  In gene expression, various kinds of proteins need to bind to specific locus
of DNA. It is still not clear how these proteins find their target locus. In
this study, the mean first-passage time (FPT) of protein binding to its target
locus on DNA chain is discussed by a chain-space coupled model. Our results
show that the 1-dimensional diffusion constant has a critical value, with which
the mean time spent by a protein to find its target locus is almost independent
of the binding rate of protein to DNA chain and the detachment rate from DNA
chain. Which implies that, the frequency of protein binding to DNA and the
sliding time on DNA chain have little influence on the search efficiency, and
therefore whether or not the 1-dimensional sliding on DNA chain increases the
search efficiency depends on the 1-dimensional diffusion constant of the
protein on DNA chain. This study also finds that only protein bindings to DNA
loci which are close to the target locus help to increase the search
efficiency, while bindings to those loci which are far from the target locus
might delay the target binding process. As expected, the mean FPT increases
with the distance between the initial position of protein in cell space and its
target locus on DNA chain. The direct binding probability, which can be
regarded as one index to describe if the 1-dimensional sliding along DNA chain
is helpful to increase the search efficiency is calculated. Our results show
that the influence of 1-dimensional sliding along DNA chain on the search
process depends on both diffusion constants of protein in cell space and on the
1-dimensional DNA chain.
