The regulation of cleavage plane orientation is one of the key mechanisms driving epithelial morphogenesis. Still, many aspects of the relationship between local cleavage patterns and tissue-level properties remain poorly understood. Here we develop a topological model that simulates the dynamics of a 2D proliferating epithelium from generation to generation, enabling the exploration of a wide variety of biologically plausible cleavage patterns. We investigate a spectrum of models that incorporate the spatial impact of neighboring cells and the temporal influence of parent cells on the choice of cleavage plane. Our findings show that cleavage patterns generate “signature” equilibrium distributions of polygonal cell shapes. These signatures enable the inference of local cleavage parameters such as neighbor impact, maternal influence, and division symmetry from global observations of the distribution of cell shape. Applying these insights to the proliferating epithelia of five diverse organisms, we find that strong division symmetry and moderate neighbor/maternal influence are required to reproduce the predominance of hexagonal cells and low variability in cell shape seen empirically. Furthermore, we present two distinct cleavage pattern models, one stochastic and one deterministic, that can reproduce the empirical distribution of cell shapes. Although the proliferating epithelia of the five diverse organisms show a highly conserved cell shape distribution, there are multiple plausible cleavage patterns that can generate this distribution, and experimental evidence suggests that indeed plants and fruitflies use distinct division mechanisms.
The spatial and temporal regulation of cell shape and cell proliferation are key mechanisms that direct tissue morphogenesis during development. Much of our knowledge of tissue morphogenesis comes from the study of simple epithelial monolayers, 2D planar sheets of strongly adhering cells in which division occurs in the plane of the epithelium. The strong structural constraints and developmental importance of epithelia have inspired a multitude of theoretical and computational models since the early 20th century [1]–[6]. Of these, an important class is topological models, where an epithelium is represented as a planar network (topology). The topology of an epithelium is defined as the network of connectivity between cells (Figure 1A and 1B). Some important topological properties include a cell' s polygonal shape, defined as its number of neighbors, and the overall distribution of cell shapes within an epithelium. There are several reasons for considering these properties. First, empirical evidence from our recent work [5] shows that the distribution of cell shapes is conserved in the proliferating epithelia of several diverse organisms, including the Drosophila larval wing disc and the Xenopus tadpole tail epidermis (Figure 1C and Table S1). Second, polygonal cell shape is linearly correlated with cell surface area (Figure 1D), a longstanding empirical observation known as Lewis' Law [2], [3]. Third, important developmental processes such as cell division, migration, and intercalation fundamentally alter topology by creating and breaking connections between cells. For these reasons, topological models have been useful both experimentally and theoretically in understanding proliferating epithelia [4]–[8] and other non-biological lattices [9], [10]. As early as the 1920s, F. T. Lewis documented the connection between cell proliferation and tissue topology, arguing that spatial control of cell divisions could affect the overall distribution of polygonal cell shapes [2], [3]. Since that time, the relationships between cell shape, proliferation and epithelial topology have been further investigated using both topological models [4], [5], [9], [10] and mechanical models [11]–[13], exploring a wide variety of phenomena including differential rates of division, adhesion forces, and stochastic divisions. However, due to unknown parameters and simplifying approximations, the specific mechanisms by which global tissue morphology emerges from the local control of cell divisions in epithelial monolayers still remains poorly understood. To better understand proliferation within the larval wing disc of Drosophila melanogaster, we recently developed a stochastic topological model of cell division [5]. Our model mathematically predicts the emergence of a specific equilibrium distribution of polygonal cell shapes (p*), revealing how local stochastic cellular processes can give rise to predictable global tissue properties. The predicted distribution p* was empirically confirmed in the larval imaginal wing disc of Drosophila melanogaster, but also closely matched in the tadpole tail epidermis of Xenopus laevis, the outer epidermis of the cnidarian Hydra vulgaris, and also Lewis' s cucumber epidermis (Figure 1C). A common characteristic across these diverse examples is that the epithelia-like tissue undergoes rapid proliferation with minimal cell rearrangement. The apparent conservation of p* across these systems is surprising. Is p* the consequence of a conserved process of cell division across these proliferating 2D epithelia? Or is it possible that distinct processes of cell division converge upon the same final distribution of shapes? More generally, how do widely varying division strategies impact global epithelial organization? Despite much experimental and theoretical progress, previous models have limitations that make it difficult to address these questions. The major difficulty lies in modeling the neighborhood and lineage dependence in cleavage plane choice. For example, our previous model encodes a mean-field approximation that ignores the variability in the number of neighbors gained via the division of neighboring cells [5]. The mathematical models of Cowan et al. [9] do not account for neighboring divisions at all: a cell never gains sides from its dividing neighbors. These models cannot be used to study modes of cell division with any spatial or temporal dependence, both of which are biologically relevant. For example, cleavage patterns with mother-daughter or neighbor-neighbor correlations in cleavage plane choice are common [14]. To explore and characterize the space of plausible cleavage patterns, a more expressive model is required. Here we present a computational model of cell division that enables us to explore a much larger class of biologically plausible division models by directly simulating the topology of a proliferating epithelium from generation to generation. This includes division schemes with spatial and temporal dependence between neighboring cells and mother-daughter cells. Given a division model, we can compute the equilibrium distribution of polygonal cell shapes, along with other tissue-level topological properties. Our findings show that the fraction of hexagons and the variability in cell shape are both important global indicators of local division parameters, and we propose that it may be possible to infer these parameters from empirical data. Furthermore, we describe several division schemes that can reproduce with high accuracy the cell shape distribution seen in five diverse organisms. We use this modeling framework to formulate and explore some of the central theoretical and empirical questions regarding the local-to-global regulation of cell shape in proliferating epithelia.
This model describes a generic proliferating epithelium with no/minimal cell rearrangement. The assumptions are based on experimental evidence from the larval stage wing disc of Drosophila melanogaster, where the absence of cell rearrangement, roughly uniform cell division rates, and cleavage plane restrictions, appear to hold [5]. These assumptions also appear to be approximately valid for the other proliferating epithelia presented in Figure 1, for example in plants, where rearrangement does not occur [2], [3]. However in some cases, rearrangement may occur more frequently and there may be a higher occurrence of tetravalent vertices and three-sided cells; for those systems the model can be modified to include those aspects, although this is beyond the scope of the current paper.
The wide spectrum of shape distributions produced by different CPMs raises the intriguing possibility of inferring the CPM based solely on empirical observations of global epithelial topology. For example, a hypothesis for a cell division strategy in a given epithelium might be rejected simply by comparing the empirical cell shape distribution with the one predicted by the CPM. Here we present the results of comparing our simulated CPMs to cell shape distribution data from natural proliferating epithelia in a diverse array of organisms. We use data, collected and published previously by our group [5], on Drosophila melanogaster (larval wing disc, arthropod), Xenopus laevis (tadpole tail epidermis, vertebrate), and Hydra (adult outer epidermis, cnidarian). In addition, we have included previously published data from two plants, Cucumis (cucumber epidermis) from the paper by F. T. Lewis [3] and Anagallis arvensis (meristem) courtesy of J. Dumais [16]. These natural epithelia show a strongly conserved cell shape distribution with between 42–48% hexagons and a standard deviation of 0. 83–0. 98 sides (Figure 1C and Table S1).
Cell division is one of the key mechanisms driving organismal growth and morphogenesis. Yet many aspects of the relationship between local cell division (how a cell chooses an orientation to divide) and global tissue architecture (e. g. , regular versus irregular cells) remain poorly understood. We present a computational framework for studying topological networks that are created by cell division; this framework reveals how certain tissue statistics can be used to infer properties of the cell division model. Recently it has been observed that five diverse organisms show almost identical cell shape distributions in their proliferating epithelial tissues, yet how this conservation arises is not understood. Using our model we show that the low variation observed in nature requires a strong correlation between how neighboring cells divide and that although the statistics of plants and fruitflies are almost identical, it is likely that they have evolved distinct cell division methods.
developmental biology/morphogenesis and cell biology
biophysics/theory and simulation
mathematics/statistics
Modeling and Inferring Cleavage Patterns in Proliferating Epithelia