The term “model” is used in a variety of fields in the natural and applied sciences to describe a mathematical or computational representation of a physical system. In molecular biology, the term usually refers to a “word model” or narrative description accompanied by a diagram, although it can also refer to a cell line or genetically engineered mouse that recapitulates aspects of a human disease. In this Review, we restrict use of the term “model” to describe an executable set of rules or equations in mathematical form. We are primarily interested in models that are built and tested using detailed cellular or biochemical experiments. Models of cellular biochemistry can be based on different mathematical formalisms, from Boolean logic to differential equations, depending on the degree of detail and the scope of the modeling effort. Most models of apoptosis have been encoded using ordinary differential equations (ODEs), which describe the evolution of a system in continuous time. ODEs are the mathematical representation of mass action kinetics, the familiar biochemical approximation in which rates of reaction are proportional to the concentrations of reactants () (Chen et al., 2010
). Diffusion, spatial gradients, or transport can be modeled explicitly using partial differential equations (PDEs), which represent biochemical systems in continuous time and space. For example, Rehm et al. (2009)
used PDEs to model the spread of mitochondrial permeabilization through a cell following an initial, localized MOMP event. Using sets of differential equations it is possible to encode a complex network of interacting biochemical reactions and then study network dynamics under the assumption that protein concentrations and reaction rates can be estimated from experimental data. Differential equation models often increase rapidly in complexity as species are added, as each new protein can give rise to a large number of model species differing in location, binding state, and degree of posttranslational modification. This problem has effectively limited data-dependent ODE/PDE models to fewer than ~20 gene products (and on the order of 50–100 model species), although efforts are underway to increase this limit.
In addition to differential equations, several other formalisms have been used to model apoptosis. Stochastic models make it possible to represent reactions as processes that are discrete and random, rather than continuous and deterministic. Stochastic models are advantageous when the number of individual reactants of any species is small (typically fewer than ~100) or reaction rates very slow (Zheng and Ross, 1991
). In these cases, a Monte Carlo procedure is used to represent the probabilistic nature of collisions and reactions among individual molecules (Gillespie, 1977
). For example, stochastic cellular automata have been used to model the movement of molecules on the mitochondrial outer membrane (Chen et al., 2007
). When sufficient time-resolved quantitative data are lacking, a less precise modeling framework is usually advantageous, and logic-based models have proven particularly popular. Boolean models, for example, are discrete two-state logical models in which each node in a network is represented as a simple on/off switch. Boolean models have been used to represent the interplay among survival, necrosis, and apoptosis pathways and to predict the likelihood that each phenotype would result following changes in the levels of regulatory proteins (Calzone et al., 2010
). However, the more qualitative and phenomenological the modeling framework, the less mechanistic the insight.
Regardless of modeling framework, a trade-off exists between model tractability and model detail or scope. The inclusion of more species makes it possible to analyze biochemical processes in greater detail or to represent the operation of large networks involving many gene products, but larger models are more difficult to constrain with experimental data, and excess detail can mask underlying regulatory mechanisms. A Jorge Luis Borges story comes to mind in which the art of cartography achieved such a perfection of detail that cartographers built a map of their empire with 1:1 correspondence to the empire itself, rendering the map useless (Borges and Hurley, 1999
). On the other hand, although small models have the advantage of relative simplicity and even analytical tractability (i.e., capable of being solved exactly without simulation), they run the risk of grossly simplifying the underlying biochemistry and of including an insufficient number of regulatory processes. As yet, no clear principles exist to guide decisions about model scope and complexity, and most studies remain constrained by the relative immaturity of modeling software and a paucity of experimental data.
Estimating values for rate constants and initial protein concentrations (the parameters in differential equation models) remains extremely challenging both computationally and experimentally. Each reaction in an ODE model is associated with one or more “initial conditions” (the concentrations of reactants at time zero) and rate constants, usually a forward and reverse rate constant. Some of these parameters are available in the literature, typically from in vitro biochemical experiments, and these values may hold true in the context of a cell. In many other cases, however, no estimates of rate constants are available and parameters must be estimated directly from experiments (Chen et al., 2010
). In addition, protein concentrations vary from cell type to cell type and should be measured directly in the cell type under investigation, although this is often not done because it is time consuming. The estimation of unknown parameter values based on data (typically, time-dependent changes in the abundance or localization of proteins in the model) is called model calibration, model training, or model fitting. Almost all realistic models of biological systems are too large for all parameters to be fully constrained by experimental data, and the models are therefore “nonidentifiable.” Thus far, the process of model calibration has been approached rather informally, but more rigorous approaches are in development (e.g., Kim et al., 2010
). Careful analysis is expected to confirm the common-sense view that solid conclusions can be reached even in the case of partial knowledge.
Modeling biological processes requires the collection and analysis of quantitative experimental data. An ODE model, which assumes that each compartment is well mixed, necessarily represents a single cell, and calibrating and testing ODE models therefore require collecting data on single cells over time. However, live-cell imaging experiments usually rely on genetically modified cell lines carrying fluorescent reporters. Creating these lines is relatively time-consuming, and the extent of multiplexing is limited by phototoxicity and the availability of noninterfering fluorophores. It is not always clear that an engineered reporter correctly represents the activity or state of modification of endogenous proteins (see, for example, discrepancies regarding initiator caspase activity reporters, discussed below; Albeck et al., 2008a
; Hellwig et al., 2008
; Hellwig et al., 2010
). Flow cytometry, immunofluorescence, and single-cell PCR are also effective means to assay single cells, and biochemical experiments (immunoblotting or ELISAs for example) performed on populations of cells remain essential for quantitative biology. Although rarely addressed, effective integration of data arising from multiple measurement methods is an area in which computational models are likely to play a key role (Albeck et al., 2006
The construction and parameterization of even a well designed model do not lead directly to a better understanding of the system—model analysis is required. The dependence of the system on parameter values is of particular interest and can be approached using sensitivity analysis. Sensitivity analysis involves systematically varying parameters (initial conditions or rate constants) while monitoring the consequences for model output (the time at which a cell undergoes apoptosis, for example). Sensitivity analysis reveals which outputs are sensitive to variation in which parameters and can be viewed as the computational equivalent of experiments that knock down or overexpress proteins while monitoring phenotype. For example, Hua et al. (2005)
created an ODE model of Fas signaling and performed sensitivity analysis by varying the initial concentration of each protein species 10- or 100-fold above or below a baseline value. Using the half-time of caspase-3 activation as an output, they predicted (and confirmed experimentally) that increases but not decreases in Bcl-2 levels would alter sensitivity to FasL. From a practical perspective, sensitive parameters must be estimated with particular care if a model is to be reliable, but from a biological perspective, they represent possible means of regulation. Points in a network that exhibit extreme sensitivity to small perturbations are often referred to as “fragile” (the converse of “robust”), and considerable interest exists in the idea that fragility analysis, a concept borrowed from control theory, might be applied to biological pathways. In this view, fragile points might identify processes frequently mutated in disease or potentially modifiable using therapeutic drugs (Luan et al., 2007
Stability analysis is another commonly used method of model analysis. Some models of biochemical networks have the interesting property of converging at equilibrium to a small set of stable states known as fixed points, where the rate of change in the concentrations of all model species is zero. Identification and characterization of fixed points can provide valuable insight into the dynamics of a system, its responses to perturbation, and the nature of regulatory mechanisms. Of particular interest in biology is bistability, a property in which a system of equations has two stable fixed points separated by an unstable fixed point. Bistability has obvious appeal in the case of apoptosis, in which cells are either alive or dead, and has been proposed to underlie a variety of binary fate decisions such as maturation of Xenopus
oocytes (Ferrell and Machleder, 1998
) and lactose utilization in E. coli
(Ozbudak et al., 2004
). From the perspective of control, many bistable systems have two valuable properties: (1) they are insensitive to minor perturbations because the system is “attracted” to the nearest stable state (in apoptosis, a bistable system would be resistant to spontaneous activation of proapoptotic proteins, for example), and (2) they exhibit “all-or-none” transitions from one stable state to another in response to small changes in the level of a key regulatory input (a property known in biochemistry as “ultrasensitivity”). Bistable processes often exhibit hysteresis (path dependence): once in the on
state, they do not readily slip back to off
. It is often assumed that the regulatory machinery for apoptosis must be bistable in the mathematical sense with one equilibrium state corresponding to caspases off
and “alive” and the other to caspases on
and “dead” (). Although bistability remains the favorite framework for thinking about the switch between life and death, bistability is not strictly necessary for a switch-like transition between two distinct states (Albeck et al., 2008b
). A monostable system in which the landscape changes through time can create a temporal switch between two states; in this case, the change in the landscape involves the creation, destruction, or translocation of precisely those proteins (caspases, cytochrome c
, etc.) that are known to regulate apoptosis. In this regard, it should be noted that the “sharpness” of a switch in a conventional bistable system refers to the steepness of the dose-response curve (to a change in the concentration of a regulatory protein, for example), not necessarily sharpness in time. In contrast, the “all-or-nothing” switch observed by time-lapse microscopy of cells undergoing apoptosis refers to a switch from alive to dead that is sharp in a temporal
sense. These considerations do not imply that the biochemical pathways controlling apoptosis are not bistable systems, but rather that bistability is not necessary a priori.