A system was devised that enables quantitative, ligand-dependent exponential amplification for various ligands that can be recognized by an RNA aptamer. The aptamer is linked to an RNA enzyme that catalyzes the joining of two oligonucleotide substrates. The product of this reaction is another RNA enzyme that undergoes self-sustained replication at constant temperature, increasing in copy number exponentially. The concentration of the ligand determines the amount of time required for the replication products to reach a threshold concentration. A standardized plot of time to threshold versus ligand concentration can be used to determine the concentration of ligand in an unknown sample. This system is analogous to quantitative polymerase chain reaction (PCR), linking rare recognition events to subsequent exponential amplification, but unlike PCR can be applied to the quantitative detection of non-nucleic acid ligands.