Coupling intercellular molecular signalling with multicellular deformation for simulating three-dimensional tissue morphogenesis

An agent-based modelling framework to study growth mechanisms in EGFR-L858R mutant cell alveolar type II cells

Mutations in the epidermal growth factor receptor (EGFR) are common in non-small cell lung cancer (NSCLC), particularly in never-smoker patients. However, these mutations are not always carcinogenic, and have recently been reported in histologically normal lung tissue from patients with and without lung cancer. To investigate the outcome of EGFR mutation in healthy lung stem cells, we grow murine alveolar type II organoids monoclonally in a three-dimensional Matrigel. Our experiments show that the EGFR-L858R mutation induces a change in organoid structure: mutated organoids display more 'budding', in comparison with non-mutant controls, which are nearly spherical. We perform on-lattice computational simulations, which suggest that this can be explained by the concentration of division among a small number of cells on the surface of the mutated organoids. We are currently unable to distinguish the cell-based mechanisms that lead to this spatial heterogeneity in growth, but suggest a number of future experiments which could be used to do so. We suggest that the likelihood of L858R-fuelled tumorigenesis is affected by whether the mutation arises in a spatial environment that allows the development of these surface protrusions. These data may have implications for cancer prevention strategies and for understanding NSCLC progression.

Systems of pattern formation within developmental biology

Applications of mathematical models to developmental biology have provided helpful insight into various subfields, ranging from the patterning of animal skin to the development of complex organ systems. Systems involved in patterning within morphology present a unique path to explain self-organizing systems. Current efforts show that patterning systems, notably Reaction-Diffusion and specific signaling pathways, provide insight for explaining morphology and could provide novel applications revolving around the formation of biological systems. Furthermore, the application of pattern formation provides a new perspective on understanding developmental biology and pathology research to study molecular mechanisms. The current review is to cover and take a more in-depth overlook at current applications of patterning systems while also building on the principles of patterning of future research in predictive medicine.

Patient-Specific Organoid and Organ-on-a-Chip: 3D Cell-Culture Meets 3D Printing and Numerical Simulation

The last few decades have witnessed diversified in vitro models to recapitulate the architecture and function of living organs or tissues and contribute immensely to advances in life science. Two novel 3D cell culture models: 1) Organoid, promoted mainly by the developments of stem cell biology and 2) Organ-on-a-chip, enhanced primarily due to microfluidic technology, have emerged as two promising approaches to advance the understanding of basic biological principles and clinical treatments. This review describes the comparable distinct differences between these two models and provides more insights into their complementarity and integration to recognize their merits and limitations for applicable fields. The convergence of the two approaches to produce multi-organoid-on-a-chip or human organoid-on-a-chip is emerging as a new approach for building 3D models with higher physiological relevance. Furthermore, rapid advancements in 3D printing and numerical simulations, which facilitate the design, manufacture, and results-translation of 3D cell culture models, can also serve as novel tools to promote the development and propagation of organoid and organ-on-a-chip systems. Current technological challenges and limitations, as well as expert recommendations and future solutions to address the promising combinations by incorporating organoids, organ-on-a-chip, 3D printing, and numerical simulation, are also summarized.

A "Numerical Evo-Devo" Synthesis for the Identification of Pattern-Forming Factors

Animals display extensive diversity in motifs adorning their coat, yet these patterns have reproducible orientation and periodicity within species or groups. Morphological variation has been traditionally used to dissect the genetic basis of evolutionary change, while pattern conservation and stability in both mathematical and organismal models has served to identify core developmental events. Two patterning theories, namely instruction and self-organisation, emerged from this work. Combined, they provide an appealing explanation for how natural patterns form and evolve, but in vivo factors underlying these mechanisms remain elusive. By bridging developmental biology and mathematics, novel frameworks recently allowed breakthroughs in our understanding of pattern establishment, unveiling how patterning strategies combine in space and time, or the importance of tissue morphogenesis in generating positional information. Adding results from surveys of natural variation to these empirical-modelling dialogues improves model inference, analysis, and in vivo testing. In this evo-devo-numerical synthesis, mathematical models have to reproduce not only given stable patterns but also the dynamics of their emergence, and the extent of inter-species variation in these dynamics through minimal parameter change. This integrative approach can help in disentangling molecular, cellular and mechanical interaction during pattern establishment.

Topological graph description of multicellular dynamics based on vertex model

Vertex models are generally powerful tools for exploring biological insights into multicellular dynamics. In these models, a multicellular structure is represented by a network, which is dynamically rearranged using topological operations. Remarkably, the topological dynamics of the network are important in guaranteeing the results from the models and their biological implications. However, it remains unclear whether the entire pattern of multicellular topological dynamics can be accurately expressed by a set of operators in the models. Surprisingly, vertex models have been empirically used for several decades without any mathematical verification. In this study, we propose a rigorous two-/three-dimensional (2D/3D) vertex model to describe multicellular topological dynamics. To do this, we classify several types of vertex models from a graph-theoretic perspective. Based on the classification, mathematical analyses reveal several conditions that enable us to apply the operators accurately without topological errors. Under these conditions, the operators can completely express the entire pattern of multicellular topological dynamics. From these results, we newly propose rigorous 2D/3D vertex models that can be applied to general multicellular dynamics, and we clarify several points to verify the results obtained from previous models.

Combining Turing and 3D vertex models reproduces autonomous multicellular morphogenesis with undulation, tubulation, and branching

This study demonstrates computational simulations of multicellular deformation coupled with chemical patterning in the three-dimensional (3D) space. To address these aspects, we proposes a novel mathematical model, where a reaction–diffusion system is discretely expressed at a single cell level and combined with a 3D vertex model. To investigate complex phenomena emerging from the coupling of patterning and deformation, as an example, we employed an activator–inhibitor system and converted the activator concentration of individual cells into their growth rate. Despite the simplicity of the model, by growing a monolayer cell vesicle, the coupling system provided rich morphological dynamics such as undulation, tubulation, and branching. Interestingly, the morphological variety depends on the difference in time scales between patterning and deformation, and can be partially understood by the intrinsic hysteresis in the activator-inhibitor system with domain growth. Importantly, the model can be applied to 3D multicellular dynamics that couple the reaction–diffusion patterning with various cell behaviors, such as deformation, rearrangement, division, apoptosis, differentiation, and proliferation. Thus, the results demonstrate the significant advantage of the proposed model as well as the biophysical importance of exploring spatiotemporal dynamics of the coupling phenomena of patterning and deformation in 3D space.

Geometric constraints alter cell arrangements within curved epithelial tissues

Organ and tissue formation are complex three-dimensional processes involving cell division, growth, migration, and rearrangement, all of which occur within physically constrained regions. However, analyzing such processes in three dimensions in vivo is challenging. Here, we focus on the process of cellularization in the anterior pole of the early Drosophila embryo to explore how cells compete for space under geometric constraints. Using microfluidics combined with fluorescence microscopy, we extract quantitative information on the three-dimensional epithelial cell morphology. We observed a cellular membrane rearrangement in which cells exchange neighbors along the apical-basal axis. Such apical-to-basal neighbor exchanges were observed more frequently in the anterior pole than in the embryo trunk. Furthermore, cells within the anterior pole skewed toward the trunk along their long axis relative to the embryo surface, with maximum skew on the ventral side. We constructed a vertex model for cells in a curved environment. We could reproduce the observed cellular skew in both wild-type embryos and embryos with distorted morphology. Further, such modeling showed that cell rearrangements were more likely in ellipsoidal, compared with cylindrical, geometry. Overall, we demonstrate that geometric constraints can influence three-dimensional cell morphology and packing within epithelial tissues.

Mechanocellular models of epithelial morphogenesis

Embryonic epithelia achieve complex morphogenetic movements, including in-plane reshaping, bending and folding, through the coordinated action and rearrangement of individual cells. Technical advances in molecular and live-imaging studies of epithelial dynamics provide a very real opportunity to understand how cell-level processes facilitate these large-scale tissue rearrangements. However, the large datasets that we are now able to generate require careful interpretation. In combination with experimental approaches, computational modelling allows us to challenge and refine our current understanding of epithelial morphogenesis and to explore experimentally intractable questions. To this end, a variety of cell-based modelling approaches have been developed to describe cell-cell mechanical interactions, ranging from vertex and 'finite-element' models that approximate each cell geometrically by a polygon representing the cell's membrane, to immersed boundary and subcellular element models that allow for more arbitrary cell shapes. Here, we review how these models have been used to provide insights into epithelial morphogenesis and describe how such models could help future efforts to decipher the forces and mechanical and biochemical feedbacks that guide cell and tissue-level behaviour. In addition, we discuss current challenges associated with using computational models of morphogenetic processes in a quantitative and predictive way.This article is part of the themed issue 'Systems morphodynamics: understanding the development of tissue hardware'.

Contractile actin belt and mesh structures provide the opposite dependence of epithelial stiffness on the spontaneous curvature of constituent cells

Actomyosin generates contractile forces within cells, which have a crucial role in determining the macroscopic mechanical properties of epithelial tissues. Importantly, actin cytoskeleton, which propagates actomyosin contractile forces, forms several characteristic structures in a 3D intracellular space, such as a circumferential actin belt lining adherence junctions and an actin mesh beneath the apical membrane. However, little is known about how epithelial mechanical property depends on the intracellular contractile structures. We performed computational simulations using a 3D vertex model, and demonstrated the longitudinal tensile test of an epithelial tube, whose inside and outside are defined as the apical and basal surfaces, respectively. As a result, these subcellular structures provide the contrary dependence of epithelial stiffness and fracture force on the spontaneous curvature of constituent cells; the epithelial stiffness increases with increasing the spontaneous curvature in the case of belt, meanwhile it decreases in the case of mesh. This qualitative difference emerges from the different anisotropic deformability of apical cell surfaces; while belt preserves isotropic apical cell shapes, mesh does not. Moreover, the difference in the anisotropic deformability determines the frequency of cell rearrangements, which in turn effectively decrease the tube stiffness. These results illustrate the importance of the intracellular contractile structures, which may be regulated to optimize mechanical functions of individual epithelial tissues.

A Mechanistic Collective Cell Model for Epithelial Colony Growth and Contact Inhibition

We present a mechanistic hybrid continuum-discrete model to simulate the dynamics of epithelial cell colonies. Collective cell dynamics are modeled using continuum equations that capture plastic, viscoelastic, and elastic deformations in the clusters while providing single-cell resolution. The continuum equations can be viewed as a coarse-grained version of previously developed discrete models that treat epithelial clusters as a two-dimensional network of vertices or stochastic interacting particles and follow the framework of dynamic density functional theory appropriately modified to account for cell size and shape variability. The discrete component of the model implements cell division and thus influences cell size and shape that couple to the continuum component. The model is validated against recent in vitro studies of epithelial cell colonies using Madin-Darby canine kidney cells. In good agreement with experiments, we find that mechanical interactions and constraints on the local expansion of cell size cause inhibition of cell motion and reductive cell division. This leads to successively smaller cells and a transition from exponential to quadratic growth of the colony that is associated with a constant-thickness rim of growing cells at the cluster edge, as well as the emergence of short-range ordering and solid-like behavior. A detailed analysis of the model reveals a scale invariance of the growth and provides insight into the generation of stresses and their influence on the dynamics of the colonies. Compared to previous models, our approach has several advantages: it is independent of dimension, it can be parameterized using classical elastic properties (Poisson's ratio and Young's modulus), and it can easily be extended to incorporate multiple cell types and general substrate geometries.

Beyond Turing: mechanochemical pattern formation in biological tissues

Background

During embryogenesis, chemical (morphogen) and mechanical patterns develop within tissues in a self-organized way. More than 60 years ago, Turing proposed his famous reaction-diffusion model for such processes, assuming chemical interactions as the main driving force in tissue patterning. However, experimental identification of corresponding molecular candidates is still incomplete. Recent results suggest that beside morphogens, also tissue mechanics play a significant role in these patterning processes.

Results

Combining continuous finite strain with discrete cellular tissue models, we present and numerically investigate mechanochemical processes, in which morphogen dynamics and tissue mechanics are coupled by feedback loops. We consider three different mechanical cues involved in such feedbacks: strain, stress, and compression. Based on experimental results, for each case, we present a feedback loop spontaneously creating robust mechanochemical patterns. In contrast to Turing-type models, simple mechanochemical interaction terms are sufficient to create de novo patterns.

Conclusions

Our results emphasize mechanochemical processes as possible candidates controlling different steps of embryogenesis. To motivate further experimental research discovering related mechanisms in living tissues, we also present predictive in silicio experiments.

Reviewers

Reviewer 1 - Marek Kimmel; Reviewer 2 - Konstantin Doubrovinski (nominated by Ned Wingreen); Reviewer 3 - Jun Allard (nominated by William Hlavacek).

Three-dimensional vertex model for simulating multicellular morphogenesis

During morphogenesis, various cellular activities are spatiotemporally coordinated on the protein regulatory background to construct the complicated, three-dimensional (3D) structures of organs. Computational simulations using 3D vertex models have been the focus of efforts to approach the mechanisms underlying 3D multicellular constructions, such as dynamics of the 3D monolayer or multilayer cell sheet like epithelia as well as the 3D compacted cell aggregate, including dynamic changes in layer structures. 3D vertex models enable the quantitative simulation of multicellular morphogenesis on the basis of single-cell mechanics, with complete control of various cellular activities such as cell contraction, growth, rearrangement, division, and death. This review describes the general use of the 3D vertex model, along with its applications to several simplified problems of developmental phenomena.