We show that the rate of site-specific association of a protein molecule of interest with the DNA chain can be ∼ 102 times higher than that of the three-dimensional diffusion-controlled collision rate limit ∼ 108 mol-1 s-1 only when the protein molecule of interest searches for its specific site on the DNA chain in a reduced dimensional space with a dimensionality dr of dr <1. Upon considering the concurrent dynamics of the linear DNA chain that is embedded in a d -dimensional space along with the one-dimensional diffusion dynamics of the nonspecifically bound protein molecule on the DNA chain, we derive the generalized scaling law ε∼ 23 (2-d) +3, where ε is the number of times by which the rate of site-specific association of the protein molecule with the DNA chain can be enhanced over the three-dimensional diffusion-controlled collision rate limit and d is the dimensionality of the reduced search space. Using the analogy between the self-intersection loop length in the theory of random walks and the ring-closure events in the theory of site specific interactions of a protein molecule with the DNA chain, we further show that the extent of packaging and volume compression of the genomic DNA inside the living cell is designed in such a way that the efficiency of the protein molecule in the process of searching for its specific site on the genomic DNA is a maximum. Our simulation results suggest that the volume compression factor θ which is the ratio between the total volume of the living cell and the volume occupied by the DNA chain along with all the other bound protein molecules should be such that θ 100 for an efficient site specific interaction of a protein molecule of interest with the linear DNA chain that is embedded in a three-dimensional space. Our theoretical and simulation results agree well with the E. coli cellular system. © 2009 The American Physical Society.