Mathematics

This is the third of a three lecture course on Homotopy theory of classifying spaces from the Summer School on Homotopy Colimits, held at the University of Regina.

• Lecture 1
• Lecture 2
• Lecture 3

This is the first of a three lecture course on Homotopy theory of classifying spaces from the Summer School on Homotopy Colimits, held at the University of Regina.

• Lecture 1
• Lecture 2
• Lecture 3

This is the second of a three lecture course on infinity categories from the Summer School on Homotopy Colimits, held at the University of Regina.

• Lecture 1
• Lecture 2
• Lecture 3

Cell migration plays a role in many contexts including cancer cell metastasis and wound healing. Better characterizing the process can help us slow down or speed up cells and improve related conditions. To this end, we developed two different models of cell migration across a flat substrate. The first one-dimensional model represents just the protein (actin) network that pushes the front of the cell forward. It consists of nearly parallel oriented line segments. It is primarily governed by two dimensionless quantities that represent a tension that keeps the line of proteins straight and the aspect ratio (width vs height) of the network. We show how changing just these quantities can produce different behaviors suggesting potential mechanisms for controlling migration. The second model is of a three-dimensional cell represented by a collection of viscoelastic components aka damped springs. Its motion is governed by a system of ordinary differential equations and includes stretching and bending elasticities and a constrained volume. It is primarily designed for considering focal adhesion forces, which are experimentally measurable. Here we show its capabilities including typical dynamics, forces, and some comparison with experiment.

The extracellular matrix (ECM) is an umbrella term for elastic and viscoelastic materials secreted by biological cells in multicellular organisms and aggregates, such as collagens and laminins in animals, cellulose and hemicellulose in plants, and extracellular polysaccharides in bacteria. Chemical and mechanical interactions between cells and the extracellular matrix help integrate information across scales, e.g., by signaling stresses and strains in tissues to individual cells. In this talk I will show how we use hybrid cellular Potts models and vertex-based models to help unravel the role of cell-ECM interactions in animal development and (briefly) in plant development. The cellular Potts model (CPM) is a suitable approach for applications in animal development due its biologically accurate yet computationally simple representation of migrating and mixing cells [1]. In this talk, I will start by showing examples on our recent hybrid CPMs of single cells in interaction with the extracellular matrix (ECM). First I will show our models of persistent cell migration of single cells in presence of adhesion to the ECM, and show how adhesion with the ECM can give rise to subdiffusive cell migration [2]. Then I will introduce cellular Potts models of topotaxis, i.e. the tendency of the tendency of cells to migrate towards to less dense regions in obstacle forests. I will show that the biologically-accurate representation in the CPM can produce stronger topotaxis than previous, more simple models based on active Brownian particles [3]. After discussing these studies in which the ECM was represented as a static field, I will turn to mechanical reciprocity between the cells and the ECM. I will use continuum representations of the ECM to show how the strains and stresses generated in the ECM by migrating cells can determine cell shape, and how it can lead to directed cell migration against preexisting or self-generated stiffness gradients [4]. I will then show how the ECM can coordinate the migration of endothelial cells during the formation of the intrasegmental vessels in the Zebrafish, Danio rerio, demonstrating recent approaches in which we represent the ECM as a fibrous material [5], and end by showing some ongoing work in plant development, including a three-dimensional (3D) vertex-based model of plant development that will become a 3D version of our open-source software package VirtualLeaf [6].

[1] Tsuyoshi Hirashima, Elisabeth G. Rens, and Roeland M.H. Merks (2017) Cellular Potts Modeling of Complex Multicellular Behaviors in Tissue Morphogenesis. Development, Growth & Differentiation, 59(5):329-339. http://dx.doi.org/10.1111/dgd.12358
[2] Leonie van Steijn, Inge M.N. Wortel, Clément Sire, Loïc Dupré, Guy Theraulaz and Roeland M.H. Merks (2022) Computational modelling of cell motility modes emerging from cell-matrix adhesion dynamics. PLOS Computational Biology, 18(2): e1009156. https://doi.org/10.1371/journal.pcbi.1009156 (see also https://ingewortel.github.io/2021-motility-from-adhesion/ for an interactive simulation).
[3] Leonie van Steijn, Joeri A.J. Wondergem, Koen Schakenraad, Doris Heinrich, Roeland M.H. Merks (2023). Deformability and collision-induced reorientation enhance cell topotaxis in dense microenvironments Biophysical Journal, 122(13), 2791-2807. https://dx.doi.org/10.1016/j.bpj.2023.06.001.
[4] Elisabeth G. Rens and Roeland M. H. Merks (2020) Cell Shape and Durotaxis Explained from Cell-Extracellular Matrix Forces and Focal Adhesion Dynamics iScience, 23:101488. https://doi.org/10.1016/j.isci.2020.101488
[5] Joaquín Abugattas-Núñez Del Prado, Koen A.E. Keijzer, Erika Tsingos, Roeland M. H. Merks (preprint) Laminin and Fibronectin Cooperate to Guide Endothelial Self-Organization During Intersegmental Vessel Formation. bioRxiv, https://doi.org/10.64898/2026.03.13.711615
[6] Roeland M. H. Merks, Michael Guravage, Dirk Inzé, Gerrit T.S. Beemster (2011) VirtualLeaf: an Open Source framework for cell-based modeling of plant tissue growth and development. Plant Physiology 155(2): 656-666 doi:10.1104/pp.110.167619