Final answer:
The statement is false; in prokaryotes, gene regulation usually occurs near the promoter site, not thousands of nucleotides away.
Step-by-step explanation:
The statement Most gene regulatory proteins in prokaryotes can act even when they are bound to DNA thousands of nucleotide pairs away from the promoter that they influence is False. In prokaryotic cells, gene regulation typically happens at or near the promoter site of the gene they are regulating. Repressors, activators, and inducers play a key role in gene regulation by binding to specific DNA sites adjacent to the genes they control. Repressors and activators regulate gene expression by binding to operator regions and promoter sites respectively, which are usually in close proximity to the genes they regulate. Unlike eukaryotic cells where transcription factors can influence genes from distant regions through DNA looping, prokaryotic gene regulation is more local and does not involve regulatory actions from thousands of nucleotides away.