Our current understanding of cell sorting relies on physical difference, either in the interfacial properties or motile force, between cell types. But is such asymmetry a prerequisite for cell sorting? We test this using a minimal model in which the two cell populations are identical with respect to their physical properties and differences in motility arise solely from how cells interact with their surroundings. The model resembles the Schelling model used in social sciences to study segregation phenomena at the scale of societies. Our results demonstrate that segregation can emerge solely from cell motility being a dynamic property that changes in response to the local environment of the cell, but that additional mechanisms are necessary to reproduce the envelopment behaviour observed in vitro. The time course of segregation follows a power law, in agreement with the scaling reported from experiment and in other models of motility-driven segregation.
How biological order emerges in a robust manner during development is an important question, as the functionality of many tissues depends on the correct spatial arrangement of cells. In this thesis, I consider two examples of ordering, cell sorting and hexagonal packing. In several developing tissues, cells of different type spontaneously self-assemble into domains that are homogenous with respect to cell type both in vitro and in vivo. Current models of sorting assume asymmetry in the physical properties of cell types - either in adhesion, cortical tension or motility. I present a minimal model demonstrating that segregation does not require such asymmetry, but can arise solely from cell motility when this is modelled as a dynamic quantity that changes in response to the composition of the local environment of a cell. Over the course of pupal development, cells in the Drosophila notum rearrange to form a hexagonally packed tissue. How does the tissue transition from disorder to order in an effective and robust way? In particular, how do stochastic fluctuations in junction length contribute to the ordering process? I address these questions by analysing data from live-imaging of the notum using a custom software package I developed. I demonstrate that neighbour exchange events are a consequence of junction fluctuations, rather than being an explicitly regulated and stereotyped process, and I present a mathematical model for how such fluctuations are generated by the stochastic turnover of myosin. I quantify the frequency of neighbour exchange events in embryos with a reduction/overexpression of Myosin II activity and establish that actomyosin is not required for neighbour exchange. In fact, the frequency of neighbour exchange events is inversely proportional to Myosin II levels. The results suggest that the gradual increase in actomyosin during development drives a process akin to annealing that aids tissue ordering.
We study a simple model in which the growth of a network is determined by the location of one or more random walkers. Depending on walker motility rate, the model generates a spectrum of structures situated between well-known limiting cases. We demonstrate that the average degree observed by a walker is a function of its motility rate. Modulating the extent to which the location of node attachment is determined by the walker as opposed to random selection is akin to scaling the speed of the walker and generates new limiting behavior. The model raises questions about energetic and computational resource requirements in a physical instantiation.
Membrane transporters carry key metabolites across the cell membrane and, from a resource standpoint, are hypothesized to be produced when necessary. The expression of membrane transporters in metabolic pathways is often upregulated by the transporter substrate. In E. coli, such systems include for example the lacY, araFGH, and xylFGH genes, which encode for lactose, arabinose, and xylose transporters, respectively. As a case study of a minimal system, we build a generalizable physical model of the xapABR genetic circuit, which features a regulatory feedback loop via membrane transport (positive feedback) and enzymatic degradation (negative feedback) of an inducer. Dynamical systems analysis and stochastic simulations show that the membrane transport makes the model system bistable in certain parameter regimes. Thus, it serves as a genetic “on-off” switch, enabling the cell to only produce a set of metabolic enzymes when the corresponding metabolite is present in large amounts. We find that the negative feedback from the degradation enzyme does not significantly disturb the positive feedback from the membrane transporter. We investigate hysteresis in the switching and discuss the role of cooperativity and multiple binding sites in the model circuit. Fundamentally, this work explores how a stable genetic switch for a set of enzymes is obtained from transcriptional auto-activation of a membrane transporter through its substrate.
Complex systems with many degrees of freedom are typically intractable, but some of their behaviors may admit simpler effective descriptions. The question of when such effective descriptions are possible remains open. The paradigmatic approach where such "emergent simplicity" can be understood in detail is the renormalization group (RG). Here, we show that for general systems, without the self-similarity symmetry required by the RG construction, the RG flow of model parameters is replaced by a more general flow of the Fisher Information Metric on the model manifold. We demonstrate that the systems traditionally studied with RG comprise special cases where this metric flow can be induced by a parameter flow, keeping the global geometry of the model-manifold fixed. In general, however, the geometry may deform, and metric flow cannot be reduced to a parameter flow -- though this could be achieved at the cost of augmenting the manifold by one new parameter, as we discuss. We hope that our framework can clarify how ideas from RG may apply in a broader class of complex systems.
The asymmetric distribution of damaged cellular components has been observed in species ranging from fission yeast to humans. To study the potential advantages of damage segregation, we have developed a mathematical model describing ageing mammalian tissue, that is, a multicellular system of somatic cells that do not rejuvenate at cell division. To illustrate the applicability of the model, we specifically consider damage incurred by mutations to mitochondrial DNA, which are thought to be implicated in the mammalian ageing process. We show analytically that the asymmetric distribution of damaged cellular components reduces the overall damage level and increases the longevity of the cell population. Motivated by the experimental reports of damage segregation in human embryonic stem cells, dividing symmetrically with respect to cell-fate, we extend the model to consider spatially structured systems of cells. Imposing spatial structure reduces, but does not eliminate, the advantage of asymmetric division over symmetric division. The results suggest that damage partitioning could be a common strategy for reducing the accumulation of damage in a wider range of cell types than previously thought.
Abstract Homophilic interactions between E-Cadherin molecules generate adhesive interfaces or junctions (AJs) that connect neighbouring cells in epithelial monolayers. These are highly dynamic structures. Under conditions of homeostasis, changes in the length of individual interfaces provide epithelia with the fluidity required to maintain tissue integrity in the face of cell division, delamination and extrinsic forces. Furthermore, when acted upon by polarized actomyosin-based forces, changes in AJ length can also drive neighbour exchange to reshape an entire tissue. Whilst the contribution of AJ remodelling to developmental morphogenesis has been subjected to intensive study, less is known about AJ dynamics in other circumstances. Here, using a combination of experiment and computational modelling, we study AJ dynamics in an epithelium that undergoes a gradual increase in packing order without concomitant large-scale changes in tissue shape or size. Under these conditions, we find that neighbour exchange events are driven by stochastic fluctuations in junction length, which are regulated at least in part by the level of junctional actomyosin. As a result of this behaviour, the steady increase in junctional actomyosin and consequent tension that accompanies development steadily reduces the rate of neighbour exchange and orders the tissue. This leads us to propose a model in which topological transitions, that underpin tissue fluidity, are either inhibited or biased by actomyosin-based forces, to drive, respectively, tissue ordering or deformation.
Under conditions of homeostasis, dynamic changes in the length of individual adherens junctions (AJs) provide epithelia with the fluidity required to maintain tissue integrity in the face of intrinsic and extrinsic forces. While the contribution of AJ remodeling to developmental morphogenesis has been intensively studied, less is known about AJ dynamics in other circumstances. Here, we study AJ dynamics in an epithelium that undergoes a gradual increase in packing order, without concomitant large-scale changes in tissue size or shape. We find that neighbor exchange events are driven by stochastic fluctuations in junction length, regulated in part by junctional actomyosin. In this context, the developmental increase of isotropic junctional actomyosin reduces the rate of neighbor exchange, contributing to tissue order. We propose a model in which the local variance in tension between junctions determines whether actomyosin-based forces will inhibit or drive the topological transitions that either refine or deform a tissue.
Abstract The gravitationally driven flow of a dense fluid within a two-layered porous media is examined experimentally and theoretically. We find that in systems with two horizontal layers of differing permeability a competition between gravity driven flow and flow focusing along high-permeability routes can lead to two distinct flow regimes. When the lower layer is more permeable than the upper layer, gravity acts along high-permeability pathways and the flow is enhanced in the lower layer. Alternatively, when the upper layer is more permeable than the lower layer, we find that for a sufficiently small input flux the flow is confined to the lower layer. However, above a critical flux fluid preferentially spreads horizontally within the upper layer before ultimately draining back down into the lower layer. This later regime, in which the fluid overrides the low-permeability lower layer, is important because it enhances the mixing of the two fluids. We show that the critical flux which separates these two regimes can be characterized by a simple power law. Finally, we briefly discuss the relevance of this work to the geological sequestration of carbon dioxide and other industrial and natural flows in porous media.
While biological studies suggest that motility of cells is involved in cell segregation, few computational models have investigated this mechanism. We apply a simple Schelling model, modified to reflect biological conditions, demonstrating how differences in cell motility arising exclusively from differences in the composition of the local environment can be sufficient to drive segregation. The work presented here demonstrates that the segregation behavior observed in the original Schelling model is robust to a relaxation of the requirement for global information and that the Schelling model may yield insight in the context of biological systems. In the model, the time course of cell segregation follows a power law in accord with experimental observations and previous work.
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