# Limits of noise for autoregulated gene expression

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## Abstract

Gene expression is influenced by extrinsic noise (involving a fluctuating environment of cellular processes) and intrinsic noise (referring to fluctuations within a cell under constant environment). We study the standard model of gene expression including an (in-)active gene, mRNA and protein. Gene expression is regulated in the sense that the protein feeds back and either represses (negative feedback) or enhances (positive feedback) its production at the stage of transcription. While it is well-known that negative (positive) feedback reduces (increases) intrinsic noise, we give a precise result on the resulting fluctuations in protein numbers. The technique we use is an extension of the Langevin approximation and is an application of a central limit theorem under stochastic averaging for Markov jump processes (Kang et al. in Ann Appl Probab 24:721–759, 2014). We find that (under our scaling and in equilibrium), negative feedback leads to a reduction in the Fano factor of at most 2, while the noise under positive feedback is potentially unbounded. The fit with simulations is very good and improves on known approximations.

## Keywords

Intrinsic noise Langevin approximation Quasi-steady-state assumption Chemical reaction network Auto-regulated gene expression## Mathematics Subject Classification

92C40 60J60 60F05## 1 Introduction

It is now widely accepted that gene expression is a stochastic process. The reason is that a single cell is a system with only one or two copies of each gene and of the order tens for mRNA molecules (Swain et al. 2002; Elowitz et al. 2002; Raj and van Oudenaarden 2008). Experimentally, this stochasticity can even be observed directly by single-cell measurements such as flow cytometry and fluorescence microscopy, which show the inherent fluctuations of protein numbers arising from cell to cell (Li and Xie 2011).

Usually, noise in gene expression is divided into an intrinsic and an extrinsic part (Swain et al. 2002; Raser and O’Shea 2005). While the intrinsic part leads to variation of protein numbers from cell to cell in the same environment, the extrinsic part is attributed to the different environmental conditions of the cell. In practice, ensemble averages eliminate intrinsic noise, while single-cell measurements over time can be thought of having a constant environment, thus eliminating extrinsic noise (Singh and Soltani 2013; Singh 2014).

Stochasticity in gene expression is not only interesting per se. Today, its role in evolution, development and cell fate decisions is under discussion (Kaern et al. 2005; Maamar et al. 2007; Fraser and Kærn 2009; Eldar and Elowitz 2010; Balázsi et al. 2011; Silva-Rocha and Lorenzo 2010; Wang and Zhang 2011). For instance, noisy gene expression can be detrimental for the survival of cells under harsh conditions (Mitosch et al. 2017; Fraser and Kærn 2009). Still, many cells have to function constantly. Therefore, mechanisms reducing and controlling the level of noise are beneficial for most of real systems.

Under the central dogma of molecular biology, modeling stochasticity of gene expression is straight-forward (see Paulsson 2005 for a review). A gene, which is either turned *on* or *off*, is transcribed into mRNA, which is translated into protein. Both, mRNA and protein are degraded at constant rates. Since the resulting chemical reaction network is linear, the master equation can be solved and all moments can be derived analytically. Most interestingly, the variance can be decomposed into the effects of switching the gene *on* and *off*, noise due to the finite life-time of mRNA, and random fluctuations in the production of protein (Paulsson 2005). It is often stated that gene expression tends to occur in bursts, which occur due to the short life-time of the *on*-state of the gene and due to the short life-time of mRNA (Kumar et al. 2015).

We are interested in the effect of self-regulation on gene expression noise. It is known that a negative feedback loop, i.e. a protein suppressing its own transcription (or translation) leads to a reduced noise, while positive feedback is attributed to increase noise (Lestas et al. 2010; Hornung and Barkai 2008). Although these findings are wide-spread, a complete mathematical analysis is lacking. At least, for negative feedback, Thattai and Oudenaarden (2001) and in more generality Swain (2004) quantify the effect of negative feedback using a linearization argument. The latter paper further analyzes different feedback models differing between translational and transcriptional autoregulation. Moreover, Dessalles et al. (2017) derive the equilibrium distribution using a multi-scale approach under negative feedback.

Most analyses of noise in unregulated gene expression rely on the master equation (e.g. Paulsson 2005). By the linearity of this equation, a solution can be given explicitly. Using the approximation that gene switching is so fast that it is effectively constantly transcribed to mRNA, this linearity can as well be used under negative feedback (Thattai and Oudenaarden 2001; Swain 2004). Our approach and also the one performed in Dessalles et al. (2017) differs in two ways. First, we are using martingale methods from stochastic analysis in order to describe the chemical system (Ethier and Kurtz 1986). Second, we can relax the assumption that the gene is transcribed effectively constantly, and therefore derive a more general result. Consequently, we are able to analyze noise in a truly non-linear system under a quasi-steady-state assumption.

The explicit expression we derive for the noise in the number of proteins is also the main difference between our findings and the results obtained in Dessalles et al. (2017). Since the authors of that paper are interested in the case of not very abundant proteins they compute a stationary distribution for the protein using martingale techniques in the context of birth–death processes as opposed to our stochastic diffusion setting.

While the full model of regulated gene expression (or any other chemical reaction network) is usually hard to study, considering an ODE approach instead, which approximates the full model, leads to new insights. Formally, a law of large numbers—usually referred to as a fluid limit—can be obtained connecting the stochastic and deterministic model (Kurtz 1970b; Darling 2002). While such a law of large numbers gives a deterministic limit, fluctuations are studied using central limit results; see Kurtz (1970a). The special situation for gene expression is that the gene and mRNA only have a few copies, while the protein is often in large abundance. Such multi-scale models are often studied under a quasi-steady-state assumption (Seegel and Slemrod 1989). Here, the species in low abundance are assumed to evolve fast, such that the slow, abundant, species only sense their time-average. For such a stochastic averaging, not only a law of large numbers is given e.g. by Ball et al. (2006), but also a central limit result has recently been obtained by Kang et al. (2014).

While a multi-scale approach to stochastic gene expression is not new (see Bokes et al. 2012; Dessalles et al. 2017), the analysis of fluctuations for such systems is not finished yet. In the case of multi-scale diffusion systems, Pardoux and Veretennikov (2001, 2003) derive a limit result for the slow components using a Poisson-equation. The results by Kang et al. (2014) are similar but are based on Markov jump processes instead of a diffusion limit framework. We apply the techniques of Kang et al. (2014) on the chemical reaction network of (un-)regulated gene expression. As our results show, fluctuations take into account all sources of noise and we give explicit formulas for the reduction of noise under negative feedback and the increase in noise under positive feedback.

## 2 The model

*neutral*model) Here,

*off*and

*on*refer to an inactive and an active gene, respectively. The mRNA is given by

*R*, and the protein by

*P*. While the first line of chemical reactions models gene switching from

*off*to

*on*and back, the second line encodes transcription and degradation of mRNA, and the third line gives translation and degradation of proteins. Exchanging the first line by then models a negative feedback and models a positive feedback. In all cases, we number the equations from left to right and from top to bottom by 1–6, so \({\mathcal {K}} = \{1,\ldots ,6\}\) is the set of chemical reactions. The species counts are given by \(X_i\) for \(i\in {\mathcal {S}}:=\{\text {off}, \text {on}, R, P\}\) for inactive and active gene, mRNA and protein, respectively. In the following we will scale the rates such that the gene switching and the mRNA production happens on a fast time-scale whereas the protein which is also present in higher abundances than the mRNA is evolving on a slower time-scale. This time-scale separation is frequently used in quantitative analyses of gene expression (see for instance Thattai and Oudenaarden 2001; Swain 2004; Ball et al. 2006; Bokes et al. 2012; Dessalles et al. 2017) since it allows to employ a quasi-steady-state assumption for the species evolving on the fast time-scale, cf. Kuehn (2015). It basically means that one first solves for the stationary points in the fast sub-system which are then used to describe the dynamics in the slow sub-system.

*N*, we will make use of the following scaling for the abundances of chemical species

*N*. We use the scaled rates \(\kappa _2, \nu _2, \kappa _3, \nu _3 = O(1)\), which are given through

*M*denoting the total copy number of genes, we have in the neutral case for independent, rate 1 Poisson processes \(Y_1,\ldots ,Y_6\). (See (Anderson and Kurtz 2015) for a general introduction on theoretical chemical reaction networks.) The first equation changes in the case of negative feedback to and in the case of positive feedback to In the sequel, we will refer to the model without, with negative and positive feedback simply as Open image in new window , Open image in new window and Open image in new window , respectively. We understand all equations ( Open image in new window ), ( Open image in new window ), ( Open image in new window ) as the bases for Open image in new window , equations ( Open image in new window ), ( Open image in new window ), ( Open image in new window ) as the bases for Open image in new window and all equations ( Open image in new window ), ( Open image in new window ), ( Open image in new window ) as the bases for Open image in new window .

We use *N* as a scaling parameter throughout. Reaction rates either come in unscaled (\(\lambda \)’s and \(\mu \)’s) or in scaled (\(\kappa \)’s and \(\nu \)’s) versions

Parameter | Model | Meaning | Scaling |
---|---|---|---|

| Scaling parameter | ||

\(\lambda _1^+, \kappa _1^+\) | Rate of switching genes on | \(\lambda _1^+=\kappa _1^+ N\) | |

\(\lambda _1^\oplus , \kappa _1^\oplus \) | \(\lambda _1^\oplus = \kappa _1^\oplus \) | ||

\(\lambda _1^-, \kappa _1^-\) | Rate of switching genes off | \(\lambda _1^-=\kappa _1^- N\) | |

\(\lambda _1^\ominus , \kappa _1^\ominus \) | \(\lambda _1^\ominus = \kappa _1^\ominus \) | ||

\(\lambda _2, \kappa _2\) | Rate of mRNA production | \(\lambda _2 = \kappa _2 N\) | |

\(\mu _2, \nu _2\) | Rate of mRNA degradation | \(\mu _2 = \nu _2 N\) | |

\(\lambda _3, \kappa _3\) | Rate of protein production | \(\lambda _3 = \kappa _3 N\) | |

\(\mu _3, \nu _3\) | Rate of protein degradation | \(\mu _3 = \nu _3 \) | |

| Total number of genes | \(M=O(1)\) |

## 3 Results

The following results are all stated in terms of the scaled parameters (\(\kappa \)’s and \(\nu \)’s and \(V_P^N\)). For the corresponding formulas using unscaled parameter notation, see Appendix E.

### 3.1 A limiting process for the amount of protein

*t*, and the scaled parameter set we find

### Theorem 1

### Proof

*N*of the corresponding equation in ( Open image in new window ), we obtain that

*F*as in (4). Computation of the equilibria is standard by solving \(F(v_P)=0\). In particular, we have to solve

### 3.2 Approximate variance and Fano factor for the amount of protein

*F*from Theorem 1, we assume that \(V_P^N \approx v_P + \frac{1}{\sqrt{N}}U\) for some stochastic process

*U*. The random process

*U*will then account for the fluctuations which are not captured by the deterministic approximation above. Hence,

*F*, and the number of mRNA, which is approximated by its mean in order to derive

*b*. Consequently, fluctuations arising from these two mechanisms cannot be accounted for in the resulting variance. As a result, fluctuations read off from (7) will be too small.

In contrast, as an application of Kang et al. (2014) (see also Appendix A), we derive the following central limit result, which takes into account all fluctuations in leading order. Precisely, our next goal is to show that \(\sqrt{N}(V_P^N - v_P)\) converges and to determine the limiting process. This limit will then provide the error due to noise between the deterministic approximation \(v_P\) and the stochastic process \(V_P^N\) of order \(\sqrt{N}\). In the proof, we will make use of the method developed by Kang et al. (2014).

### Theorem 2

*F*be as in Theorem 1 and assume further weak convergence of the initial conditions:Then, for the models Open image in new window and Open image in new window , Open image in new window , where

*U*solves

*W*the one-dimensional standard Brownian motion and

Hence, we see that, in contrast to the Langevin approach (7) above, fluctuations arising from gene switching and RNA dynamics are also accounted for in Theorem 2; see also Sect. 3.3 for an interpretation of the individual terms. For the difference between the Langevin approximation and our result and its implications see also Sect. 4.3.

The proof of the Theorem is given in Appendix B. Briefly, we apply the stochastic averaging principle on multiple time scales developed in Kang et al. (2014). The whole approach is revisited in Appendix A. There we also state the conditions which need to be satisfied for the theory to apply. Amongst others these include solving a certain Poisson equation which enables a clean time-scale separation.

### Remark 1

(*Deriving the Fano factor in equilibrium*) While (8) provides a dynamical result along paths of \(X_P\), we can also use this approximation and study the process in equilibrium by setting \(X_P(0)=v_P^*N\), where \(v_P^*\) is the unique solution of \(F(v_P)=0\) given by (5).

*N*appears on the right hand side, some authors call the Fano factor dimensionless. Empirically, it was found e.g. by Bar-Even et al. (2006), that for all classes of genes and under all conditions, the variance in protein numbers was approximately proportional to the mean, which is again reminiscent of the lacking

*N*in the Fano factor above.

We note here that this approach of computing the Fano factor of \(X_P^N\) in equilibrium was achieved by an unjustified exchange of limits. Namely, for the approximate Fano factor of \(X_P^N\) in equilibrium, we would have to perform \(t\rightarrow \infty \) first, and only then compute \(N\rightarrow \infty \), but our approach exchanged this limit.

### 3.3 Interpretation of the Fano factor

Adjusted explanations hold in the cases of no or positive feedback.

### 3.4 Comparing the noise in Open image in new window , Open image in new window and Open image in new window

It is frequently reported that a negative feedback in gene expression results in a reduced variance (noise) of protein levels, whereas a positive feedback enhances noise (Lestas et al. 2010; Hornung and Barkai 2008). These observations can be made precise by our results from above. Here, we report some consequences on the equilibrium variance and the Fano factor, \({\mathbb {V}}[X_P]/{\mathbb {E}}[X_P]\).

Additionally, we see from Eq. (16) that the change in noise is maximal if the gene is *off* most of the time, while still having the same amount of protein as in the unregulated (neutral) case. This finding is reminiscent of the fact that gene expression comes in bursts. The burstiness is most extreme if the gene is *on* only for a short time, producing a large amount of mRNA, and afterwards *off* for a long period. Especially, we see that for \(\kappa _1^\ominus \rightarrow \infty \) the maximal reduction due to negative feedback is twofold while the increase in noise is unbounded for \(\kappa _1^- \rightarrow \infty \) in case of positive auto-regulation of the protein.

### 3.5 Negative feedback in a simpler model

### 3.6 Refining the Fano factor

Here, we again consider the original model given by equations ( Open image in new window ), ( Open image in new window ) and ( Open image in new window ), but use a different scaling for the mRNA. We assume that not only the protein evolves on the slow time-scale but also the mRNA production and degradation. This is a slightly more complex model since we cannot average the number of mRNA molecules when analyzing the protein fluctuations. This model allows us to compare our results in a more straightforward way with results obtained previously in Thattai and Oudenaarden (2001), Swain (2004); see Sects. 4.2 and 4.3.

## 4 Comparison to previous results

Here, we compare our results in the neutral case with those obtained in Paulsson (2005), and in the case of negative feedback with the formulas for the Fano factor derived in Dessalles et al. (2017), Ramos et al. (2015), Swain (2004) and Thattai and Oudenaarden (2001).

### 4.1 The neutral case, Paulsson (2005)

### 4.2 Negative feedback, Thattai and Oudenaarden (2001)

*N*is large and \(\phi \) is small by (8))

Recalling our exact result for the Fano factor from Eq. (14), we note that due to the linearization of the mRNA expression and thus a basically constant mRNA production, the noise emerging from the random gene switches is not adequately represented in the formula obtained in Thattai and Oudenaarden (2001). To be more precise, in contrast to our formula in (14), the effect of mRNA noise due to gene switching (first term in first bracket) is not taken into account at all. Additionally, the negative feedback (last bracket in (22)) does not affect the noise in the same way as it does in the exact formula (14). As can be seen in Fig. 2, when comparing the solid and the dashed lines, these effects lead to an underestimation of the actual noise which is produced by an exact simulation of Open image in new window .

### 4.3 Negative feedback, Swain (2004)

For a comparison of the two results we refer to the solid and dotted lines in Fig. 2 where we see that due to the missing term in the approximation obtained by Swain, his result slightly underestimates the Fano factor resulting from simulations of Open image in new window . However, the difference becomes smaller for a lower number of expected proteins, see Fig. 2b. This can be explained by the difference between Eqs. (23) and (18). The missing term in Swain’s derivation can be related to noise emerging from the gene switching. These processes however are given by the overall parameter configuration. Hence the larger difference between the solid and dotted lines in (a) when compared to (b) simply emerges from the corresponding term in (a) being larger than in (b). Thus, it therefore has a stronger effect on the overall fluctuations in protein numbers.

### 4.4 Negative feedback, Dessalles et al. (2017)

*O*(1)) abundance of protein in the system. Assuming the same time-scale separation as in our model, i.e. gene switching and mRNA processes evolve on a fast time-scale, the resulting birth–death process for

*P*has a stationary distribution which they compute explicitly. Moreover, setting

### 4.5 Negative feedback in a simpler model, Ramos et al. (2015)

In Ramos et al. (2015), the authors derive in their equation (11) the Fano factor for the simplified model, which we introduced in Sect. 3.5. (We note that their model slightly differs since the protein binds to the gene and therefore cannot degrade in this state.) Their results connect the Fano factor to the covariance of the number of active genes and the number of proteins in equilibrium. Since they also give the limiting distribution (in terms of a confluent hypergeometric function), they can evaluate this covariance and also the Fano factor numerically. From their Figure 2, one can see that—if proteins are somewhat abundant—the Fano factor stabilizes around 1/2 for various parameter combinations; a result reminiscent of Dessalles et al. (2017) as discussed above.

### 4.6 Negative feedback for small amounts of protein

Theorems 1 and 2—and all subsequent calculations—only hold under the scaling described in \(\Box _*\) or in Appendix D. In these scalings, we have that \(X_P = O(N)\), i.e. there the protein is abundant. In this case, we see from (14) that the Fano factor is at least 1/2. In Sect. 4.4, we discussed the results by Dessalles et al. (2017), where \(X_P = O(1)\) is used, but the limiting result for \(\rho \rightarrow \infty \) implies that proteins are abundant and leads to a Fano factor of at least 1/2.

*n*proteins. For this process, they compute the equilibrium distribution

*Z*as a normalizing constant. In this case, one can see that for \(\lambda _1^\ominus \gg \lambda _1^+\), \(\pi \) is concentrated around 1, i.e. there is a single molecule of the protein and hence, the Fano factor becomes arbitrarily small. Since this in particular means a Fano factor below 1/2, Ramos et al. (2015) call this the infra-Fano regime.

## 5 Conclusion

Quantifying noise in gene expression is essential for understanding regulatory networks in cells (Thattai and Oudenaarden 2001). Our results capture most of the previously derived results. While negative feedback is known to reduce noise under auto-regulated gene expression, we improve on the quantification of this effect, i.e. our results account for all possible sources of noise due to gene activation, mRNA fluctuations and the protein processes itself. We note that our results require that proteins are abundant. Since the infra-Fano regime described in Sect. 4.6 relies on small amounts of protein, our results do not recover this regime.

In addition, we provide the same quantification of noise also for positive feedback, where noise is increased. In particular, (14) shows that the average time the gene is *off* determines the reduction of noise in all cases relative to unregulated genes; see also Grönlund et al. (2013). As we saw earlier, for both, negative and positive feedback, noise difference between the non-regulated and the model with feedback is largest if the gene is *off* most of the time. This can be interpreted by the burstiness of gene expression. It is largest for genes which are *off* for long times and then turned *on* for a short time in which mRNA is produced. Interestingly, previous approaches mostly gave approximations for noise for negative feedback if switching the gene on and off is very fast (Thattai and Oudenaarden 2001; Swain 2004) and if the gene is *on* most of the time (Thattai and Oudenaarden 2001) or in a simplified model (Ramos et al. 2015). Hence, all previous papers could not have seen the effects of gene activation switching on protein noise. As in previous results also obtained in Dessalles et al. (2017), we find that in the limit where the gene is *off* most of the time, the negative feedback reduces noise at most by a factor of two. Additionally we find that noise can increase unboundedly for positive feedback.

Today, quasi-steady-state assumptions are frequently used when analyzing chemical reaction networks. While the intuition suggests the correct approach when approximating the system by a deterministic path, studying fluctuations is apparently much less obvious. In Kim et al. (2015), some special cases are studied when a straight-forward approximation of the fluctuations works. In our analysis, we use a new approach by Kang et al. (2014) and can also interpret all terms arising in (14), see Sect. 3.3.

Due to taking into account all potential sources of fluctuations the fit of simulations and theory (see e.g. Fig. 2) is excellent and improves on previous studies. There, noise arising from the gene switching its state has been averaged out, and only the recent approach of Kang et al. (2014) reveals the impact of these stochastic processes on the noise in protein numbers.

In their paper, Kang et al. (2014) gave as an example an approximation of noise for Michaelis–Menten kinetics and a model for virus infection. Their method relies mostly on solving a Poisson equation \(L_2h=F_N-F\), where \(L_2\) is the generator of the fast subsystem (gene and RNA in our example), \(F_N\) and *F* describe the evolution of the slow system (protein) including all fluctuations and in the limit using the quasi-steady-state assumption, respectively. We stress that this approach is not only useful for equilibrium situations, but also for understanding noise if the slow system has not reached equilibrium yet, e.g. after a cell division.

It was argued that complexity of gene regulatory networks leads to a reduction in the level of noise, while certain network motifs always lead to increased levels of noise (Becskei and Serrano 2000; Cardelli et al. 2016). Experimentally, gene expression noise can be used to understand the dynamics of gene regulation (Munsky et al. 2012). Our analysis should provide an approach for distinguishing between different models of gene regulation based on measurements of noise levels.

## Notes

### Acknowledgements

Open access funding provided by Max Planck Society. We thank Jens Timmer, Freiburg, for discussion. Additionally, we thank three anonymous referees for carefully reading the manuscript and helpful comments which improved the manuscript.

## Supplementary material

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