Decay times in turnover statistics of single enzymes

The first passage times for enzymatic turnovers in nonequilibrium steady state display a statistical symmetry property related to nonequilibrium fluctuation theorems, which makes it possible to extract the chemical driving force from single molecule trajectories in nonequilibrium steady state. Below, we show that the number of decay constants needed to describe the first passage time distribution of this system is not equal to the number of states in the first passage problem, as one would generally expect. Instead, 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 submatrix 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.

Figure 1

(a) A multistep enzyme reaction converting A to B. The turnover time distributions $w_{\pm }\left(t\right)$ are given by the first exit time problem in (b). The enzyme starts in state $E_0$ and is absorbed in $E_n$ or $E_{-n}$. The rate constants $m_{10}$ and $m_{n-1,n}$ are pseudo-first-order, i.e., proportional to the concentrations of A and B, respectively. In the more general scheme studied in Ref. 3, arbitrary transitions within a cycle are allowed. An example of this, with $n=5$ different enzymatic states, is illustrated in (c). By periodicity, state $E_k$ is equivalent to $E_{k+n}$, and $m_{ij}=m_{i+n,j+n}$.