We next investigated underlying factors that determine the number of significant co-operative and antagonistic interactions for each metabolite in MPCA. The number of protein co-variates with each metabolite did not correlate with the degree of metabolite abundance variation across the DO cohort (Extended Data Fig. 3a). Instead, the number of recapitulated metabolite–protein relationships was positively associated with the number of established biochemical reactions linked to each metabolite (Extended Data Fig. 3b). This suggests that in MPCA, metabolites with higher numbers of protein correlates participate in more biological reactions, serving as substrates, products or cofactors. A prominent example is NAD⁺, which is a critical redox equivalent and electron carrier used in many biochemical reactions (Fig. 1d,e). We further analysed co-operativity at the pathway level (Extended Data Figs. 3c–i and 4a–e and Supplementary Table 4) and statistical properties of covariation derived from different forms of metabolite–protein relationships (Extended Data Fig. 4f–n). These analyses are provided in the Supplementary Discussion.