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. 2006 Sep 15;2(9):e120.
doi: 10.1371/journal.pcbi.0020120. Epub 2006 Jul 28.

Mathematical Modeling Identifies Inhibitors of Apoptosis as Mediators of Positive Feedback and Bistability

Free PMC article

Mathematical Modeling Identifies Inhibitors of Apoptosis as Mediators of Positive Feedback and Bistability

Stefan Legewie et al. PLoS Comput Biol. .
Free PMC article


The intrinsic, or mitochondrial, pathway of caspase activation is essential for apoptosis induction by various stimuli including cytotoxic stress. It depends on the cellular context, whether cytochrome c released from mitochondria induces caspase activation gradually or in an all-or-none fashion, and whether caspase activation irreversibly commits cells to apoptosis. By analyzing a quantitative kinetic model, we show that inhibition of caspase-3 (Casp3) and Casp9 by inhibitors of apoptosis (IAPs) results in an implicit positive feedback, since cleaved Casp3 augments its own activation by sequestering IAPs away from Casp9. We demonstrate that this positive feedback brings about bistability (i.e., all-or-none behaviour), and that it cooperates with Casp3-mediated feedback cleavage of Casp9 to generate irreversibility in caspase activation. Our calculations also unravel how cell-specific protein expression brings about the observed qualitative differences in caspase activation (gradual versus all-or-none and reversible versus irreversible). Finally, known regulators of the pathway are shown to efficiently shift the apoptotic threshold stimulus, suggesting that the bistable caspase cascade computes multiple inputs into an all-or-none caspase output. As cellular inhibitory proteins (e.g., IAPs) frequently inhibit consecutive intermediates in cellular signaling cascades (e.g., Casp3 and Casp9), the feedback mechanism described in this paper is likely to be a widespread principle on how cells achieve ultrasensitivity, bistability, and irreversibility.

Conflict of interest statement

Competing interests. The authors have declared that no competing interests exist.


Figure 1
Figure 1. Mathematical Model of the Intrinsic Apoptosis Pathway
(A) Schematic representation of intrinsic and extrinsic apoptosis pathways. Dotted lines indicate positive (green) or negative (red) regulation, and the solid lines refer to release of Smac and cyto c from mitochondria. The regulatory interactions considered in the model are highlighted in gray. The numbers 1–4 refer to additional feedbacks described in the Discussion. (B) Kinetic scheme of the model. The reactions depicted in gray, which are involved in Casp3-mediated feedback cleavage of Casp9, were eliminated in the Casp9-mutant model in order to dissect the role of XIAP-mediated feedback. A*, activated Apaf-1; C3, Casp3; C9, Casp9.
Figure 2
Figure 2. Dynamic and Steady-State Behaviour of the Caspase Cascade
(A) Time course of Casp3 activation (wild-type model) upon a step-like increase in the amount of active Apaf-1 (A*tot) at t = 0 from zero to the concentration indicated. (B) Steady-state stimulus-response curves of the wild-type model (black line) and of the Casp9-mutant model (gray line), where Casp3-mediated feedback cleavage of Casp9 does not occur. Stable and unstable steady states are indicated by solid and dashed lines, respectively.
Figure 3
Figure 3. Schematic Representation of XIAP-Mediated Feedback
At resting state (top left) Casp9 is efficiently inhibited by XIAP, so that Casp3 is inactive. Upon stronger stimulation (top right) some Casp9 escapes XIAP-mediated inhibition and activates Casp3, which then sequesters XIAP away from Casp9 (redistribution). This XIAP redistribution finally results in strong activation of both Casp9 and Casp3 (bottom right), and retains the system in an active state even if the stimulus is reduced (bottom left). The numbers on the top of each scheme correspond to those indicated next to the stimulus-response in Figure 2B (black line).
Figure 4
Figure 4. Determinants for Bistability and Irreversibility I
The dose-response curves of the Casp9-mutant model (F), those of the wild-type model (G), and those obtained for noncompetitive caspase binding to XIAP (H) were analyzed for varying Casp3 and Casp9 expression levels. Five types of qualitative behaviour, which are schematically depicted in (A–E), could be distinguished in the physiological range of Apaf-1 expression levels (0–200 nM). The light and dark gray areas in (F–H) correspond to the bistable regions of the model (BR, BI), and the abbreviations MN, MG, and MB indicate the qualitative behaviour outside the bistable region. Experimentally measured caspase concentrations (see Table 1) are highlighted by dashed lines in (F–H).
Figure 5
Figure 5. Determinants for Bistability and Irreversibility II
The qualitative behaviour of caspase activation according to Figure 4A–4E is shown as a function of the XIAP level, and of the competition ratio α. The competition ratio α (Protocol S1) equals the fold-change in XIAP's affinity for Casp9 brought about by Casp3 binding to XIAP (and vice versa), and thereby quantifies the degree of competitive caspase binding to XIAP as indicated on the top.
Figure 6
Figure 6. Binary Integration of Multiple Inputs
The threshold stimulus, A*tot,T, where the bistable system switches from the lower to the higher steady state (point 2 in Figure 2B), is plotted as a function of Casp3 (gray dotted line), Casp9 (gray solid line), or XIAP (black dashed line) expression. In addition, the impact of simultaneous alterations of Casp3 and Casp9 (black solid line) or of Casp3, Casp9, and XIAP (black dash-dotted line) to the same relative extent is shown. The intersection of the graphs corresponds to the default protein concentrations (see Table 1). The terms “linear,” “quadratic,” and “quartic” indicate the relationship between protein expression and the apoptotic threshold, A*tot,T.

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