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.

Padé SSA: A frequency domain method for estimating the dynamics of stochastic reaction networks

“…These eigenvalues do not depend on x 0 and f and we assume that all eigenvalues are distinct and arranged in descending order of their real parts (which are negative due to ergodicity). The eigen-decomposition of the resolvent operator (see the supplement of [13]) provides the expression (5), where each α j captures the contribution in the dynamics of the eigenmode corresponding to σ j . Under the assumption of exponential ergodicity, the resolvent operator is compact and this ensures that it can be accurately approximated by a finite-rank operator [16] and hence we would have α j → 0 as j → ∞.…”

“…, β q−1 , which is necessary to ensure the stability of the dynamics. The exact expressions for A and b were derived in [13] and they can be found in the Appendix of this paper. Now let us consider the scenario that the coefficient vector y = (κ 0 , .…”

“…Observe that 𝔸 m f ( x 0 ) can be easily computed recursively (see [13]) and so we do not need to estimate it with simulations, unlike the Padé derivatives for finite values of s .…”

“…We add positivity constraints on the denominator coefficients β 0 , …, β q −1 , which is necessary to ensure the stability of the dynamics. The exact expressions for A and b were derived in [13] and they can be found in the Appendix of this paper.…”

“…Finally we would like to point out that our Padé SSA method is inspired by a similar method called Padé PSD that we have recently developed for the estimation of power spectral density (PSD) of stochastic single-cell trajectories from reaction network models (see [13]).…”