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Here we used a series of CTCF mutations to explore CTCF's relationship with chromatin and its contribution to gene regulation. CTCF's impact depends on the genomic context of bound sites and the unique binding properties of WT and mutant CTCF proteins. Specifically, CTCF's signal strength is linked to changes in accessibility, and the ability to block cohesin is linked to its binding stability. Multivariate modeling reveals that both CTCF and accessibility contribute independently to cohesin binding and insulation, but CTCF signal strength has a stronger effect. CTCF and chromatin have a bidirectional relationship such that at CTCF sites, accessibility is reduced in a cohesin-dependent, mutant-specific fashion. In addition, each mutant alters TF binding and accessibility in an indirect manner, changes which impart the most influence on rewiring transcriptional networks and the cell's ability to differentiate. Collectively, the mutant perturbations provide a rich resource for determining CTCF's site-specific effects.