This invention covers processes for the isothermal amplification of DNA molecules having a preselected sequence. It is based on the unexpected discovery that primers having, at some positions, adenine substituted by 2-aminopurine or diaminopurine, guanine by inosine, thymine by 2-thiothymine, and cytosine by N4-ethylcytosine (“SAMRS nucleotides”) were accepted by enzymes used in the standard helicase-dependent amplification (HDA). Further unexpected was the discovery that target nucleotides are efficiently amplified in an HDA-like process (hereinafter abbreviated as simply HDA) using substituted primers. Also discovered was the diminution of spurious products through the use of SAMRS-substituted primers.