Identifying the sources of noise synergy and redundancy in the gene expression of feed-forward loop motif

The propagation of noise through parallel pathways is a characteristic feature of feed-forward loops (FFLs) in genetic networks. Although the contributions of the direct and indirect pathways to output variability have been well characterized, the impact of their joint action arising from their shared input and output remains poorly understood. Here, we identify a cross-interaction noise emerging specifically from this pathway convergence. Using inter-gene correlations, we reveal the regulatory basis of the cross-interaction noise and interpret it as synergy or redundancy in noise propagation. Positive values of cross-interaction noise reflect synergy (noise amplification), while negative values reflect redundancy (noise suppression); a zero value indicates that the parallel pathways act independently. Synergy typically arises in coherent FFLs, whereas redundancy is common in incoherent ones. To quantify this effect, we introduce relative synergy noise, a dimensionless quantity, which captures the magnitude and sign of synergy and redundant noise relative to other noise sources. Further, by systematically tuning intrinsic noise strengths through effective gene expression burst, we find that when the intermediate node exhibits the highest intrinsic noise, it results in a relative synergy noise value approaching zero, indicating pathway independence. In contrast, when intrinsic noises follow a hierarchy in which the input is the most noisy, the intermediate is the least noisy, and the output is in between them, FFLs exhibit the strongest synergy in coherent motifs and the strongest redundancy in incoherent motifs. Furthermore, by relating these synergies and redundancies to dynamical properties such as sign-sensitive delay or response acceleration, the framework offers a statistical lens to interpret the functional roles in cellular decision-making. Our framework, thus, advances the mechanistic understanding of noise propagation in FFLs by quantifying pathway coupling as a measurable and biologically interpretable quantity.