Oxygen-reducing enzyme electrodes are prepared from laccase of Trametes versicolor and a series of osmium-based redox polymer mediators covering a range of redox potentials from 0.11 to 0.85 V. Experimentally obtained current density generated by the film electrodes is analyzed using a one-dimensional numerical model to obtain kinetic parameters. The bimolecular rate constant for mediation is found to vary with mediator redox potential from 250 s(-1) M-1 when mediator and enzyme are close in redox potential to 9.4 x 10(4) s(-1) M-1 when the redox potential difference is large. The value of the bimolecular rate constant for the simultaneously occurring laccase-oxygen reaction is found to be 2.4 x 10(-5) s(-1) M-1. The relationship between mediator-enzyme overpotential and bimolecular rate constant is used to determine the optimum mediator redox potential for maximum power output of a hypothetical biofuel cell with a planar cathode and a reversible hydrogen anode. For laccase of T. versicolor (E-e(o) = 0.82), the optimum mediator potential is 0.66 V (SHE), and a molecular structure is presented to achieve this result.