Research topics of the Biomathematics Group
The research of the Biomathematics Group of the Faculty of Mathematics focuses on the following areas:
Population genetics and evolution
Charles Darwin's 1837 sketch of an evolutionary tree
Deeper understanding of biological evolution requires quantitative prediction of the consequences of fundamental processes such as inheritance, mutation, recombination, migration, and selection. As in other natural sciences, mathematical models are essential for this purpose. Already some 100 years ago, the first basic models were developed to explain Darwin's theory of evolution using Mendel's theory of inheritance.
The field that studies this interaction of genetic and evolutionary mechanisms is called population genetics. Animal and plant breeding, and more recently human medicine, are the main areas of practical application. The focus of evolutionary ecology is the influence of ecological interactions on evolutionary processes.
There are two complementary but interrelated approaches for the study of evolution. In the first, evolutionary processes, i.e., the temporal changes of traits or gene frequencies, are assessed for given assumptions about genetic mechanisms and forms of selection.
Mathematically, this is done using differential equations, difference equations, or stochastic processes.
In the second approach, one starts from real data representing the genetic composition of living species. With modern sequencing methods, it is possible to sequence whole genomes of hundreds or thousands of individuals of a population. The primary interest is in the differences in DNA sequences between individuals, called polymorphisms.
From the patterns in these polymorphisms along the genome, one can infer evolutionary processes in the history of species (e.g., which ones are most closely related and how long ago the separation occurred, whether a population underwent phases of massive growth or contraction, or which genes are adaptive and have recently become established in a population by positive selection). Once again, stochastic processes are the main mathematical tool, complemented by statistical methods.
Kinetic theory, self-organisation and multi-scale phenomena
3D simulation of epithelial tissue and collective motion of cells in a confined domain
Birds organise into flocks, cells organise into tissues, pedestrians organise into lanes. How do individual rules give rise to group order? Kinetic theory gives us the mathematical tools to understand the emergence of self-organisation rigorously by linking individual and group behaviour.
At the heart of our approach lies coarse-graining: the rigorous derivation of macroscopic equations directly from microscopic dynamics. Kinetic equations are the crucial intermediate step in this process, bridging the gap between the individual and the collective. Working out this connection means analyzing partial differential equations and designing efficient numerical methods.
This matters far beyond physics. In the biological sciences, most models still operate at a single scale, either large or small. But real biological systems are inherently multi-scale: population dynamics depend on individual behavior, tissue function depends on cellular processes, and epidemic spread depends on individual contacts. Understanding how large-scale patterns emerge from lower-scale phenomena is key to many biological and social systems — and mathematics is the only known tool able to build this bridge rigorously, across scales and across disciplines.
PDE models of cell movements
From the paper: N. Sfakianakis, D. Peurichard, A. Brunk, C. Schmeiser, Modelling cell-cell collision and adhesion with the Filament Based Lamellipodium Model, Biomath 7 (2018), Article ID: 1811097.
The crawling motion of many cell types is based on the lamellipodium, a flat cell protrusion supported by a network of actin filaments.
The Filament Based Lamellipodium Model (FBLM) is a two-dimensional continuum model for the chemo-mechanics of the lamellipodium, including descriptions of the homeostatic balance of filament polymerization and depolymerization, the dynamics of cross-linking and substrate adhesions, the mechanical response to cell membrane stretching, and more.
The figure shows a snapshot from a simulation of a cluster of 14 cells described by the FBLM, incorporating the effects of a chemotactic signal (level sets in yellow and red), of a substrate with varying adhesiveness (darkness of the background), and of cell-cell adhesion (red arrows).
Mathematics of cell biology
Groups of cells move through a deformable environment.
Mathematical models provide a framework for understanding how
cellular processes and interactions give rise to
complex behaviour across different spatial and temporal scales.
Research in mathematical cell biology addresses questions ranging from
intracellular organisation and single-cell motility to
Of particular interest is how mechanical interactions, cell-to-cell variability and the surrounding environment shape the dynamics of individual cells and cell populations.
Ordinary and partial differential equations, agent-based models, analytical methods and numerical simulations are used to connect biological mechanisms with experimentally observed behaviour and data, and test biological hypotheses.
If you are interested in any of these topics, you can further explore our website, in particular: