Funded Grants

Evolution of topological features in complex biological systems

Research in my group focuses on developing broadly applicable mathematical tools that can help us understand the functional principles of multicellular organisms. Our work is guided by three fundamental interconnected biological questions: Can we identify generic rules that allow us to predict the gene-expression patterns in developing organisms? How do intra- and inter-cellular force networks coordinate their actions to realize highly reproducible tissue-folding processes during embryonic development? Can we develop robust algorithms to reconstruct from single-cell sequencing data the spatial position of individual cells, their temporal position in the differentiation cycle, and ultimately their biological function in multicellular organisms?

To tackle these problems quantitatively, we are investigating new mathematical and computational approaches that combine traditional modeling techniques based on deterministic and stochastic differential equations with modern statistical, graph-theoretic and topological concepts. Our efforts are aided by the close collaboration with experimentalists in the fields of biology and biophysics. We are specifically interested in integrating advanced image and data analysis methods with predictive modeling techniques that are applicable to broad classes of complex biological systems and can be rigorously tested in experiments.

Development of innovative mathematical approaches is necessitated by the current revolution in single-cell microfluidics and optimal imaging techniques, which will offer access to unprecedented high-resolution data over the next decade. Detailed simultaneous measurements of the individual states of thousands of cells will produce high-dimensional data matrices and tensors that need to be compressed and analyzed efficiently with minimal loss of biologically relevant information. The size and complexity of these data pose severe conceptual and practical challenges to traditional bio-physical and bio-mathematical theory: The relevant macro-state variables of cells are a priori unknown, the data may be noisy and incomplete, and the dimensionality may be so large that conventional models require a prohibitively large number of parameters and, hence, cannot be tested conclusively and become difficult to interpret.

To help overcome these limitations, our research program integrates elements from applied mathematics, biophysics and quantitative biology to realize four main goals: (1) Design and implementation of efficient data-completion and dimensionality-reduction algorithms to reconstruct gene expression profiles with high accuracy and to identify relevant macroscopic state-variables from large-scale RNA sequencing data. (2) Robust identification of persistent topological features in optical imaging and sequencing data by combining information-theoretic concepts with Morse-Smale complex analysis to reconstruct intercellular force networks during morphogenesis and detect pathological tissue deformations. (3) Design and implementation of network-based and continuum-based field theories that focus on predicting the spatio-temporal dynamics of experimentally observed topological features. (4) Systematic verification of model predictions against data provided by our collaborators from sequencing experiments in single cells as well as gene expression and force measurements in multicellular systems.

The successful completion of this program will establish a new class of experimentally validated topological and statistical analysis tools that promise profound insights into the regulatory mechanisms underlying evolution and function of complex biological systems.