  The diffusive arrival of transcription factors at the promoter sites on the
DNA sets a lower bound on how accurately a cell can regulate its protein
levels. Using results from the literature on diffusion-influenced reactions, we
derive an analytical expression for the lower bound on the precision of
transcriptional regulation. In our theory, transcription factors can perform
multiple rounds of 1D diffusion along the DNA and 3D diffusion in the cytoplasm
before binding to the promoter. Comparing our expression for the lower bound on
the precision against results from Green's Function Reaction Dynamics
simulations shows that the theory is highly accurate under biologically
relevant conditions. Our results demonstrate that, to an excellent
approximation, the promoter switches between the transcription-factor bound and
unbound state in a Markovian fashion. This remains true even in the presence of
sliding, i.e. with 1D diffusion along the DNA. This has two important
implications: (1) minimizing the noise in the promoter state is equivalent to
minimizing the search time of transcription factors for their promoters; (2)
the complicated dynamics of 3D diffusion in the cytoplasm and 1D diffusion
along the DNA can be captured in a well-stirred model by renormalizing the
promoter association and dissociation rates, making it possible to efficiently
simulate the promoter dynamics using Gillespie simulations. Based on the recent
experimental observation that sliding can speed up the promoter search by a
factor of 4, our theory predicts that sliding can enhance the precision of
transcriptional regulation by a factor of 2.
