Maximum likelihood estimation of log-affine models using detailed-balanced reaction networks

TL;DR

This paper investigates the use of reaction networks to compute maximum likelihood estimates for log-affine models, demonstrating how the choice of basis affects network dynamics and stability.

Contribution

It extends Gopalkrishnan's construction to arbitrary spanning sets, showing Markov bases ensure global stability of the MLE steady state.

Findings

Markov bases guarantee global stability

Choice of spanning set influences network dynamics

Extended construction allows for flexible network design

Abstract

A fundamental question in the field of molecular computation is what computational tasks a biochemical system can carry out. In this work, we focus on the problem of finding the maximum likelihood estimate (MLE) for log-affine models. We revisit a construction due to Gopalkrishnan of a mass-action system with the MLE as its unique positive steady state, which is based on choosing a basis for the kernel of the design matrix of the model. We extend this construction to allow for any finite spanning set of the kernel, and explore how the choice of spanning set influences the dynamics of the resulting network, including the existence of boundary steady states, the deficiency of the network, and the rate of convergence. In particular, we prove that using a Markov basis as the spanning set guarantees global stability of the MLE steady state.