This paper presents a method for controlling DNA strand displacement kinetics using toehold exchange. The authors derive the BM (branch migration) rate constant using a quasi-steady state (QSS) approximation, which assumes that intermediate concentrations change slowly compared to the overall reaction. They validate the QSS assumption by comparing the timescale of the overall reaction with that of intermediate equilibration, finding that the reaction timescale is much longer than the intermediate equilibration time. The BM rate constant is derived from a three-step model of toehold exchange, and the authors show that the derived rate constant accurately models the kinetics under certain conditions. The authors also calculate the critical concentration below which the BM rate constant remains a valid predictor of kinetics. They find that when the concentrations of intermediates I and J are low, the BM rate constant accurately models the kinetics. However, when these concentrations are high, the BM rate constant overestimates the kinetics. The critical concentration is calculated based on the ratio of intermediate concentrations to the initial concentrations of the reactants. The authors also calculate the binding energies of the toehold using thermodynamic data. They use a step-by-step method to calculate the standard free energy of two complexes, S and I(0,3), which is used to infer the binding energy of the toehold. They compare different energy models, including those based on Pyshnyi, NUPACK, and Owczarzy, and find that the binding energy of the toehold is approximately -2.95 kcal/mol. The authors also provide a Matlab script for calculating the BM rate constant and critical concentration based on toehold energies, branch migration length, temperature, and energy model. The script uses different energy models to calculate the BM rate constant and critical concentration for different toehold exchange reactions. The results show that the BM rate constant and critical concentration depend on the energy model and the parameters of the toehold exchange reaction.This paper presents a method for controlling DNA strand displacement kinetics using toehold exchange. The authors derive the BM (branch migration) rate constant using a quasi-steady state (QSS) approximation, which assumes that intermediate concentrations change slowly compared to the overall reaction. They validate the QSS assumption by comparing the timescale of the overall reaction with that of intermediate equilibration, finding that the reaction timescale is much longer than the intermediate equilibration time. The BM rate constant is derived from a three-step model of toehold exchange, and the authors show that the derived rate constant accurately models the kinetics under certain conditions. The authors also calculate the critical concentration below which the BM rate constant remains a valid predictor of kinetics. They find that when the concentrations of intermediates I and J are low, the BM rate constant accurately models the kinetics. However, when these concentrations are high, the BM rate constant overestimates the kinetics. The critical concentration is calculated based on the ratio of intermediate concentrations to the initial concentrations of the reactants. The authors also calculate the binding energies of the toehold using thermodynamic data. They use a step-by-step method to calculate the standard free energy of two complexes, S and I(0,3), which is used to infer the binding energy of the toehold. They compare different energy models, including those based on Pyshnyi, NUPACK, and Owczarzy, and find that the binding energy of the toehold is approximately -2.95 kcal/mol. The authors also provide a Matlab script for calculating the BM rate constant and critical concentration based on toehold energies, branch migration length, temperature, and energy model. The script uses different energy models to calculate the BM rate constant and critical concentration for different toehold exchange reactions. The results show that the BM rate constant and critical concentration depend on the energy model and the parameters of the toehold exchange reaction.