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Dynamical analysis of mCAT2 gene models with CTN-RNA nuclear retention

This content has been downloaded from IOPscience. Please scroll down to see the full text. 2015 Phys. Biol. 12 016010 (http://iopscience.iop.org/1478-3975/12/1/016010) View the table of contents for this issue, or go to the journal homepage for more Download details: IP Address: 193.140.216.7 This content was downloaded on 25/04/2017 at 11:22 Please note that terms and conditions apply.

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Phys. Biol. 12 (2015) 016010

doi:10.1088/1478-3975/12/1/016010

PAPER

RECEIVED

19 August 2014

Dynamical analysis of mCAT2 gene models with CTN-RNA nuclear retention

ACCEPTED FOR PUBLICATION

18 December 2014 PUBLISHED

21 January 2015

Qianliang Wang1 and Tianshou Zhou1,2 1 2

School of Mathematics and Computational Science, Sun Yat-Sen University, Guangzhou 510275, People’s Republic of China Guangdong Province Key Laboratory of Computational Science and School of Mathematics and Computational Science, Sun Yat-Sen University, Guangzhou 510275, People’s Republic of China

E-mail: [email protected] Keywords: mCAT2 gene, CTN-RNA nuclear retention, sequestration model

Abstract As an experimentally well-studied nuclear-retained RNA, CTN-RNA plays a significant role in many aspects of mouse cationic amino acid transporter 2 (mCAT2) gene expression, but relevant dynamical mechanisms have not been completely clarified. Here we first show that CTN-RNA nuclear retention can not only reduce pre-mCAT2 RNA noise but also mediate its coding partner noise. Then, by collecting experimental observations, we conjecture a heterodimer formed by two proteins, p54nrb and PSP1, named p54nrb-PSP1, by which CTN-RNA can positively regulate the expression of nuclear mCAT2 RNA. Therefore, we construct a sequestration model at the molecular level. By analyzing the dynamics of this model system, we demonstrate why most nuclear-retained CTN-RNAs stabilize at the periphery of paraspeckles, how CTN-RNA regulates its protein-coding partner, and how the mCAT2 gene can maintain a stable expression. In particular, we obtain results that can easily explain the experimental phenomena observed in two cases, namely, when cells are stressed and unstressed. Our entire analysis not only reveals that CTN-RNA nuclear retention may play an essential role in indirectly preventing diseases but also lays the foundation for further study of other members of the nuclear-regulatory RNA family with more complicated molecular mechanisms.

1. Introduction The central dogma of molecular biology provides only a framework of genetic information flow from DNA to protein through mRNA. With the advent of single-cell and single-molecule measurement technologies as well as inexpensive DNA sequencing methods, moredetailed molecular mechanisms of gene expression have been uncovered, e.g., chromatin remodeling [1], alternative splicing [2], and RNA nuclear retention [3]. Biological experiments have verified that transcription occurs often in a burst manner (by burst we mean that mRNAs are produced in a period of high activity followed by a long refractory period) [4], and single-cell experimental measurements have also provided evidence for transcriptional bursting both in bacteria [5] and in eukaryotic cells [6]. This kind of burst can result in substantial variations (or noise) in mRNA and further protein numbers. In addition, experiments have shown that RNA nuclear retention has significant influence on the dynamics of gene © 2015 IOP Publishing Ltd

expression, especially in eukaryotic cells [3, 7–10], but relevant mechanisms are not clear. An important task in the post-genomics era is to reveal the contributions of different sources of noise to the phenotypic variability of genetically identical cells in homogeneous environments. More and more experiments have confirmed that RNA nuclear retention is not an exceptional but a ubiquitous phenomenon occurring during the process of gene expression. A subset of RNAs that carry out functions without being translated into proteins includes housekeeping RNAs and almost non-coding RNAs (ncRNAs). These ncRNAs may be short (100 pt or 200 pt) [11, 12]. In general, long ncRNAs, denoted lncRNAs, can perform regulatory roles in many aspects of gene expression [13, 14]. Especially in eukaryotic cells, a majority of ncRNA is nuclear-retained [15–18]. It has been shown that the percentage of ncDNAs in prokaryotes is less than 25% of the total genome, but this percentage in humans can be up to 98% and most of these ncRNAs

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Q Wang and T Zhou

contain active transcription units [19]. The increasing number of non-coding genes found in multi-cellular organisms makes the transcriptome complex. An earlier study indicated that, in many cell types, as much as 30% of the poly(A+) RNA is nuclear-retained and undetectable in the cytoplasm [15]. Another study, using an oligo deoxythymidine probe in mammalian cells, showed that a population of poly(A+) RNAs known as inter-chromatin granule clusters (IGCs) is enriched in nuclear speckles [19–22]. However, IGCs are not transcription sites but are thought to be involved in the assembly and modification and/or storage of the pre-mRNA processing machinery [23]. The poly(A+) RNAs in IGCs do not appear to be transported to the cytoplasm [22], as would be the case if they represented protein-coding mRNAs, implying that the corresponding RNAs constitute new members of the nuclear-regulatory RNA family. CTN-RNA, a mouse tissue–specific ∼8 kb nuclear-retained poly(A+) RNA, regulates the level of its protein-coding partner and is transcribed from the protein-coding mouse cationic amino acid transporter 2 (mCAT2) gene through alternative promoter and poly(A+) site usage [3]. An experiment has verified that CTN-RNA is diffusely distributed in the nucleus and is also localized at paraspeckles [3, 24]. A very long 3′ UTR of CTN-RNA contains some elements for adenosine-to-inosine editing, involved in CTN-RNA nuclear retention. Interestingly, knockdown of CTNRNA also down-regulates mCAT2 mRNA. In addition, upon cellular stress, CTN-RNA is post-transcriptionally cleaved to produce protein-coding mCAT2 mRNAs. These experimental facts reveal a hidden role of the cellular nucleus in harboring RNA molecules that generally are not needed to produce proteins but whose cytoplasmic presence is required quickly upon physiologic stress. Such a mechanism of action highlights an important role of a nuclearretained RNA transcript in regulating gene expression [3, 8, 9, 25]. To help understand the molecular mechanism related to mCTA2 gene expression, which is important for our mathematical modeling and analysis, we give more details here. CTN-RNA is an alternative transcript generated from the gene mCAT2 that encodes the protein mCAT2. CTN-RNA differs from canonical mCAT2 mRNA in that it uses a different promoter and a distal poly(A+) site (a very long 3′ untranslated region (UTR)) and is nuclear enriched. However, similar to mCAT2, CTN-RNA is spliced and contains the entire open reading frame of the mCAT2 protein. The fate of the two RNA species is quite different: mCAT2 RNA is normally exported and translated, whereas CTN-RNA is retained in the nucleus and near paraspeckles for some cell types [3, 9]. The key both to locating at the periphery of paraspeckles and to choosing a nuclear retention form lies in the long 3′ UTR of CTN-RNA, which contains double-stranded RNA hairpins formed by inverted repetitive elements. These 2

RNA hairpins are shown to be A to I hyper-edited and associated with DBHS proteins in vivo, similar to the case of linking nuclear retention of inosine-containing RNA to DBHS protein [26]. The fate of the CTN-RNA does not end in paraspeckles since the long 3′ UTR may be cleaved off, potentially via the paraspeckleassociated cleavage factor CFIm68 [27], which can respond to a variety of stress signals. The cleavage event is associated with a concomitant rise at a lower mCAT2 RNA level in the cytoplasm and with a pulse of protein production [3]. As the mCAT2 protein mediates the uptake of precursors through the nitrous oxide response pathway [28, 29], the retentionreleased mechanism allows the cell to rapidly mount a nitrous oxide response to stress. In spite of this, questions such as how CTN-RNA regulates its proteincoding partner, how the mCAT2 gene can maintain a stable expression, why most nuclear-retained CTNRNAs stabilize at the periphery of paraspeckles, and how CTN-RNA nuclear retention controls noise in mCAT2 gene expression remain unanswered from the standpoint of biophysics. In this paper, we address these questions. First we construct and analyze a simplified model for mCAT2 gene expression at the transcription level, which in particular considers the effect of CTN-RNA nuclear retention. We derive analytical expressions for probability distributions of the number of RNAs in three representative ranges of remaining probability from the chemical master equation (CME) and obtain a biologically reasonable estimator for remaining probability, which can be conveniently used to make predictions. It is significant that we find that CTNRNA nuclear retention always reduces the noise in pre-mCAT2 RNA. In addition, we find that CTNRNA nuclear retention is well able to control or mediate the noise in its coding partner (i.e., mCAT2 RNA): if the mean burst size defined as the number of transcripts per time unit is at a low level, then the retention probability pr changes in a small interval, implying that CTN-RNA nuclear retention plays a role in increasing the cytoplasmic mCAT2 mRNA noise in this case; if the mean burst size is at a high level, then pr changes in a large range, implying that CTN-RNA nuclear retention plays a role in attenuating the noise. It should be pointed out that the amount of CTN-RNA is essentially determined by its combination with other complexes, which results in a much slower degradation rate. More precisely, the nuclear-retained CTN-RNA forms a very stable complex by combining with an unspecified substance, which accumulates at the periphery of paraspeckles and accompanies the entire transcriptional process of the mCAT2 gene. This is a main reason why the amount of CTN-RNA is much greater than that of mCAT2 RNA [3, 8–10]. We conjecture that this substance is a stable heterodimer, named p54nrb-PSP1 throughout this paper, which most likely binds to the site of the CTN-RNA.

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Figure 1. Schematic diagram for a basic mCAT2 gene model incorporating the effect of CTN-RNA nuclear retention. It is assumed that the gene promoter has one ON state, where transcription is highly efficient, and one OFF state, where transcription is inhibited; the mature mRNAs formed from pre-mRNAs are divided into two parts in a probabilistic manner: CTN-RNA and mCAT2 RNA. The former, which is ∼8 kb, uses the poly(A+) site, and has a much longer 3′ untranslated region (UTR), is kept in the nucleus; whereas the latter, which is ∼4.3 kb, uses the nearer poly(A+) site, and has a shorter 3′ UTR, is first exported to the cytoplasm and then translated into proteins.

Based on this conjecture, we construct a dynamical sequestration model by which we can easily interpret how CTN-RNA regulates its protein-coding partner (i.e., mCAT2 RNA) and obtain results in good accord with experimental observations [3, 10]. In addition, based on the structures of CTN-RNA and mCAT2 RNA as well as our numerical simulation, we predict two possible ways for binding sequestering proteins, demonstrating that the mixed way (i.e., one CTNRNA may alternatively choose one or two heterodimers) may have better flexibility when cells confront fluctuating environments. Finally, we analyze a full model of mCAT2 gene expression that considers not only dynamic transcription but also the dynamic translation process, by distinguishing two cases: unstressed (or normal) and stressed cells. We first show that CTN-RNA nuclear retention can indirectly regulate the noise level of mCAT2 protein in the normal case. Then, for the case where cells are subject to high stress, we characterize the dynamics of the CTN-RNA; that is, the cleavage of the CTN-RNA by CFIm68 and its fast release to the cytoplasm can substitute for mCAT2 RNA in generating those proteins that can keep the mCAT2 gene at a stable level. We argue that such a latent mechanism of CTN-RNA may play an essential role in indirectly preventing diseases. Our model and analysis may lay a foundation for further studying other members of the nuclear-regulatory RNA family with more complicated molecular mechanisms.

2. A simplified model reveals the essential role of CTN-RNA nuclear retention in controlling mCAT2 expression noise To show clearly how CTN-RNA nuclear retention regulates mCAT2 expression noise, here we analyze a simplified mCAT2 gene model at the transcriptional 3

level (see figure 1). First, based on the range of the mean mCAT2 gene expression level combined with model analysis, we derive a reasonable estimator for CTN-RNA nuclear retention probability, which can be used to make a prediction. Then we reveal the mechanism of how CTN-RNA nuclear retention controls the mCAT2 RNA noise. In the next section, we introduce a more realistic model, which considers not only CTN-RNA nuclear retention but also the role of relevant proteins around paraspeckles. 2.1. An estimator for CTN-RNA nuclear retention probability Let M p represent pre-mCAT2 RNA transcribed from the mCAT2 gene (D ). Assume that the pre-mCAT2 RNA generates two mature mRNAs, denoted Mr and Mc , where the former is retained in the nucleus, whereas the latter is first exported to the cytoplasm and then translated into the mCAT2 protein Pc [3, 24]. Refer to figure 1. First we focus on the transcription process. The related biochemical reactions take the form k0

ki

di

D⟶D + B × M p , M p ⟶Mi , Mi ⟶ϕ ;

(1)

where i = r , c . The first reaction describes how the active mCAT2 gene (D) is transcribed into pre-mRNA with a burst frequency (k 0 ) and a burst size (B ), both of which together characterize bursting dynamics. The second reaction represents that one part of the premRNAs still stays in the nucleus but finally forms CTN-RNAs (Mr ) at a rate k r = k1 pr , whereas the other part forms mCAT2 RNAs (Mc ), which are finally exported to the cytoplasm from the nucleus at a rate k c = k 2 (1 − pr ), where pr represents the probability that pre-mRNAs remain in the nucleus (called the remaining or retention probability throughout this paper; pr ∈ [0,1) ), k1 represents the net rate at which M p forms Mr , and k 2 is the total rate, which consists of

Phys. Biol. 12 (2015) 016010

Q Wang and T Zhou

Figure 2. Feasible ranges of CTN-RNA nuclear-retention probability: (a) the change in the remaining probability pr with two parameters, where different colors represent different values of pr . (b) The influence of ωr on pr , where the inset shows the dependence of pr on dr . The parameter values are set as k1 = 0.8, k 2 = 1.6 ∼ 3.2, dr = 0.04 ∼ 0.02, d c = 0.4 .

two parts: the rate of M p forming Mc and the rate of Mc exported from the nucleus to the cytoplasm. The last reaction describes the degradation process, with d i representing the degradation rate of Mi . Assume that transcriptional burst size (B) is a discrete random variable obeying a discrete distribution Prob{B = i} = αi with i ∈ {0,1, ⋯}, where αi are constants valuated in the interval [0,1]. If α1 = 1 and αk ≡ 0 for all k ≠ 1, implying that B ≡ 1 holds, where the symbol ⋅ denotes expectation, then this case corresponds to so-called constitutive expression [30–33]. The other cases correspond to bursty expression [34–41]. Thus, we can perform model analysis in a unified framework. Let P (m⃗ ; t ) represent the probability that M p has m p copies of nuclear pre-mRNA, Mr has mr copies of nuclear CTN-RNA, and Mc has mc copies of cytoplasmic mCAT2-RNA at the moment t, where m⃗ = (m p , mr , mc ). Then the chemical master equation (CME) for the entire system takes the form

(

∂P m⃗ ; t ∂t

m ⎞ ) = k ⎛⎜ ∑ α E −i − I ⎟ P p

0⎜

⎝ i=0

+

∑ j ∈ {r , c }

i mp

{k (E j

⎟ ⎠

−I

PM p ( m p ) =

)

× ⎡⎣ m p P m⃗ ; t ⎤⎦ + d j Em j − I

( ) ( × ⎡⎣ m j P ( m⃗ ; t ) ⎤⎦ }

) (2)

where Em j denotes a shift operator with the inverse Em−j1 ( j = p, r , c) and I is the identity operator. Note that in the constitutive expression case, it is not difficult to show that three kinds of mRNAs obey the following Poisson distributions in the steady state: 4

mp!

e −λ p,

λ rmr −λ r e , mr ! λ mc PMc ( m c ) = c e −λ c mc ! PMr ( m r ) =

(

(3)

)

where λ p = k 0 k r + k c , λ r = λ p ⋅ k r d r and λc = λ p ⋅ k c d c . Thus, the steady-state averages and variances for three mRNAs are easily given, with each being expressed as a function of the parameters λ p , λr , and λC . In the bursty expression case, however, we in general cannot derive the analytical mRNA distributions but can give steady-state mRNA means of the form

m¯ p =

k0 B

( kc + kr )

=

k0 B ⎡k − k − k p ⎤ 2 1 r⎦ ⎣ 2

(

)

and m¯ j =

( m⃗ ; t)

−1 m p Em j

mp

λp

kj dj

m¯ p

(4)

where j = r , c . From experimental data in references [3, 42], we find that there is an approximate relation between the mean amount of CTN-RNA and that of mCAT2 RNA, that is, 〈m¯ r 〉 ≈ 2 〈m¯ c 〉. In addition, it is reasonable to assume k 2 > k1 and d r > d c , implying that the nuclear CTN-RNA is more stable than mCAT2. Thus, if denoting k 2 = αk1 and d c = βd r as well as combining the experimentally estimated relation 〈m¯ r 〉 ≈ 2 〈m¯ c 〉 with equation (4), where α is 2 ∼ 4 and β is 10 ∼ 20 according to references [3, 8, 9], then we arrive at the following approximate expression for remaining probability:

Phys. Biol. 12 (2015) 016010

pr ≈

Q Wang and T Zhou

2 2 = 2 + (β α) 2 + ωr / ωc

(

(5)

)

where ωr = k1/d r , ωc = k 2/d c . According to the experimental range of α and β, we thus conclude that the maximal remaining probability is not more than 66.7%. On the other hand, we find numerically that pr is between 16.7% and 44.4%, which is consistent with previous statistical results [43–45]. Refer to figure 2(a), where different colors represent different values of pr . Therefore, equation (5) not only gives a reasonable estimator for remaining probability but also can help experimental biologists make predictions. In general, it is not easy to experimentally measure remaining probability for in vivo organisms, but equation (5) provides a method for estimating this probability since equation (5) involves only two experimentally measureable parameters. Experiments and our calculations both indicate that merely a small part of the pre-mCAT2 RNAs stays in the nucleus (about 10% ∼ 40% of the average level for most organisms [43–45]), although the number of CTN-RNAs is about twice that of the mCAT2 RNA number in the steady state. In general, the degradation of CTN-RNA is much slower in contrast with that of complexes formed by the association of nuclear CTN-RNA with other chemical substances. In fact, it has been experimentally verified that mCAT2 RNAs decay out within 6 h, whereas only a small part of the complexes decays out after 9 h [3, 9, 10]. This indicates that the larger number of CTNRNAs in the nucleus in the steady state does not necessarily imply that the remaining probability is at a high level. From figure 2(a), we observe that if β is more than 16 and α is less than 2.5, the remaining probability will change little, implying that a slower degradation rate results in a more reliable remaining probability. This may be advantageous for producing a more stable number of mCAT2 RNAs. However, if β is fixed between 10 and 16, the remaining probability will increase as α increases. From the standpoint of biology, this is reasonable since an increase in α is often followed by an abrupt increase in burst size during a certain time interval, which leads to the production of more pre-mCAT2 RNAs. In turn, increasing α can speed the processing of these pre-mCAT2 RNAs, and at the same time, increasing the remaining probability can increase the amount of nuclear-retained CTNRNAs, thus helping the mCAT2 RNA take on part of the superfluous pre-mCAT2 RNAs. From equation (5), we see that the remaining probability pr is determined by the ratio of the CTNRNA birth/death rate over the mCAT2 RNA birth/ death rate. In general, ωc changes less than ωr , so we can set ωc as a constant. Thus, it follows from equation (5) that pr is approximately in inverse proportion to ωr . The numerical simulation also verifies such an approximate relation; see figure 2(b), where the solid line represents the theoretically estimated relation between pr and ωr , whereas the red circles 5

represent the dependence of pr on ωr obtained under a stochastically fluctuating condition. To show how the degradation rate d r affects the remaining probability, we provide a subfigure, figure 2(c), where the arrow represents the change trend of the percentage of nuclear retention transcripts as the relevant complex involving CTN-RNA nuclear retention gradually becomes stable. From this subfigure, we observe that the remaining probability decreases with slackening d r . A slower d r implies that the amount of the complex formed by the combination of CTN-RNA with other chemical substances becomes more stable, whereas the percentage of RNAs retained in the nucleus becomes lower. This is reasonable since a larger ωr corresponds to either a larger birth rate or a smaller death rate, indicating that the CTN-RNA number gradually increases with increasing ωr . Thus, suitably reducing the remaining probability can guarantee not only that the mCAT2 RNA produces adequate proteins but also that the nuclear-retained CTN-RNA number is not so large that it influences its normal function. We point out that the degradation rate of CTN-RNA, d r , is small in most cases (referring to an experimental result in reference [3]). However, the d r used here is not the natural degradation rate of CTNRNA but the mean value of all the CTN-RNA degradation rates in the cellular nucleus. 2.2. Effect of CTN-RNA nuclear retention on the mCAT2 RNA noise From the CME (2), we can readily derive the following ordinary differential equations (ODEs) for secondorder moments [46]: d m p2 = k 0 B 2 + 2k 0 B m p + kr + kc dt (6a) × m p − 2 kr + kc m p2

(

(

)

)

d m j2 = k j m p + 2k j m p m j + d j m j dt (6b) − 2d j m j2 d m p m j = k 0 B m j + k j m p2 − m p dt (6c) − kr + kc + d j m p m j

(

(

)

)

where j = r , c . From equation (6), we can further derive analytical expressions for steady-state moments and for mRNA noise (by noise we mean that it is defined as the ratio of variance over mean). The results are 1

ηm2 p =

m¯ p 1

ηm2 j =

where Be =

m¯ j

( 1 + Be ) , ⎛ ⎞ k j Be ⎜⎜ 1 + ⎟ kr + kc + d j ⎟⎠ ⎝

( 〈B 〉 − B ) 2

(7)

(2 B ) and j = r , c . Note that the total nuclear noise is quantified by ηm2 p + ηm2r

Phys. Biol. 12 (2015) 016010

Q Wang and T Zhou

whereas the cytoplasmic noise is quantified by ηm2c . Also note that the relation 2 〈m¯ c 〉 ≈ 〈m¯ r 〉 implies the B estimation ηm2r < 1 + eB 1 +1 α ηm2 p < 13 ηm2 p . Thus, we cone

clude that accumulating CTN-RNAs in the nucleus can attenuate the total nuclear noise level. This conclusion can be simply explained as follows: The nuclear retention of CTN-RNA always reduces the noise in pre-mCAT2 RNA. The total nuclear noise consists of two parts: the pre-mCAT2 RNA noise and the CTN-RNA noise. However, the noise generated from the nuclear retention of CTN-RNA makes only a slight contribution to the total nuclear noise since the former level is below 30% of the latter level. In the extreme case where CTN-RNA nuclear retention does not exist, i.e., pr = 0, the noise intensities for the pre-mRNA (m p ) in the nucleus and for the mature mRNA (Mc ) in the cytoplasm can be calculated according to η˜m2 p = η˜m2 c =

1 ˜p m 1 ˜c m

( 1 + Be ) , ⎛ k 2 Be ⎞ ⎜1 + ⎟ k2 + dc ⎠ ⎝

(8)

˜ p = k 0 B k 2 and m ˜ c = k 0 B dc . where m Similar formulas were also given in references [39, 47]. By calculation, we find that the inequality ηm2 p η˜m2 p < 1 always holds for any values of system parameters. This indicates that CTN-RNA nuclear retention always reduces the nuclear CTN-RNA noise. Since RNA nuclear retention occurs in most eukaryotic cells and in many prokaryotic cells, we conclude that previous results obtained using those gene models that did not consider RNA nuclear retention overestimated the pre-mRNA noise [34–41]. Note that transcriptional noise consists of two parts: so-called promoter noise, generated mainly by stochastic switching between promoter activity states [38, 39, 41], and intrinsic noise, due to synthesis and degradation of mRNAs. Thus, the noise in the qualitative conclusion that RNA nuclear retention always reduces the pre-mRNA noise is actually the promoter noise. Therefore, the transcriptional noise in real cells might not be as large as we thought. However, it is not easy to determine which of ηm2c and η˜m2 c is larger in general. From equation (7) combined with equation (8), we can see that the noise level of mature mRNA mc relies on both mean burst size B and remaining probability pr , which in general takes values between 0.1 and 0.6 for most eukaryotic cells [43–45]. Figure 3(a) shows this dependence relationship for a fixed yet representative ratio, k 2/k1 = 4, where color changes represent changes in the ratio ηm2c η˜m2 c . Note that the ratio ηm2c η˜m2 c = 1 is naturally a boundary for distinguishing whether RNA nuclear retention exists. We observe from figure 3(a) that if the mean burst size is at a low level (e.g., B = 1 ∼ 5 ),

6

then the retention probability should change in a small interval (e.g., from 0.2 to 0.4). This is because, at the beginning of transcription, the number of two mature mRNAs formed from the same pre-mCAT2 RNAs is not very large due to a small mean burst size. Therefore, only a small part of the CTN-RNA is kept in the nucleus, implying that pr cannot take a large value. In this case, we have ηm2c > η˜m2 c , which implies that CTNRNA nuclear retention plays a role in increasing the cytoplasmic mCAT2 mRNA noise. If the mean burst size is at a high level (e.g., B ≫ 5 ), then pr should change in a large range (e.g., from 0.4 to 0.65). This is because in this situation, the number of all the mature mRNAs formed from pre-mCAT2 RNAs becomes large due to a large mean burst size. Therefore, there may be many nuclear-retained mRNAs that have a greater chance to form CTN-RNAs. In this case, CTNRNA nuclear retention plays a role in attenuating the cytoplasmic mCAT2 RNA noise. The foregoing analysis actually reveals an essential mechanism of how CTN-RNA nuclear retention mediates the cytoplasmic mature mRNA noise. To better understand this point, here we further specify the CTN-RNA nuclear retention process according to the size of the remaining probability pr : the region that pr is in between 0 and 0.15 is defined as ‘slow retention’, the region that pr is in between 0.15 and 0.4 is defined as ‘moderate retention’, and the region that pr is in between 0.4 and 0.65 is defined as ‘fast retention’. Thus, figure 3(b) demonstrates the foregoing mechanism in a more intuitive fashion. Based on this mechanism, we conclude that the CTN-RNA nuclear retention process can effectively mediate the noise level of its protein-coding partner mCAT2 RNA to a suitable range that is different from the one adjusted by the burst size since the latter is wider than the former. We will see that this mechanism has a great impact on the downstream mCAT2 protein noise as well.

3. Construct and analysis of a sequestration model Experiments [3, 10] have verified that CTN-RNA can positively regulate the expression of nuclear mCAT2 RNA in a sequestration manner, but what the sequestrating molecule is, as well as what the molecular mechanism for sequestration is, is not clear. Here, based on another experimental report [48] combined with discussions in reference [3], we speculate that this sequestrating molecule is a heterodimer formed by two proteins, p54nrb and PSP1, and named p54nrbPSP1 (see figure 4). Thus, we construct a sequestration model on the basis of the previous model. In particular, we analyze the CTN-RNA dynamics and obtain results that are able to explain the experimental phenomena.

Phys. Biol. 12 (2015) 016010

Q Wang and T Zhou

Figure 3. The dependence of the ratio ηm2c η˜m2 c on both the mean burst size B and the remaining probability pr : (a) the quantitative result for a fixed ratio k 2 k1; (b) the qualitative result, where the CTN-RNA nuclear retention process is specified by three regions divided according to the size of pr .

Figure 4. Schematic diagram for a sequestration model that shows that nuclear-retained CTN-RNAs can positively regulate the expression of the mCAT2 gene by sequestering a heterodimer (named p54nrb-PSP1): (a) the normal case; (b) the case where most of the nuclear CTN-RNAs are knocked down. See the main text for an explanation of the relevant molecular mechanisms.

3.1. Model construct and simple analysis As is well known, the mCAT2 gene produces premCAT2 RNAs, part of which are kept in the nucleus (named CTN-RNAs) and the other part of which form mature mCAT2 RNAs, which are first exported to the cytoplasm and then translated into functional proteins. Note that both CTN-RNA and mCAT2 RNA have a common 3′ UTR, which has an important influence on the stability of RNAs [3, 8–10]. In the case where cells are unstressed, both nuclear-retained CTN-RNAs and heterodimers near paraspeckles can form stable complexes. These complexes accumulate on the outside of the paraspeckles so that the 7

heterodimers cannot be freely diffused inside the cells, resulting in fewer chances to bind to the site of nuclear mCAT2 RNAs (see figure 4(a)). In this case, mCAT2 RNAs are normally exported to the cytoplasm and carry out their translational function therein. In contrast, in the case that cells are stressed, if most of the nuclear CTN-RNAs are knocked down by experimental means, then the heterodimers formed around the paraspeckles can be freely diffused in the cells and combine with those mCAT2 RNAs that are not exported to the cytoplasm, leading to the proteincoding transcripts quickly degrading out (see figure 4(b)). In this case, the number of those mCAT2

Phys. Biol. 12 (2015) 016010

Q Wang and T Zhou

RNAs that carry out translational functions in the cytoplasm can be greatly reduced. We will show that in each case, the mCAT2 RNA expression level is reduced in a manner in agreement with experimental observation [3], mainly due to the effect of sequestration. In this way, CTN-RNA positively regulates the mCAT2 RNA level. Upon considering the sequestration effect of the conjectured heterodimers, the preceding simplified mCAT2 gene model should be modified to the following more general model: k0

kr

D⟶D + B × M p , M p ⟶Mr , k c res

dr

dc

M p ⟶Mc , Mr ⟶ϕ , Mc ⟶ϕ v1

d r*

Mr + n1 E ⇌ En1 Mr ⟶ϕ , v−1 v2

d c*

(9)

Mc + n 2 E⟶En 2 Mc ⟶ϕ

In equation (9), pre-mCAT2 RNA (M p ) produces two species: CTN-RNA (Mr ) and mCAT2 RNA (Mc ). From an experiment [3], we know that the 3′ UTR sequence is common to both mCAT2 mRNA and CTN-RNA and is a ∼1.5 kb region. Thus, it is reasonable to assume that there exists a complex, denoted by E, which can influence or regulate the stability of mCAT2 mRNA since both CTN-RNA and mCAT2 RNA compete for binding to a factor or factors. According to the literature, this complex is most likely a heterodimer, formed by two proteins, p54nrb and PSP1, but there is not a large amount of it since those PSP1s that form heterodimers with a subset of the total cellular pool of p54nrb occupy only a small fraction of the total cellular pool [48]. More precisely, p54nrb is expressed only at a higher level than PSP1 [48]. In general, CTN-RNA forms a relatively stable complex with the heterodimer due to its stronger competition for binding to the heterodimer, represented by En1 Mr , which is located at the periphery of paraspeckles [3, 8–10]. In equation (9), the reversible reaction represents that one CTN-RNA may combine with n1 heterodimers in the nucleus to form a complex with a much slower degradation rate d r* (i.e., d r ≫ d r* ). In contrast, one mCAT2 RNA may rapidly combine with n2 heterodimers in the cytoplasm to form another complex. If most of the CTN-RNAs are knocked down, the heterodimers can diffuse from the paraspeckles and seize mCAT2 RNA to form first one complex, denoted by En2 Mc , and then another complex that is likely quite unstable and has a much faster degradation rate d c* (i.e., d c* ≫ d c ). In general, n1 and n2 are unknown due to a lack of supporting experimental data. Based on the framework of the foregoing sequestration model, we will show that there are two feasible yet representative cases for the ratio n1 n2 that occur most likely in real biological systems. 8

Note that we are interested mainly in regulation of mCAT2 RNA by CTN-RNA in the nucleus. Therefore, the corresponding sequestration model, which is shown schematically in figure 4(a), can be described by the following reactions: k0

kr

D⟶D + B × M p , M p ⟶Mr , kc

d1

dc

M p ⟶Mc , Mr ⟶ϕ , Mc ⟶ϕ , d r*

v1

Mr + n1 E ⇌ En1 Mr ⟶ϕ

(10)

v−1

We argue that this model is able to characterize the dynamics of nuclear CTN-RNA that positively mediate mCAT2 RNA. In fact, since the nuclear CTNRNA sequesters most heterodimers in the cell, the mCAT2 RNA can be effectively exported to the nucleus and further translated into functional proteins. Thus, the dynamics of the complex En1 Mr can be approximately described by d ⎡ ⎤ ⎡ ⎤ n ⎣ En1 Mr ⎦ = v1 ⎣ Mr ⎦ [E] 1 dt

(

− v−1 +

d r*

)

⎡ En Mr ⎤ ⎣ 1 ⎦

(11)

where Mr represents the free nuclear CTN-RNA, [.] denotes the number of species in the cell, and En1 Mr satisfy a conservation condition ⎡ Mr ⎤ ⎡ ⎤ ⎡ ⎤ ⎣ ⎦total = ⎣ Mr ⎦ + ⎣ En1 Mr ⎦

(12)

Assuming that this reaction quickly reaches equilibrium, we have ⎡ Mr ⎤ ⎣ ⎦total = m¯ r ,

(v

−1

+ d r* ⎡⎣ En1 Mr ⎤⎦ = v1 ⎡⎣ Mr ⎤⎦ [E]n1

)

Furthermore, if v−1 ≫ d r*, then v

⎡M ⎤ ⎣ r⎦ [Mr ]total

≈

(13) 1 , 1 + κ r [E]n1

where v 1 = κr . Because of k r ≫ 1, E ≫ 1, and n1 ⩾ 1 −1 [3, 10], we thus conclude that the free nuclear CTNRNA occupies only a small fraction of the total nuclear CTN-RNA, implying [En1 Mr ] ≈ [Mr ]total = m¯ r . In other words, most of the CTN-RNAs bind to the heterodimers and stably locate at the periphery of the paraspeckles. This result is in agreement with experimental observation [3]. On the other hand, according to reference [42], we are given to know [En1 Mr ] ≈ 2 m¯ c . Note that the amount of mCAT2 RNA product is more than that of CTN-RNA product per time unit since the remaining probability pr is usually less than 1/2 [43–45] during nearly the entire transcription process. Or likewise, the amount of CTN-RNA is about twice (or even more than twice) that of mCAT2 RNA since the stable complex (En1 Mr ) accumulates near the paraspeckles, causing it to be less depleted than the mCAT2 RNA. Here we simply explain ‘less depleted’ as follows: the mean CTN-RNA degradation rate is in general a function of two variables, d r and d r*, that is, d˜r = f (d r , d r* ). Because d r*

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Q Wang and T Zhou

is close to zero, it is a given that the degradation of CTN-RNA is expected to be a very slow process, implying d˜r ≪ d c . Also, based on the experimental data [3, 8, 9], we know that if ∼80% of the nuclear CTN-RNAs are knocked down, the amount of mCAT2 RNA will decrease by ∼50%. Since our interest is mainly in the process of releasing the heterodimers that are around the paraspeckles and seizing mCAT2 RNAs, from figure 4(b) we have d ⎡ # ⎤ ⎣ En 2 Mc ⎦ = v2 Mc rb [E] dt

(

n2

)

− d c* ⎡⎣ En 2 Mc ⎤⎦ (14)

which characterizes the complex dynamics of this processes, where Mc# = (1 − res )[Mc ], in which res represents the probability that mCAT2 RNA escapes successfully from the nucleus to the cytoplasm. Owing to the interaction of E with other RNAs in the process in which the heterodimers are released from the paraspeckles [48], the mean diffusion rate of the heterodimers may not be faster than the exporting rate of the mCAT2 RNA from the nucleus to the cytoplasm [45, 47, 48], implying that not all the heterodimers releasing from the paraspeckles bind to the nuclear mCAT2 RNAs. Assume that the rate for this binding is rb, where 0 < rb < 1. In the steady state, we can see from equation (14) that the number of nuclear En2 Mc is so small that it can be omitted. Since equations (11)– (13) hold for ∼20% CTN-RNA survival, this implies * [Mr ]total ≈ m¯ r 5. In addition, most mCAT2 RNAs exist in the cytoplasm, and the ∼80% amount of E sequestrated by CTN-RNAs is in theory equal to the ∼50% amount of the depleted mCAT2 RNA, so we have an approximate relationship * [Mc ]total ≈ m¯ c 2. The combination of these implies that the approximate ratio in the steady state is given by n1 n2 ≈ 50% m¯ c 80% m¯ r , which yields

n1/n2 ≈ 1 3 due to m¯ r ≈ 2 m¯ c . On the other hand, based on the structures of CTN-RNA and mCAT2 RNA combined with feasible values of the parameter rb [3, 10], we can first show that there are two possible ratios, n1/n2 = 1/2 and n1/n2 = 2/2, and then give possible sites for the heterodimer binding to CTN-RNA or mCAT2 RNA. Refer to figure 5(a). In fact, if rb ≈ 60% ∼ 65%, then we have n1/n2 = 1/2. This means that one heterodimer (in fact, p54nrb-PSP1) binds to a certain site either at the 3′ UTR of the CTN-RNA in the case of unstressed cells or at the 3′ UTR common to the CTNRNA and the mCAT2 RNA. Refer to case (1) in figure 5. If most of the nuclear CTN-RNAs are knocked down, the heterodimers are first released from the paraspeckles and then become so free that they can seize nuclear mCAT2 RNAs and bind to the common 3′ UTR of the mCAT2 RNAs near the 5′ UTR, leading to the fast degradation of the mCAT2 RNAs. If rb ≈ 30% ∼ 35%, then we have n1/n2 = 2/2. Refer to case (2) in figure 5. In this case, one 9

heterodimer may bind to a certain site at the longer 3′ UTR, similar to case (1), whereas the other heterodimer may bind to the 5′ UTR or to the same longer 3′ UTR. Thus, figure 5(a) gives an accurate interpretation of how the common 3′ UTR governs the stability of mCAT2 RNA quantitatively. Moreover, the most likely scenario is that two kinds of rates exist simultaneously. The reason is that the CTN-TNA is well able to sequester the heterodimers according to its own quantity and can regulate its protein-coding partner well. Figures 5(b) and (c) show numerical results for the time evolution of the numbers of some species of molecules (e.g., RNA and complex molecules). In these two subfigures, we assume that the transcription products of about 60 genes sequester the heterodimers in each cell, whose mean number is from 350 ∼ 400; the transcriptional process for each gene follows a geometric distribution, with the mean burst size being 8. Under condition (b) in figure 5, the Gillespie stochastic simulation shows that the total amount of all the CTN-RNAs, including the complexes and the free CTN-RNAS, in the nucleus is about 240, whereas the number of transcription products in the cytoplasm is about 120. The rate between them is in agreement with an experimental result [3]. To model another experimental result in reference [3], we allowed the CTNRNAs to degrade rapidly and allowed the combining rate between the CTN-RNAs and the heterodimers as well as the generating rate of the mCAT2 RNA in the cytoplasm to be reduced simultaneously. As a result, the number of all the CTN-RNAs is about 40, whereas the number of mCAT2 RNAs in the cytoplasm is about 70. The rate between them is also in agreement with experimental results [3, 10]. Similar results hold for condition (c) in figure 5. Finally in this subsection, we describe from the standpoint of dynamics how the amount of mCAT2 RNA is affected by that of CTN-TNA. If most of the CTN-RNAs are knocked down, then according to our proposed sequestration model, we know that the transcriptional noise, i.e., the noise in the mCAT2 RNA in the cytoplasm, is increased. This is mainly because the mean number of mCAT2 RNAs in the cytoplasm is greatly reduced. Such a reduction would result in abnormal noise in the downstream nitric oxide pathway, which may induce some diseases [49, 50]. Therefore, the fluctuation in the number of nuclear CTNRNAs in general cannot be very large. Our model analysis implies that CTN-TNA nuclear retention would play a significant role in mediating the protein-coding partner mCAT2 RNA noise. 3.2. Nuclear-retained CTN-RNA can dynamically control mCAT2 gene expression As is seen from the foregoing analysis, nuclear-retained CTN-RNA can regulate not only nuclear mCAT2 RNA but also cytoplasmic mCAT2 RNA. In this subsection,

Phys. Biol. 12 (2015) 016010

Q Wang and T Zhou

Figure 5. (a) Schematic diagram for possible sites and ways for binding the predicted heterodimer, based on two kinds of structures of CTN-RNA. Two cases are shown: in case (1), the CTN-RNA permits the binding of one heterodimer but has two possible binding regions; in case (2), the CTN-RNA permits the binding of two heterodimers. Here label 1 represents a longer 3′ UTR region but there are two choices for binding, whereas label 2 represents a shorter 3′ UTR region for binding at different regions. (b) and (c) Timeevolutional molecule numbers of some species (including RNAs and complexes), obtained by the Gillespie algorithm [51, 52]. Here (b) is for unstressed cells. In this case, the CTN-RNA sequesters almost all the heterodimers, so there is little chance for the heterodimers to bind to the site of the mCAt2 RNA. Figure (c) is for stressed cells. In this case, most of the CTN-RNAs are knocked down, so the heterodimers easily bind to the site of the mCAT2 RNA but rapidly degrade out. The parameter values are set as follows: k 0 = 1.2, D = 60; B = 8, k1 = 1.5, pr = 0.4, d c = 0.6, E = 350 ∼ 400; (b) k 2 = 3, d1 = 0.24, v1 = 3.6, v−1 = 0.06, dr* = 0.004; (c) k 2 = 1.8, d1 = 3, v1 = 0.6, v−1 = 0.03, dr* = 0.0001, v2 = 6, d c* = 1.6 .

Figure 6. Schematic diagram for an mCAT2 gene model that considers CTN-RNA nuclear retention and paraspeckles in the normal case, i.e., a cell is not invaded by virus, where the transcriptional and translational processes of the mCAT2 gene are also shown.

we will show how the expression level of the mCAT2 gene is qualitatively and quantitatively controlled by CTN-RNA. To this end, we will consider two cases separately: unstressed and stressed cells, since the corresponding molecular mechanisms are different. 10

3.2.1. No-stress case Figure 6 shows molecular details for the transcriptional and translational processes of the mCAT2 gene in the normal case (i.e., the cell is unstressed). Based on this figure, the dynamics of the mCAT2

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Q Wang and T Zhou

gene expression can be described by the following reactions: k0

D⟶D + B × M p , M p ⟶Mr , dc

d1

M p ⟶Mc , Mc ⟶ϕ , Mr ⟶ϕ , d r*

v1

v−1

δc

Mc ⟶Mc + Pc , Pc ⟶ϕ ;

(15)

where Pc represents mCAT2 protein. The parameters νc and δc are translation and degradation rates, respectively. Compared with equation (10), this model changes only the form of those reactions that describe CTN-RNA dynamics in the nucleus since the ‘form’ has a tiny effect on the number of CTN-RNAs (data are not shown here). Here we use kr

dr

two simple reactions, M p ⟶Mr and Mr ⟶ϕ, to describe the CTN-RNA nuclear retention process, where d r ≪ d c . Let P x ⃗; t represent the probability of possessing m p molecules of nuclear premCAT2 RNA, mr molecules of nuclear CTN-RNA, mc molecules of cytoplasmic mCAT2 RNA, and nc molecules of mCAT2 protein at a time t, and denote x ⃗ = ( m p , mr , mc , nc ). The CME for the full system takes the form

( )

( )

∂P x ⃗ ; t ∂t

⎛ mp ⎞ = k 0 ⎜⎜ ∑α i Em−pi − I ⎟⎟ P x ⃗; t ⎝ i=0 ⎠ ⎡ −1 + νc En c − I P ⎣ m c P x ⃗; t ⎤⎦ + δc En c − I ⎡⎣ nc P x ⃗; t ⎤⎦ +

) )

∑

{k (E j

−I

ηn2c =

+ δc ⎡⎣ δc + d c m¯ c ⎤⎦ (19) 

(

1 n¯ c  protein fluctuations

)

mRNA fluctuations

which consists of two parts: one is the internal fluctuation of mCAT2 protein, characterized by 1 n¯c ; the other results from mCAT2 RNA fluctuation, which is not equal to zero, implying that the mCAT2 protein number does not follow a Poisson distribution. In the bursty expression case that corresponds to Be ≠ 0, since the condition k r + k c > d c ≫ δc in general holds due to the fact that the half-life of mCAT2 protein is in general longer than that of mCAT2 RNA, we obtain an approximate formula for calculating the protein noise: ηn2c ≈

(1n¯ ) c

protein fluctuation

(

)(

)(

)

(20)

mRNA fluctuation

( ) ( )

−1 m p Em j

m¯ c

+ δc d c 1 + γBe 1 m¯ c 

( )

( (

n¯ c

δc δc + d c

In particular, in the constitutive expression case that corresponds to Be = 0, we have

Mr + n1 E ⇌ En1 Mr ⟶ϕ , vc

1

+

⎡ ⎛ ⎞⎤ Be k c dc × ⎢1 + ⎜1 + ⎟ ⎥ (18) kr + kc + d c ⎝ kr + kc + δc ⎠ ⎦ ⎣

kr

kc

1

ηn2c =

where γ = α ⎡⎣ α + pr (1 − pr ) ⎤⎦ . If the effect of CTNRNA nuclear retention is not considered, i.e., pr = 0, then equation (20) becomes ηn2c ≈

)

1 n¯ c

+

δc 1 + Be , dc m¯ c

which is consistent with the result from the previous theory [47]. Note that pr in general is no more than 2 3, implying α (α + 2) < γ < 1. In particular, we × ⎡⎣ m j P x ⃗; t ⎤⎦ (16) have 2 3 < γ < 1 since α is between 2 and 4. Now we state qualitative results for how CTNfrom which we can directly derive the following ODEs RNA nuclear retention affects the mCAT2 RNA noise and, further, the mCAT2 protein noise. It is [46]: easy to see from the preceding analytical expression d nc dt = νc m c − δc nc (17a) that CTN-RNA nuclear retention always reduces the mCAT2 RNA noise and, further, the mCAT2 prod nc2 dt = νc m c + 2νc m c nc tein noise. This implies that (i) previous gene mod+ δc nc − 2δc nc2 (17b) els [47, 53–57] overestimate the protein noise; (ii) one advantage of RNA nuclear retention is that it d m c nc dt = kc m p nc + νc m c2 can attenuate expression noise. This is a reason why RNA nuclear retention exists widely in most eukar− d c + δc m c nc (17c) yotic cells. Next, we numerically explore the latent mechand m p nc dt = k 0 B nc + νc m p m c ism of how CTN-RNA nuclear retention influences − kr + kc + δc m p nc (17d) expression noise. As is well known, most (∼90%) genes in eukaryotic cells are expressed in a bursty manBy combining equation (17) with equation (4), we ner [3, 9], and the stochastic switching between prothus obtain a formula for calculating the coefficient of moter activity states is a main source for generating variation for the mCAT2 protein level (i.e., the protein gene expression noise [38, 39, 41, 58]. Here we use the noise intensity) in the steady state: earlier simple ON–OFF gene model to annotate the j ∈ {r , c }

× ⎡⎣ m p P x ⃗; t ⎤⎦ + d j Em j − I

( ) ( )

(

(

}

)

(

)

11

)

Phys. Biol. 12 (2015) 016010

Q Wang and T Zhou

Figure 7. Phase diagrams for effects of CTN-RNA nuclear retention on the noise in the mCAT2 gene expression: (a) different colors represent different values of γ; (b) different colors represent different noise intensities of the mCAT2 protein. Other parameter values are set as 〈m ¯ c 〉 = 150, 〈n¯c 〉 = 1200, d c = 0.4, δc = 0.03.

relationship between bursting expression and CTNRNA nuclear retention. The numerical results are shown in figure 7. From this figure, we observe that for a fixed α, increasing the probability of CTN-RNA nuclear retention can make the lumped parameter γ slowly approach its minimum, whereas decreasing the remaining probability can make γ gradually approach 1. Refer to the color bar in figure 7(a), which describes the change in γ. On the other hand, figure 7(b) shows how the mean burst size B and the parameter γ affect the mCTA2 protein noise. We observe that at a low mean burst size (between 1 and 4), pre-mCAT2 RNA increases the mCAT2 protein noise by indirectly increasing the value of γ due to the decrease in remaining probability. In contrast, at a high mean burst size (between 12 and 16), pre-mCAT2 RNA reduces the mCAT2 protein noise by indirectly decreasing the value of γ due to the increase in remaining probability. Two arrows in figure 7(b) represent possible but simple strategies that pre-mCAT2 RNA takes according to different mean burst sizes. In a word, figure 7 reveals that the mCAT2 protein noise level can be well controlled by nuclear CTN-RNA retention. This point was ignored in most previous studies. 3.2.2. Stress case In some cases, cells need to be under high stresses such as viral infection and wound healing and respond suitably to such stresses. In general, there are two main kinds of stress sources: one is that the cells first sense the corresponding signals via IFN-γ -IGR and LPSTLR4 and then transmit them to paraspeckles through a still-unspecified network pathway [3, 9, 25, 39, 59]; the other is that an α-amanitin protein inhibits RNA polymerase II transcription and generates high stresses on the cells, thus causing the mCAT2 gene to stop producing transcripts. In the latter case, the 3′ UTR of the nuclear CTN-RNA is first quickly cleaved by 12

CFIm68 [60], and the relatively abundant CTN-RNAs are then exported to the cytoplasm to generate sustained mCAT2 proteins. In both cases of cellular stress, nuclear CTN-RNA can produce mCAT2 protein via a series of complicated but fast processes so that mCAT2 gene expression is kept relatively stable, implying that nuclear-retained CTN-RNA acts as a firefighter. Here we are interested in the cellular stress generated in the latter case (see figure 8). For the case that we are interested in, the full biochemical process can be divided into three stages: kj

dj

M p ⟶M j , M j ⟶ϕ , vc

δc

Mc ⟶Mc + Pc , Pc ⟶ϕ (j = r , c) v3

(

En1 Mr + E * ⇌ E * En1 Mr v−3

)

k3

⟶Mc* + E * + En1 vc*

d c*

(21a)

(21b) δc

Mc* ⟶Mc* + Pc , Mc* ⟶ϕ , Pc ⟶ϕ

(21c)

Stage (I), described by the reactions (21a), represents that a fraction of the nuclear pre-mCAT2 RNAs still produce CTN-RNA and mCAT2 RNA, even though the mCAT2 gene is repressed; and the cytoplasmic mCAT2 RNAs still generate mCAT2 proteins, even though their quantity may be very low before the cleaved CTN-RNAs are exported. On the other hand, the stressed cell may emancipate the signal or signals to paraspeckles via a yet-unspecified pathway so that the CFIm68 factor is rapidly cleaved from the longer 3′ UTR of the CTN-RNA. After that, the mCAT2 RNA is released to the cytoplasm from the nuclear paraspeckles. This is simply described by reactions of stage (II), i.e., equation (21b). At stage (III), described by equation (21c), the rising amount of shorter mCAT2 RNA in the cytoplasm, generated by the cleavage event, can result in a pulse of mCAT2 protein

Phys. Biol. 12 (2015) 016010

Q Wang and T Zhou

Figure 8. Schematic diagram for a gene model that additionally shows the dynamic process of how a cell corresponds to stress signals except for dynamic transcriptional and translational processes of the mCAT2 gene. Although a cell receives signals in many ways, here we show only two: (i) stress signals are transferred to the receptor proteins LPS-TLR4 and IFN-γ -IGR through certain invading viruses; (ii) stress signals inhibit RNA polymerase II transcription and further the transcription of the mCAT2 gene by α-amanitin or DRB.

production. Moreover, the large number of cleaved CTN-RNAs is required to sustain the normal expression of the mCAT2 gene, although stresses still exist in this case. Following we analyze the dynamics of the model systems corresponding to these stages separately. At stage (I), since the persistent time is not very long, we can simply use ODEs to model the transcriptional dynamics of the mCAT2 gene, for the three following reasons: (i) the number of mCAT2 genes is not large; (ii) one mCAT2 gene generates 20 ∼ 50 transcripts at most, but there are possibly hundreds of such transcripts in a cell since each cell has 10 ∼ 20 paraspeckles on average [26, 60]; (iii) CTN-RNAs sequester most of the heterodimers, so the number of the former approximates that of the latter. Based on the reactions (21a), we can derive analytical expressions for the time-evolutional noise levels of the premCAT2 and nuclear CTN-RNA from moment equations of the corresponding CME (see the appendix for details):

(

ηm2 p (t ) = exp(At )

)

m¯ ) ( (t ) ≈ ( exp ( d t ) m¯ ) + C + 1 + Be

ηm2 c

m¯ p

c

(22a)

p

c

(

where A = k r + k c ≫ d c , C1 ≈ 1 +

(22b)

1

4d c A

)

γBe m¯ c

+

2d c , A

and A ≫ d c is assumed to hold in most cases. Note that the total mRNA noise in the nucleus, 2 represented by ηTotal (t ), results mainly from the pre13

mCAT2 RNA fluctuations, represented by ηm2 p (t ), due 2 to k r + k c ≫ d c . The dynamic change of ηTotal (t ) is plotted in figure 9(a), showing that before stress signals arrive, the total mRNA fluctuates around its mean, represented by the solid black line obtained from equation (7), where the red star points represent the actual values of the pre-mCAT2 RNA noise. When the cell senses a high-stress signal, the noise level characterized by equation (22a) undergoes a rapid increase according to the property of an exponential function. Since the pre-mCAT2 RNA needs to keep a stable amount of mCAT2 RNA according to the above analysis, increasing the remaining probability will increase neither the total nuclear noise nor the nuclear mCAT2 RNA noise in this case. Similar to the dynamics of pre-mCAT2 RNA as shown in figure 9(b), however, the mCAT2 RNA noise level may increase when the mCAT2 gene is inhibited due to high stress. Compared with the noise level of pre-mCAT2 RNA, the protein-coding mRNA noise level may not be obviously increased until after a certain time. This has an apparent advantage since the downstream mCAT2 protein noise cannot go up without constraint. In fact, we can show that the time-dependent protein noise level is given by (see the appendix for the derivation):

ηn2c (t ) ≈ ⎡⎣ exp δc t

( )

n¯ c ⎤⎦ + C 2

where δc ≪ d c ≪ k c + k r and C 2 ≈

(

νc 3 + γBe dc n¯ c

(23)

).

This

indicates that the mCAT2 protein noise can increase in

Phys. Biol. 12 (2015) 016010

Q Wang and T Zhou

Figure 9. Characteristics of the pre-CAT2 RNA noise and the generating rate of the cleaved CTN-RNA at three different stages described in the main text. Figures (a) and (b) describe how the pre-mCAT2 RNA noise and the CTN-RNA noise vary, where (b) shows characteristics of the protein-coding mRNA noise at the second stage and a particular generating rate of the cleaved CTN-RNAs is indicated. Figure (c) shows the time-dependent relationship of this generation rate at stage (II), demonstrating that it increases with time in a nonlinear manner. Figure (d) shows the time-evolutional course of the change rate for the cytoplasmic noise during the ¯ c = 100, d c = 0.6, d c* = 0.27, m entire process. The parameter values are set as B = 5, m ¯ c* = 200 ∼ 230, C1 = 0.018, C3 = 0.0075.

a manner similar to the mCAT2 RNA case (referring to equation (22)). Next, we construct a dynamical model for the stage (II) described by the reactions (21b). We first make some rational interpretations using a well-known Michaelis–Menten model at this stage. Based on previous studies [3, 8–10, 60], we understand that multiple closely positioned points occur in the long 3′ UTR region in a cleavage event. The cleavage factor CFIm68 can rapidly join the action of the complex consisting of CTN-RNA and p54nrb-PSP1. However, reverse reactions can also frequently occur due to the joint occurrence of several unspecified factors, e.g., the CFIm68 drops from the complex or the CFIm68 is not completely combined due to the need for the adjustment of complexes and their space structure. Note that the final reaction in equation (21b) describes the entire process, including the cutting off of the long tails of the CTN-RNAs and the exporting of those RNAs from the nucleus to the cytoplasm. This process is slower than the previous reaction, being limited by several factors; e.g., the process of cutting the CFIm68 protein requires time (although not much), and the process of releasing the CTN-RNA also requires time. 14

Specifically, the CTN-RNA needs to find its exact position in the nuclear pore, and many transcripts need to be exported to the cytoplasm from the nuclear pore one by one. Based on the foregoing analysis, we know that the transcriptional dynamics of the mCAT2 gene can be modeled after the following Michaelis–Menten kinetics: d ⎡ ⎤ ⎡ ⎤ E * En1 Mr ⎦ = v3 ⎣ E *⎦ ⎡⎣ En1 Mr ⎤⎦ dt ⎣ ⎡ ⎤ − v−3 + k3 ⎣ E * En1 Mr ⎦ (24a)

(

)

(

)

(

)

d ⎡ *⎤ ⎡ ⎤ ⎣ Mc ⎦ = k3 ⎣ E * En1 Mr ⎦ dt

(

)

(24b)

where v3, v−3 > k 3, and the amount of CFIm68 represented by [E *] needs to satisfy the following conservation condition: ⎡ ⎤ ⎡ ⎤ * ⎣ E *⎦ + ⎣ E * En1 Mr ⎦ = ETotal

(

)

(25)

Since CFIm68 binds to the site of the CTN-RNA quickly and since the complex E * En1 Mr reaches equilibrium rapidly, this results in the following

(

)

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Q Wang and T Zhou

approximation: ⎡ ⎤ * ⎣ E * En1 Mr ⎦ ≈ ETotal m¯ r

(

(

)

m¯ r + Η3

) (26)

where Η3 = (v−3 + k 3 )/v3. Thus, the generating rate of the cleaved CTN-RNA is given by v treated = d ⎡⎣ Mc* ⎤⎦ dt = Vmax m¯ r

(

m¯ r + Η3

)

(27)

* . where Vmax = k 3 ETotal It is well known that the Michaelis–Menten curve has two key points: Vmax 2 and Vmax . Now we give interpretations for the biological meaning of these two points. Note that that when the entire process of stages (I) and (II) is considered, the pre-mCAT2 RNA noise first increases and then decreases as the amount of the cleaved CTN-RNAs in the cytoplasm goes up, as demonstrated by the solid blue line in figure 9(a). The chief reason is that the transcription of pre-mCAT2 RNA becomes slower and slower in this case. >From a dynamical point of view for the sake of illustration, it means that the parameter A in the formula equation (22a) is not a constant but may decrease in a linear or nonlinear manner as time goes on (although the exact dependence is not known). On the other hand, the treated CTN-RNA begins to replace the role of the pre-mCAT2 RNA by supplying protein-coding transcripts before the treated generating rate reaches Vmax 2 (see figure 9(c)). At this moment, the premCAT2 RNA likely stops working, and its noise level may be quite low (see figure 9(a)). A reasonable explanation for this is as follows: The change in the growth rate (i.e., the derivative vtreated ̇ ) is extreme between 0 and Vmax 2 but slight between Vmax 2 and Vmax . The corresponding dynamical process is similar to the exporting process of the cleaved CTN-RNA. As the amount of cleaved CTN-RNA in the cytoplasm continues to increase, we find without difficulty that the rate of the cleaved CTN-RNA being exported to the cytoplasm goes from slow to fast. When the value of vtreated is between 0 and Vmax 2, the most likely case is that the basic protein-coding transcripts are adequate enough to maintain important processes and that the pre-mCAT2 RNA begins to stop working (see figure 9(C)). It should be noted that it is possible that the total number of transcripts rises when the rate vtreated is close to Vmax 2. After that, the aggregate of all the transcripts in the cytoplasm appears as a visible pulse when the rate vtreated is close to Vmax or reaches Vmax (see figures 10(a) and (b). The main reason is that the amount of nuclear CTN-RNA is usually about twice the mCAT2 RNA amount. When most of the nuclear CTN-RNAs are released to the cytoplasm, the amount of all the cytoplasmic transcripts is at most two times that of the normal case. After most of the cleaved CTN-RNAs execute translation work and if m¯ c* m¯ c ≈ 2 ∼ 2.5, the cytoplasmic noise

15

becomes dominant. Although we cannot characterize the precise dynamic change of this noise during the entire process of releasing the cleaved CTN-RNAs, we can conjecture about the possible trend of this change by introducing the index called the change rate of noise, as shown in figure 9(d). Interestingly, we can obtain from equation (22b) the analytical expression for this change rate: d 2 f1 = dt ηmc = d c exp d c t mc . The solid green line in figure 9(d) describes the change of this function before the cleaved CTN-RNAs are released. Next we analyze the dynamic noise for the entire process, including the foregoing three stages. Note that the last stage can be depicted essentially by a group of ODEs when most of the cleaved CTN-RNAs start to implement translation processes after the releasing process is finished. The amount of nuclear CTN-RNA is approximately equal to that of executing transcripts, although it might undergo a series of complicated processes. In other words, we can neglect the death number of CTN-RNAs during such a series of processes, which has been verified by an experiment [18]. Therefore, at stage (III), a number of cleaved CTN-RNAs can substitute mCAT2 RNAs to execute translation functions. In addition, we can show that the dynamics of cytoplasmic transcript noise as well as protein noise are given respectively by (see the appendix for the derivation):

( )

( )

η 2 (t ) ≈ m c*

ηn2c (t ) ≈

( ) + (1 − γ) B

exp d c* t m¯ c*

e

exp δc t

( )+

1

n¯ c*

n¯ c*

⎡δ ⎛ ν* ⎞⎤ c γBe − 1 + 2 ⎜⎜ c − 1⎟⎟ ⎥ ×⎢ ⎢⎣ d c* ⎝ d c* ⎠ ⎥⎦

(

(28a)

m¯ c*

)

(28b)

where m¯ c* and n¯c* represent the initial values for cytoplasmic transcripts and proteins after the cleaved CTN-RNAs are completely released. Based on the preceding approximate formulas for the cytoplasmic transcript noise at two different stages (I) and (III), we can predict the trend regarding how this noise varies dynamically in the process during which the cleaved nuclear CTN-RNAs are released. In fact, we can see from equation (28a) that the initial value of the noise in the cytoplasmic transcript (i.e., η 2 (0)) is only ηm2c 2 higher than that given by the secmc*

ond part of equation (7) (see figure 9(b)). This implies that the cytoplasmic transcript noise may decrease as the amount of the cleaved CTN-RNA becomes greater and greater. We point out that the releasing process would be in essence the one optimizing the cytoplasmic noise, which plays a latent yet key role in maintaining the mCAT2 protein noise level in a small range or the mCAT2 gene expression level at a relatively stable interval for a long time. We can give a

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Q Wang and T Zhou

Figure 10. Numerical simulation of transcripts and proteins for normal and stressed cells, showing that there are differences in timedependent numbers of components (e.g., pre-CAT2 RNA, CTN-RNA, mCAT2 protein, and cytoplasmic transcript) between the two cases. Here, in the initial 300 min, all the gene products work normally. The parameter values are set as follows: in the normal case, k 0 = 1.2, B = 5, k1 = 1.8, pr = 0.3, k 2 = 4.1, d c = 0.6, dr = 0.05, v = 0.4, and δc = 0.06. In this case, the burst size follows a geometry distribution; in the stressed case, k1 = 1.2, k 2 = 3.1, d c = 0.2, pr = 0.1 ∼ 0.25, v3 = 0.6 ∼ 2.1, v−3 = 0.4 ∼ 1.4, k 3 = 0.15 ∼ 0.9, vc* = 0.36, and d c* = 0.08.

biologically reasonable interpretation for the results shown in figure 9(d). In fact, we can derive from equation (28a) the analytical expression for the change rate of the cytoplasmic noise after the releasing process is finished, that is, ⎛ ⎞ d Combining f2 = dt ⎜ η 2 * ⎟ = d c* exp d c* t m¯ c* . ⎝ mc ⎠

( )

this with m¯ c* m¯ c ≈ 2 ∼ 2.5 and d c > d c*, we find not only that there is a large difference between the two initial values (e.g., f2 (0) < f1 (0)/2) but also that the change rate for the noise described by the red line in figure 9(d) increases more slowly after the releasing process is finished. The entire persistent time can only be about 25 min [3, 9, 10]. After that, the mCAT2 gene recovers to a normal work state. Therefore, the cytoplasmic noise during the releasing process can be roughly shown by the dashed blue-circle line in figure 9(d). In a word, the primary function of the CTN-TNA releasing process is to keep the mCAT2 gene expression at a relatively stable level for a long time. Figure 9 indicates that after cells sense stress signals, transcription stops and the existing pre-mRNA and the cytoplasmic mCAT2 RNA still work as usual, although their amounts decrease more and more. This should gain time for releasing the CTN-RNAs from the nucleus to the cytoplasm. On the other hand, the CTN-RNA complex remaining in the nucleus reaches dynamic equilibrium through fast binding to the CFlm68 and begins to produce CTN-RNAs and is then exported to the cytoplasm. This is a dynamic process from slow to fast. We point out that when the generating rate of the cleaved CTN-RNAs arrives at or approaches a maximum, the pre-mRNA stops working so that a basic amount is guaranteed. With the mCAT2 RNA, the case is similar, and a large number of the cleaved CTN-RNAs begin to carry out their 16

translation function after the releasing process is finished. Correspondingly, the number of downstream mCAT2 proteins and their noise may change with the nuclear change (figure 10). There seems to be no exact description for this dependence relationship, but we can imagine that in the stressed case, the potential mechanism of the nuclear-retained mRNAs is that the related mRNAs act as a firefighter to regulate the relative stability of the downstream proteins through a series of changes in the CTN-RNA. We point out that CTN-RNA is only a kind of nuclear-retained RNA that can be exported to the cytoplasm from the nucleus under different conditions, but there should be a family of other nuclear-retained mRNAs that play the role of indirectly regulating the downstream components through some unspecified signal pathways or protein–protein networks under different conditions. In contrast with the simple means of the CTN-RNA action specified herein, however, the corresponding means would be more complex.

4. Conclusion and discussion Under normal conditions, various types of RNAs accumulate in the nucleus and carry out their functions therein. These RNAs include small nuclear RNAs (sn-RNAs) and small nucleolar RNAs (sno-RNAs), which are involved in the processing of a variety of nuclear precursor RNAs, including pre-mRNAs, pretRNAs, and pre-rRNAs. Based on the mode of nuclear accumulation, RNAs can be divided into two categories: One class is the so-called U sn-RNAs [61] involved in the processing of pre-mRNA, which are transcribed by RNA polymerase II. These RNAs leave the nucleus after transcription. Another class fundamentally different from the former class is those that

Phys. Biol. 12 (2015) 016010

Q Wang and T Zhou

are retained in the nucleus but consist of mutant versions of RNAs that are normally exported to the cytoplasm [62]. In this paper, we have constructed mCAT2 gene models with CTN-RNA nuclear retention. By analyzing the dynamical behaviors of these models, we have shown that CTN-RNA nuclear retention can not only reduce pre-mCAT2 RNA noise but also mediate its coding partner noise. In addition, by collecting experimental observations, we have conjectured a heterodimer formed by two proteins, p54nrb and PSP1, named p54nrb-PSP1, by which CTNRNA can positively regulate the expression of nuclear mCAT2 RNA. On the basis of this conjecture, we have further constructed a sequestration model for CTNRNA regulation at the transcriptional level. By dynamical analysis of this model, we have demonstrated why most nuclear-retained CTN-RNAs stabilize at the periphery of paraspeckles, how CTN-RNA regulates its protein-coding partner, and how the mCAT2 gene can maintain a stable expression. The obtained results are in agreement with experimental observations. The entire analysis not only provides insights into the mechanism of how RNA nuclear retention mediates the expressions of gene products, including premRNA, mature mRNA, and protein, but also lays the foundation for further studying other members of the nuclear-regulatory RNA family with more complicated molecular mechanisms. It should be pointed out that our results were obtained for simplified cases; e.g., our models of mCAT2 gene expression did not consider the regulation of transcription factors [63] and the effect of promoter architectures [38]. Besides these factors, mCAT2 gene expression also involves other, more complex, biochemical processes. In fact, with more indepth studies of regulatory systems of gene expression, a number of new molecular mechanisms based on the central dogma in biology have been elucidated, e.g., chromatin remodeling [64, 65], alternative splicing of different mechanisms [66], stochastic transitions between multiple states of the mCAT2 gene promoter [65, 67, 68], DNA methylation [69], histone modification [30, 70], and multi-step transcription as well as multi-step translation [39]. When all these molecular mechanisms associated with mCAT2 gene expression are considered, the role of CTN-RNA nuclear retention in regulating gene expression is not clear. It is possible that different mechanisms result in different results, but we can expect that the qualitative conclusion obtained here still remains invariant. We have revealed the essential mechanism of how CTN-RNA nuclear retention regulates mCAT2 gene expression and have described the process of how CTN-RNAs are released from the nucleus to the cytoplasm, but our model neglected some molecular details associated with CTN-RNA nuclear retention, which would be difficult to describe with a mathematical model. However, our qualitative conclusion obtained here should not be completely invalidated if 17

other more complex biochemical processes are considered since our models have captured some essential processes of mCAT2 gene expression. Finally, we point out that, to the best of our knowledge, the study of CTN-RNA nuclear retention seems to be still in its early stages. In fact, there are many unsolved problems involving the role of CTN-RNA nuclear retention in gene expression. More in-depth studies are expected.

Acknowledgments This work was supported by NSF (91230204), 973 Plan (2014CB964703), and Guangdong’s Distinguished Youth Plan (20110171120045), P.R. China.

Appendix A. Analytical results for noise in the simplified model (1)Transcription process The corresponding reactions read k0

ki

di

D⟶D + B × M p , M p ⟶Mi , Mi ⟶ϕ

(A1)

where j = r , c . From the corresponding CME, we can directly derive the following ODEs:

(

d mp

dt = k 0 B − kc + kr

)

mp

d mj

dt = k j m p − d j m j

d m p2

dt = k 0 B 2 + 2k 0 B

(

+ kr + kc

)

(

− 2 kr + kc d m j2

(A2b) mp

mp

)

m p2

dt = k 0 B + kj

(

(A2d)

mj m p2 − m p

(

− kr + kc + d j d mr mc

(A2c)

dt = k j m p + 2k j m p m j + d j m j − 2d j m j2

d mpm j

(A2a)

)

)

mpm j

(A2e)

dt = kr m p m c + kc m p m r

(

− dr + dc

)

mr mc

(A2f )

In the steady state, we can obtain analytical expressions for means, variances, and covariances: m¯ p = k 0 B / A

(A3a)

m¯ j = k j m¯ p / d j

(A3b)

m¯ p2 = m¯ p

2

(

)

+ Be + 1

m¯ p

(A3c)

Phys. Biol. 12 (2015) 016010

m pm j = m¯ p

m¯ j2 = m¯ j

Q Wang and T Zhou

m¯ j +

dj A + dj

Be m¯ j

⎛ k j Be ⎞ ⎟ m¯ j + ⎜⎜ 1 + A + d j ⎟⎠ ⎝

2

(A3e)

n¯ c +

(

k r k c Be dc + dr ⎛ 1 1 ⎞ ×⎜ + ⎟ m¯ p A + dr ⎠ ⎝ A + dc k r k c Be ≈ m¯ r m¯ c + dc ⎛ 1 1⎞ ×⎜ + ⎟ m¯ p A⎠ ⎝ A + dc

m r m c = m¯ r

δc n¯ c d c + δc ⎡ k c Be ⎛ dc ⎞ ⎤ × ⎢1 + ⎜1 + ⎟⎥ A + dc ⎝ A + δc ⎠ ⎦ ⎣ δc 1 + γBe n¯ c ≈ m¯ c n¯ c + dc

m c nc = m¯ c

(A3d)

or

m¯ c +

m¯ c

(A3f )

n¯ c +

νc 1 + γB e dc

(

(

(2)Translation process considering mCAT2 RNA only The corresponding reactions read dc

νc

δc

Mc ⟶Mc + Pc , Mc ⟶ϕ , Pc ⟶ϕ

(A4)

From the corresponding CME, we can straightforwardly derive the following ODEs: d nc

dt = νc m c − δc nc

d nc2

dt = νc m c + 2νc m c nc

+ δc nc − 2δc nc2

d m p nc

)

dt = k 0 B

m c nc

d m r nc

)

m p nc

(

)

(A7d)

× (1 − γ) Be n¯ c

Thus, we obtain the approximate expression for noise intensity: ηn2c ≈

1

δc 1 + γB e dc

1

+

n¯ c

(

m¯ c

d mp

(

dt = − kc + kr

)

mp

dt = kc m p − d c m c

d m p2

dt = kr + kc

(

)

mp

(

− 2 kr + kc d m p mc

dt = kc

d m c2

)

(

m p2 − m p

(

)

− kr + kc + d c

(A5e)

In the steady state, we obtain

)

d mc

(A5d)

m r nc

2δc d r + δc

m p2

(B1a) (B1b)

(B1c)

) m p mc

(B1d)

dt = kc m p + 2kc m p m c + d c m c − 2d c m c2

n¯ c = νc m¯ c / δc

(A8)

(1)Transcription process before CTN-RNA is releasedThe corresponding ODEs take the form

dt = kr m p nc + νc m r m c − d r + δc

(A7c)

(A5b)

nc + νc m p m c

(

n¯ c +

)

Appendix B. Analytical results for noise when pre-mCAT2 RNA is repressed

(A5c)

− kr + kc + δc

(A7b)

(A5a)

dt = kc m p nc + νc m c2

(

m¯ c

2

m r nc ≈ m¯ r

− d c + δc

)

⎛ νc ⎞ + ⎜1 + ⎟ n¯ c d c + δc ⎠ ⎝ d c ⎞ k c Be νc ⎛ n¯ c + ⎜1 + ⎟ d c + δc ⎝ A + δc ⎠ A + d c ⎡ ⎤ 2 νc 1 + γBe ⎥ n¯ c ≈ n¯ c + ⎢ 1 + dc ⎣ ⎦

n¯ c2 = n¯ c

where A = k r + k c ≫ d r .

d m c nc

)

(B1e)

(A6)

where the initial values are set as follows: Owing to d c ≫ δc and d c ≫ d r , we have m pnc = m¯ p

n¯ c

≈ m¯ p

n¯ c

dc δc + Be n¯ c A + d c A + δc d c δc (A7a) + Be n¯ c A + dc A

18

m p (0) = m¯ p

(B2a)

m c (0) ≈ m¯ c

(B2b)

m p2 (0) = m¯ p

2

(

+ 1 + Be

)

m¯ p

(B2c)

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Q Wang and T Zhou

m p (0) m c (0) ≈ m¯ p + m c2 (0)

m¯ c

From the following ODEs: (B2d)

2d c Be m¯ c A

d nc

d nc2

⎛ 2 2d c ⎞ ≈ ⎜1 + ⎟ m¯ c ⎝ A ⎠ ⎡ ⎛ 4d c ⎞ ⎤ + ⎢1 + ⎜1 + ⎟ γBe ⎥ m¯ c ⎝ A ⎠ ⎦ ⎣

dt = νc m c − δc nc

dt = νc m c + 2νc m c nc + δc nc − 2δc nc2

(B2e)

d m c nc dt = kc m p nc + νc m c2

(

Thus, we can obtain m p (t ) = exp(−At ) m¯ p

(B3a)

A m c (t ) = exp −d c t m¯ c A − dc dc − exp(−At ) m¯ c A − dc A ≈ exp −d c t m¯ c A − dc

d m p nc

)

(

d m p mc

(

(B3b)

(

)

2

+ Be m¯ p

p

)(B3c)

⎛ A m p (t ) m c (t ) ≈ exp ⎡⎣ − A + d c t ⎤⎦ ⎜ m¯ p ⎝ A − dc ⎞ 2Ad c B m ⎟ × m¯ c + 2 ¯ e c A − d c2 ⎠ ≈ exp ⎡⎣ − A + d c t ⎤⎦ m¯ p ⎞ 2d c Be m¯ c ⎟ (B3d) × m¯ c + ⎠ A

(

)

) ( (

m¯ c + exp −2d c t

⎡⎛ 2d c ⎞ × ⎢⎜1 + ⎟ m¯ c ⎣⎝ A ⎠

)

⎛ exp(At ) ⎜ 2 ηm p (t ) = + 1+ ⎜ m¯ p ⎝

( ) + ⎛⎜ 1 + 4dc ⎞⎟

m¯ c

⎝

(B5e)

m¯ c

(B6b) 3d c Be m¯ c A 2δc n¯ c + Be n¯ c (B6c) A

nc2 (0) ≈ n¯ c + n¯ c 2 νc + 3 + γB e dc

(

(B6d)

) n¯c

(B6e)

n¯ c

(B7a)

we can derive

(

nc (t ) ≈ exp −δc t

)

m p (t ) m c (t ) ≈ exp(−At ) m¯ c + exp ⎡⎣ − A + d c t ⎤⎦ × ⎡⎣ m¯ p m¯ c

(

(B3e)

Furthermore, we can derive the analytical expression for time-dependent transcriptional noise levels under the condition k r + k c ≫ d c :

exp d c t

m p mc

m c (0) nc (0) ≈ n¯ c

2

⎤ ⎛ 4d c ⎞ + ⎜1 + ⎟ γBe m¯ c ⎥ ⎝ ⎦ A ⎠

ηm2 c ≈

m p (0) nc (0) ≈ m¯ p

)

(

m c2 (t ) ≈ exp −d c t

)

(B6a)

m p (0) m c (0) ≈ m¯ p +

( m¯

Be m¯ p γB e

A ⎠ m¯ c

⎞ ⎟ ⎟ ⎠

(B4a)

+

2d c (B4b) A

(2)Translation process before CTN-RNA is released to the cytoplasm

)

⎤ ⎛ 3d c ⎞ +⎜ Be − 1⎟ m¯ c ⎥ ⎝ A ⎠ ⎦

(B7b)

m p (t ) nc (t ) ≈ exp(−At ) n¯ c + exp ⎡⎣ − A + δc t ⎤⎦ × ⎡⎣ m¯ p n¯ c ⎤ ⎛ 2δc ⎞ +⎜ Be − 1⎟ n¯ c ⎥ ⎝ A ⎠ ⎦

(

(

)

) n¯c ≈ exp ( −δc t ) n¯ c + exp ( −2δc t ) ⎡ ⎤ 2 νc × ⎢ n¯ c + ( 3 + γBe ) n¯ c ⎥ dc ⎣ ⎦

m c (t ) nc (t ) ≈ exp −d c t nc2 (t )

19

(B5c)

where the initial values are set as follows:

= exp(−At ) m¯ p + exp(−2At )

m p nc

nc (0) ≈ n¯ c m¯ p2 − m¯ p

(B5b)

(B5d)

)

dt = kc m p2 − A + dc

m p2 (t ) = exp(−At ) m¯ p + exp(−2At )

dt = νc m p m c − A + δc

)

(

(

) m c nc

− d c + δc

(

(B5a)

(B7c)

(B7d)

(B7e)

Phys. Biol. 12 (2015) 016010

Q Wang and T Zhou

Furthermore, we can obtain the analytical expression for the translational noise: ηn2c (t )

dc

n¯ c

(

)

(

( ) + νc ( 3 + γBe )

exp δt t

≈

⎛ 2δc ⎞ * ⎟ n¯ c nc2 (t ) ≈ exp −δc t ⎜ 1 + d c* ⎠ ⎝

(B8)

n¯ c

+ exp −2δc t

)

⎡ × ⎢ n¯ c* ⎣

δc

2

+

d c*

( γBe − 1)

n¯ c*

⎤ ⎛ νc ⎞ + 2⎜ − 1⎟ n¯ c* ⎥ ⎥⎦ ⎝ d c* ⎠

(3)Translation process after most CTN-RNAs are released to the cytoplasm

Thus, we can obtain analytical expressions for the time-dependent noise levels:

The corresponding ODEs take the form d mr

ηr2 (t ) ≈

dt = −d c* m r

(C1a) ηn2c (t ) ≈

d m r2

dt = d c* m r − 2d c* m r2

(C1b)

dt = νc* m r − δc nc

(C1c)

d nc2

dt = νc* m r + 2νc* m r nc (C1d)

dt = νc* m r2 − d c* m r

(

− d c* + δc

)

m r nc

(C1e)

where νc ≈ νc*; d c > d c* ≫ δc . If the initial values are approximated by steady states, then we can derive

(

m r (t ) = exp −d c* t

(

m r2 (t ) ≈ exp −d c* t

)

)

(

×

( m¯

2 r

m¯ r

(C2a)

m¯ r

+ exp −2d c* t

)

)

+ γBe m¯ r

⎛ 2δc ⎞ ⎟ exp −δc t n¯ c* n c (t ) ≈ ⎜ 1 + d c* ⎠ ⎝ 2δc − exp −d c* t n¯ c* d c*

(

(

)

(

)( 1 − d ν ) + exp ⎡⎣ −( d + δ ) t ⎤⎦ * c

* c

(C2b)

)

m r (t ) nc (t ) = 2 exp −d c* t

* c

(C2c)

n¯ c*

c

⎡⎛ 2δc ⎞ ⎟ m¯ c + 2 γBe − 1 × ⎢⎜1 + ⎢⎣ ⎝ d c* ⎠

(

+

)

δc 2d * ⎤ 1 + γBe + c ⎥ n¯ c* (C2d) dc νc* ⎥⎦

(

m¯ r

( )+

1

n¯ c*

n¯ c*

⎡δ ⎛ νc ⎞⎤ c ×⎢ γB e − 1 + 2 ⎜ − 1⎟ ⎥ ⎢⎣ d c* ⎝ d c* ⎠ ⎥⎦

)

(C3a)

(C3b)

References

+ δc nn − 2δc nc2

d m r nc

( )

exp d c* t + γBe

exp δc t

(

d nc

(C2e)

)

20

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