Noise in gene expression, either due to inherent stochasticity or to varying inter- and intracellular environment, can generate significant cell-to-cell variability of protein levels in clonal populations. We present a theoretical framework, based on stochastic processes, to quantify the different sources of gene expression noise taking cell division explicitly into account. Analytical, time-dependent solutions for the noise contributions arising from the major steps involved in protein synthesis are derived. The analysis shows that the induction level of the activator or transcription factor is crucial for the characteristic signature of the dominant source of gene expression noise and thus bridges the gap between seemingly contradictory experimental results. Furthermore, on the basis of experimentally measured cell distributions, our simulations suggest that transcription factor binding and promoter activation can be modelled independently of each other with sufficient accuracy.
Within a genetically identical population, individual cells show significant phenotypic heterogeneity (1; 2; 3). This variability directly affects the cell's ability to respond to environmental factors like changes in ligand concentration. Especially, reactions underlying protein synthesis are often based on small numbers of molecules, like transcription factors or ribosomes, such that stochastic fluctuations have to be taken into account.
A lot of effort has been undertaken to quantify the origins of gene expression noise experimentally and theoretically. Stochasticity or noise inherent to gene expression seems to be one of the main driving forces for the observed cell-to-cell variability in several experiments which have measured the variance in protein abundances in different cellular systems (4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 18). Considerable confusion stems from diverging experimental results which have identified different origins for the main contribution to gene expression noise (19) such that a complete picture is still missing. For prokaryotes, translational efficiency was identified as the main source of variability of expression levels consistent with a stochastic model in which proteins are produced in sharp and random bursts (20). However, later experimental observations in individual living cells either by measuring mRNA levels or by real-time observations at single molecule level indicated that promoter activation predominantly causes gene expression noise (21; 22). Furthermore, extrinsic factors, like the cellular state, were also identified to give the main contribution to phenotypic variations in a clonal population (16). Similar contradictory results have been found in eukaryotes, where in the budding yeast Saccharomyces cerevisiae a two-reporter system, expressing two fluorescent proteins from identical promoters, identified switching between active and inactive promoter states due to slow stochastic chromatin-remodelling events as the by far largest source of noise (5). In later in-vivo experiments it was shown for a large set of genes at their native expression levels that the noise has a clear sign of transcriptional origin due to low-copy mRNA molecules (7; 14). Moreover, a direct monitoring of mRNA production from a gene at the resolution of single molecules in mammals revealed strong mRNA bursts dominating gene expression noise (9). In contrast, for human cells, genes at native induction level showed significant noise contribution from long-term variations of the cellular state (15). It seems on first sight that no general rule can be given to determine the main sources of gene expression noise. Protein levels, however, are often strongly optimized, because they have to allow for precise and reliable information processing within the cell. Any significant deviation from the optimal level would result in reduction of fitness and an evolutionary disadvantage. Thus, random fluctuations are in general detrimental for cellular systems and several regulatory mechanisms have evolved to minimize them. Only in rare cases noise can be used to drive phenotypic switching providing a non-genetic mechanism to population heterogeneity, as found for bacterial persistence against antibiotics (23) and competence for DNA uptake from the environment (24).
In order to track down the individual contributions of the molecular mechanisms involved in protein synthesis several mathematical models have been introduced (25; 26; 27; 28; 29; 30; 31). Some of these models ignore the effect of binomial partitioning by cell division which will lead to strong discrepancies to experiments whenever cellular mRNA is long-lived and appears in low copy number (29; 30; 31), whereas others lack the dynamic description of mRNA bursts (25; 27). In the present work we develop an analytical framework which allows for a time-dependent description of gene expression and accounts for effects of symmetric cell division. We consider a one-gene-system consisting of activator/transcription factor (TF) binding (repressor unbinding), promoter activation, transcription, and translation (Fig. 1).
All gene specific events contribute to the so-called intrinsic noise. Differences between cells, either in global cellular state or in the concentration or activity of any factor that affects gene expression are referred to as extrinsic noise (4). Therefore, the cell-to-cell variability of a specific protein in a large clonal population with fixed generation time is characterized by the two contributions of intrinsic and extrinsic noise, summing up to the overall variance of the protein. Assuming no specific feedback of a produced protein on upstream processes, the intrinsic noise contribution decomposes into partial contributions stemming from activator binding, promoter activation, transcription and translation. In deriving analytical expressions for these partial contributions to gene expression noise, we discuss limiting ca
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