Portal de Periódicos da CAPES

Bistability Analyses of a Caspase Activation Model for Receptor-induced Apoptosis

2004; Elsevier BV; Volume: 279; Issue: 35 Linguagem: Inglês

10.1074/jbc.m404893200

1083-351X

Thomas Eißing, H. Conzelmann, Ernst Dieter Gilles, Frank Allgöwer, Eric Bullinger, Peter Scheurich,

Protein Kinase Regulation and GTPase Signaling

Apoptosis is an important physiological process crucially involved in development and homeostasis of multicellular organisms. Although the major signaling pathways have been unraveled, a detailed mechanistic understanding of the complex underlying network remains elusive. We have translated here the current knowledge of the molecular mechanisms of the death-receptor-activated caspase cascade into a mathematical model. A reduction down to the apoptotic core machinery enables the application of analytical mathematical methods to evaluate the system behavior within a wide range of parameters. Using parameter values from the literature, the model reveals an unstable status of survival indicating the need for further control. Based on recent publications we tested one additional regulatory mechanism at the level of initiator caspase activation and demonstrated that the resulting system displays desired characteristics such as bistability. In addition, the results from our model studies allowed us to reconcile the fast kinetics of caspase 3 activation observed at the single cell level with the much slower kinetics found at the level of a cell population. Apoptosis is an important physiological process crucially involved in development and homeostasis of multicellular organisms. Although the major signaling pathways have been unraveled, a detailed mechanistic understanding of the complex underlying network remains elusive. We have translated here the current knowledge of the molecular mechanisms of the death-receptor-activated caspase cascade into a mathematical model. A reduction down to the apoptotic core machinery enables the application of analytical mathematical methods to evaluate the system behavior within a wide range of parameters. Using parameter values from the literature, the model reveals an unstable status of survival indicating the need for further control. Based on recent publications we tested one additional regulatory mechanism at the level of initiator caspase activation and demonstrated that the resulting system displays desired characteristics such as bistability. In addition, the results from our model studies allowed us to reconcile the fast kinetics of caspase 3 activation observed at the single cell level with the much slower kinetics found at the level of a cell population. Apoptosis is a genetically defined major form of programmed cell death enabling the organism to remove unwanted cells, e.g. during embryonal development and after immune responses, to select educated immune cells and to eliminate virally infected and transformed cells (1Leist M. Jaattela M. Nat. Rev. Mol. Cell. Biol. 2001; 2: 589-598Crossref PubMed Scopus (1389) Google Scholar, 2Hengartner M.O. Nature. 2000; 407: 770-776Crossref PubMed Scopus (6296) Google Scholar). Enhanced or inhibited apoptotic cell death can be involved in severe pathological alterations, including developmental defects, autoimmune diseases, neurodegeneration, or cancer. Extrinsic and intrinsic apoptotic pathways can be distinguished, although partly employing overlapping signal transduction pathways. A hallmark of the ongoing apoptotic process is the activation of a family of aspartate-directed cysteine proteases, the caspases. Caspases are produced as proenzymes and become activated upon cleavage (3Thornberry N.A. Lazebnik Y. Science. 1998; 281: 1312-1316Crossref PubMed Scopus (6182) Google Scholar). Activation of caspases finally dismantles the cells via the cleavage of important regulatory and structural proteins and enables phagocytic removal of the dying cell (4Savill J. Fadok V. Nature. 2000; 407: 784-788Crossref PubMed Scopus (1281) Google Scholar). A simplified outline of the extrinsic pathway of apoptosis induction after death receptor stimulation is depicted in Fig. 1. Mathematical modeling and systems theory can provide valuable tools to get insight into complex dynamical systems, to test hypotheses, and to identify weak points (5Kitano H. Nature. 2002; 420: 206-210Crossref PubMed Scopus (1596) Google Scholar, 6Csete M.E. Doyle J.C. Science. 2002; 295: 1664-1669Crossref PubMed Scopus (905) Google Scholar). Previous modeling approaches in apoptosis focused on the extrinsically triggered pathways, resulting in complex models (7Fussenegger M. Bailey J.E. Varner J. Nat. Biotechnol. 2000; 18: 768-774Crossref PubMed Scopus (208) Google Scholar, 8Schöberl B. Gilles E.D. Scheurich P. Yi T.-M. Hucka M. Morohashi M. Kitano H. Proceedings of the International Congress of Systems Biology, Pasadena, CA, November 4–7, 2001. Omnipress, Madison, WI2001: 158-167Google Scholar). The model parameters were fitted to data derived from cell population studies showing caspase activation in a range from 30 min to several hours. These models can describe and nicely illustrate certain aspects of the signal transduction pathway. However, more recent experimental results performed at the single cell level show that the majority of caspases are activated within a very short time interval (<15 min) (9Rehm M. Dussmann H. Janicke R.U. Tavare J.M. Kogel D. Prehn J.H. J. Biol. Chem. 2002; 277: 24506-24514Abstract Full Text Full Text PDF PubMed Scopus (269) Google Scholar, 10Goldstein J.C. Waterhouse N.J. Juin P. Evan G.I. Green D.R. Nat. Cell Biol. 2000; 2: 156-162Crossref PubMed Scopus (886) Google Scholar, 11Luo K.Q. Yu V.C. Pu Y. Chang D.C. Biochem. Biophys. Res. Commun. 2003; 304: 217-222Crossref PubMed Scopus (53) Google Scholar, 12Tyas L. Brophy V.A. Pope A. Rivett A.J. Tavare J.M. EMBO Rep. 2000; 1: 266-270Crossref PubMed Scopus (226) Google Scholar). Obviously, the single cell level is relevant for a mechanistic understanding. With the focus on receptor-induced apoptosis, we used Monte Carlo methods to look for parameter domains that enable an appropriate description of apoptosis induction in a single cell (model based on Fig. 1, data not shown). The obtained results revealed an unexpected responsiveness of the system toward minute initiator caspase activation if required to act rapidly. This behavior of the model was caused by the caspase cascade that represents the main signaling route in so-called type I cells (13Scaffidi C. Fulda S. Srinivasan A. Friesen C. Li F. Tomaselli K.J. Debatin K.M. Krammer P.H. Peter M.E. EMBO J. 1998; 17: 1675-1687Crossref PubMed Scopus (2633) Google Scholar) (see Fig. 1, yellow background). We therefore translated the current picture of the extrinsically triggered caspase cascade in a very elementary form into a mathematical model enabling a thorough investigation through the application of analytical methods. Our results showed that within large parameter ranges, including values from the literature, this straightforward model structure is unable to appropriately describe the expected behavior that can be deduced from experimental data. We then showed a way of extending our model structure to reconcile these observed differences and presented a model now able to describe key characteristics like a fast execution phase and bistability. In addition, results from our model studies show a way to reconcile the fast kinetics of caspase 3 activation observed at the single cell level with the much slower kinetics found at the level of a cell population in terms of understanding and modeling. The Mathematical Model—For each reaction considered under "Results" one reaction rate can be deduced (v1–v13). The cleavage reactions (1, 2, and 4) are treated as being irreversible, and it is assumed that the intermediary cleavage products ("enzyme-substrate complexes") only achieve very low levels and can thus be eliminated as reasonable estimations that have been confirmed by simulation experiments (data not shown). The reaction rate equations are deduced according to the law of mass action, which we here consider prior to other kinetic approaches, like Michaelis-Menten kinetics, although theoretical considerations show that the results would be very similar in our case. From this, molecular balances can be derived for each considered molecular species resulting in a system of ordinary differential equations (see Equations 1, 2, 3, 4, 5, 6, 7, 8). Two different model structures are considered in greater detail during this study. The basic model includes the reactions 1–10 and translates into the Equations 1, 2, 3, 4, 5, 6 (not including v11 in Equation 2). The extended model hypothesis includes all reactions and translates into the complete equation system where v1 = k1[C8*]·[C3], v2 = k2[C3*]·[C8], v3 = k3[C3*]·[IAP] – k–3[iC3* ∼ IAP], v4 = k4[C3*]·[IAP], v5 = k5[C8*], v6 = k6[C3*] v7 = k7[iC3* ∼ IAP], v8 = k8[IAP] – k–8, v9 = k9[C8] – k–9, v10 = k10[C3] – k–10, v11 = k11[C8*]·[BAR] – k–11[iC8* ∼ BAR], v12 = k12[BAR] – k–12, and v13 = k13[iC8* ∼ BAR].d[C8]dt=−v2−v9(Eq. 1) d[C8*]dt=v2−v5(−v11)(Eq. 2) d[C3]dt=−v1−v10(Eq. 3) d[C3*]dt=v1−v3−v6(Eq. 4) d[IAP]dt=−v3−v4−v8(Eq. 5) d[C3*~IAP]dt=v3−v7(Eq. 6) d[BAR]dt=−v11−v12(Eq. 7) d[C8*~BAR]dt=v11−v13(Eq. 8) The models were implemented in both Matlab (for simulation experiments) and Mathematica (for analytical analysis). Initial Conditions, Parameters, and Units—The average concentrations in an unstimulated cell (i.e. initial conditions) of caspase 8 and 3 were quantified in HeLa cells to be 130,000 and 21,000 molecules/cell, respectively, using quantitative Western blot analyses. 1T. Eissing, H. Conzelmann, E. D. Gilles, F. Allgöwer, E. Bullinger, and P. Scheurich, unpublished data. The average concentration of IAP(s) 2The abbreviations used are: IAP, inhibitor of apoptosis protein; BAR, bifunctional apoptosis regulator; CARP, caspase 8- and 10-associated RING proteins; Cn, pro-caspase n; Cn*, activated caspase n. was estimated to be 40,000 molecules/cell. Other reported concentrations are 30, 200, and 30 nm for caspase 8, caspase 3, and XIAP, respectively (14Stennicke H.R. Jurgensmeier J.M. Shin H. Deveraux Q. Wolf B.B. Yang X. Zhou Q. Ellerby H.M. Ellerby L.M. Bredesen D. Green D.R. Reed J.C. Froelich C.J. Salvesen G.S. J. Biol. Chem. 1998; 273: 27084-27090Abstract Full Text Full Text PDF PubMed Scopus (647) Google Scholar, 15Sun X.M. Bratton S.B. Butterworth M. MacFarlane M. Cohen G.M. J. Biol. Chem. 2002; 277: 11345-11351Abstract Full Text Full Text PDF PubMed Scopus (203) Google Scholar). Estimating a cell volume of 1 picoliter shows that 600 molecules/cell = 1 nm. Accordingly, these values are roughly in the same order of magnitude and were used as initial concentrations. The other compounds were considered not to be present in the absence of a stimulus. In the extended model, the concentration of the newly introduced molecule BAR was assumed to be 40,000 molecules/cell. We consider the unit molecules/cell more illustrative for cellular concentrations than the unit molar, but on the other hand we prefer and use units such as m–1 s–1 for the Km/kcat ratios. Table I lists the parameters as used in the "single set" simulations (unless indicated otherwise). The respective values are also provided in more common units (in brackets). For the reactions 3 and 5–10 the parameter values were taken from literature as stated in the text. The respective references are summarized in Table II. For the reactions 1, 2, and 4 values were chosen that are in accordance with the desired kinetics and the requirement for bistability (as deduced from bifurcation analyses). The values for reaction 11 were fixed under the assumption of a similar binding affinity as reported for reaction 3. The values for the reactions 12 and 13 represent estimated turnover rates.Table ISimulation parametersValueUnitValueUnitk15.8·10-5 (5.8·105)cell·min-1·mo-1 (m-1 s-1)k-10k210-5 (105)cell·min-1·mo-1 (m-1 s-1)k-20k35·10-4 (5·106)cell·min-1·mo-1 (m-1 s-1)k-30.21 (0.035)min-1 (s-1)k43·10-4 (3·106)cell·min-1·mo-1 (m-1 s-1)k-40k55.8·10-3 (120)min-1 (min)k-50k65.8·10-3 (120)min-1 (min)k-60k71.73·10-2 (40)min-1 (min)k-70k81.16·10-2 (60)min-1 (min)k-8464 (1.3·10-11)mo·cell-1·min-1 (m·s-1)k93.9·10-3 (180)min-1 (min)k-9507 (1.4·10-11)mo·cell-1·min-1 (m·s-1)k103.9·10-3 (180)min-1 (min)k-1081.9 (2.3·10-12)mo·cell-1·min-1 (m·s-1)k115·10-4 (5·106)cell·min-1·mo-1 (m-1 s-1)k-110.21 (0.035)min-1 (s-1)k1210-3 (693)min-1 (min)k-1240 (1.1·10-12)mo·cell-1·min-1 (m·s-1)k131.16·10-2 (60)min-1 (min)k-130 Open table in a new tab Table IIUnstability rangesk+mink+maxk-Explanations and refs.m-1 s-1m-1 s-1m-1 s-1v13·1045·106NoIn vitro KM/kcat = 106m-1 s-1 (14Stennicke H.R. Jurgensmeier J.M. Shin H. Deveraux Q. Wolf B.B. Yang X. Zhou Q. Ellerby H.M. Ellerby L.M. Bredesen D. Green D.R. Reed J.C. Froelich C.J. Salvesen G.S. J. Biol. Chem. 1998; 273: 27084-27090Abstract Full Text Full Text PDF PubMed Scopus (647) Google Scholar, 18Stennicke H.R. Salvesen G.S. Cell Death Differ. 1999; 6: 1054-1059Crossref PubMed Scopus (156) Google Scholar)v22·1045·106NoC3* faster than C8* using fluorogenic substrates (18Stennicke H.R. Salvesen G.S. Cell Death Differ. 1999; 6: 1054-1059Crossref PubMed Scopus (156) Google Scholar, 21Van de Craen M. Declercq W. Van den brande I. Fiers W. Vandenabeele P. Cell Death Differ. 1999; 6: 1117-1124Crossref PubMed Scopus (175) Google Scholar, 45Garcia-Calvo M. Peterson E.P. Rasper D.M. Vaillancourt J.P. Zamboni R. Nicholson D.W. Thornberry N.A. Cell Death Differ. 1999; 6: 362-369Crossref PubMed Scopus (195) Google Scholar)v31·1055·106YesTo obtain: in vitro Ki = 0.7 nm (23Ekert P.G. Silke J. Vaux D.L. Cell Death Differ. 1999; 6: 1081-1086Crossref PubMed Scopus (391) Google Scholar, 38Deveraux Q.L. Takahashi R. Salvesen G.S. Reed J.C. Nature. 1997; 388: 300-304Crossref PubMed Scopus (1724) Google Scholar)v41·1035·106NoEstimationt1/2min (min)t1/2max (min)k- (m·s-1)k+ (min-1) = ln2/t1/2v530300Nov630300Not1/2 ∼ 180 min for caspases; t1/2 = 30-40 min for DIAP (31Ditzel M. Meier P. Trends Cell Biol. 2002; 12: 449-452Abstract Full Text Full Text PDF PubMed Scopus (21) Google Scholar, 46Yoo S.J. Huh J.R. Muro I. Yu H. Wang L. Wang S.L. Feldman R.M. Clem R.J. Muller H.A. Hay B.A. Nat. Cell Biol. 2002; 4: 416-424Crossref PubMed Scopus (319) Google Scholar)v730300Nov830300Yesv960500YesProduction rate to establish the initial concentrationv1060500Yes Open table in a new tab Steady State Derivation—The steady states were derived under the steady state condition dy/dt = 0 (for all compound concentrations y). A consecutive elimination of variables leads to a polynomial in C3*, whose solutions present the steady state concentrations of C3* from which the steady state concentrations of the other molecular species can be derived. The life steady state can be factored out leaving a quadratic equation of the general form ax2 + bx + c = 0 for the basic model and a fourth-order polynomial for the extended model. The quadratic formula was used to construct the green (b2 = 4ac; saddle-node bifurcation manifold) and the blue (–b/a = 0) area in Fig. 3. Interestingly, the coefficient c is the same as the coefficient c derived in the stability analysis (see below). Stability Analysis—For the life steady state one can construct the characteristic polynomial det(λI – A) = 0. Here, det refers to the determinant, λ represents the eigenvalues, I represents the identity matrix, and A represents the Jacobian matrix evaluated at the life steady state. For the non-linear ordinary differential equation system to be locally (asymptotically) stable, all eigenvalues need to have negative real parts. The Hurwitz criterion provides conditions for stability based on the coefficients of the characteristic polynomial. The most restrictive for the basic model is that the coefficient c (below) is positive, which was also used to construct the red area shown in Fig. 3 (transcritical bifurcation manifold). The stability of the steady states other than the life steady state were evaluated numerically, c = k5(IAP k3k7 + k6(k7 + k–3)) – C3 C8 k1k2(k7 + k–3). Deriving an Input Distribution—We assume a population behavior as depicted in Fig. 5A, which can be interpreted as a probability distribution. We further assume that 100% of caspase activation corresponds to 100% cell death, i.e. caspase 3 becomes significantly activated in every cell within the population. Furthermore, based on the simulations of the deterministic single cell model described in Fig. 4, we can describe the maximal caspase 3 activation as a function of C8* input. Hereby we assume the maximal caspase activation to define the time point of cell death. This correlates the stochastic time point of cell death to a stochastic input signal for the single cells within a population. From the original distribution of Fig. 5A we thus obtain a distribution of cell death probability as a function of input activation. The corresponding probability density function can be derived by differentiation as shown in Fig. 5B.Fig. 4Bistable behavior of the extended model. Simulation experiments with varying inputs (initial C8* concentrations) and C3* over time as output are shown. Above a certain input threshold (∼75 molecules of C8*) the system becomes fully activated, whereas subthreshold activation results in recovery. The two stable steady states can be envisioned, with the life steady state corresponding to the blue area and the apoptotic steady state corresponding to the green area achieved after longer time periods.View Large Image Figure ViewerDownload (PPT) The Biology of the Model System—The type I cell-like (13Scaffidi C. Fulda S. Srinivasan A. Friesen C. Li F. Tomaselli K.J. Debatin K.M. Krammer P.H. Peter M.E. EMBO J. 1998; 17: 1675-1687Crossref PubMed Scopus (2633) Google Scholar) model (called basic model hereafter) was constructed with the purpose of being as uncomplicated as possible without neglecting essential steps concerning our analyses (see below), i.e. simplifications represent conservative estimations. As a model input we use a pulse of activated caspase 8, which is produced by the death-inducing signaling complex formed at the membrane after death receptor stimulation (16Lavrik I. Krueger A. Schmitz I. Baumann S. Weyd H. Krammer P.H. Kirchhoff S. Cell Death Differ. 2003; 10: 144-145Crossref PubMed Scopus (61) Google Scholar) (although the initial steps seem to be more complex in the case of TNFR1 (17Micheau O. Tschopp J. Cell. 2003; 114: 181-190Abstract Full Text Full Text PDF PubMed Scopus (2026) Google Scholar)). The model is outlined in Fig. 1 (yellow background) and contains the following reactions:C8*+C3→C8*+C3*REACTION 1 C8+C3*→C8*+C3*REACTION 2 C3*+IAP↔iC3*∼IAPREACTION 3 C3*+IAP→C3*REACTION 4 Pro-caspase 3 (C3, standing for the executioner caspases in general, e.g. caspases 3, 6, and 7) is cleaved and activated by activated caspase 8 (14Stennicke H.R. Jurgensmeier J.M. Shin H. Deveraux Q. Wolf B.B. Yang X. Zhou Q. Ellerby H.M. Ellerby L.M. Bredesen D. Green D.R. Reed J.C. Froelich C.J. Salvesen G.S. J. Biol. Chem. 1998; 273: 27084-27090Abstract Full Text Full Text PDF PubMed Scopus (647) Google Scholar, 18Stennicke H.R. Salvesen G.S. Cell Death Differ. 1999; 6: 1054-1059Crossref PubMed Scopus (156) Google Scholar) (C8*; standing for both initiator caspases, caspases 8 and 10) (Reaction 1). Activated caspase 3 (C3*) acts in terms of a positive feedback loop onto pro-caspase 8 (C8) (19Slee E.A. Harte M.T. Kluck R.M. Wolf B.B. Casiano C.A. Newmeyer D.D. Wang H.G. Reed J.C. Nicholson D.W. Alnemri E.S. Green D.R. Martin S.J. J. Cell Biol. 1999; 144: 281-292Crossref PubMed Scopus (1687) Google Scholar, 20Cowling V. Downward J. Cell Death Differ. 2002; 9: 1046-1056Crossref PubMed Scopus (209) Google Scholar, 21Van de Craen M. Declercq W. Van den brande I. Fiers W. Vandenabeele P. Cell Death Differ. 1999; 6: 1117-1124Crossref PubMed Scopus (175) Google Scholar) (Reaction 2). Here we neglect the presumably amplifying effect of caspase 6 within this feedback loop. Activated caspase 3 binds to and is inactivated by XIAP, here for simplicity termed IAP, as cIAP-1 and cIAP-2 also have the capacity to block caspase 3, although with less efficiency (22Salvesen G.S. Duckett C.S. Nat. Rev. Mol. Cell. Biol. 2002; 3: 401-410Crossref PubMed Scopus (1581) Google Scholar, 23Ekert P.G. Silke J. Vaux D.L. Cell Death Differ. 1999; 6: 1081-1086Crossref PubMed Scopus (391) Google Scholar). IAP-bound activated caspase 3 may form a pool (Reaction 3), but, in parallel, IAP molecules can be cleaved by the activated caspase 3 (Reaction 4). The cleavage products of XIAP have been described to exert only minor effects on caspase 3 (24Deveraux Q.L. Leo E. Stennicke H.R. Welsh K. Salvesen G.S. Reed J.C. EMBO J. 1999; 18: 5242-5251Crossref PubMed Scopus (682) Google Scholar), so these are neglected. Also, the two cleaved forms of caspase 3 are not distinguished, as both have been described to possess similar catalytic activities (15Sun X.M. Bratton S.B. Butterworth M. MacFarlane M. Cohen G.M. J. Biol. Chem. 2002; 277: 11345-11351Abstract Full Text Full Text PDF PubMed Scopus (203) Google Scholar). Furthermore, activated caspases, as well as activated caspase 3 complexed with IAPs, are continuously degraded and pro-caspases and IAPs are subjected to a turnover (Reactions 5–10, degradation and turnover reactions detailed under "Experimental Procedures"). We thus obtain a system of six ordinary differential equations as detailed under "Experimental Procedures." Bistability and Apoptosis—Several signal transduction pathways governing cell fate decisions have experimentally and theoretically been shown to display a bistable behavior (25Ferrell J.E. Xiong W. Chaos. 2001; 11: 227-236Crossref PubMed Scopus (308) Google Scholar, 26Ferrell Jr., J.E. Curr. Opin. Cell Biol. 2002; 14: 140-148Crossref PubMed Scopus (867) Google Scholar, 27Xiong W. Ferrell Jr., J.E. Nature. 2003; 426: 460-465Crossref PubMed Scopus (614) Google Scholar). Bistability is also an obvious and mandatory property of the apoptotic machinery, as the status "alive" must be stable and resistant toward minor accidental trigger signals (i.e. "noise") (28Tyson J.J. Chen K.C. Novak B. Curr. Opin. Cell Biol. 2003; 15: 221-231Crossref PubMed Scopus (1174) Google Scholar). Also, caspases are known to possess zymogenicity (18Stennicke H.R. Salvesen G.S. Cell Death Differ. 1999; 6: 1054-1059Crossref PubMed Scopus (156) Google Scholar) and partial activation is observed in some physiological processes (29Newton K. Strasser A. Genes Dev. 2003; 17: 819-825Crossref PubMed Scopus (69) Google Scholar). However, if the apoptotic initiation signal is beyond a certain threshold, the cell must irreversibly enter the pathway to develop apoptosis. In the following, we use this information in a reverse engineering manner (6Csete M.E. Doyle J.C. Science. 2002; 295: 1664-1669Crossref PubMed Scopus (905) Google Scholar) and take bistability as an "essential condition" to evaluate possible models with respect to this expected behavior. The Basic Model Shows an Unstable Life Steady State— Solving the equation system of our basic model under steady state conditions reveals three steady states. One of these, the "life steady state," corresponds to the initial conditions in which the system remains without an external trigger, i.e. without initial caspase 8 activation. This was expected because our model parameters defining the new synthesis of molecules had been chosen to balance their degradation, and additional influences were neglected. The stability of the steady state provides information about the system behavior close to that steady state, indicating the response to very minor activating input signals. If the steady state is stable, the system will return to its original steady state, provided the perturbations are small enough, which is a situation expected for our model. The stability of a steady state can be evaluated using the Hurwitz criterion, which is well established in systems theory (see "Experimental Procedures"). The point where the stability properties change, i.e. the bifurcation point, provides insight into the qualitative system behavior for certain parameter ranges. We introduced fixed parameter values for all reaction rates except for Reaction 1 (Table I) and found that the life steady state is only stable for k1 values below ∼3.2 × 103m–1 s–1, a value more than 300 times lower than reported in literature (14Stennicke H.R. Jurgensmeier J.M. Shin H. Deveraux Q. Wolf B.B. Yang X. Zhou Q. Ellerby H.M. Ellerby L.M. Bredesen D. Green D.R. Reed J.C. Froelich C.J. Salvesen G.S. J. Biol. Chem. 1998; 273: 27084-27090Abstract Full Text Full Text PDF PubMed Scopus (647) Google Scholar, 18Stennicke H.R. Salvesen G.S. Cell Death Differ. 1999; 6: 1054-1059Crossref PubMed Scopus (156) Google Scholar) (the situation is illustrated in Fig. 2). Fig. 3 shows the bifurcation point (red area) in dependence of three parameter classes with fixed ratios in each class. The parameter combination that can be deduced from literature (Fig. 3, yellow dot) is far away from those combinations enabling a stable life steady state (Fig. 3, below the red area). To further confirm our results, we conducted several million simulations with small inputs and random sets of parameters taken from the parameter ranges shown in Table II. All combinations resulted in significant caspase activation with very small input signals (i.e. an unstable life steady state), although the onset time varied greatly (data not shown). Bistability within a Small Parameter Domain—The two additional steady states besides the life steady state provide further information. Theoretically, one additional steady state within the positive concentration range is sufficient to achieve bistability (because phenomena other than an unstable steady state could separate the two stable steady states). However, in our model this configuration is only possible above the red area in Fig. 3 (see "Experimental Procedures") and for that setting the life steady state is unstable. Accordingly, bistability is only possible if both additional steady states are within the positive concentration range (Fig. 3, above the blue area). Another restriction is imposed by our biological system, because the solutions must contain real numbers (Fig. 3, above the green area). Thus, bistability is only possible in a very restricted parameter area far away from values reported in literature (Fig. 3, yellow dot). In accordance with general considerations on this topic (25Ferrell J.E. Xiong W. Chaos. 2001; 11: 227-236Crossref PubMed Scopus (308) Google Scholar, 30Angeli D. Ferrell Jr., J.E. Sontag E.D. Proc. Natl. Acad. Sci. U. S. A. 2004; 101: 1822-1827Crossref PubMed Scopus (763) Google Scholar) the feedback from caspase 3 onto caspase 8 is necessary for bistability (data not shown). IAPs and Their Cleavage—Interestingly, by and large the stability of the life steady state seems to be independent of the IAP cleavage reaction (Fig. 3). This is not expected and indeed dynamic simulations show that, upon faster IAP cleavage, the onset of caspase activation is achieved more rapidly for parameter combinations where the life steady state is unstable (data not shown). However, for parameter combinations where the life steady state is stable, a slower reaction stabilizes the life steady state by enlargement of its area of attraction (globally stable without this reaction). This can be explained by the fact that we assumed higher concentrations of IAPs than that of caspase 3 to make a conservative estimation concerning the stability of the life steady state. If we assume lower numbers of IAP molecules, we can achieve bistability (which would also require the turnover of IAPs not to exceed that of caspases significantly) even in the absence of this cleavage reaction. However, the parameters in which a stable life steady state is possible would be even further away from those values reported in literature (data not shown). Together, the model indicates that the IAP cleavage reaction is important for a decisive switching. Further analysis of the model reveals that apoptosis can only proceed after the IAP pool is exhausted, as otherwise most of the active caspase 3 molecules become neutralized (data not shown). As the binding of active caspase 3 to IAPs is a reversible reaction, a slower degradation of the complexes would elevate the levels of free active caspase 3 and therefore promote apoptosis. Thus, our results also argue for the view of IAPs as altruistic proteins sacrificing themselves to prevent cell death (31Ditzel M. Meier P. Trends Cell Biol. 2002; 12: 449-452Abstract Full Text Full Text PDF PubMed Scopus (21) Google Scholar). In our investigations we did not change the initial concentrations to restrict the number of free parameters. However, in the mathematical formulas it can be seen that changing a parameter value has similar effects as changing an initial concentration, and so we have indirectly evaluated these influences as well. Bistability through Model Extension—An inherent problem of the described basic model of caspase activation is that relatively fast activation kinetics must be realized to be consistent with parameter values from literature and observations in various experimental setups (9Rehm M. Dussmann H. Janicke R.U. Tavare J.M. Kogel D. Prehn J.H. J. Biol. Chem. 2002; 277: 24506-24514Abstract Full Text Full Text PDF PubMed Scopus (269) Google Scholar, 10Goldstein J.C. Waterhouse N.J. Juin P. Evan G.I. Green D.R. Nat. Cell Biol. 2000; 2: 156-162Crossref PubMed Scopus (886) Google Scholar, 11Luo K.Q. Yu V.C. Pu Y. Chang D.C. Biochem. Biophys. Res. Commun. 2003; 304: 217-222Crossref PubMed Scopus (53) Google Scholar, 12Tyas L. Brophy V.A. Pope A. Rivett A.J. Tavare J.M. EMBO Rep. 2000; 1: 266-270Crossref PubMed Scopus (226) Google Scholar, 32Krippner-Heidenreich A. Tubing F. Bryde S. Willi S. Zimmermann G.