  A growing amount of evidence points to the fact that many enzymes exhibit
fluctuations in their catalytic activity, which are associated with
conformational changes on a broad range of timescales. The experimental study
of this phenomenon, termed dynamic disorder, has become possible due to
advances in single-molecule enzymology measurement techniques, through which
the catalytic activity of individual enzyme molecules can be tracked in time.
The biological role and importance of these fluctuations in a system with a
small number of enzymes such as a living cell have only recently started being
explored. In this work, we examine a simple stochastic reaction system
consisting of an inflowing substrate and an enzyme with a randomly fluctuating
catalytic reaction rate that converts the substrate into an outflowing product.
To describe analytically the effect of rate fluctuations on the average
substrate abundance at steady-state, we derive an explicit formula that
connects the relative speed of enzymatic fluctuations with the mean substrate
level. We demonstrate that the relative speed of rate fluctuations can have a
dramatic effect on the mean substrate, and lead to large positive deviations
from predictions based on the assumption of deterministic enzyme activity. Our
results also establish an interesting connection between the amplification
effect and the mixing properties of the Markov process describing the enzymatic
activity fluctuations, which can be used to easily predict the fluctuation
speed above which such deviations become negligible. As the techniques of
single-molecule enzymology continuously evolve, it may soon be possible to
study the stochastic phenomena due to enzymatic activity fluctuations within
living cells. Our work can be used to formulate experimentally testable
hypotheses regarding the magnitude of these fluctuations, as well as their
phenotypic consequences.
