Quantitative estimation of death receptor abundance and clustering

Apoptosis inducing DRs such as Tumour Necrosis Factor Receptor 1 (TNFR1) and Death Receptors 4 and 5 (DR4/5) are expressed at comparable protein levels in HeLa cells [39]. Additionally, it is known that their protein expression level is correlated with the receptor abundance on the cell surface [8]. High variation in TNFR1 surface abundance were estimated in previous studies ranging from 300 to 3000 molecules per single HeLa cell [40,41]. To get more accurate estimates, we performed the single cell quantification of TNFR1 membrane expression in HeLa cells employing the QuantiBRITE phycoerythrin beads based assay (see S1 File). We determined that the average number of TNFR1 does not exceed 905 receptors per cell. We further used this quantity as the reference in our comparative quantification of DR4/5 receptors based on MS data set (S1 File). Thus, we calculated that DR4 and DR5 receptors are present on HeLa cell surface in an average amount of 769 and 926 monomeric receptors, respectively (Table A in S1 File).

Next, we estimated the amount of the DR complexes associated with DL at the single cell level. Due to the fact that the DR-DL association is generally much quicker [42] than the downstream processes such as ProCasp8 dimerization and subsequent Casp8 activation [43], we applied the rapid equilibria approximation to calculate the amount of DL bound receptors. According to the law of mass action the time evolution of the amount of DR-DL complexes is (1)

Where [R_total] is the total number of receptors per cell (Table A in S1 File), [RL] is the number of DR-DL complexes and [L] is death ligand concentration (Table B in S1 File).

Setting the RL to the rapid equilibrium (2)

From (1) we calculated the average number of DR-DL complexes per cell as a function of L, R_total and the DL dissociation constant K_d (3)

The minimal unit of the active DR-DL complex is the trimer [44]. The trimeric DR-DL complex gives birth to a single DISC platform which internalizes within the subsequent 10–15 minutes [45,46]. If the DISC has not bound to cellular Inhibitor of Apoptosis Proteins (cIAPs), cIAP1 or cIAP2, then it either releases active RIP1 protein into the cytosol [47] where it can form RIPoptosome or Necrosome platforms (Fig 2B) (as in case of TNFR1), or it makes active RIP1 protein accessible for further RIP1/3 and ProCasp8 proteins accumulation on the DISC itself (as in case of DR4/5 activation) [5,17]. Therefore, in the modelling routine each activated DISC was translated into a single RIP1 protein molecule which is available immediately after DL introduction to the cell culture.

Trimeric DR-DL complexes tend to organise high order clusters in cellular membranes [44,48] and bring several associated DISC/RIPoptosomes into close proximity. Such clustering stimulates more efficient signalling [49] and enables ProCasp8 activation not only by dimerization on the single DISC/RIPoptosome but also by synchronised binding of two ProCasp8 monomers with two independent DISC/RIPoptosomes within one cluster. To introduce DISC/RIPoptosomes clustering processes in the model, we estimated the number and the size of the DR-DL clusters based on the experimentally derived DR-DL probability distribution from a study published earlier by Fricke and co-workers [44]. We calibrated probability redistribution from the total pool of activated trimeric DR-DL complexes, calculated in the previous step, to the clusters of different size (see S1 File). Using these probabilities, we assigned for each random DISC/RIPoptosomes complex formed its associated cluster. The final algorithm assumes that DISC/RIPoptosomes complexes within one cluster are able to first encourage the activation of ProCasp8 by direct dimerization (cis-activation) and subsequently activate ProCasp8 via simultaneous binding within closed proximity (trans-activation) (Fig 2A). Thus, this information about the amount of the activated DR-DL complexes and their clustering conformation served as an important input for the model. The scenario of non-clustering DR signalling was studied as well by setting the probability of trans-activation of Casp8 within DISC/RIPoptosomes complexes cluster to zero. This scenario is hereafter referred to as disrupted clustering.

Mathematical modelling of the initiation of the extrinsic apoptosis pathway

We have developed a core model capturing the cascade of intracellular reactions that are essential for the initiation of the apoptosis. The model reactions are partitioned into two modules: a stochastic and a deterministic module (Fig 2).

The first stochastic module represents the process of stochastic assembly of DR-induced DISC/RIPoptosome multiprotein platform which facilitates initiation of ProCasp8 dimerization and self-activation by cleavage (Casp8*; activated Casp8 dimer in Fig 2E). We implemented this module with the direct Gillespie stochastic simulation algorithm [50,51] which accounts for molecular fluctuations and slow association and dissociation rates following each component of the platform individually. It assigns the reaction propensities in probabilistic terms. The binding propensities of ProCasp8 together with its binding partner protein, Fas-associated death domain protein (FADD), and competitor protein RIP1/RIP3 that comprise the core scaffold of RIPoptosome are calculated from the concentrations that we quantified experimentally in HeLa cell culture (Table D in S1 File). FADD protein is crucial for apoptotic initiation [35,52]. This protein consists of both a Death Effector Domain (DED) and Death Domain (DD) which are specific motifs for ProCasp8 [53] and RIP1 [16,54] self-oligomerization respectively. Through these domains, ProCasp8 and RIP1 are bridged via FADD (as shown in grey in Fig 2B). RIP3 protein can form homo-oligomers, but can also associate with RIP1 scaffolds through the RIP homotypic interaction motif (RHIM), forming amyloid structures [27,28] (Fig 2B). Intensive recruitment of RIP3 molecules to the amyloid triggers transphosphorylation of RIP3 by RIP1 with consequent transmission of phosphate groups to the MLKL pseudokinase. Phosphorylated MLKL executes necroptosis [25,30]. Therefore, in the absence of FADD and joint Casp8 activation platforms these structures spontaneously trigger necroptosis [35,55,56] (necrosome complex; purple in Fig 2B). Additionally, we quantified the concentrations of the cellular FLICE (FADD-like IL-1β-converting enzyme)-inhibitory protein (c-FLIP). As a DED-containing protein, cFLIP in its short (cFLIPs) and long (cFLIPl) form, can be recruited to the ProCasp8 platform abrogating or restricting activation of Casp8 [53,57,58] (cFLIP molecules; light and dark blue in Fig 2B). In addition to this suppression, Casp8 activation can be disrupted by binding its own processed DEDs which may remain in the cytosol (DED1-DED2; white in Fig 2B).

The second deterministic module mimics the activation of two effector caspases, Caspase 3 (Casp3) and Caspase 6 (Casp6) which is triggered by stochastically activated Casp8. Pro-forms of both caspases form stable dimers at physiological concentrations [59]. By cleavage, Casp8 activates Casp3 (Casp3*; activated dimer of Casp3 in Fig 2E) [60]. Casp3* activates Casp6 (Casp6*; activated dimer of Casp6 in Fig 2E) [61,62] and has autocatalytic function cleaving ProCasp3 [63,64]. Finally, Casp6* can cleave free ProCasp8 (Casp8*; cleaved monomer of Casp8 in Fig 2E) [64–67] however Casp8 becomes active only after a very slow dimerization (Casp8*) [19,21]. Previous models suggest that this effector caspase feedback upon weak DR stimulation probably can accelerate Casp8 activation which was initially started at the DISC or RIPoptosome platform [68]. However, the feedback can be inhibited by X-linked IAP (XIAP) which tightly binds Casp3 and, further, marks Casp3 with ubiquitin that leads to its proteasomal degradation [69,70]. The overall dynamics of Casp8 activation can be tracked quantitatively with a Casp8-specific FRET cleavage probe (FRET, Fig 2E). The fixed threshold rate of this FRET probe cleavage accurately determines the moment of MOMP in HeLa cells [2]. Based on the mass action and conservation laws, the time evolution of the variables that comprise this module were modelled by a deterministic system of ordinary differential equations (ODE) (details in S1 File).

All protein concentrations and parameters used in the model are provided in Tables D and E of Materials and Methods file (S1 File).

Stochastic initiation of Casp8 activation through DISC/RIPoptosome assembly

The estimated weight of the RIPoptosome after short DR-targeted stimulation may exceed 2MDa [10,24,29]. To reproduce the RIPoptosome growth and composition we first employed the stochastic modelling module simulating the assembly of the individual RIPoptosomes at the single cell level. RIP1 on its own forms unlimited filaments in vitro [28], however, in the cell the long-term RIPoptosomal filament growth is limited by the cell volume and the stiffness of the cellular components. We followed unlimited filament growths without implementation of these physical limits, focusing on the initial dynamics of RIPoptosome progression. Fig 3 and S2 Fig illustrates the simulated molecular composition of RIPoptosome in the single HeLa cell treated with a dose of 5 ng/mL of the DL (rhTRAIL).

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Fig 3. Simulated by model stochastic molecular assembly of RIPoptosome and Casp8, RIP1/RIP3 activation in a single cell.

(A) Model generated individual RIPoptosome growth over first 20 min after DR stimulation with 5ng/mL of DL assuming receptor clustering scenario. (B,C) RIP1 filaments formation triggered immediately after DR stimulation which results in RIPoptosomes of size 2 MDa and higher within first 5 min. (B) ProCasp8 recruitment is mostly abundant in the filaments of the lower molecular weight. (C) FADD recruitment is proportional to the filament molecular weight and persistently increases with the filament progression. (D) Single cell stochastic traces of RIPoptosome driven Casp8 activation, (E) accumulation of DEDs (DED1-DED2) and (F) RIP1-RIP3 heterodimers induced by treatment with 5 ng/mL of the DL. 100 traces shown generated assuming DR clustering scenario. (G) Time dependant variability in Casp8 activation over population of 100 cells stimulated with DL doses of 5, 7.5, 50 and 250 ng/mL for regular (solid line) and disrupted (dashed line) receptor clustering order.

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The composition and the time evolution of individual RIPoptosomes within single cell differed from one to another. Consequently, the size and, therefore, molecular weight of those RIPoptosomes varies as well. As an example, we display the composition change in a few randomly chosen RIPoptosomes over the first 20 min with 1 min step interval (Fig 3A, S2 Fig). Next, we calculated the progression of the molecular weight of a complete cellular pool of RIPoptosomes as simulated by the model. Interestingly, we found a high degree of variation between the RIPoptosomes formed within the same cell (Fig 3B). Our simulations confirmed that in HeLa cells, the most populated protein within each RIPoptosome is RIP1 through its highly stable association mechanism. This is explained by the RHIM domain binding property that shares homology with β-amyloids assembly domains. Simulation of the model revealed that the RIP1 filaments formation is triggered immediately after DR stimulation (Fig 3A). The model also predicts that it would be possible to observe RIPoptosomes of size 2 MDa only 5 minutes after DR stimulation (Fig 3B).

FADD recruitment to the fraction of the high molecular weight complexes is persistently increasing with post treatment time [29]. Our simulations show as well that the abundance of FADD within a single RIPoptosome increases linearly with time progression (Fig 3C) in conjunction with the filament growth. As a result, the abundance of FADD on average will not exceed the amount of 10 molecules per origin within the first two hours. Moreover, this abundance is independent of DL dose. Thus, a low dose of 5 ng/mL of the DL and a high dose of 50 ng/mL will result in similar FADD abundance (S4A and S4B Fig).

On the contrary, ProCasp8 recruitment in the single cell is most abundant in the RIPoptosome of the lower molecular weight (Fig 3B). The binding of the ProCasp8 or its DEDs domain to the end of the filament blocks the RIP1 recruitment and therefore also blocks intensive filament growth by competition. The population average over 600 cells shows that ProCasp8 abundance per RIPoptosome (origin) saturates after 2 hours of stimulation (S4C and S4D Fig). This relative abundance does not vary significantly for doses of 5 or 50 ng/mL of the DL and is unaffected by the clustering or non-clustering assumption in the model. These rapid saturation dynamics of ProCasp8 compared to linear FADD translocation has been observed earlier in experiments where no co-binding of FADD and Casp8 has been observed after 1 hour of stimulation but has become apparent at the second hour [29].

Single cell stochastic RIPoptosome assembly is the source of variation in Casp8 and RIP1/3 protein activation

Molecular fluctuations in the RIPoptosome composition within single cells cause the fluctuations in the active Casp8 abundance (Fig 3D). Stochastic single cell Casp8 activation traces for 5 ng/mL dose simulation with the corresponding per cell accumulation of Casp8 Pro domain (DED1-DED2) are shown in Fig 3D and 3E. Interestingly, limited expression of RIP3 [28] protein in HeLa cell gives rise to very low and therefore heterogeneous distribution of RIP1-RIP3 heterodimers among the cells (Fig 3F) making the spontaneous event of the necroptosis less probable to overtake the apoptotic course of the cell death.

Averaged over the population the Casp8 activation time course demonstrated high dependence on the dose of the DL as well as the clustering capacity (S3 Fig). Thus, even low doses of the DL with enhanced clustering property can activate Casp8. This result confirms the established success in the application of combinational therapeutics where the DL has been combined with the ligand specific cross-linking antibodies that enhance receptor clustering [49].

As expected, the overall variability in the Casp8 activation is a function of the treatment dose (Fig 3G). Despite the coefficient of variation being within the limits of low-variance (less than 1), the early Casp8 initiation dynamics can bring significant stochasticity into triggering the downstream death pathway. Interestingly, the enhanced receptor clustering did not reduce the variability in the individual HeLa cell Casp8 activation dynamics significantly. We observed only a minor decrease in the coefficient of variation over all tested conditions (Fig 3G).

Deterministic modelling of effector caspases feedback into Casp8 activation

Next we studied the downstream caspase cleavage cascade, the second deterministic modelling module (Fig 2E), which feedbacks to the DISC/RIPoptosome based Casp8 production and is potentially capable of boosting cell apoptotic capacity especially following treatment of low doses of DL [68]. As an input we used the population average of the stochastic traces (Fig 2D, S3 Fig) we simulated for the first module of the DISC/RIPoptosome based network initiation assuming DR clustering (Fig 2B). Thus, we merged two modules into one complete deterministic system (Fig 2C) which enabled us to adjust undetermined parameters and estimate parameter sensitivity, hence avoiding computationally expensive parameter scans of the full stochastic formalism (see Materials and methods, S1 File).

The first undetermined parameter is the rate constant of Casp3 ubiquitin dependent degradation (kcat). Ubiquitination of active Casp3, which is set by XIAP, will attract proteasomal complex leading to Casp3 degradation. However, application of proteasome inhibitors does not stabilise the pool of active Casp3 and consequently does not result in reduced Casp3 proteasomal degradation. Instead, Casp3 catalytic activity is absolutely required for its own proteasomal degradation [71,72]. Therefore, dynamics of Casp3 degradation triggered by XIAP will not match the general degradation dynamics triggered by ubiquitin ligases for other types of proteins and this specific rate constant needed to be identified. We estimated that kcat needs to be significantly higher (1.75 min^-1) from the general (basal) ubiquitin-dependent degradation rate (0.04 min^-1) [73] (Table E in S1 File). Again, low doses of the DL bring into play a switch-like sensitive response to the change in kcat value (Fig 4A). In this case the cell death delay can be initiated in a spontaneous fashion if the Casp3 degradation mechanism is perturbed.

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Fig 4. Cell death delay sensitivity to the change in ProCasp6 concentration and Casp3 degradation kinetics in the clustering scenario.

(A) Dramatic increase in the cell death delay upon the increased Casp3 degradation rate constant (kcat) by XIAP mediated ubiquitination which is a function of proteasomal protein degradation capacity as well as Casp3 self-cleavage for the cells stimulated with lower DL doses (5, 7.5 ng/mL). (B) High sensitivity to the minor decrease in XIAP concentration from its mean estimated concentration in HeLa cell 63 nM will lead to dramatic decrease in cell death delay for the cells stimulated with lower DL doses.

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Furthermore, the similar steep ultra-sensitive response can be also initiated by the mild fluctuations in the XIAP concentration. We found that slight deviations from the mean XIAP level, 63 nM, quantified earlier for HeLa [1] could speed up the cell death by more than 3-fold in the case of low DL doses (Fig 4B). This decrease could be very sudden through this switch-like type of response. Indeed, XIAP specific inhibitors such as Embelin, Mithramycin A are able to overcome the DL resistance in different cancer types [74,75].

Semi-stochastic hybrid modelling of the complete initiation network

Finally, with the fully identified parameter set we formulated the new semi-stochastic hybrid model of apoptotic pathway initiation in a single cell with the fixed partitioning of the whole network into discrete (Fig 2B) and continuous reactions (Fig 2E). The slow discrete reactions are the DISC/RIPoptosome assembly. The fast continuous reactions capture the caspase cleavage cascade.

The simulation results for a single cell response on the addition of low and high amounts of the DL are demonstrated in Fig 5. We observed a prolonged ramp effect for all variables of the network before the system switched to the rapid response. The ramp duration for the displayed example exceeded 10 hours after treatment with the low dose of the DL (Fig 5A). Whereas the high dose treatment stimulates the ramp for shorter times, around one hour for a shown example (Fig 5C). In a similar manner to the simulations with our entirely deterministic model, the delay for the switch in the single cell response is a function of DL dose (Fig 4).

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Fig 5. Simulated by semi-stochastic model single cell network activation.

(A-D) Plots represent single cell responses to the low, 5 ng/mL, and high, 50 ng/mL, DL treatment doses. Responses begin with the slow ramp regime (in grey) characterised by high stochastic noise explained by random assembly of RIPoptosome. Once a sufficient rate of Casp8 activation that can overcome XIAP dependent Casp3 inactivation is reached, a rapid switch transition is triggered through the positive caspase feedback loop. The molecular number of active Casp8 dimers; active Casp3, Casp6 and Casp8 monomers; XIAP and ubiquitin targeted Casp3 molecules per HeLa cell are reflected (A, C) as well as theirs percentage from the total initial pool (B, D). (E-J) Plots represent cell death delay driven by intrinsic noise. Single cell stochastic ramp traces of FRET probe cleavage are shown for the five randomly chosen cells simulated with the low, 5 ng/mL, (E) and high, 50 ng/mL, (H) DL treatment doses upon receptor clustering. The trajectories were terminated at the point of death threshold rate (see S1 File). Scatter plots represent the relation between individual cell ramp noise and time of cell death delay within the population of 600 cells for the low, 5 ng/mL, (F-G) and high, 50 ng/mL, (I-J) DL doses with the normal (F, I) and disrupted (G, J) receptor clustering order.

https://doi.org/10.1371/journal.pcbi.1007374.g005

However, for both high and low doses we also observed very high dynamic noise in the ramp (Fig 5A and 5C). This noise characterises the time course of dimeric Casp8 and active Casp3 accumulation. In experiments both proteins are very unstable and hardly detectable in the pre MOMP period of apoptotic initiation [19,71]. As we have shown earlier initial formation of new Casp8 dimer species can be limited by the vulnerability in molecular assembly of the DISC/RIPoptosome platform (Fig 3). Moreover, active Casp8 dimer is unstable due to high dissociation rate in the cytoplasm [19,20]. Indeed, Casp8 under physiological concentrations is found mainly in monomeric form [18,20,59] (Fig 5B and 5D). Therefore, this process prevents accumulation of the excess catalytically active pool of Casp8 for further downstream apoptotic signalling in the pre MOMP period.

Casp3, as the main Casp8-dependent effector caspase [60], follows the noise in the dynamic course of Casp8 dimer during the ramp. Besides, Casp3 is sacrificed in the pre MOMP period due to the excess amount of XIAP which effectively [1,76] blocks Casp3 activity by binding and subsequent ubiquitination which leads to Casp3 degradation.

To study how the ramp noise property in individual cells influences the cell death delay we have performed 600 independent simulations of the semi-stochastic model mimicking the overall cell culture response. These simulations were repeated for four different scenarios: low and high dose treatment scenarios with or without receptor clustering order. The coefficient of variation in Casp8 dependent FRET probe cleavage calculated over ramp period for each cell was considered as the measure of the noise strength. As earlier, the moment of the individual cell death was recorded once the rate of FRET probe cleavage exceeded the expected experimental threshold rate [2] (Fig 5E and 5H). For the individual cells treated with low dose the cell death delay varied from 1 to 10 hours if we integrated the receptor clustering order. Even higher variability was observed when the clustering was absent. In this case the cell death time could vary from 1 to 22 hours. Examples of FRET time traces for five individual cells are shown in Fig 5E. By visualising the relationship between the single cell death delay and dynamic ramp noise strength over a population, we found out that noise was an important determinant of the delay. For both clustering and non-clustering scenarios this relationship follows the same trend (Fig 5F and 5G). Moreover, this trend was independent of the treatment dose (Fig 5I and 5J). Furthermore, for all tested scenarios coefficient of variation higher than 0.5 strictly characterised early dying cells which commit apoptosis within the first 2 hours. Interestingly, receptor clustering enhanced ramp noise resulting in higher values of coefficient of variation (Fig 5F and 5I).

Fractional cell killing was observed in DR-targeted treatments especially when applied in low amounts [2]. As we have shown, the high dispersion of the death delays was the main reason for fractional cell killing. What we found more interesting is that dispersion of the delays could exhibit strong bimodality clearly distinguishing between the fraction of early and late dying cells. Clear bimodality was predicted by our model particularly for the low ligand dose upon receptor clustering order (Figs 6 and 5A). Taking this fact together with the ramp noise analysis (Fig 5F) we can conclude that the high noise in the ramp sensitises cells for early death which will take place within the first five hours at the latest. This fluctuation-enhanced sensitivity has been called ‘stochastic focusing’ and allows quicker system relaxation to the stationary state when the noise is high. The bimodality breaks when receptor clustering is interrupted (Fig 6C, see S1 File) and most of the cells would die only after 10 hours. On the population average cell dynamics receptor clustering provides slightly quicker Casp8 activation for the low dosage of the DL (Fig 6E). This may enable better coupling of this stochastic process with the continuous positive caspase feedback loop. Thus, stochastic focusing coupled with the positive feedback facilitates a more robust bimodal response without the need of multi-stability encoded in the system itself. Finally, the overall cell survival can be dramatically reduced by enhancing the receptor clustering mechanisms (Fig 6G).

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Fig 6. Low DL dose induced bimodal distribution of death delay within the cell culture.

(A-D) Plots demonstrate the frequency distribution of death delays over the population of 600 cells simulated under low (A,C) and high (B,D) DL treatment doses under receptor clustering (A,B) and disrupted receptor clustering (C,D) assumption. (E-H) Casp8 activation characteristics calculated from the population average are not indicative for the apoptotic rate. (E) and (F) show population average dynamics of the Casp8-dependent FRET probe cleavage for the population of 600 cells simulated under low and high DL dose scenarios under receptor clustering (red line) and disrupted receptor (blue line) clustering assumption. (G) and (H) show the cumulative apoptosis event occurrence in the cell population.

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DR amount, proteins concentrations and kinetic parameters determination in HeLa

We quantified TNFR1 in HeLa cells by QuantiBRITE phycoerythrin beads based assay. The amount of DR4/5 was calculated from TNFR1 level by comparative MS data analysis (Table A in S1 File). Receptor clustering conformation was calculated from experimentally derived cluster size probability distributions (S1 Fig). Initial protein concentrations were taken from the literature (Table D in S1 File). Except FADD and RIP1, which we quantified with recombinant protein comparative Western Blot and ProCasp6 concertation that we adjusted using the complete deterministic model (S1 File). Most binding kinetics and catalytic enzymes activity parameters were retrieved from the literature (Table E in S1 File). Hence FRET probe cleavage rate and Casp3 degradation rate were adjusted in the simulations.

Mathematical model

Modelling formalism of Gillespie stochastic simulation algorithm (SSA) and ODE integration as well as semi-stochastic hybrid model was implemented in the MATLAB 2017b environment (see also S1 File).