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# Boolean modeling: a logic‐based dynamic approach for understanding signaling and regulatory networks and for making useful predictions

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The biomolecules inside or near cells form a complex interacting system. Cellular phenotypes and behaviors arise from the totality of interactions among the components of this system. A fruitful way of modeling interacting biomolecular systems is by network‐based dynamic models that characterize each component by a state variable, and describe the change in the state variables due to the interactions in the system. Dynamic models can capture the stable state patterns of this interacting system and can connect them to different cell fates or behaviors. A Boolean or logic model characterizes each biomolecule by a binary state variable that relates the abundance of that molecule to a threshold abundance necessary for downstream processes. The regulation of this state variable is described in a parameter free manner, making Boolean modeling a practical choice for systems whose kinetic parameters have not been determined. Boolean models integrate the body of knowledge regarding the components and interactions of biomolecular systems, and capture the system's dynamic repertoire, for example the existence of multiple cell fates. These models were used for a variety of systems and led to important insights and predictions. Boolean models serve as an efficient exploratory model, a guide for follow‐up experiments, and as a foundation for more quantitative models. WIREs Syst Biol Med 2014, 6:353–369. doi: 10.1002/wsbm.1273

Illustration of the transition between a continuous and a Boolean description of a process through which node Y positively regulates node X. It is assumed that the concentration of the node Y increases linearly in time, [Y] = t. The diagram indicates the time‐course of the concentration [X] or activity X of node X as a function of time at intervals of 0.05. Blue diamonds correspond to the case when the regulation is described by a Hill function, i.e., $dXdt=tntn+0.5n−X$, with n = 5. Red triangles correspond to regulation described by a step function, i.e., $dXdt=Bt−X$, where B(t) = 1 if t ≥ 0.5 and is zero otherwise. Green squares correspond to a Boolean function X(t) = B(t).
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Illustration of the dynamics of a piecewise linear model. (a) The directed network associated with the model forms a negative feedback loop. The Boolean update functions are completely determined by the network. (b) The state transition graph of a synchronous Boolean model of the network. The symbols correspond to the states of the system indicated in the order ABC. The Boolean model has two limit cycle attractors (sustained oscillations). (c) The time‐course of the continuous variables associated with nodes A (green triangles), B (blue circles), and C (red squares) according to a piecewise linear model with γ = 1 and θ = 0.5. The sustained oscillations agree with the six‐state limit cycle of the synchronous Boolean model.
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Illustration of methods that integrate the structure and logic of regulatory interactions. (a) A hypothetical signal transduction network and Boolean model corresponding to it. The Boolean update rules that are not completely determined by the network are indicated next to the relevant nodes. (b) The expanded representation of the network, which includes three complementary nodes, indicated by prepending the node name by ∼, and two composite nodes, indicated by small black filled circles. (c) The stable motifs of the expanded network in the case of a sustained input signal (SI = 1). The first stable motif corresponds to the state 11101 (in the order I, A, B, C, O), while the second stable motif corresponds to the state 11011.
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State transition graphs corresponding to the Boolean model presented in Figure . The symbols correspond to the states of the system, indicated in the order A, B, C, thus 000 represents SA = 0, SB = 0, and SC = 0. A directed edge between two states indicates the possibility of transition from the first state to the second by updating the nodes in the manner specified by the updating scheme. An edge that starts and ends at the same state (a loop) indicates that the state does not change during update. (a) The state transition graph corresponding to synchronous update, when all nodes are updated simultaneously. The two states that have loops are the fixed points of the system. (b) The state transition graph corresponding to updating one node at a time. While several states have loops, indicating that at least one of the nodes does not change state during its update, only the states that have no outgoing edges are fixed points of the system.
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Illustration of the main steps of constructing a Boolean dynamic model of a biomolecular interaction network. The directed edges among steps indicate the order in which they may be tackled. The dashed edges mark complementary analysis that is currently not routinely undertaken but we expect will be increasingly used.
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A simple Boolean network model. (a) The directed network associated with the Boolean model. The edges with arrows represent positive effects. Note that the network does not uniquely determine the Boolean updating function for node B. (b) The Boolean updating functions in the model. Note that the network can also support an alternative updating function, BB = SA AND SC. (c) The truth tables of the Boolean updating functions given in (b).
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