How do I computationally predict the binding position of a DNA binding protein on the promoters of differentially expressed genes?
You could run FIMO on your entire genome for TFs (transcription factors; DNA-binding proteins) of interest, which gives you binding sites: genomic intervals where those TFs bind.

https://bioinformatics.stackexchange.com/a/2491/776

You can then map or intersect promoter regions with those TF binding sites with BEDOPS bedmap or bedops, respectively:

https://bedops.readthedocs.io/en/latest/content/reference/statistics/bedmap.html

https://bedops.readthedocs.io/en/latest/content/reference/set-operations/bedops.html

The use of bedmap is recommended to assign TFs to a promoter of interest. This tool will return both the promoter region and any TFs that overlap it by the specified overlap criteria.