REPRESENTING BIOCHEMICAL NETWORKS WITH LOGIC-BASED MODELS What Is a Logic-Based Model ?

Computational models are increasingly used to analyze the operation of complex biochemical networks, including those involved in cell signaling networks. Here we review recent advances in applying logic-based modeling to mammalian cell biology. Logic-based models represent biomolecular networks in a simple and intuitive manner without describing the detailed biochemistry of each interaction. A brief description of several logic-based modeling methods is followed by six case studies that demonstrate biological questions recently addressed using logic-based models and point to potential advances in model formalisms and training procedures that promise to enhance the utility of logic-based methods for studying the relationship between environmental inputs and phenotypic or signaling state outputs of complex signaling networks. With accelerating pace, molecular biology and biochemistry are identifying complex patterns of interactions among intracellular and extracellular biomolecules.With respect to cell signaling in eukaryotes, the focus of this review, complex multicomponent networks involving many shared components govern how a cell will respond to diverse environmental cues. Powerful experimental approaches now exist for identifying components of these networks and for determining their biochemical activities, but understanding the networks as an integrated whole is difficult using intuition alone. Thus, mathematical and computational modeling is increasingly playing a role in data interpretation and attempts to extract general biological understanding (1, 2). Depending on the network studied, the data available, and the questions posed, a diverse spectrum of modeling approaches exists, ranging from the highly abstract to the highly specified (3, 4). The goal of this review is to discuss logic-based modeling, an approach lying midway between the complexity and precision of differential equations on one hand and data-driven regression approaches on the other. Within the spectrum of modeling methods currently being applied to cellular biochemistry, models involving differential equations bear the closest relationship to underlying biochemical rate laws. Sets of coupled ordinary differential equations (ODEs) can effectively represent chemical reactions when the number of molecules is large and mass action approximations are appropriate. Partial differential equations (PDEs) add the ability to represent spatial gradients (5), and stochastic methods make it possible to analyze systems in which the number of molecules is small (6). Networks of differential equations can model the temporal and spatial dynamics of biochemical processes in considerable detail, making it possible to study chemical mechanism and predict network dynamics under various conditions. However, the topology of ODEand PDE-based models (that is, patterns of interaction among the species) must be specified in advance, andmodel output is strongly dependent on the values of free parameters (typically initial protein concentrations and rate constants). Estimating these parameters is a computationally intensive task requiring substantial data. As networks get larger, ODEmodeling becomesmore andmore challenging, andmodels that attempt to capture real biological data are currently limited to a few dozen components. At the other extreme, a very active field for computing graphical representations of biological networks through literature analysis or identification of correlations in high-throughput data has emerged. In these graphs, termed protein interaction networks (PINs or interactomes) or protein signaling networks (PSNs), genes and proteins are represented by nodes and potential interactions by edges (links). The edges can be directional or not and signed (inhibitory/activating) or not and typically represent a wide range of interaction modes from direct physical binding to correlated gene expression (7) or integrated database entries (8). Graphs are an attractive way to summarize diverse relationships among large numbers of biomolecules across multiple organisms, but they are not executable per se and cannot be used to compute input-output relationships. Moreover, network graphs rarely take into account dynamic changes in signaling activities, cell type-specific biochemistry, or context-dependent variations (9). Here, we review logic-based models, which represent a compromise between highly specified differential equation models and protein interaction graphs. Using logic-based methods, it is possible to model interactions among large numbers of protein species and perform model training, model validation, and model-based prediction. The first application of logic-based modeling to biological pathways is credited to Kauffman, who used discrete logic to model the biological process of gene regulation (10). Subsequent work focused on delineating theoretical properties of logic-based models of gene regulation (11, 12). Huang and Ingber were among the first to apply logic-based modeling to cell signaling networks, demonstrating that specific cell phenotypes might correspond to dynamic steady states of a This work was funded by National Institutes of Health Grants P50GM68762 andU54-CA112967 and theDepartment of Defense Institute of Collaborative Biotechnologies. *To whom correspondence should be addressed. Phone: (617) 2521629. Fax: (617) 258-0204. E-mail: lauffen@mit.edu. Current Topic/Perspective Biochemistry, Vol. 49, No. 15, 2010 3217 logic-based model of intracellular signaling species (13). This example of linking environmental inputs to phenotypic outputs via a logic-based model of a biochemical signaling network has sparked considerable interest in the possibility of harnessing logic-based models to understand the relationship between biochemical signaling network and cell state, reflected in a large number of studies over the past few years (13-33). This review is divided into two sections. In the first, we describe the fundamentals of logic-based modeling; in the second, we discuss six applications of logic-based modeling to eukaryotic biology.We focus on logic-basedmodels of biochemical signaling networks and refer the reader to the literature for amore in-depth explanation of theoretical considerations (34), applications of logic-based models to gene regulatory networks (11), andmodels of intercellular communication (35, 36). REPRESENTING BIOCHEMICAL NETWORKS WITH LOGIC-BASED MODELS What Is a Logic-Based Model? Consider the graphical representation of a signaling network common to protein interaction networks (Figure 1a): the nodes in the graph represent proteins, and the edges represent interactions. Such a graph depicts nodes that interact physically or have correlated expression or genetic profiles (depending on the underlying data source) but do not allow us to explicitly compute the state of activity of individual nodes given different inputs or initial network states. Performing such a calculation requires information about how each node reacts to the activities of its input nodes. In logic-based models, these dependencies are specified by “gates” (Figure 1b) which, in Boolean logic, are specified by “truth tables” that list output states for all possible combinations of input states (Figure 1c). Figure 1d shows the truth tables of the OR, AND, andNOTBoolean logic gates as well as a small network in which gates are assembled to create the AND-NOT logic gate. To illustrate how logic-based modeling can be applied to a biological network, consider a hypothetical representation of epidermal growth factor receptor (EGFR) and several downstream proteins (Figure 1e). This toy network is too simple to be realistic but demonstrates several issues of importance when building a logic-based model. Either epidermal growth factor (EGF) or heregulin (HRG) can bind to and activate EGFR (Figure 1d,e). EGFR then stimulates the Raf/ERK and PI3K/ AKTpathways (themultitude of knownbiochemical interactions in this case are modeled as a single “activating” edge). ERK activity inhibits EGFR-dependent PI3K activation, whereas AKT positively regulates the Raf/ERK pathway (Figure 1d,e). With this information, it is possible to compute the response of the unperturbed network to a given input as well as responses resulting from inhibition of a node (by a drug for example). However, under all simulated conditions [EGF or HRG alone or in combination (Figure 1f)], the network response is the same. This is to be expected because binary logic cannot encode the differential sensitivities of EGFR to EGF and HRG, a point to which we return below. Modeling Nondiscrete Processes Using Logic-Based Approaches. The assumption in Boolean logic that all species are either on or off (state 1 or 0, respectively) is clearly an unrealistic way to represent binding curves or catalytic reactions. Fortunately, logic-based modeling provides several approaches for modeling intermediate states of activity (Figure 2a). Multistate discrete models specify additional levels between 0 and 1, whereas fuzzy logic allows for continuous node states. In fuzzy logic, which has foundwide utility in industrial control systems, a set of user-defined functions transforms discrete logic statements into relationships between continuous inputs and output levels. Other methods of describing discrete or Boolean logic models as continuous or mixed discrete continuous have also been implemented successfully [Figure 2a (dashed lines)] (28, 37, 38). How is a prototypical biological interaction approximated using discrete and nondiscrete logic formalisms? In Figure 2b, a sigmoidal relationship between input and output level [e.g., a protein kinase acting on a substrate (black solid line)] is approximated by binary (red solid line), ternary (green dashed line), and quaternary (blue dashed-dotted line) discrete logic functions. Fuzzy logic and mixed discrete continuous logic can closely approximate the real respo