Time-varying causal inference from phosphoproteomic measurements in macrophage cells

Abstract

Cellular signaling circuitry in eukaryotes can be studied by analyzing the regulation of protein phosphorylation and its impact on downstream mechanisms leading to a phenotype. A primary role of phosphorylation is to act as a switch to turn "on" or "off" a protein activity or a cellular pathway. Specifically, protein phosphorylation is a major leit motif for transducing molecular signals inside the cell. Errors in transferring cellular information can alter the normal function and may lead to diseases such as cancer; an accurate reconstruction of the "true" signaling network is essential for understanding the molecular machinery involved in normal and pathological function. In this study, we have developed a novel framework for time-dependent reconstruction of signaling networks involved in the activation of macrophage cells leading to an inflammatory response. Several signaling pathways have been identified in macrophage cells, but the time-varying causal relationship that can produce a dynamic directed graph of these molecules has not been explored in detail. Here, we use the notion of Granger causality, and apply a vector autoregressive model to phosphoprotein time-course data in RAW 264.7 macrophage cells. Through the reconstruction of the phosphoprotein network, we were able to estimate the directionality and the dynamics of information flow. Significant interactions were selected through statistical hypothesis testing ( t-test) of the coefficients of a linear model and were used to reconstruct the phosphoprotein signaling network. Our approach results in a three-stage phosphoprotein network that represents the evolution of the causal interactions in the intracellular signaling pathways.

Fig. 1

Schematic to show the stacking of the data matrices. Each column corresponds to the time series data of each of the k variables.

Fig. 2

Heat-map of the correlation matrix between the input and output variables. This matrix contains the pairwise correlation coefficient between columns of matrix X and Y for the whole time series [–10] minutes.

Fig. 3

The reconstructed network for the underlying signaling network in RAW 264.7 macrophages. This network represents the cross-talk between phosphoproteins considering the whole time-series for [1]–[10] minute period. The pink connections are common edges in the underlying network and the timevarying network (Fig. 5). Different edge-widths are used to represent low (0.4 ≤ r < 0.5), medium (0.5 ≤ r < 0.75) and high (r ≥ 0.75) correlation coefficients corresponding to the edges. Inhibitory connections are shown with a blunt end instead of an arrow.

Fig. 4

Histogram of the p-values (t-test on the model coefficients) for the underlying network generated from 17×17 p-value numbers.

Fig. 5

Time-dependent cascade of the phosphoprotein signaling network in RAW 264.7 macrophages in three stages, (a) Reconstructed network in stage 1 related to [1]–[4] minute interval. (b) Reconstructed network in stage 2 related to [3]–[7] minute interval, (c) Reconstructed network in stage 3 related to [6]–[10] minute interval. The pink connections are common to all the three networks as well as the underlying network (Fig. 3). Different edge-widths are used to represent low (0.4 ≤ r < 0.5), medium (0.5 ≤ r < 0.75) and high (r ≥ 0.75) correlation coefficients corresponding to the edges. Inhibitory connections are shown with a blunt end instead of an arrow.