Numerical algebraic geometry for model selection and its application to the life sciences

TL;DR

This paper introduces a numerical algebraic geometry framework for model selection in life sciences, enabling efficient parameter estimation and validation for polynomial models despite challenges like non-linearity and partial data.

Contribution

It presents a novel application of polynomial homotopy continuation methods to analyze polynomial models, improving global optimization in model selection tasks.

Findings

Successfully applied to cell signaling models

Effectively identifies global optima in complex models

Demonstrates utility in synthetic biology and epidemiology

Abstract

Researchers working with mathematical models are often confronted by the related problems of parameter estimation, model validation, and model selection. These are all optimization problems, well-known to be challenging due to non-linearity, non-convexity and multiple local optima. Furthermore, the challenges are compounded when only partial data is available. Here, we consider polynomial models (e.g., mass-action chemical reaction networks at steady state) and describe a framework for their analysis based on optimization using numerical algebraic geometry. Specifically, we use probability-one polynomial homotopy continuation methods to compute all critical points of the objective function, then filter to recover the global optima. Our approach exploits the geometric structures relating models and data, and we demonstrate its utility on examples from cell signaling, synthetic biology, and epidemiology.