Pathway dynamics can delineate the sources of transcriptional noise in gene expression

TL;DR: This work demonstrates mathematically that, in some cases, the same noise decomposition can be achieved at the transcriptional level with non-identical and not-necessarily independent reporters, and uses the result to show that generic reporters lying in the same biochemical pathways can replace dual reporters, enabling the Noise decomposition to be obtained from only a single gene.

Abstract: Single-cell expression profiling is destructive, giving rise to only static snapshots of cellular states. This loss of temporal information presents significant challenges in inferring dynamics from population data. Here we provide a formal analysis of the extent to which dynamic variability from within individual systems ("intrinsic noise") is distinguishable from variability across the population ("extrinsic noise"). Our results mathematically formalise observations that it is impossible to identify these sources of variability from the transcript abundance distribution alone. Notably, we find that systems subject to population variation invariably inflate the apparent degree of burstiness of the underlying process. Such identifiability problems can be remedied by the dual-reporter method, which separates the total gene expression noise into intrinsic and extrinsic contributions. This noise decomposition, however, requires strictly independent and identical gene reporters integrated into the same cell, which can be difficult to implement experimentally in many systems. Here we demonstrate mathematically that, in some cases, the same noise decomposition can be achieved at the transcriptional level with non-identical and not-necessarily independent reporters. We use our result to show that generic reporters lying in the same biochemical pathways (e.g. mRNA and protein) can replace dual reporters, enabling the noise decomposition to be obtained from only a single gene. Stochastic simulations are used to support our theory, and show that our "pathway-reporter" method compares favourably to the dual-reporter method.

Summary

Introduction

• Sometimes it can afford evolutionary advantages, for example, in the context of bet-hedging strategies.
• Such sources of variability contribute extrinsic noise, and reflect the variation in gene expression and transcription activity across the cell population.
• Here the authors develop a widely applicable generalisation (and simplification) of the original dual-reporter approach [4].
• The Telegraph Model The Telegraph model was first introduced in [21], and since then has been widely employed in the literature to model bursty gene expression in eukaryotic cells [22–25].
• Throughout, the authors will refer to the probability mass function p̃T (n; θ) as the Telegraph distribution with parameters θ.

Identifiability Considerations

• Decoupling the effects of extrinsic noise from experimental measurements has been notoriously challenging.
• In Fig. 2A (middle panel), the authors compare the representation obtained in (4) with the corresponding fixed-parameter negative binomial distribution for two different sets of parameters.
• Thus, the distribution of any instantaneously bursty system with mean burst intensity b can be obtained from one with greater burst frequency, by varying the mean burst intensity θ according to a shifted beta prime distribution.
• Noise on the transcription rate will invariably produce copy number data that is suggestive of a more bursty model.
• The truncated normal distribution is not chosen on the basis of biological relevance, but rather to demonstrate that even a symmetric noise distribution (except for truncation at 0) produces qualitatively similar results to the distributions used in the precise non-identifiability results.

Resolving Non-identifiability

• The results of the previous section show that additional information, beyond the observed copy number distribution, is required to constrain the space of possible dynamics that could give rise to the same distribution.
• The decomposition applies to dynamic noise [39], and generalises to higher moments in [40].
• The dual-reporter method requires distinguishable measurements of transcripts or proteins from two independent and identically distributed reporter genes integrated into the same cell.
• As the authors show in the next section, there are many situations where the random variable E(X;Z) is precisely the common part of E(Y ;Z) and E(X;Z) (i.e., h(Z′) = E(X;Z)), and the normalised intrinsic contribution to the covariance is either zero or negligible.
• In these cases, the normalised covariance of X and Y will identify precisely the extrinsic noise contribution η2ext to the total noise η2X .

The Pathway-Reporter Method

• The authors show that for some reporters X and Y belonging to the same biochemical pathway, the covariance of X and Y continues to identify the extrinsic, and subsequently intrinsic, noise contributions to the total noise.
• Thus, again the NDP holds, and the normalised covariance of E(XN ;Z) and E(XP ;Z) will identify the noise on the transcriptional component KN λ(λ+µ) .
• The time series of copy numbers for each of nascent mRNA, mature mRNA and protein broadly follow each other, each with delay from its predecessor (Fig. 4B).
• The parameter KN is given the noise distribution Beta(3, 6), which has a slightly higher coefficient of variation η2 = 0.2.
• The results for the nascent mRNA–protein reporters, case (c), given in Table 4 show comparable performance to dual reporters, with only modest overshoot; even in the worst performing case of λ = 0.5, µ = 1 the result of the pathway reporters is within one standard deviation, in a very tight distribution.

Discussion

• The ability to extract transcriptional dynamics from measured distributions of mRNA copy numbers is limited.
• It is therefore necessary to collect further information, beyond measurements of the transcripts alone, in order to constrain the number of possible theoretical models of gene activity that could represent the system.
• The authors have developed a theoretical framework for estimating levels of extrinsic noise, which can assist in resolving the non-identifiability problems.
• The dual reporter method of Swain et al. [4] already provides one such approach; but it is experimentally challenging to set up in many systems, and requires strictly identical and independent pairs of gene reporters.
• The authors have exploited this to yield reliable estimates of noise strength, which they are confident will assist in setting better practices for model fitting and inference in the analysis of single-cell data.

Acknowledgments

• The authors gratefully acknowledge Rowan D. Brackston helpful discussions in the early stages of this research.
• The authors also wish to thank Arjun Raj for providing valuable feedback on this work.
• L.H. and M.P.H.S. were supported by the University of Melbourne DVCR fund.

Data Availability

• Simulations of the models used in the paper are performed using Gillespie’s Stochastic Simulation Algorithm (SSA) implemented in Julia.
• The simulation code is available in the GitHub repository https://github.com/leham/PathwayReporters.
• The data used in the paper are provided in the supplementary datasets.

Author Contributions

• L.H. and M.J. conceptualised the research, with support from M.P.H.S.
• All authors provided critical feedback and helped shape the research.

Figures

Figure 2: (A) Accuracy of our representations for the Telegraph and negative binomial distribution. For each of the results in (3)-(5), we compare the (fixed-parameter) Telegraph and negative binomial distributions with their respective compound representations for two different sets of parameter values. The top panel (shown in pink) is shows comparisons for (3). Referring to Table 1, the parameter values for the top panel are (left) λ = 2, µ′ = 12, K ′ = 100, µ = 3, and K ∼ BetaK′(5, 9), and (right) λ = 1, µ′ = 20, K ′ = 100, µ = 2 and K ∼ BetaK′(3, 18). The middle panel (green) gives comparisons for (4), with parameter values (left) λ = 10, β = 2, µ = 2 and K ∼ Gamma(12, 2) and (right) λ = 1, β = 1, µ = 2 and K ∼ Gamma(3, 1). The bottom panel (coral) gives comparisons for (5). The parameter values (left) are λ′ = 10, λ = 15 and c = 2 and (right) are λ′ = 2, λ = 5 and c = 3. (B) The top figure compares a Telegraph(2, 4, 60) distribution with samples from a compound Telegraph distribution with normal noise Norm(37, 10) on the transcription rate parameter. The middle figure compares a NegBin(5, 0.5) with samples from a compound Telegraph distribution with normal noise Norm(5.5, 2.3) on the transcription rate parameter. The bottom figure compares a NegBin(5, 1) distribution with samples from a compound negative binomial distribution with normal noise Norm(2.3, 0.6) on the burst intensity parameter.

Table 2: A comparison of the pathway-reporter method and the dual-reporter method for constitutive expression under the model M1. Here PR gives the results of the nascent and mature pathway reporters, while DR (Mat) gives the results of dual reporters calculated from the mature mRNA. We considered noise on both the transcription rate (KN) and the maturation rate (KM). The decay rate is fixed at one, with the other parameters scaled accordingly. In each case, the maturation rateKM is varied according to aGamma(8, 1.25) distribution, which has coefficient of variation 0.125. The values given are the average of 100 simulations, each calculated from 500 copy number samples, and the errors are ± one standard deviation. Our theory predicts that pathway-reporters will identify the noise on the nascent transcription rate KN (η2ext). The noise distribution parameters are chosen to produce an average nascent mRNA copy number of approximately 5 and an average mature mRNA copy number of approximately 50.

Table 4: A comparison of the pathway-reporter method and dual-reporter method for bursty expression. Here PR (NP) gives the results of the nascent and protein pathway reporters, PR (MP) gives the results of the mRNA and protein reporters, while DR (Mat) gives the results of the dual reporters calculated from the mature mRNA. We consider noise on all of the parameters except for δM and KM ; see discussion in main text. The values given are the average of 100 simulations, each calculated from 500 copy number samples, and the errors are ± one standard deviation. Our theory predicts that pathwayreporters will identify the noise at both the promoter level (λ, µ) and transcriptional level (KN , δm); the total extrinsic noise in each case is given by η2ext. As before, the noise distribution parameters are chosen to produce an average nascent mRNA copy number of 5 and an average mature mRNA copy number of 50, and an average number of 1000 proteins.

Figure 4: (A) Stochastic models of gene expression. The model M1 is the simplest model of mRNA maturation. Here nascent (unspliced) mRNA are shown in red/blue wavy lines; the blue segments represent introns and the red segments represent the exons. Nascent mRNA are synthesized at the rate KN , and spliced into mature mRNA (blue wavy lines) at rate KM . Degradation of the mature mRNA occurs at rate δM . The model M2 is the well-known two-stage model of gene expression. The model M3 is the extension of the two-stage model to include promoter switching. The nodes A (active) and I (inactive) represent the state of the gene, with transitions between states occurring at rates λ and µ. The remaining parameters are the same as those in the model M2. The model M4 extends the model M3 by incorporating mRNA maturation. Here KN is the transcription rate parameter, and KM is the maturation rate. All other parameters are the same as in M3. (B) Time series simulation of the copy number and activity state of a gene modelled by M4. For ease of visualisation, the parameters were artificially chosen as λ = 2, µ = 2.5, KN = 40, KM = 4, Kp = 4 and δp = 1, with all parameters scaled relative to δm = 1. (C) As λ approaches 0, we see a higher correlation in the copy numbers of nascent mRNA, mature mRNA and protein. Again, the parameters are artificially chosen to be λ = 0.1, µ = 2.5, KN = 80, KM = 4, Kp = 4 and δp = 1, with all parameters scaled relative to δm = 1.

Figure 3: A comparison of joint distributions in the case of moderate extrinsic noise and no extrinsic noise. The plots are generated from a three-stage model of gene transcription, incorporating the production of nascent mRNA, mature mRNA and protein. Details of the model can be found in Fig. 4 (model M4) and the associated text. The top panel shows nascent-mature, nascent-protein and mature-protein joint distributions in the case of extrinsic noise, while the bottom panel displays the corresponding plots in the case of no extrinsic noise. Extrinsic noise produces a visibly more correlated joint distribution, which forms the basis of the pathway-reporter method.

Table 1: Summary of the non-identifiability results. The results given in lines 1, 3 and 5 are our contributions, while the remaining representations (lines 2 and 4) are known and can be obtained as special cases of our results. Note that here we use Tele(λ, µ,K) to denote a Telegraph distribution with parameters λ, µ,K. In lines 3 and 4, the parameter β > 0 can be chosen freely and determines the mean burst intensity in the resulting compound system. In line 5 the parameters b, θ > 0 are again mean burst intensities, and b can be chosen freely in the determination of the distribution of θ.

Figure 1: Modelling the effects of both intrinsic and extrinsic noise. (A) A schematic of the Telegraph process, with nodes A (active) and I (inactive) representing the state of the gene. Transitions between the states A and I occur stochastically at rates µ and λ, respectively. The parameter K is the mRNA transcription rate, and δ is the degradation rate. (B) The compound model incorporates extrinsic noise by assuming that parameters θ of the Telegraph model vary across an ensemble of cells, according to some probability distribution f(θ; η). (C) Variation in the parameters across the cell population leads to greater variability in the mRNA copy number distribution.

Figure 5: Heatmaps for the intrinsic contribution to the covariance, which estimates the level of overshoot in the pathway-reporter approach for the nascent-protein and matureprotein reporters; blue regions show an overshoot of less than ≃ 0.05. Here the intrinsic contribution is calculated using stochastic simulations of the model M4. In the left panels, the parameter µ = 2 is fixed, while δp and the on-rate λ are varied between 0.01 and 1 and 0.5 and 4, respectively. In the right panels, the parameter µ = 20 is fixed, while δp and the on-rate λ vary between 0.01 and 1, and 1 and 10, respectively. In all cases, the parameters are scaled so that δM = 1. The maturation rate is fixed at 20, with the parameters KN and KP are chosen to produce a mean protein level of 1000, a mean nascent mRNA level of 5 and a mean mature mRNA level of 50.

Table 3: A comparison of the pathway-reporter method and the dual-reporter method for constitutive expression under the model M2. Here PR gives the results of the mRNAprotien pathway reporters, while DR (Mat) gives the results of dual reporters calculated from the mature mRNA. We considered noise on the transcription rate (Km), the protein synthesis rate (Kp), and the protein decay rate (δp). The mRNA decay rate is fixed at one. In each case, we varied Kp according to a Gamma(5, 0.4) distribution and δp according to a Gamma(8, 0.125) distribution; the corresponding noise strengths are 0.20 and 0.125, respectively. We considered different noise distributions on Km, which produce a range of noise strengths. The noise distribution parameters are selected to produce a mean mRNA of approximately 50 and a mean number of approximately 1000 proteins in each simulation. The values given are the average of 100 simulations, each calculated from 500 copy number samples, and the errors are ± one standard deviation. As our theory predicts, the mRNA-protein reporters identify the noise on the transcription rate parameter Km (η2ext).

Full text

Pathway dynamics can delineate the sources of
transcriptional noise in gene expression
Lucy Ham,
1∗
Marcel Jackson,
2
Michael P.H. Stumpf
1,3∗
1
School of BioSciences, University of Melbourne, Australia
2
Department of Mathematics and Statistics, La Trobe University, Australia
3
School of Mathematics and Statistics, University of Melbourne, Australia
∗
E-mail: lucy.ham@unimelb.edu.au, mstumpf@unimelb.edu.au
Single-cell expression proling has opened up new vistas on cellular processes.
Among other important results, one stand-out observation has been the con-
rmation of extensive cell-to-cell variability at the transcriptomic and pro-
teomic level. Because most experimental analyses are destructive we only
have access to snapshot data of cellular states. This loss of temporal infor-
mation presents signicant challenges in inferring dynamics, as well as causes
of cell-to-cell variability. In particular, we are typically unable to separate
dynamic variability from within individual systems (“intrinsic noise”) from
variability across the population (“extrinsic noise”). Here we mathematically
formalise this non-identiability; but we also use this to identify how new
experimental set-ups coupled to statistical noise decomposition can resolve
this non-identiability. For single-cell transcriptomic data we nd that sys-
tems subject to population variation invariably inate the apparent degree
of burstiness of the underlying process. Such identiability problems can,
in principle, be remedied by dual-reporter assays, which separates total gene
expression noise into intrinsic and extrinsic contributions; unfortunately, how-
ever, this requires pairs of strictly independent and identical gene reporters
to be integrated into the same cell, which is dicult to implement experimen-
tally in most systems. Here we demonstrate mathematically that, in some
cases decomposition of transcriptional noise is possible with non-identical and
not-necessarily independent reporters. We use our result to show that generic
reporters lying in the same biochemical pathways (e.g. mRNA and protein)
can replace dual reporters, enabling the noise decomposition to be obtained
1
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from only a single gene. Stochastic simulations are used to support our the-
ory, and show that our “pathway-reporter” method compares favourably to
the dual-reporter method.
Introduction
Noise is a fundamental aspect of every cellular process [1]. Frequently it is even of func-
tional importance, for example in driving cell-fate transitions. Sometimes it can aord
evolutionary advantages, for example, in the context of bet-hedging strategies. Some-
times, it can be a nuisance, for example, when it makes cellular signal processing more
dicult. But noise is nearly ubiquitous at the molecular scale, and its presence has pro-
foundly shaped cellular life. Analysing and understanding the sources of noise, how it
is propagated, amplied or attenuated, and how it can be controlled, has therefore be-
come a cornerstone of modern molecular cell biology. Noise arising in gene expression has
arguably attracted most of the attention so far (but see e.g. [2] and [3] for the analysis
of noise at the signalling level). Generally speaking, gene expression noise is separable
into two sources of variability, as pioneered by Swain et al. [4]. Intrinsic noise is gen-
erated by the dynamics of the gene expression process itself. The process, however, is
often inuenced by other external factors, such as the availability of promoters and of
RNA polymerase, the inuence of long noncoding RNA as a transcriptional regulator [5],
as well as dierences in the cellular environment. Such sources of variability contribute
extrinsic noise, and reect the variation in gene expression and transcription activity
across the cell population. As such, understanding extrinsic noise lies at the heart of
understanding cell-population heterogeneity.
So far, elucidating the sources of gene expression noise from transcriptomic measure-
ments alone has proven dicult [6, 7]. The fundamental hindrance lies in the fact that
single-cell RNA sequencing, which provides most of the available data, is destructive, so
that datasets reect samples from across a population, rather than samples taken repeat-
edly from the same cell. As temporal information is lost in such measurements [8], it may
be impossible to distinguish temporal variability within individual cells (e.g. burstiness),
from ensemble variability across the population (i.e. extrinsic noise). A number of nu-
merical and experimental studies have suggested this confounding eect [9–11], showing
that systems with intrinsic noise alone exhibit behaviour that is indistinguishable from
systems with both extrinsic and intrinsic noise. This is examined more formally in [12],
where we show that the moment scaling behaviour and transcript distribution may be
indistinguishable from situations with purely intrinsic noise. The limitations in infer-
ring dynamics from population data are becoming increasingly evident, and a number of
studies that seek to address some of these problems have emerged [13, 14].
Here we provide a detailed analysis of the extent to which sources of variability are
identiable from population data. We are able to prove rigorously that it is in general
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impossible to identify the sources of variability, and consequently, the underlying tran-
scription dynamics, from the observed transcript abundance distribution alone. Systems
with intrinsic noise alone can always present identically to similar systems with extrinsic
noise. Moreover, we demonstrate that extrinsic noise invariably distorts the apparent de-
gree of burstiness of the underlying system: data which seems “bursty” is not necessarily
generated by a bursty process, if there is cell-to-cell variability in e.g. the transcriptional
machinery across cells. This has important ramications for our analysis of experimental
data: we cannot assess causes of transcriptional variability, if we do not simultaneously
assess cell-to-cell variability in the transcriptional machinery. Our results highlight (in
fact prove mathematically) the requirement for additional information, beyond the ob-
served copy number distribution, in order to constrain the space of possible dynamics
that could give rise to the same distribution.
This seemingly intractable problem can at least partially be resolved with a brilliantly
simple approach: the dual-reporter method [4]. In this approach, noise can be separated
into extrinsic and intrinsic components, by observing correlations between the expression
of two independent, but identically distributed uorescent reporter genes. Dual-reporter
assays have been employed experimentally to study the noise contributed by both global
and gene-specic eects [15–17]. A particular challenge, however, is that dual reporters
are rarely identically regulated [17, 18], and are not straightforward to set up in ev-
ery system. More recently, it has been shown that the independence of dual reporters
cannot be guaranteed in some systems [
19]. As a result, there have been eorts in de-
veloping alternative methods for decomposing noise [18, 20]. Here we develop a widely
applicable generalisation (and simplication) of the original dual-reporter approach [4].
We demonstrate that non-identical and not-necessarily independent reporters can provide
an analogous noise decomposition. Our result shows that measurements taken from the
same biochemical pathway (e.g. mRNA and protein) can replace dual reporters, enabling
the noise decomposition to be obtained from a single gene. This completely circumvents
the requirement of strictly independent and identically regulated reporter genes. The
results obtained from our “pathway-reporter” method are also borne out by stochastic
simulations, and compare favourably with the dual-reporter method. In the case of con-
stitutive expression, the results obtained from our decomposition are identical to those
obtained from dual reporters. For bursty systems, we show that our approach provides a
satisfactorily close approximation, except in extreme cases.
Modelling Intrinsic and Extrinsic Noise
A simple model for stochastic mRNA dynamics is the Telegraph model: a two-state
model describing promoter switching, transcription, and mRNA degradation. In this
model, all parameters are xed, and gene expression variability arises due to the inherent
stochasticity of the transcription process. As discussed above, this process will often
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Figure 1: Modelling the eects of both intrinsic and extrinsic noise. (A) A schematic of
the Telegraph process, with nodes A (active) and I (inactive) representing the state of
the gene. Transitions between the states A and I occur stochastically at rates µ and λ,
respectively. The parameter K is the mRNA transcription rate, and δ is the degradation
rate. (B) The compound model incorporates extrinsic noise by assuming that parameters
θ of the Telegraph model vary across an ensemble of cells, according to some probability
distribution f(θ; η). (C) Variation in the parameters across the cell population leads to
greater variability in the mRNA copy number distribution.
be inuenced by extrinsic sources of variability, and so modications to account for this
additional source of variability are required.
The Telegraph Model
The Telegraph model was rst introduced in [21], and since then has been widely employed
in the literature to model bursty gene expression in eukaryotic cells [22–25]. In this model,
the gene switches probabilistically between an active state and an inactive state, at rates
λ (on-rate) and µ (o-rate), respectively. In the active state, mRNAs are synthesized
according to a Poisson process with rate K, while in the inactive state, transcription does
either not occur, or possibly occurs at some lower Poisson rate, K
0
≪ K. Degradation of
RNA molecules occurs independently of the gene promoter state at rate δ. Fig. 1A shows
a schematic of the Telegraph model. Throughout the discussion here, we will rescale all
parameters of the Telegraph model by the mRNA degradation rate, so that δ = 1. The
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steady-state distribution for the mRNA copy number can be explicitly calculated as [26],
˜p
T
(n; θ) =
K
n
λ
(n)
n!(µ + λ)
(n)
1
F
1
(λ + n, λ + µ + n, −K). (1)
Here θ denotes the parameter vector (µ, λ, K, δ), the function
1
F
1
is the conuent hyper-
geometric function [27], and, for real number x and positive integer n, the notation x
(n)
abbreviates the rising factorial of x (also known as the Pochhammer function). Through-
out, we will refer to the probability mass function ˜p
T
(n; θ) as the Telegraph distribution
with parameters θ. Constitutive gene expression is just a limiting case of the Telegraph
model, which arises when the o-rate µ is 0, so that the gene remains permanently in the
active state. In this case, the Telegraph distribution simplies to a Poisson distribution
with rate K; Pois(K).
At the opposite extreme is instantaneously bursty gene expression in which mRNA are
produced in very short bursts with potentially prolonged periods of inactivity in between.
This mode of gene expression has been frequently reported experimentally, particularly
in mammalian genes [17,22,24,25]. Transcriptional bursting may be treated as a limit of
the Telegraph model, where the o-rate, µ, has tended toward innity, while the on-rate
λ remains xed. In this limit, it can be shown [9, 12] that the Telegraph distribution
converges to the negative binomial distribution NegBin(λ,
K
µ+K
).
The Compound Distribution
We can elegantly account for random cell-to-cell variation across a population by way of
a compound distribution [28]
˜q(n; η) =
Z
˜p(n; θ)f (θ; η) dθ, (2)
where ˜p(n; θ) is the stationary probability distribution of a system with xed parameters θ
and f(θ; η) denotes the multivariate distribution for θ with hyperparameters η. Often we
will take ˜p(n; θ ) to be the stationary probability distribution of the Telegraph model ((1)),
and refer to (2) as the compound Telegraph distribution. Sometimes ˜p(n; θ) will be the
Poisson distribution or the negative binomial distribution, depending on the underlying
mode of gene activity.
The compound distribution is valid in the case of ensemble heterogeneity, that is, when
parameter values dier between individual cells according to the distribution f(θ; η), but
remain constant over time [2]. This model is also a valid approximation for individ-
ual cells with dynamic parameters, provided these change suciently slower than the
transcriptional dynamics [29]. Fig. 1B gives a pictorial representation of the compound
distribution.
In general, the compound Telegraph distribution ˜q(n; η) will be more dispersed than
a Telegraph distribution to account for the uncertainty in the parameters; see Fig. 1C.
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Frequently asked questions

Q1. What contributions have the authors mentioned in the paper "Pathway dynamics can delineate the sources of transcriptional noise in gene expression" ?

Such identifiability problems can, in principle, be remedied by dual-reporter assays, which separates total gene expression noise into intrinsic and extrinsic contributions ; unfortunately, however, this requires pairs of strictly independent and identical gene reporters to be integrated into the same cell, which is difficult to implement experimentally in most systems. Here the authors demonstrate mathematically that, in some cases decomposition of transcriptional noise is possible with non-identical and not-necessarily independent reporters. The authors use their result to show that generic reporters lying in the same biochemical pathways ( e. g. mRNA and protein ) can replace dual reporters, enabling the noise decomposition to be obtained