Combined mathematical methods in the description of the FoF1-ATP synthase catalytic cycle

•ATP synthase mathematical modelling.•Theoretical description of ATP synthesis and hydrolysis.•Computer ATP synthase simulator.

The FoF1-ATP synthase is one of the key enzymes in supplying energy production in almost all living systems. In this paper, we provide a theoretical description of its catalytic cycle using combined mathematical methods. These methods include Langevin dynamics for the rotation of the central protein core and the Monte–Carlo method to model nucleotide and proton binding. This model is the first in which ATP synthesis and hydrolysis can occur depending on the nucleotide concentration and system conditions. The main advantage of the presented model is the possibility of obtaining results for both single-molecular protein-machines and large ensembles of proteins. The calculated rates are close to the experimentally measured rates for a single enzyme. The model has been formalised as a computer simulation that allows researchers to evaluate ATP production in different types of living cells.