Quantifying mRNA synthesis and decay rates using small RNAs
QQuantifying mRNA synthesis and decay ratesusing small RNAs
Vlad Elgart, Tao Jia, Rahul Kulkarni,Department of Physics, Virginia Tech,E-mail: [email protected], [email protected] 20, 2018
Abstract
Regulation of mRNA decay is a critical component of global cellu-lar adaptation to changing environments. The corresponding changes inmRNA lifetimes can be coordinated with changes in mRNA transcrip-tion rates to fine-tune gene expression. Current approaches for measuringmRNA lifetimes can give rise to secondary effects due to transcription in-hibition and require separate experiments to estimate changes in mRNAtranscription rates. Here, we propose an approach for simultaneous deter-mination of changes in mRNA transcription rate and lifetime using regula-tory small RNAs to control mRNA decay. We analyze a stochastic modelfor coupled degradation of mRNAs and sRNAs and derive exact resultsconnecting RNA lifetimes and transcription rates to mean abundances.The results obtained show how steady-state measurements of RNA levelscan be used to analyze factors and processes regulating changes in mRNAtranscription and decay.
Cellular adaptation to changing conditions is critically dependent on pro-cesses that enable rapid responses to environmental fluctuations. While consid-erable research has focused on changes in transcription, research over the pastseveral years has demonstrated that control of mRNA decay plays an increas-ingly important role in cellular responses [1, 2, 3]. Correspondingly, there issignificant interest in understanding factors and processes which govern regula-tion of mRNA decay.The traditional approach for measuring mRNA lifetimes involves quantifica-tion of mRNAs remaining at different times following inhibition of transcription,e.g., by the addition of rifampicin [2]. This procedure requires multiple mea-surements during time intervals of the order of the mRNA lifetime, hence hightemporal resolution is required for short-lived mRNAs. More significantly, theprocedure for inhibition of transcription can give rise to secondary effects whichinfluence mRNA decay [2], hence it is of interest to consider alternative ap-proaches. In the following, we outline a novel proposal for quantifying mRNAdecay. 1 a r X i v : . [ q - b i o . CB ] O c t igure 1: Panel A: Kinetic scheme for mRNA and sRNA production and decayincluding coupled mutual degradation. Panel B: The proposed setup involvessteady-state measurements for three strains: ∆(sRNA), WT and ∆(mRNA).Unregulated steady-state mean levels of mRNAs and sRNAs along with regu-lated levels of these molecules are measured. The measured quantities allowdetermination of the average mRNA transcription rate k m and decay rate τ m relative to the sRNA production rate k s . If k s is held fixed, and the conditionsare varied, the proposed scheme leads to simultaneous determination of fold-changes in the rate of transcription and the rate of mRNA decay. Note that themRNA/sRNA interaction parameter can be arbitrary.A naturally occurring process which regulates turnover for many bacterialmRNAs involves interactions with regulatory small RNAs (sRNAs). Experi-ments have shown that post-transcriptional regulation of gene expression canoccur via binding and subsequent coupled degradation of the mRNA and regu-latory sRNA which occurs rapidly [10, 17]. The corresponding coarse-grainedmodel of gene expression has been analyzed by several groups [9, 8, 11, 12, 13]and shown to be consistent with experimental observations [9]. We will showthat the kinetic scheme for this mode of regulation leads to an exact mathemat-ical result relating RNA decay rates. The obtained result is valid even in thepresence of large fluctuations that are typical for low abundance mRNAs.Our analysis considers a generalized stochastic model for regulation by sR-NAs as illustrated in Fig. 1. The complexity of processes leading to transcriptionsuggests that RNA synthesis in many instances is not adequately modeled asa Poisson process [7, 14]. Hence, we model mRNA and sRNA production by arbitrary stochastic processes with mean arrival rates k m and k s respectively.Degradation of a mRNA and a sRNA can occur either independently (with con-stant probability per unit time τ m and τ s respectively) or through the coupleddegradation process.The experimental setup in our proposal for quantifying mRNA decay is asfollows. Consider three different strains as shown in Fig. 1B: two unregulatedstrains (i.e., with either sRNA or mRNA deleted) and the wild type (WT) strain.In the WT strain, both mRNA and sRNA are present and regulate each other.In steady state, we derive the following exact relations (Appendix) connecting2RNA/sRNA lifetimes and transcription rates to the mean abundances k m τ m = (cid:104) m (cid:105) k s τ s = (cid:104) s (cid:105) (1)for the unregulated strains, and τ m τ s = (cid:104) m (cid:105) − (cid:104) ˜ m (cid:105)(cid:104) s (cid:105) − (cid:104) ˜ s (cid:105) . (2)Here (cid:104) m (cid:105) and (cid:104) s (cid:105) are the mean mRNA(sRNA) abundances in strains lackingthe sRNA(mRNA), and (cid:104) ˜ m (cid:105) and (cid:104) ˜ s (cid:105) are the mean mRNA and sRNA levels inWT strain where both are present.The above relations suggest an alternative approach (Fig. 1B) for quantify-ing decay times for mRNAs that either have a naturally occurring small RNAregulator or for which an antisense RNA regulator can be designed. Consideran experimental setup expressing the sRNA from an inducible promoter suchthat its transcription rate is primarily controlled by inducer concentration. Themean transcription rate k s can, in principle, be determined using single-moleculemethods [18]. Now the basic parameters for the coupled system are k m , k s , τ m and τ s . If k s is known, then the values of the other parameters can be deter-mined using experimental measurements of (cid:104) m (cid:105) , (cid:104) s (cid:105) , (cid:104) ˜ m (cid:105) and (cid:104) ˜ s (cid:105) in combinationwith equations Eqns. 1-3 above. Alternatively, experiments can be designed tokeep k s fixed while factors regulating mRNA decay are changed e.g., by dele-tion of a protein known to play a role in mRNA decay. The above equationscan be used to simultaneously determine the corresponding fold-changes of themRNA/sRNA lifetimes and the mean mRNA transcription rate.The proposed approach can be used to address several important questionsof current interest, some of which are highlighted in the following. By targetedmutagenesis of specific mRNA sequence elements, the induced fold-change inmRNA lifetime, as well as the corresponding change in the transcription rate k m ,can be determined using the same experimental setup. This is an important fea-ture, given that recent experiments have observed coordination between changesin transcription and changes in mRNA degradation [16]. Quantifying the changein mRNA lifetimes induced by mutations to different components of cellulardegradation pathways can address such issues as the role of polyadenylation inmRNA decay [6]. It would also be of interest to design high-throughput ex-periments for different mRNAs which are regulated by corresponding antisenseRNAs, all of which are expressed from identical inducible promoters and thushave the same k s . The proposed procedure can then be used for genome-widedetermination of relative transcription rates and lifetimes of mRNAs. Theseeffective parameters, in turn, serve as critical inputs to systems-level models ofcellular processes [15].In summary, we have proved an exact relation for a nonlinear stochasticmodel of cellular post-transcriptional regulation. The derived results suggesta novel procedure for simultaneous determination of mRNA production ratesand mRNA lifetimes. While the focus was on bacterial mRNAs, the procedure3utlined can also be applied to higher organisms and used to systematically ex-plore the sequence determinants and processes involved in regulation of mRNAdecay. Appendix
The proposed experimental setup involves measurements of mRNA/sRNA abun-dances for three strains, the sRNA deleted ∆(sRNA) strain, mRNA deleted∆(mRNA) strain, and the wild type (WT) strain (both species are presentand regulate each other.) The regulation model (with constant creation ratesfor mRNA and sRNA) has been analyzed by several groups [9, 11, 13] and issummarized in the reaction scheme described in Fig.1.Recent experiments which provide evidence for transcriptional bursting [14]point towards the need to go beyond the Poisson process as a model for RNAsynthesis. Accordingly, we take both mRNA and sRNA creation events as arbitrary stochastic processes. Degradation of RNA is assumed to be a Poissonprocess with rate τ − x , where τ x is the mean lifetime of x RNA , x = { m, s } . Inthe WT strain, mRNA and sRNA undergo a coupled degradation process. Weassume that this process is symmetric with respect to the number of mRNAand sRNA molecules involved, e.g., M + S → ∅ .Let us choose a particular realization of the system evolution during timeinterval t = [0 , T ]. For large values of T , we derive x ( T ) − x (0) = C x ( T ) − Y ( T ) − τ − x (cid:90) T d t x ( t ) , (3)where x ( t ) is the number of molecules of the species x = { m, s } at the time t .In Eq. 3, C x ( t ) is the total number of molecules of the species x created duringsystem evolution until time T , and Y ( T ) is the total number of molecules ofeither species that is mutually degraded within the time interval [0 , T ]. Finally,using the law of large numbers, the number of molecules degraded naturally in[0 , T ] is given by the last term in the Eq. 3.Dividing both sides of Eq. 3 by T and taking a limit T → ∞ we obtainlim T →∞ x ( T ) − x (0) T = k x − τ − x (cid:104) ˜ x (cid:105) − lim T →∞ Y ( T ) T , (4)where (cid:104) ˜ x (cid:105) is average number of molecules in the system. We also defined k x asa mean arrival rate of the species x = { m, s } . The limit on the left hand side ofEq. 4 vanishes in the case of finite degradation rates τ − x (number of moleculesat any time is finite.) Note that Y ( T ) is an extensive quantity (it is monotonicincreasing function of T ) and therefore, the limit on the right hand side of Eq. 4is finite. 4ence, we derive k m − τ − m (cid:104) ˜ m (cid:105) − lim T →∞ Y ( T ) T = 0 ,k s − τ − s (cid:104) ˜ s (cid:105) − lim T →∞ Y ( T ) T = 0 , (5)which immediately yields the following expression k m − τ − m (cid:104) ˜ m (cid:105) = k s − τ − s (cid:104) ˜ s (cid:105) . (6)In the unregulated case Y ( T ) ≡ T , since one of the RNA speciesis deleted and there is no coupled degradation. In this situation one gets0 = k m − τ − m (cid:104) m (cid:105) , k s − τ − s (cid:104) s (cid:105) , (7)where (cid:104) x (cid:105) , x = { m, s } are the average number of molecules during unregulatedsystem evolution. Combining the set of equations above with Eq. 6, we derivethe results in the main text Eqns 1-3. We note that the derived results arevalid even if the binding of mRNA and sRNA is taken to be reversible and thelifetime of the mRNA-sRNA complex is finite. Finally, the time average can bereplaced by the ensemble average in the steady state.We have validated the derived results using stochastic simulations basedon the Gillespie algorithm [4]. Production of RNA molecules was taken tooccur in transcriptional bursts [14] i.e. each burst corresponds to the arrival ofa random number of RNAs drawn from a geometric distribution, conditionalon the production of at least 1 RNA molecule [5]. The waiting-time betweenbursts was a random variable drawn from exponential or Gamma distributions.As expected, the results from the simulations were in excellent agreement withthe derived analytical results. References [1] J.A. Bernstein, A.B. Khodursky, P.H. Lin, S. Lin-Chao, and S.N. Cohen.Global analysis of mRNA decay and abundance in Escherichia coli at single-gene resolution using two-color fluorescent DNA microarrays.
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