Geometric and Topological Data Analysis of Enzyme Kinetics

The research we plan to conduct in this research project is concerned with Extracellular Signal Regulated Kinase (ERK) kinetics. Mutations affecting ERK are associated with diseases such as human cancer and developmental defects, making them of significant interest in contemporary biological research. While the effects of such mutations have been observed in vivo, their effects on the mechanism of ERK activation, i.e. quantitive changes to mathematical models describing such phenomena, have remained unknown so far. The aim is to derive novel techniques for quantifying the effect genetic perturbations, such as mutations, have on ERK kinetics.
To this end, we will investigate families of ODE models based on such kinetics and derive parameter inferences for various mutants. In this light, we are interested in model reduction techniques and characterising their utility and practical identifiability in the context of real-world data through tools of algebra, geometry and topology. The relative strength of the various models arising can then be tested by means of model comparison. The objective is to then go on to study the shape of distributions resulting from Bayesian parameter inferences, using methods of Topological Data Analysis (TDA), in order to distinguish between various genetic perturbations at the level of cells. We identify three directions of research we aim to investigate in order to achieve this goal: The algebra of Quasi-Steady-State Approximations, geometric characterisations of their goodness of approximation, and theoretical results from TDA guaranteeing recoverability of relevant topological information. The proposed directions of research are motivated by successful results obtained from analysing one model assumption during my MSC thesis, which has subsequently been written into a manuscript been submitted and undergoing revision, here we plan to analyse a family in a more mathematically principles manner. Moreover, we will look into investigating what conclusions can be made at the level of genetic perturbations in cells by studying the shape of parameter inferences through running simulations involving synthetic data. To the best of our knowledge, this would constitute a novel achievement. Furthermore, we aim to generalise the inference pipeline used in the MSc thesis to more complex models of ERK mechanisms involving a larger number of sites.
On the more theoretical side, open questions are to strengthen and extend existing results on algebraic and geometric characterisations of model reductions, such as Quasi-Steady-State Approximation, aiming to understand better the accuracy of these approximations. Moreover, it would be worthwhile to investigate how the discriminative power of tools of Topological Data Analysis compares between different models.
Possible collaborators are the Shvartsman Lab in Princeton, who motivated the MSc project mentioned above and supplied the measurement data.
This project falls within the EPSRC Research areas, Algebra, Geometry & Topology, Statistics and Applied Probability and Mathematical biology.