Algebra, geometry and topology of ERK kinetics (Q2092836)

The paper presented here substantiates and corroborates mathematically the Bayesian parameter study of the publication [\textit{E. Yeung} et al., ``Inference of multisite phosphorylation rate constants and their modulation by pathogenic mutations'', Current Biology 30, No. 5, 877--882 (2020; \url{doi:10.1016/j.cub.2019.12.052})]. It centers around a polynomial mass action ODE model describing the so-called MEK/ERK signalling pathway important in cell division, specialisation, survival and death. The starting point consists of experimental data in form of time-course measurements from the above-mentioned publication. The focus of attention lies in variations of the notion of identifiability. Structural identifiability means that ``parameter recovery is possible with perfect data'', ``practical identifiability can be defined in terms of the boundedness \dots of the confidence regions of a likelihood test''. As it turns out, the so-called linear ERK model is structurally and practically identifiable (Theorem 1). Algebraic field theory is invested to find sufficient conditions for other versions of identifiability. As for practical identifiability, a necessarily rather technical algorithmic test is developed. Next, Bayesian inference on parameter values is applied to the linear ERK model. Parameters are estimated to fit the experimental data. Bayes yields a posterior probability density upon specifying an observation. Finally, to study posterior distributions, topological data analysis is invoked. After reviewing the basic facts of persistent homology, the persistence module of a filtration of a space or a simplicial complex is defined, together with the notion of barcode of a persistence module, and an upper estimate of the bottleneck distance of two barcodes in the context of functions defining the filtrations is derived (Theorem 15). The bandwidth is a measure of how the probability mass is concentrated near a sample point. As a biological conclusion of the theory, one arrives at quantitatively comparing parameter posteriors in the linear ERK model for phosphorylation in so-called wild type kinetics and pathological mutations. Mathematical proofs are provided in the appendix.