Figure 1.
Network Inference Using RS-HDMR.
(A) Heat Map of RS-HDMR sensitivity indices. The indices shown are the maximum
values among the nine individual data sets and thirteen pairwise comparison data sets. Network species on the ordinate describe the output
, and species on the abscissa represent the inputs
. (B) Sensitivity Indices
of First Order RS-HDMR Component Functions
. First-order RS-HDMR component functions were calculated from all nine individual data sets, using each variable as the output
. The first (
) and second (
) most significant functions were consistent across all nine data sets, and their average sensitivity index values
are reported. (C) RS-HDMR Identified Significant Network Connections. Significant network interactions (
) from individual and pairwise RS-HDMR analysis. (D) Bayesian Network Analysis Identified Network Topology. Reproduced from Sachs et al., 2005.
Figure 2.
A Comparison of ARACNE, CLR, and RS-HDMR Network Inference.
Network inference results both excluding (A–G) and including (H–Q) connections identified through pairwise comparison datasets. (M–O) Network connections with normalized edge weights . (A,H,M) RS-HDMR sensitivity indices,
. (H,M) Network species on the ordinate describe the output
, and species on the abscissa represent the inputs
, for the connections identified through pairwise-comparison. (B,I,N) ARACNE network inference results. (C,J,O) CLR network inference results. (D–G, K–Q) Venn diagrams comparing network connections identified with normalized edge weights above a defined threshold of 0, 0.2, 0.4, or 0.6. Circle areas scaled by the number of connections.
Figure 3.
First and Second Order RS-HDMR Component Functions Describing PKC.
(A-B) First-order RS-HDMR component functions describing interaction between inputs and an output, PKC, were averaged over corresponding RS-HDMR functions describing the same network connections under various experimental conditions. The thick line describes the mean function, and thin lines are one standard deviation above and below the mean function. (C) With PKC as the output variable , the heat map indicates
values as a function of
(p38) and
(Jnk) shown on the ordinate and abscissa, calculated from data set
. (D-E) The correlation coefficient
(D) and scatter plot (E) describe RS-HDMR fitting accuracy for predicting PKC in data set
, with or without including higher-order component functions. (A-C) p38 and Jnk are normalized to [0,1], and component function outputs are the same scale as (E).
Figure 4.
RS-HDMR Component Functions and the Predictive Capability of RS-HDMR Compared to Amelia II.
RS-HDMR-generated FEOMs predict the values of network nodes (shown in columns) in a single cell based on the other node values in that cell, using data set . (A) Fitting accuracy described by the correlation coefficient, R, of the predicted vs. observed values of the test data for RS-HDMR (blue) and Amelia II (red). (B–D) Fitting accuracy scatter plots, where a higher density of data points is indicated by warmer color. Observed values are normalized to the maximum for each network node. (B,C) Observed vs. RS-HDMR inferred values of the training (B) and test (C) data. (D) Observed vs. Amelia-inferred values of the test data. (E–F) RS-HDMR component functions of the first (E) and second (F) order. Only the most significant second order function is shown, and the heat map indicates
values as a function of
and
shown on the ordinate and abscissa. Inputs
and
are linearly normalized to [0,1]. Component function outputs
and
are normalized to the same scale as in B–D.
Table 1.
Portion of Total Variance Accounted for by First-Order RS-HDMR Expansions for Akt and Relative Errors of First-Order RS-HDMR IO-mappings.
Table 2.
Dream3 Phosphoprotein Prediction Results.
Figure 5.
Inverse FEOMs Infer Experimental Conditions.
Inverse FEOMs were constructed between data sets describing the network under general stimulatory (“Control”) and specifically perturbative (“Activated” or “Inhibited”) experimental conditions. The perturbed node was used as the output, whose values were digitized according to experimental conditions, being either relatively high (1) shown in black or low (0) shown in white. (A–B, D–H) These histograms describe RS-HDMR-fitting ability for data observed under activating or inhibiting conditions. Although the experimental perturbations are approximated as discrete, RS-HDMR expresses the model output as continuous, thus the distribution of RS-HDMR fitted results approximately resembles two Gaussian distributions. Clear separation of the two distributions for a given plot indicates good RS-HDMR prediction of the corresponding perturbation. (C) Inverse FEOM accuracy using a SVM classifier and RS-HDMR. RS-HDMR accuracy corresponds to histograms in A–B, D–H.
Table 3.
Dream3 Cytokine Release Prediction Results.
Figure 6.
Data for the Dream3 (A) and Dream4 (B) challenges are presented as heat maps, where lighter color indicates higher value (generally concentration). For the “Condition” arrays, columns may represent whether a condition (e.g., a growth factor or inhibitor) is present (white) or absent (black). For Dream3 data, the “+/− Cancer” column describes whether the cell-type is normal (black) or cancer (white). Each row in either the “Phosphoproteins” or “Cytokines” array corresponds to the adjacent row in the “Condition” array. Arrays labeled “Training” were used to identify RS-HDMR component functions, which then served as FEOMs to predict network behavior in the “Test” arrays.
Figure 7.
Dream Challenge Prediction Accuracy.
Scatter plots describing the observed vs. RS-HDMR predicted values of the training (A,C,E) and test (B,D,F) data from the three Dream challenges. (A,B) Dream3 challenge phosphoprotein prediction. (C,D) Dream3 challenge cytokine release prediction. (E,F) Dream4 phosphoprotein prediction.
Table 4.
Dream4 Signaling Prediction Results.