Stochastic models for cell polatization and crawling: Journey from external fluctuations in reaction-diffusion system to internal noise effects

The plasma membrane hosts a diverse set of proteins whose attachment, detachment, and lateral diffusion give rise to complex spatiotemporal patterns that regulate essential cellular processes. In motile eukaryotic cells, localized protein patterns at the membrane control actin organization and thereby establish a front–rear axis that drives locomotion. While persistent migration following a chemotactic signal can be captured by deterministic reaction–diffusion models coupled to a phase-field description of cell shape, amoeboid motion in cells such as neutrophils and {\it Dictyostelium discoideum} is markedly more irregular and requires the inclusion of stochastic effects. Existing modeling approaches typically introduce external noise into pattern-forming mechanisms or, alternatively, although much less frequently, they account for intrinsic fluctuations arising from the small number of participating molecules. Here, we compare and connect these two complementary modeling strategies for membrane protein dynamics driving cell locomotion in {\it Dictyostelium discoideum}. We present a stochastic reaction–diffusion model coupled to a phase field, previously calibrated to experimental data, alongside a related formulation in which fluctuations emerge intrinsically from molecular discreteness. Through systematic numerical simulations, we characterize the resulting membrane patterns and dynamical behaviors. Our results test the hypothesis that externally imposed fluctuations can be effectively reduced to intrinsic noise derived from first principles, and we discuss the similarities, differences, and limitations of both approaches.