Interaction of a protein molecule with a specific-site on the DNA lattice can be modeled as an unbiased random jump process. Here we show that there exists a critical jump size (kc) beyond which site-specific association of a protein molecule with a DNA lattice cannot be facilitated. The maximum achievable association rate is predicted to be ∼1010 mol- 1 s- 1. This critical jump size scales with the total length of DNA lattice (N) as kc ∝ N2 / 3. Beyond kc the mean first passage time MFPT (denoted as T) required for the protein molecule to target the specific-site follows a linear scaling law as T ∝ N rather than the usual T ∝ N2 scaling law. On the basis of these results we argue that the evolution of the super coiled structures of the genomic DNA must be a consequence of the existence of this critical jump sizes. We finally show that the random jump method of searching for the specific-site by the protein molecule on the DNA lattice itself introduce an abstract linear type potential favoring the site-specific association rate.