersection in our numerical simulations (right-most image). https://doi.org/10.1371/journal.pcbi.1010762.s002 (PDF) S1 Video. Comparison of the mesh relaxation method for different values of the tolerance for normal motion (left column: 10−3, right column: 10−4) and different mesh sizes (top row: 4 × 10−2, bottom row: 2 × 10−2). The colormap represents the distance to the original surface, which is in all cases below 5 × 10−3. https://doi.org/10.1371/journal.pcbi.1010762.s003 (M4V) S2 Video. Growth of a cell aggregate driven by cell growth and cell divisions in the case where the cell lifetime t D is ten times larger than the shape relaxation time-scale τ. https://doi.org/10.1371/journal.pcbi.1010762.s004 (MP4) S3 Video. Growth of a cell aggregate driven by cell growth and cell divisions in the case where the cell lifetime t D is ten times smaller than the shape relaxation time-scale τ. https://doi.org/10.1371/journal.pcbi.1010762.s005 (MP4)

Acknowledgments We thank Quentin Vagne and Guillermo Vilanova for comments on the manuscript.

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[1] Url: https://journals.plos.org/ploscompbiol/article?id=10.1371/journal.pcbi.1010762

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