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Optimal sensor placement is an important yet unsolved problem in controltheory. In biological organisms, genetic activity is often highly nonlinear,making it difficult to design libraries of promoters to act as reporters of thecell state. We make use of the Koopman observability gramian to develop analgorithm for optimal sensor (or reporter) placement for discrete timenonlinear dynamical systems to ease the difficulty of design of the promoterlibrary. This ease is enabled due to the fact that the Koopman operatorrepresents the evolution of a nonlinear system linearly by lifting the statesto an infinite-dimensional space of observables. The Koopman framework ideallydemands high temporal resolution, but data in biology are often sampledsparsely in time. Therefore we compute what we call the temporally fine-grainedKoopman operator from the temporally coarse-grained Koopman operator, thelatter of which is identified from the sparse data. The optimal placement ofsensors then corresponds to maximizing the observability of the fine-grainedsystem. We demonstrate the algorithm on a simulation example of a circadianoscillator.

Optimal reporter placement in sparsely measured genetic networks using the Koopman operator