Boolean network-based analysis of the apoptosis network: Irreversible apoptosis and stable surviving
Introduction
Apoptosis or programmed cell death has been subjected to extensive experimental studies for more than two decades. In the apoptosis process, a range of cellular signals are integrated by a complex molecular network (Falschlehner et al., 2007; Henson and Gibson, 2006). This network as a whole, not the status of individual signaling molecules controls when a cell can (or should) no longer survive and enter programmed death irreversibly (Janes et al., 2005). To understand how the network operates, it is necessary not only to study what are its composing components or what molecular interactions exist between the components, but also to analyze the network as a whole or at the systems level to identify which of its properties are essential for biological functions, and how these properties emerge from the specific way by which the interacting components are organized into an integrated system. In the latter regard, computer models formulating the various molecular components and their interactions can in many aspects complement experimental studies. One main advantage of using computer simulations for systems level analysis is that components or interactions can be systematically varied in silico, either individually or jointly, to probe their roles in any general or specific cellular contexts (Stelling, 2004; Wolkenhauer, 2002).
Depending on the states of its components and the signals it receives, the outcome of the apoptosis network can be either surviving or apoptosis. Essential systems level properties of the network should include the robust stability of the surviving state as well as the irreversibility of the survival to apoptotic transition. The former property means that apoptosis can be triggered only by some definite cellular signals, but not by spontaneous fluctuations in the states of individual molecular components. The latter property means that apoptosis, once started, cannot be abolished either by withdrawing of the triggering signals or by spontaneous fluctuations.
How multi-stability and irreversibility generally arise in cellular networks and what their biological implications are have been analyzed theoretically and in some cases demonstrated experimentally (Ferrell, 2002; Ferrell and Xiong, 2001). In these studies, irreversibility and multi-stability have often been attributed to nonlinear dynamics associated with feedback loops in the respective networks.
As for the apoptosis network in particular, a number of computer modeling studies based on ordinary differential equations (ODE) have been reported (Bentele et al., 2004; Albeck et al., 2008; Rehm et al., 2006; Bagci et al., 2006; Eissing et al., 2004; Legewie et al., 2006). These ODE models are based on detailed knowledge or assumptions of kinetics parameters associated with all molecular processes considered in the respective models, and provide detailed dynamics of these processes. Thus an advantage of ODE models is that they can not only be used for theoretical analyses (Bagci et al., 2006; Eissing et al., 2004; Legewie et al., 2006), but also be calibrated based on and then tested against quantitative experimental measurements when such measurements are available (Albeck et al., 2008; Rehm et al., 2006). However, most previous ODE models have focused on limited parts of the apoptosis network. To obtain a more complete picture, it is of interest to investigate more extensive networks, especially those involving the interactions of apoptotic pathways with surviving pathways such as the growth factor (GF) pathway. As more molecular components are included, it becomes increasingly challenging to develop mutually compatible ODEs to cover all the different types of molecular processes. Moreover, in the higher-dimensional continuous state space of the ODE framework, it is difficult to formulate and extract systems properties from the complex dynamics. It is also difficult to perform systematic explorations in the model space, kinetic parameter space as well as in the initial condition space.
Boolean network (BN) is an alternative way to model complex cellular networks. A BN is a qualitative mathematical model composed of nodes having discrete states and rules governing the temporal evolution of the states. Since introduced by Kauffman in late 1960s (Kauffman, 1969), BN has been used to model gene regulatory networks and signaling pathways (Huang and Ingber, 2000; Li et al., 2004). Although BN does not model dynamics in terms of continuous variables, it has been argued that BN may provide good approximations to a variety of nonlinear behaviors of biological systems (Amaral et al., 2004). The state space of BN is discrete by definition, and is thus more accessible to statistical analyses. Such analyses readily enable the formulation and extraction of systems level properties for relatively large networks of higher dimensional variable spaces. BN can also afford extensive explorations in the model space and the initial condition space in such systems.
In this work, we construct a 40-node Boolean model of the cellular apoptosis network to analyze its systems properties and how they can be linked to specific structural features of the network. Because of the simplicity of BN, we are able to consider a relatively complete set of known molecular processes, including both anti-apoptosis (or pro-survival) and pro-apoptosis pathways, and the integration of intrinsic and extrinsic signals. One major purpose of our study is to understand the emergence of systems properties. These include the stability of the surviving states and the irreversibility of the apoptotic process. Through extensive analyses of the state space, we try to identify key network components that lead to these properties. In silico experiments are performed with a large number of random initial states at different combinations of input cellular signals. Critical pro-survival and pro-apoptotic nodes are identified based on statistical correlations between initial states and final outcomes. Irreversibility of the apoptotic process is characterized using time-dependent input signals. Stability of the surviving states is analyzed based on perturbations in the state space. The roles of specific feedback structures in the network are investigated using simulated knockout experiments.
Section snippets
Structure of the model
Our BN model (Fig. 1) includes 37 internal nodes representing states of signaling molecules, 2 input nodes representing extracellular signals, and 1 output node corresponding to the DNA damage event. The nodes are interconnected, connections corresponding to a set of rules defining how the states of each node evolve in time depending on the states of other nodes (see below). This model is based on extensive surveys of literature as well as on expert-curated databases, especially the reactome
Initial states that lead to apoptosis independent of external signals
From the simulations, we found that some of the initial states (lethal initial states) led to “apoptosis” under all the four different input conditions. These states would not be observed in real experiments but provide basis to analyze the roles of individual network components. Table 1 lists the percentages of lethal initial states obtained using the complete model and two alternative models with selected connections removed. These include Model-B in which connection B (see Fig. 1, the
Conclusions
In summary, we have analyzed the apoptosis network using a Boolean model. Based on simulations started from randomly sampled initial states, the model is able to reproduce well-known pro-apoptotic and pro-survival molecular components (nodes). The effects of the initial states of some but not all of these nodes depend on the presence/missing of certain key feedback loops. The model suggested that although irreversibility of the apoptosis process indeed depends on the presence of feedback loops
Acknowledgments
This work has been supported by the Chinese Natural Science Foundation (Grant no. 90403120) and the Chinese Ministry of Science and Technology (Grant nos. 2006AA02Z303 and 2006CB910700).
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