A logic-circuit framework for mapping long-range interaction networks to 3D chromatin conformations

Eukaryotic genomes self-organize into diverse three-dimensional (3D) chromatin conformations through intranuclear long-range interactions, yet it remains challenging to interpret how concurrent interactions combine along a chromatin polymer to shape measurable conformational readouts. Building on a minimal harmonic polymer model and the Gaussian covariance formalism, we present a 3D genome circuit representation that reorganizes the covariance-derived effective interaction strength (EIS) of a target locus pair into motif-level interaction patterns. In this representation, the graph topology of a chromatin interaction network determines the arrangement of circuit operations that summarize how local loop motifs contribute to EIS through series-like, parallel-like, or more general coupled-network combinations. The resulting EIS is then connected to experimentally accessible statistics, including pairwise contact probabilities and spatial-distance distributions from the Gaussian equilibrium ensemble, and loop stability through a first-passage description. Beyond single-pair readouts, we derive a closed-form Pearson correlation coefficient between two inter-locus spatial distances in a general harmonic network, providing an analytical way to quantify coordinated proximity changes between two locus pairs. We apply the framework to (1) estimate the effective Shh-ZRS interaction strength in a preformed CTCF-dependent loop configuration and (2) quantify enhancer-promoter proximity coordination in a shared-enhancer configuration. This motif-level circuit representation provides physics-grounded design rules for interpreting how local interaction topology, interaction strength, and perturbations such as anchor deletion or tether addition alter EIS and associated 3D conformational readouts.