Reduction of dimensionality in a diffusion search process and kinetics of gene expression

In order to activate a gene in a DNA molecule a specific protein (transcription factor) has to bind to the promoter of the gene. We formulate and partially answer the following question: how much time does a transcription factor, which activates a given gene, need in order to find this gene inside the nucleus of a cell? The estimate based on the simplest model of diffusion gives a very long time of days. We discuss various mechanisms by which the time can be reduced to seconds, in particular, the reduction of dimensionality, in which diffusion takes place, from three-dimensional space to two-dimensional space. The potential needed to keep the diffusing particle in 2D (i.e, at the surface of size L2 in a volume of size L3) should scale as U kBT ln L. For aL=1 μm and a target size a=10 Å we find U=8kBT, i.e., it is a potential strength of the order of the strength of ionic interactions in water.