Without exceptions, genomic DNA of living organisms is replicated using the leading and the lagging strand. In a conventional idea of mutagenesis accompanying DNA replication, mutations are thought to be introduced stochastically and evenly into the two daughter DNAs. Here, however, we hypothesized that the fidelity of the lagging strand is lower than that of the leading strand. Our simulations with a simplified model DNA clearly indicated that, even if mutation rates exceeded the so-called threshold values, an original genotype was guaranteed in the pedigree and, at the same time, the enlargement of diversity was attained with repeated generations. According to our lagging strand-biased-mutagenesis model, mutator microorganisms were established in which mutations biased to the lagging strand were introduced by deleting the proofreading activity of DNA polymerase. These mutators (disparity mutators) grew normally and had a quick and extraordinarily high adaptability against very severe circumstances. From the viewpoint of the fidelity difference between the leading and the lagging strand, the basic conditions for the acceleration of evolution are examined. The plausible molecular mechanism for the faster molecular clocks observed in birds and mammals is discussed, with special reference to the accelerated evolution in the past. Possible applications in different fields are also discussed.