Periodic molecular dynamics simulations are developing to a routine tool for the investigation of complex, polymeric materials. A typical application is the simulation of the curing reaction of covalently cross-linked polymers, which provides detailed understanding of network formation at the molecular scale, with examples including gelation and glass transitions. In this article, we delineate the connection between percolation theory and gel-point detection in periodic polymeric networks. Specifically, we present an algorithm that can detect the onset of percolation during cross-linking of polymers in periodic molecular dynamic simulations. A sample implementation is provided at . As an example, we apply the algorithm to simulations of an epoxy resin undergoing curing with an amine hardener. We also compare results with indirect gel point measurements obtained from monitoring the growth of the largest mass and the onset of secondary cycles.

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Despite the primary role of cell proliferation in tissue development and homeostatic maintenance, the interplay between cell density, cell mechanoresponse, and cell growth and division is poorly described in theoretical models of tissues. As a consequence, the predictive power of such models is largely undermined, and the lack of systematic statistical and experimental analysis in controlled experiments does not allow to calibrate and validate already existing tools. In this article, we first report an experimental investigation of cell proliferation on all time- and length-scales and quantify the role of cell-cell and cell-matrix interactions. Based on these results, we build a minimal density-driven mean-field model of cell proliferation within 2D epithelia which can account for mechanoresponse and which is based on a separation of cell population into growing and dividing cells. We show that the model can capture the in-vitro experimental results and is further validated by in-silico experiments, and we highlight the importance of the separation of time scale and populations in the description of the cell life cycle. Additionally, we show that the mechanoresponse observed in the proliferation patterns is responsible of large-scale density structures across the tissues that we can capture by nonlinear delayed Fisher-Kolmogorov-like equations. This work sets a first building stone in a theoretical description of tissues on long time- and length-scale while accounting for proliferation.

We investigate the formation and maintenance of the homeostatic state in the case of 2D epithelial tissues following an induction of hyperosmotic conditions, using media enriched with 80 to 320 mOsm of mannitol, NaCl, and urea. We characterise the changes in the tissue immediately after the osmotic shock, and follow it until the new homeostatic state is formed. We characterise changes in cooperative motility and proliferation pressure in the tissue upon treatment with the help of a theoretical model based on the delayed Fisher-Kolmogorov formalism, where the delay in density evolution is induced by the the finite time of the cell division. Finally we explore the adaptation of the homeostatic tissue to highly elevated osmotic conditions by evaluating the morphology and topology of cells after 20 days in incubation. We find that hyperosmotic environments together with changes in the extracellular matrix induce different mechanical states in viable tissues, where only some remain functional. The perspective is a relation between tissue topology and function, which could be explored beyond the scope of this manuscript.