This day event is part of the collaborative workshop "Mathematical Biology in Saxony" organized jointly by the University of Oxford, the Algebraic Systems Biology of Heather Harrington in the Center for Systems Biology in Dresden (CSBD) and the Nonlinear Algebra group of Bernd Sturmfels at MPI-MiS in Leipzig.
This workshop offers a unique opportunity for researchers from these different institutions to explore current research areas in mathematical biology. Attendees will have the opportunity to learn about exciting research topics and establish valuable collaborations.
A longstanding goal of biology is to identify the key genes and species that critically impact evolution, ecology, and health. Network analysis has revealed keystone species that regulate ecosystems and master regulators that regulate cellular genetic networks. Yet these studies have focused on pairwise biological interactions, which can be affected by the context of genetic background and other species present generating higher-order interactions. The important regulators of higher-order interactions are unstudied. To address this, we apply methods from polyhedral geometry to quantify epistasis in a fitness landscape to ask how individual genes and species influence the interactions in the rest of the biological network.
Joint work with Holger Eble, Lisa Lamberti, and William B. Ludington.
The main goal of Phylogenetics is to find evolutionary relations between biological entities.These relations are inferred from data and are encoded in a graph called a phylogenetic tree. Available data can come from DNA sequences or aminoacid sequences of different species but, once we have the data, how to determine which phylogenetic tree accurately represents the hidden evolutionary relations? In this talk we explore how understanding the so called phylogenetic variety helps us with model selection and phylogenetic reconstruction. We will start by introducing phylogenetic varieties and time-reversible models, then we present a general framework to study the phylogenetic reconstruction process for time-reversible models. Our main example will be the Tamura-Nei model for DNA, for which we give phylogenetic invariants for three and four species. This is joint work with Marta Casanellas and Roser Homs Pons.
The Euler characteristic transform (ECT) is an operation which takes shapes as input and returns integer valued functions. The ECT is used in practice to study shapes for two reasons. First, no two well-behaved shapes have the same ECT, meaning that a shape is described entirely by its ECT. Second, it is straightforward to compare the ECTs of shapes quantitatively. It is important to verify that the ECT is stable, i.e. that changing an input shape slightly will not drastically affect the resulting ECT. We present a novel stability result for the ECTs of curves and show how our results can be leveraged to estimate the ECT of an underlying curve in the presence of noisy data.
Geometric deep learning extends deep learning to incorporate information about the geometry and topology of data, especially in complex domains like graphs. Many existing methods in this field rely on message passing. However in this talk we will take a new approach by combining the theory of differential k-forms in euclidean space with the geometric information of graphs and complexes given by embeddings of node coordinates. This method offers interpretability and geometric consistency without the use of message passing.
In topological data analysis, persistence barcodes record the persistence of homological generators in a one-parameter filtration built on the data at hand. In contrast, computing the pointwise Euler characteristic (EC) of the filtration merely records the alternating sum of the dimensions of each homology vector space.
In this talk, we will show that despite losing the classical "signal/noise" dichotomy, EC tools are powerful descriptors, especially when combined with new integral transforms mixing EC techniques with Lebesgue integration. Our motivation is fourfold: their applicability to multi-parameter filtrations and time-varying data, their remarkable performance in supervised and unsupervised tasks at a low computational cost, their satisfactoryproperties as integral transforms (e.g., regularity and invertibility properties) and the expectation results on the EC in random settings. Along the way, we will give an insight into the information these descriptors record.
The evolution of gene families involves duplications, losses, and transfers of genes to other organisms. To represent this history, a gene tree is embedded in a species tree, which creates homology relations between genes. Despite its key role in assigning functions to novel discovered genes, estimating gene family histories (GFHs) is still challenging, and several methods have been developed to address it. In particular, tree-free methods aim to infer homologous relationships between genes using sequence similarity without prior knowledge of the underlying trees. A novel approach based on graph theory falls into this category, and it has been partially implemented in a pipeline. In this presentation, I will give an overview of the family of graphs used in the pipeline, along with preliminary results and future directions.
Spatial transcriptomics technologies produce gene expression measurements at millions of locations within a tissue sample. An open problem in this area is the inference of spatial information about single cells. Here we present a multiscale method to pinpoint the locations of individual sparsely dispersed cells from subcellular spatial transcriptomics data. We integrate this approach with multiparameter persistence landscapes, a state of the art tool in topological data analysis, to identify a loop structure in infiltrating glomerular immune cells in a mouse model of lupus nephritis. Joint work with colleagues in the Mathematical Institute and Wellcome Centre for Human Genetics at the University of Oxford, and the Beijing Genomics Institute.
There are two ways of thinking about mathematics in biology. (1) How do real world biological problems motivate mathematical problems. (2) How does mathematics allow us to advance biological and biomedical challenges. In this talk I will address both. In the first part of the talk I will start with applied problems in time series modeling in microbial ecology and move to the question of when can a dynamical system be learned. In the second part I will start with geometric and topological representations of shape and move to how we use these for biomedical applications and medical interventions. I probably should have stated the title as zeit und form.
