# Modeling and analysis of melanoblast motion

## Abstract

Melanoblast migration is important for embryogenesis and is a key feature of melanoma metastasis. Many studies have characterized melanoblast movement, focusing on statistical properties and have highlighted basic mechanisms of melanoblast motility. We took a slightly different and complementary approach: we previously developed a mathematical model of melanoblast motion that enables the testing of biological assumptions about the displacement of melanoblasts and we created tests to analyze the geometric features of cell trajectories and the specific issue of trajectory interactions. Within this model, we performed simulations and compared the results with experimental data using geometric tests. In this paper, we developed the associated mathematical model and the main focus is to study the crossings between trajectories with new theoretical results about the variation of number of intersection points with respect to the crossing times. Using these results it is possible to study the random nature of displacements and the interactions between trajectories. This analysis has raised new questions, leading to the generation of strong arguments in favor of a trail left behind each moving melanoblast.

## Keywords

Cell migration Melanoblast Spatial prey predator Reduced model Geometric probability## Mathematics Subject Classification

92C17## Notes

### Acknowledgements

We are grateful to Laura Machesky (Beatson Institute, UK) for the original observation of the movement of melanoblasts ex vivo and to provide the experimental movies, WT1 to WT9. We are grateful to Véronique Letort-Le Chevalier and to the students of the Ecole Centrale for useful contributions to the first part of this study and Florian De Vuyst for his pioneering works about reduction of models. This work was supported by the Ligue Contre le Cancer, ARC, INCa, and ITMO Cancer.

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