Final answer:
The statement is False. Eukaryotic gene expression involves complex interactions between transcription factors, which include repressors and activators, and does not solely depend on competition with RNA polymerase.
Step-by-step explanation:
The statement 'Like bacterial repressors, eukaryotic gene expression proteins act by directly competing with the RNA polymerase for access to the DNA' is False. In eukaryotic cells, gene expression regulation involves a complex system of transcription factors that do not simply compete with RNA polymerase for binding sites on the DNA. Instead, these transcription factors can include both activators and repressors. Repressors can bind to specific DNA regions such as promoters or enhancers and block transcription by preventing the binding of activators necessary for the recruitment of RNA polymerase. Eukaryotic gene expression is regulated by various mechanisms, such as chromatin remodeling, transcription factor binding, and mRNA transcript degradation, offering a nuanced level of control that extends beyond the actions of repressors alone.