IEEE TRANSACTIONS ON NANOBIOSCIENCE 1 Negative feedback through mRNA provides the best control of gene-expression noise

Abhyudai Singh Member, IEEE

Abstract—Genetically identical cell populations exposed to the same environment can exhibit considerable cell-to-cell variation in the levels of specific proteins. This variation or expression noise arises from the inherent stochastic nature of biochemical reactions that constitute gene-expression. Negative feedback loops are common motifs in gene networks that reduce expression noise and intercellular variability in protein levels. Using stochastic models of gene expression we here compare different feedback architectures in their ability to reduce stochasticity in protein levels. A mathematically controlled comparison shows that in physiologically relevant parameter regimes, feedback regulation through the mRNA provides the best suppression of expression noise. Consistent with our theoretical results we find negative feedback loops though the mRNA in essential eukaryotic genes, where feedback is mediated via intron-derived microRNAs. Finally, we find that contrary to previous results, protein mediated translational regulation may not always provide significantly better noise suppression than protein mediated transcriptional regulation.

Index Terms—Gene-expression noise, negative feedback, noise suppression, microRNAs, linear noise approximation !

Protein 1 INTRODUCTION He inherent probabilistic nature of biochemical re- T actions that constitute gene-expression together with II Translation low copy numbers of mRNAs can lead to large stochastic IV fluctuations in protein levels [1], [2], [3]. Intercellular I variability in protein levels generated by these stochastic mRNA fluctuations is often referred to as gene-expression noise. III Increasing evidence suggests that gene-expression noise Transcription can be detrimental for the functioning of essential and housekeeping proteins whose levels have to be tightly maintained within certain bounds for optimal performance Promoter Gene [4], [5], [6]. Moreover, many diseased states have been attributed to an increase in expression noise in particular Fig. 1. The process of gene-expression where mRNAs genes [7], [8], [9]. Given that stochasticity in protein levels are transcribed from the gene and proteins are trans- can have significant effects on biological function and phe- lated from individual mRNAs (red arrows). Different notype, cells actively use different regulatory mechanisms feedback mechanisms in gene-expression where the to minimize expression noise [10], [11], [12], [13], [14], rate of transcription or translation is dependent on the [15], [16]. mRNA or protein count (dashed lines). Negative feedback loops are key regulatory motifs within cells that help reduce stochasticity in protein levels. A com- mon and well characterized negative feedback mechanism sophisticated negative feedback loops where the protein is protein mediated transcriptional regulation where the inhibits the translation of its own mRNA [29], [30] or protein expressed from a gene inhibits its own transcription mRNA inhibits the transcription of its gene [31], [32]. We [17], [18], [19], [20]. For example, it is estimated that here compare and contrast the noise suppression ability of over 40% of Escherichia coli transcription factors regulate these different feedback mechanisms in gene-expression. their own expression through this feedback mechanism [21]. Both theoretical and experimental studies have shown Gene-expression is typically modeled by assuming that that such a negative feedback at the transcriptional level mRNA transcription and protein translation from individual reduces noise in protein numbers [22], [23], [24], [25], [26], mRNAs occurs at fixed constant rates. Feedback mecha- [27], [28]. Recent work has provided evidence of more nisms can be incorporated in this model by assuming that the transcriptional rate or translation rate is a monotonically • A. Singh is with the Department of Electrical and Computer Engineer- decreasing function of either the protein count or the mRNA ing, University of Delaware, Newark, DE 19716. count. This procedure results in four different negative E-mail: [email protected] feedback architectures, which are illustrated in Figure 1. For example, feedback architecture I corresponds to protein IEEE TRANSACTIONS ON NANOBIOSCIENCE 2

TABLE 1 Frequency of different expression/degradation events and the corresponding reset maps.

Event Reset in population count Probability event will occur in (t,t + dt]

Transcription m(t) → m(t) + B kmdt

mRNA degradation m(t) → m(t) − 1 γmm(t)dt

protein translation p(t) → p(t) + 1 kpm(t)dt

protein degradation p(t) → p(t) − 1 γp p(t)dt mediated transcriptional regulation where the transcription Moreover, whenever a particular event occurs, the mRNA rate is a decreasing function of the protein count. Similarly, and protein population count is reset accordingly. Let m(t) feedback architecture IV corresponds to a scenario where and p(t) denote the number of molecules of the mRNA and the protein translation rate per mRNA is a decreasing protein at time t, respectively. Then, the reset in m(t) and function of the mRNA count. p(t) for different gene-expression and degradation events We derive analytical expressions for the protein noise is shown in the second column of Table 1. The frequency levels for each of these different feedback architectures. with which different events occur is determined by the Using these expressions we determine which feedback third column of Table 1, which lists the probability that provides the best noise suppression, and how does its a particular event will occur in the next infinitesimal time performance depend on gene-expression parameters such interval (t,t + dt]. as mRNA and protein half-life. It is important to point out that comparisons between different feedback architectures To quantify noise in protein levels we first write the are done keeping the mean protein and mRNA count differential equations that describe the time evolution of fixed. Furthermore, we assume that different feedbacks the different statistical moments of the mRNA and protein also have the same feedback strength, which is measured count. The moment dynamics can be obtained using the by the sensitivity of the transcription/translation rate to following result: For the above gene-expression model, the the mRNA/protein count. Such a form of comparison is time-derivative of the expected value of any differentiable also referred to in literature as a mathematically controlled function ϕ(m, p) is given by equation (2) [37], [38]. Here, comparison [33]. and in the sequel we use the symbol h.i to denote the The paper is organized as follows: In Section 2 we expected value. Using (2) with appropriate choices for quantify the extent of stochasticity in protein levels in a ϕ(m, p) we obtain the following moment dynamics: gene-expression model with no negative feedback. Protein dhmi dhpi noise levels for feedback architectures I − IV are computed = k hBi − γ hmi, = k hmi − γ hpi (3a) dt m m dt p p in Section 3. In Section 4 we compare the noise suppression 2 dhm i 2 2 abilities of the different feedback architectures. Finally, a = kmhB i + γmhmi + 2kmhBihmi − 2γmhm i (3b) discussion of our results is provided in Section 5. dt dhp2i = k hmi + γ hpi + 2k hmpi − 2γ hp2i (3c) dt p p p p 2 GENEEXPRESSIONMODELWITHNOREG- dhmpi = k hm2i + k hBihpi − γ hmpi − γ hmpi. (3d) ULATION dt p m p m We consider a gene-expression model where transcriptional As done in many studies we quantify noise in protein levels events take place at rate km with each event creating a burst through the coefficient of variation squared defined as of B mRNA molecules, where B is an arbitrary discrete 2 2 ¯ 2 random variable with probability distribution CV = σ¯ /hpi , (4) where σ¯ 2 is the steady-state variance in protein levels Probability{B = z} = αz, z = {1,2,3,...}. (1) and hp¯i denotes the steady-state mean protein count [39], Typically B = 1 with probability one. However, many genes [40]. Quantifying the steady-state moments from (3) and encode promoters that allow for transcriptional bursting substituting in (4) we obtain where B > 1 and many mRNAs can be made per tran- (hB2i + hBi)γ 1 scriptional event [34], [35], [36]. Protein molecules are CV 2 = p + ¯ ¯ (5) translated from each single mRNA at rate kp. We assume 2hBi(γp + γm)hmi hpi that mRNAs and proteins degrade at constant rates γm and where γp, respectively. In the stochastic formulation of this model, hBik hm¯ ik transcription, translation and degradation are probabilistic hm¯ i = m , hp¯i = p (6) events that occur at exponentially distributed time intervals. γm γp IEEE TRANSACTIONS ON NANOBIOSCIENCE 3

* + dhϕ(m, p)i ∞ = ∑ kmαz[ϕ(m + z, p) − ϕ(m, p)] + γmm[ϕ(m − 1, p) − ϕ(m, p)] + kpm[ϕ(m, p + 1) − ϕ(m, p)] dt z=1

+ γp p[ϕ(m, p − 1) − ϕ(m, p)] . (2)

denote the steady-state mean mRNA and protein count, determines the sensitivity of the transcription rate to the respectively. The first term on the right-hand-side of (5) protein count and can be interpreted as the strength of the corresponds to noise in protein levels that arises from negative feedback. stochastic production and degradation of mRNA molecules, ¯ and is inversely proportional to the mean mRNA count hmi. To obtain the time evolution of the statistical moments The second term in (5) represents Poissonian noise arising we use (2), with km now replaced by (8). This results in from random birth-death of individual protein molecules. the following moment dynamics: Given that mRNA population counts are typically or- ders of magnitude smaller than protein population counts dhmi = hkm(p)ihBi − γmhmi (10a) hm¯ i/hp¯i ≈ 10−3 from [2]
, we ignore the second term in dt (5) and approximate CV 2 as dhpi = kphmi − γphpi (10b) 2 dt 2 (hB i + hBi)γp 2 CV ≈ . (7) dhm i 2 2 ¯ = hkm(p)ihB i + γmhmi + 2hkm(p)mihBi − 2γmhm i 2hBi(γp + γm)hmi dt (10c) This approximation implies that gene-expression noise pri- 2 marily arises from fluctuations in mRNA counts that are dhp i 2 = kphmi + γphpi + 2kphmpi − 2γphp i (10d) transmitted downstream to the protein level. In summary, dt (7) represents the steady-state noise in protein level when dhmpi 2 = kphm i + hkm(p)pihBi − γphmpi − γmhmpi. there is no feedback in gene-expression. Next, we quantify dt protein noise levels for different feedback mechanisms. (10e) Quantifying the steady-state moments from (10) and sub- 3 INTRODUCINGREGULATORYMECHA- stituting in (4) gives the following protein noise level for NISMSINGENE-EXPRESSION feedback architecture I: We first consider protein mediated transcriptional regulation 2 2 γp(hBi + hB i) which corresponds to feedback architecture I in Figure 1. CVI = , (11) 2hBi(γp + γm)(1 + κ)hm¯ i 3.1 Protein mediated transcriptional regulation where the steady-state mean protein count is the unique Transcriptional regulation is incorporated in the model by solution to the equation assuming that the transcription rate is dependent on the pro- ¯ hBikpkm(hpi) ¯ tein levels. More specifically, transcriptional events occur = hpi (12) γmγp at rate km(p), which is a monotonically decreasing function of the protein count p(t). This corresponds to a negative and the steady-state mean mRNA count is given by feedback mechanism where any increase (decrease) in hp¯iγp protein numbers is compensated by a decrease (increase) hm¯ i = . (13) k in the transcription rate. To quantify the protein noise p levels we use the linear noise approximation [41], which As done in the previous section, to obtain the noise level involves linearizing the transcription rate km(p) about the (11) we assumed that the protein population count is steady-state average number of protein molecules hp¯i. This much larger than the mRNA population count, and hence approximation is valid as long as the stochastic fluctuations ignored expression noise arising from random birth and in protein counts are small, which is likely to be true for death of individual protein molecules. Throughout the paper 2 tightly regulated essential proteins. Towards this end, we we use CVX, X ∈ {I,II,III,IV} to denote the steady-state assume protein noise level corresponding to feedback architecture X. Moreover, CV 2, given by equation (7) represents the   p(t) − hp¯i k (p) ≈ k (hp¯i) − noise level when there is no feedback. Comparing the m m 1 κ ¯ (8) hpi analytical expressions in (7) and (11) one can see that for > where km(hp¯i) is the average transcription rate. The dimen- κ 0, the noise level with protein mediated transcriptional sionless constant regulation is always smaller than the noise level with no feedback. As expected, when κ = 0 (i.e., the transcription hp¯i dk (p) m rate is independent of the protein count and there is no κ = − |p=hp¯i > 0 (9) km(hp¯i) dp 2 2 feedback), CVI = CV . IEEE TRANSACTIONS ON NANOBIOSCIENCE 4

Feedback type Description Expression noise (CV2) Ref. No feedback, feedback strength !=0 Fixed transcription and translation rates

Feedback architecture I, feedback strength !>0 Transcription rate

decreasing function of [19,20] protein count

Feedback architecture II, feedback strength !>0 Translation rate decreasing function of [29,30] protein count

Feedback architecture III, feedback strength !>0 Transcription rate decreasing function of [31,32] mRNA count

Feedback architecture IV, feedback strength 0

Fig. 2. Gene expression noise measured by the coefficient of variation squared (CV 2) of the protein count corresponding to different feedback circuits. Here the random variable B denotes the transcriptional burst size, hm¯ i denotes the steady-state mean mRNA count, γp and γm represent the protein and mRNA degradation rate, respectively. The dimensionless constant κ represents the strength of the negative feedback. The last column provides references of genes that encode the corresponding feedback architecture.

3.2 Protein mediated translational regulation be obtained from (2), with kpm replaced by the right-hand- side of (14). Following steps similar to those in the previous We now consider a more sophisticated form of negative section, we obtain the following protein noise level for feedback where proteins regulate gene-expression at the feedback architecture II: translational level (feedback architecture II in Figure 1). 2 We model protein mediated translational regulation by 2 γp(hBi + hB i) CVII = . (16) assuming that the protein translation rate per mRNA is 2hBi(γm + γp(1 + κ))(1 + κ)hm¯ i a monotonically decreasing function k (p) of the protein p Finally, we consider feedback architectures III and IV. count p(t). Thus, the total protein production rate k (p)m p The procedure for quantifying protein noise levels for III is dependent on both the mRNA and protein population and IV is very similar to that used for feedback architec- count. tures I and II, except that the transcription and translation As before, we assume that the stochastic fluctuations rates are now monotonically decreasing functions of the p t m t p¯ in ( ) and ( ) around their respective means h i and mRNA count rather the protein count. Due to space con- m¯ h i are sufficiently small and approximate the total protein siderations we only present the final result in Figure 2, production rate as which lists the protein noise levels for different feedback   p(t) − hp¯i architectures. kp(p)m ≈ kp(hp¯i) m(t) − κ hm¯ i , (14) hp¯i 4 COMPARISONS BETWEEN DIFFERENT where kp(hp¯i) is the average protein translation rate per mRNA and the dimensionless constant GENEREGULATORYARCHITECTURES Analytical expressions in Figure 2 show two general trends hp¯i dkp(p) κ = − | ¯ > 0 (15) across different feedback architectures. Firstly, the protein k (hp¯i) dp p=hpi p noise levels are inversely proportional to the steady-state is the strength of the negative feedback architecture II. The mean mRNA count. Secondly, increasing the feedback moment dynamics corresponding to this feedback can again strength κ, i.e., making the transcriptional/translational IEEE TRANSACTIONS ON NANOBIOSCIENCE 5 rates more sensitive to the protein/mRNA count, results in We here analyzed the noise suppression properties of four lower protein noise levels. To assess the noise suppression different negative feedback loops within gene-expression abilities of different feedback architectures we perform a (Figure 1). Assuming that stochastic fluctuations in the pop- mathematically controlled comparison where all circuits are ulations of the protein and the mRNA are sufficiently small, assumed to have the same feedback strength and steady- we derived explicit analytical formulas for the protein noise state mean mRNA count. level for each of the four feedback mechanisms. These We first compare the noise suppression abilities of pro- formulas reveal that some feedback architectures are inher- tein mediated transcriptional and translational regulation ently better at noise suppression, while the performance of (i.e., feedback architectures I and II). Towards that end, others is dependent on the parameters of gene-expression. we compute the ratio 2 5.1 Which feedback provides the best noise sup- CVII γp + γm 2 = < 1, (17) pression? CVI γm + γp(1 + κ) Our results indicate that in a mathematically controlled which shows that feedback architecture II always provides comparison, feedback architecture IV provides the best better noise suppression than I. For a fixed feedback gain suppression of gene-expression noise. More specifically, for κ, the above ratio monotonically decreases with increasing a fixed feedback strength, a feedback mechanism where γ /γ and p m the protein translation rate per mRNA is a monotonically CV 2 CV 2 1 decreasing function of the mRNA count, provides the least lim II = 1, lim II = < 1. (18a) 2 2 amount of statistical fluctuations in the protein count. This γp/γm→0 CVI γp/γm→∞ CVI 1 + κ result raises an interesting question of how this feedback Thus, when the protein half-life is much smaller than the architecture is implemented within genes. mRNA half-life (γp/γm  1) then the noise suppression In recent years microRNAs, which represent a class ability of feedback architecture II is far superior to that of short non-coding RNAs, have been recognized as key of I. However, when the mRNA half-life is much smaller regulators of gene expression [42]. These microRNAs typ- than the protein half-life (γp/γm  1), the difference in the ically regulate gene-expression at the translational level noise suppression abilities of the two feedback mechanisms by directly binding to a mRNA transcript and inhibiting is small. protein translation [43]. In eukaryotes, genes contain non- Comparing the noise suppression abilities of protein coding segments called introns, which are removed from the mediated translation regulation and mRNA mediated tran- transcribed mRNAs through splicing. Removal of introns scriptional regulation (i.e., feedback architectures II and III) is a necessary step before mRNAs can start translating we find proteins and microRNAs can be derived from the excised 2 CVII γm(1 + κ) + γp introns [44], [42]. Recent studies have shown that many 2 = . (19) CVIII γm + γp(1 + κ) microRNAs are contained within the intronic regions of the same gene, whose translational activity is regulated by The above ratio shows that in genetic circuits where protein that microRNA [45], [46]. In other words, both the mi- molecules are more stable than the mRNA (γ < γ ), p m croRNA and its target mRNA originate from the same gene, mRNA mediated transcriptional regulation provides better and hence, are coexpressed. This structural relationship noise suppression than protein mediated translation regula- between them creates a negative feedback circuit, where tion. On the other hand, if a gene encodes a very unstable any random increase in mRNA counts automatically in- protein such that γ > γ , then protein mediated translation p m creases microRNA levels, which in turn reduces the mRNA regulation outperforms mRNA mediated transcriptional reg- translation rate. Given that disruption of these microRNA ulation. mediated feedback loops have been associated with cancer Finally, our results indicate that feedback architecture progression and neurodegenerative diseases [45], feedback IV provides the best noise suppression, irrespective of the IV type architectures play a critical role in these genes to mRNA and protein degradation rates. This is illustrated 2 minimize protein level fluctuations about desired set points. in Figure 3 which plots CVX, X ∈ {I,II,III,IV} as a function of the negative feedback strength κ. As can be seen from the figure, feedback architecture IV provides 5.2 Feedback architecture II versus III the best noise suppression while feedback architecture I A mathematically controlled comparison between protein is the least effective in reducing gene-expression noise. mediated translation regulation (feedback architecture II) Moreover, depending on the mRNA and protein half-life, and mRNA mediated transcriptional regulation (feedback feedback architecture II or III will provide the second best architecture III) showed that their relative noise suppression suppression of expression noise. abilities is dependent on the protein and mRNA half-life (Figure 3). Reference [47] does a survey of about 2000 5 DISCUSSION genes in budding yeast and shows that for most genes the What regulatory mechanisms control stochasticity in pro- ratio γp/γm is much smaller than one, i.e., genes encode tein levels such that cellular process can occur with suf- short-lived mRNAs but long-lived proteins. In this physio- ficient high fidelity is a fundamental question in biology. logically relevant parameter regime, feedback architecture IEEE TRANSACTIONS ON NANOBIOSCIENCE 6

II I IV III

mRNA half-life > Protein half-life mRNA half-life < Protein half-life No feedback No feedback

I I III II II III

Gene expression noise Gene II IV IV

Negative feedback strength Negative feedback strength Fig. 3. Decrease in gene expression noise with increasing negative feedback strength (κ) for different circuit architectures when mRNA half-life is smaller than protein half-life (right) and when mRNA half-life is larger than protein half-life (left). The y-axis is normalized by the gene-expression noise when there is no feedback. These plots correspond to a 5-hour protein half-life with a 1-hour (right) and 10-hour (left) mRNA half-life.

III is predicted to provide better noise suppression than Consistent with the observations of [16], we find that architecture II, and second best noise suppression among feedback architecture II always provides better noise sup- all the different feedback architectures. pression than architecture I. However, the difference in the In addition to inhibiting translation, increasing evidence protein noise level with feedback architectures I and II criti- suggests that microRNAs can also modulate the transcrip- cally depends on γp/γm. In particular, increasing γp/γm, i.e., tional activity of genes [48]. Thus, like IV, feedback archi- making the protein half-life much shorter than the mRNA tecture III can also be implemented using intron-derived half-life, increases this difference, and enhances the noise microRNAs. An example of feedback architecture III is suppression ability of feedback architecture II compared present in the endothelial nitric-oxide synthase (eNOS) to I. On the other hand, decreasing γp/γm, i.e., making gene. This gene encodes an essential protein required for the protein half-life much longer than the mRNA half-life, generating nitric oxide and its levels need to be tightly decreases this difference, and diminishes the advantage of regulated as its over-expression and under-expression have using feedback architecture II over I for noise reduction. been related to diseased states [32]. Data suggests that small In physiologically relevant parameter regime where γp/γm RNAs derived from introns within the eNOS gene repress is much smaller than one [47], our analysis predicts that the transcriptional activity of the eNOS gene [31], [32]. protein mediated translational regulation may not provide Thus, any increase in eNOS mRNA transcripts would in- significantly better noise suppression than protein mediated hibit further transcriptional events from the gene creating an transcriptional regulation. effective negative feedback circuit for eNOS homeostasis. In summary, we have developed analytical formulas that connect stochasticity in protein levels to the negative 5.3 Protein mediated feedback circuits feedback architecture. These formulas reveal that mRNA Above analysis suggests that in physiologically relevant mediated feedback circuits are superior to protein mediated parameter regime, mRNA mediated feedback circuits (feed- feedback circuits for noise reduction under physiologically back architecture III & IV) provide better noise suppression relevant parameter regimes. Consistent with this prediction than protein mediated feedback circuits (feedback architec- we find many essential genes encoding mRNA dependent ture I & II). However, these comparisons where done at a feedback circuit through intro-derived microRNAs. Our fixed feedback strength. Typically, protein molecules do not analysis not only make experimentally testable predictions function as monomers but bind together to form dimers, but also will be helpful in designing precise synthetic gene tetramers, etc. Such protein multimerization can induce networks with minimal fluctuations in protein levels. cooperativity in a feedback circuit that makes transcrip- tion/translation rates ultra sensitive to protein levels, and hence, substantially increase the negative feedback strength ACKNOWLEDGMENTS [26], [27]. Thus protein mediated feedback loops can also provide effective noise suppression by operating at much The author would like to thank Joao Hespanha, Mustafa higher values of κ compared to mRNA mediated feedback Khammash, Leor Weinberger and members of the Wein- loops. berger lab for many discussion on this topic. IEEE TRANSACTIONS ON NANOBIOSCIENCE 7

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