Protein-nucleic acid interactions exhibit varying degrees of specificity. Relatively high affinity, sequence-specific interactions, can be studied with structure determination, but lower affinity, non-specific interactions are also of biological importance. We report simulations that predict the population of nucleic acid paths around protein surfaces, and give binding constant differences for changes in the protein scaffold. The method is applied to the non-specific component of interactions between eIF4Es and messenger RNAs that are bound tightly at the cap site. Adding a fragment of eIF4G to the system changes both the population of mRNA paths and the protein-mRNA binding affinity, suggesting a potential role for non-specific interactions in modulating translational properties. Generally, the free energy simulation technique could work in harness with characterized tethering points to extend analysis of nucleic acid conformation, and its modulation by protein scaffolds.