Constraints on Fluctuations in Sparsely Characterized Biological Systems

Biochemical processes are inherently stochastic, creating molecular fluctuations in otherwise identical cells. Such “noise” is widespread but has proven difficult to analyze because most systems are sparsely characterized at the single cell level and because nonlinear stochastic models are analytically intractable. Here, we exactly relate average abundances, lifetimes, step sizes, and covariances for any pair of components in complex stochastic reaction systems even when the dynamics of other components are left unspecified. Using basic mathematical inequalities, we then establish bounds for whole classes of systems. These bounds highlight fundamental trade-offs that show how efficient assembly processes must invariably exhibit large fluctuations in subunit levels and how eliminating fluctuations in one cellular component requires creating heterogeneity in another.

Figure 1

Efficient complex assembly implies variability. We consider all reaction systems in which subunits $X_0$ and $X_1$ form stable complexes as defined in Eq. (7), where the cloud indicates that all system components can arbitrarily affect the shared production rate. The inset time traces illustrate the dramatic fluctuations in subunit levels, suggesting that small noise approximations are inaccurate. The solid red line is the exact lower bound on subunit fluctuations Eq. (9) as it applies to the average of the subunit noise levels [11] . As the efficiency approaches 100%, the subunit fluctuations diverge. We plot the average of ${\eta }_{00}$ and ${\eta }_{11}$ normalized by the noise that subunits would exhibit in the absence of feedback and complex formation. Dots indicate simulation results for different realizations of assembly processes across a range of parameters, feedback functions, and embedding networks. Many systems deviate greatly from the isolated linear noise prediction (dashed yellow line), but no system can beat the exact bound. The simulation data should not be interpreted in terms of density but are presented to illustrate what is possible and show by construction that the bound is virtually tight.

Figure 2

Effective control implies variability. (a) We consider all possible feedback control systems in which fluctuations in component $X_1$ are controlled via $X_0$ as defined in Eq. (10). The specific type of feedback control is left unspecified (cloud of components). The exact bound of Eq. (11) establishes how much rate variability is needed to reduce $X_1$ fluctuations when $X_0$ and $X_1$ have equal lifetimes (solid red line). We plot here the normalized standard deviation of $X_1$ relative to its uncontrolled noise levels $\sqrt{⟨s_{11}⟩/⟨x_1⟩}$. The dashed line indicates a marginally tighter but algebraically more involved bound (see Supplemental Material [11]) that is virtually indistinguishable from Eq. (10) in this range. Dots correspond to numerical solutions to various instantiations of the feedback systems for a range of response functions and parameters. (b) We consider any component $X_1$ undergoing first order degradation while embedded in an arbitrarily complicated control network (cloud). Equation (12) bounds the relative noise suppression in such components in terms of the scaled variation of their production rate $R$ (red line). This bound is provably tight (see Supplemental Material [11]).