In this paper we presented a comprehensive analysis of the T-LGL survival signaling network to unravel the unknown facets of this disease. By using a reduction technique, we first identified the fixed points of the system, namely the normal and T-LGL fixed points, which represent the healthy and disease states, respectively. This analysis identified the T-LGL states of 54 components of the network, out of which 36 (67%) are corroborated by previous experimental evidence and the rest are novel predictions. These new predictions include RAS, PLCG1, IAP, TNF, NFAT, GRB2, FYN, SMAD, P27, and Cytoskeleton signaling, which are predicted to stabilize at ON in T-LGL leukemia and GAP, SOCS, TRADD, ZAP70, and CREB which are predicted to stabilize at OFF. In addition, we found that the node P2 can stabilize in either the ON or OFF state, whereas two nodes, TCR and CTLA4, oscillate. We have experimentally validated the prediction that the node SMAD is over-active in leukemic T-LGL by demonstrating the predominant phosphorylation of the SMAD family members Smad2 and Smad3. The predicted T-LGL states of other nodes provide valuable guidance for targeted experimental follow-up studies of T-LGL leukemia.
Among the predicted states, the ON state of Cytoskeleton signaling may not be biologically relevant as this node represents the ability of T cells to attach and move which is expected to be reduced in leukemic T-LGL compared to normal T cells. This discrepancy may be due to the fact that the network contains insufficient detail regarding the regulation of the cytoskeleton, as there is only one node, FYN, upstream of Cytoskeleton signaling in the network. While the network is able to successfully capture survival signaling without necessarily capturing the cytoskeleton signaling, this discrepancy suggests that follow-up experimental studies should be conducted to determine the relationship between cytoskeleton signaling and survival signaling in the T-LGL network. We note that in the case of perturbation of TBET, PI3K, NFκB, JAK, or SOCS, the node Cytoskeleton signaling exhibits oscillatory behavior induced by oscillations in TCR. At present it is not known whether this predicted behavior is relevant.
Using the general asynchronous (GA) Boolean dynamic approach, we analyzed the basins of attraction of the fixed points. We found that the basin of attraction of the normal fixed point is larger than that of the T-LGL fixed point. The trajectories starting from each initial state toward the T-LGL fixed point () may be indicative of the accumulating deregulations that lead to the disease-associated stable survival state. Although the fixed points, being time independent, are the same for all update methods or implementations of time, the update method may affect the structure of the state transition graph of the system and the basins of attraction of the fixed points. We note that the GA method assumes that each node has an equal chance of being updated. If quantitative or kinetic information becomes available in this system, unequal probabilities may be implemented by grouping the nodes into several “priority classes” and assigning a weight to each class where higher weights indicate more probable transitions 
. Incorporating such information into the state space may prune the allowed trajectories and give further insights into the accumulation of deregulations.
We took one step further by performing a perturbation analysis using dynamical and structural methods to identify the interventions leading to the disappearance of the disease fixed point. We note that our study has a dramatically larger scope than the previous key mediator analysis of Zhang et al 
. For the dynamical analysis, we employed the GA approach instead of the random order asynchronous method and considered all possible initial conditions as opposed to performing numerical simulations using a specific initial condition. Zhang et al
only focused on the node Apoptosis, and identified as “key mediators” the nodes whose altered state increases the frequency of ON state of Apoptosis. An increase in Apoptosis' ON state does not necessarily imply that apoptosis is the only possible final outcome of the system. In this work, after finding the fixed points, which completely describe the state of the whole system, we performed dynamic perturbation analysis by fixing the state of each node to its opposite state in the T-LGL fixed point and determining which fixed points were obtained and what their basins of attraction were. This way we were able to identify and distinguish the key mediators whose altered state completely eliminates the leukemic outcome, and those whose altered state reduces the basin of attraction of the leukemic outcome. Moreover, numerical simulations, as done in 
, may not be able to thoroughly sample different timing. In this study, using a reduction technique, we found the cases when timing does not matter with certainty (where there is only one fixed point), and also the cases in which timing and initial conditions may matter (where there are two reachable fixed points). For the perturbation analysis using the structural method, we used the simple path (SP) measure to identify important mediators of the disease outcome and observed consistent results with the dynamic analysis. Our dynamical and structural analysis led to the identification of 19 therapeutic targets (the first 19 nodes in the first column of ), 53% of which are supported by direct experimental evidence and 15% of which are supported by indirect evidence.
Multi-stability (having multiple steady states) is an intrinsic dynamic property of many disease networks 
, which is related to the presence of feedback loops in the network. In a graph-theoretical sense, a feedback loop is a directed cycle whose sign depends upon the parity of the number of negative interactions in the cycle. A positive/negative feedback loop has an even/odd number of negative interactions. It was conjectured that the presence of positive feedback loops in the network is necessary for multi-stability whereas the existence of negative feedback loops is required for having sustained oscillations 
. From a biological point of view, the former dynamical property is associated with multiple cell types after differentiation while the latter is related to stable periodic behaviors such as circadian rhythms 
. We note that the T-LGL signaling network consists of both positive and negative feedbacks and thus has a potential for both multi-stability and oscillations. Indeed, the negative feedback in the top sub-graph of causes the complex attractor shown in . In contrast, the negative feedback on the node P2 of the bottom sub-graph is counteracted by the positive self-loop on the same node, thus no complex attractor is possible for the bottom sub-graph of . The two mutual inhibition-type positive feedback loops present in the bottom sub-graph and the self-loop on P2 generate the three fixed points, while the positive self-loop on Apoptosis maintains the normal fixed point once Apoptosis is turned ON.
Negative feedback loops can be a source of oscillations 
, homeostasis 
, or excitation-adaptation behavior 
. Especially, when the activation is slower than the inhibitory interaction in the negative feedback, it can lead to sustained oscillations 
. In the T-LGL network, the negative feedback loop between the T cell receptor TCR and CTLA4 modulates stimulus-induced activation of the receptor in such a way that CTLA4 is indirectly activated after prolonged TCR activation, whereas the inhibition of TCR by CTLA4 is a direct interaction 
. That is, activation is slower than inhibition in the negative feedback and thus an oscillatory behavior reminiscent of that obtained by our asynchronous Boolean model would also be observed in continuous modeling frameworks as well. Although no time-measurements of the T cell receptor activity in T-LGL exist, it has been reported that there is variability for TCR activation in different patients (
and unpublished observation by T.P. Loughran), supporting the absence of a steady state behavior.
Our study revealed that both structural and dynamic analysis methods can be employed to identify therapeutic targets of a disease, however, they differ in implementation efficiency as well as the scope and applicability of the results. The structural analysis does not require mapping of the state space and thus is less computationally intensive and is more feasible for large network analysis, but it may not capture all the initial states and thus may miss or inaccurately identify some important features. The dynamic analysis method, while computationally intensive, yields a comprehensive picture of the state transition graph, including all possible fixed points of the system, their corresponding basins of attraction, as well as the relative frequency of trajectories leading to each fixed point. We demonstrated that the limitations related to the vast state space of large networks can be overcome by judicious use of the network reduction technique that we developed in our previous study 
. We conclude that the structural method incorporating the cascading effects of node disruptions is best employed for quick exploratory analysis, and dynamic analysis should be performed to get a thorough and detailed insight into the behavior of a system. Overall, the combined analysis presented in this study opens a promising avenue to predict dysregulated components and identify potential therapeutic targets, and it is versatile enough to be successfully applied to a large variety of signal transduction and regulatory networks related to diseases.