A geometrically controlled rigidity transition in a model for confluent 3D tissues

Abstract

The origin of rigidity in disordered materials is an outstanding open problem in statistical physics. Previously, a class of 2D cellular models has been shown to undergo a rigidity transition controlled by a mechanical parameter that specifies cell shapes. Here, we generalize this model to 3D and find a rigidity transition that is similarly controlled by the preferred surface area S 0 : the model is solid-like below a dimensionless surface area of s S V 5.413 0 0 2 3 º » with V being the average cell volume, and fluid-like above this value. We demonstrate that, unlike jamming in soft spheres, residual stresses are necessary to create rigidity. These stresses occur precisely when cells are unable to obtain their desired geometry, and we conjecture that there is a well-defined minimal surface area possible for disordered cellular structures. We show that the behavior of this minimal surface induces a linear scaling of the shear modulus with the control parameter at the transition point, which is different from the scaling observed in particulate matter. The existence of such a minimal surface may be relevant for biological tissues and foams, and helps explain why cell shapes are a good structural order parameter for rigidity transitions in biological tissues.

Tissue rheology in embryonic organization

“…Fluidlike tissues, in contrast, are typically associated with low cell density, asymmetry in cell shape, and random cell motion and/or frequent cell rearrangements (Szabó et al, 2006;Angelini et al, 2011;Bi et al, 2015Bi et al, , 2016Yang et al, 2017;Merkel & Manning, 2018). Fluidlike tissues, in contrast, are typically associated with low cell density, asymmetry in cell shape, and random cell motion and/or frequent cell rearrangements (Szabó et al, 2006;Angelini et al, 2011;Bi et al, 2015Bi et al, , 2016Yang et al, 2017;Merkel & Manning, 2018).…”

“…The EMBO Journal 38: e102497 | 2019 ª 2019 The Authors work suggested that cell geometry might be a more reliable parameter determining the phase state of confluent cell layers and tissues, as it also provides information not only about cell cohesion but also the mobility state of the individual cells (Yang et al, 2017;Merkel & Manning, 2018). For instance, asymmetric cell shapes with long cell-cell contacts offer more degrees of freedom to promote cell rearrangements compared to isotropic cell shapes (Bi et al, 2016) ( Fig 2C).…”

“…At high cell density, cell motion slows down and becomes more coordinated and directional (small black arrowheads, right panel), characteristic for a solid-like tissue. A fluid-to-solid phase change of a tissue that reaches a critical value of cortical tension, adhesion and diffusive motility has been described as a densityindependent rigidity transition (Bi et al, , 2016Yang et al, 2017;Merkel & Manning, 2018). B At constant density, a tissue can acquire a fluid-like state when its cells have small and weak cell-cell contacts (left panel) and a solid-like state when its cells have large, mature and strong cell-cell contacts (right panel).…”

“…This critical point corresponds to a specific value of a control parameter, which once it is reached causes a discontinuous change in an order parameter (Fig 3 and Glossary). Control parameters other than cell density, such as cell-cell adhesion, cortical tension (Angelini et al, 2011;Bi et al, 2015Bi et al, , 2016 and geometry (Yang et al, 2017;Atia et al, 2018;Merkel & Manning, 2018), were also theoretically predicted to trigger fluidto-solid phase transitions in cell assemblies once reaching a critical value. Other properties that a system may exhibit at the transition point is gain or loss of symmetry, e.g.…”

“…When a tissue transits from a gas to a fluid-like state, or from a fluid to a solid-like state, its deformability decreases. From theoretical work and in vitro experiments, specific cellular parameters (control parameters, Glossary) have been identified that not only quantitatively characterize the phase state of a tissue, but upon fine-tuning and regulation can also trigger abrupt transitions between the different phase states (Angelini et al, 2011;Bi et al, 2015Bi et al, , 2016Park et al, 2015;Merkel & Manning, 2018). Such cellular parameters can thus be used to provide a universal description of the rheological state of a tissue and reveal the biological mechanisms underlying this state.…”