  It is hard to identify nonlinear biological models strictly from data, with
results that are often sensitive to experimental conditions. Automated
experimental workflows and liquid handling enables unprecedented throughput, as
well as the capacity to generate extremely large datasets. We seek to develop
generalized identifiability conditions for informing the design of automated
experiments to discover predictive nonlinear biological models. For linear
systems, identifiability is characterized by persistence of excitation
conditions. For nonlinear systems, no such persistence of excitation conditions
exist. We use the input-Koopman operator method to model nonlinear systems and
derive identifiability conditions for open-loop systems initialized from a
single initial condition. We show that nonlinear identifiability is
intrinsically tied to the rank of a given dataset's power spectral density,
transformed through the lifted Koopman observable space. We illustrate these
identifiability conditions with a simulated synthetic gene circuit model, the
repressilator. We illustrate how rank degeneracy in datasets results in
overfitted nonlinear models of the repressilator, resulting in poor predictive
accuracy. Our findings provide novel experimental design criteria for discovery
of globally predictive nonlinear models of biological phenomena.
