Regulatory Activity Revealed by Dynamic Correlations in Gene Expression Noise

Mary J. Dunlop1, Robert Sidney Cox III2, Joseph H. Levine1, Richard M. Murray1, Michael B. Elowitz2*

1Division of Engineering and Applied Science and 2Division of Biology California Institute of Technology 1200 E. California Blvd. M/C 114-96 Pasadena, CA 91125 * Corresponding author

Abstract

Gene regulatory interactions are context–dependent, active in some cell types or cellular states but not in others. Stochastic ‘noise’ in gene expression propagates through active, but not inactive, regulatory links [1, 2]. Thus, noise could provide a non-invasive means of interrogating the activity state of a regulatory link in different contexts. However, global, or ‘extrinsic’, noise partially correlates fluctuations in gene expression even in the absence of explicit regulatory connections [2, 3], complicating static analysis. Here we show that single-cell time-lapse microscopy, by revealing time lags due to regulation, can discriminate between correlations due to extrinsic noise and those due to active regulatory connections.

We demonstrate quantitative differences between active and inactive regulation in a simple synthetic gene circuit, which are consistent with predictions from a stochastic mathematical model of the underlying circuit. This approach is also effective on natural networks:

Dynamic noise correlations in the galactose metabolism network genes in E. coli revealed that the CRP-GalS-GalE feed-forward loop is inactive in standard conditions. More generally, these results provide a method to analyze context-dependent function in endogenous regulatory networks without explicit perturbations.

Introduction

Cells use circuits composed of interacting genes and proteins to implement diverse cellular and developmental programs. A major goal of systems biology is to connect the regulatory architecture of these circuits to the dynamic behavior of individual cells. Several problems confound attempts to predict cellular behaviors from gene circuit maps. Quantitative information about biochemical parameters describing the system is often minimal. In some cases, it may even be unclear in which direction regulation occurs (who regulates whom). Additionally, regulatory links that are active in some cellular states may not be active in the context being investigated. Identifying the subset of regulatory links that are active in a given state would simplify analysis of the circuit as a whole [4].

Regulatory links may be present, but inactive, for several reasons: In the simplest case, the concentration of a regulatory factor may be outside its effective range (Fig. 1A). Transcription factors may also be inactive due to post-translational modification or the absence of necessary co-factors [5, 6].

Recent work has shown that noise in gene expression can lead to substantial cell-cell variability

[1-3, 7, 8], which can propagate through networks [1, 2]. Noise in a regulatory circuit can only propagate through one or more active regulatory links. Thus, noise correlations may provide information about the activity of regulatory connections without explicit perturbation of cellular components.

However, gene expression correlations arise not just from regulation, but also from global variations, or ‘extrinsic noise,’ in the overall rate of gene expression [2, 3]. For example, 2

fluctuating numbers of ribosomes, polymerase components, and cell size can affect the expression of many genes in a cell, positively correlating expression. In practice, the definition of extrinsic noise depends on how the regulatory system is defined [9], here we assume extrinsic noise is global to all measured genes [2, 9, 10]. At the same time, a target gene can fluctuate independently of its regulator due to stochasticity, or ‘intrinsic noise,’ in its own expression. Fig.

1B illustrates how these conflicting effects prevent discrimination between noise and regulation as a source of correlation in static measurements.

Gene regulation occurs with a delay; it takes time for protein concentrations to build up sufficiently to have a regulatory effect on the downstream genes they control (Fig. 1C) [11]. The sign of the delay between a fluctuation in regulator concentration and its effect on target protein levels provides information about the direction of the link. Note that no such delay occurs for global extrinsic noise, which affects all genes simultaneously. Thus, by following the expression of multiple genes over time in an individual cell, one can decouple noise from regulatory correlations. This effect can be analyzed using the temporal cross correlation function, which describes how well two signals are correlated when one of them is shifted in time relative to the other. Similar approaches have been used to infer connectivity of in vitro metabolic networks

[12, 13]. Since these experiments were not conducted in living cells they used a prescribed time- varying input to perturb the system and it was unnecessary to consider many of the details particular to actual cellular noise sources.

We implemented a stochastic model of gene regulation [14, 15], incorporating values for biochemical and noise parameters from a previous study [2] (Methods). We simulated expression of a regulatory protein, A, which represses a target gene, B, and an additional unregulated gene,

C. With only extrinsic noise the three signals are positively correlated, though A also represses B with a delay (Fig. 2A). With only intrinsic noise, repression of B by A generates a delayed anti- correlation in the expression of these two genes. The cross correlation functions (Fig. 2B) were calculated using simulated time-series data. In addition, a linearized model was used to calculate analytic solutions that approximate the full system well (Supplementary Text). The cross correlations show several features: (1) Repression appears as a dip at an effective regulation delay time, denoted !reg. (2) The direction of regulation is given by the sign of !reg. (3) Extrinsic noise causes a positive peak in the cross correlation function close to !=0, both with and without regulation. (4) The relative balance of intrinsic and extrinsic noise affects the magnitude of !reg.

We found that !reg is most sensitive to the protein degradation time, while the magnitude of the dip is determined primarily by how switch-like the gene regulation function is (Supplementary

Text).

We tested this approach in vivo by measuring active and inactive regulatory links in a well- controlled genetic circuit. We built a synthetic gene circuit with three fluorescent proteins (Fig.

3A). A protein fusion of the transcription factor " cI and yellow fluorescent protein (YFP) represses production of red fluorescent protein (RFP), which is under the control of a variant of the ! PR promoter, OR2* [16]. Cyan fluorescent protein (CFP) is controlled by a strong independent constitutive promoter (Methods).

Cells were grown and imaged in three colors using automated time-lapse fluorescence microscopy (Methods). A filmstrip from one movie is shown in Fig. 3B. Here, strong anti- correlation is visible between RFP and YFP, while CFP is expressed at a more homogeneous level across all cells. The appearance of spatially grouped sub-populations of cells that display

similar fluorescence states occurs because !reg exceeds the cell cycle time, consistent with simulation predictions (Fig. 2) and previous measurements [10].

To analyze these data quantitatively, we used semi-automated image analysis software to extract fluorescence intensities for individual cells (Methods). A typical time trace of fluorescence data from a single cell lineage is shown in Fig. 4A. The same data shifted in time reveal the temporal anti-correlation between CI-YFP and RFP signals. To properly handle the branching nature of the data, we introduced a modified formula for the cross correlation function (Methods). Fig. 4C shows the resulting cross correlations for cases of active repression (CI-YFP and RFP) and no regulation (CI-YFP and CFP). These functions displayed all features predicted by the model, including a strong dip at a negative lag time due to repression, and positive correlation at !=0 due to global noise in the unregulated case.

We next asked how the relative amplitude of intrinsic and extrinsic noise affects cross correlations. We constructed a plasmid-based variant of the synthetic circuit using the low-copy

SC101 origin of replication (~10 copies per cell [17]). Copy number fluctuations on the plasmid-borne circuit increase the effective extrinsic noise level for genes in the circuit, and reduce the relative importance of intrinsic noise, whose uncorrelated fluctuations average out.

As expected, the circuit was more variable in fluorescence intensity on the plasmid than in the chromosome (Fig. 3C). Although anti-correlation is still visible between CI-YFP and RFP, certain cells are brighter than others in all colors, and more variability is seen in the time courses

(Fig. 4B). Fig. 4D shows cross correlation functions calculated from this data. The amplitude at

!=0 is increased relative to the chromosomal construct in both the regulated and unregulated case, reflecting simultaneous correlations. As shown in Fig. 4C-D, these results confirm model 5

predictions, and demonstrate that regulation can be discriminated even when extrinsic noise amplitudes are large.

Do the methods demonstrated in the synthetic circuit apply to endogenous gene circuits? To address this, we considered the feed-forward loop network motif, which appears frequently in transcriptional gene circuits [18]. One fundamental question about specific natural instances of the feed-forward loop motif is whether they behave actively as feed-forward loops in any given cellular state or context.

Mathematical modeling of the feed-forward loop predicts that the cross correlation between Y and Z (Fig. 5A) is similar to the cross correlation function one obtains with simple repression

(Fig. 2), with an additional source of symmetric correlation due to X. Inducers that bind to and inactivate Y cause a loss of the anti-correlated features (Fig. 5B). Thus, the cross-correlation function between Y and Z appears qualitatively different when Y is actively repressing Z versus when the regulation is inactive.

One of the best-studied feed-forward loops occurs in E. coli galactose metabolism (Fig. 5C) [19].

The repressor GalS is activated by CRP in response to cyclic AMP. GalS in turn represses the galE operon, as well as itself. Finally, CRP also activates the galE promoter. Compared to the basic feed-forward loop model, this system is complicated by additional regulatory inputs from the GalR repressor. Repression by GalS and GalR is inhibited by galactose, or its non- metabolizable analog, fucose [20]. Pulse-like expression of galE in response to glucose depletion was shown to result from the feed-forward loop that regulates its expression [19]. If the feed-forward loop is actively regulating expression of galE there should be a qualitative difference between the cross correlation functions with and without fucose.

We constructed strains in which the PgalE promoter controlled expression of cfp and the PgalS promoter controlled expression of yfp. Both reporter genes were integrated in the chromosome as transcriptional reporters (promoter fusions) rather than protein fusions. This strategy reduces perturbation of endogenous circuit function, and allows for signal amplification using strong ribosome-binding sites for reporter gene expression.

When we used this strain to analyze gene expression noise in the wild type galactose metabolism network (Fig. 5C), we found that the cross correlation functions were similar both with and without fucose (Fig. 5D), showing a symmetric peak at "=0. These results suggested, surprisingly, that GalS does not play an active role in regulating GalE in this cellular context.

However, the galE operon is known to have an additional regulator, GalR, which inhibits transcription through loop formation in the promoter [20]. We reasoned that dominant repression by GalR could explain the lack of feed-forward loop behavior. To test this hypothesis, we deleted galR (Methods) and repeated the measurements (Fig. 5E). In this strain, the cross correlation function without fucose showed a signature of repression with a dip at "<0 and peak near "=0 (Fig. 5F). Addition of fucose to the !galR strain caused the cross correlation function to become symmetric, with a postive peak at "=0, consistent with inhibition of GalS by the inducer. Together, these results reveal that the activity of the GalS-mediated feed-forward loop is cell state dependent: inactive under the standard media conditions used here, but capable of functioning when GalR is disrupted. It may also be active under other environmental conditions, including transient stimuli (like glucose depletion, as in [19]).

An interesting aspect of gene expression in individual cells is the combination of correlated

(extrinsic) and uncorrelated (intrinsic) noise sources. As described above, active regulatory links

introduce characteristic features in the cross correlation function between genes. However, the converse is not true: the presence of time lags can in principle occur without direct regulation in complex gene networks. Certain network architectures can produce qualitatively similar cross correlation functions (Supplementary Material). However, the use of dynamic noise correlations in conjunction with knowledge of potential regulatory links, can provide a powerful means of determining effective activity states, as shown in galactose feed-forward loop example.

The conflicting effects of intrinsic and extrinsic noise analyzed above constrain the design of cellular regulatory systems that suppress or exploit variability. Recent work has shown that noise provides an essential source of variation in differentiation [21-23] and phenotypic switching [24-26]. Our results now suggest that noise can be used to analyze context- or cell state-dependent regulatory interactions without explicit perturbations. This approach requires monitoring gene expression levels over time in individual cells, something that has become increasingly feasible in diverse biological systems [27]. These results both demonstrate a basic principle of genetic circuit operation, that not all regulatory interactions are active all of the time, and also provide a general framework for determining these activity states in natural genetic circuits with minimal perturbations.

Methods

Plasmids and Cells

The plasmid pNS2-#VL (Fig. S1) was constructed by synthesizing a region starting with the kanamycin promoter and ending just before CI-YFP (synthesis by Blue Heron). The sequences

for CFP and the red fluorescent protein mCherry were modified as in the GFPuv116 of ref. [28] in regions immediately downstream of the start codon, and were codon optimized for expression in E. coli. The synthesized construct was cloned into the plasmid pZS21-cIYFP [2], replacing the

Tet promoter with the synthesized fragment. The resulting plasmid was transformed into

MG1655Z1, a derivative of MG1655 that over-expresses LacI, for experiments.

To integrate the construct onto the chromosome, the region from the kanamycin resistance marker through the end of CI-YFP was amplified by using PCR with 50 bp homology regions included on the end of each primer. The construct was integrated into the galK region of

MG1655Z1, using the recombineering methods described in [29]. Insertion was verified with colony PCR.

Promoter regions for galS and galE were taken from the reporter library in [30]. Promoter- fluorescent protein reporters were made with fusion PCR and verified with sequencing. The fusion PCR product was cloned into a vector with the kanamycin resistance marker and SC101 origin of replication. A region including kanamycin and both promoter-reporter fusions was amplified using PCR with homology arms for intC and integrated into the chromosome of

MG1655 using recombineering methods as in [29]. Integration was verified with colony PCR and sequencing.

galR was deleted from the MG1655 strain with chromosomally integrated PgalS-YFP/PgalE-CFP.

The chloramphenicol marker from pKD3 was amplified using the PCR primers described in [31] with the homology arms for GalR deletion from [32].

Time-Lapse Microscopy

Synthetic Circuit

Single colonies were inoculated in selective LB media and grown overnight. This culture was diluted back 1:100 in 1/4 strength LB with kanamycin and between 10-15 µM IPTG (varies for different movies). The promoter for CI-YFP can be controlled by the IPTG concentration in the media, which we adjusted to maintain the mean CI-YFP concentration in the active regulatory range. The cells were then grown to OD 0.1-0.2 and diluted back 1:100 in M9 minimal media containing 0.2% glycerol, 0.01% Casamino Acids, 0.15 µg/ml biotin, and 1.5 µM thiamine (we denote this media MGC). Cells were placed on 1.5% MGC low melting temperature agarose pads containing 10-15 µM IPTG and grown at 37°C for 3 hours to equilibrate to the inducer conditions on the pad. The pad was then placed in 200 µl of MGC and shaken to release the cells. These equilibrated cells were placed on a fresh pad for time-lapse imaging. The temperature of the microscope chamber was kept at 32°C for the duration of the movie. Images were taken every 10 minutes in phase and each of the three fluorescent color channels.

Galactose Network

Cells were grown overnight in MO (M9 salts supplemented with 1 mM MgS04, 0.1 mM MgCl2, and 30 µg/ml kanamycin) supplemented with 0.4% (w/v) glucose, 0.5% (v/v) glycerol, and 0.1%

(w/v) Casamino acids (MON in [19]). Cultures were diluted back 1:50 in MO + 0.8% mannose

(and 10 mM D-fucose, where applicable). After reaching OD 0.1-0.2, cells were further diluted and placed on a pad made of the same MO + mannose (and fucose) media as the original dilution. Cells were grown and imaged at 37°C.

Fluorescence analysis of cell lineages was done with custom MATLAB software.

Stochastic Simulations

The circuit shown in Fig. 2A (right) was modeled as modeled as follows, with A, B, and C

representing the three indicated protein concentrations:

˙ A = " A + E + IA # $A " B˙ = B + E + I # $B (1) 1+(A /K)n B ˙ C = "C + E + IC # $C

These equations include terms for protein production (#i), protein degradation and dilution ($), ! and the contributions of extrinsic and intrinsic noise sources (E and Ii, respectively). All noise

and biochemical parameters are based on [2], or values measured directly from the present

experimental data; all are listed below for completeness. E(t) represents extrinsic noise and is the

same for the three proteins; Ii(t) for i={A,B,C} models intrinsic noise, which is distinct for each

protein. Noise sources are modeled as zero-mean signals with finite correlation times using an

Ornstein-Uhlenbeck process [14]. This extrinsic noise signal has a standard deviation of

#ext=0.35, and a correlation time proportional to the cell cycle length: Tcc/log(2). Intrinsic noise

1/2 has a standard deviation #int,i=(!i) , and a correlation time of Tint/log(2). The cell cycle time is

Tcc=60 mins and Tint=5 mins. The following parameters are used, unless otherwise indicated: n =

1.7, K = 120 nM, $A = $C = 1.39 mol/cell/min (chosen so that $A/% = K), $B = 4.5 mol/cell/min,

and % = log(2)/Tcc.

Cross Correlations

The cross correlation between two signals f(t) and g(t) is defined as

1 N " # "1 ~ S ( ) f (n )g~(n) 0 f ,g # = ! +# " # N " # n=0

S f ,g (") = Sg, f (#") ! " < 0.

N"1 ˜ 1 ! Here, f = f " # f (n) and! N is the number of time points. This function is normalized to N n= 0

S (") R (") = f ,g . This function is also known as cross covariance. f ,g S (0)S (0) ! f , f g,g

We adapted this standard formula to accommodate tree-structured (branching) data. First, we ! take each cell lineage separately, and calculate the Rf,g("). However, because all cell lineages

originate from a single cell, many pairs of points are counted multiple times. Therefore, we

subtract off the extra contribution of such point pairs, so that each pair is counted only once in

the overall sum. The following modified formula incorporates this correction:

1 1 +N cells #1% N#" #1 ( Ncells #2% ki #" #1 (. S (") = - ' f˜ (n + ")g˜ (n)* # ' f˜ (n + ")g˜ (n)*0 " # 0 f ,g N N $ $ i i $ ' $ i i * # " cells ,- i= 0 & n= 0 ) i= 0 & n= 0 )/0

! S f ,g (") = Sg, f (#") " < 0. !

Ncells "1 ˜ 1 ! ! f i = f i " # f i . Ncells i= 0

Here Ncells is the number of cells at the end of the movie and ki is the branching point between fi !

and fi+1. Note that the first term is the original cross correlation function averaged over Ncells. The

second term removes over-counted data.

As in the non-branched case, the function is normalized by 12

S f ,g (") R f ,g (") = . S f , f (0)Sg,g (0)

Experimental cross correlation curves have been cropped at the ends to remove points that are ! generated by very small amounts of data.

Model Fits

Optimization software (MATLAB) was used to minimize the difference between experimental

and simulated data, taking into account standard errors for experimental data points. Three

parameters were fit: g, the derivative of the gene regulation function evaluated at the steady state

repressor concentration (see Supplementary Text); WE/WI, the ratio of extrinsic to intrinsic

noise; and #B, the rate of protein production in steady-state.

Feed Forward Loop Simulations

The cross correlation functions were calculated analytically using the methods described in the

Supplementary Material. The following linearized equations were used to describe the system:

x˙ = "#x + E + $x

y˙ = "#y + gxy x + E + $y

z˙ = "#z + gxz x + gyz y + E + $z,

where the parameter values used in these calculations are gxy = g, gxz = g, gyz = -g, where g = !

0.01, and the white noise power constants are Wx = Wy = Wz = 1 and We = 0.064. Addition of

fucose is modeled by setting gyz = 0. Here, for simplicity, we model intrinsic noise as white

noise. Extrinsic noise is modeled using an Ornstein-Uhlenbeck process, as described in the

Supplementary Material.

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Figure Captions

Figure 1: Using noise to analyze the activity of gene regulatory interactions. (A) Target gene expression versus repressor concentration (schematic). Regions where changes in repressor concentration cause changes in target gene expression are active. When repressor levels saturate

(right) or are insufficient to repress (left), then the link is inactive. Regulatory links may be inactive for other reasons as well. (B) Noise can produce different types of static correlations.

In each plot, dots represent individual cells from a hypothetical population. Top plots show correlations without an active regulatory link, while the bottom shows correlations with active repression (gene regulation function is shown as a black line). We consider noise regimes in which either intrinsic (uncorrelated) or extrinsic (correlated) noise dominates. Active repression causes negative correlations between the transcription factor and its target, intrinsic noise decorrelates the two, and extrinsic noise causes positive correlations even without active regulation. Thus, correlations derived from static snapshots are ambiguous. (C) Temporal gene expression patterns for a repressor, A, (green line) and its target, B, (magenta line) concentrations are anti-correlated at a delay time denoted !reg (schematic).

Figure 2: Dynamic cross correlations in simulated regulatory interactions. (A) Protein A (green line) represses production of protein B (magenta line). Protein C (cyan line) is expressed constitutively. Data are normalized by mean concentration. Extrinsic noise positively correlates the time traces, while fluctuations in A (green curve) produce opposite fluctuations in B

(magenta) at a delay ("reg). Simulated time traces are shown for three noise regimes, as indicated.

Note that the intrinsic noise time scale is much faster than that due to extrinsic noise. See 16

Methods for detailed model and numerical parameters. (B) Mean cross correlation functions

RA,B(!) and RA,C(!) are shown in red and blue, respectively. Dots represent simulated data, while solid lines plot analytic solutions for the linearized model. Note that active repression causes a dip at !reg, while extrinsic noise results in positive correlation near "=0. See Methods for definition of cross correlation function.

Figure 3: Time-lapse movies of gene expression fluctuations in a synthetic genetic circuit. (A)

Synthetic regulation system. LacI, produced constitutively from a chromosomal gene, regulates production of a fluorescent protein fusion to the ! CI repressor, which in turn represses production of RFP. cfp is expressed from a strong constitutive promoter. All three genes experience the same extrinsic noise, while intrinsic noise is different for each. (B) Left panels show filmstrip of cells with chromosomally integrated circuit. Cells are imaged in three colors; each filmstrip shows two colors at a time for clarity. Note strong anti-correlation between RFP

(red) and YFP (green), and the lower correlation between CFP (blue) and YFP. Scale bar, 5 µm.

Right panel shows individual colors and phase images for t = 330 mins. Note anti-correlation between RFP and YFP and the uniform expression of CFP. (C) Filmstrip of the same circuit on a low-copy plasmid shows increased variability in all colors. In particular, in the right panel note the increased variability of CFP relative to the chromosomal case. Note that yellow pixels indicate equal levels of red and green. Colors and scale bar are the same as in (B).

Figure 4: Experimental data and cross correlation analysis of regulated and unregulated target genes. (A) Typical lineage traces for the chromosomally integrated construct. RFP is shown unshifted (left) and shifted by !reg=100 mins (right) to reveal the delayed anti-correlation. Data 17

are normalized by mean intensity. (B) The plasmid-based construct also shows delayed anti- correlation with !reg =120 mins. (C) Cross correlation functions RCI-YFP,CFP(!) (blue circles) and

RCI-YFP,RFP(!) (red circles) and model fits (dashed black line) for the chromosomally integrated construct. Chromosomal data are averaged over n=5 independent movies (with 100-200 cells per microcolony upon movie completion), n=6 movies for plasmid data. Error bars show standard error of the mean. For the model fit to the chromosomal construct, local sensitivity g=-0.01, ratio of extrinsic to intrinsic noise WE/WI=4.5, and strength of target promoter, #B=1.7. See Methods for definition of parameters. All other parameters are listed in Methods. (D) Cross correlation functions for the plasmid-based construct. For the model fits, g=-0.01, WE/WI=1.7, #B=0.5.

Figure 5: The activities of regulatory interactions in the gal feed-forward loop are cell state- dependent. (A) In a model feed-forward loop, the cross correlation between Y and Z exhibits a positive peak at !=0 due to simultaneous correlation by X and extrinsic noise. In addition, repression of Z by Y produces a negative dip in the cross correlation function at !reg < 0. (B)

When repression by Y is inhibited, the cross correlation function exhibits positive correlations without a time lag due. (C) The galactose metabolism pathway is controlled by a feed-forward loop and additional regulation by GalR. (D) Cross correlation functions between PgalS-YFP and

PgalE-CFP with and without fucose show similar features (n=15 movies without fucose, n=8 movies with fucose). In particular, only positive correlations without a time delay are observed.

(E) When galR is deleted, galE is only controlled by a feed-forward loop. (F) Cross correlation functions between PgalS-YFP and PgalE-CFP without fucose show delayed anti-correlation between galS and galE. When fucose is added, this anti-correlation is eliminated. n=9 movies

both with and without fucose. Note that the peak of the cross correlation function is reduced from the galR deletion strain, suggesting that noise in GalR positively correlates GalS and GalE.

Captions for Supplementary Figures and Movies

Figure S1. Map of plasmid pNS2-%VL. This plasmid contains the synthetic circuit shown in

Fig. 3A. Features include two fluorescent protein genes (cfp and mCherry), the cI-yfp

gene fusion, a low-copy SC101 origin of replication, kanamycin resistance gene, and

regulating promoters as indicated. In particular, PVN represents the OR2* variant of the !

PR promoter.

Movie S1. Movie of YFP (false colored in green) and RFP (red) for the chromosomally

integrated synthetic circuit. 10 minutes between frames. Frames from this movie are

shown in Fig. 3B.

Movie S2. Movie of YFP and CFP for the chromosomally integrated synthetic circuit.

Figure 1

A B Intrinsic >> Extrinsic Noise Extrinsic >> Intrinsic Noise C

Inactive Active Inactive A B

No Regulation (or Inactive) Expression Level Expression Level !

Protein Concentration reg 0 Time [Repressor]

A B Active Repression Expression Level

[TF] [TF] Figure 2

A1 A2 A3 Extrinsic Noise Only Intrinsic Noise Only Extrinsic and Intrinsic Noise

1.2 1.4

1.1 1.1 1.2 A A 1 B 1 1 C B C 0.9 0.8 0.9 0.8 Normalized Concentration 0.6 0 100 200 300 400 500 0 100 200 300 400 500 0 100 200 300 400 500 Time (mins) Time (mins) Time (mins)

B1 B2 B3 Extrinsic Noise Only Intrinsic Noise Only Extrinsic and Intrinsic Noise

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No Regulation R ( ) 0.5 0.5 0.5 A,C ! Nonlinear Simulation Linearlized Exact Solution R(!) 0 0 0 Repression RA,B(!) Nonlinear Simulation !0.5 !0.5 !0.5 Linearlized Exact Solution !reg !reg !reg !1 !1 !1 !450 !300 !150 0 150 300 450 !450 !300 !150 0 150 300 450 !450 !300 !150 0 150 300 450 ! (mins) ! (mins) ! (mins) Figure 3

A Intrinsic Noise

cI-yfp PLlac0-1

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Intrinsic Intrinsic rfp cfp Noise P (O 2*) Noise R R P!70

B Chromosome Chromosome YFP RFP

Phase CFP YFP CFP

YFP RFP 60 150 240 330 420 Time = 330 mins Time (mins)

C Plasmid Plasmid YFP RFP

Phase CFP YFP CFP

YFP RFP 60 150 240 330 420 Time = 330 mins Time (mins) Figure 4

A Chromosome Chromosome, Time Shifted 2 2

1.5 1.5

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Normalized 0.5 0.5 Fluorescense

0 0 0 50 100 150 200 250 300 0 20 40 60 80 100 120 140 160 180 200 Time (mins) YFP(t) Time (mins) YFP(t) RFP(t) RFP(t+!reg) B CFP(t) Plasmid Plasmid, Time Shifted

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!1 !1 !200 !100 0 100 200 !200 !100 0 100 200 ! (mins) ! (mins) Figure 5 a b

X X RY,Z(!) RY,Z(!)

1 1 Y Y 0.75 0.75

Z 0.5 Z 0.5 0.25 0.25

-600 -400 -200 200 400 600 -600 -400 -200 200 400 600 -0.25 ! (mins) -0.25 ! (mins) -0.5 -0.5

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-1 -1 c d RYFP,CFP(!) 1 galactose/ cAMP-CRP fucose 0.5

0 GalS GalR

!0.5 0 mM Fucose 10 mM Fucose GalETKM !1 !600 !400 !200 0 200 400 600 ! (mins) e f RYFP,CFP(!) 1 galactose/ cAMP-CRP fucose 0.5

GalS 0

!0.5 0 mM Fucose 10 mM Fucose GalETKM !1 !600 !400 !200 0 200 400 600 ! (mins) 1 Analytic Solutions for Cross Correlation Func- tions Due to Noise

In this section we develop an analytic method for calculating arbitrary cross correlation functions. We consider the following system of equations

A˙ = αA + E + IA βA (1) α − B˙ = B + E + I βB (2) 1 + (A/K)n B − C˙ = α + E + I βC, (3) C C −

where E represents extrinsic noise and Ii is intrinsic noise for i = A, B, C . The noise sources have zero mean so the equilibrium point of the deterministic{ } system αB can be calculated as A = α /β, B = n , and C = α /β. We expect eq A eq β(1+(αA/(βK)) ) eq C perturbations due to noise to be small, so it is valid to linearize the system about the equilibrium point. Deﬁning a = A A , b = B B , and c = C C we obtain − eq − eq − eq the following set of linear dynamics

a˙ = E + I βa (4) A − b˙ = ga + E + I βb (5) B − c˙ = E + I βc (6) C − where g is the local sensitivity

αA n−1 αB n ( ) g = βK . αA n 2 −K(1 + ( βK ) )

If we assume that the mean value of A is in the center of the Hill function so Aeq = αB n K, as is the case with our simulated system, then this simpliﬁes to g = 4K . This analysis is still possible regardless of the steady state value of A, but certain− regimes (the middle of the Hill function, saturated edges of the nonlinearity) are better approximated by linear models. Note that the constant g is the only place that information about the nonlinearity enters the equations. The cross correlation theorem states that cross correlation in the time domain is equal to multiplication in the frequency domain

−1 ∗ Rf,g(τ)=F [f˜ g˜],

where ∗ denotes the complex conjugate and f(t) and g(t) are the two signals. However, we average over many cross correlation functions, so we need to calculate the expected value of the cross correlation function over many realizations of the noise

E R (τ) = E F −1[f˜∗g˜] . (7) { f,g } { }

1 Taking the Fourier transform of Eqns. (4)–(6) we ﬁnd 1 a˜ = (E˜ + I˜ ) (8) β + iω A 1 ˜b = (E˜ + I˜ + ga˜) (9) β + iω B 1 c˜ = (E˜ + I˜ ) (10) β + iω C θ E˜ = η˜ (11) β + iω E λ I˜ = i η˜ . (12) i κ + iω i Below, we calculate cross correlation expressions for two cases: two independent genes (A and C in Fig. 2 of the main text) and a simple regulatory link with repression (A and B). The ﬁrst, and simpler, case is worked through in detail, while the results of the second case are summarized.

1.1 Unregulated Case We substitute Eqns. (8)–(12) into Eqn. (7), dropping tildes to simplify notation

1 θ λ 1 θ λ E R (τ) = E F −1[ ( η∗ + A η∗) ( η + C η )] { a,c } { β iω β iω E κ iω a β + iω β + iω E κ + iω c } − − − 1 θ2 θλ = E F −1[ ( η∗ η + C η∗ η { β2 + ω2 β2 + ω2 E E (β iω)(κ + iω) E c − θλ λ λ + A η∗η + A C η∗η )] . (β + iω)(κ iω) a E β2 + ω2 a c } − Because the Fourier transform is a linear operation we can analyze each of the four terms individually. We use two features of white noise to simplify analysis. ∗ First, white noise has a ﬂat power spectral density ηi (ω)ηi(ω)=Wi and second E η (t)η (t) = 0 for i = j. Thus, { i j } " E η (t)η (t) = E F −1[F [η (t)η (t)]] { i j }i#=j { i j } = E F −1[η∗(ω)η (ω)] { i j } =0.

Therefore if we have a deterministic function G(ω)

−1 ∗ 1 −1 −1 ∗ E F [G(ω)η (ω)ηj(ω)]] i#=j = E F [G(ω)] ) F [η (ω)ηj(ω)] { i } {√2π i } 1 −1 −1 ∗ = F [G(ω)] ) E F [η (ω)ηj(ω)] √2π { i } =0,

2 where ! represents convolution. Due to these white noise properties the last three terms in the cross correlation expression become zero and the remaining term simpli- ﬁes to 1 θ2 E R (τ) = F −1[ W ]. { a,c } β2 + ω2 β2 + ω2 E Applying the inverse Fourier transform we ﬁnd

∞ 2 1 1 θ −iωτ E Ra,c(τ) = 2 2 2 2 WE e dω. { } 2π !−∞ β + ω β + ω This integral can be solved using Cauchy’s Residue theorem. Specifically, we need to consider two cases: τ < 0 and τ 0. In the first case we can apply Jordan’s lemma if we use a contour that encircles≥ the upper half plane (Fig. 1a). Using Cauchy’s Residue theorem we find

R lim f(z)dz + f(z)dz =2πi Res. R→∞ ! ! CR −R "

By Jordan’s lemma the contour at inﬁnity, CR, becomes zero and we are left with the integral we want to evaluate. In our integral we have a second-order pole at z = iβ and a second-order pole at z = iβ. Since we are closing the contour in the upper − half plane we need to evaluate the residue at z = iβ:

∞ 2 1 1 θ −iωτ 1 2 2 2 2 WE e dω = 2πiRes(iβ) 2π !−∞ β + ω β + ω 2π θ2W = E eβτ (1 βτ). 4β3 − For τ 0 we can use Jordan’s lemma if we choose a contour that encircles the lower half plane≥ (Fig. 1b). Since the direction of encirclement is now clockwise, the residue theorem has an additional negative sign:

∞ 2 1 1 θ −iωτ 1 2 2 2 2 WE e dω = ( 2πiRes( iβ)) 2π !−∞ β + ω β + ω 2π − − θ2W = E e−βτ(1 + βτ). 4β3 Combining these two results we ﬁnd

θ2W E R (τ) = E e−β|τ|(1 + β τ ). { a,c } 4β3 | | Note that this expression for the cross correlation used a and c, the mean subtracted versions of A and C. These expressions are consistent with those calculated directly from the nonlinear simulations because we used the mean subtracted version of the cross correlation function.

3 a b

Im(z) Im(z)

iβ iβ

-R R Real(z) Real(z) -R R

-iβ -iβ

Figure 1: Contours used for evaluating Cauchy’s Residue theorem. Pole locations shown are for the unregulated cross correlation expression.

1.2 Regulated Case Analysis of the cross correlation function due to repression is similar. After simpliﬁ- cation using the white noise properties discussed in the previous section we ﬁnd

θ2 W E R (τ) = F −1[ E ] { a,b } (β + iω)2(β iω)2 − g θ2 W + F −1[ E ] (β + iω)3(β iω)2 − g λ2 W + F −1[ A A ]. (β + iω)2(β iω)(κ + iω)(κ iω) − − These three terms have a convenient interpretation: The first term is the artificial correlation due to extrinsic noise, the second term is extrinsic noise that has propagated through the link, the third term is intrinsic noise that has propagated through the link. Cauchy’s Residue theorem and Jordan’s lemma are applied to find

2 θ WE βτ 4β3 e (1 βτ) 2 −  gθ WE βτ 2 2  + 16β4 e (3 4βτ +2β τ )  −2 2  2 βτ κ (1−2βτ)−β (5−2βτ) κτ 1  +λAgWA(e 4β2(β2−κ2)2 + e 2κ(β−κ)2(β+κ) ) τ < 0  Ra,b(τ)=  2  θ WE −βτ 4β3 e (1 + βτ) 2  gθ WE −βτ  + 16β4 e (3 + 2βτ)  − −  2 e βτ e κτ  +λ gW ( 2 2 2 + 2 ) τ 0.  A A 4β (κ −β ) 2κ(β+κ) (β−κ)  ≥ 

4 1.3 Summary The cross correlation relations are summarized as θ2W E R (τ) = N E e−β|τ|(1 + β τ ) { A,C } A,C 4β3 | | 2 θ WE βτ 2 2 NA,B( 16β4 e (2gβ τ 4β(g + β)τ +3g +4β) 2 −2  2 βτ κ (1−2βτ)−β (5−2βτ) κτ 1  +λAgWA(e 4β2(κ2−β2)2 + e 2κ(β−κ)2(β+κ) )) τ < 0  E RA,B(τ) =  { }  2  θ WE −βτ NA,B( 16β4 e (2β(g +2β)τ +3g +4β) 2 −βτ 1 −κτ 1  +λAgWA(e 2 2 2 + e 2 )) τ 0  4β (κ −β ) 2κ(β+κ) (β−κ) ≥  where the normalization factors are 1 NA,B = RA,A(0)RB,B(0) % 1 NA,C = RA,A(0)RC,C (0) θ%2W λ2 W R (0) = E + A A A,A 4β3 2βκ(κ + β) θ2(3g2 +6gβ +4β2) W λ2 g2(κ +2β) W λ2 W R (0) = E + A A + B B B,B 16β5 4κβ3(κ + β)2 2κβ(κ + β) θ2W λ2 W R (0) = E + C C . C,C 4β3 2βκ(κ + β) To compare these results to the nonlinear simulation we use the constants speciﬁed αB n in Table 1, where g = 4K . WE, Wi for i = A, B, C are treated as binary variables and set to 0 or 1 to turn− oﬀ and on extrinsic{ and intrinsic} noise for comparison to the full nonlinear simulations shown in the text. We assume WA = WB = WC for simplicity. Fig. 2 in the text shows that the analytic solutions for cross correlations match the simulated nonlinear system extremely well. We have calculated the cross correlation functions for two types of regulation. This method can be applied more generally to larger networks, provided perturbations due to noise are small enough that linearization is a valid approximation.

1.4 Sensitivity Analysis Two prominent features of the cross correlation curve for repression are the location of the dip, τreg, and the magnitude of the dip, M (shown schematically in Fig. 2). We calculate how sensitive these features are to variations in the system parameters. These results indicate which parameters play a primary role in setting the features of the cross correlation function. To ﬁnd τreg for repression we take dR (τ) a,b =0. dτ &τ=τreg & & 5 Parameter Value αA 1.39 molecules/cell/min αB 4.5 molecules/cell/min αC 1.39 molecules/cell/min β 0.0116 1/min K 120 nM n 1.7 κ 0.139 1/min θ 0.0532 (molecules/cell)1/2/min 1/2 λA 0.621 (molecules/cell) /min 1/2 λB 1.12 (molecules/cell) /min 1/2 λC 0.621 (molecules/cell) /min Tcc 60 mins Tint 5 mins Table 1: Simulation Parameters

M τ τ

τreg

Figure 2: Schematic of cross correlation function features (a) τreg and (b) M.

6 In general it is not possible to find a closed form solution for τreg, however numerical root finding methods can be applied. The nominal parameter values for sensitivity analysis are those listed in Table 1. For a feature of the cross correlation function, y, we find the normalized sensitivity

y(pi + ∆pi) y(pi ∆pi) Si = − − . 2 ∆pi y(pi)

For ∆pi =0.05pi the parameteric sensitivities are shown in Table 2. Large values indicate that the feature y is very sensitive to that parameter. The sign of the sensitivity indicates whether y will get larger (Si > 0) or smaller (Si < 0) as the parameter is increased.

Parameter τreg M β -108.5 2.39 g 18.9 -85.7 θ 11.3 5.5 κ 4.0 -12.4 λA -0.9 2.5 λB 0.0 0.0 λC 0.0 0.0 Table 2: Normalized Sensitivities

τreg is most sensitive to the parameter β, which sets the time scale of both protein decay and extrinsic noise. As the cell cycle (log(2)/β) gets longer, the location of the dip moves further away from zero. M is most sensitive to the local sensitivity g, which is negative for repression. As g becomes less negative, the repressor has less of an eﬀect on its target and the dip gets smaller. In the extreme case when g = 0, which indicates an inactive or non-existent regulatory connection, the dip disappears.

2 Degenerate Cross Correlation Functions

Diﬀerent regulatory networks can generate similar cross correlation functions. This property is referred to as the “degeneracy” of the cross correlation function, and can interfere with attempts to use the cross correlation function to uniquely identify unknown regulatory architectures. Here we explore how degeneracy appears and ways in which its limits may be partially circumvented.

2.1 Redundant Network Elements The problem is shown clearly with the network diagram in Fig. 3. Here we assume that regulatory parameters are identical for all the links (except for the sign of regulation, repression vs. activation). Proteins B and C are redundant because they are controlled by the same input, A, and extrinsic noise aﬀects them in the

7 same way. Thus, in the extreme case where only extrinsic noise is present, B(t) and C(t) will be identical. Consequently, the cross correlations of each with D are equal, RB,D(τ)=RC,D(τ), even though there is no link between B and D.

B C

Figure 3: Network with redundant components. Cross correlation functions RB,D(τ) and RC,D(τ) may look similar.

Intrinsic noise helps to discriminate between RB,D(τ) and RC,D(τ) because intrinsic noise in C propagates to D causing additional time-lagged correlation. Intrinsic noise in B, on the other hand, does not propagate to aﬀect D. Thus, one obtains an additional time-lagged component in RC,D(τ) compared to RB,D(τ). More speciﬁcally, analytic solutions for the two cross correlation functions are summarized by

R (τ)=E F −1[f ∗ f + f ∗ f ] B,D { Be De BA DA } R (τ)=E F −1[f ∗ f + f ∗ f + f ∗ f ] , C,D { Ce De CA DA CC DC }

where fi,j is the Fourier transform of the differential equations describing the dynamics of protein j in response to noise from i, ∗ is the complex conjugate, e is extrinsic noise, and E refers to the expected value. Assuming the network connections are identical for A-B and A-C we find the difference between the two cross correlations

R (τ) R (τ)=E F −1[f ∗ f ] . C,D − B,D { CC DC } This term represents propagation of noise in C to affect D. It is non-zero due to intrinsic noise in C. Fig. 4 compares RB,D(τ) and RC,D(τ). RC,D(τ) is lower because intrinsic noise in C propagates through the repression link. The difference increases when intrinsic noise in C gets larger. This simple example illustrates how redundant network elements can confound analysis. More complicated networks will have similar problems any time there are two or more network elements that are controlled by the same inputs. Intrinsic noise or other signals that affect individual genes can help to distinguish correlations due to regulation from correlations due to symmetric network elements. This degeneracy arises due to the effective redundancy of two genes, B and C, which are regulated identically and have similar function. In practice, such redundancy may be unusual. For example, in the case of the galactose network considered in the text, galR and galS have similar function, and even similar binding sites. How- ever, galS and not galR, is regulated by CRP. Thus, the two genes are not redundant.

8 1.0

R (τ) 0.5 B,D RC,D(τ)

- 600 - 400 - 200 200 400 600 τ (mins)

- 0.5

- 1.0

Figure 4: Cross correlation functions for network with redundant elements.

2.2 Parametric Degeneracies A second class of degeneracies comes from uncertainty in parameters. Consider the two cascades in Fig. 5. If we measure RA,C (τ) is it possible to determine that there is a middle element in the network or will the cross correlation function be indistinguishable from that of RX,Y (τ) in certain cases? In this example extrinsic

A X Ext. noise B Ext. noise Y C

Figure 5: Example of a network showing parametric degeneracy. Gray arrows show how extrinsic noise aﬀects the system.

noise is helpful in discriminating between the two cascades. In the two-step cascade, extrinsic noise affects A, B, and C, and thus enters into the cross correlation function in three ways, while in the one-step cascade it only enters twice. Without extrinsic noise it is possible to adjust parameters to make the two cross correlation functions look nearly identical (Fig. 6a). For example, if B degrades quickly, but the net strength of the two networks are the same then the resulting cross correlation functions are very similar. When extrinsic noise is considered, the two cross correlations separate (Fig. 6b). Extrinsic noise increases overall positive correlation near τ = 0 for both cascades, but the effect is more pronounced in the longer cascade. Additionally, extrinsic noise increases the the effective regulatory timescale of repression in the longer cascade, shown by the location of the dip at τ < 0.

9 a b Intrinsic Noise Only Extrinsic and Intrinsic Noise 1 1 R (τ) 0.75 A,C 0.75 R (τ) 0.5 X,Y 0.5

0.25 0.25

-400 -200 200 400 -400 -200 200 400 -0.25 τ (mins) -0.25 τ (mins)

-0.5 -0.5

-0.75 -0.75

-1 -1

Figure 6: Cross correlation functions for two cascades with degenerate parameters. (a) No extrinsic noise, WE = 0. (b) Extrinsic and intrinsic noise. Simulation parameters are β = β = β = β =1/60 mins−1, β =1/5 mins−1, g = g g = 0.01, x y a c b xy ab bc − Wi = 1, WE =0.064.

In this example the intrinsic noise-only dynamics are described by

a˙ = β a + η − a a b˙ = β b + η + g a − b b ab c˙ = β c + η + g b − c c bc

x˙ = β x + η − x x y˙ = β y + η + g x − y y xy where intrinsic noise has been approximated by white noise and parameters are given in the ﬁgure caption. Since biochemical parameters are often unknown or uncertain there are many possible situations where the shape of two cross correlation functions may look very similar, but knowledge of noise and network properties can help to clarify cross correlations, as described here.

10 Figure S1

P!70

CFP kanR

pNS2-!VL 7.5 kb mCherry SC101

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PLlacO1 cI YFP fusion