The kinetic mechanism of NTP binding and hydrolysis by the Escherichia coli replicative helicase, the DnaB protein, in the absence and presence of the single-stranded DNA (ssDNA), has been quantitatively examined using the rapid quench-flow technique, under single-turnover conditions. In the case of both the free helicase and the enzyme-ssDNA complexes, the mechanism is independent of the type of base of the cofactor or the DNA; the bimolecular association is followed by the reversible chemical hydrolysis and subsequent conformational transition of the enzyme-product complex. The NTP hydrolysis step is significantly faster for the purine than for the pyrimidine cofactor, both in the absence and in the presence of the DNA. The temperature effect indicates that the nature of intermediates of the purine nucleotide, ATP, is different from the nature of the analogous intermediates of the pyrimidine nucleotide, CTP. Nevertheless, both types of cofactors seem to approach a similar "exit" state at the end of the reaction. The effect of ssDNA on the kinetics of NTP hydrolysis depends on the type of nucleotide cofactor and the base composition of the DNA and is centered at the hydrolysis step. Homoadenosine ssDNA oligomers are particularly effective in increasing the hydrolysis rate. The allosteric signal from the DNA, which activates the NTP hydrolysis, comes predominantly from the strong DNA-binding subsite. The role of the weak DNA-binding subsite is to modulate the allosteric effect of the strong subsite. The significance of these results for the mechanism of the free energy transduction by the DnaB helicase is discussed.