Quantifying Intrinsic and Extrinsic Variability in Stochastic Gene Expression Models

Genetically identical cell populations exhibit considerable intercellular variation in the level of a given protein or mRNA. Both intrinsic and extrinsic sources of noise drive this variability in gene expression. More specifically, extrinsic noise is the expression variability that arises from cell-to-cell differences in cell-specific factors such as enzyme levels, cell size and cell cycle stage. In contrast, intrinsic noise is the expression variability that is not accounted for by extrinsic noise, and typically arises from the inherent stochastic nature of biochemical processes. Two-color reporter experiments are employed to decompose expression variability into its intrinsic and extrinsic noise components. Analytical formulas for intrinsic and extrinsic noise are derived for a class of stochastic gene expression models, where variations in cell-specific factors cause fluctuations in model parameters, in particular, transcription and/or translation rate fluctuations. Assuming mRNA production occurs in random bursts, transcription rate is represented by either the burst frequency (how often the bursts occur) or the burst size (number of mRNAs produced in each burst). Our analysis shows that fluctuations in the transcription burst frequency enhance extrinsic noise but do not affect the intrinsic noise. On the contrary, fluctuations in the transcription burst size or mRNA translation rate dramatically increase both intrinsic and extrinsic noise components. Interestingly, simultaneous fluctuations in transcription and translation rates arising from randomness in ATP abundance can decrease intrinsic noise measured in a two-color reporter assay. Finally, we discuss how these formulas can be combined with single-cell gene expression data from two-color reporter experiments for estimating model parameters.

Gene expression noise can be decomposed into intrinsic and extrinsic noise [35][36][37]. More specifically, intrinsic noise is the protein variability that arises from the inherent stochastic nature of biochemical reactions associated with transcription, translation, mRNA and protein degradation. Given that many mRNA species are present at low copy numbers inside cells, random birth and death of individual mRNA transcripts generates considerable intrinsic noise [38][39][40][41]. Let Z be any cell-specific factor (such as cell cycle stage, abundance of RNA polymerases/ribosomes, cellular environment, etc.) that affects expression of a given gene. Then, cell-to-cell differences in Z will create intercellular variability in gene expression, that is referred to as extrinsic noise Variations in Z induce fluctuations in model parameters (such as the transcription and translation rate), and extrinsic noise can be effectively quantified through analysis of deterministic gene expression models with corresponding parameter fluctuations [42].
We define intrinsic and extrinsic noise in the context of a twocolor experiment, where the gene of interest is duplicated inside the cell (Figure 1). Consider two identical copies of a promoter that express two different reporter proteins P 1 and P 2 . Let p 1 (t) and p 2 (t) denote the level of these proteins at time t inside the cell. Since cell-specific factor Z is common to both copies of the gene, cell-to-cell variations in Z will make p 1 (t) and p 2 (t) correlated. The contribution of Z to expression noise is quantified via the extrinsic noise defined as Extrinsic Noise~S and is related to the covariance between reporter levels. If reporter levels are perfectly correlated, and assuming Sp 1 T~Sp 2 T, which is the total noise in protein level measured by its coefficient of variation squared. Intrinsic noise is the protein variability that is not accounted for by extrinsic noise, and is defined as In summary, a two-color assay can be used to decompose the total protein noise level into intrinsic and extrinsic noise components, computed via (1) and (3), respectively.
Analytical formulas for intrinsic and extrinsic noise are derived for a class of stochastic gene expression models with fluctuations in the transcription or translation rate. Assuming mRNA production occurs in random bursts, transcription rate is represented by either the burst frequency (how often the bursts occur) or the burst size (number of mRNAs produced in each burst). Our results show that fluctuations in the transcription burst frequency enhance extrinsic noise but do not affect the intrinsic expression noise. However, fluctuations in the transcriptional burst size or mRNA translation rate increase both intrinsic and extrinsic noise. A recent study has implicated fluctuations in ATP levels as a major driver of gene expression variability [43]. Since ATP affects both transcription and translation, simultaneous fluctuations in multiple model parameters is investigated. Interestingly, simultaneous fluctuations in the transcription and translation rates decrease intrinsic noise in certain parameter regimes. Finally, usefulness of these formulas in interpreting two-color reporter experiments and estimating model parameters is discussed.

Gene Expression with Constant Parameters
We begin by introducing the standard stochastic gene expression model [44][45][46][47], where all model parameters are fixed, and expression variability arises due to the stochastic nature of transcription and translation processes.

Model Formulation
Transcription has been shown to occurs in ''bursts'' with each burst producing multiple mRNA copies [48][49][50][51][52][53]. Assume mRNAs are produced in bursts of size B m that occur at a rate k m . We refer to k m and B m as the transcriptional burst frequency and burst size, respectively. Consistent with measurements [50], B m is assumed to be a geometrically distributed random variable with probability distribution ProbabilityfB m~i g~a i~( 1{s) i s, 0vsƒ1, i~f0,1,2, . . .g ð4Þ and mean burst size SB m T :~(1{s)=s. Proteins are produced from each mRNA at a translation rate k p . Finally, mRNAs and proteins degrade at constant rates c m and c p , respectively. The stochastic model considers transcription, translation and degradation as probabilistic events that occur at exponentially-distributed time intervals [54,55]. Moreover, whenever a particular event occurs, the mRNA and protein population count is reset accordingly. Let m(t) and p(t) denote the number of molecules of the mRNA and protein at time t, respectively. Then, the reset in m(t) and p(t) for different events is shown in the second column of the table in Figure 2. The third column lists the propensity functions f (m,p) which determine how often an event occurs. In particular, the probability that a particular event will occur in the next infinitesimal time interval (t,tzdt is given by f (m,p)dt.

Computation of Intrinsic Noise
It is relatively straight forward to derive differential equations describing the time evolution of the different statistical moments of the mRNA and protein count. For the above model, the timederivative of the expected value of any differentiable function Q(m,p) is given by where DQ(m,p) is the change in Q when an event occurs, f (m,p) is Figure 1. Decomposing gene expression variability into extrinsic and intrinsic noise using a two-color reporter assay. Two identical copies of a promoter express two different reporter proteins. Correlation in reporter levels is a measure of extrinsic noise that arises from cell-to-cell differences in shared cellular factors. Intrinsic noise is the protein variability that is not accounted for by extrinsic noise, and typically originates from the inherent stochastic nature of biochemical processes. doi:10.1371/journal.pone.0084301.g001 the event propensity function, and S:T represents the expected value [56,57]. Using the resets and propensity functions in Figure dSmpT dt~k p Sm 2 TzSB m Tk m SpT{c p SmpT{c m SmpT: ð7dÞ Setting the left-hand-side of (7) to zero and solving for the moments results in the following steady-state mean protein and mRNA levels where SB m T is the mean transcriptional burst size and S:T represents the steady-state expected value. As done in previous studies of intrinsic and extrinsic noise [35,36,58], the steady-state coefficient of variation squared (variance divided by mean squared) is used as a metric for quantifying the extent of variability/noise in protein copy numbers. From the steady-state protein variance and mean we obtain which represents the total intrinsic noise in protein level for fixed parameters. As B m is geometrically distributed, SB 2 m T2 SB m T 2 zSB m T, and (9) reduces to The first term on the right-hand-side of (10) represents the noise in mRNA copy numbers that is transmitted to the protein level [15,46]. The second term is the Poissonian noise arising from random birth-death of protein molecules. Next, the noise additional to (10) that comes from fluctuations in individual model parameters (such as k m , SB m T and k p ) is quantified.

Transcription Burst Frequency Fluctuations
Consider a cell-specific factor Z at the transcriptional level (such as a transcription factor). Then, fluctuations in Z can either affect the transcriptional frequency k m or burst size B m in the model. The former case of burst frequency fluctuations is considered first.

Modeling Parameter Fluctuations
Let z(t) denote the level of a cellular factor Z inside the cell at time t. Fluctuations in z(t) are modeled through a simple birthdeath process with probabilities of formation and degradation in the infinitesimal time interval (t,tzdt given by where k z and c z represent the production and degradation rate of Z, respectively. For the process described in (11), the steady-state mean, coefficient of variation squared CV 2 z and the autocorrelation function R z (t) are given by Thus by changing k z and c z , both the extent and time-scale of fluctuations in z(t) can be independently modulated. Note the inverse relationship between SzT and CV 2 z implies Poisson statistics. Fluctuations in Z are incorporated in the model by assuming that the transcription burst frequency is no longer a constant but given by k m z(t)=SzT, making it a random process with mean k m and coefficient of variation squared CV 2 z . Throughout this manuscript, CV 2 z represents the extent of parameter fluctuations. Since Z similarly affects expression of both copies of the gene in a two-color assay, fluctuations in z(t) make reporter levels correlated in Figure 1 and induce extrinsic noise.

Computation of Total Noise
The stochastic model consists of six birth-death events that change cellular factor, mRNA and protein copy numbers by integer amounts. Using the propensity functions in Figure 2 and (11) in (5) we obtain for any differentiable function Q(m,p,z). Appropriate choices of Q(m,p,z) result in which yield the steady-state variability in protein level as The first two terms on the right-hand-side of (15) represent the noise level with fixed parameters (Eq. (10)). The third term is the additional noise due to burst frequency fluctuations. Next, (15) is decomposed into intrinsic and extrinsic noise components as measured by the two-color reporter assay ( Figure 1).

Computation of Intrinsic and Extrinsic Noise
Extrinsic noise can be approximated by the coefficient of variation squared of the protein level in a deterministic gene expression model with corresponding parameter fluctuations [42]. The deterministic counterpart to the stochastic model is the set of ordinary differential equations driven by the stochastic process z(t) defined in (11).
and leads to moment dynamics identical to (14) except for Quantification of protein noise level from (14) (with (14c)-(14d) replaced by (18a)-(18b)) gives the extrinsic noise, which is subtracted from (15) Extrinsic noise~CV 2 As expected, extrinsic noise increases with extent of parameter fluctuations CV 2 z . On the contrary, intrinsic noise is independent of CV 2 z and is equal to CV 2 fixed . An important limit considered previously is the case where parameter values (in this case transcription burst frequency) are drawn from a static distribution [36]. In our model, this corresponds to a scenario where the timescale of fluctuations in z(t) are slow compared to mRNA/protein turnover rates. When c z %c m ,c p , Eq. (19c) reduces to Extrinsic noise~CV 2 z , and this result is consistent with previous calculations of extrinsic noise for parameter values drawn from a static distribution (see Eq. 25 in [36]).

Transcription Burst Size Fluctuations
Consider an alternative scenario of a fixed transcription burst frequency but varying burst size. Assume mRNAs are produced in geometrically distributed bursts with mean SB m Tz(t)=SzT, where z(t) is the level of the cellular factor inside the cell at time t. This implies ProbabilityfB m~i g~a i~( 1{s(t)) i s(t), i~f0,1,2, . . .g, and mean burst size Computation of Total Noise where a i is given by (21). Equation (24) yields moment dynamics identical to (14) except for the time derivative of Sm 2 (t)T. For Q(m,p,z)~m 2 , Using the fact that for a geometric distribution and (21), (23) is written as Steady-state analysis of (14) (with (14c) replaced by (25)) results in the total protein noise level for transcriptional burst size fluctuations. As expected when CV 2 z~0 (no parameter fluctuations) (26) reduces to (10). Comparison of (26) with (15) reveals that for a given CV 2 z , burst size fluctuations generates larger variability in protein level than burst frequency fluctuations.

Computation of Intrinsic and Extrinsic Noise
For burst size fluctuations, the deterministic model used for quantifying extrinsic noise will be identical to (16). Since both transcriptional burst size and frequency appear together, replacing k m z(t)=SzT with k m , and SB m T with SB m Tz(t)=SzT in (16) does not alter the model. Thus, extrinsic noise is same irrespective of whether fluctuations are in the transcriptional burst size or frequency. Using (19c) and (26) Total noise~CV 2 burst{size~I ntrinsic noisezExtrinsic noise ð27aÞ Extrinsic noise~CV 2 z c m c p (c m zc p zc z ) (c m zc p )(c m zc z )(c z zc p ) : In contrast to (19), intrinsic noise linearly increases with CV 2 z for burst size fluctuations (Figure 3).

Translation Rate Fluctuations
Next, we consider mRNA translation rate fluctuations and set it equal to k p z(t)=SzT. From Figure 2, this implies that the propensity function for the translational event is now nonlinear and given by k p z(t)m(t)=SzT. Since mRNA production is no longer dependent on Z, z(t) and m(t) are independent random processes.

Computation of Total Noise
Statistical moments of z(t),m(t),p(t) are obtained from (13) with k m z(t)=SzT replaced by k m , and k p replaced by k p z(t)=SzT. Using the fact that z(t) and m(t) are independent yields Note that the moment dynamics is not closed, in the sense that, the time derivative of the second order moments Sp 2 (t)T depends on the third order moment Sm(t)p(t)z(t)T: This phenomenon occurs due to nonlinear propensity functions and typically closure methods are needed to solve for the moments [56,57]. The independence of z(t) and m(t) is exploited for moment closure. More specifically, which is dependent on the fourth order moment Sm 2 z 2 T. As equations (28)-(30) form a closed system of equations that yield total variability in protein level as Computation of Intrinsic and Extrinsic Noise Strategy for decomposing (31) into its intrinsic/extrinsic components is similar to previous sections: extrinsic noise is first computed from a deterministic model and then subtracted from (31) for the intrinsic noise. Consider the differential equation model with translation rate fluctuations. Replacing k m z(t)=SzT by k m , and k p by k p z(t)=SzT in (17), we obtain moment dynamics identical to (28) except for Steady-state analysis of (28)-(30) (with (28c)-(28d) replaced by (33a)-(33b)) yields

Intrinsic noisezExtrinsic noise ð34aÞ
Intrinsic noise~S B m Tz1 SmT As in (27), fluctuations in the translation rate enhance both intrinsic and extrinsic noise (Figure 3).

Simultaneous Model Parameter Fluctuations
Previous sections focused on expression variability generated by fluctuations in individual parameters. However, stochasticity in the abundance of certain cellular factors (such as ATP) can simultaneously affect both transcription and translation. Motivated by this scenario, we investigate how perfectly correlated fluctuations in the transcription rate (measured by either the transcriptional burst frequency or burst size) and translation rate affect intrinsic and extrinsic noise.

Transcription Burst Frequency and Translation Rate Fluctuations
Assume transcriptional bursts occur at a rate k m z(t)=SzT with a geometrically distributed burst size independent of z(t) and given by (4). Each mRNA produces proteins at a rate k p z(t)=SzT, which is perfectly correlated with burst frequency. Let m~½SzT,SmT,SpT,Sz 2 T,Sm 2 T,Sp 2 T,SmzT,SpzT,SmpT T ð35Þ be a vector containing all the first and second order moments of the population counts. Then, using (13) with k p replaced by k p z(t)=SzT, time evolution of m can be compactly represented as where vectorâ a 1 , matrices A 1 , B 1 depend on model parameters and m is a vector of third order moments. As one would expect, nonlinear propensity function for the translation event leads to unclosed moment dynamics. It turns out that incorporating certain higher order moments in m can close moment equations. More specifically, the time derivative of is closed and is given by for some vectorâ a 3 and matrix A 3 . Steady-state analysis of (38) results in an exact analytical formula for the total steady-state protein noise level. In previous sections (individual parameter fluctuations), average protein copy number was invariant of CV 2 z and given by (8). However, simultaneous transcription/translation rate fluctuations enhance mean protein level from (8) to To resolve total noise into its intrinsic/extrinsic components the following deterministic model is used For (40), the moment generator equation is obtained by replacing k p with k p z(t)=SzT in (17). Performing an identical analysis as (36)- (38) for the hybrid model (40) yields the extrinsic noise, which is subtracted from the total noise to obtain the intrinsic noise. Unfortunately, these expressions are too complex to be listed here but are illustrated in Figure 4. Interestingly, simultaneous fluctuations in the burst frequency and translation rate can either increase or decrease intrinsic noise depending on model parameters.
To further elucidate the relationship between intrinsic noise and CV 2 z , the case of slow fluctuations in z(t) compared to mRNA/ protein turnover rates (i.e., c z %c m ,c p ) is considered. In this case noise expressions reduce to Extrinsic noise~C where the mean mRNA and protein levels are given by (see (39)) Equation (41a) reveals that when

Discussion
Given the different functional roles of gene expression noise inside cells [3,32], much work has focused on understanding how variations in the level of a protein arises between otherwise identical cells. A class of models were introduced where stochasticity arises from two sources: i) Random production and degradation of individual mRNA transcripts/protein molecules stemming from the inherent probabilistic nature of biochemical reactions and ii) Fluctuations in model parameters that correspond to randomness in cell-specific factors. Exact analytical formulas for total variability in protein level were derived, in spite of the fact that in many cases parameter fluctuations lead to nonlinear propensity functions. These formulas were decomposed into intrinsic and extrinsic noise components as measured by the two-color reporter assay ( Figure 1).

Which Mechanism Generates the Largest Gene Expression Noise?
Individual-parameter fluctuations. Comparison of (19), (27) and (34) 10)). Assuming ATP affects transcriptional burst size and translation rate, 10% variability in ATP abundance (CV 2 z~0 :1) enhances noise level three-fold from 0:1 to 0.32. In comparison, burst size fluctuations of similar magnitude only increase 0.1 to 0.16. These results reinforce recent observations that intercellular variation in ATP abundance can be a major driver of gene expression noise [43]. An implicit assumption in this analysis is that protein and mRNA degradation is insensitive to ATP. Since both ATP-dependent and ATPindependent degradation pathways exist within cells, further work on ATP-sensitive degradation rates is clearly needed.
Relationship between Intrinsic Noise and CV 2 z Using Monte Carlo simulation techniques previous studies had shown that parameter fluctuations can alter intrinsic noise measurements in a two-color assay [42,60]. Building up on these results, a systematic analytical analysis of how fluctuations in both individual and multiple model parameters affect randomness in protein populations counts was performed. Main findings are as follows: N Intrinsic noise is invariant of fluctuations in the transcription burst frequency (i.e., how often mRNA bursts occur from the promoter).

f~0 (Transcription burst frequency fluctuations): ð53eÞ
Therefore, if extrinsic noise~0:5 in an experiment, from (52) and (53d), CV 2 fixed &intrinsic noise=2 for burst size fluctuations. Traditional approach of assuming CV 2 fixed~i ntrinsic noise would overestimate CV 2 fixed by 100%. Using f~0:4 for simultaneous burst frequency/translation rate fluctuations gives CV 2 fixed0 :83|intrinsic noise, and CV 2 fixed~i ntrinsic noise may not be a bad approximation in this case. It can be shown that for the physiologically relevant parameter regime (51). Thus, our results provide the necessary correction factors for accurately determining CV 2 fixed from two-color reporter experiments, which would be useful for estimating SB m T and k m .
In conclusion, our analysis reveals how stochastic synthesis and degradation of biomolecules combines with parameters fluctuations to generate heterogeneity in protein level across a clonal cell population. These results will help understand how stochastic variability is regulated inside cells, and for extracting meaningful information from single-cell gene expression measurements. Future work will consider scenarios where randomness in cellular factor levels simultaneously affects synthesis and degradation pathways, or only degradation. Unfortunately, exact solutions are unavailable in many of these cases. However, preliminary analysis has found moment closure techniques useful for obtaining closedform solutions for the statistical moments. A recent study has generalized notions of intrinsic and extrinsic noise from statistical moments to temporal correlations [61]. In particular, the autocorrelation function of p(t) can be decomposed into intrinsic and extrinsic components based on the two-color assay [61]. Future will work will derive analytical expressions for protein autocorrelation and cross-correlation functions in stochastic models with parameter fluctuations, and study how noise signature within them can be used for probing genetic systems.

Supporting Information
Text S1 Formulas for factor f in Eq. 52. (PDF)