Speaker
Description
We present a result on local well posedness for a highly nonlocal nonlinear diffusion–adhesion system [RZ]. Macroscopic systems of this type were previously obtained through upscaling [ZR] and can account for the effect of microscopic receptor binding dynamics in cell–cell adhesion. The system couples an integro PDE featuring degenerate diffusion of porous medium type and nonlocal adhesion with a novel nonlinear integral equation. The approach is based on decoupling the system and using Banach’s fixed point theorem to solve each of the two equations individually and subsequently the full coupled system. A key challenge lies in identifying a suitable functional setting. One of the main results is the local well posedness of the integral equation with a Radon measure as parameter. The analysis of this equation employs the Kantorovich–Rubinstein norm, marking what appears to be the first use of this norm in the study of a nonlinear integral equation. For details of a multiscale derivation of models of this kind, we refer to [ZR] and to the talk “A cell–cell adhesion model: a multiscale derivation” in this conference.
Bibliography
@article{rajendran_local_2025,
title = {Local {Well}-{Posedness} for a {Novel} {Nonlocal} {Model} for {Cell}-{Cell} {Adhesion} via {Receptor} {Binding}},
volume = {57},
issn = {0036-1410, 1095-7154},
url = {https://epubs.siam.org/doi/10.1137/24M1667518},
doi = {10.1137/24M1667518},
language = {en},
number = {4},
urldate = {2026-03-22},
journal = {SIAM Journal on Mathematical Analysis},
author = {Rajendran, Mabel Lizzy and Zhigun, Anna},
month = aug,
year = {2025},
pages = {4016--4067},
}
@article{zhigun_modelling_2024,
title = {Modelling non-local cell-cell adhesion: a multiscale approach},
volume = {88},
issn = {0303-6812, 1432-1416},
shorttitle = {Modelling non-local cell-cell adhesion},
url = {https://link.springer.com/10.1007/s00285-024-02079-8},
doi = {10.1007/s00285-024-02079-8},
abstract = {Abstract
Cell-cell adhesion plays a vital role in the development and maintenance of multicellular organisms. One of its functions is regulation of cell migration, such as occurs, e.g. during embryogenesis or in cancer. In this work, we develop a versatile multiscale approach to modelling a moving self-adhesive cell population that combines a careful microscopic description of a deterministic adhesion-driven motion component with an efficient mesoscopic representation of a stochastic velocity-jump process. This approach gives rise to mesoscopic models in the form of kinetic transport equations featuring multiple non-localities. Subsequent parabolic and hyperbolic scalings produce general classes of equations with non-local adhesion and myopic diffusion, a special case being the classical macroscopic model proposed in Armstrong et al. (J Theoret Biol 243(1): 98–113, 2006). Our simulations show how the combination of the two motion effects can unfold. Cell-cell adhesion relies on the subcellular cell adhesion molecule binding. Our approach lends itself conveniently to capturing this microscopic effect. On the macroscale, this results in an additional non-linear integral equation of a novel type that is coupled to the cell density equation.},
language = {en},
number = {5},
urldate = {2026-03-22},
journal = {Journal of Mathematical Biology},
author = {Zhigun, Anna and Rajendran, Mabel Lizzy},
month = may,
year = {2024},
pages = {55},
}
