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

Strain softening and hysteresis arising from 3D multicellular dynamics during long-term large deformation

Living tissues exhibit complex mechanical properties, including viscoelastic and elastoplastic responses, that are crucial for regulating cell behaviors and tissue deformations. Despite their significance, the intricate properties of three-dimensional (3D) cell constructs are not well understood and are inadequately implemented in biomaterial engineering. To address this gap, we developed a numerical method to analyze the dynamic properties of cell constructs using a 3D vertex model framework. By focusing on 3D tissues composed of confluent homogeneous cells, we characterized their properties in response to various deformation magnitudes and time scales. Stress relaxation tests revealed that large deformations initially induced relaxation in the shapes of individual cells. This process is amplified by subsequent transient cell rearrangements, homogenizing cell shapes and leading to tissue fluidization. Additionally, dynamic viscoelastic analyses showed that tissues exhibited strain softening and hysteresis during large deformations. Interestingly, this strain softening originates from multicellular structures independent of cell rearrangement, while hysteresis arises from cell rearrangement. Moreover, tissues exhibit elastoplastic responses over the long term, which are well represented by the Ramberg-Osgood model. These findings highlight the characteristic properties of cell constructs emerging from their structures and rearrangements, especially during long-term large deformations. The developed method offers a new approach to uncover the dynamic nature of 3D tissue mechanics and could serve as a technical foundation for exploring tissue mechanics and advancing biomaterial engineering.

Vertex models capturing subcellular scales in epithelial tissues

Vertex models provide a robust theoretical framework for studying epithelial tissues as a network of cell boundaries. They have been pivotal in exploring properties such as cell packing geometry and rigidity transitions. Recently, extended vertex models have become instrumental in bridging the subcellular scales to the tissue scale. Here, we review extensions of the model aiming to capture experimentally observed subcellular features of epithelial tissues including heterogeneity in myosin activity across the tissue, non-uniform contractility structures, and mechanosensitive feedback loops. We discuss how these extensions change and challenge current perspectives on observables of macroscopic tissue properties. First, we find that extensions to the vertex model can change model properties significantly, impacting the critical threshold and in some cases even the existence of a rigidity transition. Second, we find that packing disorder can be explained by models employing different subcellular mechanisms, indicating a source of stochasticity and gradual local size changes as common mesoscopic motifs in the mechanics of tissue organization. We address complementary models and statistical inference, putting vertex models in a broader methodological context and we give a brief overview of software packages utilized in increasingly complex vertex model studies. Our review emphasizes the need for more comparative, systematic studies that identify specific classes of vertex models which share a set of well-defined properties, as well as a more in-depth discussion of modeling choices and their biological motivations.

Tissue Active Matter: Integrating Mechanics and Signaling into Dynamical Models

The importance of physical forces in the morphogenesis, homeostatic function, and pathological dysfunction of multicellular tissues is being increasingly characterized, both theoretically and experimentally. Analogies between biological systems and inert materials such as foams, gels, and liquid crystals have provided striking insights into the core design principles underlying multicellular organization. However, these connections can seem surprising given that a key feature of multicellular systems is their ability to constantly consume energy, providing an active origin for the forces that they produce. Key emerging questions are, therefore, to understand whether and how this activity grants tissues novel properties that do not have counterparts in classical materials, as well as their consequences for biological function. Here, we review recent discoveries at the intersection of active matter and tissue biology, with an emphasis on how modeling and experiments can be combined to understand the dynamics of multicellular systems. These approaches suggest that a number of key biological tissue-scale phenomena, such as morphogenetic shape changes, collective migration, or fate decisions, share unifying design principles that can be described by physical models of tissue active matter.

Renormalized mechanics and stochastic thermodynamics of growing model protocells

Uncovering the rules governing the nonequilibrium dynamics of the membranes that define biological cells is of central importance to understanding the physics of living systems. We theoretically and computationally investigate the behavior of model protocells-flexible quasispherical vesicles-that exchange membrane constituents, internal volume, and heat with an external reservoir. The excess chemical potential and osmotic pressure difference imposed by the reservoir act as generalized thermodynamic driving forces that modulate vesicle morphology. We identify an associated nonequilibrium morphological transition between a weakly driven regime, in which growing vesicles remain quasispherical, and a strongly driven regime, in which vesicles accommodate rapid membrane uptake by developing surface wrinkles. This transition emerges due to the renormalization of membrane mechanical properties by nonequilibrium driving. Further, using insights from stochastic thermodynamics we propose a minimal vesicle growth-shape law that remains robust even in strongly driven, far-from-equilibrium regimes.

The Mechanics of Building Functional Organs

Organ morphogenesis is multifaceted, multiscale, and fundamentally a robust process. Despite the complex and dynamic nature of embryonic development, organs are built with reproducible size, shape, and function, allowing them to support organismal growth and life. This striking reproducibility of tissue form exists because morphogenesis is not entirely hardwired. Instead, it is an emergent product of mechanochemical information flow, operating across spatial and temporal scales-from local cellular deformations to organ-scale form and function, and back. In this review, we address the mechanical basis of organ morphogenesis, as understood by observations and experiments in living embryos. To this end, we discuss how mechanical information controls the emergence of a highly conserved set of structural motifs that shape organ architectures across the animal kingdom: folds and loops, tubes and lumens, buds, branches, and networks. Moving forward, we advocate for a holistic conceptual framework for the study of organ morphogenesis, which rests on an interdisciplinary toolkit and brings the embryo center stage.

The maximum size of cell-aggregates is determined by the competition between the strain energy and the binding energy of cells

The development of tissues and organs is affected by how cells interact with each other to form aggregates. Such an interaction is in turn determined by several different factors, such as inter-cellular attractive forces, cell motility, and the strain energy of cells. Here, we have used mathematical modelling and numerical simulations to explore how the interplay between these factors can influence the formation and stability of 2D cell aggregates. Cell aggregates were created by incrementally accumulating cells over an initial seed. The binding energy density of these aggregates was determined using the harmonic approximation and was integrated into a probabilistic model to estimate the maximum cluster size, beyond which the aggregate becomes unstable and breaks into smaller fragments. Our simulations reveal that the ratio of strain energy to internal adhesive energy (U s / U b ) critically impacts cell aggregation; smaller ratios allow for larger cluster sizes. These findings have significant implications for tissue engineering, in-vitro modeling, the study of neurodegenerative diseases, and tissue regeneration, providing insights into how physical and biological characteristics of cells influence their aggregation and stability.

SimuCell3D: three-dimensional simulation of tissue mechanics with cell polarization

The three-dimensional (3D) organization of cells determines tissue function and integrity, and changes markedly in development and disease. Cell-based simulations have long been used to define the underlying mechanical principles. However, high computational costs have so far limited simulations to either simplified cell geometries or small tissue patches. Here, we present SimuCell3D, an efficient open-source program to simulate large tissues in three dimensions with subcellular resolution, growth, proliferation, extracellular matrix, fluid cavities, nuclei and non-uniform mechanical properties, as found in polarized epithelia. Spheroids, vesicles, sheets, tubes and other tissue geometries can readily be imported from microscopy images and simulated to infer biomechanical parameters. Doing so, we show that 3D cell shapes in layered and pseudostratified epithelia are largely governed by a competition between surface tension and intercellular adhesion. SimuCell3D enables the large-scale in silico study of 3D tissue organization in development and disease at a great level of detail.

Modelling contractile ring formation and division to daughter cells for simulating proliferative multicellular dynamics

Cell proliferation is a fundamental process underlying embryogenesis, homeostasis, wound healing, and cancer. The process involves multiple events during each cell cycle, such as cell growth, contractile ring formation, and division to daughter cells, which affect the surrounding cell population geometrically and mechanically. However, existing methods do not comprehensively describe the dynamics of multicellular structures involving cell proliferation at a subcellular resolution. In this study, we present a novel model for proliferative multicellular dynamics at the subcellular level by building upon the nonconservative fluid membrane (NCF) model that we developed in earlier research. The NCF model utilizes a dynamically-rearranging closed triangular mesh to depict the shape of each cell, enabling us to analyze cell dynamics over extended periods beyond each cell cycle, during which cell surface components undergo dynamic turnover. The proposed model represents the process of cell proliferation by incorporating cell volume growth and contractile ring formation through an energy function and topologically dividing each cell at the cleavage furrow formed by the ring. Numerical simulations demonstrated that the model recapitulated the process of cell proliferation at subcellular resolution, including cell volume growth, cleavage furrow formation, and division to daughter cells. Further analyses suggested that the orientation of actomyosin stress in the contractile ring plays a crucial role in the cleavage furrow formation, i.e., circumferential orientation can form a cleavage furrow but isotropic orientation cannot. Furthermore, the model replicated tissue-scale multicellular dynamics, where the successive proliferation of adhesive cells led to the formation of a cell sheet and stratification on the substrate. Overall, the proposed model provides a basis for analyzing proliferative multicellular dynamics at subcellular resolution.

Tissue interplay during morphogenesis

The process by which biological systems such as cells, tissues and organisms acquire shape has been named as morphogenesis and it is central to a plethora of biological contexts including embryo development, wound healing, or even cancer. Morphogenesis relies in both self-organising properties of the system and in environmental inputs (biochemical and biophysical). The classical view of morphogenesis is based on the study of external biochemical molecules, such as morphogens. However, recent studies are establishing that the mechanical environment is also used by cells to communicate within tissues, suggesting that this mechanical crosstalk is essential to synchronise morphogenetic transitions and self-organisation. In this article we discuss how tissue interaction drive robust morphogenesis, starting from a classical biochemical view, to finalise with more recent advances on how the biophysical properties of a tissue feedback with their surroundings to allow form acquisition. We also comment on how in silico models aid to integrate and predict changes in cell and tissue behaviour. Finally, considering recent advances from the developmental biomechanics field showing that mechanical inputs work as cues that promote morphogenesis, we invite to revisit the concept of morphogen.