Bistability in apoptosis: roles of bax, bcl-2, and mitochondrial permeability transition pores

Abstract

We propose a mathematical model for mitochondria-dependent apoptosis, in which kinetic cooperativity in formation of the apoptosome is a key element ensuring bistability. We examine the role of Bax and Bcl-2 synthesis and degradation rates, as well as the number of mitochondrial permeability transition pores (MPTPs), on the cell response to apoptotic stimuli. Our analysis suggests that cooperative apoptosome formation is a mechanism for inducing bistability, much more robust than that induced by other mechanisms, such as inhibition of caspase-3 by the inhibitor of apoptosis (IAP). Simulations predict a pathological state in which cells will exhibit a monostable cell survival if Bax degradation rate is above a threshold value, or if Bax expression rate is below a threshold value. Otherwise, cell death or survival occur depending on initial caspase-3 levels. We show that high expression rates of Bcl-2 can counteract the effects of Bax. Our simulations also demonstrate a monostable (pathological) apoptotic response if the number of MPTPs exceeds a threshold value. This study supports our contention, based on mathematical modeling, that cooperativity in apoptosome formation is critically important for determining the healthy responses to apoptotic stimuli, and helps define the roles of Bax, Bcl-2, and MPTP vis-à-vis apoptosome formation.

FIGURE 1

Mitochondria-dependent apoptotic pathways. The dotted region indicates the interactions included in this model. Solid arrows denote chemical reactions or upregulation; those terminated by a bar denote inhibition or downregulation; and dashed arrows describe subcellular translocation. The following abbreviations are used: pro8, procaspase-8; casp8, caspase-8; pro9, procaspase-9; casp9, caspase-9; pro3, procaspase-3; casp3, caspase-3; ICAD, inhibitor of caspase activated DNase; cyt c, cytochrome c; apop, apoptosome; VAC, the complex formed by voltage dependent anion channel (VDAC), adenine nucleotide translocase (ANT), and cyclophilin D (CyP-D) at the mitochondrial permeability transition pore.

FIGURE 2

Reduced models of apoptosis (A) in the presence of cooperative interactions and positive feedback loop; (B) in the presence of positive feedback but absence of cooperative interactions; and (C) with inhibition of caspase-3 by IAP, in the absence of cooperativity.

FIGURE 3

Time evolution of caspase-3 in response to minor changes in initial caspase-8 concentration [casp8]₀. (A) Cell survival when [casp8]₀ is small (10⁻⁵ μM), indicated by the decrease in the caspase-3 concentration (initially 10⁻⁵ μM) to zero at steady state; (B) apoptosis, implied by the increase in caspase-3 concentration to a high value (4.5 × 10⁻³ μM) at steady state despite the same initial value of caspase-3 as in panel A, when [casp8]₀ is higher (10⁻⁴ μM); (C) monostable apoptosis, in the absence of cooperativity (p = 1) using the same initial concentrations of caspase-3 and caspase-8 as in panel A. Here caspase-3 concentration reaches a high value (6 × 10⁻³ μM) at long times indicative of apoptotic response. Note that the ordinate scales are different in the panels.

FIGURE 4

Effect of degradation rate of Bax (μ_Bax) on cell behavior. (A) Bifurcation diagram for μ_Bax. Limit point for saddle-node bifurcation is μ_Bax = 0.11 s⁻¹. Below this limit point, there are three steady states, two of which are stable and one unstable; above the limit point, there is only one stable state, cell survival. The arrows indicate the equilibrium concentrations reached when starting from any point in the diagram; (B) the high initial concentration of caspase-3 (0.1 μM) decreases to zero when μ_Bax is in the monostable cell survival region (0.2 s⁻¹) despite the high [casp8]₀ value; (C) bifurcation diagram for . Limit point for saddle-node bifurcation is 6.03 × 10⁻⁶ μM/s. The stable cell survival line ([casp3]_o = 0) is not shown because the y axis is logarithmic.

FIGURE 5

Effects of Bcl-2 expression rate on cell behavior in response to changes in Bax levels. (A) Effect on the bifurcation diagram for μ_Bax; (B) effect on the bifurcation diagram for .

FIGURE 6

Bifurcation diagram as a function of the concentration of mitochondrial permeability pore complex [VAC]. The limit points for saddle-node bifurcation are at [VAC] =2.3 × 10⁻⁴ μM and 9.0 × 10⁻⁴ μM.

FIGURE 7

Time evolution of (A) cyt c; (B) caspase-3 levels with (open circles) and without (solid circles) inhibition of Bax degradation. Inhibition of Bax degradation induces an accumulation in cyt c and caspase-3 levels.

FIGURE 8

Comparison of the experimental and theoretical changes in caspase-3, Bax, Bcl-2, cyt c, in response to the relative changes in p53 (that are induced by changes in the flavanoid silymarin concentrations). Panel A displays the caspase-3 levels corresponding to each selected [p53] computed by present simulations. Panels B–D display the results from computations (solid bars) and those from experiments (criss-crossed bars).