Proc Natl Acad Sci U S A 99(20): 12795-12800

Intrinsic and extrinsic contributions to stochasticity in gene expression

^{Center for Studies in Physics and Biology and}^{Laboratory for Cancer Biology, The Rockefeller University, 1230 York Avenue, New York, NY 10021}

^{To whom reprint requests should be addressed. E-mail:}ac.lligcm.dnc@niaws.

^{Present address: Department of Physiology, McGill University, 3655 Promenade Sir William Osler, Montréal, QC, Canada H3G 1Y6.}

Edited by Robert H. Austin, Princeton University, Princeton, NJ, and approved June 17, 2002

Received 2002 Jan 23

Abstract

Gene expression is a stochastic, or “noisy,” process. This noise comes about in two ways. The inherent stochasticity of biochemical processes such as transcription and translation generates “intrinsic” noise. In addition, fluctuations in the amounts or states of other cellular components lead indirectly to variation in the expression of a particular gene and thus represent “extrinsic” noise. Here, we show how the total variation in the level of expression of a given gene can be decomposed into its intrinsic and extrinsic components. We demonstrate theoretically that simultaneous measurement of two identical genes per cell enables discrimination of these two types of noise. Analytic expressions for intrinsic noise are given for a model that involves all the major steps in transcription and translation. These expressions give the sensitivity to various parameters, quantify the deviation from Poisson statistics, and provide a way of fitting experiment. Transcription dominates the intrinsic noise when the average number of proteins made per mRNA transcript is greater than ≃2. Below this number, translational effects also become important. Gene replication and cell division, included in the model, cause protein numbers to tend to a limit cycle. We calculate a general form for the extrinsic noise and illustrate it with the particular case of a single fluctuating extrinsic variable—a repressor protein, which acts on the gene of interest. All results are confirmed by stochastic simulation using plausible parameters for Escherichia coli.

Abstract

Molecules are discrete entities. When present in large numbers, addition or removal of any single molecule typically has little effect on the properties of a system. However, stochastic fluctuations can become significant in smaller systems. In living cells, many components are present at very low copy numbers, [e.g., of order one for DNA loci and of order tens for transcription factors (1)]. Therefore, stochastic effects are thought to be particularly important for gene expression and have been invoked to explain cell–cell variations in clonal populations (2–4). Indeed, cellular components interact with one another in complex regulatory networks. Thus, fluctuations in even a single component may potentially affect the performance of the entire system.

Consider a particular gene of interest. The amount of protein it produces will vary from cell to cell in a population and over time in a single cell. These fluctuations originate in two ways: First, even if all cells were in precisely the same state, the reaction events leading to transcription and translation of the gene would still occur at different times, and in different orders, in different cells. Such stochastic effects are set locally by the gene sequence and the properties of the protein it encodes and will be referred to as “intrinsic” noise.

In addition, one must consider that other molecular species in the cell, e.g., RNA polymerase (RNAP), are themselves gene products and therefore will also vary over time and from cell to cell. This variation causes additional, and corresponding, fluctuations in the expression of the gene of interest that will be referred to as “extrinsic” noise. Thus, extrinsic sources of noise arise independently of the gene but act on it. Examples of extrinsic variables are numerous. They include the number of RNAPs or ribosomes, the stage in the cell cycle, the quantity of the protein, and mRNA degradation machinery, and the cell environment. In general, the total variation in gene expression will have both intrinsic and extrinsic sources. A particular cellular component will suffer intrinsic fluctuations in its own concentration and, at the same time, will be a source of extrinsic noise for other components with which it interacts.

Although the stochastic nature of gene expression has long been postulated (2), previous theoretical research (5–11) has concentrated on intrinsic noise. Excepting studies of plasmid copy number control (12), extrinsic effects have only been added in a post hoc manner (13). It is not known which molecular properties influence noise or even how a clear measurement of intrinsic noise could be obtained in vivo.

This paper seeks to address several problems. First, we distinguish between intrinsic and extrinsic sources of noise and integrate both within a single framework. Second, we model intrinsic noise at a level that allows direct connection with biochemical parameters, including those related to cell growth. Third, we suggest an experimental method that can be used to discriminate and quantify the two components of noise in living cells. Our approach, by integration of intrinsic and extrinsic effects, is general enough to allow comparison with experimental data (14).

Time is the only extrinsic value, and there is consequently almost no extrinsic noise in the protein concentration (because this varies little during the cell cycle; see Fig. Fig.2).2). The intrinsic noise in both cases, calculated by Eq. 21 (with an appropriate expression for cell volume when needed), is a very good approximation to ∫dt $\hat{η}$ equation M44 , which is found to be 0.15 ± 0.008 from simulation and 0.15 from integrating Eq. 17. Parameter values are published as supporting information on the PNAS web site, except k₀ = 0.01 s^{. Values stated are mean results from 100 simulations and errors are ±1 SD.}

There are two extrinsic variables: time and the repressor. Noise is calculated for protein numbers, not concentrations. To find the intrinsic noise, two copies of the (repressed) gene were simulated (see text). All proteins (including the repressor) were created according to the full scheme of Fig. Fig.11 (with, for simplicity, the same rate constants, although different t_ds). Parameter values are given in the supporting information, except k₀ = 0.01 s^, $\hat{f}$ ₁ = 5 × 10^M^⋅s^,b₁ = 0.33 s^{, and}t_d = 0.7T for the repressor gene. η_rep is calculated to be 0.17 by integrating Eq. 17 over one cell cycle. Because of the repressor, expression is reduced to about 10% of its constitutive level (the expression level of Table Table1).1). Values given are mean results from 100 simulations, and errors are ±1 SD.

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Acknowledgments

We are grateful for conversations with S. Bekiranov, A. J. Levine, J. Paulsson, N. Rajewsky, B. Shraiman, N. Socci, and M. Zapotocky. P.S.S., M.B.E., and E.D.S. acknowledge support from the National Institutes of Health (GM59018), the Seaver Institute and Burroughs-Wellcome Fund, and the National Science Foundation (DMR0129848), respectively.

Acknowledgments

Abbreviation

RNAP	RNA polymerase

Abbreviation

Footnotes

This paper was submitted directly (Track II) to the PNAS office.

Footnotes

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