A problem of substantial interest is to systematically map variation in chromatin structure to gene expression regulation across conditions, environments, or differentiated cell types. We developed and applied a quantitative framework for determining the existence, strength, and type of relationship between high-resolution chromatin structure in terms of DNaseI hypersensitivity (DHS) and genome-wide gene expression levels in 20 diverse human cell lines. We show that ~25% of genes show cell-type specific expression explained by alterations in chromatin structure. We find that distal regions of chromatin structure (e.g., +/- 200kb) capture more genes with this relationship than local regions (e.g., +/- 2.5kb), yet the local regions show a more pronounced effect. By exploiting variation across cell-types, we were capable of pinpointing the most likely hypersensitive sites related to cell-type specific expression, which we show have a range of contextual usages. This quantitative framework is likely applicable to other settings aimed at relating continuous genomic measurements to gene expression variation.