Practical indistinguishability in a gene regulatory network inference problem, a case study.

Computationally inferring mechanistic insights and underlying control structures from typical biological data is a challenging pursuit. The technical reasons for this are multifaceted-and we delve into them in depth here, but they are easy to understand and involve both the data and model development. Even the highest-quality experimental data come with challenges. There are always sources of noise, a limit to how often we can measure the system, and we can rarely measure all the relevant states that participate in the full underlying complexity. There are usually sources of uncertainty in model development, which give rise to multiple competing model structures. To underscore the need for further analysis of structural uncertainty in modeling, we use a meta-analysis across six journals covering mathematical biology and show that a huge number of mathematical models for biological systems are developed each year, but model selection and comparison across model structures appear to be less common. We walk through a case study involving inference of regulatory network structure involved in a developmental decision in the nematode, Pristonchus pacificus. We first examine the practical indistinguishability of a model structure, or the ability to uniquely infer the structure given the data, across a wide range of synthetic data regimes by refitting both the true model structure and several misspecified models. We then use real biological data and compare across 13,824 models-each corresponding to a different regulatory network structure, to determine which regulatory features are supported by the data across three experimental conditions. We find that the best-fitting models for each experimental condition share a combination of features and identify a regulatory network that is common across the model sets for each condition. This model is capable of describing the data across the experimental conditions we considered and exhibits a high degree of positive regulation and interconnectivity between the key regulators, $\mathrm{eud-1}$ , $\mathrm{sult-1}$ , and $\mathrm{nhr-40}$ . While the biological results are specific to the molecular biology of development in Pristonchus pacificus, the general modeling framework and underlying challenges we faced doing this analysis are widespread across biology, chemistry, physics, and many other scientific disciplines.