Nucleosome positioning plays an essential role in cellular processes by modulating accessibility of DNA to proteins. Here, using only sequence-dependent DNA flexibility and intrinsic curvature, we predict the nucleosome occupancy along the genomes of Saccharomyces cerevisiae and Drosophila melanogaster and demonstrate the predictive power and universality of our model through its correlation with experimentally determined nucleosome occupancy data. In yeast promoter regions, the computed average nucleosome occupancy closely superimposes with experimental data, exhibiting a <200 bp region unfavourable for nucleosome formation bordered by regions that facilitate nucleosome formation. In the fly, our model faithfully predicts promoter strength as encoded in distinct chromatin architectures characteristic of strongly and weakly expressed genes. We also predict that nucleosomes are repositioned by active mechanisms at the majority of fly promoters. Our model uses only basic physical properties to describe the wrapping of DNA around the histone core, yet it captures a substantial part of chromatin's structural complexity, thus leading to a much better prediction of nucleosome occupancy than methods based merely on periodic curved DNA motifs. Our results indicate that the physical properties of the DNA chain, and not just the regulatory factors and chromatin-modifying enzymes, play key roles in eukaryotic transcription.