The effect of microRNA on protein variability and gene expression fidelity

Abstract

Small regulatory RNA molecules such as microRNA modulate gene expression through inhibiting the translation of messenger RNA (mRNA). Such posttranscriptional regulation has been recently hypothesized to reduce the stochastic variability of gene expression around average levels. Here, we quantify noise in stochastic gene expression models with and without such regulation. Our results suggest that silencing mRNA posttranscriptionally will always increase, rather than decrease, gene expression noise when the silencing of mRNA also increases its degradation, as is expected for microRNA interactions with mRNA. In that regime, we also find that silencing mRNA generally reduces the fidelity of signal transmission from deterministically varying upstream factors to protein levels. These findings suggest that microRNA binding to mRNA does not generically confer precision to protein expression.

Significance

Small RNAs that temporarily silence messenger RNA molecules from being translated affect not just average gene expression levels but also stochastic variability around averages. Previous work has suggested that the role of such complex formation is to decrease detrimental noise in gene expression. Our analysis of simple gene expression models suggests that introducing small RNAs into a system will increase, rather than decrease, protein noise in the parameter regime that describes the interactions between microRNA and messenger RNA.

Introduction

Regulating gene expression is the fundamental process through which cells maintain cellular states. Binding of small regulatory RNA molecules to messenger RNA (mRNA) is a widespread mechanism to modulate gene expression in bacteria, archea, and eukaryotes (1,2,3,4,5,6). Although this mechanism is conserved across all kingdoms of life, the exact role such small RNA interactions remains a field of ongoing debate, especially regarding the functional role of microRNA (miRNA) in eukaryotes. Initial experimental results from perturbation studies reported only small effects of miRNA interactions on average levels of gene expression in human cell lines (7,8). This lack of severe effects on protein averages has led to the intriguing hypothesis that the role of small RNAs is to affect the variability of gene expression around average levels rather than the averages themselves (9,10,11).

Non-genetic cell-to-cell variability is due to the stochastic nature of individual biochemical reactions in cells that involve only a small number of molecules (12,13). Stochastic variability of gene expression around deterministic averages has been shown to be functionally relevant in cells and has been the subject of much experimental and theoretical research (13,14,15). Recent work reported that introducing miRNA interactions decreased gene expression variability in mouse embryonic stem cells and concluded that “miRNAs confer precision to protein expression” by reducing undesirable gene expression variability (11).

Here, we quantify theoretically how variability in protein levels is affected by mRNA-miRNA complex formation. We find that miRNA interactions can increase or decrease protein variability depending on the biochemical rate constants. However, in the biologically relevant regime, we show that introducing miRNA will increase both “intrinsic” noise in gene expression due to small mRNA numbers as well as “extrinsic” noise due to fluctuating transcription rates.

Furthermore, we quantify the effect miRNA has on the fidelity of signal transmission when transcription rates vary due to a deterministically varying signal rather than stochastic noise. We find that the signal-to-noise ratio in gene expression is not generally improved through the introduction of miRNA.

Finally, we quantify how miRNA interactions affect the interpretation of experimentally reported mRNA-protein correlations when bound complexes of mRNA and small RNAs are not experimentally distinguished from unbound mRNA that is being actively transcribed.

Results and discussion

Protein variability due to “intrinsic” low-copy-number mRNA noise

To analyze the effect of mRNA-miRNA interactions on gene expression variability, we consider a simple model of gene expression (see Fig. 1) that includes the following stochastic reactions for transcription and translation:

$$\underset{\text{transcription}\text{and}\text{mRNA}\text{degradation}}{\underbrace{\begin{array}{ccc}m& \stackrel{\lambda }{\to }& m+1\\ m& \stackrel{{\beta }_{m}m}{\to }& m-1\end{array}}}\underset{\text{translation}\text{and}\text{protein}\text{degradation}}{\underbrace{\begin{array}{ccc}p& \stackrel{\alpha m}{\to }& p+1\\ p& \stackrel{{\beta }_{p}p}{\to }& p-1\end{array}}}\text{.}$$ (1)

Figure 1.

Basic model of gene expression in the presence of miRNA. We model a stochastic process in which proteins are made at a rate proportional to the abundance of mRNA, which alternates between an actively translated free state and a silenced miRNA bound state. All chemical species are assumed to undergo first-order degradation, and we consider the regime in which miRNA abundances are large and approximately constant compared with mRNA variability. To see this figure in color, go online.

Here, $m$ denotes the number of mRNA molecules, which are made with a constant transcription rate, and $p$ denotes the number of protein molecules, whose birth rate is proportional to mRNA levels. Both mRNA and protein are assumed to undergo first-order degradation. While this model ignores a lot of complications, it has become a commonly used reference model to illustrate stochastic effects in gene expression due to low numbers of molecules (12,13,16,17).

To quantify the effect of small RNAs binding to mRNA and inhibiting translation, we introduce the following additional reactions to describe miRNA binding and unbinding to mRNA molecules as well as miRNA-mRNA complex degradation:

$$\underset{\text{miRNA}-\text{mRNA}\text{complex}\text{formation}\text{and}\text{disassociation}}{\underbrace{\begin{array}{ccc}\left(m,c\right)& \stackrel{\delta m}{\to }& \left(m-1,c+1\right)\\ \left(m,c\right)& \stackrel{\gamma c}{\to }& \left(m+1,c-1\right)\end{array}}}\underset{\text{complex}\text{degradation}}{\underbrace{c\stackrel{\phi c}{\to }c-1}}\text{,}$$ (2)

where $c$ denotes the molecular abundance of the miRNA-mRNA complex that is not actively translated.

Taken together, Eqs. 1 and 2 correspond to the previously discussed gene expression model (10,11,18,19) in the regime in which miRNA levels are approximately constant and abundant compared with mRNA levels, as would typically be the case in cells (20). This model differs from previous work that analyzed the effect of miRNA levels that fluctuate due to being coproduced with mRNA in a feedforward loop (21).

In the above reactions, the transcription rate $\lambda$ may itself fluctuate or be regulated by upstream factors. To highlight the effect of miRNA interactions on “intrinsic” noise due to low numbers of mRNA, we first analyze the constant transcription case and consider time-varying transcription rates in subsequent sections.

Because the reaction rates in Eqs. 1 and 2 are linear, all moments can be exactly determined from the chemical master equation. In particular, we can analytically calculate the coefficient of variation for protein levels at stationarity $C{V}_p:={\sigma }_p/⟨p⟩$, where ${\sigma }_p$ denotes the standard deviation, and here (and throughout the article) angular brackets denote stationary state averages. Following the standard linear noise approach (13), we obtain (see appendix A)

$$C{V}_p^2=\frac{1}{⟨p⟩}+\underset{\text{protein}\text{variability}\text{due}\text{to}\text{mRNA}\text{noise}}{\underbrace{\frac{1}{⟨m⟩}\frac{1}{1+\frac{{\tau }_p}{{\tau }_m}\left(1-\frac{R}{1+{\tau }_c/{\tau }_p}\right)}}}\text{,}$$ (3)

where the “recycling parameter”

$$R:=\frac{\gamma ⟨c⟩}{\gamma ⟨c⟩+\lambda }$$ (4)

denotes the relative importance of the complex dissociation versus de novo production of mRNA through transcription, and ${\tau }_p,{\tau }_m,\mathrm{a}\mathrm{n}\mathrm{d}{\tau }_c$ denote the average lifetimes of protein, unbound mRNA, and mRNA-miRNA complexes, respectively.

Here, the average lifetime of a chemical species has the usual model independent meaning, i.e., the average time a molecule spends in the system between its appearance and disappearance in production, degradation, or conversion events. In our model, average lifetimes are related to the rate constants by

$${\tau }_m=\frac{1}{{\beta }_m+\delta },{\tau }_c=\frac{1}{\phi +\gamma },{\tau }_p=\frac{1}{{\beta }_p}\text{.}$$ (5)

We strictly distinguish between mRNA and the miRNA-bound complex as different chemical species with different lifetimes. According to this definition, an miRNA-mRNA binding event removes an mRNA molecule from the system and generates an miRNA-mRNA complex. In contrast, colloquially, and experimentally, mRNA bound to miRNA is often included in the definitions of mRNA abundances and lifetimes. Both definitions are equally valid and lead to the same physical conclusions as long as they are applied consistently.

Time-averaging fluctuations with and without miRNA

The protein variance of the above model has been reported previously in terms of rate constants together with the protein number distribution in terms of generalized hypergeometric functions (22). Here, we analyze the implications of Eq. 3 by relating protein variability in systems with and without miRNA.

First, we clarify the effect of time-averaging stochastic fluctuations with and without mRNA: in the absence of miRNA interactions (or when complexes never dissociate and complex formation simply corresponds to an additional mRNA degradation channel), $R=0$, and Eq. 3 reduces to the well-known result (13,17,23,24) for stochastic gene expression in which Poisson fluctuations in mRNA levels get transmitted to protein levels with a time-averaging factor that only depends on the ratio of mRNA and protein lifetimes and is an increasing function of ${\tau }_m/{\tau }_p$ as depicted in Fig. 2 A.

Figure 2.

Effect of miRNA on protein variability due to intrinsic mRNA noise. (A) In the absence of miRNA interactions, short-lived mRNAs transmit less of their variability to proteins according to the time-averaging factor ${\left(1+{\tau }_p/{\tau }_m\right)}^{-1/2}$ indicated by the black line (13,17,23,24). In systems with miRNA interactions, a larger fraction of intrinsic mRNA noise is transmitted to protein levels than without miRNA interactions for given molecular lifetimes (blue area). Inset: mRNA fluctuations decay more slowly in the presence of miRNA (blue line) than without miRNA (black line) for a given mRNA lifetime. Introducing miRNA into a given system decreases the fraction of mRNA noise transmitted to proteins because miRNA interactions decrease mRNA lifetimes as indicated by hypothetical experimental transitions (arrows) in four different representative parameter regimes that characterize the miRNA-mRNA complex dynamics: $\phi =3,\gamma =3$ (red disc), $\phi =1.5,\gamma =5$ (red square), $\phi =1.25,\gamma =0$ (yellow disc), and $\phi =0,\gamma =6$ (yellow square), and ${\beta }_m=1.5,\delta =4.5$ with all rates expressed per average protein lifetime ${\tau }_p=1/{\beta }_p$. (B) In the biologically expected parameter regime, miRNA tag mRNA for increased degradation or leave mRNA degradation unaffected, which corresponds to $\phi \ge {\beta }_m$. In this regime, introducing miRNA always increases protein variability due to intrinsic mRNA noise regardless of all other rate constants (blue region). While the fraction of mRNA noise transmitted to protein levels decreases upon introduction of miRNA (see A), the amount of inherited mRNA noise increases due to lower mRNA averages. Plotted is the protein variability $C{V}_p$ due to stochastic mRNA noise with miRNA relative to the case without miRNA (see Eq. 3). Dotted lines indicate the red transitions of (A) with varying miRNA binding rates but otherwise identical parameters. In the hypothetical scenario in which miRNAs protect mRNA from degradation such that $\phi <{\beta }_m$, introducing miRNA can increase or decrease protein variability. When the miRNA-mRNA complex rarely dissociates, miRNAs act simply as an irreversible mRNA degradation channel, and the noise increases (yellow disc). Only when the miRNA-mRNA complex dissociates frequently but is rarely degraded can introducing miRNA decrease protein noise (yellow square). To see this figure in color, go online.

When comparing systems with miRNA interactions with systems without miRNA interactions but with equal molecular lifetimes, Eq. 3 shows that systems with miRNA interactions inherit a larger fraction of the mRNA variability. This holds regardless of all other biochemical details as indicated by the accessible blue region for systems with $R\ge 0$ in Fig. 2 A.

Larger noise transmission in the presence of miRNA for a given mRNA lifetime can be intuitively understood by considering fluctuating mRNA levels $m\left(t\right)$ as a time-varying translation rate. In the absence of feedback, standard linear response methods (14,25) establish protein variability in terms of mRNA variability through

$$C{V}_p^2=\frac{1}{⟨p⟩}+C{V}_m^2{\int }_0^{\infty }\frac{e^{-t/{\tau }_p}}{{\tau }_p}{A}_m\left(t\right)dt\text{,}$$ (6)

where $A_m\left(t\right)$ is the autocorrelation of mRNA fluctuations normalized by the mRNA variance (see appendix A.2).

In the above model of gene expression, mRNA variability exactly follows Poisson statistics with normalized fluctuations $C{V}_m^2=1/⟨m⟩$ regardless of miRNA interactions (see appendix A.2), which has been reported previously for this model (22) as well as closely related models of stochastic gene expression with miRNA interactions (21).

However, temporal correlations of mRNA levels decay as $A_m\left(t\right)=e^{-t/{\tau }_m}$ only when $R=0$ (see appendix A.2). With miRNA interactions, the lifetime of mRNA molecules no longer determines the timescale of mRNA fluctuations, and instead we find $A_m\left(t\right)\ge e^{-t/{\tau }_m}$ (see Fig. 2 A, inset). For a given mRNA lifetime, mRNA autocorrelations thus decay more slowly in the presence of miRNA interactions, which leads to more of the variability passed on to protein levels because fast fluctuations will get averaged out more than slow fluctuations due to the fact that protein levels are a lagged readout of mRNA levels.

Equation 3 does not rely on quasi-steady-state approximations to eliminate the complex from the stochastic dynamics and thus differs from a previously published solution for the transmitted mRNA variability (11). While this difference does not affect the above analysis, we note that, in general, quasi-steady-state approximations can be qualitatively misleading when used to describe stochastic fluctuations in systems with reversible complex formation (26,27,28).

The above results describe the transmission of noise from mRNA to protein levels with and without miRNA. By establishing a general relation between systems with equal molecular lifetimes, they mathematically summarize the principles of noise transmission. However, they do not directly determine the effect of introducing miRNA into a given experimental system because such a perturbation reduces mRNA lifetimes (see Fig. 2 A).

Effect of introducing miRNA into a system

Introducing miRNA interactions into a given system reduces mRNA averages and lifetimes, which results in opposing effects on protein variability: lower averages increase the magnitude of mRNA noise, while shorter lifetimes reduce how much of that mRNA noise is transmitted to protein levels. To determine the net result, we mimic the previous experimental approach in which miRNA interactions were introduced by inserting a miRNA binding site into the mRNA of a synthetic gene expression reporter (11). This corresponds to taking a system from $\delta =0$ to some $\delta >0$ while keeping all other rate constants fixed. Such an experimental perturbation affects mRNA abundances and recycling according to

$$⟨m⟩=\frac{\lambda }{{\beta }_m+\frac{\delta \phi }{\gamma +\phi }},R=\frac{\gamma }{\gamma +\phi }\frac{\delta }{{\beta }_m+\delta }\text{,}$$ (7)

and lifetimes according to Eq. 5.

Understanding the effect of introducing miRNA interactions on protein noise thus corresponds to substituting Eqs. 5 and 7 into Eq. 3 to determine the overall effect of the additional complexing reaction. This shows that miRNA interactions can increase or decrease protein noise depending on the biochemical parameters describing the miRNA-mRNA dynamics (see Fig. 2 B). However, in the relevant biological scenario, miRNAs are expected to increase mRNA degradation (29,30), i.e., the miRNA-mRNA complex should be degraded with a higher rate than unbound mRNA. In our model, this regime corresponds to $\phi >{\beta }_m$, for which protein noise is an increasing function of $\delta$ regardless of all other parameters (see appendix A.3). In this regime, protein variability thus always increases upon introduction of miRNA as indicated by the blue region in Fig. 2 B (left panel).

In the scenario when miRNA only inhibits translation without affecting degradation rates (31) the miRNA-mRNA complex is degraded with the same rate as unbound mRNA, and we have $\phi ={\beta }_m$, for which we find that introducing miRNA will also always increase protein noise. One specific example of such a transition is illustrated in Fig. 2 B (left panel), but the result holds in general regardless of the other rate constants (see appendix A.3).

Only if miRNAs were to actively protect mRNA from degradation, i.e., the miRNA-mRNA complex has a lower degradation rate than naked mRNA such that $\phi <{\beta }_m$, can miRNA interactions potentially decrease protein noise. Even in this unexpected regime, noise suppression is not guaranteed, and protein variability can go up, down, or remain unchanged depending on the details of the miRNA-mRNA complex: when miRNAs never dissociate, we have $R=0$, and protein noise always increases even when $\phi <{\beta }_m$ because the decrease in noise transmission due to shorter ${\tau }_m$ is always more than offset by the increased mRNA noise due to lower averages (see yellow disc in Fig. 2 B, right panel). In contrast, when miRNA-mRNA complex dissociation dominates over complex degradation such that $\gamma >{\gamma }^{\ast }$ where

$${\gamma }^{\ast }=\phi \frac{{\beta }_p+\phi }{{\beta }_m-\phi }\text{,}$$ (8)

introducing miRNA interactions leaves mRNA averages almost unaffected because mRNAs eventually come back to the active state after binding to miRNA (see Eq. 7). Thus, the net effect is only the increased time averaging of mRNA fluctuations and a decrease of protein noise with miRNA interactions (see yellow square in Fig. 2 B, right panel). In the intermediate parameter regime when $\gamma ={\gamma }^{\ast }$, the two effects perfectly cancel, and introducing miRNA leaves protein noise exactly unaffected while reducing mRNA averages.

To summarize, introducing miRNA can only suppress noise when two conditions are met: binding of miRNA has to protect mRNA from degradation, and secondly, the miRNA binding has to be reversible such that complex dissociation dominates over complex degradation. Both conditions are not expected in typical biological systems given the current understanding of miRNA-mRNA interactions (29,30,31), although it is an intriguing possibility that some miRNAs could potentially operate in that regime.

All results so far apply to noise in gene expression due to small numbers of mRNA molecules. Additional variability that originates upstream of mRNA, e.g., through transcription factor variability, has been shown to be relevant in many, if not all, quantitative studies of gene expression (32,33,34,35). We next analyze whether the above conclusions generalize when protein noise is not just due to low-copy-number mRNA noise.

Protein variability due to “extrinsic” noise in transcription rates

We consider the previous model with a generic “extrinsic” noise source in the form of a stochastically varying upstream variable that causes fluctuations in the transcription rate, i.e., in Eq. 1, we replace the constant transcription rate $\lambda$ with $\lambda \times z$, where the fluctuating variable $z$ is subject to the following stochastic reactions for $b=\mathrm{1,2,3},\dots$,

$$\begin{array}{ccc}z& \stackrel{\mu \left(b\right)}{\to }& z+b\\ z& \stackrel{{\beta }_{z}z}{\to }& z-1\end{array}\text{.}$$ (9)

For full generality, we here allow for arbitrary bursting dynamics in transcription with arbitrary distribution of bursts $P\left(b\right)=\mu \left(b\right)/{\sum }_b\mu \left(b\right)$, which leads to transcriptional fluctuations with magnitude and timescale (36)

$$C{V}_z^2=\frac{1+\left(C{V}_b^2+1\right)⟨b⟩}{2}\frac{1}{⟨z⟩},{\tau }_z=\frac{1}{{\beta }_z}\text{,}$$ (10)

where $⟨b⟩$ and $C{V}_b:={\sigma }_b/⟨b⟩$ denote the average and coefficient of variation of the burst size distribution $P\left(b\right)$, respectively. The above model of extrinsic noise allows for transcriptional bursting dynamics (37) but includes the nonbursting dynamics as a special case.

Solving the model of Eqs. 1 and 2 subject to this fluctuating transcription rate is straightforward using standard methods (13,23). The resulting total protein variability is algebraically involved but can be written as

$$C{V}_{p,\text{tot}}^2=C{V}_{p,\mathrm{int}}^2+C{V}_{p,\text{ext}}^2\text{,}$$ (11)

where the intrinsic noise contribution $C{V}_{p,\mathrm{int}}^2$ is given by Eq. 3, and the general result for $C{V}_{p,\text{ext}}^2$ is presented in appendix B.1.

In the absence of miRNA interactions, the extrinsic noise term is given by

$$C{V}_{p,\text{ext}}^{\text{no}\text{miRNA}}=C{V}_{\lambda }\sqrt{\frac{1+\frac{{\tau }_p}{{\tau }_m}+\frac{{\tau }_p}{{\tau }_z}}{\left(\frac{{\tau }_p}{{\tau }_m}+1\right)\left(\frac{{\tau }_p}{{\tau }_z}+1\right)\left(\frac{{\tau }_m}{{\tau }_z}+1\right)}}\text{,}$$ (12)

which is a decreasing function of ${\tau }_m$ for fixed ${\tau }_z,{\tau }_p$ as indicated by the black line in Fig. 3 A. In the absence of miRNA, shorter mRNA lifetimes thus increase transmission of extrinsic transcription rate fluctuations to protein level.

Figure 3.

Effect of miRNA on protein variability due to extrinsic transcriptional noise. (A) In the absence of miRNA interactions, reducing mRNA lifetimes increases how much transcription rate variability is transmitted to protein levels. The quantitative behavior depends on the timescale of extrinsic fluctuations but is independent of all other details. The black line indicates the specific example of ${\tau }_z={\tau }_p$. When comparing systems with equal mRNA lifetimes, a smaller fraction of extrinsic noise is transmitted to protein levels with miRNA interactions than without miRNA interactions. (B) In the biologically expected parameter regime $\phi \ge {\beta }_m$ in which miRNA tag mRNA for increased degradation or do not affect degradation at all, introducing miRNA increases protein noise due to transcription rate variability regardless of all other details, analogous to the constraint on intrinsic fluctuations illustrated in Fig. 2B. This generic effect is not due to variability in miRNA levels but is caused by the reduction of mRNA lifetimes. In the hypothetical scenario in which miRNAs protect mRNA from degradation such that $\phi <{\beta }_m$, introducing miRNA can increase or decrease protein variability. When the miRNA-mRNA complex rarely dissociates, the noise will increase (yellow disc). Only when the miRNA-mRNA complex dissociates frequently but is rarely degraded could introducing miRNA decrease protein noise (yellow square). Arrows in this figure indicate the same representative transitions as parameterized in Fig. 2 A. To see this figure in color, go online.

Equation 12 has been reported previously to model gene expression noise in the absence of miRNA interactions (23). Our generalization of Eq. 12 to gene expression with miRNA interactions shows (appendix B.2) that systems with miRNA interaction always inherit less “extrinsic” noise when comparing systems of equal molecular lifetimes, as illustrated by the accessible blue region in Fig. 3 A.

Effect of introducing miRNA into a system

The above result relates extrinsic noise contributions in systems with equal molecular lifetimes but different miRNA dynamics. Next, we answer the complementary question of how introducing miRNA interactions affects how much extrinsic noise is inherited by the protein levels.

We find that the fraction of inherited extrinsic noise can increase, decrease, or remain unaffected depending on the biochemical rate constants. However, analogous to the result for intrinsic result, in the biologically relevant scenario in which miRNA-mRNA complexes are degraded with at least the same rate as unbound mRNA such that $\phi \ge {\beta }_m$, miRNA interactions always increase the amount of inherited extrinsic noise, as indicated by the accessible blue region and representative example transitions in Fig. 3 B (left panel) (see appendix B.3).

In the hypothetical scenario when miRNAs protect mRNA from degradation such that $\phi <{\beta }_m$, introducing miRNA can, in principle, decrease extrinsic noise in protein variability. However, that can only happen when miRNA-mRNA complex dissociation is significant, as indicated in Fig. 3 B (right panel, yellow square). When miRNA-mRNA complexes rarely dissociate such that $R\approx 0$, the inherited extrinsic noise increases even when $\phi <{\beta }_m$ due to the shorter mRNA lifetime as quantified by Eq. 12 and indicated in Fig. 3 B (right panel, yellow disc). While the exact effect of miRNA on extrinsic noise differs from that on intrinsic noise, Fig. 3 B illustrates that the same constraint applies to the effect of miRNA on both intrinsic and extrinsic noise.

An increase in extrinsic noise when introducing miRNA has been reported before but was ascribed as being due to the stochastic variability of miRNA levels (11). The generic effect we report here is fundamental to changing mRNA dynamics through miRNA and is not caused by variability introduced into the system through variability in miRNA abundances. Variability in miRNA levels due to finite abundances will further increase extrinsic noise in experimental systems beyond the unavoidable timescale effect presented here. While the effect of miRNA on mRNA fluctuations due to “extrinsic” noise has been considered previously (38), we here specifically quantify its subsequent effect on protein levels.

The above conclusions apply not only to extrinsic noise processes as defined in Eq. 9 but also exactly hold for a gene stochastically switching between two transcriptional states

$${\lambda }_{\text{off}}\stackrel{k_{\text{on}}}{\underset{k_{\text{off}}}{\rightleftharpoons }}{\lambda }_{\text{on}}\text{.}$$ (13)

Such a stochastic switching process leads to an extrinsic noise contribution with identical mathematical form in terms of the transcription rate variability and timescale as detailed in appendix B.4.

Theoretical results are consistent with previous experimental observations

The above results show that neither intrinsic nor extrinsic noise will be filtered by miRNA interactions unless miRNA binding actively protects mRNA from degradation. Because miRNA-mRNA complexes are presumed to be degraded at least as much as unbound mRNA (29,30), we thus conclude that miRNAs are not expected to reduce gene expression noise.

While this finding challenges the previous conclusion that “the reduction of intrinsic noise is a generic property of miRNAs,” (11) it does not contradict the previously reported experimental data. That is because the previous conclusion was based on comparing systems with equal abundance but unequal transcription rates (11). In such a comparison, the increase in protein variability due to miRNA interactions is counteracted by a decrease in protein variability due to increased transcription rates as indicated by the solid lines in Fig. 4 A. The experimental observation that systems with miRNA exhibit lower intrinsic noise when conditioned on protein abundance (11) is thus entirely consistent with our result that introducing miRNA increases gene expression noise in a given system with otherwise fixed kinetic parameters as illustrated by Fig. 4 A, which resembles the previously reported experimental plot. Note, in the experiment, different transcription rates are due to differences in copy numbers of the plasmid encoding for the protein of interest. See appendix B.5 for the explicit calculations of how changing transcription and translation rates affect protein abundance and variability.

Figure 4.

Noise increase when introducing miRNA is consistent with previously reported experimental data. (A) Arrows indicate the effect of introducing miRNA for three different systems with respective transcription rates $\lambda ⟨z⟩=\mathrm{1,5,30}$ per protein lifetime and remaining parameters as given in Fig. 2A (red disc) with a translation rate per mRNA $\alpha =100$ per protein lifetime. For any given system, introducing miRNA increases intrinsic protein noise while reducing protein averages. Increasing the transcription rate while leaving all other rate constants unchanged increases protein abundances and decreases protein variability with or without miRNA (solid lines) (see appendix B.5). A previous interpretation of experimental data was based on comparing systems “vertically” in this plot, which compares systems with equal protein abundance but unequal transcription rates. Dashed lines illustrate how changing the translation rate per mRNA can adjust protein abundances to desired levels without affecting intrinsic noise due to stochastic mRNA fluctuations. Plotted is the protein variability due to intrinsic mRNA noise as defined in Eq. 3 in the cellular regime in which $1/⟨p⟩\ll C{V}_{p,\mathrm{int}}^2$ (see appendix B.5 for the general case and explicit calculations). (B) Introducing miRNA increases extrinsic noise in protein levels. Arrows indicate the same hypothetical experimental transition as before (Fig. 3B, left panel, red disc) for transcriptional rate variability of magnitude $C{V}_z=0.2$ and timescale ${\tau }_z={\tau }_p$. For a given average transcription rate, introducing miRNA increases extrinsic protein noise while reducing protein averages. The quantitative details depend on the exact values of parameters, but in the regime $\phi \ge {\beta }_m$, a noise increase is unavoidable even when miRNA themselves do not exhibit any stochastic variability. Solid lines indicate the effect of changing either transcription or translation rates on protein variability due to extrinsic noise in transcription. To see this figure in color, go online.

While our results establish that introducing miRNA into a given system is expected to increase protein variability, they do not establish whether miRNA could be of evolutionary advantage for cells to control noise. The latter question can only be answered by considering the physiological constraints under which a cell must operate. For example, it seems reasonable that cells are required to keep protein abundances at a preferred physiological level. However, this does not constrain mRNA levels because the cellular translation rate per mRNA is subject to adaptation and varies over several orders of magnitude (39,40). In other words, eliminating miRNA interactions from a given system would reduce noise and increase mRNA abundances, which can be ensured to achieve the same protein abundance by lowering the translation rate per mRNA as indicated by dashed lines in Fig. 4 A. Note, such a change will not affect the energetic cost of gene expression, which is determined by the average rates of translation and transcription (40) that remain unchanged in such a scenario.

The above arguments do not consider all possible costs and constraints under which cells operate. For example, on cellular timescales, it might be biochemically impossible for cells to adjust translation rates per mRNA quickly and gene specifically. In such a scenario, cells may be forced to adjust protein abundances without adjusting $\alpha$. As a consequence, mRNA abundances may become constrained such that a comparison between equal mRNA abundances and different transcription rates might become physiologically relevant.

This illustrates that any argument about “the role of miRNA” requires explicit consideration of the constraints under which cells operate. Here, we do not make any claims about “the role of miRNA” but simply point out that introducing miRNA into any given gene expression system is expected to increase noise in the biologically relevant regime. We do not claim that there are no possible scenarios under which adjusting protein abundances through miRNA interactions could be advantageous for cells. However, current analyses have not established evidence for the advantage of utilizing miRNA to achieve low variability in gene expression.

Effect of miRNA on signal transmission

The above results establish that introducing miRNA increases both intrinsic noise due to low-copy-number mRNA as well as extrinsic noise due to transcription rate variability. However, not all transcription rate variability is noise: the desired effect of gene regulation is to adjust protein levels in response to cellular signals.

We next analyze whether miRNA could improve the transmission of an upstream signal through a noisy mRNA channel. Based on the previous sections, we expect both intrinsic noise as well as the response to extrinsic variability to increase upon introduction of miRNA. However, the relative magnitude of the two effects is unknown.

To analyze the effect of miRNA on signal transmission, we consider a generic signal in the form of periodic transcription rate, i.e., in the reactions defined in Eq. 1, we replace the constant transcription rate $\lambda$ with $\lambda \times z\left(t\right)$ where

$$z\left(t\right)=A\left(1+\sin \left(\omega t\right)\right)\text{.}$$ (14)

All moments and correlations for this model of gene expression can still be exactly determined analytically, although the process is nonstandard (see appendix C). The resulting total protein variability can be written as

$$C{V}_{p,\text{tot}}^2=C{V}_{p,\mathrm{int}}^2+C{V}_{p,\text{signal}}^2\text{,}$$ (15)

where the intrinsic noise contribution $C{V}_{p,\mathrm{int}}^2$ is still given by Eq. 3 (and thus increases with the introduction of miRNA in the biologically relevant regime). In the absence of miRNA interactions, the protein variability due to the upstream signal is given by

$$C{V}_{p,\text{signal}}^{2\text{no}\text{miRNA}}=C{V}_z^2\frac{1}{1+{\omega }^2{\tau }_m^2}\frac{1+\frac{{\tau }_p}{{\tau }_m}\frac{1-{\omega }^2{\tau }_m{\tau }_p}{1+{\omega }^2{\tau }_p^2}}{1+\frac{{\tau }_p}{{\tau }_m}}\text{.}$$ (16)

The general result for $C{V}_{p,\text{signal}}$ in the presence of miRNA is algebraically involved and is detailed in appendix C.

Because introducing miRNA can increase both signal and noise, we next consider the ratio of the two. We formally define the signal-to-noise ratio (SNR) as the variability of protein levels due to the time-varying upstream signal $z\left(t\right)$ relative to the protein variability in the absence of any signal variability:

$$\text{SNR}:=\frac{C{V}_{p,\text{signal}}}{C{V}_{p,\mathrm{int}}}\text{.}$$ (17)

In general, many different definitions of the SNR have been utilized (41), and for gene expression variability in particular, other information theoretic measures have been proposed to quantify the fidelity of signal transmission (42).

The general result for the SNR of protein levels subject to periodic driving is algebraically involved and detailed in appendix C. To analyze the effect of miRNA, we first determine the SNR in the absence of miRNA:

$${\text{SNR}}^{\text{no}\text{miRNA}}=\sqrt{\frac{C{V}_z^2⟨m⟩}{1+{\omega }^2{\tau }_m^2}\left(1+\frac{1-{\omega }^2{\tau }_m{\tau }_p}{1+{\omega }^2{\tau }_p^2}\frac{{\tau }_p}{{\tau }_m}\right)}\text{.}$$ (18)

By plotting the SNR relative to its value in the absence of any miRNA interactions given by Eq. 18, we can graphically illustrate the effect of miRNA on the SNR (see Fig. 5 B). We find that in the biologically relevant regime of $\phi \ge {\beta }_m$, introducing miRNA typically decreases the SNR, although this behavior is not strictly general (see supporting material). For example, for high-frequency oscillations varying on timescales faster than protein lifetimes, the SNR can go up, but the effect is small. Red discs in Fig. 5 B correspond to the same experimental transition parameterized in Fig. 2 for periodic driving on three different timescales. These results show that introducing miRNA does not generally increase the fidelity of signal transmission from upstream transcriptional signal to protein levels.

Figure 5.

The effect of miRNA interactions on signal transmission in gene expression. (A) We consider a periodic signal as an example of a deterministically varying transcription rate. (B) Quantifying gene expression fidelity in terms of the signal-to-noise ratio (SNR) as defined in Eq. 17. The effect of miRNA is illustrated through plotting the SNR normalized by its value in the absence of miRNA interactions $\left(\delta =0\right)$. In principle, miRNA interactions can increase or decrease the SNR depending on system parameters. However, the SNR will generally decrease unless the signal varies on very short timescales, which can lead to a marginal increase in the SNR (see supporting material). Arrows indicate the same hypothetical experimental transition as in Figs. 2, 3, and 4 for three different signal periods. To see this figure in color, go online.

Effect of miRNA on mRNA-protein correlations

Previous work (43) established that mRNA-protein correlations must satisfy

$${\rho }_{mp}=\frac{C{V}_p}{C{V}_m}$$ (19)

in any gene expression system in which proteins are made at a rate proportional to mRNA levels and proteins are undergoing first-order degradation as long as $C{V}_p^2\gg 1/⟨p⟩$, which is the typical regime of gene expression. This relationship has been used (43) to rule out a large class of gene expression models given the experimentally observed mRNA-protein correlations (17) because no type of mRNA regulation can affect the above relation. Equation 19 thus directly applies to our models of gene expression with miRNA interactions. However, this theoretical prediction will not be observed experimentally if miRNA-mRNA complexes are not experimentally distinguished (44) from unbound mRNA molecules.

Such an experiment reports a “total amount of mRNA”

$$\overline{m}:=m+c$$ (20)

rather than only the actively translated mRNA. Considering the fraction of this “total mRNA” silenced in miRNA-mRNA complexes

$$F:=\frac{⟨c⟩}{⟨m⟩+⟨c⟩}\text{,}$$ (21)

we find that Eq. 19 does not hold for ${\rho }_{\overline{m},p}$ but instead is replaced with the following inequalities,

$$\frac{C{V}_p}{C{V}_{\overline{m}}}\left(1-F\right)\le {\rho }_{\overline{m},p}\le \frac{F}{2-F}+\frac{C{V}_p}{C{V}_{\overline{m}}}\frac{1-F}{1-F/2}\text{,}$$ (22)

even in the absence of feedback. Experimentally reported mRNA-protein correlations ${\rho }_{\overline{m}p}$ can thus deviate significantly from the “expected” straight line relation for gene expression models with linear translation rates. Such deviations can be significant even for modest fractions of mRNA bound in miRNA-mRNA complexes, as illustrated in Fig. 6 B. Experimentally observed correlations in organisms with a “silent” pool of mRNA may thus not agree with the prediction of Eq. 19 even when translation is perfectly proportional to actively translated mRNA. The bound of Eq. 22 is achievable as illustrated by construction through the example systems depicted in Fig. 6 B. The lower bound is achieved when the lifetime of the miRNA-mRNA complex is much larger than all other timescales in the system. The upper bound is achieved when the complex lifetime is very short and the transcription rate is periodic with high frequencies (see appendix D.2).

Figure 6.

Previously established bounds on mRNA-protein correlations do not apply when free mRNA and miRNA-mRNA complexes are not experimentally distinguished. (A) We consider the impact of experimentally reporting $\overline{m}=m+c$, i.e., the “total amount of mRNA,” which includes both the actively translated mRNA as well as silenced miRNA-mRNA complexes. (B) The black line indicates the expected relation Eq. 19,which applies to the model defined in Eqs. 1 and 2 in the typical cellular regime even for time-varying transcription rates and feedback control. This relationship does not hold for ${\rho }_{\overline{m}p}$ and $C{V}_p/C{V}_{\overline{m}}$. Even in the absence of any feedback, the straight line relation of Eq. 19 expands to the trapezoid described by Eq. 22, illustrated here for $F=0.3$. Example systems for different parameter values demonstrate that periodically driven systems (blue dots) can access the entire trapezoid. Randomizing parameter values over a wide range suggests that stochastically driven systems (purple dots) do not approach the upper bound. See appendix D.2 and supporting material for corresponding parameter regimes. (C) The correlation coefficient ${\rho }_{\overline{m}p}$ can be significantly larger than ${\rho }_{mp}$ when complexes are short lived even though $m$, rather than $\overline{m}$, sets the protein production rate. To see this figure in color, go online.

Interestingly, the “total mRNA,” which includes the silenced complexes, can be more correlated with protein levels than the actively translated pool of mRNA when miRNA-mRNA complexes are short lived. Already for the special case of constant transcription, we find (appendix D.3) that the correlation coefficients violate the naive expectation that ${\rho }_{\overline{m}p}<{\rho }_{mp}$ when the mRNA-miRNA complex is short lived. In particular, we can derive the following asymptotic behavior:

$$\frac{{\rho }_{\overline{m}p}}{{\rho }_{mp}}\sim \{\begin{array}{c}\sqrt{1-F}\text{when}{\tau }_c\gg {\tau }_p\\ \frac{1}{\sqrt{1-F}}\text{when}{\tau }_c\ll {\tau }_p\end{array}$$ (23)

(see Fig. 6 C).

This highlights that correlation coefficients can be misleading when naively interpreted: even in simple three-component systems, we cannot infer “who talks to whom” by comparing correlation coefficients. Indirectly connected molecular components can be more correlated than the molecules through which the components are connected.

Conclusion

Our theoretical analysis clarifies how miRNA interactions affect noise in gene expression. We find that introducing miRNA interactions increases or decreases protein variability depending on the biochemical details of the miRNA-mRNA interactions. However, when miRNAs increase mRNA degradation, as is biologically expected, our analysis predicts that introducing miRNA will always increase stochastic protein variability.

This result is generally consistent with prior experimental observations (11) but challenges the previous interpretation of the data that miRNAs function as a generic noise reduction mechanism in cells (11). This highlights that the interpretation of noise in gene expression data depends crucially on the reference point: conditioning data on average abundances can be misleading because miRNA interactions affect both protein noise as well as abundances. Similar effects are well known in medical statistics. For example, the “birth weight paradox” is caused by inferring the effect of maternal smoking through analyzing infant mortality conditioned on low birth weight (45). This leads to the paradoxical observation that smoking during pregnancy appears to have a protective effect on infants when the exact opposite is true (46). This situation is analogous to our results of Fig. 4 A in which systems with miRNA have lower noise than systems without miRNA when conditioned on average abundances even though introducing miRNA increased protein noise for any given system.

Our analysis further establishes that for cells that are constrained in protein abundances, transcription rates, or translation rates, miRNA does not confer an advantage in noise suppression. Only when cells have to operate at fixed mRNA abundances could miRNA interactions become advantageous from a stochastic noise perspective. The latter scenario could, e.g., be relevant when mechanistic constraints force cells to adjust protein levels on cellular timescales only through mRNA abundances.

On a technical level, we analyze the same model discussed previously (11) but without making a quasi-steady-state assumption for miRNA-mRNA complexes. Our work goes beyond the existing results on protein noise in gene expression models with miRNA (11,22) because we clarify how the competing effects of reducing mRNA lifetimes and abundances lead to a net increase in protein noise. Throughout, we consider the limit to which miRNA abundances themselves do not fluctuate and are approximately constant. This approximation does not affect our conclusion that introducing miRNA into a system increases protein variability because fluctuating miRNA will only further increase protein variability.

Our theoretical analysis is based on a simple model of stochastic gene expression that ignores many relevant effects. Much recent work has shown that feedback control (47,48), cellular growth, cell division, and variability in cell division times (49,50,51,52), and competition for RNA polymerases and ribosomes (53) all significantly affect protein variability. While we cannot exclude the possibility that taking such complications into account will affect our conclusions, our results based on an oversimplified model seem to capture the essence of prior experimental observations. Taken together with the pioneering experimental work of Schmiedel et al. (11), this suggests that in some cases, a basic understanding of noise in gene expression might be achievable through analyzing “toy models” that ignore a lot of details.

All our analytical results are derived using exact methods. To verify that our derivations do not include algebraic errors, we additionally ran numerical simulations for different parameter sets, showing that our analytical results agree with the protein variability observed in numerical simulations (see supporting material).

Author contributions

R.F. and A.H. derived the results and wrote the article.

Editor: Ramon Grima.

Appendices

Appendix A: Protein variability for constant transcription

For chemical reaction systems with linear rates, all moments and temporal correlation functions can be exactly derived from the chemical master equation. In particular, the matrix of normalized (co)variances ${\eta }_{ij}:=\text{Cov}\left(x_i,x_j\right)/\left(⟨x_i⟩⟨x_j⟩\right)$ satisfies the Lyapunov equation (13,23,36,54)

$$M\eta +{\left(M\eta \right)}^T+D=0\text{,}$$ (24)

where $M$ and $D$ for the stochastic reaction system defined by Eqs. 1 and 2 are given by (enumerating the variables as $m=x_1,c=x_2,p=x_3$)

$$M=\left(\begin{array}{ccc}-\frac{1}{{\tau }_m}& \frac{R}{{\tau }_m}& 0\\ \frac{1}{{\tau }_c}& -\frac{1}{{\tau }_c}& 0\\ \frac{1}{{\tau }_p}& 0& -\frac{1}{{\tau }_p}\end{array}\right)$$ (25)

$$D=\left(\begin{array}{ccc}\frac{2}{{\tau }_m⟨m⟩}& -\frac{R}{{\tau }_m⟨c⟩}-\frac{1}{{\tau }_c⟨m⟩}& 0\\ -\frac{R}{{\tau }_m⟨c⟩}-\frac{1}{{\tau }_c⟨m⟩}& \frac{2}{{\tau }_c⟨c⟩}& 0\\ 0& 0& \frac{2}{{\tau }_p⟨p⟩}\end{array}\right)\text{.}$$

For general stochastic processes, $M$ can be determined from the molecular lifetimes and logarithmic sensitivities of the rate functions (13,23). Molecular lifetimes and reaction event step sizes determine the diffusion matrix $D$ (36).

Solving Eq. 24 subject to Eq. 25 then gives the following normalized (co)variances

$$\begin{array}{cc}& {\eta }_{mm}=\frac{1}{⟨m⟩},{\eta }_{cc}=\frac{1}{⟨c⟩},{\eta }_{mc}=0\\ & {\eta }_{mp}=\frac{1}{⟨m⟩}\frac{1}{1+\frac{{\tau }_p}{{\tau }_m}\left(1-\frac{R}{1+{\tau }_c/{\tau }_p}\right)}\\ & {\eta }_{pc}=\frac{1}{⟨m⟩}\frac{1}{1+\frac{{\tau }_p}{{\tau }_m}\left(1-\frac{R}{1+{\tau }_c/{\tau }_p}\right)}\frac{1}{1+\frac{{\tau }_c}{{\tau }_p}}\\ & {\eta }_{pp}=\frac{1}{⟨p⟩}+\frac{1}{⟨m⟩}\frac{1}{1+\frac{{\tau }_p}{{\tau }_m}\left(1-\frac{R}{1+{\tau }_c/{\tau }_p}\right)}\text{,}\end{array}$$ (26)

and thus the result of Eq. 3 in the main text for $C{V}_p^2={\eta }_{pp}$.

Note, obtaining ${\eta }_{pp}$ through Eqs. 24 and 25 is a matter of computational convenience. Alternatively, one can determine all second-order moments directly from the chemical master equation that defines the stochastic process of Eqs. 1 and 2 from first principles.

A.1 Stationary mRNA distribution

Considering only the mRNA $m$ and mRNA-miRNA complexes $c$, the chemical master equation for their joint probability distribution is given by

$$\begin{array}{cc}\frac{dP\left(m,c\right)}{dt}& =-\left(\lambda +{\beta }_{m}m+\delta m+\gamma c+\phi c\right)P\left(m,c\right)\\ & +\lambda P\left(m-1,c\right)+{\beta }_m\left(m+1\right)P\left(m+1,c\right)\\ & +\delta \left(m+1\right)P\left(m+1,c-1\right)\\ & +\gamma \left(c+1\right)P\left(m-1,c+1\right)\\ & +\phi \left(c+1\right)P\left(m,c+1\right).\end{array}$$ (27)

Direct substitution of the Poissonian ansatz

$$P\left(m=n_m,c=n_c\right)=\frac{{⟨m⟩}^{n_m}{e}^{-⟨m⟩}}{n_m!}\frac{{⟨c⟩}^{n_c}{e}^{-⟨c⟩}}{n_c!}\text{,}$$ (28)

with averages

$$⟨m⟩=\frac{\lambda }{\delta \frac{\phi }{\phi +\gamma }+{\beta }_m},⟨c⟩=\frac{\lambda \frac{\delta }{\delta +{\beta }_m}}{\gamma \frac{{\beta }_m}{\delta +{\beta }_m}+\phi }\text{,}$$ (29)

into Eq. 27 leads to $0=dP\left(m,c\right)/dt$. By uniqueness of the stationary state, the stationary-state distribution for mRNA and mRNA-miRNA complexes is thus simply the product of independent Poissonians.

Note, joint probability distributions that are products of independent Poissonians occur more generally: for any system in which reactions only add molecules at a constant rate, remove molecules with a first-order rate or convert molecules from one species to another at a first-order rate; the resulting joint distributions are a product of independent Poissonian distributions (55,56). Reaction networks of this form are known as Jackson networks in the queueing theory literature.

A.2 mRNA autocorrelations

Because our model does not include any feedback, the protein variability due to “upstream” variability of mRNA-levels is determined by

$$\text{Cov}\left(m,p\right)=\text{Var}\left(m\right){\int }_0^{\infty }I\left(t\right)A_m\left(t\right)dt\text{,}$$ (30)

where $I\left(t\right)$ is the impulse response of protein levels to mRNA-levels and $A_m\left(t\right)$ is the normalized autocorrelation of mRNA fluctuations (57), defined as

$$A_m\left(t\right)=\frac{⟨m\left(0\right)m\left(t\right)⟩-{⟨m⟩}^2}{\text{Var}\left(m\right)}\text{.}$$ (31)

Regardless of the mRNA dynamics, protein variability for our model can be written as (43)

$${\eta }_{pp}=\frac{1}{⟨p⟩}+{\eta }_{mp}\text{,}$$ (32)

and we thus obtain Eq. 6 in the main text because the protein impulse response is given by

$$I\left(t\right)=\frac{⟨p⟩}{⟨m⟩}\frac{e^{-t/{\tau }_p}}{{\tau }_p}\text{,}$$ (33)

obtained through solving the system of ordinary differential equations governing protein dynamics for the case of a mRNA Dirac delta input. This can be done via taking the matrix exponential (58)

$$C\left(t\right)=e^{Mt}\text{,}$$ (34)

where $M$ is determined by the logarithmic sensitivities of the rate functions and the lifetimes of the molecules (58).

For our model, $M$ is given by Eq. 25, and solving Eq. 34 leads to the following autocorrelation for mRNA fluctuations in the general $R\ne 0$ case:

$$A_m\left(t\right)=\frac{\left({\mu }_1+\frac{1}{{\tau }_m}\right)e^{{\mu }_{2}t}-\left({\mu }_2+\frac{1}{{\tau }_m}\right)e^{{\mu }_{1}t}}{{\mu }_1-{\mu }_2}\text{,}$$ (35)

where

$${\mu }_{\mathrm{1,2}}=-\frac{1}{2}\left(\frac{1}{{\tau }_m}+\frac{1}{{\tau }_c}\pm \sqrt{{\left(\frac{1}{{\tau }_m}-\frac{1}{{\tau }_c}\right)}^2+\frac{4R}{{\tau }_m{\tau }_c}}\right)\text{.}$$ (36)

In the absence of miRNA when $R=0$, the autocorrelation is simply a single exponential as intuitively expected:

$$A_m^{\ast }\left(t\right)=e^{-t/{\tau }_m}\text{.}$$ (37)

We enumerate the eigenvalues such that ${\mu }_1$ is the larger eigenvalue and thus ${\mu }_2\le -1/{\tau }_m\le {\mu }_1$. This allows us to compare the general autocorrelation to the $R=0$ case as follows:

$$\begin{array}{c}\frac{A_m\left(t\right)}{A_m^{\ast }\left(t\right)}=\frac{\left({\mu }_1+\frac{1}{{\tau }_m}\right)e^{\left({\mu }_2+\frac{1}{{\tau }_m}\right)t}-\left({\mu }_2+\frac{1}{{\tau }_m}\right)e^{\left({\mu }_1+\frac{1}{{\tau }_m}\right)t}}{{\mu }_1-{\mu }_2}\\ \ge \frac{\left({\mu }_1+\frac{1}{{\tau }_m}\right)\left(1+{\mu }_{2}t+\frac{t}{{\tau }_m}\right)-\left({\mu }_2+\frac{1}{{\tau }_m}\right)\left(1+{\mu }_{1}t+\frac{t}{{\tau }_m}\right)}{{\mu }_1-{\mu }_2}\\ =\frac{{\mu }_1-{\mu }_2}{{\mu }_1-{\mu }_2}=1,\end{array}$$ (38)

where the inequality follows because ${\mu }_2+1/{\tau }_m\le 0$ and $e^x\ge 1+x$. This completes the proof of the autocorrelation result presented in the main text.

A.3 Effect of introducing miRNA

Substituting Eq. 7 into Eq. 3 yields

$$\begin{array}{c}{\eta }_{pp}=\frac{\delta \phi +{\beta }_m\left(\phi +\gamma \right)}{\lambda \left(\gamma +\phi \right)}\times \\ \frac{{\beta }_p\left(\gamma +\phi +{\beta }_p\right)}{\delta \phi +{\beta }_m\left(\gamma +\phi +{\beta }_p\right)+{\beta }_p\left(\gamma +\delta +\phi +{\beta }_p\right)}\text{,}\end{array}$$ (39)

in the regime in which $1/⟨p⟩\ll C{V}_p^2$. In the absence of miRNA, this expression becomes

$${\eta }_{pp,\delta =0}=\frac{{\beta }_m}{\lambda }\frac{{\beta }_p}{{\beta }_p+{\beta }_m}\text{.}$$ (40)

The inequality

$${\eta }_{pp}<{\eta }_{pp,\delta =0}$$ (41)

reduces to

$$\gamma {\beta }_m>\phi \left(\gamma +\phi +{\beta }_p\right)\text{,}$$ (42)

as proven through the symbolic software package Wolfram Mathematica (v.11) using the “reduce” command under the assumption that all parameters are positive.

When $\phi \ge {\beta }_m$ the condition of Eq. 42 cannot be satisfied, and noise suppression is impossible. This holds exactly also when the $1/⟨p⟩$ term is not negligible in Eq. 3, as introducing miRNA interactions will decrease protein abundances and thus only further increase protein variability through that term.

Appendix B: Protein variability with stochastically varying transcription rates

B.1 Stationary-state fluctuations

For the stochastically varying transcription rate defined in Eq. 9, the normalized (co)variance matrix satisfies the Lyapunov equation of Eq. 24 for

$$M=\left(\begin{array}{cccc}-\frac{1}{{\tau }_z}& 0& 0& 0\\ \frac{1-R}{{\tau }_m}& -\frac{1}{{\tau }_m}& \frac{R}{{\tau }_m}& 0\\ 0& \frac{1}{{\tau }_c}& -\frac{1}{{\tau }_c}& 0\\ 0& \frac{1}{{\tau }_p}& 0& -\frac{1}{{\tau }_p}\end{array}\right)$$ (43)

$$D=\left(\begin{array}{cccc}\frac{2{\text{CV}}_z^2}{{\tau }_z}& 0& 0& 0\\ 0& \frac{2}{{\tau }_m⟨m⟩}& -\frac{R}{{\tau }_m⟨c⟩}-\frac{1}{{\tau }_c⟨m⟩}& 0\\ 0& -\frac{R}{{\tau }_m⟨c⟩}-\frac{1}{{\tau }_c⟨m⟩}& \frac{2}{{\tau }_c⟨c⟩}& 0\\ 0& 0& 0& \frac{2}{{\tau }_p⟨p⟩}\end{array}\right)\text{,}$$

with ${\tau }_z$ and $C{V}_z^2$ as given in Eq. 10. Solving Eq. 24 subject to Eq. 43 gives the following normalized (co)variances:

$$\begin{array}{l}{\eta }_{mz}=C{V}_z^2\frac{1-R}{1-\frac{R}{1+{\tau }_c/{\tau }_z}+\frac{{\tau }_m}{{\tau }_z}}\\ {\eta }_{pz}=\frac{{\eta }_{mz}}{1+\frac{{\tau }_p}{{\tau }_z}},{\eta }_{cz}=\frac{{\eta }_{mz}}{1+\frac{{\tau }_c}{{\tau }_z}}\\ {\eta }_{mm}=\frac{1}{⟨m⟩}+C{V}_z^2\frac{1-R}{1-\frac{R}{1+{\tau }_c/{\tau }_z}+\frac{{\tau }_m}{{\tau }_z}}\left(1+\frac{R}{1+\frac{{\tau }_m}{{\tau }_c}}\frac{1}{1+\frac{{\tau }_z}{{\tau }_c}}\right)\\ {\eta }_{cc}=\frac{1}{⟨c⟩}+C{V}_z^2\frac{1-R}{1-\frac{R}{1+{\tau }_c/{\tau }_z}+\frac{{\tau }_m}{{\tau }_z}}\left(\frac{1}{1+\frac{{\tau }_c}{{\tau }_z}}\frac{1}{1+\frac{{\tau }_m}{{\tau }_c}}+\frac{1}{1+\frac{{\tau }_c}{{\tau }_m}}\right)\\ {\eta }_{mc}=C{V}_z^2\frac{1-R}{1-\frac{R}{1+{\tau }_c/{\tau }_z}+\frac{{\tau }_m}{{\tau }_z}}\left(\frac{1}{1+\frac{{\tau }_c}{{\tau }_z}}\frac{1}{1+\frac{{\tau }_m}{{\tau }_c}}+\frac{1}{1+\frac{{\tau }_c}{{\tau }_m}}\right)\\ {\eta }_{mp}=\frac{1}{⟨m⟩}\frac{1}{1+\frac{{\tau }_p}{{\tau }_m}\left(1-\frac{R}{1+{\tau }_c/{\tau }_p}\right)}+\frac{1}{1+\frac{{\tau }_p}{{\tau }_m}\left(1-\frac{R}{1+{\tau }_c/{\tau }_p}\right)}\times C{V}_z^2\frac{1-R}{1-\frac{R}{1+{\tau }_c/{\tau }_z}+\frac{{\tau }_m}{{\tau }_z}}(1+\frac{R}{1+\frac{{\tau }_m}{{\tau }_c}}\frac{1}{1+\frac{{\tau }_z}{{\tau }_c}}+\frac{1-R}{1+\frac{{\tau }_p}{{\tau }_z}}\frac{{\tau }_p}{{\tau }_m}\\ +R\frac{{\tau }_p}{{\tau }_m}\left(\frac{1}{1+\frac{{\tau }_m}{{\tau }_c}}\frac{1}{1+\frac{{\tau }_c}{{\tau }_z}}\frac{1}{1+\frac{{\tau }_p}{{\tau }_c}}+\frac{1}{1+\frac{{\tau }_c}{{\tau }_m}}\frac{1}{1+\frac{{\tau }_p}{{\tau }_c}}\right))\\ {\eta }_{pc}=\frac{1}{⟨m⟩}\frac{1}{1+\frac{{\tau }_p}{{\tau }_m}\left(1-\frac{R}{1+{\tau }_c/{\tau }_p}\right)}+\frac{1}{1+\frac{{\tau }_p}{{\tau }_m}\left(1-\frac{R}{1+{\tau }_c/{\tau }_p}\right)}\times C{V}_z^2\frac{1-R}{1-\frac{R}{1+{\tau }_c/{\tau }_z}+\frac{{\tau }_m}{{\tau }_z}}(1+\frac{R}{1+\frac{{\tau }_m}{{\tau }_c}}\frac{1}{1+\frac{{\tau }_z}{{\tau }_c}}+\frac{1-R}{1+\frac{{\tau }_p}{{\tau }_z}}\frac{{\tau }_p}{{\tau }_m}\\ +R\frac{{\tau }_p}{{\tau }_m}\left(\frac{1}{1+\frac{{\tau }_m}{{\tau }_c}}\frac{1}{1+\frac{{\tau }_c}{{\tau }_z}}\frac{1}{1+\frac{{\tau }_p}{{\tau }_c}}+\frac{1}{1+\frac{{\tau }_c}{{\tau }_m}}\frac{1}{1+\frac{{\tau }_p}{{\tau }_c}}\right))\frac{1}{1+\frac{{\tau }_c}{{\tau }_p}}\\ +\frac{C{V}_z^2}{1+\frac{{\tau }_p}{{\tau }_c}}\frac{1-R}{1-\frac{R}{1+{\tau }_c/{\tau }_z}+\frac{{\tau }_m}{{\tau }_z}}\left(\frac{1}{1+\frac{{\tau }_c}{{\tau }_z}}\frac{1}{1+\frac{{\tau }_m}{{\tau }_c}}+\frac{1}{1+\frac{{\tau }_c}{{\tau }_m}}\right)\\ {\eta }_{pp}=\frac{1}{⟨p⟩}+{\eta }_{mp}\text{.}\end{array}$$ (44)

Breaking up the protein noise in contributions from the mRNA and the varying transcription rate as described in Eq. 11 results in Eq. 26 for $C{V}_{p,\mathrm{int}}^2$ and

$$\begin{array}{c}C{V}_{p,\text{ext}}^2=[1-R+\frac{R}{1+\frac{{\tau }_c}{{\tau }_m}}\frac{1+\frac{{\tau }_p}{{\tau }_m}+{\tau }_p/{\tau }_c}{1+{\tau }_p/{\tau }_c}\\ +R\frac{1}{1+\frac{{\tau }_c}{{\tau }_z}}\frac{1}{1+\frac{{\tau }_m}{{\tau }_c}}\frac{1+\frac{{\tau }_p}{{\tau }_m}+{\tau }_p/{\tau }_c}{1+{\tau }_p/{\tau }_c}\\ +\frac{1}{1+\frac{{\tau }_p}{{\tau }_z}}\left(1-R\right)\frac{{\tau }_p}{{\tau }_m}]\\ \times C{V}_z^2\frac{1-R}{1-\frac{R}{1+{\tau }_c/{\tau }_z}+\frac{{\tau }_m}{{\tau }_z}}\frac{1}{1+\frac{{\tau }_p}{{\tau }_m}\left(1-\frac{R}{1+{\tau }_c/{\tau }_p}\right)}\text{.}\end{array}$$ (45)

B.2 Comparing extrinsic noise in systems with equal lifetimes

Comparing extrinsic noise with miRNA interactions (Eq. 45) and without miRNA interactions (Eq. 12) for systems with equal lifetimes shows that the following inequality,

$$C{V}_{p,\text{ext}}<C{V}_{p,\text{ext}}^{\text{no}\text{miRNA}}\text{,}$$ (46)

is equivalent to

$$\begin{array}{c}\left(R-2\right){\tau }_c^3\left({\tau }_z+{\tau }_m\right)\left({\tau }_m+{\tau }_p\right)\left({\tau }_m\left({\tau }_z+{\tau }_p\right)+{\tau }_z{\tau }_p\right)\\ <{\tau }_m({\tau }_c^2({\tau }_z{\tau }_m\left(-\left(R-6\right){\tau }_m{\tau }_p+{\tau }_m^2-\left(R-6\right){\tau }_p^2\right)\\ +{\tau }_z^2\left({\tau }_m+{\tau }_p\right)\left(2{\tau }_m-\left(R-4\right){\tau }_p\right)+{\tau }_m^2{\tau }_p\left({\tau }_m+2{\tau }_p\right))\\ +{\tau }_c{\tau }_m\left(4{\tau }_z{\tau }_m{\tau }_p\left({\tau }_z+{\tau }_p\right)+{\tau }_m^2{\left({\tau }_z+{\tau }_p\right)}^2+\left(4-R\right){\tau }_z^2{\tau }_p^2\right)\\ +{\tau }_z{\tau }_m^2{\tau }_p\left({\tau }_m\left({\tau }_z+{\tau }_p\right)+\left(2-R\right){\tau }_z{\tau }_p\right)).\end{array}$$ (47)

We were unable to prove this inequality analytically, but we confirmed the inequality computationally for ${10}^6$ randomly chosen parameter sets by sampling $R$ uniformly from $\left(\mathrm{0,1}\right)$ and ${\tau }_m,{\tau }_z,{\tau }_c$ log uniformly from $\left({10}^{-3},{10}^3\right)$ relative to ${\tau }_p$. We found that the inequality was satisfied for all numerical values (see supporting material), strongly suggesting that the inequality holds at least over the above (wide) range of relative timescales.

B.3 Effect of introducing miRNA

The extrinsic miRNA noise can be rewritten in terms of the rate constants by substituting Eq. 7 into Eq. 45:

$$\begin{array}{c}C{V}_{p,\text{ext}}^2=-({\tau }_z{\beta }_p\left(\delta +{\beta }_m\right)\left(\frac{\gamma \delta }{\left(\gamma +\phi \right)\left(\delta +{\beta }_m\right)}-1\right)({\left(\gamma +\phi \right)}^3\\ -{\left(\gamma +\phi \right)}^2{\tau }_z^2\left(\gamma +\delta +{\beta }_m+\phi \right)(-\delta \phi -{\beta }_m\left(\gamma +{\beta }_p+\phi \right)\\ -{\beta }_p\left(\gamma +\delta +{\beta }_p+\phi \right))+{\tau }_z\left(\gamma +\delta +{\beta }_m+{\beta }_p+\phi \right)({\left(\gamma +\phi \right)}^3\\ +{\left(\gamma +\phi \right)}^2\left(\delta +{\beta }_m+{\beta }_p\right)+\left(\gamma +\phi \right){\beta }_p\left(\delta +{\beta }_m\right)-\gamma \delta {\beta }_p)\\ +{\left(\gamma +\phi \right)}^2\left(\delta +{\beta }_m+{\beta }_p\right)+{\beta }_p\left(\delta \phi +\left(\gamma +\phi \right){\beta }_m\right)))\\ \times (\left(\gamma +\phi \right)\left({\tau }_z{\beta }_p+1\right)\left(\gamma +\delta +{\beta }_m+\phi \right)(\delta \phi \\ +{\beta }_m\left(\gamma +{\beta }_p+\phi \right)+{\beta }_p\left(\gamma +\delta +{\beta }_p+\phi \right))({\tau }_z(\gamma +\delta \\ +{{\tau }_z\left(\delta \phi +\left(\gamma +\phi \right){\beta }_m\right)+{\beta }_m+\phi )+1))}^{-1}.\end{array}$$ (48)

For $\phi ={\beta }_m$, we established that Eq. 48 always increases for $\delta >0$ relative to $\delta =0$ using the “reduce” command of Wolfram Mathematica (v.11) under the assumption that all parameters are positive.

For $\phi \ne \beta$, we were unable to establish the general behaviour of Eq. 48 analytically. However, we numerically established that Eq. 48 cannot decrease with $\delta$ when $\phi >{\beta }_m$. We did so by evaluating Eq. 48 for ${10}^6$ randomly chosen parameter sets, i.e., we sampled $R$ uniformly from $\left(\mathrm{0,1}\right)$ and ${\beta }_m,\gamma ,\phi ,\delta ,{\tau }_z$ log uniformly from $\left({10}^{-3},{10}^3\right)$ relative to ${\tau }_p$. We found that Eq. 48 was always larger for $\delta >0$ relative to $\delta =0$ for all numerical values (supporting material), strongly suggesting that the inequality holds at least over the above (wide) range of relative timescales. In contrast, for $\phi <{\beta }_m$, we numerically found that it is possible for the extrinsic noise to decrease (see supporting material).

B.4 Genes with on-off transcriptional states

Instead of the transcription rate defined in Eq. 13, we consider an on-off model for the transcription rate given by

$${\lambda }_{\text{off}}\stackrel{k_{\text{on}}}{\underset{k_{\text{of}f}}{\rightleftharpoons }}{\lambda }_{\text{on}}\text{.}$$ (49)

Standard techniques (13) applied to the master equation of the system defined in Eqs. 1 and 2 subject to Eq. 49 show that the (co)variance matrix for this system satisfies a Lyapunov equation with exactly the same matrices as defined in Eq. 43 with the only difference that ${\tau }_z$ and $C{V}_z^2$ are replaced by ${\tau }_{\lambda }=1/\left(k_{\text{on}}+k_{\text{off}}\right)$ and $C{V}_{\lambda }^2=k_{\text{off}}/k_{\text{on}}$, which describe the magnitude and timescale of the stochastic switching process defined in Eq. 49. All results of the previous section thus directly generalize to genes with upstream fluctuations due to switching between discrete on-off states.

B.5 Changing transcription or translation rates while fixing all other rate constants

The general protein abundance is given by

$$⟨p⟩=\frac{\alpha }{{\beta }_p}⟨m⟩$$

for the mRNA abundance

$$⟨m⟩=\frac{\lambda }{{\beta }_m+\frac{\delta \phi }{\gamma +\phi }}⟨z⟩\text{.}$$

Protein abundances thus change linearly with $\lambda$ or $\alpha$ when all other rate constants are kept fixed.

For simplicity, we first consider protein variability due to intrinsic fluctuations given by Eq. 3 when $C{V}_{p,\mathrm{int}}\gg 1/⟨p⟩$. In that regime, all variables in Eq. 3 are independent of the translation rate $\alpha$. Adjusting the translation rate thus leaves the protein variability unchanged while changing abundances, corresponding to the dashed lines in Fig. 4 A, which illustrate that systems without miRNA interactions exhibit lower noise than those with miRNA interactions when compensating their higher mRNA abundance with a decreased translation rate per mRNA such that the protein abundance agrees between systems with and without miRNA. The same argument also holds when $1/⟨p⟩$ is not negligible, and decreasing the translation rate will increase $C{V}_{p,\mathrm{int}}$. That is because when adjusting $\alpha$ until the two systems have equal protein abundance, the $1/⟨p⟩$ term in Eq. 3 cancels out, and only the $1/⟨m⟩$ term matters, which is independent of $\alpha$ and larger for systems with miRNA when leaving all other rate constants unchanged (in the regime of interest $\phi \ge {\beta }_m$).

Increasing the transcription rate affects $⟨p⟩,⟨m⟩$ linearly. $C{V}_{p,\mathrm{int}}$ given by Eq. 3 thus decreases with ${\lambda }^{-1}\sim {⟨p⟩}^{-1}$, as illustrated by the solid lines in Fig. 4 A.

The extrinsic noise contribution $C{V}_{p,\text{ext}}$ given by Eq. 45 is independent of both $\alpha$ and $\lambda$, as illustrated by the solid lines in Fig. 4 B.

Appendix C: Protein variability with periodic upstream transcription rates

For systems with the periodic upstream signal defined in Eq. 14, we cannot directly apply the standard approach to determine the (co)variance matrix from the Lyapunov equation. Instead, we utilize the fluctuation balance equations that must be satisfied by any pairs of components X ${}_i$ and X ${}_j$ in a system with stationary probability distributions (36)

$$\text{Cov}\left(x_i,R_j^{-}-R_j^{+}\right)+\text{Cov}\left(x_j,R_i^{-}-R_i^{+}\right)=\sum _{k=1}^N{s}_{ki}{s}_{kj}⟨r_k⟩\text{,}$$ (50)

where $x_i$ is the abundance of molecule X${}_i$, $R_j^{\pm }$ is the total flux of production or degradation for molecule X${}_j$, and $⟨r_k⟩$ is the average reaction rate for reaction $k$ in which levels of X${}_i$ are changed by $s_{ki}$.

A system subject to a periodically varying transcriptional input will not reach stationarity. However, we can determine the (co)variances of the system when considering time averages over multiples of one period (36). Intuitively, such (co)variances correspond to population averages over nonasynchronously oscillating cells.

Using the balance relations of Eq. 50, one can solve the model defined in Eqs. 1 and 2 for an arbitrary transcription rate, which yields the following covariance equations:

$$\begin{array}{c}{\eta }_{mm}=\frac{1}{⟨m⟩}+R{\eta }_{mc}+\left(1-R\right){\eta }_{mz}\\ {\eta }_{cc}=\frac{1}{⟨c⟩}+{\eta }_{mc}\\ {\eta }_{mc}=\frac{{\eta }_{cz}}{1+\frac{{\tau }_m}{{\tau }_c}}+\frac{{\eta }_{mz}}{1+\frac{{\tau }_c}{{\tau }_m}}\\ {\eta }_{mp}=\frac{{\eta }_{mm}+{\eta }_{pz}\left(1-R\right)\frac{{\tau }_p}{{\tau }_m}+R\frac{{\tau }_p}{{\tau }_m}\left(\frac{{\eta }_{mc}}{1+{\tau }_p/{\tau }_c}\right)}{1+\frac{{\tau }_p}{{\tau }_m}\left(1-\frac{R}{1+{\tau }_c/{\tau }_p}\right)}\\ {\eta }_{pc}=\frac{{\eta }_{mp}}{1+\frac{{\tau }_c}{{\tau }_p}}+\frac{{\eta }_{mc}}{1+\frac{{\tau }_p}{{\tau }_c}}\\ {\eta }_{pp}=\frac{1}{⟨p⟩}+{\eta }_{mp}.\end{array}$$ (51)

For the periodically varying transcription rate, we can determine the covariances ${\eta }_{mz},{\eta }_{pz},{\eta }_{cz}$ from the impulse response functions through

$$\text{Cov}\left(x_i,z\right)=\text{Var}\left(z\right){\int }_0^{\infty }{I}_{x_i}\left(t\right)A_z\left(t\right)dt$$ (52)

for $m=x_1,c=x_2,p=x_3$. The impulse response functions can be obtained through solving the system of ordinary differential equations that determines the time-dependent ensemble averages conditioned on the history of the transcriptional input $z\left(t\right)$ defined as (59)

$$\tilde{m}\left(t\right):=⟨m|z\left[-\infty ,t\right]⟩\text{,}$$ (53)

$$\tilde{c}\left(t\right):=⟨c|z\left[-\infty ,t\right]⟩,$$ (54)

$$\tilde{p}\left(t\right):=⟨p|z\left[-\infty ,t\right]⟩\text{,}$$ (55)

where angular brackets denote ensemble averages at a given time $t$. For a transcriptional impulse in the form of a Dirac delta function $D_{\text{Dirac}}\left(t\right)$, their dynamics obey

$$\begin{array}{c}\frac{d\tilde{m}\left(t\right)}{dt}=D_{\text{Dirac}}\left(t\right)-\left({\beta }_m+\delta \right)\tilde{m}\left(t\right)+\gamma \tilde{c}\left(t\right)\\ \frac{d\tilde{c}\left(t\right)}{dt}=\delta \tilde{m}\left(t\right)-\left(\gamma +\phi \right)\tilde{c}\left(t\right)\end{array}$$ (56)

$$\frac{d\tilde{p}\left(t\right)}{dt}=\alpha \tilde{m}\left(t\right)-{\beta }_p\left(t\right)\tilde{p}\text{.}$$ (57)

In the absence of miRNA interactions in the system, the covariances are given by

$$\begin{array}{c}{\eta }_{mz}=\frac{1}{2}\frac{1}{1+{\omega }^2{\tau }_m^2}\\ {\eta }_{pz}=\frac{1}{2}\frac{1-{\omega }^2{\tau }_m{\tau }_p}{\left(1+{\omega }^2{\tau }_m^2\right)\left(1+{\omega }^2{\tau }_p^2\right)}\text{,}\end{array}$$ (58)

and the upstream contribution to the protein noise is

$$C{V}_{p,\text{signal}}^2=\frac{1}{2}\frac{1}{1+{\omega }^2{\tau }_m^2}\frac{1+\frac{1-{\omega }^2{\tau }_m{\tau }_p}{1+{\omega }^2{\tau }_p^2}\frac{{\tau }_p}{{\tau }_m}}{1+\frac{{\tau }_p}{{\tau }_m}}\text{.}$$ (59)

The addition of miRNA complicates things significantly. The impulse response functions are given by

$$\begin{array}{c}{I}_m\left(t\right)=\frac{⟨m⟩}{A}\frac{1-R}{{\tau }_m}\frac{\left(\left({\mu }_1+\frac{1}{{\tau }_m}\right)e^{{\mu }_{2}t}-\left({\mu }_2+\frac{1}{{\tau }_m}\right)e^{{\mu }_{1}t}\right)}{{\mu }_1-{\mu }_2}\\ I_p\left(t\right)=\frac{⟨p⟩}{A}\frac{1-R}{{\tau }_p{\tau }_m}(\frac{{\mu }_2+\frac{1}{{\tau }_m}}{\left({\mu }_1-{\mu }_2\right)\left({\mu }_1+\frac{1}{{\tau }_p}\right)}\left(e^{-t/{\tau }_p}-e^{{\mu }_{1}t}\right)\\ -\frac{{\mu }_1+\frac{1}{{\tau }_m}}{\left({\mu }_1-{\mu }_2\right)\left({\mu }_2+\frac{1}{{\tau }_p}\right)}\left(e^{-t/{\tau }_p}-e^{{\mu }_{2}t}\right))\\ I_c\left(t\right)=\frac{⟨c⟩}{A}\frac{1-R}{{\tau }_c{\tau }_m}(\frac{{\mu }_2+\frac{1}{{\tau }_m}}{\left({\mu }_1-{\mu }_2\right)\left({\mu }_1+\frac{1}{{\tau }_c}\right)}\left(e^{-t/{\tau }_c}-e^{{\mu }_{1}t}\right)\\ -\frac{{\mu }_1+\frac{1}{{\tau }_m}}{\left({\mu }_1-{\mu }_2\right)\left({\mu }_2+\frac{1}{{\tau }_c}\right)}\left(e^{-t/{\tau }_c}-e^{{\mu }_{2}t}\right))\text{,}\end{array}$$ (60)

with ${\mu }_1,{\mu }_2$ as defined in Eq. 36. Substituting the impulse responses of Eq. 60 into Eq. 52 then yields

$$\begin{array}{c}{\eta }_{mz}=\frac{1-R}{{\tau }_m}\frac{1}{2}\left(\frac{{\mu }_2+\frac{1}{t_m}}{{\mu }_1-{\mu }_2}\frac{{\mu }_1}{{\mu }_1^2+{\omega }^2}-\frac{{\mu }_1+\frac{1}{t_m}}{{\mu }_1-{\mu }_2}\frac{{\mu }_2}{{\mu }_2^2+{\omega }^2}\right)\\ {\eta }_{pz}=\frac{1-R}{2{\tau }_p{\tau }_m}(\frac{{\mu }_2+\frac{1}{{\tau }_m}}{\left({\mu }_1-{\mu }_2\right)\left({\mu }_1+\frac{1}{{\tau }_p}\right)}\left(\frac{1/{\tau }_p}{1/{\tau }_p^2+{\omega }^2}+\frac{{\mu }_1}{{\mu }_1^2+{\omega }^2}\right)\\ -\frac{{\mu }_1+\frac{1}{{\tau }_m}}{\left({\mu }_1-{\mu }_2\right)\left({\mu }_2+\frac{1}{{\tau }_p}\right)}\left(\frac{1/{\tau }_p}{1/{\tau }_p^2+{\omega }^2}+\frac{{\mu }_2}{{\mu }_2^2+{\omega }^2}\right))\\ {\eta }_{cz}=\frac{1-R}{2{\tau }_c{\tau }_m}(\frac{{\mu }_2+\frac{1}{{\tau }_m}}{\left({\mu }_1-{\mu }_2\right)\left({\mu }_1+\frac{1}{{\tau }_c}\right)}\left(\frac{1/{\tau }_c}{1/{\tau }_c^2+{\omega }^2}+\frac{{\mu }_1}{{\mu }_1^2+{\omega }^2}\right)\\ -\frac{{\mu }_1+\frac{1}{{\tau }_m}}{\left({\mu }_1-{\mu }_2\right)\left({\mu }_2+\frac{1}{{\tau }_c}\right)}\left(\frac{1/{\tau }_c}{\frac{1}{{\tau }_c^2}+{\omega }^2}+\frac{{\mu }_2}{{\mu }_2^2+{\omega }^2}\right))\text{,}\end{array}$$ (61)

where we have made use of ${\eta }_{zz}=1/2$, which is the variability of the periodic signal of Eq. 14. The (co)variances of mRNA, miRNA-mRNA complexes, and protein can be obtained by substituting Eq. 61 into Eq. 51. In particular, we find the protein variability due to the periodic signal is given by

$$\begin{array}{c}C{V}_{p,\text{signal}}^2=\frac{1}{2}\frac{1-R}{{\tau }_m}\frac{1}{1+\frac{{\tau }_p}{{\tau }_m}\left(1-\frac{R}{1+{\tau }_c/{\tau }_p}\right)}\times \\ [1-R+\frac{R}{1+\frac{{\tau }_c}{{\tau }_m}}\frac{1+\frac{{\tau }_p}{{\tau }_m}+{\tau }_p/{\tau }_c}{1+{\tau }_p/{\tau }_c}\\ \times \left(\frac{{\mu }_2+\frac{1}{t_m}}{{\mu }_1-{\mu }_2}\frac{{\mu }_1}{{\mu }_1^2+{\omega }^2}-\frac{{\mu }_1+\frac{1}{t_m}}{{\mu }_1-{\mu }_2}\frac{{\mu }_2}{{\mu }_2^2+{\omega }^2}\right)\\ +\frac{1}{{\tau }_p}(\frac{{\mu }_2+\frac{1}{{\tau }_m}}{\left({\mu }_1-{\mu }_2\right)\left({\mu }_1+\frac{1}{{\tau }_p}\right)}\left(\frac{1/{\tau }_p}{1/{\tau }_p^2+{\omega }^2}+\frac{{\mu }_1}{{\mu }_1^2+{\omega }^2}\right)\\ -\frac{{\mu }_1+\frac{1}{{\tau }_m}}{\left({\mu }_1-{\mu }_2\right)\left({\mu }_2+\frac{1}{{\tau }_p}\right)}\left(\frac{1/{\tau }_p}{1/{\tau }_p^2+{\omega }^2}+\frac{{\mu }_2}{{\mu }_2^2+{\omega }^2}\right))\\ \times \left(1-R\right)\frac{{\tau }_p}{{\tau }_m}+\frac{R}{1+\frac{{\tau }_m}{{\tau }_c}}\frac{1}{{\tau }_c}\frac{1+\frac{{\tau }_p}{{\tau }_m}+{\tau }_p/{\tau }_c}{1+{\tau }_p/{\tau }_c}\times \\ (\frac{{\mu }_2+\frac{1}{{\tau }_m}}{\left({\mu }_1-{\mu }_2\right)\left({\mu }_1+\frac{1}{{\tau }_c}\right)}\left(\frac{1/{\tau }_c}{1/{\tau }_c^2+{\omega }^2}+\frac{{\mu }_1}{{\mu }_1^2+{\omega }^2}\right)\\ -\frac{{\mu }_1+\frac{1}{{\tau }_m}}{\left({\mu }_1-{\mu }_2\right)\left({\mu }_2+\frac{1}{{\tau }_c}\right)}\left(\frac{1/{\tau }_c}{1/{\tau }_c^2+{\omega }^2}+\frac{{\mu }_2}{{\mu }_2^2+{\omega }^2}\right))]\text{.}\end{array}$$ (62)

Appendix D: Correlations between protein levels and “total” mrna pool

For $\overline{m}=m+c$, the definition of correlations implies that

$${\rho }_{\overline{m}p}={\rho }_{mp}\frac{1+\frac{\text{Cov}\left(c,p\right)}{\text{Cov}\left(m,p\right)}}{\sqrt{1+2\frac{\text{Cov}\left(m,c\right)}{\text{Var}\left(m\right)}+\frac{\text{Var}\left(c\right)}{\text{Var}\left(m\right)}}}\text{.}$$ (63)

In the absence of feedback, we can consider the time-dependent average values of the stochastic variables conditioned on the history of the upstream variability (59) as defined in Eq. 55. These conditional averages obey

$$\begin{array}{c}\frac{d\stackrel{\sim }{m}\left(t\right)}{dt}=\lambda \times z\left(t\right)-\left({\beta }_m+\delta \right)\stackrel{\sim }{m}\left(t\right)+\gamma \stackrel{\sim }{c}\left(t\right)\\ \frac{d\stackrel{\sim }{c}\left(t\right)}{dt}=\delta \stackrel{\sim }{m}\left(t\right)-\left(\gamma +\phi \right)\stackrel{\sim }{c}\left(t\right)\end{array}$$ (64)

$$\frac{d\tilde{p}\left(t\right)}{dt}=\alpha \tilde{m}\left(t\right)-{\beta }_p\left(t\right)\tilde{p}\text{.}$$ (65)

Taking time averages of Eq. 64 multiplied with $\tilde{m}\left(t\right),\tilde{c}\left(t\right)$, and $\tilde{p}\left(t\right)$ then yields the following (co)variances for the conditional averages:

$$\begin{array}{c}{\eta }_{\tilde{m}\tilde{m}}=R{\eta }_{\tilde{m}\tilde{c}}+\left(1-R\right){\eta }_{\tilde{m}z}\\ {\eta }_{\tilde{c}\tilde{c}}={\eta }_{\tilde{m}\tilde{c}}\\ {\eta }_{\tilde{m}\tilde{c}}=\frac{{\eta }_{cz}}{1+\frac{{\tau }_m}{{\tau }_c}}+\frac{{\eta }_{\tilde{m}z}}{1+\frac{{\tau }_c}{{\tau }_m}}\\ {\eta }_{\tilde{m}\tilde{p}}=\frac{{\eta }_{\tilde{m}\tilde{m}}+{\eta }_{pz}\left(1-R\right)\frac{{\tau }_p}{{\tau }_m}+R\frac{{\tau }_p}{{\tau }_m}\left(\frac{{\eta }_{\tilde{m}c}}{1+{\tau }_p/{\tau }_c}\right)}{1+\frac{{\tau }_p}{{\tau }_m}\left(1-\frac{R}{1+{\tau }_c/{\tau }_p}\right)}\\ {\eta }_{\tilde{p}\tilde{c}}=\frac{{\eta }_{\tilde{m}\tilde{p}}}{1+\frac{{\tau }_c}{{\tau }_p}}+\frac{{\eta }_{\tilde{m}\tilde{c}}}{1+\frac{{\tau }_p}{{\tau }_c}}\\ {\eta }_{\tilde{p}\tilde{p}}={\eta }_{\tilde{m}\tilde{p}}.\end{array}$$ (66)

Note, these covariance equations hold for any time-varying transcription rate in the absence of feedback. Because the upstream variability $z\left(t\right)$ is unspecified, the covariances ${\eta }_{\tilde{m}z}$, ${\eta }_{\tilde{c}z}$, and ${\eta }_{\tilde{p}z}$ are unknown, and the problem is underdetermined. However, we can utilize Eq. 66 to constrain the correlations of our model system for all possible upstream variability. Furthermore, since our model is completely linear with an additive upstream variability, we have (59)

$$\text{Cov}\left(\tilde{m},z\right)=\text{Cov}\left(m,z\right)$$ (67)

and similarly $\text{Cov}\left(\stackrel{\sim }{p},z\right)\text{=Cov}\left(p,z\right),\text{Cov}\left(\stackrel{\sim }{c},z\right)\text{=Cov}\left(c,z\right)$.

We consider ${\rho }_{\overline{m}p}$ in the biologically relevant regime where protein abundances are large, and their intrinsic noise is negligible compared with the low-copy-number noise of mRNAs and variability of transcription dynamics. In that regime,

$$\begin{array}{c}{\rho }_{\overline{m}p}=\frac{\text{Cov}\left(\overline{m},p\right)}{\sqrt{\text{Cov}\left(\overline{m},\overline{m}\right)\text{Cov}\left(p,p\right)}}\\ =\frac{C{V}_p}{C{V}_{\overline{m}}}\frac{⟨p⟩}{⟨\overline{m}⟩}\left(\frac{\text{Cov}\left(m,p\right)+\text{Cov}\left(p,c\right)}{\text{Cov}\left(p,p\right)}\right)\\ =\frac{C{V}_p}{C{V}_{\overline{m}}}\left(1-F+F\frac{{\eta }_{pc}}{{\eta }_{mp}}\right).\end{array}$$ (68)

Constraining ${\rho }_{\overline{m}p}$ then boils down to finding the minimum value for ${\eta }_{pc}$.

D.1 Bounds on correlations

The positivity of variances in Eq. 66 implies that ${\eta }_{\tilde{m}\tilde{p}},{\eta }_{\tilde{m}\tilde{c}}>0$ and thus ${\eta }_{pc}\ge 0$, which establishes the the lower bound of Eq. 68:

$${\rho }_{\overline{m}p}\ge \frac{C{V}_p}{C{V}_{\overline{m}}}\left(1-F\right)\text{,}$$ (69)

which is presented in Eq. 22 of the main text.

The proof for the upper bound in Eq. 22 is more involved. To proceed, we first consider the regime in which all abundances are large, which leads to the stochastic covariance equations equivalent to Eq. 66. From the definition given in Eq. 63, one can show via substituting in Eq. 51 and using the definition of the correlation coefficient that

$${\rho }_{\overline{m}p}=\frac{C{V}_p}{C{V}_{\overline{m}}}\left(1-F\right)+\frac{{\rho }_{mc}{\rho }_{pc}F}{\sqrt{{\left(1-F\right)}^2+\left(2F-F^2\right){\rho }_{mc}^2}}\text{.}$$ (70)

Because any possible correlation matrix must be positive semidefinite, we have

$${\rho }_{pc}\le {\rho }_{mp}{\rho }_{mc}+\sqrt{\left(1-{\rho }_{mp}^2\right)\left(1-{\rho }_{mc}^2\right)}\text{.}$$ (71)

Maximizing ${\rho }_{\overline{m}p}$ for a given $C{V}_p/C{V}_{\overline{m}}$ subject to the above inequality and the identity

$${\rho }_{mp}=\frac{C{V}_p}{C{V}_{\overline{m}}}\sqrt{{\left(1-F\right)}^2+\left(2F-F^2\right){\rho }_{mc}^2}$$ (72)

yields

$${\rho }_{\overline{m}p}\le \frac{F}{2-F}+\frac{C{V}_p}{C{V}_{\overline{m}}}\frac{1-F}{1-F/2}\text{.}$$ (73)

If the abundances are finite, then ${\rho }_{\overline{m}p},C{V}_p/C{V}_{\overline{m}}$ decrease by the same factor, so they obey the same limitations. For the details of the mathematical manipulations and the optimization, see supporting material.

D.2 Achieving the bounds

The dots in Fig. 6 B denote example systems that correspond to a density-adjusted subset of random parameter values to illustrate the range that is achievable by systems. To obtain those systems, we considered our model with periodic upstream signal defined in Eq. 14 and sampled $R$ uniformly from $\left(\mathrm{0,1}\right)$, $\overline{m},p,{\tau }_m,{\tau }_p,{\tau }_c$ log uniformly from $\left({10}^{-2},{10}^2\right)$, and $A,\omega$ uniformly from $\left(\mathrm{0,100}\right)$, with $F=0.3$ and without transcriptional bursts. For systems with the stochastic upstream variability as defined in Eq. 9, we sampled $R$ uniformly from $\left(\mathrm{0,1}\right)$, and $\overline{m},p,z,{\tau }_m,{\tau }_p,{\tau }_c,{\tau }_z$ log uniformly from $\left({10}^{-2},{10}^2\right)$, with $\lambda =1$, $F=0.3$ without transcriptional bursts.

For this range of tested parameters, we did not find stochastic systems that approached the upper bound of Eq. 22. In contrast, the upper bound of Eq. 22 was approached by systems with periodic driving at high frequencies and short-lived miRNA-mRNA complexes. See supporting material for more detailed numerical evidence of this fact.

For either type of transcriptional variability, the lower bound of Eq. 22 is achieved for long-lived mRNA-miRNA complexes. This can be shown by considering Eq. 51, which applies to both stochastic and periodic transcriptional variability and implies that ${\eta }_{pc}\to {\eta }_{cz}$ when ${\tau }_c\gg {\tau }_m,{\tau }_p$. Furthermore, for stochastically varying transcription rates, ${\eta }_{cz}\to 0$ when ${\tau }_c\gg {\tau }_z$, which follows from Eq. 44. Analogously, for periodically varying transcription rates, ${\eta }_{cz}\to 0$ when ${\tau }_c\gg 1/\omega$, which follows from Eq. 61. In either case, we thus approach ${\eta }_{pc}=0$, which corresponds to the lower bound of Eq. 68.

D.3 Ratio of correlations

For our miRNA model with a constant mRNA production rate, we have

$$\text{Cov}\left(m,c\right)=0,\frac{\text{Cov}\left(c,p\right)}{\text{Cov}\left(m,p\right)}=\frac{⟨c⟩}{⟨m⟩}\frac{{\eta }_{pc}}{{\eta }_{mp}}=\frac{⟨c⟩}{⟨m⟩}\frac{1}{1+\frac{{\tau }_c}{{\tau }_y}}\text{,}$$ (74)

which allows us to rewrite Eq. 63 as

$$\frac{{\rho }_{\overline{m}p}}{{\rho }_{mp}}=\sqrt{1-F}+\frac{F}{\sqrt{1-F}}\frac{1}{1+\frac{{\tau }_c}{{\tau }_p}}\text{,}$$ (75)

where we have substituted the definition of $F$ as given in Eq. 21 of the main text. Equation 75 is minimized for ${\tau }_c\gg {\tau }_p$ when the ratio becomes $\sqrt{1-F}$ and maximized for ${\tau }_c\ll {\tau }_p$ when the ratio becomes $1/\sqrt{1-F}$, which proves Eq. 23 of the main text.