arXiv subjects

Carles Falcó

Publications and source records attributed to Carles Falcó.

14 recordsLinked to original sources

arXiv2026

Speed and stability of segregated waves in a pressure-based model of heterogeneous cell populations

We consider a minimal pressure-based model of heterogeneous cell populations consisting of proliferative and non-proliferative cells with different mobilities. The model is formulated as a system of reaction--cross--diffusion equations describing the spatio-temporal dynamics of the cell densities. The model is known to admit one-dimensional travelling wave solutions with strictly segregated components: non-proliferative cells occupy a finite region at the leading edge, while proliferative cells remain at the rear. However, the speed, parameter dependence, and stability of these waves remain poorly understood. In this work, we derive an almost explicit variational bound on the wave speed by reformulating the problem as a free-boundary problem for a generalised porous--Fisher equation. The estimates we obtain apply to general pressure laws and growth kinetics, agree closely with the results of numerical simulations, and become sharp in the incompressible limit, where we formally recover a fully explicit characterisation of the wave speed. We then analyse the stability of the waves to show that segregated waves are stable only when non-proliferative cells are more mobile than proliferative cells. Finally, motivated by numerical observations of finger-like protrusions, we investigate the stability of incompressible segregated circular waves through asymptotic shape-perturbation analysis. This yields explicit expressions for the pressure, interface velocity, and growth rates of angular modes, thereby making evident the destabilisation mechanisms that may lead to the emergence of fingering instability. Interestingly, we find that, in contrast with the one-dimensional case, the stability of such circular waves is not determined solely by the relative value of the mobility coefficients, and thus instabilities may arise irrespective of which cell type has the larger mobility.

math.AP↗

arXiv2026

Coupled chemotactic fronts in heterogeneous sensor-consumer cell mixtures

Chemotaxis underlies the collective migration of cell populations in developmental processes and immune responses. While the theoretical investigation of single-cell-type collective chemotaxis has received considerable attention, heterogeneous chemotaxis involving multiple interacting cell types remains poorly understood. Here, we generalise a model of heterogeneous self-generated chemotaxis and analyse the resulting collective migration patterns. We show that coupled migration between two cell types gives rise to \emph{propagating terraces} -- coupled travelling fronts moving at different speeds. While a sensor-only population leads the migrating collective, a slower mixed sensor-consumer population follows. Our analysis reveals that these fronts are coupled via the dynamics of the self-generated chemoattractant gradients. We derive analytical expressions for the migration speeds of the two fronts just in term of model parameters and experimentally measurable quantities. Our analytical results reveal that heterogeneity can enhance long-range migration via self-generated chemotaxis for sensor cells. While sensor cells can leverage benefit from mixing with consumer cells, the latter migrate more efficiently when mixing with cells of the same type. Together, our results provide a comprehensive theoretical framework for understanding heterogeneous self-generated chemotaxis.

q-bio.CB↗

arXiv2026

A nonlinear theory for chemotactic fronts of mixed populations

Collective migration of heterogeneous cell populations is central to many biological and physiological processes, including development and immune response. Recent experimental and theoretical advances have shown how asymmetric interactions with self-generated chemical gradients shape the spatial distribution of distinct cell types within migrating collectives. However, the principles governing robust spatial organisation of heterogeneous cell populations remain poorly understood. Here, we use asymptotic analysis to systematically derive a nonlinear analytical theory for heterogeneous cell collectives guided by self-generated chemotaxis. Our theory disentangles how heterogeneity in cell diffusivity, chemoattractant consumption, and chemotactic sensitivity shape the density profiles of migrating heterogeneous collectives, revealing four distinct dynamical behaviours that together capture all possible regimes. We calibrate our framework to experimental data on the co-migration of dendritic and T cells. We predict that this system operates in a parameter regime that balances intercellular mixing with T-cell localisation at the leading front of the migrating collective. Our theory reveals that this behaviour is enabled by intermediate long-range chemoattractant signalling generated through strong chemoattractant consumption by dendritic cells. Overall, our framework provides general principles for understanding how non-reciprocal chemical interactions shape robust collective migration in heterogeneous cell populations.

q-bio.CB↗

arXiv2026

Theory of adhesion-driven self-organisation in growing tissues

Cell invasion and spatial pattern formation are two distinct manifestations of cellular self-organisation in development, regeneration, and disease. Here, we develop and analyse a unified theoretical framework that links these two seemingly different behaviours within a single mechanistic model for adhesion-mediated self-organisation in growing cell populations. Using a multiscale analysis, we show that the balance between cell-cell adhesion, self-diffusion, and proliferation controls the emergence of distinct collective dynamics. We find that for weak adhesion, tissues invade through stable monotone fronts. As adhesion increases, invasion slows, fronts become unstable, leading to aggregates and spatial patterns emerging behind the advancing edge. In two spatial dimensions, these instabilities generate fingering morphologies reminiscent of dysregulated invasion in cancer. Crucially, we show that density-dependent regulation of adhesion suppresses these instabilities and restores cohesive tissue expansion. Together, our results identify adhesion strength and its regulation as key determinants of whether tissues invade cohesively or fragment into patterns, and provide a unified framework for understanding collective migration, morphogenesis, and dysregulated growth.

q-bio.TO↗

arXiv2025

A nonlocal-to-local approach to aggregation-diffusion equations

Over the past decades, nonlocal models have been widely used to describe aggregation phenomena in biology, physics, engineering, and the social sciences. These are often derived as mean-field limits of attraction-repulsion agent-based models, and consist of systems of nonlocal partial differential equations. Using differential adhesion between cells as a biological case study, we introduce a novel local model of aggregation-diffusion phenomena. This system of local aggregation-diffusion equations is fourth-order, resembling thin-film or Cahn-Hilliard type equations. In this framework, cell sorting phenomena are explained through relative surface tensions between distinct cell types. The local model emerges as a limiting case of short-range interactions, providing a significant simplification of earlier nonlocal models, while preserving the same phenomenology. This simplification makes the model easier to implement numerically and more amenable to calibration to quantitative data. Additionally, we discuss recent analytical results based on the gradient-flow structure of the model, along with open problems and future research directions.

q-bio.CB↗

arXiv2024

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 modelling, 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 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 analysing cell cycle data, providing mechanistic insights into how individual cells regulate proliferation, based on population-based experimental measurements.

q-bio.QM↗

arXiv2024

Travelling waves in a minimal go-or-grow model of cell invasion

We consider a minimal go-or-grow model of cell invasion, whereby cells can either proliferate, following logistic growth, or move, via linear diffusion, and phenotypic switching between these two states is density-dependent. Formal analysis in the fast switching regime shows that the total cell density in the two-population go-or-grow model can be described in terms of a single reaction-diffusion equation with density-dependent diffusion and proliferation. Using the connection to single-population models, we study travelling wave solutions, showing that the wave speed in the go-or-grow model is always bounded by the wave speed corresponding to the well-known Fisher-KPP equation.

math.AP↗

arXiv2024

Structural identifiability analysis of linear reaction-advection-diffusion processes in mathematical biology

Effective application of mathematical models to interpret biological data and make accurate predictions often requires that model parameters are identifiable. Approaches to assess the so-called structural identifiability of models are well-established for ordinary differential equation models, yet there are no commonly adopted approaches that can be applied to assess the structural identifiability of the partial differential equation (PDE) models that are requisite to capture spatial features inherent to many phenomena. The differential algebra approach to structural identifiability has recently been demonstrated to be applicable to several specific PDE models. In this brief article, we present general methodology for performing structural identifiability analysis on partially observed reaction-advection-diffusion (RAD) PDE models that are linear in the unobserved quantities. We show that the differential algebra approach can always, in theory, be applied to such models. Moreover, despite the perceived complexity introduced by the addition of advection and diffusion terms, identifiability of spatial analogues of non-spatial models cannot decrease in structural identifiability. We conclude by discussing future possibilities and the computational cost of performing structural identifiability analysis on more general PDE models.

q-bio.QM↗

arXiv2023

Competing effects in fourth-order aggregation-diffusion equations

We give sharp conditions for global in time existence of gradient flow solutions to a Cahn-Hilliard-type equation, with backwards second order degenerate diffusion, in any dimension and for general initial data. Our equation is the 2-Wasserstein gradient flow of a free energy with two competing effects: the Dirichlet energy and the power-law internal energy. Homogeneity of the functionals reveals critical regimes that we analyse. Sharp conditions for global in time solutions, constructed via the minimising movement scheme, also known as JKO scheme, are obtained. Furthermore, we study a system of two Cahn-Hilliard-type equations exhibiting an analogous gradient flow structure.

math.AP↗

arXiv2023

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.

q-bio.TO↗

arXiv2022

A local continuum model of cell-cell adhesion

Cell-cell adhesion is one the most fundamental mechanisms regulating collective cell migration during tissue development, homeostasis and repair, allowing cell populations to self-organize and eventually form and maintain complex tissue shapes. Cells interact with each other via the formation of protrusions or filopodia and they adhere to other cells through binding of cell surface proteins. The resulting adhesive forces are then related to cell size and shape and, often, continuum models represent them by nonlocal attractive interactions. In this paper, we present a new continuum model of cell-cell adhesion which can be derived from a general nonlocal model in the limit of short-range interactions. This new model is local, resembling a system of thin-film type equations, with the various model parameters playing the role of surface tensions between different cell populations. Numerical simulations in one and two dimensions reveal that the local model maintains the diversity of cell sorting patterns observed both in experiments and in previously used nonlocal models. In addition, it also has the advantage of having explicit stationary solutions, which provides a direct link between the model parameters and the differential adhesion hypothesis.

q-bio.CB↗

arXiv2022

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.

physics.soc-ph↗

arXiv2022

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 random number of secondary cases. It has been claimed that the so-called superspreading events in COVID-19 are governed by a power-law tailed distribution of secondary cases, with no finite variance. Using a continuous-time branching process, we show that for such power-law superspreading the survival probability of an outbreak as a function of 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 quantify the counterintuitive hazards infinite-variance superspreading poses and conclude that superspreading only leads to new phenomenology in the infinite-variance case.

physics.soc-ph↗

arXiv2022

Bulk-Boundary eigenvalues for Bilaplacian problems

We initiate the study of a bulk-boundary eigenvalue problem for the Bilaplacian with a particular third order boundary condition that arises from the study of dynamical boundary conditions for the Cahn-Hilliard equation. First we consider continuity properties under parameter variation (in which the parameter also affects the domain of definition of the operator). Then we look at the ball and the annulus geometries (together with the punctured ball), obtaining the eigenvalues as solutions of a precise equation involving special functions. An interesting outcome of our analysis in the annulus case is the presence of a bifurcation from the zero eigenvalue depending on the size of the annulus.

math.AP↗