Quantifying fluctuations in reversible enzymatic cycles and clocks

Biochemical reactions are fundamentally noisy at a molecular scale. This limits the precision of reaction networks, but it also allows fluctuation measurements that may reveal the structure and dynamics of the underlying biochemical network. Here, we study nonequilibrium 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.

Figure 1

State transition diagrams for three related Markov chains with $N=4$. (a) The circular reversible Markov chain $C\left(t\right)$ with state space $\left\{0,1,...,N-1\right\}$ and transition rates $k_{i\pm }$. We assume there is a drift in the clockwise direction. The average period can be defined as the reciprocal of the steady-state flux. (b) The circular chain can be unfolded onto the integer line, resulting in the Markov chain $Z\left(t\right)\in \mathbb{Z}$ with initial state $Z\left(0\right)=0$. The stochastic period corresponds to the first passage time at state $N$, $T_{0,N}$, which has a well-defined probability distribution because of the forward drift. The winding number $W\left(t\right)$ counts the net number of completed cycles of the particle. (c) To study the backward fluctuations, it is helpful to consider the current position $Z\left(t\right)$ minus the maximum value of the winding number so far, defined as the stochastic variable $V\left(t\right)\in {\mathbb{Z}}_{<N}$. The maximum winding number can be increased by 1 with a rate $k_{N-1+}$ from the state $N-1$, causing $V\left(t\right)$ to be reset to 0. The winding number relative to its maximum value is given by $\mathrm{\Delta }\left(t\right)$.

Figure 2

Example time traces of the current cycle number of the particle [winding number $W\left(t\right)]$, and the furthest position so far [maximum $M\left(t\right)]$. The data are produced by kinetic Monte Carlo simulations of a system with $N=5$ states per cycle and uniform rates, such that on average one forward substep is made per $1.0\mathrm{s}$ and one backward substep per $1.5\mathrm{s}$. As indicated in the main text, the stochastic period $T_N=T_{0,N}$ corresponds to the interval between two increments of $M\left(t\right)$, and it has an average of $15\mathrm{s}$ in this example.

Figure 3

Dissipation suppresses fluctuations. The randomness parameter is calculated in the optimal case of uniform rates as a function of the number of steps per cycle $N$ and the dissipation per cycle $\mathrm{\Delta }F$. It is evident that for low dissipation, increasing the number of steps cannot lower the randomness parameter. However, when the free-energy drop per cycle becomes large ($\mathrm{\Delta }F\ge 2N{k}_{B}T$), $N$ becomes limiting and the precision increases only with increasing the number of substeps. The dotted lines represent the final inequality in Eq. (25) for the case $N=5$.