We thank the Reviewers for their valuable comments which helped us to improve the manuscript. These comments allowed us to view our study from a different angle, and for this reason we modified (and shortened) the title which now emphasizes the result of analysis of NF-κB information channel capacity shown in Fig. 5. Additionally, we reformulated the image analysis section in Methods hoping to improve its clarity. Below, we include our responses, which indicate also how the manuscript has been modified to address Reviewers’ concerns. We hope that the revised manuscript is now suitable for publication in Biology Direct.
Reviewer’s report 1
Marek Kimmel, Rice University, Houston, TX, USA
This is an interesting paper, contributing to the understanding of the mechanistic details of the active and passive transport between cytoplasm and nucleus using an important example of NF-kB. The message of the paper is very well documented, both experimentally and by simulations. I have three remarks or rather discussion item, which in my opinion are worthy of clarification.
Reviewer’s recommendations to authors
Naively, one obvious thing to do is to compare functioning of the system with importins and exportins to the system without these molecules. It seems easy computationally, but more difficult experimentally. It would be good to know the authors’ opinion on this and maybe some simulations carried out.
Authors’ response: Knocking out or silencing importins presents an experimental difficulty, since it would disturb the functioning of an entire cell. However, in the case of NF-κB family interactions with importins are well established due to experiments in which NLS sequences in p65, and other NF-κB proteins were mutated, see [19, 20]. Wolynes group  and others [13, 14] have also shown that IκBα mask these NLS. Therefore the question we consider is not whether importins are indispensable for NF-κB regulation, but rather whether explicit modelling of importins–NF-κB interactions adds to the understanding of kinetics of the NF-κB pathway. We address this question in response to reviewer 2, by comparing models in which the importin binding step is modelled explicitly or lumped with NF-κB nuclear import step.
In several places, the possibility of the cell optimizing this or that is mentioned. There are two caveats to such hypotheses. One is that most real-life systems (not only biological) are clearly suboptimal, but linger for long periods regardless (trilobites and human genome, being ad hoc examples). Why should cells be different? Second, optimization of two different processes may be contradictory (consider cancer cells dividing slower than normal cells). The NF-κB system is a multifunctional hub, so how to optimize such an object? Authors’ insights are welcome.
Authors’ response: We agree that there are many systems that appear to be far from optimal. However, since NF-κB pathway has been evolutionarily conserved from Drosophila to mammals (Ghosh et al., 1998, Annu. Rev. Immunol. 16:225–260), and is on the first lines of defence against pathogens, we expect that it is optimized for fast responses. Additionally, since excessive inflammation can be harmful we hypothesize that it is also designed for fast shut-off.
I like the “digitized” pulse-series experiment. However, what I would expect is that if a “1” is succeeded by a “0”, there should be some low-level transients observed, while there is a complete absence of signal. Might you explain how it is possible? Is this exactly what is also expected in an experiment (no such experiment has been performed, correct?)?
Authors’ response: The model predicts that pulses lasting 15 min or less produce a single pulse of nuclear NF-κB with no tail (see Additional file 3: Figure S4) and this prediction is in agreement with experimental data at the population level [30, 64]. See also Additional file 3: Figure S5 and Ashall et al.  for single cell analysis of responses to repeated 5 min TNFα pulses, also showing no tail in NF-κB activity.
DETAILS: Background: Please explain if exportins help export mRNA and proteins, while importins only help import proteins, or is the distinction more complex.
Authors’ response: Importins and exportins, together termed karyopherins, are involved in the classical pathway of nuclear transport of proteins. On top of that, pre-microRNA and tRNA are also exported by members of the exportin family. In contrast, mRNA do not use karyopherins but instead are exported by a heterodimer of NXF1 and NXT1 proteins . The distinctions amongst karyopherins, importins and exportins are now explained better in the Background section.
Results: “NF-κB is transported back to the cytoplasm complexed with IκBα, which passes through nuclear pores after association with exportin 1”. Do you mean that exportin pulls IkBa, which in turn pulls NF-kB out of the nucleus?
Authors’ response: IκBα binds NF-κB. After exportin 1 binds the NES of IκBα, it enables the free IκBα and IκBα:NF-κB complex to cross the nuclear pore. This is now clarified in the main text.
Is dynamics of exporting of such large complex different from say, exporting of pure IkBa?
Authors’ response: IκBα, having a mass of 32 kDa, is at least partially independent of importins and exportins, i.e. it can cross nuclear pores alone, see Fagerlund et al. . Since IκBα is mostly cytoplasmic, we assume in the model that IκBα translocates to the nucleus independently of importins but uses exportins to translocate out of the nucleus.
“For freely diffusing molecules the ratio of nuclear export to nuclear import …” Do you mean the ratio of rates? The reasoning outlined in this paragraph is pivotal for the paper, so it should be carefully phrased.
Authors’ response: Yes, we mean the ratio of rates, we have now clarified it in revised text.
Responses to Reviewer 2
James Faeder, University of Pittsburgh, Pittsburgh, PA, USA
This paper presents an extended computational model of NF-κB signaling that specifically considers the interactions of NF-κB and IκBαlpha (IκBα) with the importin and exportin proteins that mediate nuclear import and export respectively. Experimental data from fibroblasts in the form of Western blots and fixed cell immunofluorescence staining is obtained that shows that at times between about 60 and 90 min the apparent concentration of IκBα in the cytoplasm exceeds that of NF-κB and yet NF-κB is still able to translocate in substantial amounts to the nucleus. It is claimed that this phenomenon can only be accurately described by the extended model that includes NF-κB interaction with importin. It is further argued that the ability of NF-κB to translocate to the nucleus even when the cytoplasmic concentration of IκBα exceeds that of NF-κB allows the system to reset more quickly following pulsatile stimulation than would otherwise be the case, which thus increases the maximum possible rate of information transmission, which is estimated to be 1 bit (based on the ability to detect only the presence or absence of TNF) times one over the reset time, which is estimated to be 60 min based on the simulations shown in Fig. 5. Overall, I see this work as a significant contribution to the ongoing effort to model and understand the mechanisms that influence NF-κB dynamics. I have some reservations about the claimed importance and novelty of the mechanisms being considered here, which I would like the authors to address prior to publication. In particular, the authors claim but do not demonstrate that the proposed model uniquely captures a key finding in their experimental data, which is that NF-κB translocation can continue even when the level of IκBα exceeds that of NF-κB. I would like to see a more conclusive demonstrate that the nuclear import mechanism the authors have explicitly added to their model is required to capture this effect.
Reviewer’s recommendations to authors
The main issue I have with this paper is that it is not clear from the presentation in the paper that the explicit consideration of karyopherin (importin/exportin) mediated transport actually results in a novel mechanism. It seems like any model that explicitly considers free cytoplasmic NF-κB will include a competition between binding to free IκBα and nuclear transport, although the parameters governing that competition could be different from those proposed in the current model. On p. 4 it is stated, however, that “although previous models assumed that free NF-κB may translocate to the nucleus, importin alpha binding precluding sequestration by IκBα was not explicitly considered. As a result, in these models efficient NF-κB translocation was possible only after the IκBα level dropped below that of NF-κB.” So, the authors are claiming the rate in these models was set so low that translocation could not compete with IκBα rebinding until IκBα levels fell below those of cytoplasmic NF-κB, but no references are given so it’s hard to tell which models the authors are referring to. I check one model that I’m familiar with, that of Lee et al. (2014), and this didn’t seem to be the case. Also, the model already has the competition mechanism built in, and it’s just a matter of varying the import rate relative to the binding rate to capture the mechanism that is described here, so although it’s useful to identify a mechanistic basis for this competition, it is not clear that the present model uniquely captures this mechanism. I think that to demonstrate that this competition parameter is indeed important for providing a correct description of the NF-κB/IκBα dynamics, they should show how varying the import rate affects the overall dynamics and also demonstrate how previous models do not capture this effect correctly.
Authors’ response: The role of importins and exportins in NF-κB regulation is well documented. The questions is whether explicit implementation of the NF-κB–importin binding step is important in modelling. We think it is, and it can be explained as follows. In the model without importins, in the presence of free IκBα, released NF-κB may translocate to the nucleus if the expected entry time is shorter (or at least comparable) with expected binding time with IκBα. The Reviewer is right that this can be assured in the model by assuming sufficiently fast nuclear translocation of NF-κB. However, in the reality the NF-κB translocation time is controlled by the size of the cell and its nucleus, diffusion coefficient, binding with importins, and the translocation through nuclear pores. It is therefore possible that imposing constrains on NF-κB translocation time (in order to make it shorter than NF-κB–IκBα binding time) we obtain the wrong picture of spatial regulation of the system. This has important implications for the more detailed reaction-diffusion models that are emerging in recent years for NF-κB and other regulatory systems (see Terry & Chaplain  and Sturrock et al. ).
In the model with importins it is NF-κB may enter the nucleus despite elevated levels of IκBα provided that expected NF-κB–importin binding time is shorter than the expected binding time with IκBα. We expect that this condition holds until concentration of free cytoplasmic IκBα is smaller than that of importins.
Therefore, we think that explicitly accounting for NF-κB–importin interactions is important in order to bring modelling to a more precise mechanistic description. This does not mean that the models that lumped together various reactions may not serve as an reasonable description. We now formulate presentation of our results in a more modest way.
We also supplement the Results section with a new figure (Fig. 6), in which we compare our model with its variant that lumps together the processes of NF-κB–importin binding and NF-κB nuclear translocation. The analysis is performed in correspondence to pulsed stimulation considered in Fig. 5 (in response to a suggestion by Reviewer 3, we introduce this figure in the Results section). Fig. 5 shows that the system can transmit information about NF-κB pulses as long as their frequency is not larger than 1/h. From the stochastic time profiles shown in Fig. 5 one can see that the response to second pulse TNFα is critical, i.e. the amplitude of the response to the second pulse is the lowest. Therefore in Fig. 6 we compare two model variants analysing the ratio of the second to the first peak amplitude and the difference between levels of IκBα in its second minimum and at t = 0. The comparison is done as a function of IκBα–NF-κB binding rate and NF-κB nuclear import coefficient. Generally in the novel model, the NF-κB translocation at the second peak is higher and is accompanied by a smaller decrease in the level of IκBα. The difference between two models is pronounced for small NF-κB import coefficient and for high IκBα–NF-κB binding rate, and as expected by the Reviewer, it vanishes when NF-κB import coefficient is large.
Additionally, we consider a more detailed model in which the three processes of NF-κB binding, complex translocation to the nucleus and importin dissociation are considered separately. As a reminder, in the original model we lumped processes of NF-κB nuclear translocation and importins dissociation in the nucleus. We demonstrate that this more detailed description enhances the effect of importins. Nonetheless, one should keep in mind that this is also a simplified picture, as in reality NF-κB is first bound by importin-α3 or α4, which are in turn bound by importin-β, the ternary complex diffuses into the vicinity of the nucleus and passes through nuclear pores. Next, in the nucleus importin β dissociates in response to RanGTP binding, and only then importin-α may dissociate from NF-κB. Since the affinity between importin-α and NLS is high (typically 10 nM) this process must be also somehow induced .
Regarding the values that the coefficients mentioned above take in existing NF-κB models, in Lee et al.  the authors assume IκBα–NF-κB binding rate equal 0.5 (μM s)-1 and NF-κB nuclear import rate equal 0.0026 s-1. Assuming NF-κB concentration equal 0.1 μM (Lee et al. scan the range 0.04–0.4 μM; the value 0.1 μM corresponding to roughly 105 was estimated by Carlotti et al. [53, 54], and assuming that IκBα concentration exceeds that of NF-κB by 50 % (i.e., assuming that there is 0.05 μM of free IκBα) we obtain the pseudo-first order NF-κB binding rate equal 0.025 s-1 (versus NF-κB nuclear import rate 0.0026 s-1). This implies that a released NF-κB molecule has about a 10 times higher chance of binding a free IκBα molecule than of translocating to the nucleus. In our earlier model  the NF-κB nuclear import rate was equal 0.01 s-1, while IκBα–NF-κB binding rate was equal 5 × 10-7 s-1. These values mean that NF-κB nuclear translocation is 2.5 times less probable than IκBα binding. The substantially different import coefficient is assumed/fitted in the models developed by Levchenko and Hoffmann (see Werner et al.  and Werner et al., 2005, Science 309:1857–61). In these models NF-κB import rate is 0.09 s-1, while the IκBα–NF-κB binding rate is also equal 0.5 (μM s)-1. Therefore in these models the NF-κB translocation outcompetes IκBα binding. We expect however, that NF-κB import rate is rather of order of 0.01 s-1 (or smaller) than of order of 0.1 s-1 (which would imply average translocation time of 10 s). To our knowledge, the NF-κB import rate was never measured directly.
Another possible problem with the modeling and inference here is the discrepancy between model and experiment that is displayed in Fig. 3c, about which I could not find any comment in the manuscript. The issue is this: in the experiment the IκBα level continues to rise between 60 and 90 min while at the same time the amount of nuclear NF-κB rises and both IκBα and NF-κB remain elevated at 180 min. The model, on the other hand, exhibits a decrease in the IkB level on the same time interval. This result begs the question how in the experiment the NF-κB level can rise as the IkB level also rises, but the model doesn’t display this behavior and hence can’t provide an explanation. The model clearly shows that IκBα and NF-κB oscillate out of phase, whereas the measured levels do not. It seems likely that some other mechanism is at play here, which is not being captured by the model. Another issue that concerns the modeling and also the interpretation of the experimental data is the basal level of NF-κB in the nucleus. On p. 6 it is stated that “WB analysis indicate[s] that in unstimulated cells the nuclear NF-κB level is very low.” That is indeed what is shown in Fig. 2d, but it is contradicted by the fixed cell imaging data shown in Fig. 2a
and in Fig. 3a, c, which indicate than an average of about 20 % of NF-κB is in the nucleus prior to stimulation. The model does not capture this effect, which calls into question whether is it also missing some key aspects of the IkB/NF-κB interaction dynamics. Something curious about the initial conditions of the model is also revealed by looking at the black points in Fig. 4a and b
showing the initial conditions in individual cells for the experiments and model respectively. Whereas the experiments exhibit considerable variability in the fractional amount of nuclear NF-κB and relatively little variation in the ratio of IkB/NF-κB, the model shows little nuclear NF-κB but considerable variability of in the relative amount of IkB. How might this discrepancy affect the observed results?
Authors’ response: Since there is no Fig. 3c, we think that the Reviewer means Fig. 4c. Indeed, there is a discrepancy between the model and single cell data, as observed by the Reviewer. The model was fitted to the population data obtained in the form of Western blots (see Fig. 2d). The immunofluorescence single cell data were provided to show that also at single cell level NF-κB translocation is possible even when IκBα exceeds initial levels. By analysing single cell images we rule out the possibility that IκBα level is very high only in the fraction of cells that do not exhibit second NF-κB translocation.
In fact this effect is more pronounced when analysed at the level of immunostaining images. As shown in Figs. 2d and 4c the average IκBα level (between 100 and 180 min) calculated based on immunofluorescence images is higher than obtained in Wester blots, and surprisingly it increases between 60 and 100 min apparently in phase with nuclear NF-κB.
The discrepancy between the fraction of nuclear NF-κB in unstimulated cell obtained by quantified Western blots and immunofluorescence images follows possibly from overshadowing of nuclei by cytoplasm, which is hard to avoid even when using confocal microscopy. This overshadowing depends on cell morphology and we failed to fully correct it by our quantification method (see Methods).
Considering the above, we think that population data better represent the average NF-κB and IκBα levels, while image-based single-cell quantification can give some insight into heterogeneity of the response. In the revised manuscript we mention and briefly discuss these discrepancies.
Reviewer’s Report 3
William Hlavacek, CNLS, Los Alamos, NM, USA
Korwek et al. report results from a study that involved both experimentation and modeling. The study was focused on understanding oscillations in nuclear localization of the transcription factor NF-kappaB in response to stimulation by an endotoxin (lipopolysaccharide, LPS) or a cytokine (TNFalpha). These signals induce the degradation of IkappaB, which is responsible for sequestering NF-kappaB in the cytosol. Degradation of IkappaB allows NF-kappaB to concentrate in the nucleus, which leads to new synthesis of IkappaB. The authors explain how, after an initial pulse of nuclear localization, NF-kappaB is able to concentrate in the nucleus a second time even though the overall abundance of its inhibitor IkappaB rises above its baseline abundance before the second pulse of nuclear localization. The explanation is that IkappaB must compete for binding to NF-kappaB with importin alpha proteins, which are karyopherins that mediate transport of NF-kappaB into the nucleus. It seems that this report offers an answer to a puzzling question about the dynamics of NF-kappaB nuclear localization. I think this report would be rather interesting to other researchers working on regulation of NF-kappaB. I suppose the major weakness of this report would be that the conclusions of the authors about the influence of karyopherins on NF-kappaB dynamics have not been directly tested, for example, by modulation of the strength of interaction between RELA and KPNA2.
Reviewer’s recommendations to authors
There are some points in the report of Korwek et al. that could be clarified. I wonder if the authors could present more illustrative simulations or introduce a simplified model to more clearly explain how the competition between IkappaB and importin alpha gives rise to the faster-than-expected oscillations in NF-kappaB nuclear localization. I’m not confident that I was able to fully appreciate the authors’ insights.
Authors’ response: In the revised manuscript we provide a comparison between the model with and without explicitly accounting for importins (see Fig. 6 and the response to Reviewer 2).
I think that competition alone is not the only deciding factor but rather it is the competition in combination with the fact that there are two different compartments where NF-kappaB can be found (cytosol and nucleus). In any case, I would appreciate a clearer explanation of the role of karyopherins in NF-kappaB nuclear localization dynamics.
Authors’ response: Yes, the Reviewer is indeed right that the discussed effect is the competition of IκBα and importin-α in combination with the fact that there are two different cellular compartments where NF-κB can be found. In the revised manuscript we clarified role of karyopherins in NF-kappaB nuclear localization dynamics.
It is not entirely clear from the manuscript as written if the above-baseline level of IkappaB during the second pulse of NF-kappaB nuclear localization is a novel observation of the authors being reported for the first time here, or rather a previously observed phenomenon.
Authors’ response: To our knowledge, we are the first to show that in single cells nuclear NF-κB translocation coincides with above-baseline levels of IκBα. Although in a report by Fagerlund et al. (2015)  NF-κB translocation and elevated IκBα are also shown by Western blotting at 90-120 min after stimulation, only our immunofluorescence single-cell data demonstrate that this effect cannot be explained by high accumulation of IκBα in some cells and nuclear translocation in others.
The authors make several assumptions about protein copy numbers. I think these assumptions could be bolstered by referring to protein copy numbers reported by the Mann group for various mammalian cell lines, such as the report by Geiger T et al. (2012) [Mol Cell Proteomics DOI
Authors’ response: The use of data detailing the exact protein copy number is indeed very compelling. Concerning the work of Geiger et al. (2012), however, we found the values included in the paper unsuitable for our model. First of all, all eleven cell lines screened for proteins by Geiger et al. (2012) were of human origin and mostly of epithelial phenotype (we are aware that it does not necessarily preclude this data from use in modelling of MEFs). Secondly, in most of these cell lines copy numbers for all of the pertinent proteins like RelA, IκBα and A20 were not quantified simultaneously. Only three lines (GAMG, Jurkat and LnCap) had iBAQ values quantified for all three of these proteins and some of these values come from only single replicate or exhibit quite significant intra-replicate variance. Furthermore, these data suggest that copy number of RelA exceeds that of IκBα by one order of magnitude, or in some cases even two (e.g. log-transformed IBAQ values for RelA and IκBα in the GAMG cell line are around 7 and 5.2, respectively). Although we admire the scope of the cited paper, we find these values hard to reconcile with our current understanding of the role of IκBα as the main RelA inhibitor.
We have decided to use the estimations of NF-κB copy number included in the works of Carlotti et al. [53, 54], as stated in the manuscript, while the values for other proteins were mostly predicted by the model.
It would be appreciated if the authors provided the HGNC names for proteins (e.g., NF-KBIA for IkappaB).
Authors’ response: We now provide HGNC names for genes in revised manuscript.
The description of nuclear trafficking is incomplete. For example, RAN is never mentioned. A more complete description of nuclear trafficking would be helpful.
Authors’ response: We include more detailed description of nuclear trafficking and discuss the role of RAN.
Furthermore, the authors may wish to acknowledge that modeling of nuclear trafficking has been considered in the past, as in the work of Zilman A., Effects of multiple occupancy and interparticle on selective transport through narrow channels: theory versus experiment. Biophysical Journal Volume 96 February 2009 1235–1248
Authors’ response: We refer to the work of Zilman et al.  and the recent work of Lolodi et al.  in the Background section.
In the abstract, the authors assume that readers will know that importins and exportins are karyopherins. The word “karyopherin” should probably be defined upon first use.
Authors’ response: We define the term karyopherins in the revised manuscript.
The authors state that BioNetGen implements the Gillespie algorithm. It would be more precise to state that BioNetGen implements an efficient variation of Gillespie’s direct method.
Authors’ response: This is now corrected it in the revised manuscript.
It is said that TNFalpha is unstable in vivo but stable “under experimental conditions.” Could the authors say more about how conditions affect TNFalpha stability? Does “resynthesized” mean “newly synthesized?”
Authors’ response: We discussed TNFα stability in vivo and in vitro studies in revised manuscript in the section “Formulation of the computational model”.
Formulation of the computational model. The abbreviation “WB” should be defined upon first use.
Authors’ response: This is now corrected it in the revised manuscript.
In the abstract, the authors make a point about information transmission rate, but this issue is next considered only in the Discussion section. It’s odd that Fig. 5
is not cited in the Results section. The authors claim that information channel capacity depends on n and tau without explaining or citing a source. It would be helpful if the authors could say more about this point and cite appropriate supporting references.
Authors’ response: We now include Fig. 5 in the Results section and clarify what we mean by information channel capacity and how it is estimated.
The authors claim that the rates of transcription and translation for IkappaB and A20 are near their maximum values. How are the maximum values estimated? Could the authors cite appropriate supporting references for the estimates of the maximum rates?
Authors’ response: We discussed how these rates are estimated and included appropriate references in the section “Formulation of the computational model” in revised manuscript.