Kinetic Uncertainty Relations for the Control of Stochastic Reaction Networks

Nonequilibrium stochastic reaction networks are commonly found in both biological and nonbiological systems, but have remained hard to analyze because small differences in rate functions or topology can change the dynamics drastically. Here, we conjecture exact quantitative inequalities that relate the extent of fluctuations in connected components, for various network topologies. Specifically, we find that regardless of how two components affect each other’s production rates, it is impossible to suppress fluctuations below the uncontrolled equivalents for both components: one must increase its fluctuations for the other to be suppressed. For systems in which components control each other in ringlike structures, it appears that fluctuations can only be suppressed in one component if all other components instead increase fluctuations, compared to the case without control. Even the general $N$-component system—with arbitrary connections and parameters—must have at least one component with increased fluctuations to reduce fluctuations in others. In connected reaction networks it thus appears impossible to reduce the statistical uncertainty in all components, regardless of the control mechanisms or energy dissipation.

Figure 1

Trade-off for mutual control systems. (a) We consider the following generic class of mutual control systems: each component is assumed to decay with some (unspecified) half-life but the way in which the two components affect each other’s production rates $R_i^{+}$ is left completely unspecified, see Eq. (1). (b) Analytical results suggest that no such control system can simultaneously suppress noise in both components below their respective Poisson fluctuations (denoted by 100% in the diagram). We find numerically that the bounds derived in the high copy number regime (red lines) in fact constrain all tested systems (black dots) regardless of noise levels and copy numbers. This numerical confirmation suggests that no mutual control system—regardless of chosen control functions or parameters—can simultaneously exhibit sub-Poisson fluctuations in both components as indicated by the lack of data points in the lower left. (c) Here, we present the analytical bounds (colored lines) of Eq. (4) for different lifetimes together with exact numerical realizations of systems with nonlinear control systems (dots). The colors correspond to fixed relative lifetimes, and each dot corresponds to a different system with a different set of control functions and parameters [see Supplemental Material (SM) for numerical details [19] ]. For each ratio of relative lifetimes none of the tested systems beat the limit of Eq. (4).

Figure 2

Trade-offs between pairs of components within a feedback ring. Two-component mutual control systems can be generalized to a multicomponent control feedback loop in which $N$ components affect the production rate of the next component within a ring as defined by Eq. (6). (a) Schematic illustration of a ringlike connection feedback loop for $N=6$. (b) Numerical support that feedback rings can only suppress noise in at most one component. Considering stochastic realizations of control systems within the class defined by Eq. (6) for $N=3$ we find numerically that the bounds derived in the high copy number regime constrained all systems regardless of noise levels and copy numbers. Each dot corresponds to the numerical data for a given system with a specified set of control functions and parameters (see SM for numerical details [19]). We find numerical confirmation that no three-component feedback ring can suppress fluctuations in more than one component. Note, that lower-dimensional feedback loops are special cases of the higher-dimensional ones in which some components are infinitely fast with ${\tau }_i\to 0$. (See Movie S1 in the SM [19].)

Figure 3

Trade-off in $N$-component control networks. Here, we consider $N$-component control structures of arbitrary topology where each component is allowed to affect any other component’s production rate (but not its own) as defined in Eq. (11). (a) Schematic illustration of a maximally connected control network for $N=6$. Since we leave all reaction rates unspecified (here and throughout the Letter) any specific realization of this class may have rates that depend on only a subset of the other components and may thus be much more sparsely connected than the above illustration—in particular, the ringlike feedback systems of Fig. 2 are a subset of the more general class considered here. (b) Here, we present numerical support for the constraint of Eq. (13) in the $N=3$ case. We generated an extensive set of numerical realizations of arbitrarily nonlinear three-component feedback systems with arbitrary lifetimes and abundances. Again, we could not find any system that violated the bound derived in the high copy number limit, and conclude that the “Poisson cube” is inaccessible by any three component feedback systems.