We develop a lattice model of site-specific DNA-protein interactions under in vivo conditions where DNA is modelled as a self-avoiding random walk that is embedded in a cubic lattice box resembling the living cell. The protein molecule searches for its cognate site on DNA via a combination of three dimensional (3D) and one dimensional (1D) random walks. Hopping and intersegmental transfers occur depending on the conformational state of DNA. Results show that the search acceleration ratio (= search time in pure 3D route/search time in 3D and 1D routes) asymptotically increases towards a limiting value as the dilution factor of DNA (= volume of the cell/the volume of DNA) tends towards infinity. When the dilution ratio is low, then hopping and intersegmental transfers significantly enhance the search efficiency over pure sliding. At high dilution ratio, hopping does not enhance the search efficiency much since under such situation DNA will be in a relaxed conformation that favors only sliding. In the absence of hopping and intersegmental transfers, there exists an optimum sliding time at which the search acceleration ratio attains a maximum in line with the current theoretical results. However, existence of such optimum sliding length disappears in the presence of hopping. When the DNA is confined in a small volume inside the cell resembling a natural cell system, then there exists an optimum dilution and compression ratios (= total cell volume/volume in which DNA is confined) at which the search acceleration factor attains a maximum especially in the presence of hopping and intersegmental transfers. These optimum values are consistent with the values observed in the Escherichia coli cell system. In the absence of confinement of DNA, position of the specific binding site on the genomic DNA significantly influences the search acceleration. However, such position dependent changes in the search acceleration ratio will be nullified in the presence of hopping and intersegmental transfers especially when the DNA is confined in a small volume that is embedded in an outer cell.