Homeostasis of mRNA concentrations through coupling transcription, exportation, and degradation

Many experiments showed that eukaryotic cells maintain a constant mRNA concentration upon various perturbations by actively regulating mRNA production and degradation rates, known as mRNA buffering. However, the underlying mechanism is still unknown. Here, we propose a mechanistic model of mRNA buffering: the releasing-shuttling (RS) model. The model incorporates two crucial factors, X and Y, which play key roles in the transcription, exportation, and degradation processes. The model explains the constant mRNA concentration under genome-wide genetic perturbations and cell volume changes. Moreover, it quantitatively explains the slowed-down mRNA degradation after Pol II depletion and the temporal transcription dynamics after Xrn1 depletion. The RS model suggests that X and Y are likely composed of multiple molecules possessing redundant functions. We also present a list of X and Y candidates, and an experimental method to identify X. Our work uncovers a possible coupling mechanism between transcription, exportation, and degradation.


INTRODUCTION
Maintaining an appropriate mRNA level is vital for all living cells.Surprisingly, many eukaryotic cells adjust the mRNA production rate and the mRNA degradation rate to achieve a constant mRNA concentration, an elegant but mysterious phenomenon called mRNA buffering [1][2][3].Multiple experiments carried out in the budding yeast Saccharomyces cerevisiae showed that the total mRNA concentration and also the mRNA concentrations of most individual genes are invariant against various genetic perturbations [4][5][6][7][8][9][10][11][12][13].In cases where transcriptionrelated genes are perturbed, the mRNA production rate decreases; interestingly, the mRNA degradation rate decreases accordingly so that the mRNA concentration is invariant.In cases where mRNA degradation-related genes are perturbed, the mRNA degradation rate drops, and the production rate decreases accordingly to maintain a constant mRNA concentration.Beyond the steady-state results, Chappleboim et al. recently showed that after an acute depletion of mRNA degradation factors, the mRNA degradation rate dropped immediately.However, the production rate decreased after a delay, so the total mRNA concentration temporarily accumulated and then gradually returned to its original level [14].mRNA buffering phenomenon has been reported in mammalian cells as well [15,16], suggesting its universality across eukaryotes.Strikingly, Berry et al. subjected mammalian cells to genome-wide genetic perturbation screening and found that the total mRNA concentration in both the nucleus and cytoplasm remained virtually unchanged despite a significant variation in the mRNA production rate [17].
Meanwhile, the total mRNA concentration and the mRNA concentrations of most genes are also constant as the cell volume increases [17][18][19][20].The homeostasis of mRNA concentration in a growing cell volume is often 54 considered a consequence of an mRNA production rate 55 proportional to cell volume with a constant mRNA degra-2 FIG. 1. Schematic of the releasing-shuttling model.One copy of protein X and Y in the released states (Xn and Yn) are released after each transcription initiation.kn is the mRNA production rate, which is also the rate of transcription initiation.Xn transitions to the cytoplasm with a constant rate αx and becomes Xc in the DF state, which degrades mRNAs with a rate proportional to the constant βm.Xc has a constant rate βx to shuttle back to the nucleus and becomes Xp in the TF state.Yn exports nuclear mRNAs to the cytoplasm with a rate proportional to the constant αm.Yn transitions to Yp in the TF state with a constant rate αy.
both degradation machinery and transcription factors [31], and Rpb4/7, which are subunits of Pol II and participate in both transcription and degradation of certain mRNAs [4,[32][33][34].All these factors share two common features: they play a role in both transcription and mRNA degradation and can shuttle between nucleus and cytoplasm to transmit information along the mRNA metabolic pipeline.These features are necessary to connect transcription and degradation in mRNA buffering.
Despite extensive knowledge of molecular details that may contribute to mRNA buffering, its mechanism remains mysterious.In this work, we provide the first mechanistic model of mRNA buffering (as far as we realize) that can explain and unify multiple experimental observations of mRNA buffering.In this work, we mainly study the total concentration of all mRNAs, which we call the mRNA concentration in the following unless otherwise mentioned.Our conclusions regarding mRNA buffering do not necessarily apply to all individual genes since they can be under specific regulations [8,9,17,35,36].Although most genes exhibit constant mRNA and protein concentration as the cell volume increases, we have shown before that genes with strong (weak) promoters may exhibit sublinear (superlinear) volume-scaling of mRNA and protein concentra-tions [24].In the following, we show that models in-116 volving mRNA feedback to its production or degrada-117 tion cannot achieve robust mRNA buffering.We then 118 introduce the minimal model of mRNA buffering, the 119 releasing-shuttling (RS) model.The critical ingredients back models generally cannot lead to robust buffering.
Changing the mRNA production or degradation rates always leads to a significant change in the mRNA concentration (Supplementary Information and Figure S1).In contrast, the mRNA concentration was largely invariant to these perturbations in experiments [8,9].In particular, the mRNA concentration and production rate appeared uncorrelated across genome-wide perturbations in mammalian cells [17].
In the following, we introduce a minimal model of mRNA buffering with two essential proteins, X and Y (Figure 1).X is a degradation factor that shuttles between the nucleus and the cytoplasm.Y is an exportation factor and is localized in the nucleus.X can be in three states: the transcription-factor (TF) state in the nucleus that is part of the preinitiation complex (PIC), X p ; the released state separated from the PIC right after transcription initiation and ready to be exported to the cytoplasm, X n ; the decay-factor (DF) state in the cytoplasm responsible for mRNA degradation, X c .Y can be in two states: the transcription-factor (TF) state in the nucleus that is part of the preinitiation complex (PIC), Interestingly, transcription is necessary for mRNA export [41], supporting our model assumption.We hypothesize that protein Y in the state Y n is an export regulator of mRNA.
The mathematical equations of the RS model are the following, Intriguingly, the numbers of nuclear and cytoplasmic

The RS model leads to mRNA buffering as the cell volume increases
We then sought to test if the RS model can achieve homeostasis of mRNA concentration during cell growth (Figure 3a).For simplicity, we assumed that the total copy numbers of proteins X and Y are proportional to the cell volume, which is valid for most proteins [46].
We first fit the experimental data of mRNA production rate vs. cell volume from Ref.
[26] using Eq. 13 assuming a constant ratio between the total cell volume and nuclear volume (the blue line in Figure 3b).From this fitting, we obtained the value of the MM parameter K v 359 for WT cells (Eq.13).K v and other parameters in-360 ferred from the fitting of kinetic data of budding yeast 361 (see the next section) constitute the basic parameters in 362 this work (Methods and Table II).In the above fitting,   Due to the rapid depletion of the DF state X c in the 460 cytoplasm, the replenishment of X p (Eq. 6) significantly 461 decreases (Figure 4b).However, we argue that a delay 462 between the decrease of the mRNA production rate and 463 the reduction of X p can emerge because the transcrip-464 tion factor concentration is still much higher than the 465 Michaelis-Menten constant K x (Eq.13).In this case, 466 the mRNA production rate will be maintained at a con-467 stant value for a finite duration, in agreement with ex-468 periments [14].Therefore, the TF state X p initially de-469 creases linearly because of the conversion from X p to X n 470 (Eq.6).The linear reduction of X p holds until X p hits 471 K x , which leads to a significant decrease in the mRNA 472 production rate (Eq.13) (Figure S6).Therefore, we can 473 approximate the duration of the accumulation phase as 474 T acc ≈ (X p − K x )/(k n,bf f dep ) according to Eq. 6.Here, 475 k n,bf is the mRNA production rate before perturbation, 476 which is also the replenishment rate from X c to X p .We (80% in this case).The values of X c , X p , and the mRNA production rate then quickly reach the new steady-state values, after which the mRNA concentration gradually returns to its original value with a relaxation time determined by the mRNA degradation rate, δ m = β m X c /V c according to Eq. 4. Therefore, the duration of the re- 4c shows a schematic of this process.

487
In the above discussion, the initial value of the tran-488 scription factor concentration x p must be far above the 489 parameter K x (Figure S6).This condition suggests that 490 for WT cells, X p is generally non-limiting for transcrip-491 tion, although it is still necessary to initiate transcrip-492 tion.We also confirmed that the three phases in the Protein X in the cytoplasm will gradually shuttle to the nucleus until most of its copies are localized in the nucleus after a complete transcription inhibition, which can be used to detect the candidates of protein X.
transcription dynamics following an acute depletion are robust against different choices of the parameters (Figure S7).In conclusion, the predictions of the RS model not only perfectly aligned with the experimental data from which we inferred the set of basic parameters (Figure 4a and Methods) but also provided valuable insights into the underlying mechanism (Figure 4b and c).
Next, we sought to systematically investigate the transcription dynamics after acute depletion of X c according to the RS model, in particular, considering cells with different concentrations of Pol II or protein X. Future experiments can test our predictions.We compared a WT cell (solid lines in Figure 4d) and a mutant cell with a lower Pol II concentration (dashed lines in Figure 4d).

507
The mutant with a lower Pol II concentration, equivalent 508 to a larger K v in the RS model (Eq.13), has a mildly 509 lower mRNA production rate than the WT cell.There-510 fore, the mutant has fewer X n , X c and more X p (Eqs.For the mutant (dashed lines in Figure 4e) with a lower protein X concentration, it has fewer TF state X p than the WT cell (solid lines in Figure 4e).However, its mRNA production rate remains close to the WT cell as long as the concentration of X p is still significantly larger than K x .Because of the fewer X p , the mutant with a lower X concentration exhibits a shorter accumulation phase than the WT cell because the duration of the accumulation phase depends on the initial X p , T acc ≈ (X p −K x )/(k n,bf f dep ).Therefore, the mutant also has a lower number of X c during the reversion phase, so the mRNA concentration recovers slower in the mutant than in the WT cell.We also investigated the temporal transcription dynamics for other mutants with different modified parameters (Figure S7).

Candidates of X and Y
Experiments showed that the mRNA concentration was buffered even when the global degradation factors such as Dcp2 and Xrn1 were knocked out [8,14].These factors play vital roles in 5'-3' mRNA degradation, the predominant pathway of mRNA degradation [31].These observations suggest that the protein X and Y in the RS model may not represent a single protein, and they may be a combination of multiple proteins.To demonstrate this idea explicitly, we investigated the response of mRNA concentration to an abrupt drop in the mRNA production rate by considering cells with different total numbers of protein X.We found that the recovery time of mRNA concentration diverges in the limit of X t → 0 (upper panel of Figure 5a).This result suggests that if protein X is only Xrn1, cells with Xrn1 knocked out should not recover from a slight noise in the mRNA production rate, so it cannot achieve mRNA buffering, contradicting experiments.We analyzed protein Y similarly and obtained the same results (lower panel of Figure 5a).
Therefore, we proposed that X and Y represent groups of multiple proteins that perform similar functions (Figure 5b).In summary, while the RS model provides a powerful framework for understanding mRNA buffering, its simplicity also allows for straightforward extensions and adaptations to capture the complexity of mRNA regu- representing protein PABPC1 (Figure 6a).The dynamics where p n is the concentration of P n in nucleus, and K p 780 is a constant.14 the value of K v (Figure 3b).
where m c,rev is the cytoplasmic mRNA number during

892
We focused on the accumulation and adaptation 893 phases to estimate the remaining parameters' values.We 894 rewrote Eq. 13 as: where X p,3 = X p,2 − T adp (⟨k n,adp ⟩ − ⟨v trans,adp ⟩), where X p,2 and X p,3 are the numbers of X p at the end of the accumulation and adaptation phases, respectively.
Based on experimental data, the mRNA production rate at the end of the accumulation phase (k n,2 ) was approximately 0.95k n,bf , while at the end of the adaptation phase, k n,3 was around 0.4k n,bf .Using Eq. 24, we wrote following equations: Given that k n,bf = 500 min −1 , we solved the above equations and obtained the values of k ′ 0 , K ′ x , and X p,bf .
Next, we numerically calculated the amount of mRNA accumulated during the accumulation and adaptation Vc , we rewrote Eq.21 as Using the linear approximations of k n and X c in the accumulation and adaptation phase, we numerically integrated the equation to obtain the total cytoplasmic mRNA m c,3 at the end of the adaptation phase, where the total mRNA is approximately 1.3m t,bf (Figure 4a). Therefore, from which we solved for the value of m n .
We have so far obtained the following parameters from the experimental data of [14]: x , and δ m (δ m,bf ) for a cell without Xrn1 depletion.We next estimated all the parameters as the starting point of the parameter search process.We set k 0 = 6000 min −1 , K y = 100 f L −1 , and Y p = 1 × 10 4 to solve K m with the definition of k ′ 0 .We set X n = 5000 to solve α x with Eq. 10.We set X c = 2 × 10 5 to solve β x with Eq. 11 and β m with the definition of δ m .We set Y p = Y n = 1000 and solve α y with Eq. 12.We then solve α m with Eq. 8. Summing up X n , X c , and X p , we obtained X t and similarly for Y t .
Finally, we optimized these 11 parameters to fit the experimental data by minimizing the weighted mean squared error (MSE) between the model predictions and the experimental data of the temporal fold changes of mRNA concentration and the mRNA production rates.
This optimized set of parameters is the basic set of parameters used in our simulations (Table II).For the simulations examining temporal changes of mRNA with different concentrations of X or Y (Figure 5a), we monitored the changes of cytoplasmic and nuclear mRNA levels after inhibiting transcription, compared to their values before perturbation.We inhibited transcriptional by instantaneously decreasing the value of k 0 to its 50% at time 0.
For the simulations of temporal changes after complete transcription inhibition (Figure 5c), we introduced a very large K v to completely shut off transcription at time zero, and then calculated the temporal changes of mRNA and X.
For the simulation of the modified model with additional protein P (Figure 6b), we simulated the temporal changes of the total number of mRNA, the cytoplasmic P (P c + P c,b ), the nuclear P (P c + P c,b ), the mRNA production rate, and the mRNA degradation rate per mRNA using Eqs.14-18 after increasing the degradation factor β m by a factor of 10 at time 0, mimicking the globally accelerated degradation of the host mRNA during the viral infection.In these simulations, we modified α m to be 20 times of the value from the set of basic parameters.
To simulate the extended model involving multiple distinct subsets of mRNAs (Figure 6d), we simulated the transcription of two groups of genes regulated by their own Xs and Ys.The parameters of these two RS models are set the same.We perturbed the expression of one group of genes by decreasing the number of its X c to its 10% at time zero.We then monitored the temporal changes of the mRNA concentrations of both groups and the total mRNA concentration.

Finding candidates of X and Y
The RS model suggests that X and Y are multifunctional molecules involved in transcription, exportation, and degradation.To identify candidate genes in these categories, we utilized various databases, including the Gene Ontology (GO) database through AmiGO (version 2.5.17I).

Y
p ; the export-factor (EF) state released from the PIC right after transcription initiation and ready to export nuclear mRNAs to the cytoplasm, Y n .We assume that the DF state X c has a finite rate to shuttle back to the nucleus and become the TF state X p , supported by multiple pieces of evidence showing that mRNA decay factors shuttle in and out of the nucleus and appear critical for transcription [8, 9, 30-32, 37].Similarly, we assume the EF state Y n has a finite rate to transition back to the TF state Y p .The release of proteins X and Y from the PIC is justified by the fact that transcription initiation is a stepby-step process propelled by different functional groups, which sequentially bind and leave the transcription machinery.In particular, the carboxyl-terminal domain (CTD) region of Pol II is a versatile harbor providing numerous docking points for proteins functioning in diverse processes such as initiation-to-elongation transformation and mRNA processing [38].We suppose proteins X and Y are two of the signaling molecules that are released during transcription initiation.Recent studies show that mRNA export is under sophisticated regulations [39, 40].

202
mRNAs are proportional to the nuclear and cytoplasmic 203 volumes, respectively.These are necessary conditions for 204 any model to be biologically valid: the model must pre-205 dict a constant mRNA concentration.Surprisingly, both 206 the nuclear and cytoplasmic mRNA concentrations are 207 independent of the mRNA production rate k n : the RS 208 model successfully explains the robust mRNA concentra-209 tion homeostasis against the change of mRNA produc-210 tion rate.This is because the information on the mRNA 211 production rate is conveyed through the amounts of Y n 212 and X c , synchronizing the speed of mRNA exportation 213 and degradation.To ensure the number of DF state X FIG. 3. The RS model predicts mRNA buffering during cell growth and slowed mRNA degradation after Pol II depletion.Exp., experiment; degr., degradation; pred., prediction; conc., concentration.(a) Schematic showing the homeostasis of mRNA concentration during cell growth.(b) Experimental data from Ref. [26] for the mRNA production rate, degradation rate, and the mRNA number as a function of cell volume during cell growth.The blue line is a fitting of the experimental data (r 2 = 0.98).The red line is the predicted mRNA degradation rate per mRNA without any fitting parameters from the RS model.The yellow line is the predicted mRNA copy number.(c) mRNA buffering occurs even if the mRNA production rate exhibits a superlinear scaling with cell volume.In this case, the mRNA degradation rate increases with the cell volume accordingly to make the mRNA concentration constant.(d) mRNA buffering occurs even if the mRNA production rate oscillates with cell volume.In this case, the mRNA degradation rate also oscillates, and the mRNA concentration remains constant.(e) The normalized mRNA concentration for a particular gene following transcription shut-off.The blue circles are for cells without Pol II depletion, and the red triangles are for cells with half Pol II depleted.The experimental data are from Ref. [26].The blue line is a fitting of the experimental data by the RS model (E 2 = 3.87 × 10 −11 ).The red line is the prediction from the RS model for cells with half Pol II depleted without further fitting.

363
we neglected the volume dependence of the concentra-364 tions of transcription factors, x p and y p and confirmed 365 that this approximation was valid (FigureS5).The nu-366 clear mRNA concentration c m,n is also constant in Eq.

367 13 due
Figure3e, and Methods).We then doubled K v to model 419

423
Temporal transcription dynamics after rapid 424 degradation of protein X 425 In Ref. [14], Chappleboim et al. rapidly depleted Xrn1 426 in budding yeast and monitored the temporal dynamics 427 of the total mRNA concentration and the level of re-428 cently transcribed mRNA, which is a good proxy for the 429 mRNA production rate.Experimental observations re-430 vealed three phases of mRNA dynamics upon the sudden 431 removal of the protein Xrn1 (Figure 4a).In the accu-432 mulation phase, the mRNA concentration increased, and 433 the mRNA production rate remained almost constant.434 In the adaptation phase, the mRNA production rate 435 dropped rapidly, and the mRNA concentration ceased to 436 increase.In the reversion phase, the mRNA production 437 rate reached the new steady-state value, and the mRNA 438 concentration gradually reduced to its original value be-439 fore the depletion of Xrn1.To verify whether the RS 440 model can explain the temporal dynamics of mRNA as 441 a more stringent test, we studied the RS model after a 442 rapid depletion of protein X. 443 First, we remark that in the RS model, protein X is 444 presumably a coarse-grained combination of several pro-445 teins.This idea is supported by the experimental obser-446 vations that mRNA buffering was still valid even if Xrn1 447 was knocked out [8, 14], which we discuss in detail later.448 Therefore, the depletion of Xrn1 should correspond to a 449 partial depletion of protein X in the RS model.Because 450 Xrn1 is the primary degradation factor in budding yeast 451 [8] and more than 90 percent copies of Xrn1 are local-452 ized in the cytoplasm [47], we modeled the depletion of 453 Xrn1 as a rapid reduction of X c to its 20% value in the 454 cytoplasm.We neglected the depletion of protein X in 455 the nucleus, including the TF state X p and the released 456 state X n for simplicity.This assumption can be relaxed 457 as long as the nuclear X does not decrease significantly.458459

FIG. 4 .
FIG.4.The RS model predicts the temporal change of mRNA concentration after an acute perturbation of the degradation factor.Exp., experiment.(a) The temporal changes in the total mRNA concentration (total mRNA) and recently-transcribed mRNA concentration (new mRNA) following acute depletion of the degradation factor X. The data points represent experimental measurements from[14].The dashed lines are fitting of the experimental data by optimizing parameters in the RS model (M SE = 2.10 × 10 −3 ).We used the mRNA production rate as a proxy for the recently-transcribed mRNA concentration.(b) Simulation results of the temporal dynamics of mRNA, kn, Xp, and Xc.(c) The schematic illustrating the dynamics of X after an acute depletion of Xc and its influence on transcription and degradation at different times.(d) Impact of a lower Pol II concentration (a larger Kv) on the temporal changes following depletion of Xc.The solid lines are WT cells, and the dashed lines are for cells with a larger Kv.Both the solid and dashed lines are relative to the values of the WT cells before depletion.(e) Influence of a lower X concentration (a lower Xt) on the temporal changes after depleting Xc.The lines have the same meaning as (d).

FIG. 5 .
FIG.5.Proteins X and Y should represent groups of proteins with similar functions.(a) The recovery of cytoplasmic mRNA (upper panel with varying copy numbers of X) and nuclear mRNA (lower panel with varying copy numbers of Y) after a rapid perturbation to the mRNA production rate.The y-axis is shifted and normalized so that the value before perturbation is one and the minimum value is 0. (b) Schematic illustrating X and Y can be groups of different proteins executing similar functions.(c) The time course of the mRNA concentration, Xc, Xp, and Xn after the complete shutoff of transcription.(d) Protein X in the cytoplasm will gradually shuttle to the nucleus until most of its copies are localized in the nucleus after a complete transcription inhibition, which can be used to detect the candidates of protein X.

511 10 and
11).Consequently, it takes longer for the mu-512 tant cell to reach the adaptation phase since it needs a 513 longer time for the TF state X p to decrease and reach 514 the MM constant K x .Because of the longer duration of 515 the accumulation phase for the mutant, which also has a 516 similar accumulation rate of X c as the WT cell, the mu-517 tant has a higher X c in the new steady state.Therefore, 518 the mutant recovers more rapidly during the reversion 10 phase than the WT cell.

604 plasm gradually shuttles back to the nucleus to become 605 X
p (Figure5c).During this process, mRNA continues 606 to be degraded and eventually stabilizes at a lower level 607 (Figure5c).By monitoring the cytoplasmic proteins that 608 are predominantly transported back to the nucleus after 609 transcription shut-off, one can identify the possible con-610 stituents of X (Figure 5d).This prediction of the RS 611 model provides an experimental protocol to examine the 612 potential candidates for X. 613 Extensions of the RS model 614 The RS model provides a fundamental understanding 615 of transcription regulation by revealing the coordination 616 of mRNA production, exportation, and degradation rates 617 to maintain a constant mRNA concentration.However, 618 it is essential to recognize that the real-world dynamics of 619 mRNA regulation can be more complex than the current 620 RS model.This section discusses some of the complexi-621 ties that may go beyond the simplified scenario of mRNA 622 buffering and how the RS model can be extended to ad-623 dress them.624 We first discuss the mRNA dynamics during viral in-625 fection.When cells are infected, the mRNA degradation

FIG. 6 .
FIG. 6.The extensions of the RS model.Degr., degradation; prod., production.(a) Schematic of the modified model in which a third protein P is added.Xv represents the viral endonuclease that increases the factor βm. (b) Simulations of the dynamics of the total mRNA concentration, nuclear P (Pn + P n,b ), cytoplasmic P (Pc + P c,b ), degradation rate per mRNA, and production rate after a sudden increment of βm at time 0. (c) Schematic of the generalized model in which different sets of genes are regulated by different groups of X and Y, e.g., RS1, to buffer mRNA concentration separately.(d) Simulations of the dynamics of two groups of mRNA regulated by distinct groups of X and Y and their sum after a sudden decrement in Xc of group 1 at time 0.

Here
P n,b = m n and P c,b = m c because P binds to mRNA tightly.To model the inhibition of PABPC1 to transcrip-777 tion, we included the free protein P, P n , to Eq. 13, so 778 that 779

869
the reversion phase, k n,rev is the mRNA production rate 870 during the reversion phase (which is constant in the re-871 version phase), and C is a constant.δ m,rev is the mRNA 872 degradation rate per mRNA so that δ m,rev = β m Xc,rev Vc , 873 where X c,rev is the number of X c during the reversion 874 phase (which is also constant).Given that the amount 875 of nuclear mRNA remains constant, we expressed the fold 876 change of total mRNA as877F C mt = m n + m c,rev m n + m c,bf ,(23)where m n is the nuclear mRNA number and m c,bf is 878 the cytoplasmic mRNA number before Xrn1 depletion.879UsingEqs.20-23, it is easy to find that δ m,rev = b.880Experimentaldata showed that k n,rev decreased to ap-881 proximately 0.4 times the mRNA production rate before 882 perturbation k n,bf : k n,rev /k n,bf = 0.4 (Figure4a).Be-883 cause the cytoplasmic mRNA numbers are the same be-884 tween the steady states before and after the perturbation, 885 k n,rev /δ m,rev = k n,bf /δ m,bf .Therefore, δ m,rev /δ m,bf = 886 0.4, and m c,bf = k n,bf /δ m,bf = 0.4k n,bf /δ m,rev .Typi-887 cally, the mRNA production rate of budding yeast cells 888 ranges from about 180 to 2300 mRNA per minute [64].889 Hence, we set the mRNA production rate of WT cells 890 before perturbation k n,bf = 500 min −1 , from which we 891 determined the value of m c,bf .

900
Calculating the analytical solutions for the time de-901 pendence of X c and k n was challenging, particularly for 902 k n as it depends on X p .To simplify the problem, we 903 made the following approximations.Simulations in Fig-904 ure 4b showed that X c increases approximately linearly 905 in the accumulation and adaptation phases.Therefore, 906 the transformation speed v trans = β x X c from X c to X p is 907 also a linear function of time during the accumulation and 908 adaption phase, v trans (t) = v trans,1 + vtrans,3−vtrans,1 Tacc+T adp t. 909 Here, T acc and T adp are the durations of the accumula-910 tion and adaptation phases.In the steady state before 911 perturbation, v trans,bf = k n,bf = β x X c,bf .At the begin-912 ning of the accumulation phase, we assumed that 80% 913 of X c are depleted so that X c,1 ≈ 0.2X c,bf , therefore 914 v trans,1 ≈ 0.2v trans,bf .Because the mRNA production 915 rate and the transformation rate must balance as well 916 in the steady after perturbation, v trans,rev = k n,rev = 917 β x X c,rev .According to the experimental data, k n,3 = 918 k n,rev ≈ 0.4k n,bf at the end of the adaptation phase, so 919 that v trans,3 = v trans,rev ≈ 0.4v trans,bf .Thus, we ob-920 tained an approximate expression for the transformation 921 speed.We also calculated the time-averaged transforma-922 tion rates in the accumulation phase (⟨v trans,acc ⟩) and 923 the adaptation phase (⟨v trans,adp ⟩).924 Similarly, we approximated the change in k n in the 925 experimental data with a combination of two linear re-926 ductions: a small slope in the accumulation phase and a 927 large slope in the adaptation phase.Likewise, we com-928 puted the time-averaged mRNA production rates durlinear approximations, we calculated the changes in X p 931 during the accumulation and adaptation phases: X p,2 = X p,bf − T acc (⟨k n,acc ⟩ − ⟨v trans,acc ⟩),

977 10 − 5 V 1. 5 n
Figure 4d, e, and Figure S7, two simulations are shown together in each plot.All parameters used in these simulations are the same except for the one indicated in each plot.
) [65-67], the Kyoto Encyclopedia of Genes and Genomes (KEGG) database [68-70], and the Saccharomyces Genome Database (SGD) [71].In our search for transcription-related genes, we focused on those annotated as transcription factors or involved in the reg-ulation of transcription, as well as the subunits of Pol 1062 II.For exportation-related genes, we looked for annota-1063 tions related to nuclear transportation factors and the 1064 subunits of the nuclear pore.In the context of degrada-1065 tion, we identified genes involved in mRNA catabolism.1066 We marked genes with dual functions in transcription 1067 and exportation as Y and genes with dual functions in 1068 transcription and degradation as X (Table

)
Here m n is the number of nuclear mRNAs, and m c is 187 the number of cytoplasmic mRNAs.kn is the mRNA 188 production rate, which depends on multiple factors, in-189 cluding the concentrations of Pol II, X p , and Y p .V c is the 190 cytoplasmic volume, and V n is the nuclear volume that 191 is proportional to V c [42].The binding rate of mRNA to 192 the EF state Y n is proportional to its concentration with 193 a factor α m , which is the limiting step of mRNA export.194αx quantifies how fast X n escapes the nucleus, and α y 195 quantifies how fast Y n transforms back to Y p .Similarly, 196 the binding rate of mRNA to the DF state X c is propor-197 tional to its concentration with a factor β m .βx quantifies 198 how fast X c shuttles back to the nucleus.199It is straightforward to find the steady-state solution 200 of the RS model:

TABLE I .
Candidates of X and Y in S. cerevisiae.Gene name Functions as Gene name Functions as Gene name Functions as Gene name Functions as CAF40 RS model in capturing complex regulatory scenarios.
lation.By incorporating additional regulatory information, the RS model can be tailored to address specific biological contexts and shed light on the diverse regula-tion mechanisms of the mRNA level.the and k ′ 0 = k 0

TABLE II .
The basic set of parameters.
975randomly sampled all parameters from lognormal distri-976 butions for the blue points in Figure2d, e, f, and S4.