A mathematical tool for exploring the dynamics of biological networks

*Courant Institute, New York University, 251 Mercer Street, New York, NY 10012;
^†Laboratory of Molecular and Cellular Neuroscience, The Rockefeller University, 1230 York Avenue, New York, NY 10021; and
^‡Department of Psychiatry, Yale University School of Medicine, 34 Park Street, New Haven, CT 06508

Contributed by Leslie Greengard, October 18, 2007 (received for review September 24, 2007)

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Fig. 1.
DARPP-32 regulatory pathways. (A) Simplified view of DARPP-32 pathway. The dopamine and glutamate cascades are shown on Left and Right (red and green), respectively. A partial view of the states of phosphorylation of DARPP-32 is shown in Center (blue). (B) Phosphorylation and dephosphorylation states of DARPP-32. DARPP-32 is phosphorylated at Thr-34 by PKA, at Thr-75 by cdk5, at Ser-97 by CK2, and at Ser-130 by CK1. Phospho-Thr-34 is preferentially dephosphorylated by PP-2B (or calcineurin); phospho-Thr-75 is preferentially dephosphorylated by PP-2A; phospho-Ser-130 is preferentially dephosphorylated by PP-2C; the phosphatase for phospho-Ser-97 is not yet characterized. Green arrow indicates positive effect; red arrows indicate negative effects. [Adapted from Nairn et al. (1).] (C) Detailed view of DARPP-32 subnetwork. Shown are the various states of phosphorylation allowed in our model.
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Fig. 2.
Dimensionality reduction of a one-parameter family of data (a space curve). In A, a three-dimensional data plot reveals that the points seem to lie on a curve. In B, a two-dimensional “visualization” of the data is obtained by linear projection onto the x-y plane. The geometry of the data is difficult to see. In C, Laplacian eigenmaps are used to project the data onto a two-dimensional space in a much more revealing manner. The fact that the data lie on a curve is quite apparent. For realistic systems, like the DARPP-32 network, where the original data are high-dimensional and inaccessible to simple visualization, this is a critical tool.
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Fig. 3.
Response manifolds for numerical simulations. (A) In the absence of perturbations, both the DARPP-32 subnetwork and the regulatory cascade follow (exactly) a simple one-parameter path, determined by the dopamine/glutamate ratio. The high-dimensional space curves are visualized in three dimensions by using Laplacian eigenmaps. (B) With modest perturbation, the response of the DARPP-32 subnetwork continues to follow a one-dimensional (but slightly noisy path). The regulatory cascades, however, no longer reflect a clear signal. (C) With sufficient noise (the large perturbation data set), the DARPP-32 response also loses its coherence, at least over some range of the dopamine/glutamate ratio. (D) Modified network 1 (see Fig. 4). cAMP is assumed to activate PP2A directly. Both PKA and PP2A are still activated in a coordinated fashion, and the DARPP-32 subnetwork clearly retains its coherence under modest perturbation. (E) Modified network 2 (see Fig. 4). All PP2A species are removed from the system (by setting their values to zero during the simulation). Clearly, the DARPP-32 subnetwork loses its coherence. (F) Modified network 3 (see Fig. 4). The regulatory cascades are eliminated by assuming that dopamine directly converts PKA to its active form (PKA-cAMP) and that glutamate directly converts PP2B to its active form (PP2B-Ca). As a result, the DARPP-32 subnetwork becomes much more sensitive to perturbations. That is, the removal of the regulatory cascades significantly affects the robustness of the signal transduction pathway.
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Fig. 4.
Three simulated modified networks. In modification 1, cAMP bypasses PKA and is assumed to directly catalyze the phosphorylation of PP2A (indicated by labels 1). In modification 2, all PP2A species are deleted from the system (indicated by labels 2). In modification 3, dopamine is assumed to convert PKA to its active form (PKA-cAMP) directly and glutamate is assumed to convert PP2B to is active form (PP2B-Ca) directly (indicated by labels 3).