Frequency Spectra and the Color of Cellular Noise

Abstract The invention of the Fourier integral in the 19th century laid the foundation for modern spectral analysis methods. By decomposing a (time) signal into its essential frequency components, these methods uncovered deep insights into the signal and its generating process, precipitating tremendous inventions and discoveries in many fields of engineering, technology, and physical science. In systems and synthetic biology, however, the impact of frequency methods has been far more limited despite their huge promise. This is in large part due to the difficulties encountered in connecting the underlying stochastic reaction network in the living cell, whose dynamics is typically modelled as a continuous-time Markov chain (CTMC), to the frequency content of the observed, distinctively noisy single-cell trajectories. Here we draw on stochastic process theory to develop a spectral theory and computational methodologies tailored specifically to the computation and analysis of frequency spectra of noisy cellular networks. Specifically, we develop a generic method to obtain accurate Padé approximations of the spectrum from a handful of trajectory simulations. Furthermore, for linear networks, we present a novel decomposition result that expresses the frequency spectrum in terms of its sources. Our results provide new conceptual and practical methods for the analysis and design of noisy cellular networks based on their output frequency spectra. We illustrate this through diverse case studies in which we show that the single-cell frequency spectrum facilitates topology discrimination, synthetic oscillator optimization, cybergenetic controller design, systematic investigation of stochastic entrainment, and even parameter inference from single-cell trajectory data.