Frequency spectrum of chemical fluctuation: A probe of reaction mechanism and dynamics

Abstract

Even in the steady-state, the number of biomolecules in living cells fluctuates dynamically, and the frequency spectrum of this chemical fluctuation carries valuable information about the dynamics of the reactions creating these biomolecules. Recent advances in single-cell techniques enable direct monitoring of the time-traces of the protein number in each cell; however, it is not yet clear how the stochastic dynamics of these time-traces is related to the reaction mechanism and dynamics. Here, we derive a rigorous relation between the frequency-spectrum of the product number fluctuation and the reaction mechanism and dynamics, starting from a generalized master equation. This relation enables us to analyze the time-traces of the protein number and extract information about dynamics of mRNA number and transcriptional regulation, which cannot be directly observed by current experimental techniques. We demonstrate our frequency spectrum analysis of protein number fluctuation, using the gene network model of luciferase expression under the control of the Bmal 1a promoter in mouse fibroblast cells. We also discuss how the dynamic heterogeneity of transcription and translation rates affects the frequency-spectra of the mRNA and protein number.

Frequency Spectra and the Color of Cellular Noise

“…We show how this expression can be alternatively derived via the resolvent connection and we also generalise this result to allow for arbitrary time-varying inputs. This generalisation yields a novel PSD decomposition result that is similar to what was found in previous SDE-based studies [20] and it extends the recent results in [26].…”

“…In this section we present a novel PSD decomposition result for linear networks, that extends a similar result recently reported in [26]. A reaction network is called linear if all its propensity functions are affine functions of the state variables.…”

“…In this context, the aim of this paper is to develop a computational method for reliably estimating the PSD for single-cell trajectories from CTMC models. Existing approaches for PSD estimation for stochastic network models, are either applicable to a particular class of networks [26,17], or they are based on dynamical approximations that are known to be inaccurate over large time-intervals and in situations where low abundance species are present [20,19]. The method we develop in this paper, called Padé PSD, especially pertains to the low abundance regime.…”

“…The aim of this paper is to develop such a method. In a recent paper [26], the analytical relationship between the PSDs of the output species and its time-dependent production rate was derived for CTMC models of certain reaction networks including birth-death and simple geneexpression. While this analysis enables investigation of the dynamics of the protein creation process from experimentally measured protein time-traces, it does not extend to nonlinear networks, such as gene expression networks with transcriptional feedback, for which some analytical results exist for simplified models [27].…”

“…A recurring theme in the existing literature is that typically the autocovariance function is well-approximated by the sum of a few exponential functions [26,20,18], and consequently the PSD is a rational function of a special form. This low dimensional feature can be theoretically explained by appealing to the compactness of the resolvent operator [28] associated with the CTMC, which we as prove, is connected to the PSD.…”