A Novel Cell Vertex Model Formulation that Distinguishes the Strength of Contraction Forces and Adhesion at Cell Boundaries

The vertex model is a useful mathematical model to describe the dynamics of epithelial cell sheets. However, existing vertex models do not distinguish contraction forces on the cell boundary from adhesion between cells, employing a single parameter to express both. In this paper, we introduce the rest length of the cell boundary and its dynamics into the existing vertex model, giving a novel formulation of the model that treats separately the contraction force and the strength of adhesion between cells. We apply this vertex model to the phenomenon of compartment boundary in the fruit fly pupa, recapturing the observation that increasing the strength of adhesion between cells straightens the compartment boundary, even though contraction forces at cell boundaries remain unchanged. We also discuss possibilities of the novel vertex models by considering the stretching of a cell sheet by external forces.

The Hippo pathway coactivator Yorkie can reprogram cell fates and create compartment-boundary–like interactions at clone margins

During development, tissue-specific patterns of gene expression are established by transcription factors and then stably maintained via epigenetic mechanisms. Cancer cells often express genes that are inappropriate for that tissue or developmental stage. Here, we show that high activity levels of Yki, the Hippo pathway coactivator that causes overgrowth in Drosophila imaginal discs, can also disrupt cell fates by altering expression of selector genes like engrailed (en) and Ultrabithorax (Ubx). Posterior clones expressing activated Yki can down-regulate en and express an anterior selector gene, cubitus interruptus (ci). The microRNA bantam and the chromatin regulator Taranis both function downstream of Yki in promoting ci expression. The boundary between Yki-expressing posterior clones and surrounding wild-type cells acquires properties reminiscent of the anteroposterior compartment boundary; Hedgehog signaling pathway activation results in production of Dpp. Thus, at least in principle, heterotypic interactions between Yki-expressing cells and their neighbors could activate boundary-specific signaling mechanisms.

Differential cell adhesion implemented by Drosophila Toll corrects local distortions of the anterior-posterior compartment boundary

Maintaining lineage restriction boundaries in proliferating tissues is vital to animal development. A long-standing thermodynamics theory, the differential adhesion hypothesis, attributes cell sorting phenomena to differentially expressed adhesion molecules. However, the contribution of the differential adhesion system during tissue morphogenesis has been unsubstantiated despite substantial theoretical support. Here, we report that Toll-1, a transmembrane receptor protein, acts as a differentially expressed adhesion molecule that straightens the fluctuating anteroposterior compartment boundary in the abdominal epidermal epithelium of the Drosophila pupa. Toll-1 is expressed across the entire posterior compartment under the control of the selector gene engrailed and displays a sharp expression boundary that coincides with the compartment boundary. Toll-1 corrects local distortions of the boundary in the absence of cable-like Myosin II enrichment along the boundary. The reinforced adhesion of homotypic cell contacts, together with pulsed cell contraction, achieves a biased vertex sliding action by resisting the separation of homotypic cell contacts in boundary cells. This work reveals a self-organizing system that integrates a differential adhesion system with pulsed contraction of cells to maintain lineage restriction boundaries.

Membrane potential regulates Hedgehog signaling and compartment boundary maintenance in the Drosophila wing disc

The Drosophila wing imaginal disc is composed of two lineage-restricted populations of cells separated by a smooth boundary. Hedgehog (Hh) from posterior cells activates a signaling pathway in anterior cells near the boundary which is necessary for boundary maintenance. Here, we show that membrane potential is patterned in the wing disc. Anterior cells near the boundary, where Hh signaling is most active, are more depolarized than posterior cells across the boundary. Elevated expression of the ENaC channel Ripped Pocket (Rpk), observed in these anterior cells, requires Hh. Antagonizing Rpk reduces depolarization and disrupts the compartment boundary. Using genetic and optogenetic manipulations, we show that membrane depolarization promotes membrane localization of Smoothened and augments Hh signaling. Thus, membrane depolarization and Hh-dependent signaling mutually reinforce each other in this region. Finally, clones of depolarized cells survive preferentially in the anterior compartment and clones of hyperpolarized cells survive preferentially in the posterior compartment.

Kinetic modelling of compartmentalised reaction networks

This paper presents a comprehensive treatment of kinetic modelling of compartmentalised reaction networks in the context of systems biology. There is still a lot of confusion about how to go about constructing compartment models, and many published models are flawed with respect to how they handle compartmentation. The modelling framework described here answers two key questions: Which rate laws should be used to describe the rates of reactions in compartmentalised systems? How should these rate laws be incorporated in the ordinary differential equations (ODEs) that describe the dynamics of the compartmentalised system? The framework rests on the fundamental definition of reaction rate as the number of reaction events per time, which is related to the time derivative of mole amount of reactant or product, an extensive property that is directly proportional to the size of the compartment in which the reaction events occur. This means that the rates of reactions that occur in a 3-dimensional compartment are proportional to the volume of the compartment, while the rates of transfers over a 2-dimensional compartment boundary or interface between compartments are proportional to the area of the boundary. Transfer rates are often incorrectly scaled with a volume instead of an area, and the reasons why this is wrong are extensively discussed. I also show how `textbook' rate equations, which I term canonical rate equations, should be modified for compartmental modelling and how they should be incorporated into either amount-change or concentration-change ODEs.

A refutation to 'A new A-P compartment boundary and organizer in holometabolous insect wings.'

We respond to a recent report by Abbasi and Marcus who present two main findings: first they argue that there is an organiser and a compartment boundary within the posterior compartment of the butterfly wing. Second, they present evidence for a previously undiscovered lineage boundary near wing vein 5 in Drosophila, a boundary that delineates a "far posterior" compartment. Clones of cells were marked with the yellow mutation and they reported that these clones always fail to cross a line close to vein 5 on the Drosophila wing. In our hands yellow proved an unusable marker for clones in the wing blade and therefore we reexamined the matter. We marked clones of cells with multiple wing hairs or forked and found a substantial proportion of these clones cross the proposed lineage boundary near vein 5, in conflict with their findings and conclusion. As internal controls we showed that these same clones respect the other two well established compartment boundaries: the anteroposterior compartment boundary is always respected. The dorsoventral boundary is mostly respected, and is crossed only by clones that are induced early in development, consistent with many reports. We question the validity of Abbasi and Marcus' conclusions regarding the butterfly wing but present no new data. Arising from: R. Abbasi and J. M. Marcus Sci. Rep. 7, 16337 (2017); https://doi.org/10.1038/s41598-017-16553-5 .