= Solution
An ATP-binding step with pseudo-first-order rate $k_1c$ has <survival probability> $S_1=e^{-k_1ct}$ and waiting <density> $P_1=-S_1'=k_1ce^{-k_1ct}$. It decreases from a finite value at zero, integrates to one and has mean $1/(k_1c)$. Its concentration dependence suggests binding-limited waiting before the large 80-degree substep.
Two independent sequential exponential steps of rates $k_a,k_b$ give the convolution
$$
\boxed{P_2(t)=\int_0^t k_ae^{-k_as}k_be^{-k_b(t-s)}ds=\frac{k_ak_b}{k_a-k_b}(e^{-k_bt}-e^{-k_at}).}
$$
It begins at zero, rises to a maximum at $\log(k_a/k_b)/(k_a-k_b)$ and decays; its mean is $k_a^{-1}+k_b^{-1}$. At equal rates it becomes $k^2te^{-kt}$. This supports at least two kinetic events during the waiting associated with the 40-degree substep, plausibly chemical transformation/product release and conformational changes. It does not uniquely identify which rate belongs to hydrolysis or a particular product: different hidden-state networks can have the same waiting distribution. Three 120-degree <Catalytic cycle>[catalytic cycles] per revolution, partitioned into 80 and 40 degrees in the assay, are consistent with three catalytic sites.
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