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An effective network reduction approach to find the dynamical repertoire of discrete dynamic networks
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Image of FIG. 1.
FIG. 1.

Operations for the creation of the expanded network. (a) Node has the update function . The addition of complementary nodes introduces a new node with update function . It also introduces a complementary node for node , which makes the update function of take the form . (b) Node has the update function . The addition of composite nodes introduces a new node that represents the cooperative effect of B and C on A. (c) Node has the update function . The two expansion operations introduce complementary nodes for , , and , and a composite node for the AND relation between B and .

Image of FIG. 2.
FIG. 2.

Identification of stable motifs from the expanded network. (a) An example of a Boolean network. (b) The expanded network representation of the Boolean network in (a). (c) The two stable motifs in the expanded network, that is, the two smallest SCCs in the network that satisfy the requirements of not containing both a node and its complementary node, and containing all the inputs of every included composite node. Each stable motif indicates the fixed states of the corresponding subset of nodes of the Boolean network.

Image of FIG. 3.
FIG. 3.

An example of a component that has an unstable oscillation. This network has an attractor in which all the nodes oscillate and also has a steady state attractor. (a) The network and its respective Boolean rules. (b)The state transition graph of the network. The nodes of the state transition graph are the states of the system (written in the order , ) and the edges represent allowed state transitions when only one node is updated. State 11 is a fixed point as there are no transitions going out of it. States 01, 00, and 10 form a complex attractor. (c) The expanded representation of the network. Note that {} forms a stable motif and that the whole expanded network forms an oscillating SCC.

Image of FIG. 4.
FIG. 4.

An example of a node configuration in which a node can stabilize without the influence of an input signal or a stable motif. In this example, and oscillate in a complex attractor, but they do not take all possible states of their state transition graph in this attractor. Specifically, they miss the  = 1,  = 1 state. As a consequence node stabilizes in the state  = 0. (a) The node configuration and their respective Boolean rules. (b) The state transition graph of nodes and . States 01, 00, and 10 form a complex attractor.

Image of FIG. 5.
FIG. 5.

The T-LGL leukemia survival signaling network. The shape of the nodes indicates the cellular location or the type of nodes: rectangles indicate intracellular components, ellipses indicate extracellular components, diamonds indicate receptors, and hexagons represent conceptual nodes (Stimuli, Stimuli2, P2, Cytoskeleton signaling, Proliferation, and Apoptosis). Node colors are used to distinguish input nodes (white), output nodes (black), and the rest of the nodes in the network (gray). An arrowhead or a short perpendicular bar at the end of an edge indicates activation or inhibition, respectively. This figure and its caption are adapted from Ref. .

Image of FIG. 6.
FIG. 6.

The three stable motifs of the T-LGL leukemia network found most often during the reduction process. The actual motifs found and the states in which each of these motifs can stabilize vary depending on the active signals. We also show the input signals (white nodes) that affect these motifs directly or almost directly (for the motif in (c)). (a) The PDGFR-S1P-SPHK1-Ceramide motif, which represents the ceramide/sphingomyelin pathway and the platelet derived growth factor receptor. (b) The IFNG-P2 motif, which is related to the control of the cytokine interferon gamma in CTLs. (c) The TBET motif, which represents the regulation of the T-box transcription factor.

Image of FIG. 7.
FIG. 7.

Distribution function for the fraction of stabilized nodes in an attractor for  = 100 for an ensemble of networks. Note the logarithmic scale in the vertical axis.

Image of FIG. 8.
FIG. 8.

Difference in the number of attractors found between the reduction and sampling methods. The squares represent the average difference in the number of attractors between the two methods, while the lower and higher limits of the bars represent the 20th and the 80th percentile of the distribution of attractor number difference. In all the cases, the difference is zero or positive, that is, the reduction method never finds less attractors than the sampling method. For all network sizes shown, an ensemble size of 100 networks was used.

Image of FIG. 9.
FIG. 9.

Time performance of the different methods (see also the main text). (a) The average time it takes to find the attractors of a network for each method. Both axes are shown in a logarithmic scale. The bump shown in the  = 150 case for the reduction method is the consequence of a network in the ensemble that took an unusually long time because of the large number of cycles in the network. (b) Cumulative distribution functions for the completion times in the  = 100 ensemble. Note that the horizontal axis has a logarithmic scale.

Image of FIG. 10.
FIG. 10.

Distribution function for the number of components that can display unstable oscillations in the  = 100 ensemble. Note the logarithmic scale in the vertical axis. For approximately 90% of the networks, we find no such components. For the rest, there are usually very few of them, with attractor sampling methods suggesting that none of them actually display unstable oscillations.

Image of FIG. 11.
FIG. 11.

The PDGFR-S1P-SPHK1-Ceramide motif, its allowed stable states, and the cell fates associated to them. For both set of stable states, the apoptosis cell fate can be reached depending on the signals present, the asynchronous update order, and on the initial state. On the other hand, the T-LGL leukemia cell fate can only be reached if the motif stabilizes in the {PDGFR = S1P = SPHK1 = ON, Ceramide = OFF} state, regardless of the signals present, the asynchronous update order or of the initial state.


Generic image for table
Table I.

The attractors of T-LGL leukemia survival network. This table shows the state of the nodes for all possible combinations of input signals in the presence of antigen (Stimuli = ON).


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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: An effective network reduction approach to find the dynamical repertoire of discrete dynamic networks