Mathematical Biology

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

Understanding how cells migrate through confined environments is crucial for elucidating fundamental biological processes, including cancer invasion, immune surveillance, and tissue morphogenesis. The nucleus, as the largest and stiffest cellular organelle, often limits cellular deformability, making it a key factor in migration through narrow pores or highly constrained spaces. In this work, we introduce a geometric surface partial differential equation (GS-PDE) model in which the cell plasma membrane and nuclear envelope are described as evolving energetic closed surfaces governed by force-balance equations. We replicate the results of a biophysical experiment, where a microfluidic device is used to impose compressive stresses on cells by driving them through narrow microchannels under a controlled pressure gradient. The model is validated by reproducing cell entry into the microchannels. A parametric sensitivity analysis highlights the dominant influence of specific parameters, whose accurate estimation is essential for faithfully capturing the experimental setup. We found that surface tension and confinement geometry emerge as key determinants of translocation efficiency. Although tailored to this specific setup for validation purposes, the framework is sufficiently general to be applied to a broad range of cell mechanics scenarios, providing a robust and flexible tool for investigating the interplay between cell mechanics and confinement. It also offers a solid foundation for future extensions integrating more complex biochemical processes such as active confined migration.

Advances in protein folding and structure prediction models have enabled new computational approaches to immunotherapeutic research by providing access to high-quality structural information at scale. In this talk, we present three core application areas. (1) Antigen structure prediction, where folding models are used to characterize the three-dimensional structure of viral, tumor-associated, and neoantigen targets in the absence of experimental data. (2) Antibody–antigen complex prediction, where multimeric and joint modeling approaches are leveraged to infer binding modes, paratope–epitope interactions, and structural determinants of specificity. (3) Immunogenicity prediction, where predicted structures are analyzed to assess surface accessibility, conformational stability, and geometric features that influence immune recognition. Together, these applications illustrate how protein folding models function not only as structure predictors, but as foundational components in quantitative pipelines for immunotherapeutic discovery and design.

Fix a positive integer $n$ and a graph $F$. A graph $G$ with $n$ vertices is called $F$-saturated if $G$ contains no subgraph isomorphic to $F$ but each graph obtained from $G$ by joining a pair of nonadjacent vertices contains at least one copy of $F$ as a subgraph. The saturation function of $F$, denoted $\mathrm{sat}(n, F)$, is the minimum number of edges in an $F$-saturated graph on $n$ vertices. This parameter along with its counterpart, i.e. Turan number, have been investigated for quite a long time.

We review known results on $\mathrm{sat}(n, F)$ for various graphs $F$. We also present new results when $F$ is a complete multipartite graph or a cycle graph. The problem of saturation in the Erdos-Renyi random graph $G(n, p)$ was introduced by Korandi and Sudakov in 2017. We survey the results for random case and present our latest results on saturation numbers of bipartite graphs in random graphs.