  The first passage times for enzymatic turnovers in non-equilibrium steady
state display a statistical symmetry property related to non-equilibrium
fluctuation theorems, that makes it possible to extract the chemical driving
force from single molecule trajectories in non-equilibrium steady state. Below,
we show that this system violates the general expectation that the number of
decay constants needed to fit a first passage time distribution reflects the
number of states in the escape problem. In fact, the structure of the kinetic
mechanism makes half of the decay times vanish identically from the turnover
time distribution. The terms that cancel out correspond to the eigenvalues of a
certain sub-matrix of the master equation matrix for the first exit time
problem. We discuss how these results make modeling and data analysis easier
for such systems, and how the turnovers can be measured.
