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.
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