Relationship between haptotaxis and chemotaxis in cell dynamics
Hiroshi Ishii, Hideki Murakawa, Yoshitaro Tanaka
TL;DR
This work develops a mathematical framework linking haptotaxis-driven nonlocal aggregation and chemotaxis-driven Keller–Segel dynamics in cell sorting. It demonstrates that, under suitable kernel approximations, the nonlocal advection term $-\nabla \cdot (g(u)\nabla W*u)$ can be captured by a KS-type chemotaxis system via representing $\nabla W$ as a linear combination of gradients of Green functions. It proves the existence of weak solutions for KS with nonlinear diffusion and density saturation, and establishes convergence in the parabolic–parabolic to parabolic–elliptic limit as the chemical relaxation vanishes, as well as convergence under kernel approximations. The results reveal that short-range haptotactic interactions can mimic long-range chemotactic effects, while chemotaxis provides long-range coordination with different costs, and they provide a novel analytic tool—the gradient-kernel approximation in arbitrary dimensions. These findings advance the mathematical understanding of cell sorting and offer a bridge between two fundamental biological transport mechanisms, with potential implications for multi-type cell interactions.
Abstract
The phenomenon where cells with elongated protrusions, such as neurons, communicate by contacting other cells and arrange themselves appropriately is termed cell sorting through haptotaxis. This phenomenon is described by partial differential equations involving nonlocal advection. In contrast, cell phenomena where cells communicate with other cells via chemical substances and arrange themselves appropriately are termed cell sorting through chemotaxis, typically modeled by chemotactic systems such as the Keller--Segel model. Although there are clear differences between haptotaxis and chemotaxis, similar behaviors are often observed. In this study, we investigate the relationship between haptotaxis and chemotaxis in cell sorting phenomena. Specifically, we analyze the connections between a nonlocal aggregation model for haptotaxis and a Keller--Segel type chemotaxis system. By demonstrating convergence under specific kernel approximations, we show how these distinct mechanisms can lead to comparable dynamic behaviors. In particular, we establish that the gradient of a given kernel can be approximated by linear combinations of gradients of fundamental solutions, which also provides a mathematical contribution of independent interest. This study provides a mathematical framework for understanding the interplay between haptotaxis and chemotaxis in cell sorting phenomena.