NF-κB signalling and cell fate decisions in response to a short pulse of tumour necrosis factor

In tissues and tumours, cell behaviours are regulated by multiple time-varying signals. While in the laboratory cells are often exposed to a stimulus for the duration of the experiment, in vivo exposures may be much shorter. In this study, we monitored NF-κB and caspase signalling in human cancer cells treated with a short pulse of Tumour Necrosis Factor (TNF). TNF is an inflammatory cytokine that can induce both the pro-survival NF-κB-driven gene transcription pathway and the pro-apoptotic caspase pathway. We find that a few seconds of exposure to TNF is sufficient to activate the NF-κB pathway in HeLa cells and induce apoptotic cell death in both HeLa and Kym-1 cells. Strikingly, a 1-min pulse of TNF can be more effective at killing than a 1-hour pulse, indicating that in addition to TNF concentration, duration of exposure also coordinates cell fate decisions.

Scientific RepoRts | 6:39519 | DOI: 10.1038/srep39519 TNF is not well-characterized experimentally and it is not known whether persistent vs. transient TNF exposure influences cell fate.
Here, we set out to determine the TNF pulse duration required for NF-κ B activation in single human cancer cells, and study how pulse duration affects TNF-induced apoptosis. Using single-cell data, we quantified the threshold of NF-κ B nuclear translocation required for inducing early gene transcription. We show that for a high TNF concentration, a 10-sec pulse was sufficient for NF-κ B activation but a longer pulse is required at a lower TNF concentration. Extrinsic apoptosis was also strongly activated by a short TNF pulse. In HeLa cells, a 1-min TNF pulse induced apoptosis with a potency similar to that of a 10-hr treatment whereas a 30-or 60-min pulse was less effective in cell killing. For Kym-1 cells, a 30-sec pulse of TNF induced as much death as continuous stimulation. Our study reveals that the duration of TNF exposure influences the TNF-induced cell fate decision.

Results
A brief TNF pulse induces nuclear localization of RelA. We designed and built a simple microfluidic system that uses laminar flow 24 to provide spatiotemporal control over TNF delivery in cell cultures. Our device consists of two inlets for culture medium with and without TNF, a three-chamber cell culture channel, and a single outlet ( Fig. 1a and Supplementary Fig. S1a). Hydrostatic pressure differences between inlet reservoirs in the default operating mode caused most of the channel to be exposed to 'Medium' (Fig. 1a,b, t = 0 sec), with a narrow band of positive-control cells continuously exposed to a laminar stream of 'Medium + TNF' (Fig. 1b, yellow boxes). To expose the rest of the cells to a TNF pulse, the 'Medium + TNF' reservoir was temporarily raised, increasing its hydrostatic pressure (Fig. 1a, right, Fig. 1b, blue boxes). This reservoir also contained Alexa647-conjugated BSA allowing us to track the TNF-containing stream by imaging ( Fig. 1b and Supplementary Movie S1). With this simple system, we reproducibly achieved TNF pulses as short as 10 sec.
To monitor TNF-induced NF-κ B activation, we seeded the device with EGFP-RelA-expressing HeLa cells ( Supplementary Fig. S1c) and quantified nuclear EGFP-RelA from time-lapse images as described previously 25 . The calculated flow shear forces in the device were small (< 0.2 Pa for imaged areas; Fig. 1b, yellow and blue boxes and Supplementary Fig. S1b), and therefore unlikely to stimulate the NF-κ B pathway via mechanotransduction. Indeed, a mock 1-min pulse of medium without TNF had no observable effect on EGFP-RelA nuclear abundance (Fig. 1c). In contrast, a 30-sec pulse of 100 ng/ml TNF resulted in transient and variable EGFP-RelA nuclear translocation in most cells ( Fig. 1d and Supplementary Movie S2). Single-cell nuclear RelA time courses resembled those observed by us and others in response to continuous or 5-min TNF treatments 22,[25][26][27][28][29] , showing that a very short TNF pulse induces NF-κ B nuclear translocation in HeLa cells and demonstrating that we could monitor it in our device.

Defining a transcription-inducing EGFP-RelA translocation.
Cell-to-cell variability of nuclear EGFP-RelA following a 30-sec TNF pulse was substantial. Some cells had large changes in nuclear EGFP-RelA abundance (Fig. 1d, red trace) while others had high initial nuclear EGFP-RelA and only a small TNF-induced increase (Fig. 1d, orange trace). We set out to determine which cells were truly 'responsive' to the TNF pulse, cells in which NF-κ B activation should result in target gene transcription. We leveraged our published dataset of same-cell EGFP-RelA nuclear translocation time courses and target transcript numbers 25 to establish the EGFP-RelA nuclear translocation threshold under which a cell is unlikely to induce NF-κ B-dependent gene transcription. In that study, we determined that the maximal fold change of nuclear RelA (ratio of maximal to initial nuclear RelA in a cell) is predictive of transcript number for TNF-driven and RelA-dependent early response genes 25 . We therefore evaluated the error rate in determining whether or not a cell had a transcription-inducing EGFP-RelA translocation ('responsive' vs. 'non-responsive') while varying two parameters: fold-change threshold and cut-off value for transcript number. Although our earlier data were from cells treated continuously with TNF, we reasoned that if a given fold change is unlikely to induce transcription under continuous exposure, then it is also unlikely to induce transcription after a short pulse. We found a minimum of total error at ~1.22-fold-change for both IL8 and TNFAIP3, two NF-κ B-inducible genes with no or few transcripts in unstimulated cells ( Fig. 2a and Supplementary Fig. S2; ~5% error). This fold-change threshold showed little dependence on the transcript number cut-off value, although values of 8 and 35 transcripts discriminated well between untreated and TNF-treated distributions for IL8 and TNFAIP3 respectively (Fig. 2a, and Supplementary Fig. S2a,b). Applying this threshold, we determined that the red-trace cell in Fig. 1d was responsive to the 30-sec TNF pulse (max fold change = 2.94), while the orange-trace cell was non-responsive (max fold change = 1.06) despite having overall greater nuclear EGFP-RelA abundance. Altogether, our analysis shows that the NF-κ B system can sense and respond to an increase of nuclear RelA of as little as ~20%.
The duration of TNF exposure required for NF-κB activation is concentration-dependent. We next assessed EGFP-RelA nuclear translocation in response to different TNF pulse durations (Fig. 2b,c). We found that 78% and 86% of cells were responsive when exposed to 100 ng/ml TNF for 10-sec or 30-sec respectively, versus 94% for continuous treatment. Importantly, fold-change distributions for a 30-sec pulse or continuous treatment with 100 ng/ml TNF were not significantly different (Fig. 2b). At 10 ng/ml TNF, cells required a longer pulse for transcription-inducing EGFP-RelA translocation; only 33% of cells were responsive after a 30-sec pulse (Fig. 2c). This percentage climbed to 89% for a 60-sec pulse and to 98% for continuously treated cells yet in this scenario, the nuclear EGFP-RelA fold-change distributions were still significantly different.
In contrast to response distributions for a short TNF pulse, distributions of nuclear EGFP-RelA fold change observed under continuous treatment with 10 ng⁄ml vs. 100 ng⁄ml TNF are similar (Fig. 2b,c; p = 0.30, unpaired Kolmogorov-Smirnov test). However, others have reported that TNF concentration impacts both the fraction of cells responding as well as certain response parameters such as the lag before RelA nuclear entry 28,30 . Applying our threshold for transcription-inducing EGFP-RelA translocation, we found that when we reduced the TNF    S3b). This suggests that for continuously exposed cells, TNF concentration is reflected both in the percentage of responding cells and in the NF-κ B pathway response amplitude, although this response saturates by 10 ng⁄ml TNF.
To understand how ligand-receptor interactions vary with TNF concentration and pulse duration, we simulated TNF-TNFR1 reversible binding kinetics as a general bimolecular surface reaction 31 . Simulations using previously determined binding constants for 125 I-labeled TNF with TNFR1 in HeLa cells 32 showed that receptors could be saturated within 30-60 sec with 100 ng/ml TNF, but only after 5-8 min with 10 ng/ml TNF (Fig. 2d). Only ~50% of receptors may be bound after a 60-sec pulse at 10 ng/ml TNF (Fig. 2d), explaining the observed concentration-dependence of the minimal TNF pulse duration for NF-κ B activation. By running similar simulations over a range of receptor numbers (for both TNFR1 which is nearly ubiquitously expressed and TNFR2, which is more restricted in expression and absent in HeLa cells 32 ) as well as a range of parameter values, we found that this result is not sensitive to receptor number and is robust to large variation in parameter values ( Supplementary Figs S4 and S5). Notably, in all contexts the dissociation of TNF from TNFR1 is slow, suggesting that complexes should persist long after a pulse (Fig. 2d). Because internalization of TNF-bound receptors begins as early as several minutes after exposure to TNF 17,33 , our simulations predicted that a short TNF pulse could be sufficient to activate later signalling events such as the assembly of pro-apoptotic Complex II.
Apoptosis occurs in response to a short pulse of TNF. To monitor caspase activity in single TNF-treated cells, we imaged HeLa cells expressing a FRET-based initiator caspase reporter (IC-RP 34 ; Fig. 3a) and quantified IC-RP cleavage by the CFP/YFP ratio ( Fig. 3a and Supplementary Movie S3). Cells were pre-treated with interferon-γ (IFNγ ) then TNF-treated. IFNγ is a cytokine that sensitizes many cancer cell lines, including replicate experiments, with cell numbers between 50 and 375 in each condition for each experiment. P-values are reported indicating that percentages of total cell death are not significantly different between 1-min pulse and continuous treatment (p = 0.80; paired two-tailed t-test) and that the percentage of cell death after a 60min pulse is significantly lower than during continuous treatment (p < 0.004; paired one-tailed t-test) but not significantly lower than after a 1-min pulse (p = 0.07; paired one-tailed t-test).
Scientific RepoRts | 6:39519 | DOI: 10.1038/srep39519 HeLa cells, to TNF-induced apoptosis, partly via increased initiator caspase-8 expression [35][36][37][38][39] and, in vivo, TNF and IFNγ often co-occur as IFNγ serves to activate macrophages which are a major source of TNF 40 . To limit the effects of dynamic signals activated by the immediate response to IFNγ and thus isolate the role of TNF, cells were pre-treated with IFNγ for 24 hrs before TNF addition. IC-RP cleavage accumulation varied from cell to cell, likely, at least in part, because of natural variation in the abundance of apoptotic signalling molecules [41][42][43][44] . For cells that underwent TNF-induced apoptosis, cleaved IC-RP accumulated sharply and the CFP/YFP peak corresponded with apoptotic morphology before cell detachment, revealing the 'time-of-death' . In cells that survived, CFP/ YFP fluctuations were small (Fig. 3a). In continuous TNF treatment, we observed significantly more apoptosis at 100 ng/ml than at 10 ng/ml ( Supplementary Fig. S6a) and therefore focused on 100 ng/ml TNF for our analysis of how the extent and timing of cell death responses vary with pulse duration. Because TNF-induced apoptosis occurs over many hours, we increased experimental throughput by performing wash-out experiments in 96-well plates instead of performing them under flow condition in the low-throughput microfluidic system. The shortest pulse we could reproducibly impose was 30-sec and thus we quantified dynamics of IC-RP cleavage after a TNF pulse ranging from 30-sec to 60-min and for cells treated continuously. Consistently, we observed that a 1-min pulse was at least as effective at killing cells as 10-hr continuous TNF treatment (Fig. 3b,c). By contrast, less cell death was observed after a 60-min pulse of TNF (p < 0.004 vs. continuous treatment, paired one-tailed t-test), showing that shortening exposure to high-concentration TNF does not necessarily decrease its pro-apoptotic effect. In addition, cells that died in response to a TNF pulse of 30 min or less died earlier on average than continuously treated cells ( Fig. 3c and Supplementary Fig. S6b).
For cells to die earlier in response to a short exposure to TNF, TNF-induced signalling pathways are likely to be distinctly coordinated in at least a subpopulation of cells. Indeed, we found subtle but statistically significant differences in the short-term EGFP-RelA dynamics (as quantified by the max fold change in nuclear EGFP-RelA), as well as in the endogenous RelA distributions, for IFNγ -pre-treated cells exposed to a 1-min TNF pulse versus cells exposed to TNF continuously ( Supplementary Fig. S7a,b). This is in contrast to what we observed in the absence of IFNγ pre-treatment where there was no statistically significant difference between a 1-min TNF pulse or continuous treatment, as expected based on our results for a shorter TNF pulse in the microfluidic device ( Fig. 4 and Fig. 2b). Despite similarity in the max fold change in nuclear EGFP-RelA, examination of nuclear EGFP-RelA at later time points suggested that EGFP-RelA resides in the nucleus for a longer period under continuous TNF treatment ( Fig. 4a and Supplementary Fig. S7c). We therefore compared the time-integrated nuclear EGFP-RelA signal, calculated as the area under the fold change of nuclear EGFP-RelA time course (AUC), to assess whether there are differences in longer term RelA dynamics. We found that both in the absence and presence of IFNγ -pre-treatment, continuous treatment with TNF led to prolonged residence of EGFP-RelA in the nucleus and thus a greater integrated signal ( Fig. 4 and Supplementary Fig. S7c,d). Although it is likely that additional distinctions also lie elsewhere in the TNF-induced signalling network, it is possible that the shorter activation of RelA in the context of a response to a short pulse of TNF leads to a reduced pro-survival response and accelerated cell death.

Efficient induction of apoptosis in response to a short pulse of TNF is concentration-dependent.
Our simulations of TNF-TNFR1 binding suggest that at low TNF concentrations much longer exposures might be required to achieve binding of TNF to a large fraction of TNFR1 (Fig. 2d). Therefore, we sought to test whether

whiskers). P-value indicates that there is a significant difference in AUC between continuous-and pulse-treated cells (one-tailed t-test).
Scientific RepoRts | 6:39519 | DOI: 10.1038/srep39519 a cell line sensitive to low TNF concentrations, the Kym-1 human rhabdomyosarcoma cell line, would exhibit duration-dependent cell death at a low TNF concentration. Indeed, we found that at 1 ng/ml TNF, the fraction of Kym-1 cells that die increased with exposure duration (Fig. 5, left panel). However, at 100 ng/ml TNF, a 30-sec exposure to TNF induced as much cell death as continuous treatment (Fig. 5, right panel) and, as for HeLa cells, our Kym-1 data suggests that a short pulse of high-concentration TNF induces significantly earlier cell death than continuous treatment ( Fig. 5 and Supplementary Fig. S8). Taken together, our data show that a short pulse of high-concentration TNF is sufficient to induce substantial cell death in both HeLa and Kym-1 cells. The duration of this pulse in turn affects the efficacy and timing of the pro-apoptotic effect of TNF -cells treated with a brief pulse tend to die earlier than cells treated continuously. Although the extent of TNF-induced cell death after a 1-min pulse is comparable with continuous treatment, longer pulse durations are less effective in HeLa cells suggesting that the duration of cytokine exposure is an important mediator of TNF-induced signalling and cell fate.

Discussion
In a tissue, cell may experience transient exposure to inflammatory cytokines as these are released in bursts by migrating tissue-resident immune cells and can be rapidly cleared. Using a simple microfluidic system, we characterized the cellular responses to different durations of exposure to TNF in single-cells. Our results demonstrate that the minimum pulse required for TNF-induced signal transduction is concentration-dependent, and that at high TNF concentrations, a pulse as short as 10-30 sec can induce significant NF-κ B translocation and a pulse of 30 sec or 1 min can induce caspase activation and apoptosis. A similar concentration-dependence was recently reported for the treatment duration required for LPS-induced NF-κ B activation in mouse embryonic fibroblasts and PDGF-induced PDGFR phosphorylation in NIH-3T3 23,45 suggesting that this may generalize to other cell types and extracellular ligand systems.
We note here that although the maximal TNF concentration used in our study exceeds serum concentrations from patient samples, cytokine secretion in intact tissue is spatially restricted and efficiently captured by neighbouring cells 46,47 . Cells exposed to TNF in vivo, in a tissue or tumour, should therefore experience a "puff ", or pulse, of TNF at local concentrations that far exceed those measured in blood. Therefore a locally concentrated short TNF pulse, which we showed to be sufficient to initiate cell fate decisions, is also likely to be biologically relevant.
Based on the current understanding of TNF-induced signalling, cell-to-cell differences in the formation of Complex I and Complex II should alter the relative strength of pro-survival and pro-apoptotic signals. TNF dissociates slowly from TNFR1 (ref. 32 and Fig. 2d), over a timescale longer than that of internalization of TNF-bound receptor complexes 19,20,33 . Considering this slow dissociation, two aspects of our results are surprising: 1) that cell death timing differed following a short pulse vs. continuous treatment and 2) that fewer HeLa cells died after a 60-min pulse than after a shorter pulse or continuous treatment. Our working model for these observations is that the duration of TNF exposure may alter the relative strength of crosstalk between pro-survival and pro-apoptotic signals. While a high fraction-bound of TNFR1 is effectively reached during a short exposure to TNF, sufficient for formation of Complex II and caspase activation, the lack of continued receptor-ligand interactions following the pulse cannot sustain long-term Complex I and NF-κ B activity. The short-pulse scenario therefore leads to weaker overall activation of the pro-survival pathway (Fig. 6, left). By contrast, a longer TNF exposure and the resulting sustained NF-κ B activity could yield a more efficient activation of NF-κ B-mediated pro-survival signals. These pro-survival signals could, in turn, block the slow time-scale pro-apoptotic caspase signals leading to an overall weaker flux through the pro-apoptotic pathway (Fig. 6, right). Under this working model, a pulse of a particular duration that optimizes activation of the pro-survival pathway relative to the pro-apoptotic pathway may therefore minimize cell death, akin to our observation of a local minimum of cell death after a 60-min pulse of TNF. Nevertheless, our results do not exclude the possibility that TNF-treated cells may also integrate additional pathways downstream of Complex I and Complex II to arrive at a duration-specified cell fate decision. Specifically, TNF-induced autocrine signals are known to contribute to the fate decision 48 and these secreted pro-survival and pro-apoptotic factors are likely diluted during flow and wash-out experiments. Analogously, the flow of blood and interstitial fluids could influence the extent and timing of TNF-induced cell death in vivo. The study of the interplay between intracellular and extracellular signals in the context of a TNF response will require experimental paradigms that integrate measurements and manipulations of both types of signals.
Overall, our results show that treatment duration is an important mediator of TNF-induced signalling and cell death decision and that in certain contexts reaching the highest fractional kill may not require maximizing the duration of exposure to a pro-death stimulus. These findings meet a growing body of work showing that signalling dynamics as well as the timing and sequence of drug addition can all influence cell fate decisions 36,[49][50][51][52][53][54] . It will be interesting in the future to examine the effect of exposure duration, and the interplay with ligand-receptor-affinity, in other cellular signalling networks.

Materials and Methods
Cell culture and treatment of cells with TNF. HeLa cells (ATCC, VA) stably expressing EGFP-RelA (described in ref. 25) and HeLa stably expressing IC-RP (described in ref. 55) were cultured in Dulbecco's Modified Eagle Medium (DMEM) supplemented with 10% FBS, 100 U/ml penicillin, 100 μ g/ml streptomycin and 0.2 mM L-glutamine (Invitrogen, MA) at 37 °C and 5% CO 2 . Kym-1 cells 56 were cultured in RPMI1640 supplemented with 10% FBS, 100 U/ml penicillin, 100 μ g/ml streptomycin and 0.2 mM L-glutamine (Invitrogen, MA) at 37 °C and 5% CO 2 . For experiments in microfluidic devices, TNF treatments are described below. For experiments in 96-well imaging plates (BD Biosciences, CA), on day 1, HeLa cells were seeded at ~4000 cells/well, Kym-1 cells were seeded at ~6000 cells/well. On day 2, culture medium was replaced with medium with 200 U/ml IFNγ (Roche, IN) to sensitize cells to TNF-induced apoptosis (it is necessary for HeLa cells which are not sensitive otherwise, but here we used it as well for Kym-1 for consistency). IFNγ is, like TNF, an inflammatory cytokine and it sensitizes many cancer cell lines to TNF-induced apoptosis, partly via increased initiator caspase-8 expression [35][36][37][38][39] . On day 3, two hours prior to TNF treatment, culture medium was replaced (with IFNγ ) and 24 hrs after IFNγ treatment, complete medium with or without TNF was spiked into wells to yield the indicated final TNF concentrations. After the specified duration, TNF-containing medium was removed, cells were rapidly washed three times and then incubated in the appropriate medium without TNF (with or without IFNγ ) for the duration of the experiment. To control for any effects of washes and medium exchange, these were also performed for cells  treated continuously (medium replaced to medium containing the appropriate concentration of TNF, with or without IFNγ at the start of treatment). Care was taken to use only conditioned and warmed medium for all the washes and media changes during the experiment, to minimize disturbances to the cells.
Device fabrication and operation. Microfluidic chips with two inlets and three chambers for cell culture (illustrated in Fig. 1a) were made in PDMS (Sylgard 184, Dow Corning Corporation, MI) and bonded to a No. 1.5 glass coverslip (ThermoScientific, MA). Pipet tips (200 μ l) were inserted into 1/16″ tygon tubing connected to steel tubes (16-gauge) into punched inlets and outlets to act as fluid reservoirs.
To prepare for experiments, devices were sterilized with 70% ethanol then abundantly flushed with sterile PBS. Channel surfaces were pacified by flushing the devices with complete culture medium and incubating > 12 hrs. Cells were then seeded at appropriate density and allowed to adhere for 12 hrs under no-flow conditions (all reservoirs at equal height). Medium was replaced then and again 24 hrs and 4 hrs before an experiment.
For treatment with a pulse of TNF, medium with TNF at the final desired concentration was prepared with 1 μ g/ml Alexa Fluor ® 647 conjugated-BSA (Life Technologies, MA). The device was securely mounted on a custom stage for a BD Pathway 855 BioImager and flow allowed to reach a steady state in the pre-pulse mode (Fig. 1a) while monitoring by imaging the Alexa-647-BSA epifluorescence signal. To pulse cells with TNF, the 'medium + TNF' reservoir was temporarily raised manually (Fig. 1a). Pulse duration was verified by imaging the Alexa-647-BSA epifluorescence signal. Determination of a nuclear EGFP-RelA fold-change threshold for cellular response to TNF. To estimate the error in determining whether a cell has had a transcriptionally significant response to TNF based on the fold-change of nuclear NF-κ B in a cell (Fig. 2a), we considered two parameters. The first parameter is the 'cut-off value for transcript number' (transcript cutoff ) which we defined as the minimum number of transcripts expected from a cell that has undergone a transcriptional response to TNF. We thus assumed that a cell with transcript cell ≥ transcript cutoff would have been likely to be transcriptionally activated in response to a stimulus. The second parameter is the fold-change threshold (FC thresh ), a parameter that defines the value of 'nuclear fold change' at which significant transcriptional activation should occur. Because our dataset contains a pool of paired values that describe the maximum fold change (FC cell ) as well as the transcript number (transcript cell ) for each of hundreds of single cells exposed to a range of TNF concentrations, we evaluated the total error as follows:

Live-cell imaging and analysis.
(1) Assume that a given value of transcript cutoff correctly partitions the dataset into transcriptionally responsive and non-responsive cells. From this it follows that we assumed that a cell with transcript cell ≥ transcript cutoff mRNAs has undergone a transcriptional response, whereas a cell with fewer transcripts (i.e. transcript cell < transcript cutoff ) has not. (2) For each value of FC thresh , quantify the number of false positives and false negatives. False positives were defined as a cell for which FC cell ≥ FC thresh , but transcript cell < transcript cutoff . Thus this cell had a greater than threshold response, as indicated by change in nuclear NF-κ B, but was not transcriptionally activated based on assumption in (1). False negatives were defined as a cell for which FC cell < FC thresh , and transcript cell ≥ transcript cutoff . Thus this cell had a sub-threshold response based on change of nuclear NF-κ B, yet was transcriptionally activated as per (1). (3) The total error for a value of FC thresh is the total number of false positives and false negatives divided by the total number of cells. The total error was calculated across a range of FC thresh values in increments of 0.01.
Finally, the nuclear EGFP-RelA fold-change threshold for cellular response to TNF was defined as the fold-change threshold (FC thresh ) where total error is minimized.
Simulations of liquid flow in the devices. Three-dimensional simulations were carried out using COMSOL Multiphysics 3.5 (COMSOL, Inc., MA), modelling the actual shape and dimensions of the device. Simulations solved Navier-Stokes equations with the different boundaries conditions and the following assumptions: 1) fluids similar to water (incompressible Newtonian fluid with a density of 998.2 kg.m −3 , and a dynamic viscosity of 0.001 N · s m −2 ), 2) no-slip boundary conditions on the channel walls and 3) steady-state conditions were reached. The hydrostatic pressures at each inlets were calculated from its height relative that of the outlet reservoir.
Model of bimolecular surface reaction. Binding of TNF to TNFR1 and TNFR2 was modelled as independent reversible receptor-ligand interactions with constant ligand concentration 31 : Scientific RepoRts | 6:39519 | DOI: 10.1038/srep39519 are the concentrations of TNF-bound TNFR1 and TNFR2, respectively, L 0 is the concentrations of TNF, R1 T and R2 T are the total concentrations of receptors (assuming 3,000 TNFR1 molecules per cell for HeLa and Kym-1, and 30,000 TNFR2 molecules per cell for Kym-1), k f1 and k r1 are the association (1.833 × 10 7 M −1 s −1 ) and dissociation rate (3.5 × 10 −4 s −1 ) constants for TNF-TNFR1 and k f2 and k r2 are the association (2.5 × 10 7 M −1 s −1 ) and dissociation rate (0.011 s −1 ) constants for TNF-TNFR2 (as reported in a previous study 32 ). We surmise that constant ligand concentration is an appropriate approximation as with a very high medium:cell volume ratio, the number of ligand molecules vastly surpasses that of receptors. Simulations were carried out in MatLab (MathWorks, MA) using the ode45s solver.