  Biochemical reactions are fundamentally noisy at a molecular scale. This
limits the precision of reaction networks, but also allows fluctuation
measurements which may reveal the structure and dynamics of the underlying
biochemical network. Here, we study non-equilibrium reaction cycles, such as
the mechanochemical cycle of molecular motors, the phosphorylation cycle of
circadian clock proteins, or the transition state cycle of enzymes.
Fluctuations in such cycles may be measured using either of two classical
definitions of the randomness parameter, which we show to be equivalent in
general microscopically reversible cycles. We define a stochastic period for
reversible cycles and present analytical solutions for its moments.
Furthermore, we associate the two forms of the randomness parameter with the
thermodynamic uncertainty relation, which sets limits on the timing precision
of the cycle in terms of thermodynamic quantities. Our results should prove
useful also for the study of temporal fluctuations in more general networks.
