Artigo Acesso aberto Revisado por pares

Digitalizing heterologous gene expression in Gram‐negative bacteria with a portable ON/OFF module

2019; Springer Nature; Volume: 15; Issue: 12 Linguagem: Inglês

10.15252/msb.20188777

ISSN

1744-4292

Autores

Belén Calles, Ángel Goñi‐Moreno, Vı́ctor de Lorenzo,

Tópico(s)

CRISPR and Genetic Engineering

Resumo

Article19 December 2019Open Access Transparent process Digitalizing heterologous gene expression in Gram-negative bacteria with a portable ON/OFF module Belén Calles Belén Calles Systems Biology Program, Centro Nacional de Biotecnología-CSIC, Madrid, Spain Search for more papers by this author Ángel Goñi-Moreno Ángel Goñi-Moreno Systems Biology Program, Centro Nacional de Biotecnología-CSIC, Madrid, Spain Search for more papers by this author Víctor de Lorenzo Corresponding Author Víctor de Lorenzo [email protected] orcid.org/0000-0002-6041-2731 Systems Biology Program, Centro Nacional de Biotecnología-CSIC, Madrid, Spain Search for more papers by this author Belén Calles Belén Calles Systems Biology Program, Centro Nacional de Biotecnología-CSIC, Madrid, Spain Search for more papers by this author Ángel Goñi-Moreno Ángel Goñi-Moreno Systems Biology Program, Centro Nacional de Biotecnología-CSIC, Madrid, Spain Search for more papers by this author Víctor de Lorenzo Corresponding Author Víctor de Lorenzo [email protected] orcid.org/0000-0002-6041-2731 Systems Biology Program, Centro Nacional de Biotecnología-CSIC, Madrid, Spain Search for more papers by this author Author Information Belén Calles1, Ángel Goñi-Moreno1,2 and Víctor Lorenzo *,1 1Systems Biology Program, Centro Nacional de Biotecnología-CSIC, Madrid, Spain 2Present address: School of Computing, Newcastle University, Newcastle upon Tyne, UK *Corresponding author. Tel: +34 91 585 45 36; Fax: +34 91 585 45 06; E-mail: [email protected] Molecular Systems Biology (2019)15:e8777https://doi.org/10.15252/msb.20188777 PDFDownload PDF of article text and main figures. Peer ReviewDownload a summary of the editorial decision process including editorial decision letters, reviewer comments and author responses to feedback. ToolsAdd to favoritesDownload CitationsTrack CitationsPermissions ShareFacebookTwitterLinked InMendeleyWechatReddit Figures & Info Abstract While prokaryotic promoters controlled by signal-responding regulators typically display a range of input/output ratios when exposed to cognate inducers, virtually no naturally occurring cases are known to have an OFF state of zero transcription—as ideally needed for synthetic circuits. To overcome this problem, we have modelled and implemented a simple digitalizer module that completely suppresses the basal level of otherwise strong promoters in such a way that expression in the absence of induction is entirely impeded. The circuit involves the interplay of a translation-inhibitory sRNA with the translational coupling of the gene of interest to a repressor such as LacI. The digitalizer module was validated with the strong inducible promoters Pm (induced by XylS in the presence of benzoate) and PalkB (induced by AlkS/dicyclopropyl ketone) and shown to perform effectively in both Escherichia coli and the soil bacterium Pseudomonas putida. The distinct expression architecture allowed cloning and conditional expression of, e.g. colicin E3, one molecule of which per cell suffices to kill the host bacterium. Revertants that escaped ColE3 killing were not found in hosts devoid of insertion sequences, suggesting that mobile elements are a major source of circuit inactivation in vivo. Synopsis Stringent ON/OFF expression switches are difficult to implement in vivo, as promoters use to have a degree of basal expression. Yet, the problem can be overcome by combining transcriptional signals with repressor proteins and inhibitory small RNAs. Although many prokaryotic promoters can be activated with exogenously added chemical inducers, virtually all of them have a certain level of basal expression and thus make robust circuit design difficult. By combining biological parts that cause strong transcriptional activity with repressor proteins and translation-inhibitory small sRNA one can find parameters that deliver an entirely digital ON/OFF behaviour. One particular configuration of the benzoate-inducible XylS/Pm promoter, the Lac repressor and the sRNA micC originated a genetic circuit with virtually zero expression in the absence of induction. Thereby digitalized expression systems afforded conditional production in vivo of highly toxic proteins such as colicins, one molecule of which suffices to kill a bacterial cell in various Gram-negative species. Introduction Whether naturally occurring or engineered, prokaryotic expression modules typically involve a promoter regulated by a signal-responding transcriptional factor and a downstream gene preceded by a segment encoding a 5′UTR of more or less complexity. Decoding of any gene of interest (GOI) and the making of its product is thus the result of combining translation and transcription (Blazeck & Alper, 2013; Bervoets & Charlier, 2019). That such expression modules display activated/non-activated states is the basis for a plethora of genetic circuits in which physiological or externally added signals are equalled to inputs in logic gates that deliver a connectable output ON/OFF (Wang et al, 2011; Moon et al, 2012; Brophy & Voigt, 2014; Bradley et al, 2016). Taken to an extreme, this can in turn be abstracted as digital 1/0 states, thereby providing the basis for ongoing attempts of biological computation and operative systems based on transcriptional factors (Ma et al, 2016; Nielsen et al, 2016; Urrios et al, 2016). A serious drawback of this approach is, however, that virtually no regulated expression system is really digital. While transcriptional capacities and induction rates can vary enormously among promoters, all of them—albeit to a greatly varying degree—have a measurable level of basal activity even in the absence of any inducer. This may be due to weak RNAP–DNA interactions or just be the unavoidable consequence of pervasive transcription (Creecy & Conway, 2015). One way or the other, while there are plenty of induced levels that can be properly assigned to the ON state or value 1, the OFF/0 state is often fixed by an arbitrary threshold output. This causes not only countless problems in predictability and portability of circuit performance but also makes engineering of expression systems for toxic proteins or products quite challenging (Saida et al, 2006; Balzer et al, 2013). The literature has many cases of tightly controlled expression systems that respond to given external signals through a positive or negative transcriptional regulation mechanism. Examples include devices based on, e.g. the tetA promoter/operator, araBAD and rha pBAD promoters or the XylS/Pm expression system. They all can be modulated in a wide range of output levels from low-to-high marks (e.g. PBAD-types and the XylS/Pm promoter) or medium-to-high levels of expression (e.g. the tetA system) while keeping relatively low basal levels of activity in the absence of inducing signal. Among these systems, the XylS/Pm regulator/promoter pair holds several beneficial features (Tropel & van der Meer, 2004; Brautaset et al, 2009). First, it is factually orthogonal, performing independently of the metabolic state of the cell. Furthermore, it is a genuine broad-host-range system and has been extensively used in various Gram-negative bacteria including E. coli, P. putida, Pseudomonas fluorescens, X. campestris and others. Induction is made by cheap benzoic acid derivatives that do not need a special transport system to enter the cell. Moreover, there is a very small likelihood of gratuitous induction or cross-talk with other expression systems and the basal level is very low, especially when combined with the single-copy RK2 origin of replication, which allows the industrial level production of toxic proteins (Sletta et al, 2004). In other cases, genetic devices have been designed for suppressing leaky basal expression levels through the engineering of super-repressors (Ruegg et al, 2018), exploitation of antisense RNAs (O'Connor & Timmis, 1987), or physical decoupling of regulatory elements along with conditional proteolysis (Volke et al, 2019). Another elegant approach has been the exploitation of diverse recombinases to maintain DNA segments in the OFF orientation in the absence of inducer (Ham et al, 2006), which can be fused to degradation tags to assure their transient expression, allowing the construction of synthetic gene networks capable of counting (Friedland et al, 2009). In this last case, expression of recombinases has to be stringently controlled in any case, and they often act in a single direction, resulting in non-reusable genetic devices. One way or the other, reliable circuits involve the use of tightly controlled expression systems. But can we really assert that the output of a given GOI in the absence of inducers is actually zero? In this work, we have pursued the design of a general-purpose digitalizer of heterologous gene expression which combines small RNA-mediated inhibition of translation with the translational coupling of a repressor to the GOI, all framed in a double feedback circuit that could generate a switch-like regime capable of operating between OFF and ON states in a reversible manner, ruled by the presence of an externally added inducer. When placed downstream of a strong inducible promoter, the thereby resulting architecture suppresses basal expression down to altogether non-detectable levels in E. coli and Pseudomonas putida. The results below thus pave the way for generating digitalized variants of popular promoters used in synthetic circuits. Also, they allow creation of switches in which the metabolic or physiological status of cells can be entirely changed upon exposure of cells to an external signal. Results and Discussion Benchmarking low and high expression states of the tightly regulated promoter Pm To evaluate the performance of the naturally occurring and tightly controlled XylS/Pm expression system, we adopted the standardized pSEVA238 plasmid. This is a medium-copy number vector with a pBBR1 origin of replication (30–40 copies/cell), a kanamycin resistance marker and an expression cargo composed of the xylS gene and the Pm promoter (Martínez-García et al, 2015; Fig 1A). The gene encoding a monomeric superfolder version of the green fluorescent protein (msf•GFP) was cloned downstream of Pm as a sensitive reporter of transcriptional activity (Fig 1A). The resulting plasmid (pS238M) was then transformed in E. coli CC118 strain and its behaviour analysed in individual cells by flow cytometry experiments. Figure 1B and C shows the kinetics of the expression along time at a fixed (1.0 mM) concentration of 3-methyl benzoate (3MBz) as inducer. Before induction (t = 0), cells show a fluorescence pattern which is very similar (but not identical) to that of the non-fluorescent control strain, i.e. E. coli CC118 strain transformed with pSEVA237M plasmid, containing a promoterless msf•GFP gene cloned exactly in the same genetic background (see grey peak in Fig 1B). Note that the median value of cells harbouring pS238M in the absence of induction is slightly higher that the promoterless counterparts, indicating a very low but still detectable basal level (Fig 1C). The system then showed a fast response after addition of inducer as reflected in the rapid displacement of the cell population to higher fluorescence signals (Fig 1B) and the sharp increase in the median fluorescence values (Fig 1C). Most, if not all the cells, were expressing msf•GFP at 20 min after adding the inducer, and the fluorescent output increased along time to reach a plateau around 60 min later (Fig 1B and C). The level of induction increased in a dose-dependent manner, and the system was so sensitive as to detect and respond to low micromolar concentrations of inducer (Fig 1D). Population heterogeneity could also be quantified by means of the coefficient of variation expressed as a percentage (CV* 100; Raj & van Oudenaarden, 2008; Eldar & Elowitz, 2010) at each time point of the assay (Fig 1E: the higher the CV the less homogeneous the population is). The outcome of the flow cytometry experiments showed significant differences in the level of msf•GFP expressed among individual cells, especially at short times after induction, reflected in broader population peaks and higher CV values. But in general, there was a clear, apparent breach in the fluorescence readout of cells treated or not with 3MBz, as background expression of msf•GFP was very low in the absence of exogenous inducer. Figure 1. Performance of a standardized XylS/Pm-based expression system Schematic representation of the pS238M plasmid, harbouring an msf•GFP gene as reporter. Main features of the DNA backbone are indicated. Evaluation by flow cytometry experiments of the basal (t = 0) and induced expression of GFP along time upon induction of the system with a fixed (1.0 mM) concentration of 3MBz, at the indicated time points. The same Escherichia coli strain was transformed with a promoterless GFP version of the otherwise identical plasmid backbone (grey plot), and this region was the considered as indicative no-fluorescence (indicated between red dashed lines). Representation of the fluorescence median and SD values of three independent experiments performed as described in the previous panel. Dose response of the XylS/Pm system. GFP fluorescence—normalized to OD600 in all cases—was monitored with respect to the control strain transformed with the empty vector to evaluate auto-fluorescence in the absence and in the presence of increasing concentrations of inducer (3MBz) as indicated. Asterisk indicates the baseline sensitivity of the instrument. Data represent the mean and SD values of three independent experiments with eight technical replicates each performed in microtiter plates. Cell-to-cell homogeneity was evaluated by means of the coefficient of variation (expressed as a percentage CV*100) of population samples analysed before and upon inducer addition at the indicated time points. Data correspond to mean CV*100 values with SD obtained from three independent experiments. Download figure Download PowerPoint Detection and quantification of very low basal expression levels In order to expose the level of low transcriptional activity of Pm which does not become apparent with GFP reporter technology, we resorted to the amplifying cascade that results from expressing a sequence-specific protease and visualizing the cleavage of a sensitive target in vivo. The rationale for this approach is shown in Fig 2. Plasmid pS238•NIa, which expresses the site-specific plum-pox protease NIa (García et al, 1989) under the same XylS/Pm device as before, was co-transformed in a ∆tpiA E. coli W3110 strain with plasmid pBCL3-57-NIa bearing an E-tagged variant of the TpiA protein, which had been engineered with an optimal cognate cleavage site. NIa is a highly active protease that efficiently cuts target proteins bearing recognition sites—even when they are grafted on a different peptidic context (Garcia et al, 1989). Therefore, inspection of TpiA integrity in a Western blot assay becomes an exquisite indicator of leaky protease expression. In the test shown in Fig 2, we used a protease-sensitive TpiA variant bearing the NIa target sequence at position E57 of the protein, which is known to be efficiently cleaved by the protease. The reaction can then be easily followed by means of Western blot assays using antibodies against the E-tag sequence added at the C-terminus of TpiA target enzyme. Figure 2 shows that basal expression of Pm in pS238•NIa plasmid was enough to produce sufficient protease as to cleave at least half of the protein produced by pBCL3-57-NIa in the absence of any inducer. This indicated that basal expression of NIa controlled by XylS/Pm regulatory system—which is generally invisible by other means—sufficed to trigger a biological effect. The sections below describe a feasible strategy to control this and to take non-induced gene expression to virtually zero. Figure 2. Hypersensitive test of the basal activity of the XylS/Pm deviceTwo plasmids shown on top enabled the experimental set up used to test activity of the NIa protease when expressed through the XylS/Pm system. A NIa-sensitive and E-tagged TpiA variant was cloned in a high copy pUC18 plasmid, while the cognate NIa protease was expressed in a pSEVA238 vector. Analysis of the cleaving activity of the NIa protease on the target TpiA-NIa protein is detected by means of a Western blot assay with anti E-tag antibodies, in the absence (−) and in the presence (+) of 1.5 mM of the XylS/Pm inducer 3MBz. The control strain (C) contains pUC18 harbouring the gene coding for the TpiA-NIa protein and the empty pSEVA238 plasmid (lower panel). Note a very significant cleavage in the absence of induction. Download figure Download PowerPoint Rationale of a designer digitalizer module Although background expression could be reduced by decreasing plasmid copy number, by using weaker RBS sequences or even by introducing mutations in the consensus promoter sequences, these approaches usually affect the induced expression level as well. Thus, we decided to explore the possibility of reducing basal levels while keeping induced levels as high as possible by introducing an additional cross-inhibition regulatory genetic circuit affecting other step of the protein expression process, i.e. mRNA translation. The designed circuit aimed at digitalizing the expression system, producing a regulatory device with a clear ON/OFF behaviour. For this reason, we named such a circuit a digitalizer module (see Fig 3A). The key parts of such a device include: (i) an inducible promoter with a given level of basal expression (P1), (ii) a strong, yet repressible promoter P2 for transcription of a translation-inhibitory sRNA, (iii) a transcriptional repressor (R) expressed through the inducible promoter P1 but translationally inhibited by the sRNA and (iv) a GOI translationally coupled (not fused) to the repressor gene. The functionality of the system stems from the fact that the repressor protein targets the strong promoter P2 from which the inhibitory sRNA is produced, so that R and sRNA are mutually inhibitory. The resulting cross-inhibition between the two components is expected to result in a switch-like scenario (Gardner et al, 2000; Kim et al, 2006; Lipshtat et al, 2006) reviewed in Zhou and Huang (2011). Qualitatively this means that if the system is in a state in which the concentration of repressor protein R is low and the concentration of the inhibitory sRNA is high then the system is OFF, as no translation can be produced. The same applies with the concentrations of R and sRNA being high and low, respectively. Thus, the system is expected to be OFF unless an external stimulus, i.e. induction of transcription, increases repressor R concentration, allowing translation to occur and causing a transition from the OFF to the ON state (Fig 3). To gain an insight into the predicted system behaviour and pinpoint key parameters for displacing the switch towards ON or OFF states, we developed the mathematical model—and run the ensuing simulations—described in Appendix Information. According to the model, the kinetic behaviour of the mutual inhibition switch as a function of the strength of each repressor (protein or sRNA) was such that the stronger the both inhibitions are, the more digital the device is; i.e., both ON and OFF expression states are more stable and the change between states is sharper (Fig 3B and C). However, a detailed analysis of the particular weight of each inhibitory element (Appendix Fig S1) showed that the strength of sRNA repression has more impact on system performance than that of the repressor R. Prediction is thus that changing binding parameters of repressor R to its cognate promoter may not have much influence on the digitalization of the system (Appendix Fig S1A) while the strength of sRNA repression is much more crucial (Appendix Fig S1B). Nevertheless, optimal digitalization is reached by combining high repression rates for both the transcriptional and the translational components of the system. Figure 3. Rationale and modelling of a genetically encoded digitalizer module A–C. (A) Expression of the GOI is controlled by a positively regulated promoter, the activity of which depends on a transcription factor (TF), and an additional switch placed downstream, consisting of a mutual inhibition circuit regulating the translation step (left). The main features of such cross-inhibition switch-like circuit, the so-called digitalizer module, include a sRNA that inhibits translation of a repressor (R) that in turns regulates the sRNA production. A translational coupler was added between the repressor and the GOI to secure a coordinated expression of both genes (right). Simulations of circuit performance are shown in panels (B) and (C). The double feedback loop formed by mR (the mRNA for both lacI and GFP) and mS (the inhibitory sRNA) is the key of the digitalization of the device. The stronger the both repressions are, the more digital the device is; i.e., both ON and OFF expression regimes are more separated (line is more horizontal) and the change between states is sharper (line is more vertical). In (B), each line is a single simulation that measures the level of mR (Y axis) while the concentration of active TF molecules—and thus P1 promoter strength—increases (X axis). The colour of the lines goes from dark red (both repressions very strong) to dark blue (both repressions very weak). The transition from OFF to ON seems to cross a single point at medium and strong repressions (arrow). This pivot point suggests the level of TFa needed to switch the system is specific and not dependent on the repression strength (except at very low values). The plot in (C) shows mS versus mR, i.e. the relationship between the two RNA species in the system. The stronger both repression effects are, the more mutually exclusive this connection is. As shown in the red line, the system is dominated by either mR or mS molecules, but not by both of them at the same time. Download figure Download PowerPoint Implementation and SBOL description of a stringent ON/OFF switch Based on the model for a cross-inhibition switch described in the previous section, we assembled the construct depicted in Fig 4A, the main features of which go as follows. The key player of the designed post-transcriptional control circuit is a cis-repressing sRNA, based on a naturally occurring small transcript in E. coli. This regulatory sRNA is composed of two parts, a scaffold sequence and a target-binding sequence. The scaffold is provided by the MicC consensus secondary structure that has been described to recruit the RNA chaperone Hfq, which is known to facilitate the hybridization of sRNA and target mRNA as well as mRNA degradation (Yoo et al, 2013). The second part of the sRNA, which is replacing the natural MicC target binding region, was designed against the LacI transcriptional repressor, preventing translation of the thereby generated mRNA. This specific 24-mer fragment is the antisense sequence to the translation initiation region (starting at the very first codon) of the target repressor LacI. According to the model predictions, the sRNA ensemble was tailored to have the highest possible repression capabilities, as previously described (Balzer et al, 2013; Na et al, 2013). Synthesis of this sRNA is in turn controlled by a LacI-dependent PA1/O4s promoter. This synthetic Plac promoter derivative was selected because the kinetic parameters of RNAP–promoter interaction in combination with the position of the operator lead to the highest LacI repression factor (Lanzer & Bujard, 1988). Figure 4. Evaluation of the functionality of the digitalizer module Schematic representation of the relevant features of the specific regulatory device used to evaluate the efficiency of the digitalizer module by means of a reporter msf•GFP gene. GFP was used as fluorescence reporter to analyse the OFF and the ON expression profiles of the digitalized version of XylS/Pm system in flow cytometry experiments. The plot represents the median and the SD values of fluorescence from three independent experiments without inducer (t = 0) and upon 3MBz induction along time, at the indicated time points. Representative experiment showing the distribution of fluorescence in cell population under non-inducing conditions (t = 0) and after induction at the indicated time points (t = 5 min to t = 160 min), as indicated. The region considered as negative for the fluorescence signal is marked between red dashed lines, as assessed by control cells carrying a plasmid with a promoterless GFP (grey plot). Analysis of the population with respect to cell-to-cell heterogeneity by means of the coefficient of variation (percentage CV*100) at the off state (t = 0) and along the induction of the system, as described. Data correspond to the mean and SD values obtained from three separate experiments. The NIa protease was used as a sensitive reporter to test both the basal and the induced activity driven by the digitalized XylS/Pm device. The Western blot shows the cleavage of a NIa-sensitive TpiA target protein under the indicated conditions, which are equivalent as those used for the experiment presented in Fig 2. The control lane #1 was pasted from a different gel to show the location of the intact protein (see a detailed cleavage kinetics in Appendix Fig S6). Download figure Download PowerPoint To ensure termination of transcription of both the bicistronic lacI-msf•GFP operon and the divergently expressed sRNA, two strong transcriptional terminators were added to the device. T500, an artificial terminator derived from T82, that carries a strong hairpin (Yarnell & Roberts, 1999), was placed downstream of msf•GFP, the proxy of any GOI. Accurate termination of the sRNA, which is very important to preserve its secondary structure, was secured by means of a double synthetic T1/TE terminator (MIT Registry, BBa_B0025). Control of the GOI in coordination with the upstream lacI repressor gene, which is the target of the sRNA, requires a translational coupler cassette. An efficient mechanism for coupling translation is based on the ability of translating ribosomes to unfold mRNA secondary structures. In our design, translational coupling is achieved by occluding the RBS of msf•GFP by formation of a secondary mRNA structure, containing a His-tag sequence added to the 3′end of the lacI gene. The sequence was designed so that it forms a strong hairpin (ΔG = −16.1 kcal/mol) that matches the Shine–Dalgarno sequence upstream of the GOI, preventing the ribosomal recruitment and therefore translation (Mendez-Perez et al, 2012). In contrast, when the upstream lacI mRNA is actively translated, the 70s ribosome disrupts inhibitory mRNA secondary structure in the downstream gene translation initiation region thus allowing its expression. Only when the first gene is translated does the inhibiting secondary structure open up enabling translation of the second gene, as extensively documented by Mendez-Perez et al (2012). The whole post-transcriptional regulatory circuit was inserted downstream of the XylS/Pm expression module within the same pSEVA backbone (pSEVA238) as before, including the msf•GFP gene as reporter (Fig 4A), generating pS238D•M. In order to ensure an accurate description of the circuit and the construct as a whole, the digitalizer was formatted using the Synthetic Biology Open Language (Quinn et al, 2015; Madsen et al, 2019), an open standard for precise description of in silico biological designs and uploaded to the SynBioHub repository (McLaughlin et al, 2018). SBOL improves existing formats by allowing researchers to record more information on genetic circuits and share all its features in a machine-readable form (Roehner et al, 2016). While FASTA is used for representing plain sequences and GenBank adds the ability to annotate them, SBOL allows for specifying information such as the hierarchical structure of the circuit, its provenance and the connectivity between the non-DNA components (e.g. transcription factors) of the circuit. The SBOL design provided (Appendix Information) is focused on the hierarchical structure. The representation of the circuit was divided into two main modules: the one formed by Pm and downstream parts, and PA1_04S and downstream parts. Subdivision into smaller modules is shown at Appendix Fig S2. Note that a large number of tools (currently > 50) can be used to interact with SBOL designs to various extents at user's will (Appendix Information). Experimental validation of a digitalized expression device Performance of the pS238D•M construct was analysed by means of flow cytometry. Figure 4B and C shows that—as compared to the parental XylS/Pm system devoid of the digitalizing module—the new regulatory circuit displayed a considerable decrease in the basal levels while maintaining high expression marks at the induced state. Another important feature is that the majority of cells are effectively expressing GFP protein at earlier times (t = 5 min) after induction of the system (Fig 4C), indicating a faster onset of the device upon induction. Insertion of the digitalizing module also favoured cell-to-cell homogeneity as manifested in the reduction of CV values at any time along the whole course of GFP protein production (Fig 4D). Dose response of the digitalized system was also analysed under equivalent conditions as before (Appendix Fig S3). Sensitivity in this case was similar to that detected for the non-digitalized version since around 10 μM concentration of inducer was enough to trigger GFP production. Note that the total switch-like response might be affected by growth (Tan et al, 2009), although possible growth-r

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