A self-consistent probabilistic formulation for inference of interactions

Abstract

Large molecular interaction networks are nowadays assembled in biomedical researches along with important technological advances. Diverse interaction measures, for which input solely consisting of the incidence of causal-factors, with the corresponding outcome of an inquired effect, are formulated without an obvious mathematical unity. Consequently, conceptual and practical ambivalences arise. We identify here a probabilistic requirement consistent with that input, and find, by the rules of probability theory, that it leads to a model multiplicative in the complement of the effect. Important practical properties are revealed along these theoretical derivations, that has not been noticed before.

Figure 1

Diverging performance of two measures applied to the same interaction data. In the middle plot, genetic interaction for each pair of gene is computed with the measures ${\epsilon }_{\text{M}}={\lambda }_{11}-{\lambda }_{01}{\lambda }_{10}$ (X axis), and ${\epsilon }_{\text{A}}={\lambda }_{11}+1-{\lambda }_{01}-{\lambda }_{10}$ (Y axis). The color code corresponds to the bar-chart at the upper left. The parallel lines indicate the standard deviation limits, 0.083 and 0.092, for εM and εA, respectively. The count of the pairs per each category is shown in a logarithm scale in the bar-chart. The scheme at the upper right show a typical four genomes set from where interaction data are obtained for a given pair of genes. Gray and black segment of the genome denote respectively the wildtype and perturbed variant of the genes A and B. The growth rates of the corresponding yeast isogenic cultures, ${\lambda }_{01}$ and ${\lambda }_{10}$ corresponds to the single mutants, and ${\lambda }_{11}$ to the double mutant. The histograms at the bottom right show the distribution along the magnitude of interaction computed with measures εM (red profile) and εA (blue profile). The inset zooms the tail farther than one standard deviation toward the right tail.

Figure 2

Sketch of the interaction scenarios. A and B are observed factors, Z accounts for unobserved factors and process, and E is the effect of interest. (a) Schematic representation of our limited information. The shaded area encloses the unobservable mechanism and background agents. Only factors A and B and the effect are observable. (b) No-interaction scenario. (c) Interaction scenario (there is a cross-over of the pathways from factors leading to the effect). (Draw with PowerPoint).

Figure 3

Distribution of # of interactors per genes in the Essential × Essential library. (a) At each gene, three dots (red, blue and green) are located according to the number of interactors obtained by εM, εA and by both, respectively. (b) Number of interactors (Y axis) vs. the average interaction score per genes (X axis). Red dots are computed with εM and blue dots with εA. The red and blue vertical lines are the two standard deviation limits, respectively. (c) Comparison of the interactions scores εM and εA for candidate hubs of Table 2. (d) and (e) Interaction network of the candidates’ hubs of Table 2, including the connections between the interactors. The hubs are located in the middle ring with larger dots. The interactors that are not connected to more than one hub are in the outer ring. The rest of interactors are in the inner ring. The hub-connections has the same color of the corresponding hubs. The other connections are in light gray. (d) Interaction network as computed by εM. (e) Interaction network as computed by εA. The dot colors are consistently used in (c–e).

Figure 4

Assessing association of non-causative factor A and causative factor C with the effect E. Unknown mechanism U force the correlation between A and C. (a) Spurious association of factors A to the effect E, where factors C is not observed. (b) Causative association of factors C to the effect E, where factors A is not observed. (Draw with PowerPoint).

Figure 5

Interaction data of binary factors dichotomous effect E. Unknown mechanism U force the correlation between A and C. Factors B and C are jointly causative, while A is not causative. (a) Assessing the interaction of factors A and B, where factors C is not observed. (b) Assessing the interaction of factors A and C, where factors A is not observed. (Draw with PowerPoint).