Beyond element-wise interactions: identifying complex interactions in biological processes

Abstract

Background: Biological processes typically involve the interactions of a number of elements (genes, cells) acting on each others. Such processes are often modelled as networks whose nodes are the elements in question and edges pairwise relations between them (transcription, inhibition). But more often than not, elements actually work cooperatively or competitively to achieve a task. Or an element can act on the interaction between two others, as in the case of an enzyme controlling a reaction rate. We call "complex" these types of interaction and propose ways to identify them from time-series observations.

Methodology: We use Granger Causality, a measure of the interaction between two signals, to characterize the influence of an enzyme on a reaction rate. We extend its traditional formulation to the case of multi-dimensional signals in order to capture group interactions, and not only element interactions. Our method is extensively tested on simulated data and applied to three biological datasets: microarray data of the Saccharomyces cerevisiae yeast, local field potential recordings of two brain areas and a metabolic reaction.

Conclusions: Our results demonstrate that complex Granger causality can reveal new types of relation between signals and is particularly suited to biological data. Our approach raises some fundamental issues of the systems biology approach since finding all complex causalities (interactions) is an NP hard problem.

Figure 1. A schematic plot of the complex interactions.

Each time trace (node) is the activity of a gene, protein, substance etc. A circle is a complex comprising of nodes. Left panel is the interactions among nodes. Right panel, the top complex can exert its influence on the rate between two complexes (top), or on the complexes themselves (bottom).

Figure 2. Application of Complex Granger Causality to simulated data.

(A). Time series considered in Example 1. The underlying causal relationship is represented by an arrow. (B) Comparison between the time domain pairwise Granger causality and the frequency domain pairwise Granger causality. The partial complex Granger causality and its 95 confidence interval after 1000 replications are shown in blue. The summation over a frequency range of the corresponding frequency-domain formulation is shown in red. (C) the corresponding spectra in the frequency domain.

Figure 3. Weakly connected signals can have an overall significant effect.

(A) The time courses of with . (B) The average value and its confidence interval of the Granger causality in example 2 when . There are no causal relations between and , and and , but the causal relationship between and is significant. (C) The lowest value of the confidence intervals as a function of . The inset shows the increasing, as one would expect, values of and but on too small a scale to be significant.

Figure 4. Partial Granger causality discards indirect connections.

(A). Simulated time series and the underlying causal relationships considered in example 3. and are multi-dimensional. (B) Blue and red error bars are defined as in figure 2. (C) Corresponding spectra in the frequency domain. The partial complex Granger causality and its 95 confidence intervals after 1000 replications.

Figure 5. Complexes in a regulatory network.

A) Inferred regulatory network of 12 genes known to participate in the yeast cell-cycle. Thick lines (blue if the interaction is documented, green if only potential according to YEASTRACT) are correct inferences. Thin lines are wrong inferences, with a dashed line representing a missed connection and a solid line representing a wrongly attributed connection. Yellow nodes denote target factors, green nodes complexes and blue nodes target genes. B) Improvement of the connection when complexes are considered. Blue dots represent the Granger causality from one member of the complex to the target gene, red squares represent the Granger causality from the complex to the target gene. Note that this hypergraph is not to be read as a power graph ([35]) as a connection from a complex to a target gene does not imply significant interactions from each of the subset elements to the target.

Figure 6. Group causality between brain regions

A) Distribution of Granger causality between all 110 pairs of left and right signals. B) Distribution of Granger causality between region averages for each of the 40 experiments. C) Distribution of Complex Granger causality between the two regions for each of the 40 experiments. Each distribution is summarized by a boxplot showing its median (in red), as well as its first and third quartile (box). Smallest and largest values are shown with the outer bars and outliers are represented by red crosses. D) Signals from the left and right hemispheres for 5 experiments. Areas in gray denote the presence of the stimulus.

Figure 7. An enzymatic reaction

(A): Time course of the three reactant and in Example 4. The enzyme has direct influence on the reaction rate . (B): Partial Granger causality between three reactants and in Example 4. (C): Partial Granger causality from to other reactants in Example 4. (D): Partial Granger causality from to complex of in Example 4. (E): Time course of the three reactant and in Example 5. In this example, the enzyme has direct influence on S. (F): Partial Granger causality between three substance and in example 5. (G): Partial Granger causality from to other reactants and groups in example 5.

Figure 8. The role of correlation in the complex interaction.

The Granger causality vs. units (N) for different cross-correlation coefficients with a = 0.022.( is the smallest possible value for the cross-correlations of 10 units.)