8-17 and 10-23 are the two most comprehensively studied RNA-cleaving DNAzymes to date and have the ability to carry out sequence-specific cleavage of both all-RNA or chimeric RNA/DNA substrates. Mutagenesis studies of 8-17 and 10-23 DNAzymes using alternative natural nucleotides to substitute a given nucleotide in the DNAzyme sequence have found that both DNAzymes are able to tolerate a variety of alterations at many sequence locations. Chemical modification studies employing nucleotides containing nonnatural nucleobases have led to findings that some specific entities of selected nucleobases are irreplaceable by other functional groups. In this work, we set out to carry out a mutagenesis study on both 8-17 and 10-23 by substituting individual nucleotides in their catalytic cores with a baseless (abasic) nucleotide or a baseless/sugarless nucleotide containing only acyclic C3 spacer. We observed that the substitution with an abasic nucleotide or C3 spacer at many locations within the catalytic core of both 8-17 and 10-23 was still able to support a significant level of catalytic activity of each DNAzyme, suggesting that both DNAzymes have considerable structural plasticity to maintain their catalytic functions. We also observed that almost all nucleobases in the catalytic core of each DNAzyme appeared to make either an absolutely essential contribution to the function of each DNAzyme or exhibit a "chaperone-like" activity that is important for the optimal function of each DNAzyme; in contrast, only one sugar ring in 8-17 and four in 10-23 were inferred to make some contribution to the optimal function of the relevant DNAzyme. Finally, our study also raised a possibility that the 10-23 DNAzyme might be a special structural variant of the larger 8-17 DNAzyme family.