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requency domain analysis of noise in simple gene circuitsChris D. Coxa�

Department of Civil and Environmental Engineering, University of Tennessee, Knoxville, Tennessee 37996and Center for Environmental Biotechnology, University of Tennessee, Knoxville, Tennessee 37996

James M. McCollumDepartment of Electrical and Computer Engineering, University of Tennessee, Knoxville, Tennessee 37996

Derek W. AustinSiemens Molecular Imaging, Knoxville, Tennessee 37932

Michael S. AllenMolecular-Scale Engineering and Nanoscale Technologies Research Group, Oak Ridge NationalLaboratory, Oak Ridge, Tennessee 37831-6006; Center for Environmental Biotechnology, Universityof Tennessee, Knoxville, Tennessee 37996; and Department of Materials Science, University of Tennessee,Knoxville, Tennessee 37996

Roy D. DarMolecular-Scale Engineering and Nanoscale Technologies Research Group, Oak Ridge NationalLaboratory, Oak Ridge, Tennessee 37831-6006 and Department of Physics, University of Tennessee,Knoxville, Tennessee 37996

Michael L. Simpsonb�

Molecular-Scale Engineering and Nanoscale Technologies Research Group, Oak Ridge NationalLaboratory, Oak Ridge, Tennessee 37831-6006; Department of Materials Science, University of Tennessee,Knoxville, Tennessee 37996; and Center for Environmental Biotechnology, University of Tennessee,Knoxville, Tennessee 37996

�Received 30 January 2006; accepted 21 April 2006; published online 30 June 2006�

Recent advances in single cell methods have spurred progress in quantifying and analyzing sto-chastic fluctuations, or noise, in genetic networks. Many of these studies have focused on identi-fying the sources of noise and quantifying its magnitude, and at the same time, paying less attentionto the frequency content of the noise. We have developed a frequency domain approach to extractthe information contained in the frequency content of the noise. In this article we review our workin this area and extend it to explicitly consider sources of extrinsic and intrinsic noise. First wereview applications of the frequency domain approach to several simple circuits, including a con-stitutively expressed gene, a gene regulated by transitions in its operator state, and a negativelyautoregulated gene. We then review our recent experimental study, in which time-lapse microscopywas used to measure noise in the expression of green fluorescent protein in individual cells. Theresults demonstrate how changes in rate constants within the gene circuit are reflected in thespectral content of the noise in a manner consistent with the predictions derived through frequencydomain analysis. The experimental results confirm our earlier theoretical prediction that negativeautoregulation not only reduces the magnitude of the noise but shifts its content out to higherfrequency. Finally, we develop a frequency domain model of gene expression that explicitly ac-counts for extrinsic noise at the transcriptional and translational levels. We apply the model tointerpret a shift in the autocorrelation function of green fluorescent protein induced by perturbationsof the translational process as a shift in the frequency spectrum of extrinsic noise and a decrease inits weighting relative to intrinsic noise. © 2006 American Institute of Physics.�DOI: 10.1063/1.2204354�

tochastic fluctuations, or “noise,” in gene circuits ariseue to the random timing of biochemical reactions withincell and the discrete nature of the molecular popula-

ions affected by these reactions. The noise can originaterom variations in the global resources (e.g., polymerases,ibosomes, nucleic and amino acids) utilized in transcrip-ion and translation (called “extrinsic noise”) or within

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the transcriptional and translational processes them-selves (called “intrinsic noise”). Not only have cellsevolved to maintain fidelity of gene function in the pres-ence of noise, there are many examples in which cellsexploit noise-driven phenotypic diversity to achieve a de-sired end. Further, the manner in which noise propagatesthrough a gene circuit allows certain inferences about thegene circuit architecture. In recent years a number oftheoretical and experimental studies have been under-
taken to characterize the sources and propagation of
© 2006 American Institute of Physics2-1

license or copyright, see http://chaos.aip.org/chaos/copyright.jsp

http://dx.doi.org/10.1063/1.2204354

http://dx.doi.org/10.1063/1.2204354

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026102-2 Cox et al. Chaos 16, 026102 �2006�

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oise in gene networks. Most of these studies have fo-used on quantifying and interpreting the magnitude ofhe noise as quantified by distributions or statistics of aoise-affected output within a population of cells. Weave developed a frequency domain (FD) approach thataptures information about both the magnitude of sto-hastic fluctuations and their timing within individualells and demonstrates that the timing plays a criticalole in gene circuit performance. Here we review recentdvances in characterizing noise in gene circuits, describehe analytical and experimental aspects of the FD ap-roach, and demonstrate its application to the analysis of

ntrinsic and extrinsic noise in transcription and transla-ion processes.

. INTRODUCTION

In 1940 the physicist Max Delbruck recognized thatmall populations of enzyme molecules within a cell wouldive rise to statistical fluctuations in biochemical reactionsnd that these fluctuations could have profound impacts onell physiology.1 He later proposed that fluctuations of thisype could explain the variation in the number of virusesroduced upon lysis of infected bacteria.2 In the followingecades many examples of the importance of stochastic fluc-uations in gene regulation have been reported. Prokaryoticxamples include regulation of lac expression at low levelsf induction,3 the lysis-lysogeny decision in phage-�,4,5 andhe swimming and tumbling periods of bacteria duringhemotaxis.6 Examples of stochastically driven phenotypeariability in eukaryotes include variability in the response toating pheromone in yeast7 and the notch-mediated

pidermal-neural decision in Drosophila neuro-ectoderm.8

ecent studies have also suggested that stochastically drivenhenotype diversity in a clonal population may ensure that aew cells remain poised to exploit changing environmentalonditions,9 thereby improving fitness.

Stochastic fluctuations �hereafter referred to as noise �re introduced into the populations of mRNA, protein, andther molecular species by several sources. The random tim-ng and discrete nature �i.e., integer number of molecules� ofolecular events such as transcription, translation, multim-

rization, and protein/mRNA decay processes lead to a noiseomponent that is intrinsic to a local gene circuit orathway.5,10,11 Conversely, fluctuations in RNAP, ribosomes,ranscription factors, or other cellular molecular machineryhared by gene circuits or pathways lead to an extrinsic noiseomponent.12,13 The difference between intrinsic and extrin-ic noise is not just one of definitions, as these fluctuationsave different sources, attributes, and consequences �Fig. 1�.or example, although intrinsic noise in one gene circuit isncorrelated with that in any other gene circuit, extrinsicoise is correlated across gene circuits as it arises fromhared cellular resources. Further, this sharing of resourcesnd transmission of fluctuations provides for a coupling be-ween gene circuits that have no direct regulatoryelationship.14 That is, a high demand for expression by oneathway can affect the rate at which an unrelated pathway isxpressed.


Over the last few years there has been a great deal offocus on the theory, analysis, modeling, and simulationof stochastic fluctuations in gene circuits andnetworks.5,7,10,13,15–26 The most rigorous and accurate math-ematical representation for calculating the discrete stochastictime evolution of a reacting system is the chemical masterequation �CME�, which describes how the overall probabilityof any state in the system evolves over time as a result ofvarious possible chemical reactions.18 Unfortunately, theCME approach becomes impractical for genetic circuits andnetworks of even moderate complexity. A tractable methodof dealing with discrete stochastic simulations is exact sto-chastic simulation;16,17 however, although simulation is aninvaluable aide in understanding stochastic fluctuations ingenetic systems, it is difficult to extract fundamental relation-ships between gene circuits parameters �e.g., kinetic rates�and function using simulation alone. Instead, such relation-ships have most often been predicted through mathematicalanalysis using approaches �e.g., Fokker-Planck and Lange-vin� that are simplifications of the CME.19,22,24,25,27

Coupled with the progress in gene circuit noise theory,analysis and simulation, there has been significant experi-mental progress. Elowitz et al. used the correlation of yellowand cyan fluorescent proteins under the control of identicalpromoters at different locations within the genome for theindependent measurement of the extrinsic and intrinsic noisecomponents described earlier, and showed that the contribu-tion of extrinsic noise to total noise may be larger than thatof intrinsic noise.12 Blake et al. measured gene circuit noisein eukaryotes and showed the effect of transcription reinitia-tion on the magnitude of the stochastic fluctuations.28 Raserand O’Shea extended the study of extrinsic and intrinsic

FIG. 1. �Color online� Intrinsic noise sources originate from the transcrip-tion and translation processes affecting a single gene, whereas extrinsicnoise sources affect all genes within the cell.

noise to eukaryotic gene expression by quantifying differ-


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026102-3 Frequency domain analysis: Gene circuits Chaos 16, 026102 �2006�

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nces of expression from two alleles in diploid yeast cells.29

hey demonstrated that extrinsic noise was also dominant inwo wild-type yeast genes and that slow rates of promotertate transition could dominate intrinsic noise as predicted byheoretical studies.21,25 Similarly, Becskei et al. further dem-nstrated that noise associated with low molecular numberranscripts in yeast was not the result of the low populationser se, but rather resulted from relatively rare gene activationvents.30 Rosenfeld et al. used time-lapse microscopy torack reporter gene activity response to an inducer over mul-iple cell generations.31 They resolved the autocorrelationunction of the gene response into contributions from rapidlyecaying intrinsic noise and more persistent extrinsic noisehat decays over approximately one cell cycle. Pedraza andan Oudenaarden used cross correlation of three fluorescenteporter genes to determine how intrinsic and extrinsic noiseropagates through gene cascades.14 The synthetic cascadeas tunable via the concentrations of two inducers. A cali-rated analytical model of the cascade correctly predictedhanges in noise response as a function of changing induceroncentrations. However, all of these experimental studiesocused on noise magnitudes, and like their theoretical coun-erparts, largely ignored the frequency content of this noise.

Most of these theoretical and experimental investigationsave dealt with noise magnitude using stochastic distribu-ions �e.g., means, variances, standard deviations� of the mo-ecular populations at steady state, largely ignoring the ratef the fluctuations. In contrast, we have employed a FD ap-roach to analysis and experimentation that deals with theoise spectra content. In this approach the noise sources haveflat �i.e., equal power at all frequencies or white� spectrum,ut these spectra are shaped as the noise propagates from theources to molecular populations or concentrations of inter-st. As a result, there is a mapping between the spectral con-ent of gene expression fluctuations and the structure andunction of the underlying gene circuits.

Our FD analysis has demonstrated that the frequencyontent of the noise may have important implications forene regulation and function that are not captured by analy-is or measurement of noise magnitudes.24–26 For example,ur analysis predicts that kinetic parameters are reflected inhe noise frequency range and that negative autoregulationot only reduces the magnitude of the noise, but also shiftshe remaining noise to higher frequencies where it may haveittle or no regulatory effect.

Unfortunately, it is more difficult to perform FD mea-urements as these require relatively long time series mea-urements from individual cells. Consequently flow cytom-try, which has been a very productive tool for measuringtochastic distributions, cannot be used for FD measure-ents. However, we recently measured stochastic fluctua-

ions in reporter gene expression of growing cells using time-apsed microscopy.32 We measured noise frequency contentn growing cell cultures and verified our theoreticalrediction24 that in addition to noise magnitude, gene circuitsanipulate noise spectra, impacting the fate and regulatory

ffect of the noise as it propagates through the geneetwork.32 We verified our prediction24 of the shift of noise
pectra to higher frequencies and the link between gene cir-

cuit structure and the resulting noise frequency range andshowed that changes in gene circuit parameters, such as cellgrowth and protein decay rates, modulate the noise fre-quency range distributions.32 Further, these results showedthat measured noise frequency range distributions combinedwith stochastic simulations can be used to probe mechanisticdetails of molecular interactions within gene circuits.32

Here we review FD analytical and experimental tech-niques and our previous results. In addition, we present amodel that explicitly considers extrinsic noise sources ema-nating from the transcriptional and translational level and usethe model for the interpretation of our experimental resultswhere noise was significantly affected by perturbation oftranslational processes.

II. FREQUENCY DOMAIN ANALYSIS: TOOLSFOR GENE CIRCUIT ARCHITECTURAL ANALYSIS

Our FD approach is equivalent to the chemical Langevinanalysis and is applicable to linear systems and nonlinearsystems for which linear approximations are sufficiently ac-curate over conditions of interest. In the FD approach thegenetic/biochemical system is conceptualized as a chemicalcircuit with nodes for each chemical species, and the noise isgenerated by a set of noise sources �Fig. 2�a��. The noisesources are characterized by their power spectral densities�PSDs�, which describe how the noise is distributed in fre-quency space �Fig. 2�b��. The PSD of the noise in the con-centration �or population� of chemical species i �Si�f�� isfound from the weighted sum of the noise sources according

FIG. 2. Basic concepts in frequency domain analysis. �a� Chemical speciesA–H are represented as nodes, each of which represents a noise source withits own characteristic PSD. Noise power transfer functions describe the ef-fect of upstream noise sources �A–D� on downstream output nodes �E–H�.For example, the noise source at A is characterized by its PSD SAA�f� and itseffect on output E is given by HAE

2 �f� SAA�f�. �b� Stochastic time seriesgenerated by noise sources �left� are characterized by their power spectraldensities �right�. The grey and black curves represent high and low fre-quency noise sources, respectively. The vertical arrows indicate locations ofpole frequencies.

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026102-4 Cox et al. Chaos 16, 026102 �2006�

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Si�f� = �j=1

G

�Hij2 �f��Sj�f� = �

j=1

G

�Hij2 �0��Sj�0�fN�ij , �1�

here f is the frequency �Hz�, Sj�f� is the PSD of noiseource j, G is the number of noise sources contributing tooise in chemical species i, �Hij

2 �f�� is the noise power trans-er function between the noise source j and chemical species, and fN�ij is a characteristic frequency range �known as theoise bandwidth� of the noise in species i due to source j.he noise power transfer functions are found from

�Hij2 �f�� = � ��xi�f��

��rj�f��2

, �2�

here �xi�f�� is the magnitude of fluctuations at frequency fn the concentration of species i due to fluctuations in mo-ecular populations at frequency f caused by noise source jrj�f��.

The noise sources are well described by shot noise withwide band �compared to the frequency limitations of the

ene circuit� white spectrum.24 Shot noise is the noise thatriginates from the discrete nature of the signal carrier. TheSD of shot noise arising from a Poisson process is propor-

ional to the average flux �kT� through the process such that24

Sj�f� = 2kT, �3�

here Sj�f� is the single-sided �positive frequency only� PSDf the noise source j. At steady state the noise sources areound in pairs �a source and a sink; Fig. 3�a�� at the points ofolecular transitions �synthesis, decay, polymerization, com-

lex formation, etc.�, so the total PSD for a pair of noiseources at location j is

Sj�total�f� = 4kT. �4�

The noise power transfer functions in genetic circuits

IG. 3. Model of single gene expression. �a� noise sources include synthesisf mRNA molecules from the template DNA strand at rate �R, and transla-ion of proteins at a rate kP ·mRNA�t�. Noise sinks include decay of mRNAnd protein with first-order rate constants �R and �P, respectively. �b� Theame as �a� except that the protein is negatively autoregulated. The loopransmission T is calculated by breaking the loop at any convenient point,ntroducing a perturbation �� just to the downstream side of the break,

easuring the response �� that returns to the upstream side of the break andalculating T=��f� /��f�. Figure from Ref. 24.

ill be of the form


�Hij�f��2 = Hij2 �0�

n=1

nj,k �1 + � f

fzn�ij�2�

m=1

mj,k �1 + � f

fpm�ij�2�

, �5�

where fz1,. . .,n �zeros� and fp1,. . .,m �poles� are frequencies as-sociated with the kinetic parameters of the reactions betweennoise source j and chemical species i. The transfer functiondescribes how noise is modified as it propagates through thecircuit. Below the first pole frequency, the PSD is flat and thenoise power is independent of frequency. As the frequencyincreases above the first pole, the noise power decreases with1/ f2. The noise power decreases by an additional factor of1 / f2 for each additional pole encountered �Fig. 2�b��. Mostreactions generate only poles, although zeros are seen in re-versible reactions.26 Zeros have the opposite effect of poles:noise power increases with f2 at frequencies above the zerofrequency. Overall, the change in noise power with fre-quency can be described by f2�nz−np� where nz and np are thenumber of zeros and poles with characteristic frequenciesless than f . Note that the PSD of the sources are flat andfeatureless �Eq. �4��, but the PSDs in the species concentra-tions have structure created by the kinetic parameters of thereactions �Eq. �1��.

The noise power transfer functions are derived from thelinearization and Fourier or Laplace transformation of theordinary differential equations describing the system. There-fore FD analysis has most of the same limitations and cave-ats as the Langevin approach.19 One notable exception is thatthe FD approach can be used for some cases of very lowmolecular populations where the time domain Langevin ap-proach fails.25 Further, some limitations are overstated in theliterature. As has been demonstrated with electronic circuitsfor quite some time, FD techniques can be applied to somenonlinear systems. However, such applications must behandled with care and the results scrutinized.

III. APPLICATION OF FD ANALYSIS

A. Single gene circuit

Figure 3�a� is a schematic diagram of an unregulatedsingle gene system. Considering only the intrinsic noise, theLangevin equations for this system are

dr

dt= − ��R + ��r + �R�t� + �R,

�6�dp

dt= − ��P + ��p + kPr + �P,

where r and p are mRNA and protein concentrations, �R and�P are mRNA and protein decay rate constants, � is the rateof dilution due to growth, �R is the transcription rate, kP isthe translation rate constant, and �R and �p are random vari-ables that represent the noise. At steady state the averagemRNA �r�� and protein �p�� concentrations found fromEqs. �1� and �2� are �R / ��R+�� and �RkP / ��R+��P+��,

33
respectively. As the typical half-life of mRNA �2–5 min� is

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uch less than the cell doubling time, we will make thepproximation �R+��R.

From Eq. �4� the PSDs of the mRNA �SRR� and proteinSPP� synthesis and decay noise are

SRR = 4�R,

�7�

SPP = 4�RkP

�R.

he signal and noise power transfer functions from the pointf mRNA �HR�f�� and protein �HP�f�� synthesis to the re-orter protein output are found by Fourier transform andolution of the Langevin equations to obtain24:

HR�f� =b

�P + �

1

�1 + if

fmRNA��1 + i

f

fprotein� ,

�8�

HP�f� = � 1

�P + ��2 1

�1 + � f

fprotein�2� ,

�HR�f��2 =� b

�P + ��2

�1 + � f

fmRNA�2��1 + � f

fprotein�2� ,

�9�

�HP�f��2 =� 1

�P + ��2

�1 + � f

fprotein�2� ,

here the pole frequencies are associated with mRNAfmRNA=�R /2�� and protein �fprotein= ��p+�� /2�� decay, andhe term b �=kP /�R; often referred to as the burst rate� is theverage number of proteins produced from each mRNA tran-cript. The power transfer functions �Eq. �9�� can be under-tood in terms of the gain �numerator� and the frequencyodification �denominator� as the noise propagates through

he circuit. The gain for transcriptional noise sources isreater by a factor of b2. Simple transcription-translation cir-uits behave as low-pass filters, as noise becomes negligiblet frequencies greater than the pole frequencies. As the effectf a noise source decreases at 1 / f2 for frequencies higherhan the first pole, other poles are often neglected in multi-ole systems. As mRNA typically decays much faster thanhe protein ��R�P�, the mRNA pole may be neglected withittle error and the single pole noise bandwidth approxima-ion can be used.24 With this simplification, the noise band-idth �range of frequencies that have a significant noise con-

ent; �fN� for both noise sources can be approximated as24

�fN ��

2fprotein =

�p + �

4, �10�

nd the variance of the output �protein population� noise is
iven by

P2 � �HR

2�0�SRR + HP2 �0�SPP��fN = p��1 + b� , �11�

where we have used the relationships for p� and b definedearlier.

The noise figures of merit are

P2

p�= 1 + b , �12�

P

p�= �1 + b�

p�. �13�

Equation �12� gives the noise strength and Eq. �13� gives thecoefficient of variation �CV� with results that are in agree-ment with previous analysis.22,23 However, the FD approachshows that this noise is spread over a frequency range relatedto �fN and controlled by the protein decay and dilution rate.

The previous analysis provides a relationship that can beused to aide the analysis of multigene systems. For a consti-tutively expressed gene circuit

Spi �f� �

4bp�� + �p� 1

1 + � 2�f

� + �p�2� , �14�

where Spi �f� is the intrinsic noise in the protein concentration,

we have neglected the mRNA pole, and we have assumedthat b+1�b.

B. Transcriptional control

In many prior analyses transcriptional control was ap-proximated using a Hill expression:23,24,34,35

I =1

1 + �d/Kh�n , �15�

where I is the induction level of the gene, d is the concen-tration of a regulatory molecule, Kh is a constant that indi-cates the value of d at which the induction level reaches 0.5,and n is the Hill coefficient. Positive and negative values ofn correspond to repression and induction, respectively. Al-though this approach may reasonably approximate the staticdeterministic behavior of transcriptional regulation, it canlead to significant errors by neglecting both the dynamics�frequency response� and the noise of transcriptional regula-tion. A more realistic description would include switchingbetween discrete high and low transcriptional rates with theaverage rate determined by the fractional amount of timespent in each of the two states. This model is consistent withtranscription controlled through protein-DNA interactions atan operator site within the gene promoter region. We per-formed a FD analysis of the case of a single operator sitewithin a single copy of the operon as shown in Fig. 4. In thisanalysis we considered two possible states, denoted as O�unbound� and O� �bound�, and transition between these
states was described by

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026102-6 Cox et al. Chaos 16, 026102 �2006�

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O ——→kfd

O�,

�16�

O� ——→kr

O ,

here kf and kr are rate constants for the forward and reverseeactions, respectively, and d is the population of the molecu-ar species that controls transcription by binding with O.quation �16� applies to both positive and negative regula-

ion as the fully induced state may be either O or O�. As ourrevious analysis treated positive regulation,25 here we con-ider only the negatively regulated case �O�→minimally in-uced�. These results can be directly applied in the analysisf negative autoregulation that follows.

As there is only one operator site, the population of thenbound operator, O�t�, is either one or zero at all times. Foregative regulation we define

��t� = �I, O = 1, �17a�

IG. 4. Model of gene regulation. �a� Gene in basal state �O� with operatornbound by molecular species d and producing transcripts at slow rate. �b�ene in induced state �O�� with molecular species d bound to operator androducing transcripts at increased rate. �c� The operator transitions betweentate O and O� as a function of time. The level of induction is given by theractional time the gene spends in state O�. Operator dynamics are charac-erized by the average time the operator remains in each state prior to tran-itioning. Figure from Ref. 25.

��t� = �0, O = 0, �17b�


�R = �0�1 − O�t�� + �IO�t� = �0 + ��I − �0�O�t� , �17c�

where ��t�, �0, �I, and �R are the instantaneous, basal, fullyinduced, and average transcription rates respectively, andO�t� is the time-averaged value of O�t�.

As individual transcription events and O�t� are uncorre-lated random processes, the autocorrelation function for thegated term in Eq. �17c�, �O��, is given by

�O�� = �O�� , �18�

where �O�� is the autocorrelation function for O�t� and�� is the autocorrelation function for a random series ofimpulse functions �average rate=�I−�0� corresponding tomRNA synthesis.25 The PSD of the noise in the mRNA syn-thesis rate is found by summation of the PSDs of mRNAdecay noise �equal in magnitude, but uncorrelated to synthe-sis shot noise24�, the wideband white noise of the constantterm �i.e., basal gene expression� in Eq. �17c�, and the gatednoise of Eq. �18� to yield:25

SRR�reg�f� = 4��0 + O�t��I − �0��

+4��I − �0�2�O�t� − �O�t��2�

�kr + kfd�

� 1

1 + � 2�f

�kr + kfd��2�

= 4�R +

4�R2� 1

O�t�− 1�

�kr + kfd� � 1

1 + � 2�f

�kr + kfd��2� , �19�

where the subscript �reg denotes the regulated case and thePSD of the gated noise term was found by direct calculationand then Fourier transformation of the autocorrelation func-tion of Eq. �18�.25 The first term in Eq. �19� is the PSD of thenoise source associated with the average transcription rate,whereas the second term accounts for variations around theaverage transcription rate caused by the operator. The opera-tor component increases at low induction levels �low valuesof O�t�� and for slow operator dynamics �low values of �kr

+kfd��. Noise from slow operator dynamics is limited to lowfrequencies by the presence of the filtering action of the polein Eq. �19�.

The dynamics of this transcriptional regulation were alsoanalyzed to show that the transfer function, Ho�f�, from con-centration of species d to the transcription rate is given by25:

HO��f� = −�krkf�/�kr + kfd� + kf�

�kr + kfd�� 1

1 + i2�f

�kr + kfd� + kf�� ,

�20�

where d� is the average concentration of species d.This analysis shows that there are at least two major

shortcomings when using a Hill function to approximatetranscription regulation for stochastic analysis. The first termon the right-hand side �rhs� of Eq. �19� is a shot noise term24

and the only noise present when the Hill expression is used.


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he second term on the rhs of Eq. �19� is an additional noiseerm �operator noise� found from this more realistic treat-

ent of transcriptional regulation21,25 and may be the domi-ant source of transcriptional noise. Further, the Hill expres-ion assumes that changes in the transcription factoroncentration are immediately reflected in the transcriptionate, whereas Eq. �19� shows that this response is limited byhe operator dynamics.

. Autoregulated gene circuit

Autoregulation, or feedback, exists when the level of anutput of a gene circuit regulates the rate at which this outputs generated. To deal with autoregulation we applied the elec-ronic feedback concept of loop transmission to gene circuitnalysis. The loop transmission, T, is the transfer functionround the loop, or the frequency dependent first derivativef the regulation strength. It may be thought of as a measuref the resistance of the circuit �electronic or biochemical� toariation from the steady state. T is calculated by breakinghe loop at any convenient point �e.g., at the point of tran-criptional regulation; Fig. 3�b��, introducing a perturbation�� just to the downstream side of the break �e.g., a smallhange in transcription rate; Fig. 3�b�� and measuring theesponse �� that returns to the upstream side of the breake.g., the change the circuit would make in transcription rate;ig. 3�b��. T�f� is given by ��f� /��f�, and the sign of T�0� isegative �resists fluctuations� for negative autoregulation ands positive �reinforces fluctuations� for positive autoregula-ion. For the gene circuit of Fig. 3�b�:

T�f� = HR�f�Ho�f�

=− HR�0�Ho�f�

�1 + if

fprotein��1 + i

f

fmRNA�

=T�0�

�1 + i2�f

�P��1 + i

2�f

�R��1 + i

2�f

�kr + kfd0 + kf�� ,

�21�

here T�0�=−HR�0�Ho�0�, and the other terms are as de-cribed previously.

Here we consider transcriptional regulation where therotein product of the gene circuit negatively regulates theranscription rate by binding to an operator site in the pro-oter. We make the simplifying assumption that the operator

ynamics are fast compared to the protein decay and dilutionates and approximate mRNA synthesis noise by modifyingq. �19� such that

SRR�reg�f� = 4�R +

4�R2� 1

O�t�− 1�

�kr + kfd�. �22�

he protein synthesis noise found by modifying Eq. �7� toccount for the negative regulation mediated decrease of the
rotein synthesis rate to obtain

Spp�reg�f� = 4kpmRNA� =4kp�R

�R= 4b�R. �23�

The noise in the protein concentration may be found by theapplication of Eq. �9�, but the transfer functions and thenoise bandwidth have been altered by the negative autoregu-lation such that24

�HR�reg�0��2 =1

�Ho�0��2� − T�0�1 − T�0��

2

=1

�Ho�0��2�HR�0�Ho�0�1 − T�0� �2

=HR

2�0��1 − T�0��2 ,

�24�

�HP�reg�f��2 = � 1

bHo�0��2� − T�0�

1 − T�0��2

=1

�bHo�0��2�HR�0�Ho�0�1 − T�0� �2

=1

b2� HR�0�1 − T�0��

2

,

and

�fN � �1 − T�0��fprotein = �1 − T�0��p

4, �25�

where �reg denotes a transfer function for the negatively au-toregulated case and the other parameters are as previouslydescribed. Comparison of Eqs. �25� and �10� reveals thatnegative autoregulation extends the noise bandwidth by afactor of �1−T�0��, as T�0� is negative. The increase in band-width occurs by shifting some of the noise to higher frequen-cies where it may subsequently be filtered out by down-stream circuit elements.24 Then,

P2 � ��HR�reg�0��2SRR�reg + �HP�0��2SPP�reg��fN

=p�

�1 − T�0��b + 1� +

b�R� 1

O�t�− 1�

�kr + kfp�� 26�

and

�P�reg2

preg�� =

� P2

p��

�1 − T�0��+

b�R� 1

O�t�− 1�

�kr + kfp��1 − T�0��. �27�

The first term in Eq. �27� shows that negative feedback de-creases the noise strength of the transcriptional and transla-tional intrinsic noise by a factor of 1 / �1−T�0�� compared tothe unregulated case �Eq. �12��, which is consistent with theearlier analysis of Thattai and van Oudenaarden.23 The sec-ond term reflects noise of operator dynamics; it is also re-duced by a factor of 1 / �1−T�0�� compared to the situationwhere the repressor molecule population is independent of
the regulated gene.

I

pscett

el

wlirwtw

T

wnStnar

026102-8 Cox et al. Chaos 16, 026102 �2006�

D

V. EXTRINSIC NOISE

To this point we have considered only the intrinsic com-onent of the noise. However, fluctuations in RNAP, ribo-omes, transcription factors, or other cellular molecular ma-hinery shared by gene circuits or pathways lead to anxtrinsic noise component.12,13 Measurements have shownhe extrinsic noise to be large for prokaryotes12,14,31,32 andhe dominant noise component in eukaryotes.29,36

We recently reported a model where we approximatedxtrinsic noise as a single source located at the point of trans-ation with a PSD given by32

SE�f� �SE�0�

�1 + �2�f

��2� , �28�

here we assumed that extrinsic noise is dominantly bandimited by dilution.31 The value of the constant term �SE�0��s set by other rates �transcription, translation, etc., of RNAP,ibosomes, proteases�. For further analysis and simulation itas convenient to also collect all the intrinsic noise terms at

he point of translation, resulting in a single noise sourceith a PSD given by

Ssource�f� �SE�0�

�1 + �2�f

��2� + SI. �29�

hese noise terms are processed by the gene circuit such that

Spm�NT �f� =

Spm�NE �0�

�1 + �2�f

��2��1 + � 2�f

� + �p�2�

+Spm�N

I �0�

�1 + � 2�f

� + �p�2� ,

�30�Spm�N

E �0� + Spm�NI �0� = 1,

here Spm�NT �f�=the normalized �Spm�N

T �0�=1� PSD of totaloise �extrinsic+intrinsic� in the protein concentration,

pm�NE �0�=the normalized PSD of extrinsic noise in the pro-ein concentration at f =0, and Spm�N

I �0�=PSD of intrinsicoise in the protein concentration at f =0. The normalizedutocorrelation function, ��, is given by the inverse Fou-ier transformation of the Spm�N

T �f� to obtain

�� = WE�� + �p

�pe−�� −

�

�pe−��+�p�� + WIe

−��+�p��,

�31�


WE =

Spm�NE �0�/Spm�N

I �0�

1 + �� + �p

��

�1 +Spm�N

E �0�/Spm�NI �0�

1 + �� + �p

��

, WI = 1 − WE.

In accordance with this analytical approach we constructed astochastic simulation model32 for use with simulators basedon variations of the Gillespie stochastic simulationalgorithm.15,17,37,38 All extrinsic noise was collected in theribosome concentration and was limited in frequency rangeby dilution. All intrinsic noise �transcription and translation�was represented by a single source at the point of translation.The reactions in the stochastic model of protein expressionfor a single gene circuit are given in reactions 1–4 in Table I.

Reactions 1 and 3 represent extrinsic noise that is filteredby the dilution rate. Reaction 2 represents the translation ofmRNA whose stochastic variation is an intrinsic noise com-ponent that was modeled in the translation noise component.The mRNA decay rate was neglected as it is usually shortcompared to the dilution rate. Reaction 4 represents dilutionand decay of protein. The weighting of extrinsic and intrinsicnoise was set by bnoise according to

SE�0�SI

= bnoise, �32�

where a value of SE�0� /SI�4 is consistent with previousreports.12,31 Note that the bnoise term used here does not rep-resent the true burst rate of the system, but rather is a mod-eling device used only to achieve the correct ratio betweenextrinsic and intrinsic noise.

V. EXPERIMENTAL MEASUREMENT OF NOISESPECTRA IN GENETIC CIRCUITS

Although there is useful information embedded withinthe spectral features of inherent gene circuit noise, spectralmeasurements are more difficult to make than noise magni-tude measurements. For example, in flow cytometry sequen-tial measurements are made on different cells and the timecourse of stochastic fluctuations in individual cells cannot bereconstructed. As a result, although used to great advantagein noise magnitude measurements,22,29,34 flow cytometry is

TABLE I. Reactions for simple stochastic simulation model of intrinsic andextrinsic noise in constitutive GFP circuit.

Reaction Rate

1. R→R+ribo k1

2. ribo→ ribo+GFP bnoise*�

3. ribo→ * �

4. GFP→ * �+�

5. ribo+ATc→ ribo-ATc kf

6. ribo-ATc→ ribo+ATc kr

7. ribo-ATc→ * �

not applicable to spectral measurements.


mcic�hacwattspitmpcf

vr

wnNncs

FcsarnbF


D

However, we recently32 used time-lapse fluorescenticroscopy12 to reconstruct continuous time histories of sto-

hastic fluctuations in growing cell cultures, thereby allow-ng the extraction of noise spectra. In these measurements theonfocal microscope settings were adjusted to collect lightwavelength=500–550 nm� in slices �thickness�celleight�. Images were acquired every 5 minutes �Ts=5 min�,nd time series of noise in green fluorescent protein �GFP�oncentrations �Xm�n ·Ts�� for individual cells �1,2 , . . . ,M�ere defined by their differences from the population mean,

nd individual noise traces �trajectories� that spanned the en-ire growth time were constructed by sequentially combininghe noise traces of cells within a common line of descent ashown in Fig. 5. Concentrations of GFP within a given ex-eriment were assumed to be proportional to fluorescencentensity per unit cell area. Noise traces were extracted fromhe images for nearly all possible trajectories in each experi-ent, and custom MATLAB �MathWorks, Inc., Natick, MA�

rograms were used to find mean fluorescence of the entireell population and to estimate population doubling timerom an exponential growth curve.

Normalized autocorrelations functions �ACFs� for indi-idual trajectories ��m� were found from the noise time se-ies �Xm�n ·Ts�� using a biased algorithm39

�m�jTs� =�n=1

N−jXm�nTs�Xm��n + j�Ts�

�n=1

NXm

2 �nTs�, �33�

here Ts was the 5 min sampling interval, n was the sampleumber �1,2 , . . . ,N�, and j had integer values from 0 to−1. As the gene circuits we studied were on high copy

umber plasmids, it was not necessary to correct for cell-ycle variations due to chromosome replication �i.e., we as-

IG. 5. �Color online� Construction of noise trajectories by sequentiallyombining the noise traces of cells within a common line of descent. Theolid curve shows a single trajectory through five generations of cell growthnd the dashed lines show alternate routes that produce other trajectories. Aepresentative noise trace is shown next to each cell in the trajectory. Theoise trace of the complete trajectory �shown at the bottom� is constructedy sequentially combining the noise traces of each cell in the trajectory.igure from Ref. 32, supplemental material.

umed that plasmid concentration remained constant�. The


composite autocorrelation function ��c� for M cell trajecto-ries was found using

�c�jTs� =�m=1

M �n=1

N−jXm�nTs�Xm��n + j�Ts�

�m=1

M �n=1

NXm

2 �nTs�. �34�

We investigated single gene circuits �pGFPasv� in E. coliTOP10 where destabilized �half-life�110 min� GFP wasconstitutively expressed �Fig. 6�a��. The average GFP fluo-rescence, which corresponded to the concentration of matureGFP protein, was measured in individual cells in growingcultures for 4–8 h periods. The noise frequency range wasdefined32 as the inverse of the time corresponding to �m

=0.5. Histograms of noise frequency ranges extracted fromthe individual trajectory autocorrelation functions �Figs. 7�a�and 7�b�� were compiled and compared with noise frequencyrange distributions found from exact stochasticsimulation17,38 using the extrinsic noise model described inreactions 1–4 in Table I.32 The simulations produced as muchdata as 500 separate experiments, and the resulting distribu-tions estimated the probability of finding a given noise fre-quency range from a randomly selected trajectory. Althoughsome measured distributions suggested a bimodal distribu-tion �Fig. 7�, this was likely due to nonrepresentative sam-pling of the rare high frequency events.32

The analysis presented earlier predicted that protein di-lution and decay rates are dominant factors defining the noisefrequency range in constitutively expressed gene circuits.24

To determine noise frequency range sensitivity to protein di-lution, we varied cell growth rate for the pGFPasv circuits bycontrolling temperature, and in a separate experiment wechanged the protein decay rate using a plasmid �pGFPaav�containing a reduced half-life ��60 min� GFP variant.40

These perturbations to gene circuit parameters were clearlyvisible in the noise spectral measurements as noise frequencyranges extended to higher frequencies as a result of fastergrowth �Fig. 7�a�� or higher protein decay rate �Fig. 7�b��.Although varying temperature changes the rates of all reac-tions, the noise frequency range of constitutively expressedcircuits is largely sensitive only to protein dilution and decayrates in contrast to noise magnitude.32

To test our prediction of increased noise frequency rangewith negative autoregulation,24 we constructed circuits withthe gene for the protein TetR inserted upstream of GFP, cre-ating a transcriptional fusion �pTetR-GFPasv; Fig. 6�b��. Thiscircuit was negatively autoregulated as its expression wasrepressed by TetR binding to operator sites in the promoter.41

A control circuit without autoregulation was also tested, inwhich a chromosomal copy of tetR was constitutively ex-pressed from the PN25 promoter. In both cases, repressionwas relieved by addition of anhydrotetracycline �ATc� to thegrowth medium and allowed the modulation of GFP expres-sion.

To determine if ATc had an effect on noise spectra inde-pendent of the autoregulation of the TetR circuit, we mea-sured the noise frequency range of the pGFPasv circuits inmedia supplemented with 100 ng/ml of ATc. There was amarked modification in the noise frequency range distribu-tion �Fig. 6�c�� indicating a change in either the processing of


tpintAgt�pmtfct

p1nt

F�pNpoau

026102-10 Cox et al. Chaos 16, 026102 �2006�

D

he noise or the nature of the noise sources. Our modelingoints to the latter with ATc inhibition of translation42 lead-ng to a reduction in the weighting and whitening of extrinsicoise, which we explored using the extrinsic noise simula-ion model in Table I. Reactions 5–7 describe the ribosome-Tc heterodimer formation and its dilution due to cellrowth.32 The frequency range distribution extracted fromhese simulations was compared to the measured distributionFig. 6�c��, with both showing a characteristic peak shift andeak broadening. Although not conclusive, this gross agree-ent between measured and simulated distributions supports

he hypothesis that the mechanism of ATc-mediated noiserequency range modulation is an increase in high frequencyontent of the global extrinsic noise associated with transla-ion and a reduction of the weighting of extrinsic noise.32

We measured noise frequency range distributions ofTetR-GFPasv and the control cells grown in media with00 ng/ml of ATc. Composite noise frequency ranges of theegatively autoregulated pTetR-GFPasv exceeded those ofhe constitutively expressed pGFPasv in 100 ng/ml of ATc

IG. 6. �Color online� Gene circuits schematics and the effect of negative a110 min half-life� gene circuit. �b� pTetR-GFPasv negatively autoregulatedGFPasv circuit �doubling time 60 min; 154 trajectories without ATc; 114egative autoregulation-mediated shift of noise frequency range �doublingTetR-GFPasv: 114 trajectories�. �e� Model of the shift of frequency range dn the left represents an unregulated circuit distribution. Negative autoregurrow. The bars in dark show portions of the distribution that are common tonaffected. Figure from Ref. 32.


by as much as �2–3 �Figs. 6�d� and 8�, whereas the con-trol circuits showed no noise frequency range increase �Fig.8�. The negative autoregulation-mediated noise remodelingwas seen as an increase of the noise frequency range �Fig. 8�and as a modification of the shape of the distribution �Fig.6�d��. Autoregulation frequency response is limited by pro-tein decay and dilution, and therefore has a larger effect onslower fluctuations than faster fluctuations. Noise trajectoriesthat would have clustered at the lower end of the frequencyrange distribution in unregulated cells are pushed to highervalues by negative autoregulation, whereas those in thehigher frequency tail of the distribution are only weakly af-fected �Fig. 6�e��. This results in frequency range distribu-tions with a shape closer to normal distributions �Fig. 6�d��.The frequency shift and the change in distribution shape areindicative of the presence of negative autoregulation. Wewould also expect a decrease in the circuit noise strength inthe autoregulated gene circuit, but the variable gain of theconfocal microscope did not allow for absolute measure-ments of noise strength.

gulation. �a� Plasmid pGFPasv containing the constitutively expressed GFPcircuit. �c� Effect of ATc on the noise frequency range of the unregulatedctories with ATc�. sim., simulated using model described in Table I. �d�60 min; pGFPasv: 154 trajectories without ATc, 114 trajectories with ATc;ution shape due to negative feedback. The right-skewed distribution shownshifts the distribution toward the center as shown by the dashed box and

the regulated and unregulated circuits. The higher frequency trajectories are

FIG. 7. Effects of cell doubling time and protein half-life on noise frequency range. Measured distributionsare shown as vertical bars and simulated distributions assolid lines. �a� Shift in noise frequency range for thepGFPasv circuit as doubling time increases from�30 min �100 trajectories; T=32 °C� to �90 min �120trajectories; T=22 °C�. �b� Shift in noise frequencyrange as protein decay time decreases from 110 min�pGFPasv; 59-min doubling time; 154 trajectories; T=26 °C� to 60 min �pGFPaav; 56-min doubling time;33 trajectories; T=26 °C�. The model described in

utoregenetraje

timeistriblationboth

Table I was used in simulations. Figure from Ref. 32.


dsagraitepniopnllt

VA

retnawmtOvorresac

Fpl�rrppt


D

These measurements validated the prediction that in ad-ition to noise magnitude gene circuits manipulate noisepectra. We verified the link between gene circuit structurend the noise frequency range and showed that changes inene circuit parameters �e.g., cell growth and protein decayates� modulate the noise frequency range. Our results showshift of noise spectra to higher frequencies and a remodel-

ng of the noise frequency range distribution that is charac-eristic of negative autoregulation. This noise spectral remod-ling may impact the regulatory effect of the noise as itropagates through the gene network, as higher frequencyoise is more easily filtered out by downstream gene circuitsn a regulatory cascade.24 One of the more intriguing aspectsf this study was that the noise frequency range distributionsrovided a means for evaluating a hypothesis of the mecha-ism of ATc-mediated remodeling of noise spectra in unregu-ated gene circuits. In the following section we take a closerook at the interaction of the ATc and the extrinsic noise ofhe gene circuit.

I. EXTRINSIC NOISE AT THE TRANSCRIPTIONALND TRANSLATIONAL LEVELS

The ATc-mediated modification of the noise frequencyange provides some insight into the relative contributions ofxtrinsic and intrinsic noise sources. Bacterial reporter pro-ein systems studied to date are often characterized by sig-ificant extrinsic noise with an autocorrelation time scalepproximately equal to the cell cycle.31,32 For convenience,e have chosen the cell machinery components RNA poly-erase �RNAP� and ribosomes to represent extrinsic noise at

he transcriptional and translational levels, respectively.ther potential sources of extrinsic noise include globalariation in the concentration of sigma factors, ribonucle-tides, amino acids, RNases, proteases, and other shared cellesources. Although these extrinsic noise sources are not di-ectly treated here, they are indirectly considered as thereffect on the reporter gene circuit is mediated by the tran-cription and translation processes associated with RNAPnd ribosomes. We further assume that the variation in the

IG. 8. �Color online� Noise frequency range vs doubling time. Measuredoints are shown with ±1 error bars estimated from simulation. The ana-ytical curve �Eq. �31�� for the pGFPasv circuit and the simulated curveTable I� for pGFPasv+100 ng ml−1 ATc are shown. Vertical black arrowsepresent regulation strength determined by the shift of the noise frequencyange. The temperature �°C� of each experiment is indicated by each dataoint. The TetR data points are for the circuit with autoregulated tetR ex-ression, whereas the TetR ctrl data points are for the circuit with constitu-ive tetR expression. Figure from Ref. 32.

oncentration of RNAP and ribosomes is driven by the syn-


thesis and growth-driven dilution processes. However, theirsynthesis also depends upon shared cell resources whichfunction as additional extrinsic noise sources, not consideredin this model. This model also does not account for the tem-porary unavailability of RNAP and ribosomes actively en-gaged in elongation reactions to initiate new transcriptionand translation reactions, which causes additional variationin their free populations. We have further assumed that theseries of maturation steps that nascent GFP must undergoprior fluorescence are sufficiently fast to be ignored, as dis-cussed in Austin et al.32 Despite these simplifying assump-tions, the model considered here is sufficiently detailed todemonstrate the effects of transcriptional and translational-level extrinsic noise sources on the expression of GFP in thepresence of ATc.

With these assumptions, we model the production ofGFP according to the reactions in Table II. The last reactionis active only in the presence of ATc and its justification isdescribed as follows. Tetracycline and many of its deriva-tives affect translation by binding to the 30S ribosomal sub-unit �reviewed in Chopra and Roberts42�. This interactionblocks the addition of aminoacyl-tRNA and prevents trans-location of the ribosome complex down the mRNA strand. Inthe presence of these types of translation inhibitors themRNA is stabilized, presumably by shielding it from degra-dation by RNases.43 Additionally, more recent work usinguntranslated RNA suggests that the increased transcription ofrRNA that occurs in the presence of translation inhibitorscreates a drain on RNA degradative capacity, resulting in anadditional indirect protection of mRNA pools.44

Based on this knowledge we can summarize the actionof ATc as having two effects: �1� it increases mRNA stabilityand �2� it causes the ribosome to stall on the transcript. In-creased mRNA stability is easily handled by decreasing themRNA decay rate �m. A closer look at the translation processis needed to understand the effect of a stalled ribosome.

TABLE II. Reactions for model of constitutive GFP expression as affectedby extrinsic noise at the transcriptional and translational level and in thepresence of ATc.

RNAP polymerase synthesis and dilution �transcriptionallevel extrinsic noise�:

——→�RNAP

RNAP ——→�

*Transcription and mRNA decay:

RNAP ——→ktc

RNAP+mRNA

mRNA ——→�m

*Ribosome synthesis and dilution �translational levelextrinsic noise�:

——→�ribo

ribo ——→�

*Translation and GFP dilution/decay:

mRNA+ribo ——→ktl

mRNA+ribo+GFP

GFP ——→�p+�

*Removal of stalled ribosomes from the cell by dilution�kstalled=0 in the absence of ATc�:

ribo ——→kstalled

*

Translation is initiated upon binding of a ribosome to a ribo-


saeissRtasbseAcpnpdhts+

Ts

wcvst

wtRtft

026102-12 Cox et al. Chaos 16, 026102 �2006�

D

ome binding site. Elongation occurs as the ribosome moveslong the mRNA and produces a growing polypeptide chain,ventually producing a complete protein. The mRNA cannitiate additional transcription events as soon as the ribo-ome clears the ribosome binding site, such that many ribo-omes can be simultaneously translating a given transcript.ibosome stalling may affect the protein production rate in

wo ways. First, the rate of transcript initiation may decreases a result of a stalled ribosome blocking access to the ribo-ome binding site. The ribosome binding site could belocked by a ribosome stalled directly in contact with it ortalled further down the transcript, causing a logjam. Thisffect can be modeled by a decrease in ktl upon addition ofTc. Second, stalled ribosomes may be removed from theell by dilution before completely translating the protein; thisrocess is represented by the last reaction in Table II. Tech-ically, the rate of stalled ribosome removal should be de-endent on �mRNA�; however, we have ignored this depen-ence to simplify our frequency domain analysis as it willave only a second order effect. The effect of this reaction iso reduce the average number of proteins produced per ribo-ome during each cell cycle according to: ktl�mRNA� / ��kstalled�.

We apply frequency domain analysis to the reactions inable II. First, the PSD of the intrinsic noise of GFP expres-ion SGFP

i �f� can be taken directly from Eq. �14� to yield:

SGFPi �f� �

4bGFP�� + �p � 1

1 + � 2�f

� + �p�2�

=4ktlribo�GFP�

�m�� + �p� � 1

1 + � 2�f

� + �p�2� , �35�

here GFP� and ribo� are the mean GFP and ribosomeoncentrations and ktl is the translation rate and the otherariables are as previously defined. Next we develop expres-ions for the PSD of the transcriptional and translational ex-rinsic noise sources �these are analogous to Eq. �28��:

Stc�f� �4RNAP�

�

1

�1 + �2�f

��2� , �36�

Stl�f� �4ribo�

� + kstalled

1

�1 + � 2�f

� + kstalled�2� , �37�

here Stc�f� and Stl�f� are the PSD of the transcriptional andranslational extrinsic noise sources, RNAP� is the meanNAP concentration, and kstalled is the rate constant for dilu-

ion of ribosome stalled on transcripts. The following trans-er functions describe the filtering of these noise sourceshrough the gene circuit:


�HRNAP−GFP�f��2 = � ktcktlribo��m��P + ��

2

1

�1 + �2�f

�m�2��1 + � 2�f

�P + ��2� ,

�38�

�Hribo−GFP�f��2 = � ktlmRNA��P + �� 2 1

�1 + � 2�f

�P + ��2� ,

where ktc is the transcription rate constant, and�HRNAP-GFP�f��2 and �Hribo-GFP�f��2 are the noise power trans-fer functions between the RNAP source and GFP output, andbetween the ribosome source and GFP output, respectively.As discussed earlier, the high frequency messenger RNApole in Eq. �38� has little effect and is ignored in the remain-der of this analysis. The PSDs of the GFP extrinsic noisecomponents are obtained by multiplying the PSDs of thenoise sources �Eqs. �36� and �37�� by the noise power trans-fer functions �Eq. �38�� to obtain:

SGFPtc−ext�f� �

4RNAP��

� ktcktlribo��m��P + ��

2

1

�1 + �2�f

��2��1 + � 2�f

�P + ��2� , �39�

SGFPtl−ext�f� �

4ribo�� + kstalled

� ktlmRNA��P + �� 2

1

�1 + � 2�f

� + kstalled�2��1 + � 2�f

�P + ��2� ,

�40�

where SGFPtc-ext�f� and SGFP

tl-ext�f� are the PSDs of the GFP noisedue to transcriptional and translational level extrinsicsources. We now take the inverse Fourier transform of Eqs.�39�, �40�, and �35� to obtain the noise-source-specific GFPautocorrelation functions:

�GFPtc−ext�� = Atc−ext� exp�− ��

�1 − � �

�p + ��2�

+��p + ��

�

exp�− ��p + ��

�1 − ��p + �

��2�� ,

�GFPtl−ext�� = Atl−ext� exp�− �� + kstalled��

�1 − �� + kstalled

�p + ��2�

+�p + �

� + kstalled

exp�− ��p + ��

�1 − � �p + � �2�� , �41�

� + kstalled


w

Tb

Tr

Aitit

TtIdt


D

�GFPi �� = Ai�exp�− ��p + �� ,

here the coefficients A are given by

Atc−ext =GFP�2

RNAP�,

Atl−ext =GFP�2

ribo�, �42�

Ai =GFP�2��p + ��

ktcRNAP�.

he total GFP autocorrelation function �GFP�� is then giveny the sum of the components:

�GFP�� = �GFPtc−ext�� + �GFP

tl−ext�� + �GFPi �� . �43�

he coefficients A reveal important insights concerning theelative importance of the three noise terms:

Atl−ext = Atc−extRNAP�ribo�

,

Ai = Atc−ext��p + ��ktc

, �44�

Atl−ext = Ai ktc

��p + ��RNAP�

ribo�.

n increase in the RNAP�/ribo� ratio increases the relativemportance of translational level extrinsic noise relative tohe other two sources. An increase in the ratio ��p+�� /ktc

ncreases the relative importance of intrinsic noise relative tohe two extrinsic sources.


Figure 9 shows normalized autocorrelation functions ofGFP expression under conditions of similar doubling timeboth with �Fig. 9�a�� and without �Fig. 9�b�� ATc addition.Also shown in Fig. 9 are the normalized autocorrelationfunctions for transcriptional-extrinsic, translational-extrinsicand intrinsic noise sources, given by

FIG. 9. Comparison of the experimental and source-specific normalizedautocorrelation functions. The source-specific normalized autocorrelationfunctions were calculated from Eq. �45�. �a� No ATc addition, doubling time59 min and �b� 100 ng ml−1 ATc addition, doubling time 55 min.

�GFP�Next−tc �� =

�GFPext−tc��

�GFPext−tc�0�

=�1 − ��p + �

��2�exp�− �� +

��p + ��

�1 − � �

�p + ��2�exp�− ��p + ��

�1 − ��p + �

��2� +

��p + ��

�1 − � �

�p + ��2� ,

�GFP�Next−tl �� =

�GFPext−tl��

�GFPext−tl�0�

=�1 − � �p + �

� + kstall�2�exp�− �� + kstall�� +

��p + �� + kstall

�1 − �� + kstall

�p + ��2�exp�− ��p + ��

�1 − � �p + �

� + kstall�2� +

��p + �� + kstall

�1 − �� + kstall

�p + ��2� , �45�

�GFP�Ni �� =

�GFPint ��

�GFPint �0�

= exp�− ��p + �� .

he experimental curves can be thought of as a mixture ofhe normalized noise-source-specific ACFs given in Eq. �45�.t is also noteworthy that the noise-source-specific curves areetermined entirely by the predominant pole frequencies inhe system ��, �p+�, and �+kstalled�. In these plots, the pro-

tein half-life is taken to be 110 min and the cell doublingtimes are as indicated in Fig. 9. We also assume that kstalled inthe presence of ATc is approximately equal to the cell dou-bling rate � corresponding to the experimental observationthat at constant temperature the cell growth rate decreases by


aptdtmtfrunttalAtprm

tmwwiFwtBkdeeitd

oece

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026102-14 Cox et al. Chaos 16, 026102 �2006�

D

bout a factor of 2 upon addition of ATc. When no ATc isresent, kstalled=0 and the normalized ACFs for the two ex-rinsic sources are identical. In this case, the experimentalata closely follows the extrinsic noise autocorrelation func-ion. Upon addition of ATc, the stalling of ribosomes onRNA shifts the autocorrelation function of translational ex-

rinsic noise to the left �i.e., a shift of the PSD to higherrequencies in the frequency domain�, whereas the autocor-elation functions of the other two noise sources are largelynaffected by the slight change in �. The experimental dataow fall between the theoretical curves for the extrinsicranslational noise and the intrinsic noise. It should be notedhat only the early portions ��1/2� of the experimentalutocorrelation functions are likely to be accurate, due to theimited experimental duration �7 h without ATc and 4 h withTc�. Another factor limiting the accuracy of the experimen-

al curve is the use of the biased autocorrelation function inrocessing the experimental data, which forces the autocor-elation function to approach zero at a � equal to the experi-ental duration.

In order to compare the autocorrelation function for theotal noise to the experimental autocorrelation functions weust first assign values to the parameters that determine theeighting factors. Inspection of Eq. �44� reveals that theeighting coefficients will depend on only two additional

ndependent parameters: RNAP�/ribo� and ktc. Inspection ofig. 9�b� reveals that extrinsic translational noise should beeighted more heavily than extrinsic transcription noise and

hat intrinsic noise should play at least a moderate role.ased on these observations we set RNAP� / ribo�=10 and

tc=2 10−4 s−1, yielding reasonable fits to the experimentalata observed in Fig. 10. The model reveals two predominantffects of ATc addition. First the ACF for the translationalxtrinsic noise is shifted to the left as a result of the increasen pole frequency due to kstalled. Second, the absolute magni-ude of the contribution of this noise source to the total ACFecreases with increasing kstalled, as seen in Eq. �41�.

To model the addition of ATc in Fig. 10, only the valuesf kstalled and � were modified as described previously. How-ver, the effects of changes in other parameters should beonsidered. In the model development we mentioned that the

IG. 10. The effect of ATc addition on the normalized autocorrelation func-ion of GFP expression noise. The solid lines represent model predictionsased on Eq. �43�, whereas the symbols represent experimental data �no ATcddition, doubling time 59 min; ATc addition �100 ng ml−1�, doubling time5 min�.

ffect of increasing mRNA stability could be incorporated by


increasing �m upon addition of ATc. Further, we suggestedthat stalled ribosomes could reduce the transcription initia-tion rate ktl by blocking the ribosome binding site. However,neither of the parameters appears either directly or indirectlyin the normalized GFP ACFs. Ribosome stalling is expectedto decrease the activity of ribosomes and could act to in-crease the ratio RNAP�/ribo�. This would tend to move themodel ACF to the right �i.e., a shift in the PSD to lowerfrequencies in the frequency domain�, away from the experi-mental data. However, as RNAP synthesis is dependent uponribosome activity it can be expected that the decrease in ri-bosome activity will be at least partially offset by a decreasein RNAP activity. The ATc experiment was conducted at ahigher temperature �30 °C, compared to 26 °C without ATc�to achieve a near constant growth rate. A temperature-relatedincrease in the transcription rate constant ktc again moves themodel ACF to lower frequencies away from the experimentaldata, but the effect is relatively modest even with a twofoldincrease in the rate constant. In our earlier work, we assumedthat the protein decay rate constant �p was independent oftemperature. An increase in this rate constant with tempera-ture would tend to shift the analytical curve to higher fre-quencies and closer to the experimental data. Cumulatively,the effect of these considerations was modest and within theexperimental error of the data.

This frequency domain analysis of extrinsic and intrinsicnoise sources and their propagation through gene circuitssuggests that translational extrinsic noise is the most domi-nant noise source in the system investigated here, with in-trinsic noise playing an important secondary role. Muchmore theoretical and experimental work is needed to gain abasic characterization of extrinsic noise sources and their ef-fects. The frequency domain analysis presented here pro-vides a convenient framework for planning and interpretingexperimental efforts with this goal.

VII. CONCLUSIONS

Phenotypic variability not originating from genetic orenvironmental causes has been long observed in biology, butexperimental capabilities to investigate these stochasticmechanisms at the molecular level have only been developedin recent years. Most studies of genetic noise to date havefocused on its magnitude. In this article, we have summa-rized our recent work that demonstrates that the frequencycontent of gene noise contains additional information thatcan provide insight into the structure and function of genenetworks. We also presented an expanded model to explainthe experimentally observed ATc-mediated frequency shift ingene noise that explicitly considers extrinsic noise at thetranscriptional and translational levels. We envision futurework in this area will make significant contributions to ageneralized understanding of gene networks. Progress is ex-pected in better understanding of the sources and propaga-tion of intrinsic noise in gene circuits and the characteristicnoise signatures of various regulatory motifs. We also expectcontinued progress in understanding how specific cellularsystems exploit noise dynamics to achieve desired pheno-types. For example, a recent study has identified a core net-
work of genes that controls the spontaneous and stochastic

esqcn

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ntry of Bacillus subtlis into a competent state and its sub-equent exit from that state.45 An interesting unanswereduestion is the relationship between the frequency of theompetence initiation and exit events and the values of ki-etic parameters in the gene network.

CKNOWLEDGMENTS

This work was supported by the DARPA Bio-omputation Program, the NSF �Award EIA-0130843� and

he National Academies Keck Futures Initiative. D.W.A. ac-nowledges support from an ORNL Center for Nanophaseaterials Sciences Research Scholar Fellowship. R.D.D. ac-

nowledges support from the DOE Science Undergraduateaboratory Internship program.

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