Quantifying cell cycle regulation by tissue crowding.

The spatiotemporal coordination and regulation of cell proliferation is fundamental in many aspects of development and tissue maintenance. Cells have the ability to adapt their division rates in response to mechanical constraints, yet we do not fully understand how cell proliferation regulation impacts cell migration phenomena. Here, we present a minimal continuum model of cell migration with cell cycle dynamics, which includes density-dependent effects and hence can account for cell proliferation regulation. By combining minimal mathematical modeling, Bayesian inference, and recent experimental data, we quantify the impact of tissue crowding across different cell cycle stages in epithelial tissue expansion experiments. Our model suggests that cells sense local density and adapt cell cycle progression in response, during G1 and the combined S/G2/M phases, providing an explicit relationship between each cell-cycle-stage duration and local tissue density, which is consistent with several experimental observations. Finally, we compare our mathematical model's predictions to different experiments studying cell cycle regulation and present a quantitative analysis on the impact of density-dependent regulation on cell migration patterns. Our work presents a systematic approach for investigating and analyzing cell cycle data, providing mechanistic insights into how individual cells regulate proliferation, based on population-based experimental measurements.

Quantifying tissue growth, shape and collision via continuum models and Bayesian inference.

Although tissues are usually studied in isolation, this situation rarely occurs in biology, as cells, tissues and organs coexist and interact across scales to determine both shape and function. Here, we take a quantitative approach combining data from recent experiments, mathematical modelling and Bayesian parameter inference, to describe the self-assembly of multiple epithelial sheets by growth and collision. We use two simple and well-studied continuum models, where cells move either randomly or following population pressure gradients. After suitable calibration, both models prove to be practically identifiable, and can reproduce the main features of single tissue expansions. However, our findings reveal that whenever tissue-tissue interactions become relevant, the random motion assumption can lead to unrealistic behaviour. Under this setting, a model accounting for population pressure from different cell populations is more appropriate and shows a better agreement with experimental measurements. Finally, we discuss how tissue shape and pressure affect multi-tissue collisions. Our work thus provides a systematic approach to quantify and predict complex tissue configurations with applications in the design of tissue composites and more generally in tissue engineering.

From random walks on networks to nonlinear diffusion.

Mathematical models of motility are often based on random-walk descriptions of discrete individuals that can move according to certain rules. It is usually the case that large masses concentrated in small regions of space have a great impact on the collective movement of the group. For this reason, many models in mathematical biology have incorporated crowding effects and managed to understand their implications. Here, we build on a previously developed framework for random walks on networks to show that in the continuum limit, the underlying stochastic process can be identified with a diffusion partial differential equation. The diffusion coefficient of the emerging equation is, in general, density dependent, and can be directly related to the transition probabilities of the random walk. Moreover, the relaxation time of the stochastic process is directly linked to the diffusion coefficient and also to the network structure, as it usually happens in the case of linear diffusion. As a specific example, we study the equivalent of a porous-medium-type equation on networks, which shows similar properties to its continuum equivalent. For this equation, self-similar solutions on a lattice and on homogeneous trees can be found, showing finite speed of propagation in contrast to commonly used linear diffusion equations. These findings also provide insights into reaction-diffusion systems with general diffusion operators, which have appeared recently in some applications.

Finite-time scaling for epidemic processes with power-law superspreading events.

Epidemics unfold by means of a spreading process from each infected individual to a variable number of secondary cases. It has been claimed that the so-called superspreading events of the COVID-19 pandemic are governed by a power-law-tailed distribution of secondary cases, with no finite variance. Using a continuous-time branching process, we demonstrate that for such power-law-tailed superspreading, the survival probability of an outbreak as a function of both time and the basic reproductive number fulfills a "finite-time scaling" law (analogous to finite-size scaling) with universal-like characteristics only dependent on the power-law exponent. This clearly shows how the phase transition separating a subcritical and a supercritical phase emerges in the infinite-time limit (analogous to the thermodynamic limit). We also quantify the counterintuitive hazards posed by this superspreading. When the expected number of infected individuals is computed removing extinct outbreaks, we find a constant value in the subcritical phase and a superlinear power-law growth in the critical phase.