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.

中文翻译：

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频谱和蜂窝噪声的颜色

19 世纪傅里叶积分的发明为现代光谱分析方法奠定了基础。通过将（时间）信号分解为其基本频率分量，这些方法揭示了对信号及其生成过程的深刻见解，在工程、技术和物理科学的许多领域催生了巨大的发明和发现。然而，在系统和合成生物学中，尽管频率方法具有巨大的前景，但其影响却非常有限。这在很大程度上是由于将活细胞中的潜在随机反应网络（其动力学通常建模为连续时间马尔可夫链（CTMC））与观察到的、明显嘈杂的单-细胞轨迹。在这里，我们利用随机过程理论来开发专门用于计算和分析嘈杂蜂窝网络频谱的频谱理论和计算方法。具体来说，我们开发了一种通用方法来从少数轨迹模拟中获得准确的 Padé 近似谱。此外，对于线性网络，我们提出了一种新的分解结果，根据其来源表示频谱。我们的结果为基于输出频谱的嘈杂蜂窝网络的分析和设计提供了新的概念和实用方法。我们通过不同的案例研究来说明这一点，在这些案例研究中，我们表明单细胞频谱有助于拓扑鉴别、合成振荡器优化、网络遗传控制器设计、