A mathematical model of a biphasic DNA amplification reaction

Abstract

Isothermal DNA amplification reactions have many applications ranging from analyte detection to DNA circuits. EXPonential Amplification Reaction (EXPAR) is a popular isothermal DNA amplification method that exponentially amplifies short DNA oligonucleotides. A recent modification of this technique using an energetically stable looped template with palindromic binding regions demonstrated unexpected biphasic amplification and much higher DNA yield than EXPAR. This Ultrasensitive DNA Amplification Reaction (UDAR) shows high-gain, switch-like DNA output from low concentrations of DNA input. Here we present the first mathematical model of UDAR based on four reaction mechanisms. We show that the model can reproduce the experimentally observed biphasic behavior. Furthermore, we show that three of these mechanisms are necessary to reproduce biphasic experimental results. The reaction mechanisms are (i) positively cooperative multistep binding caused by two palindromic trigger binding sites on the template; (ii) gradual template deactivation; (iii) recycling of deactivated templates into active templates; and (iv) polymerase sequestration. Understanding of these mechanisms also illuminates behavior of EXPAR and other nucleic acid amplification reactions. For a deeper understanding of the roles these mechanisms play in DNA amplification reactions, we apply dynamical systems analysis to the model. We first consider the long term behavior of partial models that lack key reaction mechanisms described above to see how their omission impacts the system's overall behavior. Then we use perturbation theory to examine the time scales on which these mechanisms operate and how their interaction leads to biphasic growth. We find that mechanisms (i) and (ii) together create a stable equilibrium reminiscent of EXPAR reactions, but the addition of mechanism (iii) changes the stability of this equilibrium and generates UDAR's characteristic high amplification. Finally, mechanism (iv) introduces a second stable equilibrium that indicates that polymerase sequestration is the mechanism that ends the second fast amplification phase. In addition, throughout this work we identify which rate constants shape different parts of the biphasic growth. These results can guide future work in rational design of molecular detection assays.