# Mathematical modeling of the formation of apoptosome in intrinsic pathway of apoptosis

## Abstract

Caspase-9 is the protease that mediates the intrinsic pathway of apoptosis, a type of cell death. Activation of caspase-9 is a multi-step process that requires dATP or ATP and involves at least two proteins, cytochrome c and Apaf-1. In this study, we mathematically model caspase-9 activation by using a system of ordinary differential equations (an ODE model) generated by a systems biology tool Simpathica—a simulation and reasoning system, developed to study biological pathways. A rudimentary version of “model checking” based on comparing simulation data with that obtained from a recombinant system of caspase-9 activation, provided several new insights into regulation of this protease. The model predicts that the activation begins with binding of dATP to Apaf-1, which initiates the interaction between Apaf-1 and cytochrome c, thus forming a complex that oligomerizes into an active caspase-9 holoenzyme via a linear binding model with cooperative interaction rather than through network formation.

### Keywords

Model checking Model comparison and selection Apoptosis Apoptosome Caspase-9 ODE model Computational systems biology## Introduction

Systems biology aims to understand the fundamental principles governing biological systems within a mathematical framework. There are many advantages to using a mathematical framework—the most important being that it would then be relatively easy to translate biological systems into *in silico* models that can be manipulated computationally and symbolically in a manner that is often simply impossible with any in vitro or in vivo models. Furthermore, it opens up the opportunities to adapt many algorithmic and analysis techniques originally devised to study natural or engineered dynamical systems that are often represented as discrete, continuous or hybrid models and analyzed using classical or modal logical frameworks.

In engineering sciences, such a consilience of model building and model checking approaches has resulted in robust and well validated engineered systems despite the fact that their design could often evolve uncontrollably in complexity to become too incomprehensible too quickly (e.g., internet or power-grid), even to the team of engineers who may have originally designed the system. But the biological systems bring up even further obstacles—often we do not know all the components participating in the process, nor possess the absolute knowledge of all the underlying interactions, complete with their kinetic parameters. To overcome these obstacles, it is important to develop system tools capable of establishing models with symbolic and logical-inference methods, analyzing them by mathematical approaches, evaluating the models with numerical simulations, and suggesting the best model system that recapitulates the biological system of interest. Recently, our group developed an automated software platform, dubbed Simpathica Model Checker, which provides “in one package” a set of the necessary systems biology tools to carry out studies of enzymatic metabolic pathways (Mishra et al. 2005). In this study we showcase an application of this platform to tease out from small number of experimental data, many of the details of biochemical processes that are thought to be occurring during apoptosis. While the study, highlighted here, is arguably limited by the data-size, it provides many insights on the nature of the “biological model checking” and the algorithmic features that systems biology tools need to incorporate.

Apoptosis is a type of cell death that removes malfunctioning or damaged cell (Evan and Littlewood 1998; Riedl and Shi 2004). Accordingly, a failure of apoptosis can contribute to diseases, such as cancer, while uncontrolled activation of the apoptotic machinery can cause unwanted loss of cells such as that occurring in neurodegenerative diseases. Apoptosis is executed by two main pathways, intrinsic, which responds primarily to intracellular stress, such as caused by chemotherapy, or extrinsic, which is activated by agonists of so-called death receptors (Cain 2003). The proteins involved in these pathways have been characterized, but it remains unclear how these proteins interact to ensure that apoptosis is prevented until needed but then executed quickly. It is likely, however, that this regulation involves quantitative changes in concentrations and activities of these proteins.

Both pathways of apoptosis can be described as a network that regulates activation of caspases, the proteases that disassemble the cell. Caspases (cysteine aspartate-specific proteases) is a family of proteins that form the execution part of the apoptotic machinery. The extrinsic pathway activates caspase-8, while intrinsic pathway activates caspase-9; either of these proteases can activate caspase-3, which does most of cell disassembly by cleaving a set of proteins. Caspase-9 functions as a holoenzyme, in which this protease is a catalytic subunit that is regulated by an oligomer of Apaf-1, also known as the apoptosome (Jiang and Wang 2004). Apaf-1 is a 130-kDa constitutively expressed protein that includes a caspase recruitment domain (CARD), a nucleotide-binding domain and 13 WD-40 repeats (Zou et al. 1997, 1999). Apaf-1 is oligomerized into the apoptosome following binding to cytochrome c in a process that requires hydrolysis of ATP or dATP (Kim et al. 2005; Yu et al. 2005). Because of experimental difficulties involved in studying the apoptosome activation, it is uncertain what sequence of reactions leads exactly to caspase-9 activation, or even how this protease is activated.

The modeling processes leading to caspase-9 activation may help to learn how to regulate caspase-9 activation for therapeutic purposes. In particular, the modeling approach could predict the outcome in a perturbed system where the balance between cell survival and cell death is compromised. Modeling can also facilitate determination of unknown factors involving the apoptotic pathway, and provide a platform for exchanging knowledge and storing information. Finally, a model of caspase-9 activation can be eventually merged with other models modularly and hierarchically to form a functional model of the cell.

Since Varner and colleagues proposed the first mathematical model of caspase activation (Fussenegger et al. 2000), several additional details have been proposed to enhance mathematical models of apoptosis by combining other major elements thought to be involved in apoptosis (Legewie et al. 2006; Nakabayashi and Sasaki 2006; Stucki and Simon 2005). For instance, there had existed earlier models to provide frameworks analogous to ours, and account for the various interactions that can affect apoptosis; however, the present study attempts to go much further in filling in many details of the intrinsic pathway that needed a clearer and more detailed description. Moreover, there exists practically no study of mathematical modeling of detailed apoptosome formation with the sole exception being the work due to Nakabayashi and Sasaki (Nakabayashi and Sasaki 2006). Even though Nakabayashi and Sasaki had modeled apoptosome assembly with the network interaction, here, in contrast to a purely simulation study as theirs, we have aimed to combine model building with model checking using novel recombinant experimental data and automated systems biology tools.

This paper proposes a mathematical approach to study and evaluate models of caspase-9 activation; additionally, it also highlights our approach to designing a software platform that can be used even by a novice user to apply, test, extend, or modify such models. This platform is based on NYU’s Simpathica Model Checker (SMC) that is part of the VALIS software environment (Paxia et al. 2002). SMC was designed for modeling, simulation, and reasoning, and is capable of effectively manipulating large, complex, and highly detailed biochemical systems (Mishra et al. 2005). Importantly, Simpathica can generate all the necessary differential equations starting from the user-defined textual or graphical descriptions, which allows even a user who is unfamiliar with this mathematical approach to apply it effortlessly. In this system, the only required inputs are reasonable ranges of initial concentrations of individual components and related metabolic reactions with proper parameters (Mishra et al. 2005). Simpathica also allows users to modify an existing model, modularly integrate with established or hypothetical models, or search over a family of plausible models, through a simple and efficient Graphical user interface (GUI) using multi-scripting facilities of VALIS. In summary, Simpathica allows users to construct and simulate models of metabolic, regulatory, and signaling networks and then to analyze their behavior with equal ease. By focusing on a well-studied and almost completely characterized system such as intrinsic apoptosis, we hope to better understand algorithmic issues inherent to an SMC-like approach.

Below, we describe how Simpathica is used to model the intrinsic apoptosis pathway as well as to check the model by comparing model-based predictions with experimental data.

## Materials and methods

### Recombinant system of caspase-9 activation

To study the biochemical mechanism of caspase activation, we used a recombinant system with purified recombinants Apaf-1, cytochrome c and procasepase-9 and-3 and nucleotide. Caspase-3 activities were measured over a duration, lasting 3 min because previous study had shown that caspase 3 activities would increase linearly over a time interval from 0 to 5 min (Rodriguez and Lazebnik 1999), which was also supported by the simulation data that exhibited an identical linear increment of caspase 3 activity (Supplement Fig. 2). Note that, as a convenient reference for the readers, all variables used in this study have been summarized in the Supplement Table 1.

### Apaf-1 and pro-caspase-9

All reagents for recombinant system were prepared as described (Benedict et al. 2000; Fearnhead et al. 1998; Lin 2006; Rodriguez and Lazebnik 1999; Zou et al. 1999).

### Assay for caspase-9 activity

We assessed caspase-9 activity by measuring conversion of procaspase-3 to caspase-3. The assay was performed as previously described elsewhere, using DEVD-AFC as a fluorogenic substrate (Rodriguez and Lazebnik 1999). Briefly, 4 μl of the tested solution was added to 0.2 ml of the assay buffer (50 mM PIPES, 10% Glycerol, 0.1 mM EDTA, 2 mM DTT, pH7.0) containing 40 μM fluorescent substrate DEVD-AFC (Biomol, Inc). The fluorescence released by caspase-3 was measured at 37°C using a Cytofluor 4000 plate reader, and the reaction rates were calculated from the fluorescence obtained before the saturation.

### Assay for cytochrome c concentration

The ELISA was performed using human cytochrome c immunoassay kit (Quantikine, DCTC0 from R&D systems). Briefly, 1:1 series diluted of the reconstituted cytochrome c standard (between 0 and 20 ng/ml) and 100 μl 293 cell extract (Rodriguez and Lazebnik 1999) were transferred to polystyrene microplate coated with monoclonal antibody against cytochrome c and incubated for 2 h at room temperature. After aspiration and rinse with wash buffer, 0.2 ml of the cytochrome c conjugate was added to each well and incubated for 2 h at room temperature. After final wash, 0.2 ml of the substrate solution was added to each well and incubated for half hour in the dark. The development was stopped by adding 50 μl of stop solution and the absorbance was compared and used to calculate the concentration of cytochrome c.

### Simulation program, Simpathica

The model equations and a graphical user interface (GUI) were generated, using Simpathica Model Checker (developed by NYU Bioinformatics Group, New York University, NY). This program is publicly available at NYU Bioinformatics Group website: http://bioinformatics.nyu.edu/Software/index.shtml.

### Theoretical basis for simulation

_{1}, k

_{−1}and k

_{2}(Reaction Constants).

### Comparison of simulation model predictions against experimental data

In this study, our model checking approach consisted of numerical comparison between simulation (*in silico*) data, \( \Im_{S} (U,T) \), and experimental (in vitro) data, \( \Im^{\prime}_{V} (U,T) \), leading to a better characterization of correct apoptosis model. Based on this initial ODE model, a suitable Kripke structure can be created to check more complex modal logic queries within Simpathica (data not shown).

*in silico*model, expressed by a system of ODE’s

*u*characterizes the input values,

*y*the output values and

*x*, all other state variables. Thus, the input–output behavior may be expressed as

_{U}(

*T*) and E

_{T}(

*U*), as follows

_{U}and E

_{T}can be given as

The mean square error (MSE) was used to measure the goodness-of-fit for comparing the model predictions to the experimental data. In this case, the model yielding lower MSE values was deemed to provide a better fit.

### Mathematical modeling and numerical methods

*C9*), cytochrome c (C), dATP (D), and caspase-3 (

*C3*). The signaling networks regulate these components as illustrated in Fig. 1. Based on the traditional caspase-9 pathway model with a specific role for multimeric holoenzyme complex, we developed a kinetic model that was used to represent caspase-9 pathway. The differential equations, variables and default parameters used in this study are summarized in the Table 2.

### Modeling APAF-1, dATP and cytochrome c interactions

*AC*denote the intermediate molecule and Apaf-1:cytochrome c complex, respectively. Previous studies have shown that Apaf-1 also binds and hydrolyzes nucleotides such as dATP. In this reaction, cytochrome c does not affect dATP binding or hydrolysis to Apaf-1 (Zou et al. 1997). Our models treat the binding of dATP and Apaf-1:cytochrome c as reversible and the formation of complex as irreversible (Fig. 2a, c).

*ACD*denote the intermediate molecule and Apaf-1:cytochrome c:dATP complex (or Apaf-1 monomer), respectively. It is believed that cytochrome c binds to Apaf-1 and dATP stabilizes the binding of cytochrome c to Apaf-1 (Zou et al. 1999). Based on this information, we proposed an alternative model that Apaf-1 molecule binds to dATP first and thus, acts as the initiator for caspase activation (Fig. 2b, d).

*ADC*denote the intermediate molecule and Apaf-1:cytochrome c:dATP complex (or Apaf-1 monomer), respectively. We treat the binding of Apaf-1 and dATP as well as Apaf-1:dATP and cytochrome c as reversible and the formation of complex as irreversible.

### Modeling APAF-1 multimerization

*S*

_{2}denote the intermediate molecule and Apaf-1:cytochrome c:dATP dimer complex, respectively. We treat the binding between Apaf-1 complexes as reversible and the formation of complex as irreversible. Apaf-1 dimer complex binds Apaf-1 monomer and forms Apaf-1 trimer complex. Then, Apaf-1 dimer complex and Apaf-1 monomer form Apaf-1 trimer complex.

*S*

_{2}denote the intermediate molecule and Apaf-1:cytochrome c:dATP trimer complex, respectively. We treat the binding between Apaf-1 dimer complex and Apaf-1 monomer reversible and the formation of complex as irreversible. Then, Apaf-1 trimer complex and Apaf-1 monomer combine to form Apaf-1 tetramer complex. Similar sets of equations hold for Apaf-1 pentamer, hexamer, and heptamer complexes as shown below:

*n*th-Apaf-1 complex binding rate, γn

_{1}will continue to increase monotonically (γ2

_{1}≫ γ1

_{1}, etc.). However, if there is no cooperative interaction among Apaf-1 complexes, rate constants of binding between complexes, γn

_{1}will remain unaffected (γ2

_{1}= γ1

_{1}, etc.).

### Modeling the formation and activation of holoenzyme (or Apaf-1:cytochrome c:dATP:caspase-9 complex)

*H*

_{0}denote the intermediate molecule and inactive holoenzyme, respectively. We treat the binding between Apaf-1 heptamer complex and procaspase-9 reversible and the formation of complex as irreversible. During the binding process, if there is a cooperative interaction among procaspase-9, rate constants of binding between complexes,

*n*th-procaspase-9 binding rate, δ

*n*

_{1}will be increased (δ

*2*

_{1}≫ δ

*1*

_{1}). However, if there is no cooperative interaction among Apaf-1 complexes, rate constants of binding between complexes, δ

*n*

_{1}will remain constant (δ

*2*

_{1}= δ

*1*

_{1}). Procaspase-9 is rapidly autoactivated once bound to multimeric Apaf-1 complex.

*H*

_{ i }and

*H*

_{ a }denote inactive holoenzyme (multimeric Apaf-1 complex with procaspase-9 or

*S*

_{7}:

*P*

_{9}) and active form (multimeric Apaf-1 complex with caspase-9 or

*S*

_{7}:

*C*

_{9}), respectively. We treat the conversion from inactive holoenzyme to active form as irreversible. Caspase-9 is fully active only when it is bound to Apaf-1 (Rodriguez and Lazebnik 1999). The activated caspase-9 is released from Apaf-1 complex.

### Modeling dual roles of cytochrome c

*C*denotes cytochrome c. We treat the conversion from inactive holoenzyme to active form as irreversible.

### Modeling caspase-3 activation

_{ a }).

*C3*denote intermediate molecule and caspase-3, respectively. We treat the conversion of procaspase-3 into the intermediate molecule as reversible and the formation of active caspase-3, irreversible.

### Modeling DEVD-Afc activity

*F*denote intermediate molecule and fluorescent molecule Afc, respectively. We treat the conversion of DEVD-Afc into the intermediate molecule as reversible and the formation of active caspase-3, irreversible. In this study, we calculate Afc “rate” (fc/min) instead of free “Afc” concentration (nM).

### The Robustness of the models

In this study, we have focused on estimating the relative model robustness among several competing network models, and used this to reconstruct the most plausible model for a biochemical pathway. We believe that this approach builds upon one of the best metrics for estimating the reliability of computationally simulated models, especially in the context where one faces relatively small-sample (and/or non-stationary) experimental data. We have tested the robustness of the models, both extensively and exhaustively, by varying the rate constant values, since most metabolic reactions, e.g., those appearing in apoptosis process, are primarily dependent on k_{1}/k_{-1} ratios. In our perturbation study, we conducted our parameter-sweep by both increasing and decreasing k_{1} values and then simulating the models for each case to evaluate the model robustness.

## Results and discussion

This study applied Simpathica, a novel systems biology platform, to model and simulate the distal part of the intrinsic pathway of apoptosis, which includes activation of caspases-9 and 3. We also compared the results of simulations to experimental data, which allowed us to optimize the models and make some conclusions regarding mechanisms of apoptosis.

### Model derivation and parameter optimization

The element reactions and kinetic parameters used in the simulation

Step | Species or reactions (equation number) | Values observed in experiments | Values used in simulation |
---|---|---|---|

0 | Nucleotide (dATP or D) | [ATP] = 2–10 × 10 | D |

(Skoog and Bjursell 1974) | |||

Cytochrome c (C) | [CytC] = 1.1 × 10 | C | |

Determined in the 293 extract by cytochrome c ELISA kit (R&D system). | |||

Apaf-1 (A) | [Apaf-1] = 1.5–3 nM (Fearnhead et al. 1998; Zou et al. 1997) | A | |

Caspase-9 (C | [Casp9] = 15–20 nM (Stennicke et al. 1999) | P9 | |

Caspase-3 (C | [Casp3] = 15–20 μM (Stennicke et al. 1998) | P3 | |

Caspase-3 substrate (V) | [DEVD-Afc] = 40 μM (Rodriguez and Lazebnik 1999) | V | |

1 | C + A → AC (Eq. 13) | | |

| |||

| |||

2 | AC + D → ACD (Eq. 14) | | |

| |||

2-1 | ACD + ACD → 2ACD | γ1 = 50, γ − 1 = 2000, γ2 = 10 | |

3 | Postulated, half of caspase-9 can be recruited in holoenzyme in 5 min with the | δ | |

δ | |||

4 | ACDP | Postulated. 20nM of caspase-9 can be completely cleaved in 20 min. | |

5 | \( \hbox{P}_{3} {\mathop{\rightarrow}\limits^{{\hbox{ACDC}_{9}}}} \hbox{C}_{3} \) (Eq. 27) | | |

Inhibition of Caspase-9 by peptide aldehydes (Garcia-Calvo et al. 1998) | |||

| |||

6 | \( \hbox{V} {\mathop{\rightarrow}\limits^{{\hbox{C}_{3}}}} \hbox{F} \) (Eq. 28) | | |

V | |||

7 | ACDC | Postulated | |

Differential equations, variable definitions and default parameters

| |

\( \begin{gathered} \dot{A}_{F} = - a_{1} *\left[ {A_{0} } \right]\left[ {C_{0} } \right] + a_{{^{ - 1} }} \left[ {\tilde{A} \bullet C} \right] \hfill \\ \dot{C}_{F} = - a_{1} *\left[ {A_{0} } \right]\left[ {C_{0} } \right] + a_{{^{ - 1} }} \left[ {\tilde{A} \bullet C} \right] \hfill \\ \dot{D}_{F} = - b_{1} *\left[ {D_{0} } \right] + b_{{^{ - 1} }} \left[ {\tilde{A}\tilde{C} \bullet D} \right] \hfill \\ \end{gathered} \) | |

\( \left[ {\dot{\tilde{A}} \bullet \dot{C}} \right] = + a_{1} *\left[ {A_{0} } \right]\left[ {C_{0} } \right] - a_{ - 1} \left[ {\tilde{A} \bullet C} \right] - a_{2} *\left[ {\tilde{A} \bullet C} \right] \) | |

\( \left[ {\dot{\tilde{A}}\dot{\tilde{C}} \bullet \dot{D}} \right] = + b_{1} *\left[ {AC} \right]\left[ {D_{0} } \right] - b_{ - 1} \left[ {\tilde{A}\tilde{C} \bullet D} \right] - b_{2} *\left[ {\tilde{A}\tilde{C} \bullet D} \right] \) | |

\( \dot{A}\dot{C} = + a_{2} *\left[ {\tilde{A} \bullet C} \right] - b_{1} *\left[ {AC} \right] + b_{ - 1} *\left[ {\tilde{A}\tilde{C} \bullet D} \right] \) | |

\( \dot{A}\dot{D} = + a'_{2} *\left[ {\tilde{A} \bullet D} \right] - b'_{1} *\left[ {AD} \right] + b'_{ - 1} *\left[ {\tilde{A}\tilde{D} \bullet C} \right] \) | |

\( \left[ {\dot{\tilde{A}}\dot{\tilde{D}} \bullet \dot{C}} \right] = + b'_{1} *\left[ {AD} \right]\left[ {C_{0} } \right] - b'_{ - 1} \left[ {\tilde{A}\tilde{D} \bullet C} \right] - b'_{2} *\left[ {\tilde{A}\tilde{D} \bullet C} \right] \) | |

\( \dot{A}\dot{C}\dot{D} = \dot{S}_{1} = \hbox{See\; differential \;equations \;for \;holoenzyme\; formation} \) | |

| |

\( \dot{S}_{1} = + b_{2} *\left[ {\tilde{A}\tilde{C} \bullet D} \right] + b'_{2} *\left[ {\tilde{A}\tilde{D} \bullet C} \right] - \gamma_{1} 1*\left[ {S_{1} } \right]\left[ {S_{1} } \right] + \gamma 1_{ - 1} *\left[ {\tilde{S}_{1} \bullet \tilde{S}_{1} } \right] \) | |

\( \left[ {\dot{\tilde{S}}_{1} \bullet \dot{\tilde{S}}_{1} } \right] = + \gamma 1_{1} *\left[ {S_{1} } \right]\left[ {S_{1} } \right] - \gamma 1_{ - 1} *\left[ {\tilde{S}_{1} \bullet \tilde{S}_{1} } \right] - \gamma 1_{2} *\left[ {\tilde{S}_{1} \bullet \tilde{S}_{1} } \right] \) | |

\( \dot{S}_{2} = + \gamma_{1} 2*\left[ {\tilde{S}_{1} \bullet \tilde{S}_{1} } \right] - \gamma_{2} 1*\left[ {S_{2} } \right]\left[ {S_{1} } \right] + \gamma_{2}^{ - 1} *\left[ {\tilde{S}_{2} \bullet \tilde{S}_{1} } \right] \) | |

\( \left[ {\dot{\tilde{S}}_{2} \bullet \dot{\tilde{S}}_{1} } \right] = + \gamma 2_{1} *\left[ {S_{2} } \right]\left[ {S_{1} } \right] - \gamma 2_{ - 1} *\left[ {\tilde{S}_{2} \bullet \tilde{S}_{1} } \right] - \gamma 2_{2} *\left[ {\tilde{S}_{2} \bullet \tilde{S}_{1} } \right] \) | |

\( \dot{S}_{3} = + \gamma 2_{2} *\left[ {\tilde{S}_{2} \bullet \tilde{S}_{1} } \right] - \gamma 3_{1} *\left[ {S_{3} } \right]\left[ {S_{1} } \right] + \gamma 3_{ - 1} *\left[ {\tilde{S}_{3} \bullet \tilde{S}_{1} } \right] \) | |

\( \left[ {\dot{\tilde{S}}_{3} \bullet \dot{\tilde{S}}_{1} } \right] = + \gamma 3_{1} *\left[ {S_{3} } \right]\left[ {S_{1} } \right] - \gamma 3_{ - 1} *\left[ {\tilde{S}_{3} \bullet \tilde{S}_{1} } \right] - \gamma 3_{2} *\left[ {\tilde{S}_{3} \bullet \tilde{S}_{1} } \right] \) | |

\( \dot{S}_{4} = + \gamma 3_{2} *\left[ {\tilde{S}_{3} \bullet \tilde{S}_{1} } \right] - \gamma 4_{1} *\left[ {S_{4} } \right]\left[ {S_{1} } \right] + \gamma 4_{ - 1} *\left[ {\tilde{S}_{4} \bullet \tilde{S}_{1} } \right] \) | |

\( \left[ {\dot{\tilde{S}}_{4} \bullet \dot{\tilde{S}}_{1} } \right] = + \gamma 4_{1} *\left[ {S_{4} } \right]\left[ {S_{1} } \right] - \gamma 4_{ - 1} *\left[ {\tilde{S}_{4} \bullet \tilde{S}_{1} } \right] - \gamma 4_{2} *\left[ {\tilde{S}_{4} \bullet \tilde{S}_{1} } \right] \) | |

\( \dot{S}_{5} = + \gamma 4_{2} *\left[ {\tilde{S}_{4} \bullet \tilde{S}_{1} } \right] - \gamma 5_{1} *\left[ {S_{5} } \right]\left[ {S_{1} } \right] + \gamma 5_{ - 1} *\left[ {\tilde{S}_{5} \bullet \tilde{S}_{1} } \right] \) | |

\( \left[ {\dot{\tilde{S}}_{5} \bullet \dot{\tilde{S}}_{1} } \right] = + \gamma 5_{1} *\left[ {S_{5} } \right]\left[ {S_{1} } \right] - \gamma 5_{ - 1} *\left[ {\tilde{S}_{5} \bullet \tilde{S}_{1} } \right] - \gamma 5_{2} *\left[ {\tilde{S}_{5} \bullet \tilde{S}_{1} } \right] \) | |

\( \dot{S}_{6} = + \gamma 5_{2} *\left[ {\tilde{S}_{5} \bullet \tilde{S}_{1} } \right] - \gamma 6_{1} *\left[ {S_{6} } \right]\left[ {S_{1} } \right] + \gamma 6_{ - 1} *\left[ {\tilde{S}_{6} \bullet \tilde{S}_{1} } \right] \) | |

\( \left[ {\dot{\tilde{S}}_{6} \bullet \dot{\tilde{S}}_{1} } \right] = + \gamma 6_{1} *\left[ {S_{6} } \right]\left[ {S_{1} } \right] - \gamma 6_{ - 1} *\left[ {\tilde{S}_{6} \bullet \tilde{S}_{1} } \right] - \gamma 6_{2} *\left[ {\tilde{S}_{6} \bullet \tilde{S}_{1} } \right] \) | |

\( \dot{S}_{7} = + \gamma 6_{2} *\left[ {\tilde{S}_{6} \bullet \tilde{S}_{1} } \right] - \delta 1_{1} *\left[ {S_{7} } \right]\left[ {P9} \right] + \delta 1_{ - 1} *\left[ {\tilde{S}_{7} \bullet P9} \right] \) | |

| |

\( \dot{H}_{1} = + \delta 1_{2} *\left[ {\tilde{S}_{7} \bullet P9} \right] - \delta 2_{1} \left[ {\tilde{H}_{1} } \right]\left[ {P9} \right] + \delta 2_{ - 1} \left[ {\tilde{H}_{1} \bullet P9} \right] \) | |

\( \left[ {\dot{\tilde{H}}_{1} \bullet P9} \right] = + \delta 2_{1} *\left[ {H_{1} } \right]\left[ {P9} \right] - \delta 2_{ - 1} *\left[ {\tilde{H}_{1} \bullet P9} \right] - \delta 2_{2} *\left[ {\tilde{H}_{1} \bullet P9} \right] \) | |

\( \dot{H}_{2} = + \delta 2_{2} *\left[ {\tilde{H}_{1} \bullet P9} \right] - \delta 3_{1} \left[ {\tilde{H}_{2} } \right]\left[ {P9} \right] + \delta 3_{ - 1} \left[ {\tilde{H}_{2} \bullet P9} \right] \) | |

\( \left[ {\dot{\tilde{H}}_{2} \bullet P9} \right] = + \delta 3_{1} *\left[ {H_{2} } \right]\left[ {P9} \right] - \delta 3_{ - 1} *\left[ {\tilde{H}_{2} \bullet P9} \right] - \delta 3_{2} *\left[ {\tilde{H}_{2} \bullet P9} \right] \) | |

\( \dot{H}_{3} = + \delta 3_{2} *\left[ {\tilde{S}_{2} \bullet P9} \right] - \delta 4_{1} \left[ {\tilde{H}_{3} } \right]\left[ {P9} \right] + \delta 4_{ - 1} \left[ {\tilde{H}_{3} \bullet P9} \right] \) | |

\( \left[ {\dot{\tilde{H}}_{3} \bullet P9} \right] = + \delta 4_{1} *\left[ {H_{3} } \right]\left[ {P9} \right] - \delta 4_{ - 1} *\left[ {\tilde{H}_{3} \bullet P9} \right] - \delta 4_{2} *\left[ {\tilde{H}_{3} \bullet P9} \right] \) | |

\( \dot{H}_{4} = + \delta 4_{2} *\left[ {\tilde{S}_{3} \bullet P9} \right] - \delta 5_{1} \left[ {\tilde{H}_{4} } \right]\left[ {P9} \right] + \delta 5_{ - 1} \left[ {\tilde{H}_{4} \bullet P9} \right] \) | |

\( \left[ {\dot{\tilde{H}}_{4} \bullet P9} \right] = + \delta 5_{1} *\left[ {H_{4} } \right]\left[ {P9} \right] - \delta 5_{ - 1} *\left[ {\tilde{H}_{4} \bullet P9} \right] - \delta 5_{2} *\left[ {\tilde{H}_{4} \bullet P9} \right] \) | |

\( \dot{H}_{5} = + \delta 5_{2} *\left[ {\tilde{S}_{4} \bullet P9} \right] - \delta 6_{1} \left[ {\tilde{H}_{5} } \right]\left[ {P9} \right] + \delta 6_{ - 1} \left[ {\tilde{H}_{5} \bullet P9} \right] \) | |

\( \left[ {\dot{\tilde{H}}_{5} \bullet P9} \right] = + \delta 6_{1} *\left[ {H_{5} } \right]\left[ {P9} \right] - \delta 6_{ - 1} *\left[ {\tilde{H}_{5} \bullet P9} \right] - \delta 6_{2} *\left[ {\tilde{H}_{5} \bullet P9} \right] \) | |

\( \dot{H}_{6} = + \delta 6_{2} *\left[ {\tilde{S}_{5} \bullet P9} \right] - \delta 7_{1} \left[ {\tilde{H}_{6} } \right]\left[ {P9} \right] + \delta 7_{ - 1} \left[ {\tilde{H}_{6} \bullet P9} \right] \) | |

\( \left[ {\dot{\tilde{H}}_{6} \bullet P9} \right] = + \delta 7_{1} *\left[ {H_{6} } \right]\left[ {P9} \right] - \delta 7_{ - 1} *\left[ {\tilde{H}_{6} \bullet P9} \right] - \delta 7_{2} *\left[ {\tilde{H}_{6} \bullet P9} \right] \) | |

\( \dot{H}_{7} = \dot{H}_{i} = + \delta 7_{2} *\left[ {\tilde{H}_{6} \bullet P9} \right] - m_{1} *\left[ {H_{i} } \right] \) | |

\( \dot{H}_{a} = + m_{1} *\left[ {H_{i} } \right] - w_{1} *\left[ {H_{a} } \right] \) | |

| |

\( \begin{aligned} \dot{P}9_{F} & = - \delta 1_{1} *\left[ {S_{7} } \right]\left[ {P9} \right] + \delta 1_{ - 1} \left[ {\tilde{S}_{7} \bullet P9} \right] - \delta 2_{1} *\left[ {H_{1} } \right]\left[ {P9} \right] + \delta 2_{ - 1} \left[ {\tilde{H}_{1} \bullet P9} \right] \\ & \quad- \delta 3_{1} *\left[ {H_{2} } \right]\left[ {P9} \right] + \delta 3_{ - 1} \left[ {\tilde{H}_{2} \bullet P9} \right] - \delta 4_{1} *\left[ {H_{3} } \right]\left[ {P9} \right] + \delta 4_{ - 1} \left[ {\tilde{H}_{3} \bullet P9} \right] \\ & \quad - \delta 5_{1} *\left[ {H_{4} } \right]\left[ {P9} \right] + \delta 5_{ - 1} \left[ {\tilde{H}_{4} \bullet P9} \right] - \delta 6_{1} *\left[ {H_{5} } \right]\left[ {P9} \right] + \delta 6_{ - 1} \left[ {\tilde{H}_{5} \bullet P9} \right] \\ & \quad - \delta 7_{1} *\left[ {H_{6} } \right]\left[ {P9} \right] + \delta 7_{ - 1} \left[ {\tilde{H}_{6} \bullet P9} \right] \\ \dot{C}9 & = w_{1} *\left[ {H_{a} } \right] \\ \end{aligned} \) | |

| |

\( \begin{gathered} \dot{P}3_{F} = - v_{1} *\left[ {P3_{0} } \right]\left[ {H_{a} } \right] + v_{ - 1} \left[ {\tilde{P}3 \bullet H_{a} } \right] \hfill \\ \left[ {\dot{\tilde{P}}3 \bullet H_{a} } \right] = + v_{1} *\left[ {P3_{0} } \right]\left[ {H_{a} } \right] - v_{ - 1} \left[ {\tilde{P}3 \bullet H_{a} } \right] - v_{2} \left[ {\tilde{P}3 \bullet H_{a} } \right] \hfill \\ \dot{C}3 = + v_{2} *\left[ {\tilde{P}3 \bullet H_{a} } \right] \hfill \\ \end{gathered} \) | |

| |

\( \begin{gathered} \dot{V} = - z_{1} *\left[ {V_{0} } \right]\left[ {C3} \right] - z_{ - 1} \left[ {\tilde{V} \bullet C3} \right] \hfill \\ \dot{F} = z_{2} *\left[ {\tilde{V} \bullet C3} \right] \hfill \\ \end{gathered} \) | |

\( \left[ {\dot{\tilde{V}} \bullet C3} \right] = + z_{1} *\left[ {V_{0} } \right]\left[ {C3} \right] - z_{ - 1} \left[ {\tilde{V} \bullet C3} \right] - z_{2} \left[ {\tilde{V} \bullet C3} \right] \) | |

| |

Abbreviation | Meaning |

| Apaf-1 (A |

| Initial concentration of APAF-1 |

| Free Apaf-1 |

| dATP (D |

| Initial concentration of dATP |

| Free dATP |

| APAF-1:dATP complex |

| Cytochrome c (C |

C | Initial concentration of cytochrome c |

C | Free cytochrome c |

| Apaf-1:cytochrome c complex |

| Apaf-1:dATP:cytochrome c complex (monomer) |

| Apaf-1:dATP:cytochrome c complex (dimer) |

| Apaf-1:dATP:cytochrome c complex (trimer) |

| Apaf-1:dATP:cytochrome c complex (tetramer) |

| Apaf-1:dATP:cytochrome c complex (pentamer) |

| Apaf-1:dATP:cytochrome c complex (hexamer) |

| Apaf-1:dATP:cytochrome c complex (heptamer) |

| Procaspase-9 (P9 |

| Initial concentration of procaspase-9 |

| Free procaspase-9 |

| Apaf-1:dATP:cytochrome c heptamer + 1 procaspase-9 |

| Apaf-1:dATP:cytochrome c heptamer + 2 procaspase-9 |

| Apaf-1:dATP:cytochrome c heptamer + 3 procaspase-9 |

| Apaf-1:dATP:cytochrome c heptamer + 4 procaspase-9 |

| Apaf-1:dATP:cytochrome c heptamer + 5 procaspase-9 |

| Apaf-1:dATP:cytochrome c heptamer + 6 procaspase-9 |

| Apaf-1:dATP:cytochrome c heptamer + 7 procaspase-9 |

| Inactive holoenzyme ( |

| Active holoenzyme ( |

| Caspase-9 (C9 |

| Procaspase-3 (P3 |

| Initial concentration of procaspase-3 |

| Free procaspase-3 |

| Caspase-3 (C3 |

| Free DEVD-Afc |

| Initial concentration of DEVD-Afc |

| Free fluorescence molecule Afc |

| |

* For the cooperative binding of Apaf-1 complexes: γ1 | |

** For the cooperative binding of procaspase-9: δ1 |

Since any modeling requires quantitative parameters of modeled processes, we assembled published kinetic parameters of processes involved in caspase-9 activation, determined these parameters experimentally using an in vitro or a recombinant system, or postulated them (Table 1). We then compared the results of simulation with the published experimental data (Rodriguez and Lazebnik 1999), and changed the kinetic constants used in the simulation to reconcile the differences between the simulated and experimentally observed processes. We note that even though we checked the models of apoptosis by adopting the parameters from the literature, which were then validated by recombinant systems, it still remained necessary to check the robustness of the selected models, because model parameters are known to depend on environmental conditions such as buffer concentrations. To test the robustness of the model, various ranges of all parameters were simulated in the model system and were used to calculate the differences with experimental datasets by using serially measured mean square errors (MSE). We observed that in almost all cases parameter values selected from published kinetic reactions led to excellent fit with the most “plausible model,” i.e., the one characterized by the minimum MSE points (Supplement Fig. 1), thus indicating an extremely low likelihood of overfitting.

Remarkably, changing most kinetic parameters in the simulation, even by several orders of magnitude, had little effect on caspase-3 activation. However, changing some parameters, of critical significance, within a relatively narrow range significantly changed the kinetics of active caspase. Changing the *K* _{m} and *k* _{2} of the active holoenzyme (step 5) had the largest effect on caspase-3 activation (Supplement Fig. 1f), suggesting that these parameters are the key determinants of caspase-3 regulation. For example, decreasing the *K* _{m} of the active holoenzyme increases the rate of pro-caspase-3 processing. On the other hand, decreasing the dissociation rate constant of the active holoenzyme (step 7) accelerates the cleavage and activation of both caspase-3 and caspase-9 (Supplement Fig. 1e).

Using the optimized simulation, we posed several specific questions regarding intrinsic apoptosis pathway.

#### 1. Of the two molecules, cytochrome c or dATP, which one initiates the formation of apoptosome?

The current model of Apaf-1 activation assumes that Apaf-1 is bound to dATP, which is hydrolyzed upon binding of Apaf-1 to cytochrome c. The resulting dADP remains bound to the Apaf-1-cytochrome c complex and then is exchanged by an undefined mechanism for a molecule of dATP, thus producing a complex that is oligomerized into the apoptosome. Therefore, in this model the binding of cytochrome c to Apaf-1 is the triggering event for the formation of apoptosome and the consequent caspase-9 activation (Kim et al. 2005; Yu et al. 2005).

However, previous studies provided evidence that cytochrome c binds Apaf-1 independently of the presence of free dATP (Zou et al. 1997), that binding to cytochrome c induces binding of Apaf-1 to dATP (Cain et al. 2000; Jiang and Wang 2000), and that hydrolysis of dATP by Apaf-1 is continuous and precedes binding of cytochrome c (Zou et al. 1999). The possible discrepancies between these observations and the model suggest that the exact sequence of reactions that result in the active apoptosome is yet to be established. We used our simulation to compare the effect on caspase-9 and caspase-3 activation of two possible initiating steps: binding of cytochrome c induces binding of dATP (Fig. 2a), or binding of dATP induces binding of cytochrome c (Fig. 2b).

Simulation of the “CytC-bind-first” model showed no significant changes of caspase-3 activity in lower concentration of both cytochrome c and Apaf-1 (Fig. 2e, g). To test the robustness of the models, we also conducted extensive perturbation analysis by simulating the model with perturbed rate constants for the binding reaction of cytochrome c with Apaf-1. In this model, decreased *a* _{ 1 } also causes no significant changes of caspase-3 activity in lower concentration of cytochrome c and Apaf-1 (Fig. 2e). This result indicates that the formation of holoenzyme in this model is relatively independent with respect to the concentration of cytochrome c and Apaf-1. Thus, any change of cytochrome c and Apaf-1 concentration could not affect the intrinsic apoptotic system, which contradicts the experimental finding that elevation of cytochrome c induces apoptosis. However, the simulation of the second model, “ATP-bind-first” model, produced a stable increase of caspase-3 activity at even lower concentration range of cytochrome c and Apaf-1 regardless of changing rate constants. This increased activity recapitulated the shape assumed by the experimentally observed data in the recombinant system (Fig. 2f, h). Therefore, the simulation favored the model in which the first step of apoptosome formation is binding of dATP to Apaf-1—a process that then facilitates subsequent binding of cytochrome c.

#### 2. Does cytochrome c affect caspase-9 activation after the apoptosome is formed?

#### 3. Is there positive cooperative interaction during the formation of Apaf-1 complex?

A key step in activation of caspase-9 is oligomerization of Apaf-1. At present, how this oligomerization occurs is not fully understood. The rapid rate of the oligomerization and the number of the subunits in the oligomer suggests a possibility of cooperative binding among the subunits. Recently, Nakabayashi and Sasaki mathematically modeled apoptosome assembly with a quadratic network interaction framework that did not involve cooperative binding (Nakabayashi and Sasaki 2006).

To test whether cooperative binding can also explain how Apaf-1 is oligomerized, we simulated the effect of cooperative binding on the activation of caspases-9 and -3, we assumed that Apaf-1 oligomerization proceeds stepwise (Eqs (17) through (22)). We created two simulation models with a linear network interaction framework. In one of the models, each Apaf-1 complex subunit has the same binding rate as previously bound subunits (γ1_{1} = γ2_{1} = γ3_{1} ··· = γ7_{1}; γ*n* _{1} is binding rate of *n*th Apaf-1 complex), while in the other model the successive Apaf-1 complex subunits have monotonically increasing binding rates (γ1_{1} ≪ γ2_{1} ≪ γ3_{1} ··· γ7_{1}).

In the linear model the increase of caspase-3 activity was exponential at low rate constant and weakly sigmoidal at higher rate constant if no positive cooperativity was assumed (Fig. 4a), and sigmoidal if positive cooperativity was introduced (Fig. 4b). The Nakabayashi and Sasaki model produced a sigmoidal curve that was initially nearly linear (Fig. 4c). To determine which of the curves fits better the experimental results, we calculated the mean square error (MSE, see Theoretical basis for Simulation) between experimental and simulation data using a least square distance function, and concluded that the model based on a linear network with cooperative interactions (Fig. 4b) fits the data the best. We also tested the robustness of the models by varying the binding rate constants. As we expected, increased binding rates elevate the holoenzyme activity. However, the resulting exponential input–output curve at low rate constant remained unchanged and overall shape remained more or less the same. These simulation results indicate that modified parameters do not change the patterns of the dynamics. Therefore, we may reasonably conclude that there exist positive cooperative interactions during the Apaf-1 complex formation.

#### 4. Is binding of procapase-9 to the apoptosome cooperative?

Caspase-9 functions as a holoenzyme in which this protease and Apaf-1 are present in 1:1 ratio (Acehan et al. 2002). Thus, total of seven caspase-9 molecules can exist in holoenzyme complex even though it is not clear how this oligomerization occurs. The simulation suggesting a positive cooperative binding among Apaf-1 complex subunits (Fig. 4) led us to test whether binding of procaspase-9 to the oligomer is also cooperative. We compared two alternative models. In one model, each procaspase-9 molecule binds with the same binding constant as the previous procaspase-9 during the formation of the holoenzyme (δ1_{1} = δ2_{1} = δ3_{1} ··· = δ7_{1}; δ*n* _{1} is binding rate of *n*th procaspase-9 molecule), while in the other alternative model the rate of binding of each successive procaspase-9 molecules increases monotonically (δ1_{1} ≪ δ2_{1} ≪ δ3_{1} … ≪ δ7_{1}).

Based on the previous results, we proposed that seven Apaf-1 complex subunits interact cooperatively with each other while forming the apoptosome, and that this positive cooperation can explain the rapidity of caspase-9 activation during apoptosis, especially considering that assuming positive cooperativity in the binding of caspase-9 to the apoptosome failed to affect the rates of caspase-3 activation

#### 5. Is free caspase-9 active?

## Notes

### Acknowledgment

We thank Dr. Gabriel Nunez (University of Michigan) for providing Apaf-1 cDNA.

### Open Access

This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

## Supplementary material

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