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Mathematical model of mitochondria-dependent apoptosis

Our attempt is to reveal whether the concentration of Bax secreted by the bacteria invading the cells is sufficient to trigger the apoptosis of the cell. To realize this aim we are planning to use previously developed mathematical model whose short description is presented below.


The biochemical mechanism of apoptosis is an area of extensive study due to importance of maintaning the cellular homeostasis. The balance between cell proliferation and apoptosis is pivotal for the healthy functioning of the organism. Dysregulation of apoptosis is implicated in many degenerative and autoimmunisation diseases and some types of cancer. Apoptosis may be triggered by extracellular signal particles, deprivation of survival signals and genetic or toxicological damage. There are two major pathway of apoptosis induction: receptor activation, througt death receptors and mitochondrial-dependent via cytochrome c release from mitochondria. The latter path is largely mediated througt Bcl-2 family proteins, which include both proapoptotic members such as Bax or Bak that promote mitochondrial permeability, and antiapoptotic members such as Bcl-2 that cancelled their effects. Another important protein is p53, which simultaneously supressese Bcl-2 and stimulate Bax activity. Moreover mitochondrial pool of 53 is directly involved in triggering of apoptosis.

Kinetic model for mitochondria-dependent apoptotic pathways

Detailed mathematical model describind motochondria-dependend apoptosis has been presented in (1). The model revealed that kinetic cooperativity in formation of the apoptosome is a key element ensuring bistability. Simulation predict a pathological state in which cells will exhibit a monostable cell survival if Bax degradation rate is above a certain treshold value, or Bax expression is below a treshold value. The authors used formalism of ordinary differential equations. All stages of the apoptosis were treated as simple chemical reactions.

Cleavage of Bid by caspase-8

The cleavage of Bid to truncated Bid (tBid) by caspase-8 (casp8) is described by the following reactions, rate constants, and fluxes:

Caspase-8 cleavage.png

Apoptosome complex formation

The apoptosome complex (apop) is a multimeric assembly composed of seven Apaf-1 and seven cyt c molecules. To include in this model the cooperative nature of apoptosome formation, it is required to adopt the folowing reactions:

Apo 1.png

with the forward reaction rate of 7k1b 1 (cyt c_Apaf-1)p for the second reaction. A reaction order (p) higher than unity for the apoptosome complex entails a kinetic cooperativity conducive to bistability, as will be depicted in the results from calculations performed by the authors.

Initiator caspase-9 activation

This stage involves five reactions; the binding of the procaspase-9 (pro9) to the apoptosome complex, succeeded by the binding of a second pro9 to form the complex apop ∙ (pro9)2 and the cleavage of the bound procaspases to yield the holoenzyme apop ∙ (casp9)2, the dissociation of which finally leads to caspases-9 which is descripted below:

Apo 2.png

Activation of caspase-3 by caspase-9

This stage is described by two subsequent reactions: the complexation of caspase-9 with the pro3, and the succeeding cleavage of pro3 to yield the active caspase-3 (casp3) molecule:

Apo 3.png

There is also the second mechanism of activation of caspase-3 which involve the complexation of holoenzyme apop (casp9)2 with the zymogen pro3, succeeded by the cleavage of pro3 to yield casp3:

Apo 4.png

Inhibition of caspase-9 and -3 by IAPs

IAP inhibits both casp9 and casp3 molecules by the following reactions:

Apo 5.png

Cleavage of Bid by caspase-3

In addition to caspase-8 that initiates the cleavage of Bid, caspase-3 produced downstream also truncates Bid to tBid in Type II cells. The corresponding reactions are:

Apo 6.png

The activation of Bid by casp3 completes the first of two positive feedback loops present in depicted model. The second loop is ensured by the cleavage of the inhibitory protein Bcl-2 by casp3. Upon cleavage, Bcl-2 is unable to longer inhibit the channel-opening activity of Bax. Therefore, casp3 indirectly enhances the formation of channels which release the proapoptotic cyt c protein

Cyt c release from mitochondria to cytoplasm

In the one model described in the literature cyt c is released when the concentration of casp3 relative to that of Bcl-2 exceeds given threshold value. This model adopt a different mechanism based on some experimental proofs, summarized by the following reactions:

Apo 7.png

Accordingly, tBid translocates to the mitochondria (k11), and forms a complex with Bax (k12a) to initiate Bax oligomerization and transporting channel (Bax2) formation (k12b). Cyt c is then released through the channel (k14).

Inhibition of Bax by Bcl-2, and cleavage (inactivation) of Bcl-2 by casp3

The cleavage of antiapoptotic Bcl-2 by casp3 (37), which otherwise would inhibit Bax, is described by corresponding reactions:

Apo 7.5.png

p53 Regulation of Bax and Bcl-2 synthesis

p53 upregulates the synthesis of Bax and downregulates that of Bcl-2. It has been found that although p53 levels increase subject to DNA damage, only in some cells is the transcriptional activity of p53 switched on, whereas in the remainder it is switched off. To account for this switch-like response of p53 to DNA damage, the following expressions are used for the rates of formation of Bax and Bcl-2. These processes may be described by means of following equations:

Apo 8.png

The aforementioned expressions closely approximate the activity of p53 which act as molecular switch. The adoption of continuous functions as opposed to discrete threshold values permit to perform a bifurcation analysis. Bax degradation and expression rates are capable of determining the transition between bistable and monostable responses


Apoptosis can be controlled by the degradation rate of proapoptotic proteins such as Bax. It was examined whether the mathematical model confirms this behavior. The limit point for saddle-node bifurcation is found as μBax = 0.11 s-1, below this limitation, there exist three steady states, two of which are stable and one unstable. Therefore in this regime described system is bistable in. Conversely, above the limit point, there is only one stable state which means that the system is monostable. In this monostable regime, even high initial casp3 concentrations decay to zero. These findings suggests that if rate of degradation of Bax is higher than the limit point, apoptosis cannot occur, regardless of caspase levels.

This model thus predicts that cells will exhibit a monostable cell survival response to apoptotic stimuli if the degradation rate of Bax is above some threshold value or if the expression rate of Bax is below other threshold value. This monostable regime always lead to the caspase-3 concentration diminishes to zero since there is not a sufficiently high concentration of Bax to open up channels on mitochondrial membrane which release cyt c to the cytoplasm. The net result is that the activation rate of caspase-3 remain lower than its degradation rate, eventually leading to vanishingly low concentrations of this protease. However, in the bistable regime a dichotomous response favoring either cell survival or apoptosis induction is elicited, depending on the original casp3 cellular level. Overall, the model anticipate that there exists a critical point at which the amount of Bax leads to monostable cell survival that might be associated with the onset of cancer in living cells. The second conclusion is observation that Bcl-2 can counteract the proapoptotic effect of Bax which result in monostable cell survival, in line with the overexpression of Bcl-2 in many types of tumors and the known role of Bcl-2 in countering the effects of Bax and related factors.

General references

  1. Bagci EZ, Vodovotz Y, Billiar TR, Ermentrout GB, Bahar I. Bistability in apoptosis: roles of bax, bcl-2, and mitochondrial permeability transition pores. Biophys J. 2006 Mar 1;90(5):1546-59. Epub 2005 Dec 9. PMID: 16339882
  2. Eissing T, Waldherr S, Allgöwer F, Scheurich P, Bullinger E. Response to bistability in apoptosis: roles of bax, bcl-2, and mitochondrial permeability transition pores. Biophys J. 2007 May 1;92(9):3332-4. Epub 2007 Feb 2. PMID: 17277182
  3. Cui J, Chen C, Lu H, Sun T, Shen P. Two independent positive feedbacks and bistability in the Bcl-2 apoptotic switch.PLoS One. 2008 Jan 23;3(1):e1469. PMID: 18213378