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# Noise decomposition of intracellular biochemical signaling networks using nonequivalent reporters

Edited by Ken A. Dill, Stony Brook University, Stony Brook, NY, and approved October 24, 2014 (received for review June 26, 2014)

## Significance

In response to a uniform stimulus, biochemical networks display a high degree of variability when assayed across many individual cells. However, due to their complexity, the relative contributions of different pathways to the overall network variability are hard to evaluate and control. Here we introduce a technique allowing for “noise mapping” to be performed within such networks. By experimentally assaying three distinct tumor necrosis factor (TNF) activated transcription factors within a mathematical framework, we show how noise is distributed in the TNF-induced network, in the presence or absence of feedback. This framework is readily scalable to allow the noise decomposition of larger and more complex signaling networks, permitting further investigations into the dependence of biological noise on signaling system architecture.

## Abstract

Experimental measurements of biochemical noise have primarily focused on sources of noise at the gene expression level due to limitations of existing noise decomposition techniques. Here, we introduce a mathematical framework that extends classical extrinsic–intrinsic noise analysis and enables mapping of noise within upstream signaling networks free of such restrictions. The framework applies to systems for which the responses of interest are linearly correlated on average, although the framework can be easily generalized to the nonlinear case. Interestingly, despite the high degree of complexity and nonlinearity of most mammalian signaling networks, three distinct tumor necrosis factor (TNF) signaling network branches displayed linearly correlated responses, in both wild-type and perturbed versions of the network, across multiple orders of magnitude of ligand concentration. Using the noise mapping analysis, we find that the c-Jun N-terminal kinase (JNK) pathway generates higher noise than the NF-κB pathway, whereas the activation of c-Jun adds a greater amount of noise than the activation of ATF-2. In addition, we find that the A20 protein can suppress noise in the activation of ATF-2 by separately inhibiting the TNF receptor complex and JNK pathway through a negative feedback mechanism. These results, easily scalable to larger and more complex networks, pave the way toward assessing how noise propagates through cellular signaling pathways and create a foundation on which we can further investigate the relationship between signaling system architecture and biological noise.

Cells transduce extracellular information through signal transduction pathways that often exhibit complex nonlinear behaviors. Commonly, receptor–ligand interactions initiate signals that diverge into parallel pathways and then converge onto common downstream elements. These branches can have different dose dependencies creating complicated dose–response characteristics, including biphasic properties (1⇓–3). In such systems, the complexity of the network creates difficulty in identifying the sources of cell response variability, even if the topology is known. However, despite the challenges, characterizing the fidelity of biochemical networks could help identify new biochemical noise regulation mechanisms that would further our understanding of cellular decision-making processes (4). Therefore, although biological noise at the gene expression level is relatively well characterized (5⇓⇓⇓⇓⇓–11), to fully understand the origins of cell decision heterogeneity we require novel experimental and mathematical methods to quantify signaling pathway noise.

Prior efforts to understand biochemical noise have primarily focused on the utilization of the equivalent dual-reporter technique. This method involves the single-cell measurement of two distinguishable fluorescent reporter proteins under statistically equivalent conditions (e.g., identical promoters, equivalent integration sites, etc.; Fig. 1*A*) (8, 9, 12). The lack of correlation between reporter expression within a cell is thought to result from stochastic chemical kinetics that randomly and independently affect both reporters, and is referred to as intrinsic noise. The remaining reporter variability originates from the factors that simultaneously affect both reporters equally within an individual cell but vary from cell to cell, and is referred to as extrinsic noise. The extrinsic factors can include the expression levels of RNA polymerase, ribosome number, cell size, or cell cycle stage, all thought to affect both reporters similarly within a cell. The advantage of the equivalent dual-reporter method is the separation of intrinsic noise from extrinsic noise in an experimentally measurable way. This method has been extended to analyze signaling networks; however, it requires simultaneous measurement of two reporters per signaling node of interest (13). This can quickly become experimentally intractable as the system size increases.

Expanding on the success of the equivalent dual-reporter method, nonequivalent dual reporters have found success characterizing sources of cell-to-cell variability. For instance, by comparing a reporter in a signaling pathway of interest to a reporter of a constitutively expressed gene, one can separate pathway-specific noise from general gene expression noise (14, 15). Alternatively, multiple reporters placed within a serial gene expression network can facilitate decomposition of the noise propagation (6). More recently, a single-reporter variation of the original dual-reporter method enabled the decomposition of the intrinsic noise into memory and stochastic terms based upon their separate timescales (16). However, the methods described above frequently use custom-designed networks whose structure is known a priori, which allows for the construction of a specific mathematical framework. Thus, although dual-reporter methods and its variations have yielded important scientific insights, these methods routinely require reporter genes to be inserted into cells which can hamper efforts to rapidly characterize a variety of signaling networks. Furthermore, these reporter methods are limited to the analysis of biological noise at the gene expression level. Therefore, despite substantial advances in the characterization of gene expression noise, we still lack tools needed to understand noise in intracellular signaling.

Here, we present a mathematical generalization of the equivalent dual-reporter method that enables decomposition of signaling network noise using nonequivalent dual reporters. These reporters do not need to be genetically encoded, thus increasing the scope of systems that can be analyzed. Using this framework, we were able to quantify the relative noisiness of both the downstream mitogen activated protein kinase (MAPK) and NF-κB signaling pathways. We also show that this methodology can be used to identify previously unappreciated feedback mechanisms affecting both MAPK and NF-κB pathways. Overall, this methodology is revealing and experimentally facile to implement in a system where detailed knowledge of the topology is unavailable.

## Mathematical Framework for Noise Decomposition Using Nonequivalent Dual Reporters

The method for noise decomposition proposed here can be understood as a generalization of the well-known extrinsic–intrinsic noise decomposition pioneered by Elowitz et al. (9). To demonstrate this relationship, we note that by conceptualizing the additive propagation of noise through the equivalent dual-reporter system, a four-node branch motif naturally emerges (Fig. 1*A*). The input node **1**–**3** (9).*SI Text*, the total noise is identical to the average variance of the reporters (Eq. **1**), a sensible result when the reporters are equivalent. The collection of factors within a single cell that causes the two reporters to change in synchrony is defined as the extrinsic noise and is mathematically defined as the covariance between the reporters (Eq. **2**). Because the two reporters are equivalent, and assuming that the reporters do not affect their common regulators (17), the remaining uncorrelated variation between the expression of the reporters in a given cell arises primarily from stochasticity. Mathematically, intrinsic noise can be defined as the difference between the total and extrinsic noise (Eq. **3**). From these definitions, we can understand why extrinsic noise is typically depicted to be in the direction of the line *B*; see Fig. S1 for an isotropic interpretation).

In the more general case where **1**–**3** are no longer valid, and the framework must be reformulated for the more general nonequivalent case. For nonequivalent reporters, *C*). In this case, we refer to trunk noise as the noise introduced upstream of *SI Text*). Importantly, these results are independent of the distributions of

Eq. **4** reveals that the trunk noise **5** and **6**). Additionally, these equations show that the three noise components can be extracted from joint measurements of

Graphically, this system can be depicted in *D*). In the total absence of noise, for a given input

For a given

Finally, we observe that for a given level of *D* simplifies to the case in Fig. 1*B* if the reporters are equivalent. In particular, *SI Text*). Furthermore, we note that if we average the branch noise values, we will arrive at the definition of intrinsic noise as expected:

## Pathway-Specific Noise in the Tumor Necrosis Factor Signaling Network

Using the above noise decomposition framework, we sought to quantify the noise contributed by the c-Jun and NF-κB pathways when activated by tumor necrosis factor (TNF), a model system for understanding signaling heterogeneity in mammalian cells (19⇓⇓⇓⇓–24), to understand noise propagation through this signaling network (Fig. 2*A*). We exposed mouse embryonic fibroblast cells to a wide range of TNF concentrations to elicit the full dynamic response of the transcription factors. For each TNF concentration, we measured the nuclear concentrations of the transcription factors in hundreds of individual cells using quantitative immunocytochemistry (immunofluorescence) (Fig. 2*B*). We find that the immunofluorescence signal has a strong correlation with the actual protein concentration (Fig. S2). We examined the responses at the 30-min time point, because the translocations of both transcription factors reach their maximum values at this time, indicating similar operational timescales (25).

In response to a stimulus, parallel signaling branches can have different dose dependencies leading to complex overall response characteristics, including biphasic qualities resulting in complex and highly nonlinear behavior (1⇓–3). However, surprisingly, we find that NF-κB and p-c-Jun levels are proportional to each other over four orders of magnitude of TNF concentrations (Fig. S3*A*). Thus, even though the average NF-κB and p-c-Jun levels are nonlinear functions of TNF (Fig. 2*D*), they are linearly related (Fig. 2*C*).

We applied Eqs. **4–6** to decompose the observed noise into a common trunk noise and branch noises specific to the NF-κB and JNK pathways (Fig. 2*D*) and observed that for the NF-κB pathway, the trunk noise was slightly greater than NF-κB branch noise, whereas for the JNK pathway, the c-Jun branch noise was greater than the trunk noise. Therefore, although both responses are subject to the noise resulting from common upstream signaling components, the NF-κB pathway introduces less noise to the signaling output in comparison with the JNK pathway. We find that the inflection point in the dose–response of the trunk and c-Jun branch noise roughly mirrors the inflection point found in the dose–response of the p-c-Jun and NF-κB mean nuclear concentration. We further characterized the noise by calculating the square of the coefficient of variation,

## Noise Decomposition of the TNF Network

Next, we sought to demonstrate how our method could be extended to analyze larger and more complex signaling networks. Many signaling networks, including that of TNF, consist of multiple levels of branching. For instance, the TNF network branches into the NF-κB and JNK pathways, and the JNK pathway subsequently branches to activate two transcription factors: ATF-2 and c-Jun (Fig. 3*A*). To decompose the noise in this six-node system (Fig. 3*A*), we considered multiple four-node branch motifs embedded within the network (Fig. 3 *B–D*). We can decompose the noise of each motif in isolation but, because the three motifs have overlapping portions, we can assemble a more detailed noise map of the original network. To perform this decomposition, we measured, in parallel experiments, the joint pairwise TNF responses of NF-κB and p-ATF-2, NF-κB and p-c-Jun, and p-ATF-2 and p-c-Jun.

First, we found that the results for the NF-κB/p-ATF-2 pair (Fig. 3*B*) were similar to that of the NF-κB/p-c-Jun pair analyzed earlier (Fig. 3*C*). Further quantitative analysis revealed that of the noise in the fully activated TNF–NF-κB pathway, ∼90% can be ascribed to the trunk portion shared with the TNF–JNK pathway, and the remaining ∼10% can be ascribed to the NF-κB–specific branch. In comparison, in the TNF–ATF-2 pathway, only ∼30% of the noise in the ATF-2 pathway originates from the trunk, and ∼70% of the noise arises from the remaining JNK pathway. Next, examining the results for the NF-κB/p-c-Jun pair (Fig. 3*C*), we observe that ∼80% of the p-c-Jun noise originates from the c-Jun-specific branch, suggesting that there may be slightly greater noise in the TNFR–c-Jun pathway than in the TNFR–ATF-2 pathway. Indeed, when we directly decomposed the p-ATF-2/p-c-Jun pair, we observed greater noise specific to the c-Jun pathway than compared with the ATF-2 pathway at higher concentrations of TNF (Fig. 3*D*).

The pairwise analysis can be used to assign relative noise contributions to each part of the TNF signaling network (Fig. 4*A*). For instance, if as a reference we assign a noise value of 1 to the initial TNF–TNFR segment, then the noise value in the TNFR–NF-κB segment is ∼0.1 (*SI Text*). This is consistent with our above observation that the TNF–TNFR signaling segment contributes ∼90% of the total noise in the TNF–NF-κB pathway and the downstream NF-κB component contributes the remaining ∼10%. Similar calculations can be used to compute the relative noise contributions from the remaining segments (*SI Text*). This analysis yields the relative noise values shown in Fig. 4*A*. Interestingly, the total noise of the TNF–NF-κB pathway is ∼32% of the noise present in the TNF–ATF-2 pathway and 26% of the noise in the TNF–c-Jun pathway, indicating an asymmetry of pathway-specific noise between the JNK and NF-κB branches in TNF signal processing. This result corroborates our prior results, which demonstrated that the information carrying capacity of the NF-κB pathway is greater than that of the JNK pathway, with the capacity of both pathways influenced by a common TNF receptor-level bottleneck (21).

Because of the inherent scalability, this noise decomposition methodology can be easily expanded to analyze the noise propagation through larger signaling networks. For example, given a hypothetical signaling network (Fig. 4*B*) and the two downstream readouts,

## Impact of Feedback on Transcription Factor Variability

Negative feedback is a well-known mechanism that cells can use to modulate biochemical noise (29⇓–31). To quantitatively demonstrate the effect of negative feedback on noise in TNF signaling, we performed a noise decomposition in wild-type cells and cells lacking A20, an enzyme up-regulated by NF-κB that is well-known to inhibit TNF-induced NF-κB activity by destabilizing the TNF receptor complex (32⇓⇓–35) (Fig. 5*A*). Destabilization of the receptor complex has also been reported to mitigate downstream JNK activation (36); however, this mechanism is still controversial (37, 38).

We also compared the noise decompositions at the 30-min and 4-h time point in these cells (Fig. 5*B*), as induced expression of A20 is negligible at the earlier time point but maximal by the latter time point (Fig. S5) (39, 40). Importantly, there exists a consistent linear relationship between NF-κB and p-ATF-2 across two time points and across both wild-type and A20^{−/−} cells, enabling direct comparison of the noise decomposition results among all four conditions (Fig. S3*B*).

At the 30-min time point, we observed that the trunk noise was, on average, slightly greater in A20^{−/−} cells than wild-type cells, whereas there was no difference in NF-κB–specific noise (Fig. 5*B*). The difference in trunk noise was greater at the 4-h time point, likely reflecting the difference between the effects of lower basally expressed A20 versus that of highly induced A20. These results corroborate the ability of A20 to regulate both NF-κB and JNK pathways at the receptor complex level. Unexpectedly, we also observed larger ATF-2 branch-specific noise in A20^{−/−} cells compared with wild-type cells with the difference greater at the 4-h time point than at the 30-min time point. This result indicated that A20 can repress the JNK pathway independent from its effects on the TNF receptor complex. At the time that this prediction was made, there was no known direct inhibition of the JNK pathway by A20, but a later study by Won et al. verified that A20 directly represses ASK1, a kinase in the JNK pathway that has no known direct effects on the NF-κB pathway (38). We note that although negative feedback could potentially violate our assumption that the trunk and branch noise levels are independent, this experiment demonstrates that on a practical basis, our noise decomposition framework can yield sensible and even predictive results regarding the effects of negative feedback on signaling noise.

## Discussion

By using the linear relationships between downstream effectors of the TNF pathway, we developed a mathematical and experimental framework that enables noise decomposition in intracellular signal transduction networks. This method distinguishes trunk from branch noise and can be derived as a natural extension of extrinsic–intrinsic noise analysis. In addition, although these results are independent of the underlying distributions, the methodology can still serve to characterize branch noise distributions. Corroborating previous results, we showed that there is a greater amount of noise present in the JNK branch than the NF-κB branch and that both branches are subject to a sizable contribution of noise from the TNF receptor complex. More detailed noise mapping of the JNK pathway revealed that within the JNK subnetwork, p-c-Jun is subject to greater noise than p-ATF-2.

Examining the impact of negative feedback on noise expression, we found further evidence that A20 is able to suppress variability at the level of the TNF receptor. We also unexpectedly discovered an additional mechanism of JNK noise suppression consistent with a recent observation of the direct inhibition of ASK1 by A20. Although negative feedback and reporter competition over common regulators can theoretically complicate the mathematical decomposition by allowing interactions between noise parameters (17), we nonetheless observed that the nonequivalent dual-reporter method can be robust to the presence of negative feedback and can provide a useful, and even predictive, first approximation. Furthermore, at a minimum, the noise analysis presented here can be used to characterize the noise and serve as a comparison against predictions developed by computational models.

Although this method requires a linear relationship between the reporters, we believe it does not tightly constrain the general applicability. In most biological signaling systems, nonlinear dose–responses align to allow for optimal information transfer which results in responses that are approximately linearly related to each other (31). Furthermore, we expect that in the case of nonlinear relationships between reporters, this method can be easily extended by replacing the slope parameter

We envision that this method and further generalizations could enable better measurements of noise which will clarify its molecular underpinnings and aid us in understanding the nature of variability within signaling pathways.

## Materials and Methods

### Cell Culture.

Wild-type and A20^{−/−} 3T3-immortalized mouse embryonic fibroblasts (gift from A. Hoffmann, University of California, San Diego) were maintained in low-glucose DMEM (Invitrogen) supplemented with 10% calf bovine serum (American Type Culture Collection) and 10 U/mL each of penicillin and streptomycin (Invitrogen). P65-GFP cells (gift from M. Covert, Stanford University, Stanford, CA) were maintained in high-glucose DMEM (Invitrogen) supplemented with 10% FBS (American Type Culture Collection) and 10 U/mL each of penicillin and streptomycin (Invitrogen). Cells were seeded at a density of ∼150 cells per mm^{2} onto 15-mm-diameter circular coverslips (Fisher Scientific), coated with 0.1% gelatin (Sigma), placed in six well plates, and then serum starved in medium with reduced serum concentration (0.1%) overnight before experimentation.

### Immunocytochemistry.

After exposure to murine TNF (Roche) at the specified concentrations and duration, the cells were washed three times with ice-cold PBS (Invitrogen) and fixed in 4% paraformaldehyde (Electron Microscopy Sciences) for 20 min. The cells were then permeabilized in 0.1% triton X-100 (Sigma) for 5 min and blocked in 10% goat serum (Invitrogen) for 60 min. Next, the cells were incubated in primary antibody solution. Primary antibody concentrations used were 1:100 rabbit anti-p65 antibody (Santa Cruz), 1:100 mouse anti-phospho-ATF-2 antibody (Santa Cruz), 1:100 mouse anti-phospho-c-Jun (Santa Cruz), and 1:100 rabbit anti-phospho-c-Jun (Cell Signaling).

Finally, the cells were incubated in a secondary antibody solution consisting of 1:200 Alexa Fluor 488-conjugated goat anti-rabbit and 1:200 Alexa Fluor 594-conjugated goat anti-mouse antibodies (Invitrogen) for 60 min and 2 µg/mL Hoechst-33258 (Sigma) for 60 min. All solutions were made in 10% goat serum (Invitrogen) in PBS, and cells were washed with PBS in between each step. To minimize experimentally induced variability and to enable quantitative comparisons across conditions, all concentrations of TNF and all cell lines were assayed at the same time using common reagents. Finally, the stained coverslips were mounted on glass microscope slides and imaged on an Axiovert 200M inverted epifluorescence microscope (Zeiss) equipped with Slidebook 4.2 (Intelligent Imaging Innovations). On average, over 350 cells were imaged per experimental condition.

### Image and Data Analysis.

Image processing and data analysis were performed using MATLAB R2009a (MathWorks). Background correction, nucleus segmentation, and quantification of nuclear concentrations of NF-κB, phospho-ATF-2, and phospho-c-Jun were performed as described previously (21). Programs are available upon request. Top and bottom second percentiles of data were discarded to reduce the influence of outliers on the estimates of variance.

## Acknowledgments

A.R. and R.C. were supported by National Institutes of Health Grants GM072024 and RR020839. R.C. was supported by the Medical Scientist Training Program at the Johns Hopkins University. A.L. acknowledges support from the Semiconductor Research Corporation SemiSynBio program.

## Footnotes

↵

^{1}A.R. and R.C. contributed equally to this work.- ↵
^{2}To whom correspondence should be addressed. Email: andre.levchenko{at}yale.edu.

Author contributions: A.R. and R.C. designed research; A.R. and R.C. performed research; A.R. and R.C. contributed new reagents/analytic tools; A.R., R.C., and A.L. analyzed data; and A.R., R.C., and A.L. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1411932111/-/DCSupplemental.

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- Mathematical Framework for Noise Decomposition Using Nonequivalent Dual Reporters
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