Abstract
Stochastic expression of genes produces heterogeneity in clonal populations of bacteria under identical conditions. We analyze and compare the behavior of the inducible lac genetic switch using well-stirred and spatially resolved simulations for Escherichia coli cells modeled under fast and slow-growth conditions. Our new kinetic model describing the switching of the lac operon from one phenotype to the other incorporates parameters obtained from recently published in vivo single-molecule fluorescence experiments along with in vitro rate constants. For the well-stirred system, investigation of the intrinsic noise in the circuit as a function of the inducer concentration and in the presence/absence of the feedback mechanism reveals that the noise peaks near the switching threshold. Applying maximum likelihood estimation, we show that the analytic two-state model of gene expression can be used to extract stochastic rates from the simulation data. The simulations also provide mRNA–protein probability landscapes, which demonstrate that switching is the result of crossing both mRNA and protein thresholds. Using cryoelectron tomography of an E. coli cell and data from proteomics studies, we construct spatial in vivo models of cells and quantify the noise contributions and effects on repressor rebinding due to cell structure and crowding in the cytoplasm. Compared to systems without spatial heterogeneity, the model for the fast-growth cells predicts a slight decrease in the overall noise and an increase in the repressors rebinding rate due to anomalous subdiffusion. The tomograms for E. coli grown under slow-growth conditions identify the positions of the ribosomes and the condensed nucleoid. The smaller slow-growth cells have increased mRNA localization and a larger internal inducer concentration, leading to a significant decrease in the lifetime of the repressor–operator complex and an increase in the frequency of transcriptional bursts.
Highlights
Transcriptional and translational regulatory networks control the phenotype of modern cells, regulating gene expression in response to changing environmental conditions and/or biological stimuli
Measured the distributions of a fluorescent reporter protein under control of the lac operator in individual E. coli cells at various inducer (TMG) concentrations [22]. They performed the measurements in the absence of LacY’s positive feedback by replacing its gene with that of the membrane protein Tsr in the lac operon. This enabled an accurate determination of the protein distribution produced by the circuit at a given inducer concentration without any confounding non-linear effects due to enhancement of the internal inducer concentration by LacY
Xa{1e{ C(a):ba x b, where a was interpreted as the frequency of transcriptional bursts relative to the protein lifetime and b as the mean number of proteins produced per burst
Summary
Transcriptional and translational regulatory networks control the phenotype of modern cells, regulating gene expression in response to changing environmental conditions and/or biological stimuli. It has been well established that intrinsic noise in gene regulation results from the discrete biochemical nature of the process [1]. Theoretical modeling of stochasticity in gene expression has been a topic of intense study in the last decade and has greatly increased our understanding of the effect that statistical noise has on gene regulation (for reviews see [7,8,9,10,11]). Various analytical methods including moment generating functions [1,3,13], the Langevin and Fokker-Planck equations [14], linear noise approximation [4], and many-body theory [15] are used to study such models of gene expression. Usually based on a variant of Gillespie’s stochastic simulation algorithm (SSA) [16] are widely employed to analyze gene network models that are too complex to be amenable to analytical modeling [17,18]
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