Algebraic Experimental Design: Theory and Computation

TL;DR

This paper develops algebraic methods to ensure unique minimal wiring diagram identification in biological experimental design, linking algebraic geometry, combinatorics, and computational algorithms to improve model recovery.

Contribution

It introduces a new algebraic framework and algorithm for guaranteeing the uniqueness of wiring diagrams in experimental data, connecting Gr"obner bases with experimental design.

Findings

A new necessary condition for the uniqueness of reduced Gr"obner bases.

A computational heuristic for generating data with fewer minimal wiring diagrams.

Application of the theory to a tumor-suppression network.

Abstract

Over the past several decades, algebraic geometry has provided innovative approaches to biological experimental design that resolved theoretical questions and improved computational efficiency. However, guaranteeing uniqueness and perfect recovery of models are still open problems. In this work we study the problem of uniqueness of wiring diagrams. We use as a modeling framework polynomial dynamical systems and utilize the correspondence between simplicial complexes and square-free monomial ideals from Stanley-Reisner theory to develop theory and construct an algorithm for identifying input data sets $V\subset \mathbb F_p^n$ that are guaranteed to correspond to a unique minimal wiring diagram regardless of the experimental output. We apply the results on a tumor-suppression network mediated by epidermal derived growth factor receptor and demonstrate how careful experimental design decisions can lead to a unique minimal wiring diagram identification. One of the insights of the theoretical work is the connection between the uniqueness of a wiring diagram for a given $V\subset \mathbb F_p^n$ and the uniqueness of the reduced Gr\"obner basis of the polynomial ideal $I(V)\subset \mathbb F_p[x_1,\ldots, x_n]$. We discuss existing results and introduce a new necessary condition on the points in $V$ for uniqueness of the reduced Gr\"obner basis of $I(V)$. These results also point to the importance of the relative proximity of the experimental input points on the number of minimal wiring diagrams, which we then study computationally. We find that there is a concrete heuristic way to generate data that tends to result in fewer minimal wiring diagrams.