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.
Highlights
Fluctuation in the number of chemical species is ubiquitous and pronounced in small reactors such as living cells
To understand how the frequency spectrum of product number fluctuation is related to the topology of the reaction network and the dynamics of elementary processes composing the network, we derived an exact analytic result for the frequency spectrum of the product number fluctuation starting from a generalized master equation, enabling the extraction of the frequency spectrum of the reaction rate fluctuation (FSRR) from the frequency spectrum of the product number fluctuation (FSPN)
We have so far assumed the translation rate coefficient, kTL, is constant, it can be a random variable, the value of which differs from cell to cell. For this case as well, Eq 3 holds and the protein number frequency spectrum is related to the messenger RNA (mRNA) number frequency spectrum by as long as the cell-to-cell heterogeneity of kTL is much greater than the dynamic fluctuation of kTL in each cell
Summary
Recent advances in single-cell experimental techniques enable direct visualization of dynamic fluctuations of the biomolecular concentration in each cell; a robust, quantitative understanding of the stochastic dynamics of the chemical noise in living cells has yet to be achieved. To understand how the frequency spectrum of product number fluctuation is related to the topology of the reaction network and the dynamics of elementary processes composing the network, we derived an exact analytic result for the frequency spectrum of the product number fluctuation starting from a generalized master equation, enabling the extraction of the frequency spectrum of the reaction rate fluctuation (FSRR) from the frequency spectrum of the product number fluctuation (FSPN). We demonstrated our approach to frequency spectrum analysis of chemical fluctuation for generalized. Enzyme kinetic models, the gene network model of luciferase expression under the Bmal 1a promoter in mouse fibroblast cells, and a more general vibrant gene network model
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