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Negative Feedback Through Mrna Provides the Best Control of Gene-Expression Noise

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Negative Feedback Through Mrna Provides the Best Control of Gene-Expression Noise 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 GENE EXPRESSION MODEL WITH NO REG- 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.
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