Analysis of a Mathematical Model of Apoptosis: Individual Differences and Malfunction in Programmed Cell Death

Apoptosis is an important area of research because of its role in keeping a mature multicellular organism’s number of cells constant hence, ensuring that the organism does not have cell accumulation that may transform into cancer with additional hallmarks. Firstly, we have carried out sensitivity analysis on an existing mitochondria-dependent mathematical apoptosis model to find out which parameters have a role in causing monostable cell survival i.e., malfunction in apoptosis. We have then generated three healthy cell models by changing these sensitive parameters while preserving bistability i.e., healthy functioning. For each healthy cell, we varied the proapoptotic production rates, which were found to be among the most sensitive parameters, to yield cells that have malfunctioning apoptosis. We simulated caspase-3 activation, by numerically integrating the governing ordinary differential equations of a mitochondria-dependent apoptosis model, in a hypothetical malfunctioning cell which is treated by four potential treatments, namely: (i) proteasome inhibitor treatment, (ii) Bcl-2 inhibitor treatment, (iii) IAP inhibitor treatment, (iv) Bid-like synthetic peptides treatment. The simulations of the present model suggest that proteasome inhibitor treatment is the most effective treatment though it may have severe side effects. For this treatment, we observed that the amount of proteasome inhibitor needed for caspase-3 activation may be different for cells in individuals with a different proapoptotic protein deficiency. We also observed that caspase-3 can be activated by Bcl-2 inhibitor treatment only in those hypothetical malfunctioning cells with Bax deficiency but not in others. These support the view that molecular heterogeneity in individuals may be an important factor in determining the individuals’ positive or negative responses to treatments.

💡 Research Summary

The paper investigates how individual molecular differences affect the dynamics of apoptosis by extending a previously published mitochondria‑dependent mathematical model of the cell‑death pathway. First, a comprehensive sensitivity analysis—both global and local—is performed on the model’s parameters, which include production rates of pro‑apoptotic proteins (Bax, Bak, Bid, active caspase‑8) and anti‑apoptotic regulators (Bcl‑2, IAPs such as XIAP). The analysis identifies the parameters that most strongly shift the system from a bistable regime (allowing a switch from survival to death) to a monostable “survival‑only” regime. In particular, the synthesis rates of Bax, Bid, and the activation rate of pro‑caspase‑3 emerge as the most sensitive knobs.

Using these findings, three “healthy” cell models are constructed by modestly adjusting the identified sensitive parameters so that the model retains its bistability and reproduces normal caspase‑3 activation kinetics. Each healthy model is then deliberately perturbed: the production rates of the most sensitive pro‑apoptotic proteins are reduced, creating three distinct malfunctioning cells that mimic genetic deficiencies such as Bax or Bid loss. In these defective models, the apoptotic signal fails to reach the threshold needed for caspase‑3 activation, resulting in a permanent survival state.

The core of the study examines four hypothetical therapeutic interventions applied to the malfunctioning cells: (i) a proteasome inhibitor, (ii) a Bcl‑2 inhibitor, (iii) an IAP (XIAP) inhibitor, and (iv) synthetic Bid‑like peptides. Numerical integration of the ordinary differential equations reveals that the proteasome inhibitor is the most potent at restoring caspase‑3 activity, because it blocks degradation of pro‑apoptotic proteins, allowing their accumulation. However, the required inhibitor concentration varies widely depending on the severity of the pro‑apoptotic deficiency; cells with severe Bax loss need substantially higher doses, raising concerns about off‑target toxicity.

The Bcl‑2 inhibitor succeeds only in cells that retain functional Bax; in Bax‑deficient scenarios the mitochondrial outer‑membrane permeabilization step cannot be triggered, so the treatment fails. The IAP inhibitor can reactivate caspase‑3 by disrupting XIAP‑caspase‑3 binding, yet its efficacy is limited in cells with high XIAP expression, which would demand higher drug levels. Finally, the synthetic Bid‑like peptide effectively mimics tBid and activates Bax/Bak, but it is ineffective when Bax itself is absent.

Collectively, these simulations underscore the importance of molecular heterogeneity among individuals. The same drug can have dramatically different outcomes depending on the underlying expression profile of key apoptotic regulators. Consequently, the authors argue for a precision‑medicine approach: before selecting an anti‑cancer or neuro‑protective therapy that targets the apoptosis pathway, clinicians should profile patients for Bax, Bcl‑2, IAP, and related protein levels. This information would guide dosage decisions (especially for proteasome inhibitors) and help predict which therapeutic class is likely to succeed.

The paper concludes by suggesting that extending the model to incorporate additional pathways (e.g., p53, NF‑κB, extrinsic death‑receptor signaling) and validating predictions with patient‑derived data would further enhance its utility for personalized treatment design.