Single TNFα trimers mediating NF-κB activation: stochastic robustness of NF-κB signaling

Background The NF-κB regulatory network controls innate immune response by transducing variety of pathogen-derived and cytokine stimuli into well defined single-cell gene regulatory events. Results We analyze the network by means of the model combining a deterministic description for molecular species with large cellular concentrations with two classes of stochastic switches: cell-surface receptor activation by TNFα ligand, and IκBα and A20 genes activation by NF-κB molecules. Both stochastic switches are associated with amplification pathways capable of translating single molecular events into tens of thousands of synthesized or degraded proteins. Here, we show that at a low TNFα dose only a fraction of cells are activated, but in these activated cells the amplification mechanisms assure that the amplitude of NF-κB nuclear translocation remains above a threshold. Similarly, the lower nuclear NF-κB concentration only reduces the probability of gene activation, but does not reduce gene expression of those responding. Conclusion These two effects provide a particular stochastic robustness in cell regulation, allowing cells to respond differently to the same stimuli, but causing their individual responses to be unequivocal. Both effects are likely to be crucial in the early immune response: Diversity in cell responses causes that the tissue defense is harder to overcome by relatively simple programs coded in viruses and other pathogens. The more focused single-cell responses help cells to choose their individual fates such as apoptosis or proliferation. The model supports the hypothesis that binding of single TNFα ligands is sufficient to induce massive NF-κB translocation and activation of NF-κB dependent genes.


Parameters justi…cation and discussion
Cell dimensions W 0 -cell volume: we adopted value the 2 10 12 l after Carlotti et al., [1]. The value W 0 does not appear explicitly in the model, but is needed to translate concentrations into numbers of molecules per cell. The same value was used in our previous studies [9], [10], and in [12].

Coe¢ cients of the TNFR1-IKKK-IKK-I B transduction pathway
This transduction pathway transmits TNF signal downstream causing I B phosphorylation and degradation and subsequent nuclear NF-B translocation. The coe¢ cients of this pathway have to be …tted simultaneously, since in most cases the change of one of coe¢ cients may be compensated by the change of others. Since we do not have direct measurements on TNFR1 receptors and IKKK activity, the …rst constraint is IKK activity (measured most accurately by Delhase [7]) and then ubiquitous data on I B degradation and NF-B nuclear translocation. The action of the pathway is attenuated by A20, which is NF-B responsive, which makes …tting of pathway coe¢ cients di¢ cult.
Important developments in modeling of TNFR1-IKKK-IKK-I B signaling are due to Park et al. [8]. The strongest discrepancy between Park et al. [8] and our models is in the IKK activity pro…le. According to Park et al. model [8] TNF stimulation results in sharp rise of active IKK which reaches the plateau, while according to our model IKK activity is transient with high peak at about 10-15 min. of TNF stimulation followed by a very low tail. In the case of A20-de…cient cells the tail is higher, but still much lower than the peak. The transient character of IKK activity was observed …rst by Delhase et al. [7] in HeLa cells and then by Lee et al. [20] in mouse …broblasts, and then by Werner et al. [16]. This transient nature of IKK activity is possibly due not to the phosphatase dephosphorylation but rather, as shown by Delhase et al. [7] to overphosphorylation. M = 1000 -total number of TNFR1 receptors, assumed. Variation of this parameter may be compensated by k b . The number of TNFR1 receptors per cell may vary signi…cantly between cell lines [21], e.g. there are about 3000 TNFR1 per cell for HeLa [3], and about 10000 for Histiocytic lymphoma (U-937) cells [21], but much less for B-cell lymphoma (Raji) cells.
k b = 4 10 6 s 1 ml/ng -receptor activation rate. We have chosen this activation coe¢ cient to assure that 90% of cells are activated (have at least one receptor active) in …rst 10 min. of TNF stimulation with the dose of 1ng/ml, which may be considered as a saturation dose. The receptor activation rate should not be confused with TNF binding rate. Binding of TNF trimer initiates receptor TNFR1 trimerization and formation of an active receptor complex in a multistep process involving binding of RIP and TRAF2. Park et al. [8] considered several reversible processes leading to the receptor activation. k f = 6 10 4 s 1 -receptor inactivation rate, it corresponds to t 1=2 = 20 min. This is in accordance with Grell [3], who found that TNF trimers dissociate from TNFR1 receptors with a half time of 33 min., while the internalization time is of the order of 10 to 20 min. Park et al. [8] assumed k f = 2 10 3 s 1 .
KN = 10 4 -total number of IKKK molecules, assumed. There is substantial freedom in choosing this parameter, i.e. a di¤erent choice of KN may be compensated by other parameters of the transduction pathway. Park et al. [8] assumed IKKK concentration of 0.045 M (what gives 45,000 molecules).
k a = 10 4 s 1 (IKKK activation rate) and k i = 0:01s 1 (IKKK inactivation rate) are assumed. The value of k a together with that of KN implies that single receptor activates at most one IKKK molecule per second. High k i causes the IKKK-IKK transduction pathway to have small inertia. Park et al. [8] assumed TRAF IKKK association rate corresponding to k a equal to 10 M 1 s 1 (which corresponds to 10 5 s 1 ). Our estimation of parameters k a , k i and k 1 (see below) is based on values of the corresponding parameters of the well studied MAPK pathway [25].
KN N = 2 10 5 -total number of IKK molecules, assumed. In our previous study [10] the same total number of IKK molecules was maintained by the balance of production and degradation terms. In the original Ho¤mann et al. model [13], only the active IKK was considered and its initial concentration was assumed to be 0.1 M (what gives 100,000 molecules). In Cheong et al. [12] time and TNF dose dependent IKK activation/inactivation rates are used. At highest TNF dose considered, concentration of active the IKK reaches 0.2 M. In Kearns et al. [17], the initial concentration of active IKK of 0.8 M (800,000 molecules) is assumed at the start of the stimulation phase. Park et al. [8] assumed total IKK concentration of 0.06 M. k 1 = 5 10 6 s 1 -IKKn activation rate. This value was …tted, it implies that one IKKKa molecule activates at most one IKKn molecule per second. Since IKKn was activated directly by TNF in previous our models [9], [10], the meaning of k 1 is di¤erent than previously. Park et al. [8] assumed IKKK IKK association rate corresponding to k 1 equal to 10 M 1 s 1 (which corresponds to 10 5 s 1 ). k 3 = 0:003s 1 -IKKa inactivation rate, …tted. In [9] and [10] k 3 = 0:0015s 1 . k 4 = 0:0005s 1 -IKKii transformation, …tted. This coe¢ cient represents two rates: transformation from IKKi to IKKii and from IKKii to IKKn. It was …tted based on [20] and [16], showing elevation of the IKK activity at about 1h in A20-/-cells, which in our model is due to recovery of IKKn through intermediate form IKKii. k a20 = 10 4 -A20 mediated TNFR1 block; this value was …tted based on pulse-pulse and A20 knockout experiments. It implies that when the number of A20 molecules equals 10 4 the activity of TNF bound receptors is twice lower than in the absence of A20. k 2 = 10 4 -IKKa inactivation due to A20; this value was …tted based on pulse-pulse and A20 knockout experiments. It implies that when the number of A20 molecules equals 10 4 , inactivation of IKKa proceeds twice faster than in the absence of A20. The new value k 2 corresponds to former k 3 =k 2 = 3 10 4 [10]. Thus the in ‡uence of A20 onto IKKa inactivation is higher, but A20 is less abundant because it degrades faster (at the rate c 5 ) and its mRNA also degrades faster (at the rate c 3 ) than in [10]. a 2 = 10 7 s 1 (IKKa mediated I B phosphorylation) and a 3 = 5 10 7 s 1 (IKKa mediated phosphorylation of I B jNF-B complexes), …tted. I B phosphorylation proceeds through its binding to IKKa, but since these complexes are very unstable, we assumed that IKKa directly phosphorylates free I B and complexed with NF-B with rates corresponding to formation of these unstable complexes. Thus a 2 = 10 7 s 1 corresponds to IKKa-I B synthesis rate of 0:1 M 1 s 1 , while a 3 = 5 10 7 s 1 corresponds IKKa-I B jNF-B synthesis rate of 0:5 M 1 s 1 . In [13], [18], [16] these rates are respectively 0:0225 M 1 s 1 and 0:185 M 1 s 1 ; but the kinetics of active IKK was very di¤erent, as said the initial concentration was assumed to be 0.1 M, and then IKK was freely degrading with the half time of 2.3 h in the presence of TNF , or with the half time of 5 min. in the absence of TNF .
At high TNF dose IKKa pulse reaches 70,000 molecules which yields I B phosphorylation rate of 0:007 s 1 , and I B jNF-B phosphorylation rate of 0:035 s 1 , which allows for almost total I B degradation in …rst 10-15 min. of TNF stimulation.
The capability of activation of NF-B by a single TNF molecule follows from high ampli…cation of a signal by TNFR1-IKKK-IKK-I B transduction cascade. Speci…cally the magnitude of this ampli…cation is determined by coe¢ cients k f , KN , k a , k i , KN N , k 1 , k 3 , a 2 , a 3 . As already said there is substantial freedom in choice of these parameters since the change of one parameter may be compensated by the change of others. For example the expected number of IKKK molecules activated by a single receptor is k a KN=k f and expected number of IKK molecules activated by IKKKa molecule is k 1 KN N=k i ; the actual number of active IKKKa and IKKa resulting from activity of single receptor is however lower due to their rapid inactivation governed respectively by coe¢ cients k i and k 3 .
Gene activation/inactivation and transcription/translation rates q 1 = 1:5 10 7 s 1 (NF-B driven activation of I B and A20 genes), and q 2 = 10 6 s 1 (I B mediated NF-B dissociation from I B and A20 sites); these values are adopted after our previous work [10]. They imply fast gene activation (with t 1=2 of order of one minute) when most of the 100,000 NF-B molecules are in the nucleus, and almost immediate turning o¤ of NF-B dependent genes when the bulk of freshly synthesized I B moves into nucleus.
Since estimation of transcription and translation coe¢ cients is controversial, we discuss it here in detail. The total amount of synthesized protein is proportional to the product of mRNA transcription and translation rates, thus one has some freedom in determining c 1 and c 4 . We assumed a likely value of c 1 = 0:1mRNA/s (transcription speed per gene copy) and then we …tted value of c 4 = 0:5 protein/(mRNA s) trying to keep both values with accordance to current knowledge.
In previous work [10] we assumed c 1 = 0:075 s 1 , while in [9] where the transcription rate was proportional to NF-B concentration in cytoplasm we assumed the value of 0:16 s 1 as the upper limit reached when all NF-B is in the nucleus. This limit was based on following estimation: The typical transcription speed in animal cells is of the order of 40 nucleotides (nt) per second (Levin,[11] p. 129). A single gene, however, can be read by a number of RNA polymerases simultaneously (see e.g. Levin [11]). Assuming that spacing between subsequent RNA polymerases is of the order of 250 nt one obtains the transcription initiation frequency of 40 (nt/s)/250nt = 0.16s 1 . Cheong et al. [12] assumed the upper limit of transcription as 0.55s 1 based on transcription speed of 55nt/s and spacing between mRNA polymerases of 100nt [27], [28]. In our opinion Cheong et al. [12] estimation gives the highest reasonable limit. The mRNA synthesis rate has been measured by Femino et al. [26] for -actin by single RNA transcript visualization as 4 mRNA/min.
As already said even unrealistically high transcription coe¢ cients may be compensated by smaller translation coe¢ cients, so the entire model can give correct predictions of proteins kinetics.
c 5 = 5 10 4 s 1 -A20 degradation rate; this value was re-…tted mostly based on our pulse-pulse experiment. In previous works [9], [10], c 5 = 3 10 4 s 1 tp = 0:01 s 1 -degradation of P-I B and P-I B bounded to NF-B. In all previous model the immediate degradation of phospho -I B was assumed. Here we add two separate fast equations for this process. The main reason is that the inhibition of proteasome can slow down degradation rate tp and that concentration of P-I B form can be measured, which potentially may help in model validation.
c 5a = 10 4 s 1 -spontaneous degradation of I B ; and c 6a = 2 10 5 s 1 spontaneous degradation of I B bounded to NF-B, these values were adopted after Pando and Verma [5].
Substantially di¤erent values are in recent works [16], [17] and [19], c 5a = 2 10 3 s 1 and c 6a = 10 6 s 1 . Speci…cally it was found in [19] that NF-B binding slows down I B spontaneous degradation by a factor of 2000. Surprisingly they found that while IKK speeds up I B degradation when it is bounded to NF-B, it slows down degradation of free I B . In our opinion these …nding (especially the 6 min. half time for free I B protein) still deserve independent veri…cation, since they seems to be in some contradiction with I B transfection experiments in which excess of I B is observed for hours. The other problem is that even basal I B transcription rate of 1.5 mRNA/s per gene copy appears to be above physiologically plausible level.

Transport coe¢ cients
The transport characteristics of I B , NF-B and I B jNF-B complexes were examined by Carlotti et al. [6], who concluded that NF-B nuclear import is 50 fold faster than export, while nuclear import of I B jNF-B complexes is negligible. In other words they found that free NF-B quickly translocates to the nucleus and its export back to the cytoplasm is due to its binding to I B ; I B jNF-B complexes quickly migrate to cytoplasm. Based on I B overexpression studies Carlotti et al. [6] assumed that the ratio of I B transport parameters (nuclear import)/(nuclear export) =2. Ho¤mann et al. [13], based on Carlotti et al. [6] and their model …ts choose i 1a = 3 10 4 s 1 , e 1a = 2 10 4 s 1 ; i 1 = 0:09s 1 ; e 2a = 0:0138s 1 (notation as in Table 1) and NF-B nuclear export as 8 10 5 s 1 . With small modi…cations these parameters are then by followed subsequent works [16], [17], [18] and [19].
In this model we totally neglected the NF-B nuclear export and …tted, NF-B nuclear import, i 1 = 0:01s 1 , I B jNF-B nuclear export: e 2a = 0:05s 1 , I B nuclear import, i 1a = 0:002s 1 , and I B nuclear export, e 1a = 0:005s 1 . The …rst two values are in basic agreement with Carlotti et al. [6] study, while the last two are not. Our coe¢ cients i 1a = 0:002s 1 and e 1a = 0:005s 1 adopted here after [10] imply that free I B is rather cytoplasmic than nuclear. When …tting the model we realized that choosing e 1a > i 1a we obtain more accurate …ts. We expect that the source of this discrepancy is the following: Carlotti et al. [6] consider I B overexpressions for which the amount of I B is several fold higher than that of NF-B. In real situation the excess of I B over NF-B is not as signi…cant. In fact, as shown by Yang et al. [22] (Fig. 4A) at low level of pI B EGFP transfection, I B , which is then expected to be in excess over NF-B, is mostly cytoplasmic. The same can be observed in Nelson et al. [23] (Fig. 3A) experiment on Hela cells cotransfected with I B EGFP and RelA-DsRed-Express. Analyzing time series of images we may observe that for various ratios of I B EGFP:RelA-DsRed-Express, these two proteins remain in the cytoplasm. It appears that when I B is present in moderate excess over NF-B, it remains mostly in cytoplasm. One could hypothesize that additional I B molecules may weakly associate with I B jNF-B complexes, which would slow down nuclear import of these semi-free I B molecules. Experiment by Malek et al. [4] suggests that NF-B heterodimers may have some additional I B binding sites.
It is not straightforward to compare transport coe¢ cients of our model, with those of [13], [16], [17], [18], since in the Ho¤mann/Levchenko models (except [12]) the "transport of concentrations" is considered, i.e. the nuclear and the cytoplasmic volume are implicitly assumed to be equal. This can imply that in fact the discrepancy between our models is smaller than it appears; for example I B concentration ratio (nucleus to cytoplasm) 2:1 implies inverse molecule number ratio 1:2.5 (assuming k v = V =U = 5, as we did in our model).

Miscellaneous
NF-B tot = 10 5 -the total number of NF-B molecules, assumed. Carlotti et al., [1] estimated number of NF-B molecules as 60,000. It can be, however, substantially higher in Rel A transfected cells, [22]. Here we adopted the value of 100,000 to be somewhere in between those of the normal and transfected cells. In our previous works we assumed [9], [10] NF-B tot = 60,000, but in [10] we found that the best agreement with Nelson et al. [23] experiment is for NF-B tot = 120,000. In the original Ho¤mann et al. model [13], and in subsequent works [16], [18], [19] NF-B concentration is assumed to be 100nM. In Kearns et al. [17], total concentration of NF-B containing complexes is 0.125nM equal to 125,000 NF-B molecules when calculated per cytoplasmic volume. a 1 = 5 10 7 s 1 -I B -NF B association (in cytoplasm), and a 1 k v in nucleus. This high value was adopted after our previous study [10]. In corresponds to 0:5 M 1 s 1 as assumed by Ho¤mann et al. [13]. It is known [4] that I B and NF B have a¢ nity of the order of 1nM, which causes that free I B and NF-B practically can not be observed together. For simplicity, we neglected dissociation of I B jNF B complexes in the model.