# Lecture "Applied Mathematics in Biology: Single-cell omics"

*Modulbeschreibung*

### Title

Applied Mathematics in Biology: Single-cell omics

### Lecturer

Simon Anders, BioQuant Center
([e-mail])

### Date and time

summer term 2024
days of week and time still to be determined by online poll

### Target Audience

students of math-oriented subjects (mathematics, physics, computer science, engineering, scientific computing); also life-sciences students with advanced math knowledge

### Topic and Aims

Many experimental techniques of modern biology need sophisticated data analysis methods. Developing these requires, among others, mastery of methods of mathematical modelling, of stochastic models, of numerics and scientific software engineering. Therefore, such methodolical research falls outside the expertise of most biologists; and is a task for applied mathematicians, physicists or math-oriented computer scientists.

One particular topic of currently rapidly growing importance is "single-cell omics": For several tissue samples, each comprising hundreds to thousands of cells, one measures for each cell a large vector of values that represent, e.g., the activity of genes or the epigenetic state in cells. This allows to, e.g., compare, healthy and diseased tissues, understand the make-up of tumours, track the embryonic development of organs, etc. In such data, each cell is represented by a vector in a feature space (typically simply $\mathbb{R}^n$) and the aim is to explore the geometry and topology of the manifolds sampled by the cells and translate the findings into the language of biology, and to perform statistical inference to gain generalizable conclusions about the experiment.

An aim of the lecture is therefore to introduce students to this area of work for researchers in applied mathematics. However, rather than giving a general overview, we will explore a specific field, namely single-cell omics, in depth, and learn not only how to bring modern methods of mathematics and scientific computing to bear on a problems of practical importance but also how to grapple with the communication difficulties of building bridges between two disciplines, mathematics and biology, with different language, concepts and ways of thinking.

Starting with the basics of the relevant parts of biology and applied math, we will advance to the current methodological frontier of the field.

### Prerequisites

- (finite-dimensional) linear algebra (matrix calculus, eigendecomposition, etc)
- experience with at least one programming language
- interest in biology

### Topics

- basics of molecular biology, as needed
- assay techniques in single cell (sc) biology
- stochastic models for omics data
- the concept of feature space, as used for omics data and in machine learning
- methods to explore high-dimensional data
- linear and non-linear methods for dimension reductions
- clustering in high-dimensional space
- non-linear regression methods
- graph-based methods in omics data analysis
- applications of machine learning methods to sc omics
- deep learning (esp. variational autoencoders) and related methods
- technqiues for handling big data
- interactive visualization for exploratory data analysis

### Form

one 90-minutes lecture per week, plus an exercise class of ~90 min per week

### Leistungspunkte (credit points)

tbd (2?)

### Dauer (length)

1 Semester

### Angebotsturnus (regularity)

For now, as a first try, once, in summer 2024. If successful, perhaps every other summer term.

### Arbeitsaufwand (work load)
(assuming 15 weeks in term)
- lectures: 30 hours (incl problem classes)
- homework: 30 hours
- perhaps also a project: 20 hours

### Prüfungsschema (exam)

tbd; written or oral exam

### Language

English

### Literature

tba