^{ 1 }

^{ 2 }

^{3}

^{*}

GH and NB conceived and designed the experiments and wrote the paper.

The authors have declared that no competing interests exist.

Interactions between genes and proteins are crucial for efficient processing of internal or external signals, but this connectivity also amplifies stochastic fluctuations by propagating noise between components. Linear (unbranched) cascades were shown to exhibit an interplay between the sensitivity to changes in input signals and the ability to buffer noise. We searched for biological circuits that can maintain signaling sensitivity while minimizing noise propagation, focusing on cases where the noise is characterized by rapid fluctuations. Negative feedback can buffer this type of noise, but this buffering comes at the expense of an even greater reduction in signaling sensitivity. By systematically analyzing three-component circuits, we identify positive feedback as a central motif allowing for the buffering of propagated noise while maintaining sensitivity to long-term changes in input signals. We show analytically that noise reduction in the presence of positive feedback results from improved averaging of rapid fluctuations over time, and discuss in detail a particular implementation in the control of nutrient homeostasis in yeast. As the design of biological networks optimizes for multiple constraints, positive feedback can be used to improve sensitivity without a compromise in the ability to buffer propagated noise.

Cells sense and process information using biochemical networks of interacting genes and proteins. Typically, a signal is sensed at a specific point of the network (input) and is propagated to modulate the activity or abundance of other network components (output). Reliable information processing requires high sensitivity to changes in the input signal but low sensitivity to random fluctuations in the transmitted signal. Since the detection of signal is inherently stochastic [

Linear (unbranched) cascades present the simplest instance of biochemical networks. Recent studies have shown that such cascades display an interplay between sensitivity to changes in input signal and the ability to buffer stochastic fluctuations [

In general, the fine line that separates “noise” from “signal” is established functionally. Nevertheless, in many systems such as the sensing of temperature, nutrient levels, ligand concentration, etc., the signal is interpreted as a long-term change in the input, whereas noise is characterized by rapid stochastic fluctuations. In this study, we focus on this particular class of systems.

We explore for gene circuits that can buffer propagated noise while maintaining signaling sensitivity. We consider a large set of networks that are differentially designed but are equally sensitive to long-term changes in the input, and compare their ability to buffer propagated noise. Systematic analysis of all three-gene circuits revealed that negative feedback amplifies propagated noise. In contrast, positive feedback appears to be a necessary element for buffering such noise. Analytical analysis demonstrated that positive feedback contributes to noise buffering by slowing down the dynamics, thus providing a longer averaging time. A detailed analysis of a recurrent network design, found in systems controlling nutrient homeostasis, suggests that it functions as a noise-reduction device based on the principles identified in our analysis.

To begin analyzing the effect of network architecture on the interplay between sensitivity and noise buffering, we considered a two-component cascade with a negative feedback loop. This cascade is composed of an input node, _{0}, which activates an output node, _{1}. The output node feeds back to repress its own expression (_{1} denotes the maximal rate of _{1}_{1} degradation, and _{0,} _{1} are Hill coefficients. Note that _{0} and _{1} are normalized by their respective dissociation constants from the gene promoter.

(A) A system with negative feedback (pink) and without negative feedback (blue) were compared in terms of (B) the output noise in response to fluctuations in the input (green), and (C) the deterministic response of the two systems to a 2-fold change in the input.

(D) Noise amplification versus susceptibility for negative feedback with different Hill coefficients. The color-code is the level of saturation of the input promoter. Sensitivity was calculated by solving the steady-state equations after a 1% change in the input levels, and noise amplification was the result of stochastic simulations (

We consider an input signal _{0}(_{0}〉 + _{0}(_{0}〉. The fluctuating component _{0}(_{0.} _{1}_{1} = 4). The analogous dynamics for a system that lacks such feedback (_{1} = 0) is also shown. Consistent with previous studies [

To rigorously quantify the interplay between the sensitivity of the input–output relation and the buffering of propagated noise, we define two measures for the sensitivity and noise-amplification of the system

As before, all quantities are measured at steady state. Both

_{1} = 0) is compared to that of increasing feedback cooperativity (_{1} = 1 and 2). Again, a clear interplay between susceptibility and noise buffering is observed, with systems that are more sensitive to changes in the input level being also more vulnerable to noise. Notably, this interplay seems to be more severe in the presence of negative feedback. Thus, for a given level of susceptibility, propagated noise is amplified to a greater extent in the presence of a negative feedback. This result is consistent with the theoretical arguments for a two-node model [

Our analysis implies that negative feedback cannot be used to buffer against rapidly varying propagated noise in systems which require a sensitive response to long-term changes in their input. To identify network architectures that can buffer noise while maintaining sensitivity, we characterized systematically the relation between susceptibility and noise-buffering of all three-node circuits (_{0}), an output node (_{2}), and an intermediate node (_{1}), connected via activating or repressing interactions (arrows). We allowed for all incoming or outgoing arrows, with the exception of the input _{0} which could affect both _{1} and _{2} (outgoing arrows), but was not subject to feedback regulation (no incoming arrows). Each arrow was assigned a positive sign (activation) or a negative sign (repression), thus leading to a total of 324 networks (

(A) Each network with an input node 0, an output node 2, and an intermediate node 1 was assigned random sets of interaction elasticities. The susceptibility and noise amplification were calculated from each elasticity set. Results are shown for (B) linear cascade (1,000 points), (C) two-node negative feedback (1,000 points), (D) coherent FFL (1,000 points), (E) incoherent FFL (1,000 points), and (F) two-node positive feedback (230 points with a stable steady state).

(G) All networks were sorted into groups according to the fraction of parameter sets that exhibited stability (gray line). Within each group, the networks were sorted according to the fraction of low noise and high susceptibility parameter sets (relative to a linear cascade) out of the stable sets (blue line). As the number of positive feedback loops increases, stability is decreased.

Each specific circuit supports a range of dynamic behaviors, depending on the precise value of the interaction parameters. Following the formalism presented by Paulsson [

The degradation terms were assumed to be first-order, and we considered _{i}_{0} = _{1} = _{2} = 0 is not formally part of our analysis, because it renders the relative fluctuation

Following linearization, the combined effect of all interaction parameters (i.e., Hill coefficients and saturation levels) is captured by the _{i}

Noise amplification was found by solving the matrix equation [_{i}_{0}. In the construction of _{0}, and that these fluctuations die out exponentially with a time scale _{0} = 1 (autocorrelation time of one unit). The exact terms of

A particular choice of elasticity values for all arrows of the network defines a single point in the

As expected, noise amplification in linear (unbranched) cascades is precisely proportional to the susceptibility (

To further characterize the properties of all three-node circuits, we calculated for each network the fraction of parameter sets that produce a stable steady state (

To better understand the reason underlying the ability of positive feedback to buffer propagated noise for a given susceptibility, we used the analytical description of a two-component system with an input _{0} and an output _{1}, as was derived by Paulsson in [_{10}/_{11} denotes the susceptibility; _{0}, _{1} denote the degradation time scales of _{0} and _{1}, respectively; and _{10}, _{11} are the elasticities (as defined in _{11} = 1. Negative feedback of _{1} on itself implies _{11} > 1, whereas positive feedback implies 0 < _{11} < 1 (if _{11} falls below zero, instability arises). As was shown by Paulsson [

Notably, _{11} [_{11} can also be decreased if the degradation is independent of _{1} (_{1}.

While positive feedback appears to be important for buffering propagated noise (when sensitivity is controlled for), such a mechanism needs to comply with several requirements. First, the feedback loop itself should produce low internal noise because intrinsic noise is not buffered. Second, the effective elasticity _{11} (_{11} → 1) the effect of the positive feedback is negligible, but when it is too low (_{11} → 0) the system is on the verge of instability, and the steady state will no longer resist small fluctuations. Finally, to avoid decrease in susceptibility due to saturation effects, _{11} must be maintained constant over a large range of parameters.

A class of mechanisms that complies with the above requirements is based on a combination of positive and negative feedbacks. Fast-acting negative feedback functions to ensure stability, while positive feedback provides the required noise buffering. A specific example for such a network is involved in nitrogen homeostasis in yeast [_{0}, the output Gat1p by _{1} and the repressor Dal80p by _{2}, the system can be modeled by the following two differential equations:

(A) In the yeast nitrogen catabolite repression system, the transcription factor Gat1p is activated in response to Gln3p. It can then activate its own transcription, as well as DAL80, which binds to the same sequences as Gat1p and represses transcription.

(B) The mean output (Gat1p) levels and its (C) noise content are shown for different input (nuclear Gln3p) levels. Simulation results are shown for the nitrogen system (pink line) and for a system with no feedback but with similar sensitivity (dashed blue line). The input noise level is _{0} = 1, and its autocorrelation time is _{0} = 2 units. The other time constants are _{1} = _{2} =10 units. Dal80p binds its own promoter with a Hill coefficient of 2, whereas all other Hill coefficients equal 1.

Here _{i}_{i}_{1} = _{2} = 0. We will neglect this factor in subsequent analysis. The _{ij}_{2}/_{i}_{2} describing the competitive inhibition of Dal80p. The Hill coefficient of _{2} binding to its own promoter is 2 because Dal80p binds as a dimer [_{2} binding to the _{1} promoter is set to 1 to enhance noise buffering and susceptibility (although a value of 2 would still increase noise averaging).

For the system described by _{i}_{2}, must be small, and all binding constants of the activator, _{i}_{1}, must be large. In this regime, _{2} responds more rapidly than _{1} and _{0} (_{22}_{2} ≫ _{11}_{1}, 1/_{0}), then it can be assumed to be at quasi-steady state, and _{1} results in an almost-constant elasticity

Detailed simulations confirm that this system can indeed buffer propagated noise, as compared to a loop-free system with the same levels of susceptibility (

The ability to distinguish input signals from stochastic fluctuations is crucial for reliable information processing. Yet, being processed by the same computation device, signal and noise are inherently coupled. It thus comes as no surprise that increasing the ability to buffer propagated noise comes typically at the expense of reducing the sensitivity toward the input signal. We study this interplay in the context of a special class of systems where the signal is retained for long time periods, whereas the noise fluctuates rapidly. Such systems are ubiquitous in the adjustment of cells to aspects of their extracellular environment.

Previous studies reported that negative feedback buffers gene expression noise [

Analytical analysis [

Whereas our study illustrates the effect of positive feedbacks, additional mechanisms could be used for reducing propagated noise by similarly increasing the averaging time. Such mechanisms include long linear cascades; cascades with an intermediate component that has a relatively large half-life [_{11} in

Positive feedbacks did not emerge as a recurrent network motif in several of the transcriptional networks analyzed [

All simulations were based on the Gibson-Bruck [

The interaction parameters of each arrow in each network are captured by the interaction elasticities _{ij}

The values for the elasticities were randomly assigned to each network. To control for similar distribution of positive and negative interactions, we defined the synthesis elasticity _{ij}_{ii}_{ii}_{ij}_{ij}_{ij}_{ii}

Stability criteria were established via the sign of the eigenvalues of the interaction matrix (

(54 KB DOC)

(280 KB PDF)

We are grateful to Moshe Oren for support and encouragement throughout the work. We thank Andreas Doncic for helping with the screen design and critical review of the manuscript, and Nitzan Rosenfeld, Sagi Levi, Ran Kafri and members of our lab for comments and discussion.

feed-forward loop

Fluctuation Dissipation Theorem