Transient amplification limits noise suppression in biochemical networks

Cell physiology is orchestrated, on a molecular level, through complex networks of biochemical reactions. The propagation of random fluctuations through these networks can significantly impact cell behavior, raising challenging questions about how network design shapes the cell's ability to suppress or exploit these fluctuations. Here, drawing on insights from statistical physics, fluid dynamics, and systems biology, we explore how transient amplification phenomena arising from network connectivity naturally limit a biochemical system's ability to suppress small fluctuations around steady-state behaviors. We find that even a simple system consisting of two variables linked by a single interaction is capable of amplifying small fluctuations orders of magnitude beyond the levels predicted by linear stability theory. We also find that adding additional interactions can promote further amplification, even when these interactions implement classic design strategies known to suppress fluctuations. These results establish that transient amplification is an essential factor determining baseline noise levels in stable intracellular networks. Significantly, our analysis is not bound to specific systems or interaction mechanisms: we find that noise amplification is an emergent phenomenon found near steady states in any network containing sufficiently strong interactions, regardless of its form or function.

Figure 1

A portion of the transcription network of Escherichia coli. Each node is associated with a particular gene or, equivalently, a transcription factor expressed by that gene. Directed links represent interactions in which one of these genes regulates the expression of another. Activatory interactions are shown as arrows, and inhibitory interactions as lines ending in a perpendicular bar.

Figure 2

Transient amplification in a randomly generated transcriptional network. (a) The maximum amplification factor $\rho \left(t\right)$ plots the maximum possible ratio of $\parallel \mathbf{u}\left(t\right)\parallel$ to $\parallel \mathbf{u}\left(0\right)\parallel$ as a function of time. The initial increase in and subsequent decay of $\rho \left(t\right)$ show that this is an asymptotically stable system capable of transient amplification. (b) The cumulative distribution function for the peak value of $\rho \left(t\right)$, averaged over an ensemble of topologically identical copies of $M$, shows that the amplification scale in (a) is not atypical.

Figure 3

Noise amplification in simple two-node cascades. (a) A simple motif in which one variable activates a second. (b) Cumulative distribution functions for the total variance maintained by stochastic forcing $\langle u^2\rangle$ (solid curves) and the variance observed in the absence of any amplification ${\langle u^2\rangle }_0$ (dashed curves) in nonreactive and reactive systems (cyan and magenta curves, respectively). (c) Probability density functions for the fraction ${\langle u^2\rangle }_0/\langle u^2\rangle$ in nonreactive and reactive systems (cyan and magenta curves, respectively).

Figure 4

Noise amplification in simple cascades with negative feedback. (a) A motif in which one variable inhibits itself and activates a second. (b) Probability density functions for $\langle u^2\rangle$ without feedback divided by $\langle u^2\rangle$ with feedback in nonreactive and reactive systems (cyan and magenta curves, respectively). Ratios $<1$ indicate systems in which negative feedback enhances small fluctuations. (c) Probability density functions for $\langle u^2\rangle$ ratios in reactive systems that become nonreactive or remain reactive after deleting the feedback loop (cyan and magenta curves, respectively).

Figure 5

Noise amplification in feedforward loops (FFLs). (a) An FFL motif involving three activatory interactions. (b) A simpler structure lacking the interaction between the two input nodes. (c) Probability density functions for $\langle u^2\rangle$ after deleting this interaction divided by $\langle u^2\rangle$ before deleting this interaction from nonreactive and reactive FFLs (cyan and magenta curves, respectively). Ratios $\ll 1$ indicate systems in which the FFL structure significantly enhances small fluctuations. (d) Probability density functions for $\langle u^2\rangle$ ratios in reactive FFLs that become nonreactive or remain reactive after deleting this interaction (cyan and magenta curves, respectively).

Figure 6

Examples of network motifs involving feedback between two variables. (a) Negative feedback. (b) Positive feedback. (c) Double-negative feedback. These motifs, along with similar structures involving self-activation and/or self-inhibition loops, are found throughout the cell. Noise amplification is, in principle, possible in any of these structures (see text).

Figure 7

A portion of the transcription network of Escherichia coli, highlighting all interactions between a selected subset of nodes (magenta lines). As emphasized in the text, if the highlighted subsystem is capable of amplifying small fluctuations around a steady state, then none of the remaining interactions (black lines) can prevent this amplification. This insight does not depend on the subsystem chosen and holds true for any network, regardless of its form or function.