Stochastic simulations!

Stochastic simulations! Application to biomolecular networks! Literature overview

Noise in genetic networks! Origins! How to measure the noise and distinguish between the two sources of noise (intrinsic vs extrinsic)?! What are the molecular processes that produce the most of noise?"! Consequences! How is the noise propagated in gene networks?! How does the noise affect the cellular behavior?!! Control! What are the cellular mechanisms that confer robustness to noise?!

Papers! Stochasticity gene expression in a single cell!!elowitz, Levine, Siggia, Swain (2002) Science 297: 1183-86! Regulation of noise in the expression of a single gene!!ozdudak, Thattai, Kurtser, Grossman, van Oudenaarden (2002) Nat Genet 31: 69-73! Control of stochasticity in eukaryotic gene expression!!raser, O'Shea (2004) Science 304: 1811-14! Noise in eukaryotic gene expression!!blake, Kaern, Cantor, Collins (2003) Nature 422: 633-637!! Gene regulation at the single-cell level!!rosenfeld, Young, Alon, Swain, Elowitz (2005) Science 307: 1962-1965! Stochastic kinetic analysis of a developmental pathway bifurcation in phage-λ E. coli cell!!arkin, Ross, McAdams (1998) Genetics 149: 1633-48! Multistability in the lactose utilization network of E. coli!!ozbudak, Thaittai, Lim, Shraiman, van Oudenaarden (2004) Nature 427: 737-740! Noise propagation in gene networks!!pedraza, van Oudenaarden (2005) Science 307: 1965-69!! Ultrasensitivity and noise propagation in a synthetic transcriptional cascade!!hooshangi, Thilberge, Weiss (2005) PNAS 102: 3581-3586! Engineering stability in gene networks by autoregulation!!becskei, Serrano (2000) Nature 405: 590-3! Design principles of a bacterial signalling network!!kollmann, Lodvok, Bartholomé, Timmer, Sourjik (2005) Nature 438: 504-507!

Stochastic gene expression in a single cell! Elowitz, Levine, Siggia, Swain (2002) Science 297: 1183-86! Abstract Clonal populations of cells exhibit substantial phenotypic variation. Such heterogeneity can be essential for many biological processes and is conjectured to arise from stochasticity, or noise, in gene expression. We constructed strains of Escherichia coli that enable detection of noise and discrimination between the two mechanisms by which it is generated. Both stochasticity inherent in the biochemical process of gene expression (intrinsic noise) and fluctuations in other cellular components (extrinsic noise) contribute substantially to overall variation. Transcription rate, regulatory dynamics, and genetic factors control the amplitude of noise. These results establish a quantitative foundation for modeling noise in genetic networks and reveal how low intracellular copy numbers of molecules can fundamentally limit the precision of gene regulation.

Stochastic gene expression in a single cell! Elowitz, Levine, Siggia, Swain (2002) Science 297: 1183-86! Goals measuring the noise at the level of single cells discriminate between the intrinsic and extrinsic noise identify genetic factors that control the magnitude of noise E. coli Definitions intrinsic noise: inherent stochasticity of biochemical processes extrinsic noise: fluctuation in other cellular components (number of ribosomes, polymerase, transcriptional regulators, etc): global to a single cell but varies from cell to cell.

Stochastic gene expression in a single cell! Elowitz, Levine, Siggia, Swain (2002) Science 297: 1183-86! Methodology: built strains of Escherichia coli, incorporating the distinguishable cyan (cfp) and yellow (yfp) alleles of green fluorescent protein in the chromosome. Double reporter construction! In each strain, the two reporter genes were controlled by identical promoters. To avoid systematic differences in copy number, the genes are inserted at loci equidistant from, and on opposite sides of the origin of replication. The two fluorescent proteins exhibited statistically equivalent intensity distributions and thus displayed the necessary independence and equivalence to detect noise. Experiment in E. coli! To assess the effect of the strength of the promotor on the level of noise, they use artificial lac repressible promoters: IPTG - laci repressor

Stochastic gene expression in a single cell! Elowitz, Levine, Siggia, Swain (2002) Science 297: 1183-86! Intrinsic vs extrinsic noise no intrinsic noise, only extrinsic noise (A) In absence of intrinsic noise, the 2 fluorescence proteins fluctuate in a correlated manner over time. Thus in each cell the level of the two proteins is equal and appears in yellow (red+green). both intrinsic and extrinsic noise (B) Due to intrinsic noise, the expressin of the 2 proteins become uncorrelated. In this case, at a given time point, some cells will express more one protein (red) and other the other protein (green). This results in a heterogeneous cell population.

Stochastic gene expression in a single cell!! Elowitz, Levine, Siggia, Swain (2002) Science 297: 1183-86 Results WT + laci (repressed) WT + laci + IPTG (derepressed) WT + ΔRecA + laci + IPTG no intrinsic noise, only extrinsic noise both intrinsic and extrinsic noise! Control of gene expression another WT + laci LacI deleted LacI under the control of the repressilator

Stochastic gene expression in a single cell! Elowitz, Levine, Siggia, Swain (2002) Science 297: 1183-86! Quantification of noise The amount of a given protein varies from cell to cell Noise strength defined as: η = variance (gene expression) mean (gene expression) = σ x 2 < x > Orthogonal contributions: η tot 2 = η int 2 + η ext 2 M22 = quite strain (wild type)" D22 = noisy strain (deletion of reca gene)"

Stochastic gene expression in a single cell! Elowitz, Levine, Siggia, Swain (2002) Science 297: 1183-86! M22 wild type! Quantification of noise D22 fully induced" (no LacI repressor)" ΔrecA! These results show that: the contribution of extrinsic noise is larger than the contribution of intrinsic noise and suggest that: the level of noise is controlled: - by expression level - by genetic background Expression level controled by IPTG

Stochastic gene expression in a single cell! Elowitz, Levine, Siggia, Swain (2002) Science 297: 1183-86! Experiments with various genetic backgrounds The level of noise is similar in most strains, but one, D22, which displays about 2x the amount of noise. This is due to the deletion of the RecA gene, involved in DNA repair.

Stochastic gene expression in a single cell! Elowitz, Levine, Siggia, Swain (2002) Science 297: 1183-86! Conclusions Using a two-reporter method, it is possible to measure and to distinguish extrinsic vs intrinsic noise. The relative contributions of extrinsic and intrinsic components to the total noise vary with expression level, but the contribution of extrinsic noise is usually much larger than the contribution of intrinsic noise. An increase of noise may arise from transient copy number differences between parts of the chromosomes (cf. increase of noise in ΔRecA mutant). The stochastic nature of gene expression gives rise to noise in protein levels, and hence creates heterogeneity in a clonal population. For the theory see: Swain, Elowitz, Siggia (2002) Intrinsic and extrinsic contributions to stochasticity in gene expression. PNAS 99: 12795-801"

Regulation of noise in the expression of a single gene! Ozbudak, Thattai, Kurtser, Grossman, van Oudenaarden (2002) Nat Genet 31: 69-73! Abstract Stochastic mechanisms are ubiquitous in biological systems. Biochemical reactions that involve small numbers of molecules are intrinsically noisy, being dominated by large concentration fluctuations. This intrinsic noise has been implicated in the random lysis/lysogeny decision of bacteriophage-lambda, in the loss of synchrony of circadian clocks and in the decrease of precision of cell signals. We sought to quantitatively investigate the extent to which the occurrence of molecular fluctuations within single cells (biochemical noise) could explain the variation of gene expression levels between cells in a genetically identical population (phenotypic noise). We have isolated the biochemical contribution to phenotypic noise from that of other noise sources by carrying out a series of differential measurements. We varied independently the rates of transcription and translation of a single fluorescent reporter gene in the chromosome of Bacillus subtilis, and we quantitatively measured the resulting changes in the phenotypic noise characteristics. We report that of these two parameters, increased translational efficiency is the predominant source of increased phenotypic noise. This effect is consistent with a stochastic model of gene expression in which proteins are produced in random and sharp bursts. Our results thus provide the first direct experimental evidence of the biochemical origin of phenotypic noise, demonstrating that the level of phenotypic variation in an isogenic population can be regulated by genetic parameters.

Regulation of noise in the expression of a single gene! Ozbudak, Thattai, Kurtser, Grossman, van Oudenaarden (2002) Nat Genet 31: 69-73! Goals quantify the relative contribution of transcription and translation to noise B. subtilis assess the effect of efficiency of transcription and translation to noise measure fluorescent reporter gene at single cell level in Bacillus subtilis develop a stochastic model of gene expression

Regulation of noise in the expression of a single gene! Ozbudak, Thattai, Kurtser, Grossman, van Oudenaarden (2002) Nat Genet 31: 69-73! Methodology Experiment in bacillus subtilis = expression level Translational and transcriptional mutants

Regulation of noise in the expression of a single gene! Ozbudak, Thattai, Kurtser, Grossman, van Oudenaarden (2002) Nat Genet 31: 69-73! translational mutants transcriptional mutants translational + transcriptional mutants

Regulation of noise in the expression of a single gene! Ozbudak, Thattai, Kurtser, Grossman, van Oudenaarden (2002) Nat Genet 31: 69-73! Theoretical model! mrna protein average number of proteins synthesized per mrna transcript low transcription rate high translation rate => high variability mean level of protein and standard deviation (noise) high transcription rate low translation rate => low variability

Regulation of noise in the expression of a single gene! Ozbudak, Thattai, Kurtser, Grossman, van Oudenaarden (2002) Nat Genet 31: 69-73! Conclusions By genetic mutations, it is possible to control transcription and translation efficiency and to study the relative contribution of these two processes to the noise. Both experiments and theory shows that noise strength is more sensitive to variation in translational efficiency than to the rate of transcription This led the authors to suggest that fast transcription followed by inefficient translation results in lower noise in protein levels. The control of noise comes at the energetic cost of producing few proteins from numerous mrna. high noise" low noise" Fig. from Raser & O'Shea (2005)" For additional theoretical results see: Thattai, van Oudenaarden (2002) Intrinsic noise in gene regulatory networks. PNAS 98: 8614-8610"

Control of stochasticity in eukaryotic gene expression! Raser, O'Shea (2004) Science 304: 1811-14! Abstract Noise, or random fluctuations, in gene expression may produce variability in cellular behavior. To measure the noise intrinsic to eukaryotic gene expression, we quantified the differences in expression of two alleles in a diploid cell. We found that such noise is gene-specific and not dependent on the regulatory pathway or absolute rate of expression. We propose a model in which the balance between promoter activation and transcription influences the variability in messenger RNA levels. To confirm the predictions of our model, we identified both cis- and trans-acting mutations that alter the noise of gene expression. These mutations suggest that noise is an evolvable trait that can be optimized to balance fidelity and diversity in eukaryotic gene expression.

Control of stochasticity in eukaryotic gene expression! Raser, O'Shea (2004) Science 304: 1811-14! Goals measure noise in eukaryots (yeast) at the single cell level. S. cerevisiae assess gene and regulatory specificity (experiments with different promoters) develop a model for gene expression

Control of stochasticity in eukaryotic gene expression! Raser, O'Shea (2004) Science 304: 1811-14! Double reporter construction! Experiment in S. cerevisiae! use a similar methodology as Elowitz et al (2002). construct diploid strains that express both cyan and yellow fluorescence proteins from identical promoters (Here is shown PHO5 which is controlled by the level of phosphate, Pi). The 2 reporter genes are inserted at the same locus on homologous chromosomes.

Control of stochasticity in eukaryotic gene expression! Raser, O'Shea (2004) Science 304: 1811-14! time after Pi starvation YFP (red) and CFP (green) fluorescence micrographs YFP and CFP values for each cell during a time course after induction of PHO5 by phosphate starvation. time Total, extrinsic, and intrinsic noise strength as a function of the population mean These results show that extrinsic noise is the dominant source of noise. (as reported for bacteria by Elowitz et al 2002)

Control of stochasticity in eukaryotic gene expression! Raser, O'Shea (2004) Science 304: 1811-14! ADH1! Experiments with different promoters:! total ( extrinsic) noise! The level of total noise ( extrinsic noise) does not depend on the promoter. PHO84! Origin of extrinsic noise: GAL1! cell size? cell shape? PHO84! => use flow cytometry to isolate subpopulations of cells that are homogeneous in size and shape => found that extrinsic noise is a bit decreased but still the dominant source of noise.

Control of stochasticity in eukaryotic gene expression! Raser, O'Shea (2004) Science 304: 1811-14! GAL1! Experiment with different promoters:! intrinsic noise! PHO84! GAL1 promoter! PHO84 promoter" PHO5 promoter! controled by Gal4p" controled by the same activator (Pho4p)" Observation! PHO5" GAL1 promoter" PHO84 promoter" PHO5 promoter" low intrinsic noise" large intrinsic noise at low epression rate" The level of intrinsic noise depends on the promoter and on the expression rate.

Control of stochasticity in eukaryotic gene expression! Raser, O'Shea (2004) Science 304: 1811-14! Theoretical model!! Comparison of various models of gene activation! Numerical simulations done with the Gillespie algorithm Slow promoter activation/deactivation, fast transcription Ex: slow chromatin remodeling Slow activation/ fast deactivation, fast transcription Ex: nucleosome sliding, DNA looping Fast promoter activation/deactivation, slow transcription Ex: rapid binding/unbinding of a transcriptional activator

Control of stochasticity in eukaryotic gene expression! Raser, O'Shea (2004) Science 304: 1811-14! Theoretical model!! Comparison of various models of gene activation! The noise strength profile of PHO5 is similar to the prediction made for case I (see panel D) when the promoter activation rate is changed (see green curve)." " Prediction:" The noise generation in PHO5 is dependent on the rate of a slow upstream promoter transition."! Predicted consequences! - a decrease of transcriptional efficiency (k m ) leads to a reduction of the noise." - a decrease of promoter activation (k a ) leads to an increase of the noise. "

Control of stochasticity in eukaryotic gene expression! Raser, O'Shea (2004) Science 304: 1811-14! complexes involved in chromatin remodeling transcription efficiency mutations in UAS mutations in snf6, arp8, gcn mutations in TATA box This confirms that a decrease of promoter activation (k a ) leads to an increase of the noise." This confirms that a decrease transcriptional efficiency (k m ) leads to a reduction of the noise."

Control of stochasticity in eukaryotic gene expression! Raser, O'Shea (2004) Science 304: 1811-14! Conclusions The two-reporter technique used by Elowitz et al (2002) to distinguish intrinsic and extrinsic noise in bacteria can be applied to eukaryots (here: yeast). Extrinsic noise is predominant over intrinsic noise. Total noise ( extrinsic) is not gene-specific, but intrinsic noise is gene-specific. high noise" low noise" Intrinsic noise does not depend on the regulatory pathway, neither on absolute rate of expression. Fig. from Raser & O'Shea (2005)" Intrinsic noise depends on the rate of a slow upstream promoter transition, such as chromatin remodeling

Noise in eukaryotic gene expression! Blake, Kaern, Cantor, Collins (2003) Nature 422: 633-637! Abstract Transcription in eukaryotic cells has been described as quantal, with pulses of messenger RNA produced in a probabilistic manner. This description reflects the inherently stochastic nature of gene expression, known to be a major factor in the heterogeneous response of individual cells within a clonal population to an inducing stimulus. Here we show in Saccharomyces cerevisiae that stochasticity (noise) arising from transcription contributes significantly to the level of heterogeneity within a eukaryotic clonal population, in contrast to observations in prokaryotes, and that such noise can be modulated at the translational level. We use a stochastic model of transcription initiation specific to eukaryotes to show that pulsatile mrna production, through reinitiation, is crucial for the dependence of noise on transcriptional efficiency, highlighting a key difference between eukaryotic and prokaryotic sources of noise. Furthermore, we explore the propagation of noise in a gene cascade network and demonstrate experimentally that increased noise in the transcription of a regulatory protein leads to increased cell-cell variability in the target gene output, resulting in prolonged bistable expression states. This result has implications for the role of noise in phenotypic variation and cellular differentiation.

Noise in eukaryotic gene expression! Blake, Kaern, Cantor, Collins (2003) Nature 422: 633-637! Goals assess the contribution of transcriptional noise to the heterogeneity of a clonal population examine the pulsatile nature of mrna production and the transcriptional efficiency explore the propagation of noise in a gene cascade network develop a model for gene expression

Noise in eukaryotic gene expression! Blake, Kaern, Cantor, Collins (2003) Nature 422: 633-637! Experiment in S. cerevisiae! Genetic construction with 2 transcriptional controls: fluorescence GAL (direct activator) and ATc (inhibitor of the TetR inhibitor) Transient response Time induction by 0.2% gal Inductions by GAL or ATc induce differential responses. induction by 40 ng /ml ATc The mode of transcriptional control has a significant influence on the response to the inducer and on the level of noise.

Noise in eukaryotic gene expression! Blake, Kaern, Cantor, Collins (2003) Nature 422: 633-637! Experiment: effect of transcriptional and translational efficiency on the noise! 28% transcription full transcription Controled by ATc or GAL Controled by mutation in codons (by keeping the same aa sequence) The level of noise in eukaryotic gene expression is influenced by transcription and by the mode of transcriptional regulation

Noise in eukaryotic gene expression! Blake, Kaern, Cantor, Collins (2003) Nature 422: 633-637! Stochastic model for eukaryotic transcription! reinitiation PC1 = inactive (or silenced) promoter PC2 = intermediate complex (TBP + some TF bound) PC3 = preinitiation complex (TBP + all TF bound) RC1 & RC2 = different repressed promoter configurations Numerical simulations done with the Gillespie algorithm proba to produce a transcript is constant ATc (k 3f is ATc-dependent) GAL (k 1f and k 1b are GAL-dependent) Simulations of the model demonstrate that pulsatile mrna production (resulting from transcriptional reinitiation) is required to reproduce the experimental observations.

Noise in eukaryotic gene expression! Blake, Kaern, Cantor, Collins (2003) Nature 422: 633-637! Experiment: cascading noise in a gene network! fluorescence level of noise at P GAL10 high level of noise low level of noise level of noise at P ADH1 induction by high level of atc low level of noise high level of noise induction by low level of atc Downstream effects of noise can have profound phenotypic consequences, drastically affecting the stability of gene expression

Noise in eukaryotic gene expression! Blake, Kaern, Cantor, Collins (2003) Nature 422: 633-637! Conclusions In eukaryots, the mode of transcriptional control can have a marked effect on the response to the noise. In eukaryots, noise arising from transcription contributes more than noise generated at the translational level (in contrast to observations in prokaryots - cf. Ozdudak et al, 2002). Downstream effects of noise can have profound phenotypic consequences, drastically affecting the stability of gene expression.

Engineering stability in gene networks by autoregulation! Becskei, Serrano (2000) Nature 405: 590-3! Abstract The genetic and biochemical networks which underlie such things as homeostasis in metabolism and the developmental programs of living cells, must withstand considerable variations and random perturbations of biochemical parameters. These occur as transient changes in, for example, transcription, translation, and RNA and protein degradation. The intensity and duration of these perturbations differ between cells in a population. The unique state of cells, and thus the diversity in a population, is owing to the different environmental stimuli the individual cells experience and the inherent stochastic nature of biochemical processes (for example, refs 5 and 6). It has been proposed, but not demonstrated, that autoregulatory, negative feedback loops in gene circuits provide stability, thereby limiting the range over which the concentrations of network components fluctuate. Here we have designed and constructed simple gene circuits consisting of a regulator and transcriptional repressor modules in Escherichia coli and we show the gain of stability produced by negative feedback.

Engineering stability in gene networks by autoregulation! Becskei, Serrano (2000) Nature 405: 590-3! Role of auto-regulations in genetic networks: Model and simulation Evolution equation unregulated free promoter RNA pol k b G + P D G* k a "active" gene k p =k a /k b auto-regulated k b G + P D G* k a k p =k a /k b In these equations, k I is the promoter isomerization rate (i.e. fraction of promoter that can be transcribed?), a is the proportionality constant between the mrna and protein concentration, n is the gene copy number, and k deg is the degradation rate These equations are obtained by applying the quasi-steady state assumption on the binding/unbinding steps of RNA polymerase (P) and of the repressor (R) to the gene promoter (G) and by assuming that G+G*+G R =1. G + R D G R free promoter inhibitor k u k b "inactive" gene k r =k b /k u

Engineering stability in gene networks by autoregulation! Becskei, Serrano (2000) Nature 405: 590-3! Role of auto-regulations in genetic networks: Model and simulation Evolution equation Stability analysis Stochastic simulation are carried out by applied random perturbation at constant time intervals. A stochastic simulation using for ex. the Gillespie algorithm would have been more appropriate...

Engineering stability in gene networks by autoregulation! Becskei, Serrano (2000) Nature 405: 590-3! Role of auto-regulations in genetic networks: Experiment in E. coli Auto-regulatory system P TetR - TetR - GFP construction Here, the TetR repressor is mutated (Y42A) in the DNA binding domain. Here, the TetR binding sites have been replaced by the LacI binding sites. Noise can be attenuated by negative auto-regulation

Engineering stability in gene networks by autoregulation! Becskei, Serrano (2000) Nature 405: 590-3! Role of auto-regulations in genetic networks: Experiment in E. coli Time course after induction (by different concentrations of atc) High variability is observed in both short and long term after induction

Engineering stability in gene networks by autoregulation! Becskei, Serrano (2000) Nature 405: 590-3! Conclusions Negative feedback loops in gene circuits (and autoregulation in particular) provide stability (robustness to noise). The experimental results obained here are consistent with the prediction of a simple model of gene expression. (It would however be interesting to run "real" high noise" low noise" stochastic simulations of the same model...) Fig. from Raser & O'Shea (2005)"

Ultrasensitivity and noise propagation in a synthetic transcriptional cascade! Hooshangi, Thilberge, Weiss (2005) PNAS 102: 3581-3586! Abstract The precise nature of information flow through a biological network, which is governed by factors such as response sensitivities and noise propagation, greatly affects the operation of biological systems. Quantitative analysis of these properties is often difficult in naturally occurring systems but can be greatly facilitated by studying simple synthetic networks. Here, we report the construction of synthetic transcriptional cascades comprising one, two, and three repression stages. These model systems enable us to analyze sensitivity and noise propagation as a function of network complexity. We demonstrate experimentally steady-state switching behavior that becomes sharper with longer cascades. The regulatory mechanisms that confer this ultrasensitive response both attenuate and amplify phenotypical variations depending on the system's input conditions. Although noise attenuation allows the cascade to act as a low-pass filter by rejecting short-lived perturbations in input conditions, noise amplification results in loss of synchrony among a cell population. The experimental results demonstrating the above network properties correlate well with simulations of a simple mathematical model of the system.

Ultrasensitivity and noise propagation in a synthetic transcriptional cascade! Hooshangi, Thilberge, Weiss (2005) PNAS 102: 3581-3586! Experiment in E. coli: construction of 3 synthetic transcriptional cascades: atc = inducer = control parameter, prevents the repression by tetr

Ultrasensitivity and noise propagation in a synthetic transcriptional cascade! Hooshangi, Thilberge, Weiss (2005) PNAS 102: 3581-3586! (circuit 1) (circuit 2) (circuit 3) Experiment Experiment Model (3) (1) (2) (3) (2) (1) The noise is more marked during the transition, especially in circuit (3). Longer cascade amplify cell-to-cell variability in the intermediate regions.

Ultrasensitivity and noise propagation in a synthetic transcriptional cascade! Hooshangi, Thilberge, Weiss (2005) PNAS 102: 3581-3586! What is the role of long cascades? Delay in the response Low-pass filter: response to long-pulse pertubation (1) atc added (3) atc (2) short-pulse perturbation 5 min (2) (1) (3) atc removed (1) (2) long-pulse perturbation (1) (2) atc 45 min (3) (3)

Ultrasensitivity and noise propagation in a synthetic transcriptional cascade! Hooshangi, Thilberge, Weiss (2005) PNAS 102: 3581-3586! Conclusions Noise (and consequently cell-to-cell variability) is amplified at transition in long cascades. Synchronization of cell responses is diminished for longer cascades. Long cascade can induce delay in the response Long cascade act as low pass-filter => Trade-off between robustness to noise and function The results are consistent with the prediction of a simple mathematical model

...much work must be done to understand how cellular processes behave robustly in the presence of underlying stochasticity. Such work often requires a non-traditional collaboration between mathematicians, physicists, and in vivo experimentalists

References! Raj A, Van Oudenaarden A (2008) Nature, nurture, or chance: stochastic gene expression and its consequences, Cell 135:216 Äì26. Sanchez A, Golding I (2013) Genetic determinants and cellular constraints in noisy gene expression. Science. 342:1188-93. Eldar A, Elowitz MB (2010) Functional roles for noise in genetic circuits. Nature. 467:167-73 Raser JM, O'Shea EK (2005) Noise in gene expression: origins, consequences, and control. Science 309:2010-3. Kaern M, Elston TC, Blake WJ, Collins JJ (2005) Stochasticity in gene expression: from theories to phenotypes. Nat Rev Genet. 6:451-64. Munsky B, Neuert G, van Oudenaarden A (2012) Using gene expression noise to understand gene regulation. Science. 336:183-7.