Figure 1.
Overview of abstract dynamic modeling approach.
1) Inferelator 1.1 infers a parsimonious set of potential regulatory influences whose expression can explain the expression of the target cluster maximally, but does so independently for each cluster. 2) The actual structure of the inferred network would consider that the regulators are members of clusters and that the network structure is complex and cyclical. 3) The regulatory influence model can be represented by a regulatory influence matrix and used to simulate the closed system of ordinary differential equations over time. 4) The optimization process (see text) is used to improve the ability of the model to simulate the system over time (i.e. calibrate to temporal data). 5) The resulting optimized model retains much of the structure of the initial model.
Figure 2.
Functional coherence of clusters defines functional modules.
To ascertain a reasonable number of clusters to consider in our model abstraction we calculated the normalized functional modularity (Y axis) for varying numbers of clusters (X axis) from the same hiearachical tree derived from the expression data. Functional modularity was defined as the number of genes annotated with a biological process gene ontology category that was functionally enriched in the gene's parent cluster with a p-value less than the threshold indicated (colored lines). The results show that 25 clusters provides a peak of functional modularity, especially for more coherent functional categories with lower p-values.
Table 1.
Optimization results.
Figure 3.
An Inferelator-based influence model provides statistically significant performance when treated as a system of ordinary differential equations (ODEs).
The Inferelator-based influence model was treated as a system of ODEs and simulated over time. Expression levels of the simulation were compared with observed values of expression by correlation (Y axis) for the initial model (red dot) or for 100 randomized matrices. The matrices were randomized by replacing all non-zero values with other non-zero values (resampled) or from a uniform distribution (uniform), or the locations of all values in the matrix were reassigned (scramble). The results (as a box and whiskers plot) show that the Inferelator-based initial matrix produces simulation over time with a performance that is significantly better than that using random permutations of the matrix.
Figure 4.
Performance trajectories of models during simulated annealing process.
The two-stage simulated annealing (SA) process described was applied using the Inferelator-based model as a starting matrix. The performance (Y axis) of each of the 25 models in each stage are shown over the steps (X axis) in the SA process. The results show that the optimization process can dramatically improve the performance of the initial model.
Figure 5.
The optimized model performs significantly better than randomly perturbed models.
The best optimized model was simulated over time to provide predictions of expression levels for clusters. Correlation (Y axis) of the simulated versus observed data is shown for the best optimized model (red dot) and for 100 randomized matrices (boxes). The matrices were randomized by replacing all non-zero values with other non-zero values (resampled) or from a uniform random distribution (uniform), or the locations of all values in the matrix were reassigned (scramble). The results (as a box and whiskers plot) show that the optimized model is capable of simulation over time with a performance that is significantly better than randomized versions.
Figure 6.
Expression patterns for an LPS-optimized model.
The LPS-pretreatment observed (black lines) and predicted (green lines) expression patterns for clusters containing more than five genes are shown. The expression patterns from a randomized consensus model (red lines) are also shown. The X axes indicate the log2 fold-change expression for the observed pattern and the predicted and random expression patterns were scaled to this range. The Y axis shows time from 0 to 100 hours post-pretreatment.
Table 2.
Correlation of gene expression between pretreatment time courses.
Table 3.
Performance of LPS-optimized model on individual clusters.
Figure 7.
Cross-predictive performance of an LPS-optimized model.
Relative expression levels (log2 fold change expression) are plotted over time (X axis) for the predicted (green line) and observed (black line) expression levels for the indicated cluster. Several representative clusters are shown and the remaining plots are included as Figure S3.
Table 4.
Interaction datasets validate network model.
Figure 8.
Comparison between neuroprotected and injurious networks.
Clusters containing more than 5 genes are shown (green squares) as determined by our functional clustering approach. The influences between clusters are shown as directed edges with red arrows indicating a positive influence (activation) and blue T lines indicating a negative influence (repression). Dashed lines indicate relationships that are significantly different between injurious and non-injurious conditions, either absent in one or opposite sign. General functional categories chosen from statistically enriched functions are indicated in grey boxes.