Regulation of NF-κB Oscillation by Nuclear Transport: Mechanisms Determining the Persistency and Frequency of Oscillation

The activated transcription factor NF-κB shuttles between the cytoplasm and the nucleus resulting in the oscillation of nuclear NF-κB (NF-κBn). The oscillation pattern of NF-κBn is implicated in the regulation of gene expression profiles. Using computational models, we previously reported that spatial parameters, such as the diffusion coefficient, nuclear to cytoplasmic volume ratio, transport through the nuclear envelope, and the loci of translation of IκB protein, modified the oscillation pattern of NF-κBn. In a subsequent report, we elucidated the importance of the “reset” of NF-κBn (returning of NF-κB to the original level) and of a “reservoir” of IκB in the cytoplasm. When the diffusion coefficient of IκB was large, IκB stored at a distant location from the nucleus diffused back to the nucleus and “reset” NF-κBn. Herein, we report mechanisms that regulate the persistency and frequency of NF-κBn oscillation by nuclear transport. Among the four parameters of nuclear transport tested in our spatio-temporal computational model, the export of IκB mRNA from the nucleus regulated the persistency of oscillation. The import of IκB to the nucleus regulated the frequency of oscillation. The remaining two parameters, import and export of NF-κB to and from the nucleus, had virtually no effect on the persistency or frequency. Our analyses revealed that lesser export of IκB mRNA allowed NF-κBn to transcript greater amounts of IκB mRNA, which was retained in the nucleus, and was subsequently exported to the cytoplasm, where large amounts of IκB were synthesized to “reset” NF-κBn and drove the persistent oscillation. On the other hand, import of greater amounts of IκB led to an increase in the influx and the efflux of NF-κB to and from the nucleus, resulting in an increase in the oscillation frequency. Our study revealed the importance of nuclear transport in regulating the oscillation pattern of NF-κBn.


Introduction
NF-κB is a transcription factor that regulates the expression profiles of a vast number of genes. In the classical pathway of NF-κB activation, extracellular stimuli such as TNFα lead to the phosphorylation and proteasomal degradation of IκB, a negative regulator of NF-κB, which retains it in the cytoplasm in the resting state. NF-κB thus liberated from inhibition translocates to the nucleus, leading to the expression of genes, including IκB. Newly synthesized IκB protein binds to NF-κB in the nucleus causing an export of nuclear NF-κB (NF-κB n ), leading to the raise of the cytoplasmic concentration of NF-κB again. If the NF-κB activating stimulus remains, IκB in the IκB:NF-κB complex is degraded, and the liberated NF-κB translocates to the nucleus again. Thus, the oscillation of NF-κB emerges [1][2][3][4][5][6].
The biological functions of NF-κB oscillation and its mechanisms have been investigated by multiple research groups leading to the discovery of several sensitive parameters affecting of NF-κB oscillation [7][8][9][10][11][12][13][14], the importance of the transport of proteins, including NF-κB [15], importance of negative feedback and inhibitor proteins [16][17][18][19], the effects of stimulation pattern on gene expression profiles [5,20] and stimulus specificity of gene expression [21]. These analyses revealed the critical parameters that regulate the activity of NF-κB, including negative feedback loops, rate constants, concentrations of molecules, and patterns of stimulation. Previously, we constructed a three-dimensional (3D) spatio-temporal model of NF-κB oscillation, and reported the importance of spatial parameters in the regulation of the NF-κB n oscillation pattern [22]. We found several factors, such as the diffusion coefficient, nuclear transport, nuclear to cytoplasmic volume (N/C) ratio, and loci of protein synthesis, to regulate the oscillation pattern of NF-κB n . In a subsequent report, we showed why the diffusion coefficient regulated the oscillation pattern [23]. We revealed that the "reset" of NF-κB n was critical for the sustained oscillation of NF-κB n , and that a distant location in the cytoplasm acted as a "reservoir" for newly synthesized IκB, which subsequently caused the "reset" of NF-κB n . A larger diffusion coefficient of IκB helped to store a greater amount of IκB in the cytoplasm, which could then diffuse back to the nucleus to sufficiently "reset" NF-κB n . Furthermore, we identified a possible change in the effective value of the diffusion coefficient caused by a change in the crowdedness of organelles, which was observed in hypoxic cells [24].
Herein, we report how and why nuclear transport regulates the oscillation pattern of NF-κB n . Several reports are available regarding the effect of nuclear transport on the activity of NF-κB in the nucleus. An earlier study showed that an siRNA-induced knock-down of nucleoporin Nup88, which is one of the constituent proteins of the nuclear pore complex (NPC) at the cytoplasmic face, prevented the nuclear accumulation of NF-κB and reduced the expression of a reporter gene [25]. Senescence has been shown to alter the expression of nucleoporins (nups) and to decrease the number of NPCs, thus impairing the nuclear translocation of NF-κB [26]. In addition, leukemogenic Nup98 fusion proteins have been reported to cause aberrant localization of CRM1, which is required for the export of NF-κB n , and the nuclear accumulation of NF-κB [27]. This was correlated with the enhanced transcription activity of NF-κB n . Constitutively active NF-κB was found in malignant melanoma cells, and knock-down of the Nup88 gene reduced the nuclear accumulation of NF-κB in these cells [25]. Aberrant localization of CRM1 caused by over expression of Nup88 and Nup214 has also been reported [28]. Although all these reports strongly suggested the importance of nuclear transport in the regulation of NF-κB, it is possible that the reduced or increased localization of NF-κB in the nucleus resulting due to the reduced or increased nuclear transport could be a simple causal effect, while there may exist a much more complicated mechanisms for the regulation of NF-κB n by nuclear transport. To elucidate the effect of alterations in parameters affecting the nuclear transport on the oscillation pattern of NF-κB, we conducted spatio-temporal simulations.
We investigated four pathways of nuclear transport in our spatio-temporal model: NF-κB import to the nucleus, its export from the nucleus, export of IκB mRNA (mRNA IκB ) from the nucleus, and the import of newly synthesized IκB to the nucleus. We performed simulations by modifying each one of these parameters, and found that these four nuclear transport pathways altered the oscillation pattern of NF-κB n differently. While alteration in NF-κB nuclear import and its export from the nucleus resulted in no appreciable change in the oscillation pattern, alteration in the nuclear export of mRNA IκB and the import of IκB altered the persistency and the frequency of oscillation, respectively. Furthermore, reduction but not enhancement of the nuclear export of mRNA IκB increased the persistency of the oscillation, which was an unexpected result.

Results
Export of mRNA IκB from and import of IκB to the nucleus independently regulate persistency and frequency of NF-κB oscillation To examine the effect of the individual components in the nuclear transport, we employed a simple spatio-temporal model of NF-κB oscillation designed to elucidate the underlying mechanisms more clearly. For this purpose, the 3D model used originally was reduced to a 1D model, as previously described (top panels of Fig 1A [23]), which included a nuclear and a nuclear membrane compartment as shown in red. Reaction schemes and rate constants were identical to those preciously shown (bottom panel of Fig 1A and S1 Fig [22 ,23]). We referred to the rate constants of the four pathways of nuclear transport as k 1 (NF-κB import to the nucleus), k 2 (NF-κB export from the nucleus), k 3 (mRNA IκB export from the nucleus), and tp 1 (IκB import to the nucleus). We examined the changes in the oscillation pattern induced by individual changes in these rate constants. Diffusion was simulated by Fick's equation as shown in our previous report ( Fig 1A).
First, we confirmed the oscillation of NF-κB n.tot , which was the combined concentration of NF-κB n and the IκB n :NF-κB n complex as a measure of the fluorescence intensity of NF-κB in the nucleus. By using the previously set 1D parameters [23], we obtained oscillation of NF-κB n. tot similar to those obtained in the 3D model ( Fig 1B).
Next, we examined the change in the oscillation pattern of NF-κB n.tot in the 1D model induced by changing the values of k 1 , k 2 , k 3 , and tp 1 . Decay time constant τ p for the envelope of the peaks of oscillation waveform of NF-κB n.tot , which was a measure of the persistency of the oscillation, and the frequency were employed as parameters characterizing the oscillation pattern. Changing k 1 values within a range from 1/64-fold to 64-fold of control slightly decreased τ p (black line in the top left panel of Fig 1C). There was virtually no change in the frequency (gray line). Insets indicate the oscillation of NF-κB n.tot at values of k 1 indicated by blue arrows. Changing k 2 caused almost no change in τ p and frequency (top right panel of Fig 1C). Thus, the import and export of NF-κB to and from the nucleus had little or no effect on the persistency and frequency of NF-κB oscillation. In contrast, modification of k 3 or tp 1 resulted in a large change in the persistency and/or frequency of oscillation. Persistency was increased by a reduction in k 3 without any appreciable change in the frequency (bottom left panel of Fig 1C). Changing the value of tp 1 significantly modified the frequency, and this was accompanied by a change in persistency (bottom right panel of Fig 1C). Increase in tp 1 dramatically increased the frequency of oscillation. Our 3D model provides almost identical results (S2 Fig).
These changes are clearly demonstrated by the sensitivity analysis ( Fig 1D). We averaged sensitivities obtained over the entire range of parameter values. It was clear that k 3 and tp 1 independently regulated persistency and frequency, while k 1 and k 2 had only a marginal effect on these characterizing parameters. Therefore, we focused on k 3 and tp 1 in the following analyses.
Changing k 3 alters oscillation from dampened to inflating mode We found that modifying k 3 changed τ p , but the change was limited to a certain range of k 3 (Fig 1C). To investigate this further, we ran simulations at wider range of k 3 (from 10 −2 to 10 3 -fold change compared to the control as shown in Fig 2). We found that τ p gradually increased with further decrease in k 3 . At k 3 of 0.1357-fold of the control, the oscillation was still dampened, but seemed likely to persist for a considerably long time (bottom left panel in Fig 2). Further decrease in k 3 resulted in a greater increase in τ p , which is shown by black circles in the middle panel of Fig 2. If we reduced k 3 to 0.0833-fold of the control, however, the mode of the oscillation changed from dampened to inflating (top left panel of Fig 2). If we increased k 3 from this point, τ p was increased, shown by white circles in the middle panel of Fig 2. Thus, there was a dramatic change in the mode from dampened to inflating oscillation at k 3 around 0.1354-fold of the control. At a k 3 value smaller than 0.0833-fold of the control, we could not obtain a stable initial condition; that is the oscillation of NF-κB n never stopped during simulations aimed at generating stable initial conditions without an NF-κB-activating stimulus.
These results were unexpected, because we simply expected that any change in k 3 would result in a monotonic change in the persistency. It was also a surprise to find that a smaller k 3 Effects of nuclear transport on the oscillation pattern of NF-κB. (A) 3D and 1D spatio-temporal models of NF-κB oscillation. The 3D shape was identical to the one used in a previous report (top left panel, [22,23]). A spherical model cell was divided into compartments enabling reaction-diffusion computational simulation. Red compartments indicates the nuclear membrane. A simplified 1D shape was employed for detail analyses (top right panel). There was a single nuclear membrane and nuclear compartment (indicated in red). Reaction schemes shown in the middle panel are identical to those presented in a previous report [23]. The effects of four kinetic rate constants (shown in red in the middle panel) were investigated. Diffusion was calculated using Fick's equation. D and @c / @x represent the diffusion coefficient and spatial derivative of the concentration, respectively. (B) Simulated oscillation pattern of NF-κB n.tot . We obtained the same oscillation pattern from both 3D and 1D models similar to prior analyses. All parameters for the simulation were the same as previously described [23]. (C) Parameter values were changed within a range of 1/64-fold decrease to 64-fold increase of control (×1) to investigate the subsequent modification in the oscillation pattern, as characterized by τ p (a measure of persistency; black lines) and frequency (gray lines). Insets show oscillations at parameter values indicated by blue arrows. The rate constants k 1 (NF-κB import to the nucleus), k 2 (NF-κB export from the nucleus), k 3 (export of mRNA IκB from the nucleus), and tp 1 (the import of newly synthesized IκB to the nucleus) were investigated. (D) Sensitivity analyses of nuclear transport. Sensitivities over the entire range of each parameter were averaged. It is clear that k 3 affected τ p , and tp 1 affected the frequency.
doi:10.1371/journal.pone.0127633.g001 resulted in persistent oscillation, because smaller k 3 was expected to result in less de novo synthesis of IκB, leading to an incomplete "reset" of NF-κB n ultimately resulting in a dampened oscillation [23]. However, our simulation showed the opposite results. Therefore, we tried to elucidate this mechanism further.

Persistent and inflating oscillation is caused by an accumulation of mRNA IκB in the nucleus
To elucidate the reason for the change in the oscillation mode at smaller k 3 , we first investigated NF-κB n concentration at the nucleus (c0 in S3 Fig) to see whether "reset" of NF-κB n was observed during sustained oscillation as previously reported [23]. There was sufficient "reset" in the case of sustained oscillation, because the amount of NF-κB n at the troughs was smaller than the initial value during sustained oscillation (gray line in the middle panel of S3 Fig), while it was higher than the initial value under conditions of dampened oscillation (black line). These results were similar to those observed following a change in the diffusion coefficient of IκB in our previous report [23]. Next we investigated the IκB concentration at the most distant cytoplasmic compartment from the nucleus (c9 in S3 Fig) to see whether the mechanism for the sustained oscillation was identical to that previously reported for a larger diffusion coefficient [23]. We found that there was no appreciable change in the IκB concentration at c9. This result differed greatly from that observed in the diffusion coefficient experiment, where a considerable amount of IκB was stored at c9 in response to a large flux of IκB due to a large diffusion coefficient, and c9 acted as a "reservoir" for IκB [23]. Thus the mechanism for sustained and/or inflating oscillation of NF-κB n at a small k 3 value was different from that for the diffusion coefficient.
To explore the specific mechanism responsible for the effects of k 3 , we investigated time courses of related species. As the "reset" of NF-κB n is also important for the oscillation (middle panel of S3 Fig) as demonstrated by diffusion coefficient studies, we first carefully investigated nuclear IκB (IκB n ), because sufficient IκB n led to the "reset" of NF-κB n [23]. If we compared the increase in IκB n from its initial level (ΔIκB n in panel a of Fig 3A), there was virtually no difference in ΔIκB n between 0.1353-fold k 3 (gray line) and the control (black line) at the beginning of the oscillation. However, it increased steeply at 0.1353-fold k 3 (green arrow) to a value greater than control. The dashed vertical line indicates the time at which the gray and black lines crossed-over. It seemed that the delayed steep increase in ΔIκB n at 0.1353-fold k 3 was the reason for the sustained oscillation.
We investigated the time courses of the upstream species next. The increase in cytoplasmic IκB from its initial level (ΔIκB) also resembled the delayed increase at 0.1353-fold k 3 , when compared to control (panel b of Fig 3A). The same was also observed for cytoplasmic ΔmRNA IκB (panel c of Fig 3A). The crossover points shifted to earlier time points as we investigated the upstream species. If we compared these with the nuclear ΔmRNA IκB.n , however, the increase was much more steep at 0.1353-fold k 3 (gray line and green arrow in panel d of Fig  3A) than in control (black line), and there was no crossover at a time earlier than that observed for cytoplasmic ΔmRNA IκB . It is important to note that when we calculated the ratio of the increase in ΔmRNA IκB.n for both the cases, it was 1/0.1353 at the same time point as the crossover observed in conjunction with the increase of cytoplasmic ΔmRNA IκB (red curve and dashed line). We then calculated the flux of ΔmRNA IκB from the nucleus by multiplying k 3 with ΔmNRA IκB.n (panel e of Fig 3A). Fluxes at 0.1353-fold k 3 and the control crossed over at the same time point as the crossover observed when the ratio of the increase in ΔmRNA IκB.n was 1/ 0.1353. Thus, the delayed increase in IκB (panel b of Fig 3A) was caused by the sustained transcription of mRNA IκB (panel d of Fig 3A), which was caused by the small k 3 .
We were surprised to find that a smaller k 3 value resulted in the sustained oscillation. However, the mechanism thus revealed was reasonable to account for the change in the oscillation mode by k 3 . If this proposed mechanism holds true, a lower rate of de novo synthesis of IκB would also result in sustained transcription of mRNA IκB leading to the sustained oscillation of NF-κB n . This was true when tr 1 , which was the translation rate of IκB (S1 Fig), was decreased to one-tenth to control; dampened oscillation at control was rescued to sustained and inflating oscillation (top panel of Fig 3B). If D mRNA.IκB , the diffusion coefficient of IκB mRNA, was larger, mRNA IκB would diffuse to distant locations within the cytoplasm, and the de novo synthesis of IκB should be reduced, leading to the sustained transcription of mRNA IκB and a sustained oscillation. This hypothesis also held true, as shown in the bottom panel of Fig 3B. In both the cases, we observed delayed increases in IκB n and mRNA IκB.n , as in the case of decreased tr 1 , and NF-κB n level at troughs was lower than that of the control (S4 Fig). Thus, all three of the parameters tested, k 3 , tr 1 , and D mRNA.IκB contributed to the sustained oscillation by the same mechanism.
In summary, when k 3 was small, the mRNA IκB flux out of the nucleus was decreased leading to less de novo synthesis of IκB and a sustained transcription of mRNA IκB that was stored in the nucleus. This then led to the "reset" of NF-κB by an increase in de novo synthesis of IκB at subsequent time points, as well as to the sustained oscillation of NF-κB. Thus, a small k 3 resulted in the use of nucleoplasmic space as a "reservoir" for mRNA IκB (Fig 3C). Larger k 3 enabled greater mRNA IκB flux out of the nucleus, and early synthesis of IκB. This then prohibited sustained transcription of mRNA IκB by NF-κB n , leading to the dampened oscillation.
Finally, we compared the persistency of the observed oscillation following changes in k 1 , k 2 , and tp 1 , since these parameters did not appear to have a major effect on persistency ( Fig 1C). As shown in S5 Fig, neither a "reset" of NF-κB n nor a steep increase in mRNA IκB.n was observed after the start of the oscillation. In particular, no change in the time course of NF-κB n and mRNA IκB.n was induced following a change in k 2 . Thus, the change in the persistency was specific to the nuclear transport induced by k 3 in our model. The mRNA IκB flux out of the nucleus is reduced by a small amount at reduced k 3 , allowing NF-κB n to sustain mRNA IκB transcription. mRNA IκBn was stored in the nucleoplasmic space due to a small k 3 . Thus nucleoplasmic space acted as a "reservoir" for mRNA IκBn . Subsequently, accumulated mRNA IκBn were exported to the cytoplasm, where large amount of IκB was newly synthesized leading to the "reset" of NF-κB n . This resulted in the sustained oscillation of NF-κB n . Our earlier results demonstrated that changes in tp 1 affected the oscillation frequency (Fig 1C).
To investigate the effect further, we modified the tp 1 values, and the changes in the oscillation patterns for various tp 1 values are shown in Fig 4. Oscillation frequency increased with an increase in tp 1 , while the average level of NF-κB n.tot was decreased. A greater than 20-fold increase in the frequency resulted from a change in tp 1 , ranging from a 1/64-fold decrease to a 64-fold increase from the control. We then explored the reason for the change in frequency by tp 1 .
Increase in the change in slope of NF-κB n contributes to increased oscillation frequency There are two ways to change the frequency of NF-κB n oscillation, namely, the slope and the amplitude (upper panel of Fig 5A). If the slope increases without a change in amplitude, oscillation frequency increases. If the amplitude decreases without a change in the slope, the oscillation frequency increases as well. Therefore, we first compared the amplitude that resulted from a change in the four parameters tp 1 , k 1 , k 2 , and k 3 (lower panel of Fig 5 and S6 Fig). While an increase in amplitude was observed following an increase in k 1 or a decrease in k 3 , respectively, virtually no change in amplitude was observed following a change in tp 1 and k 2 . Therefore, a reduction in the amplitude caused by a change in tp 1 was not the reason for the increased oscillation frequency.
Next we investigated the slope of NF-κB n . Rising (slope+) and falling (slope-) slopes were analyzed independently (S7A Fig). Slope+ and slope-were direct consequences of the inward and outward fluxes of NF-κB and IκB n :NF-κB n , respectively, which were calculated by k 1 Ã NF-κB and k 2 Ã IκB n :NF-κB n . If we compared NF-κB n.tot , which was the summation of IκB n :NF-κB n and NF-κB n with NF-κB n alone, there was only a small difference between the two (S7B Fig). In addition, d 1n Ã NF-κB n :IκB n was negligible in comparison to a 1n Ã NF-κB n Ã IκB n (Materials and Methods). We then approximated the change in NF-κB n.tot by using NF-κB n , which was further approximated by Eq 1 (Materials and Methods). Thus we employed k 1 Ã NF-κB and a 1n Ã NF-κB n Ã IκB n as slope+ and slope-, respectively (Fig 5B). Following the 16-fold increase in tp 1 , an increase of 3.40-fold was observed both in k 1 Ã NF-κB and a 1n Ã NF-κB n Ã IκB n , indicating that both slope+ and slope-became steeper with the same magnitude ( Fig 5B). If we compared slope+ and slope-for the 16-fold increase in k 1 , increases to the same degree were observed for both slopes. However, no change in these slopes were obtained from the 16-fold increase in k 2 and the 1/7.4-fold decrease in k 3 (Fig 6A). These are summarized in S8 Fig, together with the change in the slopes as designated by tp 1 .
Since we hypothesized that the oscillation frequency could be calculated by using slopes and amplitude by a simplified oscillating waveform with triangle wave, we estimated oscillation frequency by Eq 2 (Materials and Methods, S9 Fig). Estimated frequencies for tp 1 , k 1 , k 2 , and k 3 agreed reasonably with frequencies obtained by simulations (Fig 6B). With a large k 1 , both the amplitude and the slopes were increased to almost the same extent (S9 Fig) and no change in the frequency was observed. Modification of tp 1 , however, resulted in an appreciable change only in the slopes but not in the amplitude. Therefore, it is clear that the frequency was altered by a change in tp 1 .
Under conditions of high tp 1 and k 1 , NF-κB flux into the nucleus was increased (Figs 5B and 6A). Increase in the flux was obviously caused by the increase in k 1 . However, why did high tp 1 also increase flux? As shown in S10 Fig, larger tp 1 caused greater IκB flux into the nucleus leading to an increase in IκB n . This then led to increased IκB n :NF-κB n flux out of the nucleus which subsequently reduced NF-κB n and increased NF-κB in the cytoplasm. Thus, the preferred storage site of NF-κB was relocated to the cytoplasm. In summary, slope+ and slopewere increased by an increase in cytoplasmic NF-κB and increase in the IκB flux into the nucleus, which was the direct consequence of the increase in k 1 and tp 1 , respectively. This was the major mechanism that drove higher oscillation frequency of NF-κB n.tot (Fig 6C). Under conditions of a smaller tp 1 , both the average level of cytoplasmic NF-κB and the inward flux of IκB toward the nucleus were small resulting in a lower frequency. Thus, tp 1 regulated the oscillation frequency by two different mechanisms for slope+ and slope-.

Discussion
We have been investigating the mechanisms that change the oscillation patterns of NF-κB following modification of spatial parameters ( [22,23] and this report). Over the course of these analyses we found that the "reset" of NF-κB n was important for the sustained oscillation, and larger D IκB helped to "reset" NF-κB n by storing newly synthesized IκB at a cytoplasmic space distant from the nucleus, which acted as a "reservoir" [23]. Here we report that the efflux of mRNA IκB and influx of IκB from and to the nucleus independently regulate the persistency and frequency of oscillation (Fig 1C and 1D). In addition, the reduction in the rate of translation and increase in D mRNA.IκB increase the persistency of oscillation. Increase in D IκB rescued the heavily-dampened oscillation as shown in the previous report [23]. Using these analyses we developed a model to explain the regulation of the oscillation pattern by spatial parameters, as summarized in Fig 7. The model indicates that distinct spatial parameters regulate the persistency and frequency of NF-κB n oscillation. D mRNA.IκB and D IκB and the rate of mRNA IκB efflux are spatial parameters that regulate the persistency, which are shown in green arrows, and the NF-κB Oscillation Regulated by Nuclear Transport rate of the influx of IκB is a spatial parameter that regulates the frequency of the oscillation, which is shown in a brown arrow. Transcription of mRNA IκB and the translation of IκB are non-spatial parameters regulating the persistency of oscillation. There is virtually no effect by any other nuclear membrane transport mechanisms on the persistency and the frequency, which is shown in black arrows. In summary, our results suggest that the mechanisms regulating IκB are responsible for the regulation of the oscillation pattern.
In our studies, the rate of transcription was calculated by the equation shown at the top of S11 Fig, which was identical to that used in a previous report [22,23]. If we changed the "n" from the control value of 2 to 1 or 3 at the 0.1353-fold reduced k 3 condition, the sustained oscillation was modified to a dampened or an inflating oscillation, respectively (bottom panels in S11 Fig). Thus "n", the parameter describing how many NF-κB molecules bound to the κB site of DNA were required for transcription, seems to regulate persistency.
We observed an inflating oscillation of NF-κB in the 1D model following a low flux of mRNA IκB nuclear export (Fig 2). The same inflating oscillation was observed in our 3D model when IκB kinase (IKK) degradation was ignored (S12 Fig). Thus, the inflating oscillation was not unique to the 1D model, but could also be observed in the 3D model provided that the effects of IKK degradation were negligible.
Together with our previous results [23], our current study demonstrates that the frequency and persistency of NF-κB oscillation can be regulated by its inhibitor molecule IκB and mRNA IκB . There was only a marginal effect by parameters directly regulating NF-κB, such as its rate of influx and efflux. These results might seem to be counterintuitive, as rates of the influx and efflux of NF-κB were expected to directly regulate NF-κB oscillation. However, our analyses revealed that IκB predominantly regulated the oscillation pattern. This further suggests that currently unknown mechanisms regulating the concentration and/or dynamics of IκB might also regulate frequency and/or persistency.
In the present study, we have analyzed effects of fluxes through the nuclear envelope (NE) by changing the corresponding rate constants (Fig 1C). There are two ways to alter the flux through NE: by changing the density/number of NPCs or by altering the rate of the flux Mechanisms regulating the oscillation pattern of nuclear NF-κB. The present study together with our previous one [23], showed distinct spatial parameters regulating the persistency and frequency of NF-κB n oscillation. D mRNAIκB and D IκB (thick zigzag light green arrows) and the rate of mRNA IκB efflux (thick light green arrow) are spatial parameters that regulate the persistency of the oscillation. The rate of the influx of IκB (light brown arrow) is a spatial parameter that regulates frequency. In addition, transcription of mRNA IκB and translation of IκB are non-spatial parameters regulating persistency (thin light green arrows). There was virtually no effect on persistency and the frequency by other nuclear membrane transport mechanisms (black arrows).
doi:10.1371/journal.pone.0127633.g007 through a single NPC. Senescence has been shown to decrease the number of NPCs [26], which suggests that it could decrease the outward flux of IκB mRNA and inward flux of IκB protein, together with other proteins and mRNAs. Since the inward and outward fluxes of NF-κB have virtually no effect on the oscillation pattern (Fig 1C), senescence could potentially lead to a persistent and low-frequency oscillation of NF-κB. A previous study has shown that the leukemogenic Nup98 fusion proteins caused an aberrant localization of the CRM1 protein [27]. Although the mechanism of nuclear export of IκB mRNA is largely unknown, CRM1 and NXF1 are possibly involved in this process, as shown in other mRNA export studies [29,30], and the export of IκB mRNA could be retarded in cells expressing leukemogenic Nup98 fusion proteins leading to the persistent oscillation of NF-κB. Thus, the nuclear transport could play an important role in some diseases and in aging of the cell.
It is important to address possible experimental procedures to alter the nuclear transport of IκB mRNA and IκB protein selectively. Although mechanisms of nuclear import of IκB are not well known [31], it was reported that the nuclear import of IκBα is temperature and ATP dependent and is blocked by a dominant-negative mutant of importin β [32]. Thus, the efficacy of IκBα import could be regulated by the expression level of a dominant-negative mutant of importin β. The export of mRNA and protein complex (mRNP) is not simple, but is composed of many steps, including the proper assembly of a mRNP, and its targeting and docking to NPC [29,30]. If the binding of CRM1 or NXF1 to IκB mRNP is blocked partially, its export could be selectively impaired leading to the persistent oscillation of NF-κB. In fact, it is reported that the export of mRNPs is influenced by elements within the mRNA, which is responsible for the binding of CRM1 or NXF1 to mRNA [29]. If the element in IκB mRNA responsible for this process is mutated, its export could be retarded leading to the persistent oscillation of NF-κB. The diffusion coefficient is inherent to the molecular species. However, its effective value can be modified by a change in the effective size. If IκB mRNP is enlarged by the binding of non-functional proteins, the effective value of D mRNA.IκB will be reduced because of hindered diffusion. This could lead to the dampened oscillation of NF-κB.
In conclusion, our investigations on the possible regulatory mechanisms of NF-κB n oscillation revealed that the export of IκB mRNA from the nucleus and the import of IκB to the nucleus are important in regulating the persistency and frequency of the oscillation. A decrease in the export of IκB mRNA facilitated an enhanced transcription by NF-κB n , which was retained in the nucleus, to be subsequently exported to the cytoplasm to "reset" NF-κB n and to sustain the persistency of the oscillation. Conversely, an increase in the influx of IκB led to increases in the influx and efflux of NF-κB resulting in the higher oscillation frequency. These results provide a better understanding of the mechanism of NF-κB n oscillation and the importance of the nuclear transport, indicating its relevance in the understanding of disease mechanisms.

Computational model
We constructed spatio-temporal 3D and 1D computational models of NF-κB oscillation as previously described [23]. We used the same chemical reaction model as in the previous report (S1 Fig, [23]). Briefly, the model comprised the formation of IKK:IκBα:NF-κB complex, the degradation of IκBα and the subsequent nuclear transportation of NF-κB, NF-κB transcription of IκBα mRNA, IκBα protein synthesis, and the nuclear export of the IκBα:NF-κB complex. We employed a simplified chemical reaction model excluding A20 and CYLD as our model was intended to extract phenomena and mechanisms for the regulation of the NF-κB oscillation pattern by nuclear transport.
The 3D spherical cell model with a diameter of 50 μm was divided into small cubic compartments (total 62,417) of identical size enabling reaction-diffusion simulations (top left panel of Fig 1A). We used Fick's equation for simulating diffusion, which was combined with differential equations for the chemical reactions. The central 8.3% of the compartments was assigned as the nucleus. In the 1D model, which was used for the analysis of the effect of nuclear transport, there were 10 cubic compartments with an edge length of 5 μm per cube, and the rightmost red compartment was assigned as the nucleus and nuclear membrane compartment (top right panel of Fig 1A). Reaction schemes shown in S1 Fig were embedded in the corresponding regions of the cytoplasm, nuclear membrane, and nucleus of the 3D and 1D models.
We employed the 1D model for the efficiency of analyses, because there were only 1/ 6241.7 th compartments in the 1D model compared to the 3D model. All models were constructed using A-Cell software [33,34]. Models and all parameters used in the present study can be downloaded from http://dx.doi.org/10.6084/m9.figshare.1417973. Kinetic parameters used in our simulation were the same as in the previous report [23].

Simulations
Simulation programs in c language were automatically generated by A-Cell. We used the parallelized version by openMP for a multi-core CPU. Simulations were run on a Linux computer equipped with an Intel compiler. Every time we changed parameters for nuclear transport, we first acquired an equilibrium forcing IKK = 0, which ensured a resting state. Thereafter a simulation of NF-κB oscillations was run by setting concentrations acquired by equilibration. Simulated concentrations of nuclear NF-κB were plotted as values normalized to the maximum at the control condition, unless otherwise noted.

Analyses
Frequency of NF-κB n oscillation was analyzed by FFT (Fast Fourier Transform) using Ori

Calculation of slopes in the oscillation waveform and estimation of oscillation frequency
The slope of the oscillation was defined as the rate of the increase or decrease in the total concentration of nuclear NF-κB (NF-κB n.tot ), which was the summation of NF-κB and its complex IκB:NF-κB in the nucleus. Thus, the rate change in NF-κB n.tot was calculated by d½NFkB n:tot dt ¼ dð½IkB n : NFkB n þ ½NFkB n Þ dt : Subscription n indicates species in the nucleus. Since there was only a small difference between NF-κB n.tot and NF-κB n (S7B Fig), we reduced the equation as follows: d½NFkB n:tot dt % d½NFkB n dt ¼ k 1 Á ½NFkB À a 1n Á ½IkB n ½NFkB n þ d 1n Á ½IkB n : NFkB n : If we compared the 2 nd and the 3 rd terms, the 3 rd term was negligibly small (9.04 × 10 −11 v.s. 1.59 × 10 −12 M/s). Therefore, we further simplified the equation as follows: d½NFkB n:tot dt % k 1 Á ½NFkB À a 1n Á ½IkB n ½NFkB n : Eq:1Þ The 1 st and 2 nd terms on the right hand side were the positive (slope+) and negative (slope-) slopes, respectively (Fig 5B).
If we simplified the oscillation waveform of NF-κB by triangular wave (top panel of Fig 5A), the change in the oscillation frequency was proportional and inversely proportional to the average slope of slope+ and slope-and amplitude. Therefore, we calculated the estimated fold change in the frequency byè st:fold change in the freq: ¼ fold change in fðslopeþÞ þ ðslopeÀÞg=2 fold change in amplitude : Eq:2Þ Supporting Information S1 Fig. Reaction scheme for the oscillation of NF-κB n in the 1D model showing rate constants and diffusion coefficients. The scheme is identical to that used in a previous report [23]. The diffusion process both for mRNA IκB (t.IκB) and protein IκB are explicitly shown by zigzag lines with bidirectional arrowheads, since these played an important role in the persistency of NF-κB n oscillation (see text). Other species diffused with diffusion coefficients of 10 −11 m 2 /s. Transcription of mRNA IκB was calculated by the equation shown above. There were two parameters controlling the transcription, tr 2 and n. Among them, n described the nonlinearity of the transcription in relation to the concentration of NF-κB n . n = 2 at the control condition, assuming that the binding of two NF-κB molecules to the κB site of genes was required for their regulation. If n was set to 1 or 3, the sustained oscillation at 0.1353-fold decreased k 3 resulted in a dampened or an inflating oscillation (left and right panels, respectively).