Frequency domain analysis of noise in simple gene circuits

Transcription

1 Frequency domain analysis o noise in simple gene circuits Chris D. Cox a Department o Civil and Environmental Engineering, University o Tennessee, Knoxville, Tennessee and Center or Environmental Biotechnology, University o Tennessee, Knoxville, Tennessee James M. McCollum Department o Electrical and Computer Engineering, University o Tennessee, Knoxville, Tennessee Derek W. Austin Siemens Molecular Imaging, Knoxville, Tennessee 3793 CHAOS 6, Michael S. Allen Molecular-Scale Engineering and Nanoscale Technologies Research Group, Oak Ridge National Laboratory, Oak Ridge, Tennessee ; Center or Environmental Biotechnology, University o Tennessee, Knoxville, Tennessee 37996; and Department o Materials Science, University o Tennessee, Knoxville, Tennessee Roy D. Dar Molecular-Scale Engineering and Nanoscale Technologies Research Group, Oak Ridge National Laboratory, Oak Ridge, Tennessee and Department o Physics, University o Tennessee, Knoxville, Tennessee Michael L. Simpson b Molecular-Scale Engineering and Nanoscale Technologies Research Group, Oak Ridge National Laboratory, Oak Ridge, Tennessee ; Department o Materials Science, University o Tennessee, Knoxville, Tennessee 37996; and Center or Environmental Biotechnology, University o Tennessee, Knoxville, Tennessee Received 30 January 006; accepted April 006; published online 30 June 006 Recent advances in single cell methods have spurred progress in quantiying and analyzing stochastic luctuations, or noise, in genetic networks. Many o these studies have ocused on identiying the sources o noise and quantiying its magnitude, and at the same time, paying less attention to the requency content o the noise. We have developed a requency domain approach to extract the inormation contained in the requency content o the noise. In this article we review our work in this area and extend it to explicitly consider sources o extrinsic and intrinsic noise. First we review applications o the requency domain approach to several simple circuits, including a constitutively expressed gene, a gene regulated by transitions in its operator state, and a negatively autoregulated gene. We then review our recent experimental study, in which time-lapse microscopy was used to measure noise in the expression o green luorescent protein in individual cells. The results demonstrate how changes in rate constants within the gene circuit are relected in the spectral content o the noise in a manner consistent with the predictions derived through requency domain analysis. The experimental results conirm our earlier theoretical prediction that negative autoregulation not only reduces the magnitude o the noise but shits its content out to higher requency. Finally, we develop a requency domain model o gene expression that explicitly accounts or extrinsic noise at the transcriptional and translational levels. We apply the model to interpret a shit in the autocorrelation unction o green luorescent protein induced by perturbations o the translational process as a shit in the requency spectrum o extrinsic noise and a decrease in its weighting relative to intrinsic noise. 006 American Institute o Physics. DOI: 0.063/ Stochastic luctuations, or noise, in gene circuits arise due to the random timing o biochemical reactions within a cell and the discrete nature o the molecular populations aected by these reactions. The noise can originate rom variations in the global resources (e.g., polymerases, ribosomes, nucleic and amino acids) utilized in transcription and translation (called extrinsic noise ) or within a Electronic mail: ccox9@utk.edu b Electronic mail: simpsonml@ornl.gov the transcriptional and translational processes themselves (called intrinsic noise ). Not only have cells evolved to maintain idelity o gene unction in the presence o noise, there are many examples in which cells exploit noise-driven phenotypic diversity to achieve a desired end. Further, the manner in which noise propagates through a gene circuit allows certain inerences about the gene circuit architecture. In recent years a number o theoretical and experimental studies have been undertaken to characterize the sources and propagation o /006/6/060/5/$3.00 6, American Institute o Physics Downloaded 30 Jun 006 to Redistribution subject to AIP license or copyright, see

2 060- Cox et al. Chaos 6, noise in gene networks. Most o these studies have ocused on quantiying and interpreting the magnitude o the noise as quantiied by distributions or statistics o a noise-aected output within a population o cells. We have developed a requency domain (FD) approach that captures inormation about both the magnitude o stochastic luctuations and their timing within individual cells and demonstrates that the timing plays a critical role in gene circuit perormance. Here we review recent advances in characterizing noise in gene circuits, describe the analytical and experimental aspects o the FD approach, and demonstrate its application to the analysis o intrinsic and extrinsic noise in transcription and translation processes. I. INTRODUCTION In 940 the physicist Max Delbruck recognized that small populations o enzyme molecules within a cell would give rise to statistical luctuations in biochemical reactions and that these luctuations could have proound impacts on cell physiology. He later proposed that luctuations o this type could explain the variation in the number o viruses produced upon lysis o inected bacteria. In the ollowing decades many examples o the importance o stochastic luctuations in gene regulation have been reported. Prokaryotic examples include regulation o lac expression at low levels o induction, 3 the lysis-lysogeny decision in phage-, 4,5 and the swimming and tumbling periods o bacteria during chemotaxis. 6 Examples o stochastically driven phenotype variability in eukaryotes include variability in the response to mating pheromone in yeast 7 and the notch-mediated epidermal-neural decision in Drosophila neuro-ectoderm. 8 Recent studies have also suggested that stochastically driven phenotype diversity in a clonal population may ensure that a ew cells remain poised to exploit changing environmental conditions, 9 thereby improving itness. Stochastic luctuations hereater reerred to as noise are introduced into the populations o mrna, protein, and other molecular species by several sources. The random timing and discrete nature i.e., integer number o molecules o molecular events such as transcription, translation, multimerization, and protein/mrna decay processes lead to a noise component that is intrinsic to a local gene circuit or pathway. 5,0, Conversely, luctuations in RNAP, ribosomes, transcription actors, or other cellular molecular machinery shared by gene circuits or pathways lead to an extrinsic noise component.,3 The dierence between intrinsic and extrinsic noise is not just one o deinitions, as these luctuations have dierent sources, attributes, and consequences Fig.. For example, although intrinsic noise in one gene circuit is uncorrelated with that in any other gene circuit, extrinsic noise is correlated across gene circuits as it arises rom shared cellular resources. Further, this sharing o resources and transmission o luctuations provides or a coupling between gene circuits that have no direct regulatory relationship. 4 That is, a high demand or expression by one pathway can aect the rate at which an unrelated pathway is expressed. FIG.. Color online Intrinsic noise sources originate rom the transcription and translation processes aecting a single gene, whereas extrinsic noise sources aect all genes within the cell. Over the last ew years there has been a great deal o ocus on the theory, analysis, modeling, and simulation o stochastic luctuations in gene circuits and networks. 5,7,0,3,5 6 The most rigorous and accurate mathematical representation or calculating the discrete stochastic time evolution o a reacting system is the chemical master equation CME, which describes how the overall probability o any state in the system evolves over time as a result o various possible chemical reactions. 8 Unortunately, the CME approach becomes impractical or genetic circuits and networks o even moderate complexity. A tractable method o dealing with discrete stochastic simulations is exact stochastic simulation; 6,7 however, although simulation is an invaluable aide in understanding stochastic luctuations in genetic systems, it is diicult to extract undamental relationships between gene circuits parameters e.g., kinetic rates and unction using simulation alone. Instead, such relationships have most oten been predicted through mathematical analysis using approaches e.g., Fokker-Planck and Langevin that are simpliications o the CME. 9,,4,5,7 Coupled with the progress in gene circuit noise theory, analysis and simulation, there has been signiicant experimental progress. Elowitz et al. used the correlation o yellow and cyan luorescent proteins under the control o identical promoters at dierent locations within the genome or the independent measurement o the extrinsic and intrinsic noise components described earlier, and showed that the contribution o extrinsic noise to total noise may be larger than that o intrinsic noise. Blake et al. measured gene circuit noise in eukaryotes and showed the eect o transcription reinitiation on the magnitude o the stochastic luctuations. 8 Raser and O Shea extended the study o extrinsic and intrinsic noise to eukaryotic gene expression by quantiying dier- Downloaded 30 Jun 006 to Redistribution subject to AIP license or copyright, see

3 060-3 Frequency domain analysis: Gene circuits Chaos 6, FIG.. Basic concepts in requency domain analysis. a Chemical species A H are represented as nodes, each o which represents a noise source with its own characteristic PSD. Noise power transer unctions describe the eect o upstream noise sources A D on downstream output nodes E H. For example, the noise source at A is characterized by its PSD S AA and its eect on output E is given by H AE S AA. b Stochastic time series generated by noise sources let are characterized by their power spectral densities right. The grey and black curves represent high and low requency noise sources, respectively. The vertical arrows indicate locations o pole requencies. ences o expression rom two alleles in diploid yeast cells. 9 They demonstrated that extrinsic noise was also dominant in two wild-type yeast genes and that slow rates o promoter state transition could dominate intrinsic noise as predicted by theoretical studies.,5 Similarly, Becskei et al. urther demonstrated that noise associated with low molecular number transcripts in yeast was not the result o the low populations per se, but rather resulted rom relatively rare gene activation events. 30 Roseneld et al. used time-lapse microscopy to track reporter gene activity response to an inducer over multiple cell generations. 3 They resolved the autocorrelation unction o the gene response into contributions rom rapidly decaying intrinsic noise and more persistent extrinsic noise that decays over approximately one cell cycle. Pedraza and van Oudenaarden used cross correlation o three luorescent reporter genes to determine how intrinsic and extrinsic noise propagates through gene cascades. 4 The synthetic cascade was tunable via the concentrations o two inducers. A calibrated analytical model o the cascade correctly predicted changes in noise response as a unction o changing inducer concentrations. However, all o these experimental studies ocused on noise magnitudes, and like their theoretical counterparts, largely ignored the requency content o this noise. Most o these theoretical and experimental investigations have dealt with noise magnitude using stochastic distributions e.g., means, variances, standard deviations o the molecular populations at steady state, largely ignoring the rate o the luctuations. In contrast, we have employed a FD approach to analysis and experimentation that deals with the noise spectra content. In this approach the noise sources have a lat i.e., equal power at all requencies or white spectrum, but these spectra are shaped as the noise propagates rom the sources to molecular populations or concentrations o interest. As a result, there is a mapping between the spectral content o gene expression luctuations and the structure and unction o the underlying gene circuits. Our FD analysis has demonstrated that the requency content o the noise may have important implications or gene regulation and unction that are not captured by analysis or measurement o noise magnitudes. 4 6 For example, our analysis predicts that kinetic parameters are relected in the noise requency range and that negative autoregulation not only reduces the magnitude o the noise, but also shits the remaining noise to higher requencies where it may have little or no regulatory eect. Unortunately, it is more diicult to perorm FD measurements as these require relatively long time series measurements rom individual cells. Consequently low cytometry, which has been a very productive tool or measuring stochastic distributions, cannot be used or FD measurements. However, we recently measured stochastic luctuations in reporter gene expression o growing cells using timelapsed microscopy. 3 We measured noise requency content in growing cell cultures and veriied our theoretical prediction 4 that in addition to noise magnitude, gene circuits manipulate noise spectra, impacting the ate and regulatory eect o the noise as it propagates through the gene network. 3 We veriied our prediction 4 o the shit o noise spectra to higher requencies and the link between gene circuit structure and the resulting noise requency range and showed that changes in gene circuit parameters, such as cell growth and protein decay rates, modulate the noise requency range distributions. 3 Further, these results showed that measured noise requency range distributions combined with stochastic simulations can be used to probe mechanistic details o molecular interactions within gene circuits. 3 Here we review FD analytical and experimental techniques and our previous results. In addition, we present a model that explicitly considers extrinsic noise sources emanating rom the transcriptional and translational level and use the model or the interpretation o our experimental results where noise was signiicantly aected by perturbation o translational processes. II. FREQUENCY DOMAIN ANALYSIS: TOOLS FOR GENE CIRCUIT ARCHITECTURAL ANALYSIS Our FD approach is equivalent to the chemical Langevin analysis and is applicable to linear systems and nonlinear systems or which linear approximations are suiciently accurate over conditions o interest. In the FD approach the genetic/biochemical system is conceptualized as a chemical circuit with nodes or each chemical species, and the noise is generated by a set o noise sources Fig. a. The noise sources are characterized by their power spectral densities PSDs, which describe how the noise is distributed in requency space Fig. b. The PSD o the noise in the concentration or population o chemical species i S i is ound rom the weighted sum o the noise sources according to Downloaded 30 Jun 006 to Redistribution subject to AIP license or copyright, see

4 060-4 Cox et al. Chaos 6, FIG. 3. Model o single gene expression. a noise sources include synthesis o mrna molecules rom the template DNA strand at rate R, and translation o proteins at a rate k P mrnat. Noise sinks include decay o mrna and protein with irst-order rate constants R and P, respectively. b The same as a except that the protein is negatively autoregulated. The loop transmission T is calculated by breaking the loop at any convenient point, introducing a perturbation just to the downstream side o the break, measuring the response that returns to the upstream side o the break and calculating T=/. Figure rom Re. 4. G S i = H ij j= G S j = j= H ij 0S j 0 Nij, where is the requency Hz, S j is the PSD o noise source j, G is the number o noise sources contributing to noise in chemical species i, H ij is the noise power transer unction between the noise source j and chemical species i, and Nij is a characteristic requency range known as the noise bandwidth o the noise in species i due to source j. The noise power transer unctions are ound rom H ij = x i, r j where x i is the magnitude o luctuations at requency in the concentration o species i due to luctuations in molecular populations at requency caused by noise source j r j. The noise sources are well described by shot noise with a wide band compared to the requency limitations o the gene circuit white spectrum. 4 Shot noise is the noise that originates rom the discrete nature o the signal carrier. The PSD o shot noise arising rom a Poisson process is proportional to the average lux k T through the process such that 4 S j =k T, where S j is the single-sided positive requency only PSD o the noise source j. At steady state the noise sources are ound in pairs a source and a sink; Fig. 3a at the points o molecular transitions synthesis, decay, polymerization, complex ormation, etc., so the total PSD or a pair o noise sources at location j is S jtotal =4k T. The noise power transer unctions in genetic circuits will be o the orm 3 4 H ij = H ij 0 n j,k n= m j,k m= + znij + pmij, 5 where z,...,n zeros and p,...,m poles are requencies associated with the kinetic parameters o the reactions between noise source j and chemical species i. The transer unction describes how noise is modiied as it propagates through the circuit. Below the irst pole requency, the PSD is lat and the noise power is independent o requency. As the requency increases above the irst pole, the noise power decreases with /. The noise power decreases by an additional actor o / or each additional pole encountered Fig. b. Most reactions generate only poles, although zeros are seen in reversible reactions. 6 Zeros have the opposite eect o poles: noise power increases with at requencies above the zero requency. Overall, the change in noise power with requency can be described by nz np where nz and np are the number o zeros and poles with characteristic requencies less than. Note that the PSD o the sources are lat and eatureless Eq. 4, but the PSDs in the species concentrations have structure created by the kinetic parameters o the reactions Eq.. The noise power transer unctions are derived rom the linearization and Fourier or Laplace transormation o the ordinary dierential equations describing the system. Thereore FD analysis has most o the same limitations and caveats as the Langevin approach. 9 One notable exception is that the FD approach can be used or some cases o very low molecular populations where the time domain Langevin approach ails. 5 Further, some limitations are overstated in the literature. As has been demonstrated with electronic circuits or quite some time, FD techniques can be applied to some nonlinear systems. However, such applications must be handled with care and the results scrutinized. III. APPLICATION OF FD ANALYSIS A. Single gene circuit Figure 3a is a schematic diagram o an unregulated single gene system. Considering only the intrinsic noise, the Langevin equations or this system are dr dt = R + r + R t + R, 6 dp dt = P + p + k P r + P, where r and p are mrna and protein concentrations, R and P are mrna and protein decay rate constants, is the rate o dilution due to growth, R is the transcription rate, k P is the translation rate constant, and R and p are random variables that represent the noise. At steady state the average mrna r and protein p concentrations ound rom Eqs. and are R / R + and R k P / R + P +, respectively. As the typical hal-lie o mrna 5 min 33 is Downloaded 30 Jun 006 to Redistribution subject to AIP license or copyright, see

5 060-5 Frequency domain analysis: Gene circuits Chaos 6, much less than the cell doubling time, we will make the approximation R + R. From Eq. 4 the PSDs o the mrna S RR and protein S PP synthesis and decay noise are S RR =4 R, 7 S PP =4 Rk P. R The signal and noise power transer unctions rom the point o mrna H R and protein H P synthesis to the reporter protein output are ound by Fourier transorm and solution o the Langevin equations to obtain 4 : b H R = P + +i mrna+i H P = P + + protein, b H R P + = + + mrna H P = + P + protein, protein, protein, where the pole requencies are associated with mrna mrna = R / and protein protein = p +/ decay, and the term b =k P / R ; oten reerred to as the burst rate is the average number o proteins produced rom each mrna transcript. The power transer unctions Eq. 9 can be understood in terms o the gain numerator and the requency modiication denominator as the noise propagates through the circuit. The gain or transcriptional noise sources is greater by a actor o b. Simple transcription-translation circuits behave as low-pass ilters, as noise becomes negligible at requencies greater than the pole requencies. As the eect o a noise source decreases at / or requencies higher than the irst pole, other poles are oten neglected in multipole systems. As mrna typically decays much aster than the protein R P, the mrna pole may be neglected with little error and the single pole noise bandwidth approximation can be used. 4 With this simpliication, the noise bandwidth range o requencies that have a signiicant noise content; N or both noise sources can be approximated as 4 N protein = p +, 0 4 and the variance o the output protein population noise is given by 8 9 P H R 0S RR + H P 0S PP N = p+b, where we have used the relationships or p and b deined earlier. The noise igures o merit are P p =+b, P p =+b. 3 p Equation gives the noise strength and Eq. 3 gives the coeicient o variation CV with results that are in agreement with previous analysis.,3 However, the FD approach shows that this noise is spread over a requency range related to N and controlled by the protein decay and dilution rate. The previous analysis provides a relationship that can be used to aide the analysis o multigene systems. For a constitutively expressed gene circuit S p i 4bp + p + + p, 4 where S p i is the intrinsic noise in the protein concentration, we have neglected the mrna pole, and we have assumed that b+b. B. Transcriptional control In many prior analyses transcriptional control was approximated using a Hill expression: 3,4,34,35 I = +d/k h n, 5 where I is the induction level o the gene, d is the concentration o a regulatory molecule, K h is a constant that indicates the value o d at which the induction level reaches 0.5, and n is the Hill coeicient. Positive and negative values o n correspond to repression and induction, respectively. Although this approach may reasonably approximate the static deterministic behavior o transcriptional regulation, it can lead to signiicant errors by neglecting both the dynamics requency response and the noise o transcriptional regulation. A more realistic description would include switching between discrete high and low transcriptional rates with the average rate determined by the ractional amount o time spent in each o the two states. This model is consistent with transcription controlled through protein-dna interactions at an operator site within the gene promoter region. We perormed a FD analysis o the case o a single operator site within a single copy o the operon as shown in Fig. 4. In this analysis we considered two possible states, denoted as O unbound and O bound, and transition between these states was described by Downloaded 30 Jun 006 to Redistribution subject to AIP license or copyright, see

6 060-6 Cox et al. Chaos 6, FIG. 4. Model o gene regulation. a Gene in basal state O with operator unbound by molecular species d and producing transcripts at slow rate. b Gene in induced state O with molecular species d bound to operator and producing transcripts at increased rate. c The operator transitions between state O and O as a unction o time. The level o induction is given by the ractional time the gene spends in state O. Operator dynamics are characterized by the average time the operator remains in each state prior to transitioning. Figure rom Re. 5. k d O O, k r O O, 6 where k and k r are rate constants or the orward and reverse reactions, respectively, and d is the population o the molecular species that controls transcription by binding with O. Equation 6 applies to both positive and negative regulation as the ully induced state may be either O or O. As our previous analysis treated positive regulation, 5 here we consider only the negatively regulated case O minimally induced. These results can be directly applied in the analysis o negative autoregulation that ollows. As there is only one operator site, the population o the unbound operator, Ot, is either one or zero at all times. For negative regulation we deine t = I, O =, 7a t = 0, O =0, 7b R = 0 Ot + I Ot = 0 + I 0 Ot, 7c where t, 0, I, and R are the instantaneous, basal, ully induced, and average transcription rates respectively, and Ot is the time-averaged value o Ot. As individual transcription events and Ot are uncorrelated random processes, the autocorrelation unction or the gated term in Eq. 7c, O, is given by O = O, 8 where O is the autocorrelation unction or Ot and is the autocorrelation unction or a random series o impulse unctions average rate= I 0 corresponding to mrna synthesis. 5 The PSD o the noise in the mrna synthesis rate is ound by summation o the PSDs o mrna decay noise equal in magnitude, but uncorrelated to synthesis shot noise 4, the wideband white noise o the constant term i.e., basal gene expression in Eq. 7c, and the gated noise o Eq. 8 to yield: 5 S RRreg =4 0 + Ot I I 0 Ot Ot k r + k d +k r + k d 4 R Ot =4 R + k r + k d +k r + k d, 9 where the subscript reg denotes the regulated case and the PSD o the gated noise term was ound by direct calculation and then Fourier transormation o the autocorrelation unction o Eq The irst term in Eq. 9 is the PSD o the noise source associated with the average transcription rate, whereas the second term accounts or variations around the average transcription rate caused by the operator. The operator component increases at low induction levels low values o Ot and or slow operator dynamics low values o k r +k d. Noise rom slow operator dynamics is limited to low requencies by the presence o the iltering action o the pole in Eq. 9. The dynamics o this transcriptional regulation were also analyzed to show that the transer unction, H o, rom concentration o species d to the transcription rate is given by 5 : H O = k rk /k r + k d + k k r + k d, +i k r + k d + k 0 where d is the average concentration o species d. This analysis shows that there are at least two major shortcomings when using a Hill unction to approximate transcription regulation or stochastic analysis. The irst term on the right-hand side rhs o Eq. 9 is a shot noise term 4 and the only noise present when the Hill expression is used. Downloaded 30 Jun 006 to Redistribution subject to AIP license or copyright, see

7 060-7 Frequency domain analysis: Gene circuits Chaos 6, The second term on the rhs o Eq. 9 is an additional noise term operator noise ound rom this more realistic treatment o transcriptional regulation,5 and may be the dominant source o transcriptional noise. Further, the Hill expression assumes that changes in the transcription actor concentration are immediately relected in the transcription rate, whereas Eq. 9 shows that this response is limited by the operator dynamics. C. Autoregulated gene circuit Autoregulation, or eedback, exists when the level o an output o a gene circuit regulates the rate at which this output is generated. To deal with autoregulation we applied the electronic eedback concept o loop transmission to gene circuit analysis. The loop transmission, T, is the transer unction around the loop, or the requency dependent irst derivative o the regulation strength. It may be thought o as a measure o the resistance o the circuit electronic or biochemical to variation rom the steady state. T is calculated by breaking the loop at any convenient point e.g., at the point o transcriptional regulation; Fig. 3b, introducing a perturbation just to the downstream side o the break e.g., a small change in transcription rate; Fig. 3b and measuring the response that returns to the upstream side o the break e.g., the change the circuit would make in transcription rate; Fig. 3b. T is given by /, and the sign o T0 is negative resists luctuations or negative autoregulation and is positive reinorces luctuations or positive autoregulation. For the gene circuit o Fig. 3b: T = H R H o = = +i H R 0H o protein+i mrna T0 +i +i +i P R k r + k d 0 + k, where T0= H R 0H o 0, and the other terms are as described previously. Here we consider transcriptional regulation where the protein product o the gene circuit negatively regulates the transcription rate by binding to an operator site in the promoter. We make the simpliying assumption that the operator dynamics are ast compared to the protein decay and dilution rates and approximate mrna synthesis noise by modiying Eq. 9 such that 4 R Ot S RRreg =4 R +. k r + k d The protein synthesis noise ound by modiying Eq. 7 to account or the negative regulation mediated decrease o the protein synthesis rate to obtain S ppreg =4k p mrna = 4k p R =4b R. 3 R The noise in the protein concentration may be ound by the application o Eq. 9, but the transer unctions and the noise bandwidth have been altered by the negative autoregulation such that 4 and H Rreg 0 = H o 0 T0 = H R0H o 0 H o 0 T0 H = R 0 T0, H Preg = bh o 0 T0 = H R0H o 0 bh o 0 T0, T0 = b H R0 4 N T0 protein = T0 p 4, 5 where reg denotes a transer unction or the negatively autoregulated case and the other parameters are as previously described. Comparison o Eqs. 5 and 0 reveals that negative autoregulation extends the noise bandwidth by a actor o T0, ast0 is negative. The increase in bandwidth occurs by shiting some o the noise to higher requencies where it may subsequently be iltered out by downstream circuit elements. 4 Then, and P H Rreg 0 S RRreg + H P 0 S PPreg N = b p + + R T0b Preg = p reg P p T0 + Ot k r + k p 6 b R Ot k r + k p T0. 7 The irst term in Eq. 7 shows that negative eedback decreases the noise strength o the transcriptional and translational intrinsic noise by a actor o / T0 compared to the unregulated case Eq., which is consistent with the earlier analysis o Thattai and van Oudenaarden. 3 The second term relects noise o operator dynamics; it is also reduced by a actor o / T0 compared to the situation where the repressor molecule population is independent o the regulated gene. Downloaded 30 Jun 006 to Redistribution subject to AIP license or copyright, see

8 060-8 Cox et al. Chaos 6, IV. EXTRINSIC NOISE To this point we have considered only the intrinsic component o the noise. However, luctuations in RNAP, ribosomes, transcription actors, or other cellular molecular machinery shared by gene circuits or pathways lead to an extrinsic noise component.,3 Measurements have shown the extrinsic noise to be large or prokaryotes,4,3,3 and the dominant noise component in eukaryotes. 9,36 We recently reported a model where we approximated extrinsic noise as a single source located at the point o translation with a PSD given by 3 S E S E 0 +, 8 where we assumed that extrinsic noise is dominantly band limited by dilution. 3 The value o the constant term S E 0 is set by other rates transcription, translation, etc., o RNAP, ribosomes, proteases. For urther analysis and simulation it was convenient to also collect all the intrinsic noise terms at the point o translation, resulting in a single noise source with a PSD given by S source S E S I. 9 These noise terms are processed by the gene circuit such that T S pm N = E S pm N E S pm N I S pm N0 + + p I 0 + S pm N0 =,, + p 30 T T where S pm N=the normalized S pm N0= PSD o total noise extrinsic+intrinsic in the protein concentration, E S pm N0=the normalized PSD o extrinsic noise in the protein concentration at =0, and S pm N0=PSD o intrinsic I noise in the protein concentration at =0. The normalized autocorrelation unction,, is given by the inverse Fourier transormation o the S pm N to T obtain = W E + p p e p e + p + W I e + p, 3 TABLE I. Reactions or simple stochastic simulation model o intrinsic and extrinsic noise in constitutive GFP circuit. Reaction E S pm N I Rate. R R+ribo k. ribo ribo+gfp b noise * 3. ribo * 4. GFP * + 5. ribo+atc ribo-atc k 6. ribo-atc ribo+atc k r 7. ribo-atc * 0/S pm N0 + + p W E =, E I WI = WE. S + pm N0/S pm N0 + + p In accordance with this analytical approach we constructed a stochastic simulation model 3 or use with simulators based on variations o the Gillespie stochastic simulation algorithm. 5,7,37,38 All extrinsic noise was collected in the ribosome concentration and was limited in requency range by dilution. All intrinsic noise transcription and translation was represented by a single source at the point o translation. The reactions in the stochastic model o protein expression or a single gene circuit are given in reactions 4 in Table I. Reactions and 3 represent extrinsic noise that is iltered by the dilution rate. Reaction represents the translation o mrna whose stochastic variation is an intrinsic noise component that was modeled in the translation noise component. The mrna decay rate was neglected as it is usually short compared to the dilution rate. Reaction 4 represents dilution and decay o protein. The weighting o extrinsic and intrinsic noise was set by b noise according to S E 0 = b noise, 3 S I where a value o S E 0/S I 4 is consistent with previous reports.,3 Note that the b noise term used here does not represent the true burst rate o the system, but rather is a modeling device used only to achieve the correct ratio between extrinsic and intrinsic noise. V. EXPERIMENTAL MEASUREMENT OF NOISE SPECTRA IN GENETIC CIRCUITS Although there is useul inormation embedded within the spectral eatures o inherent gene circuit noise, spectral measurements are more diicult to make than noise magnitude measurements. For example, in low cytometry sequential measurements are made on dierent cells and the time course o stochastic luctuations in individual cells cannot be reconstructed. As a result, although used to great advantage in noise magnitude measurements,,9,34 low cytometry is not applicable to spectral measurements. Downloaded 30 Jun 006 to Redistribution subject to AIP license or copyright, see

9 060-9 Frequency domain analysis: Gene circuits Chaos 6, composite autocorrelation unction c or M cell trajectories was ound using FIG. 5. Color online Construction o noise trajectories by sequentially combining the noise traces o cells within a common line o descent. The solid curve shows a single trajectory through ive generations o cell growth and the dashed lines show alternate routes that produce other trajectories. A representative noise trace is shown next to each cell in the trajectory. The noise trace o the complete trajectory shown at the bottom is constructed by sequentially combining the noise traces o each cell in the trajectory. Figure rom Re. 3, supplemental material. However, we recently 3 used time-lapse luorescent microscopy to reconstruct continuous time histories o stochastic luctuations in growing cell cultures, thereby allowing the extraction o noise spectra. In these measurements the conocal microscope settings were adjusted to collect light wavelength= nm in slices thicknesscell height. Images were acquired every 5 minutes T s =5 min, and time series o noise in green luorescent protein GFP concentrations X m n T s or individual cells,,...,m were deined by their dierences rom the population mean, and individual noise traces trajectories that spanned the entire growth time were constructed by sequentially combining the noise traces o cells within a common line o descent as shown in Fig. 5. Concentrations o GFP within a given experiment were assumed to be proportional to luorescence intensity per unit cell area. Noise traces were extracted rom the images or nearly all possible trajectories in each experiment, and custom MATLAB MathWorks, Inc., Natick, MA programs were used to ind mean luorescence o the entire cell population and to estimate population doubling time rom an exponential growth curve. Normalized autocorrelations unctions ACFs or individual trajectories m were ound rom the noise time series X m n T s using a biased algorithm 39 N j Xm nt n= s X m n + jt s m jt s = N, 33 Xm n= nt s where T s was the 5 min sampling interval, n was the sample number,,...,n, and j had integer values rom 0 to N. As the gene circuits we studied were on high copy number plasmids, it was not necessary to correct or cellcycle variations due to chromosome replication i.e., we assumed that plasmid concentration remained constant. The M c jt s = m= N j Xm n= nt s X m n + jt s M m= N Xm n= nt s. 34 We investigated single gene circuits pgfpasv in E. coli TOP0 where destabilized hal-lie0 min GFP was constitutively expressed Fig. 6a. The average GFP luorescence, which corresponded to the concentration o mature GFP protein, was measured in individual cells in growing cultures or 4 8 h periods. The noise requency range was deined 3 as the inverse o the time corresponding to m = 0.5. Histograms o noise requency ranges extracted rom the individual trajectory autocorrelation unctions Figs. 7a and 7b were compiled and compared with noise requency range distributions ound rom exact stochastic simulation 7,38 using the extrinsic noise model described in reactions 4 in Table I. 3 The simulations produced as much data as 500 separate experiments, and the resulting distributions estimated the probability o inding a given noise requency range rom a randomly selected trajectory. Although some measured distributions suggested a bimodal distribution Fig. 7, this was likely due to nonrepresentative sampling o the rare high requency events. 3 The analysis presented earlier predicted that protein dilution and decay rates are dominant actors deining the noise requency range in constitutively expressed gene circuits. 4 To determine noise requency range sensitivity to protein dilution, we varied cell growth rate or the pgfpasv circuits by controlling temperature, and in a separate experiment we changed the protein decay rate using a plasmid pgfpaav containing a reduced hal-lie 60 min GFP variant. 40 These perturbations to gene circuit parameters were clearly visible in the noise spectral measurements as noise requency ranges extended to higher requencies as a result o aster growth Fig. 7a or higher protein decay rate Fig. 7b. Although varying temperature changes the rates o all reactions, the noise requency range o constitutively expressed circuits is largely sensitive only to protein dilution and decay rates in contrast to noise magnitude. 3 To test our prediction o increased noise requency range with negative autoregulation, 4 we constructed circuits with the gene or the protein TetR inserted upstream o GFP, creating a transcriptional usion ptetr-gfpasv; Fig. 6b. This circuit was negatively autoregulated as its expression was repressed by TetR binding to operator sites in the promoter. 4 A control circuit without autoregulation was also tested, in which a chromosomal copy o tetr was constitutively expressed rom the P N5 promoter. In both cases, repression was relieved by addition o anhydrotetracycline ATc to the growth medium and allowed the modulation o GFP expression. To determine i ATc had an eect on noise spectra independent o the autoregulation o the TetR circuit, we measured the noise requency range o the pgfpasv circuits in media supplemented with 00 ng/ml o ATc. There was a marked modiication in the noise requency range distribution Fig. 6c indicating a change in either the processing o Downloaded 30 Jun 006 to Redistribution subject to AIP license or copyright, see

10 060-0 Cox et al. Chaos 6, FIG. 6. Color online Gene circuits schematics and the eect o negative autoregulation. a Plasmid pgfpasv containing the constitutively expressed GFP 0 min hal-lie gene circuit. b ptetr-gfpasv negatively autoregulated gene circuit. c Eect o ATc on the noise requency range o the unregulated pgfpasv circuit doubling time 60 min; 54 trajectories without ATc; 4 trajectories with ATc. sim., simulated using model described in Table I. d Negative autoregulation-mediated shit o noise requency range doubling time 60 min; pgfpasv: 54 trajectories without ATc, 4 trajectories with ATc; ptetr-gfpasv: 4 trajectories. e Model o the shit o requency range distribution shape due to negative eedback. The right-skewed distribution shown on the let represents an unregulated circuit distribution. Negative autoregulation shits the distribution toward the center as shown by the dashed box and arrow. The bars in dark show portions o the distribution that are common to both the regulated and unregulated circuits. The higher requency trajectories are unaected. Figure rom Re. 3. the noise or the nature o the noise sources. Our modeling points to the latter with ATc inhibition o translation 4 leading to a reduction in the weighting and whitening o extrinsic noise, which we explored using the extrinsic noise simulation model in Table I. Reactions 5 7 describe the ribosome- ATc heterodimer ormation and its dilution due to cell growth. 3 The requency range distribution extracted rom these simulations was compared to the measured distribution Fig. 6c, with both showing a characteristic peak shit and peak broadening. Although not conclusive, this gross agreement between measured and simulated distributions supports the hypothesis that the mechanism o ATc-mediated noise requency range modulation is an increase in high requency content o the global extrinsic noise associated with translation and a reduction o the weighting o extrinsic noise. 3 We measured noise requency range distributions o ptetr-gfpasv and the control cells grown in media with 00 ng/ ml o ATc. Composite noise requency ranges o the negatively autoregulated ptetr-gfpasv exceeded those o the constitutively expressed pgfpasv in 00 ng/ ml o ATc by as much as 3 Figs. 6d and 8, whereas the control circuits showed no noise requency range increase Fig. 8. The negative autoregulation-mediated noise remodeling was seen as an increase o the noise requency range Fig. 8 and as a modiication o the shape o the distribution Fig. 6d. Autoregulation requency response is limited by protein decay and dilution, and thereore has a larger eect on slower luctuations than aster luctuations. Noise trajectories that would have clustered at the lower end o the requency range distribution in unregulated cells are pushed to higher values by negative autoregulation, whereas those in the higher requency tail o the distribution are only weakly aected Fig. 6e. This results in requency range distributions with a shape closer to normal distributions Fig. 6d. The requency shit and the change in distribution shape are indicative o the presence o negative autoregulation. We would also expect a decrease in the circuit noise strength in the autoregulated gene circuit, but the variable gain o the conocal microscope did not allow or absolute measurements o noise strength. FIG. 7. Eects o cell doubling time and protein hallie on noise requency range. Measured distributions are shown as vertical bars and simulated distributions as solid lines. a Shit in noise requency range or the pgfpasv circuit as doubling time increases rom 30 min 00 trajectories; T=3 C to 90 min 0 trajectories; T= C. b Shit in noise requency range as protein decay time decreases rom 0 min pgfpasv; 59-min doubling time; 54 trajectories; T =6 C to 60 min pgfpaav; 56-min doubling time; 33 trajectories; T=6 C. The model described in Table I was used in simulations. Figure rom Re. 3. Downloaded 30 Jun 006 to Redistribution subject to AIP license or copyright, see

11 060- Frequency domain analysis: Gene circuits Chaos 6, TABLE II. Reactions or model o constitutive GFP expression as aected by extrinsic noise at the transcriptional and translational level and in the presence o ATc. RNAP polymerase synthesis and dilution transcriptional level extrinsic noise: RNAP RNAP * Transcription and mrna decay: k tc RNAP RNAP+mRNA FIG. 8. Color online Noise requency range vs doubling time. Measured points are shown with ± error bars estimated rom simulation. The analytical curve Eq. 3 or the pgfpasv circuit and the simulated curve Table I or pgfpasv+00 ng ml ATc are shown. Vertical black arrows represent regulation strength determined by the shit o the noise requency range. The temperature C o each experiment is indicated by each data point. The TetR data points are or the circuit with autoregulated tetr expression, whereas the TetR ctrl data points are or the circuit with constitutive tetr expression. Figure rom Re. 3. These measurements validated the prediction that in addition to noise magnitude gene circuits manipulate noise spectra. We veriied the link between gene circuit structure and the noise requency range and showed that changes in gene circuit parameters e.g., cell growth and protein decay rates modulate the noise requency range. Our results show a shit o noise spectra to higher requencies and a remodeling o the noise requency range distribution that is characteristic o negative autoregulation. This noise spectral remodeling may impact the regulatory eect o the noise as it propagates through the gene network, as higher requency noise is more easily iltered out by downstream gene circuits in a regulatory cascade. 4 One o the more intriguing aspects o this study was that the noise requency range distributions provided a means or evaluating a hypothesis o the mechanism o ATc-mediated remodeling o noise spectra in unregulated gene circuits. In the ollowing section we take a closer look at the interaction o the ATc and the extrinsic noise o the gene circuit. VI. EXTRINSIC NOISE AT THE TRANSCRIPTIONAL AND TRANSLATIONAL LEVELS m * mrna Ribosome synthesis and dilution translational level extrinsic noise: ribo ribo * Translation and GFP dilution/decay: k tl mrna+ribo mrna+ribo+gfp p + GFP * Removal o stalled ribosomes rom the cell by dilution k stalled =0 in the absence o ATc: k stalled * ribo The ATc-mediated modiication o the noise requency range provides some insight into the relative contributions o extrinsic and intrinsic noise sources. Bacterial reporter protein systems studied to date are oten characterized by signiicant extrinsic noise with an autocorrelation time scale approximately equal to the cell cycle. 3,3 For convenience, we have chosen the cell machinery components RNA polymerase RNAP and ribosomes to represent extrinsic noise at the transcriptional and translational levels, respectively. Other potential sources o extrinsic noise include global variation in the concentration o sigma actors, ribonucleotides, amino acids, RNases, proteases, and other shared cell resources. Although these extrinsic noise sources are not directly treated here, they are indirectly considered as there eect on the reporter gene circuit is mediated by the transcription and translation processes associated with RNAP and ribosomes. We urther assume that the variation in the concentration o RNAP and ribosomes is driven by the synthesis and growth-driven dilution processes. However, their synthesis also depends upon shared cell resources which unction as additional extrinsic noise sources, not considered in this model. This model also does not account or the temporary unavailability o RNAP and ribosomes actively engaged in elongation reactions to initiate new transcription and translation reactions, which causes additional variation in their ree populations. We have urther assumed that the series o maturation steps that nascent GFP must undergo prior luorescence are suiciently ast to be ignored, as discussed in Austin et al. 3 Despite these simpliying assumptions, the model considered here is suiciently detailed to demonstrate the eects o transcriptional and translationallevel extrinsic noise sources on the expression o GFP in the presence o ATc. With these assumptions, we model the production o GFP according to the reactions in Table II. The last reaction is active only in the presence o ATc and its justiication is described as ollows. Tetracycline and many o its derivatives aect translation by binding to the 30S ribosomal subunit reviewed in Chopra and Roberts 4. This interaction blocks the addition o aminoacyl-trna and prevents translocation o the ribosome complex down the mrna strand. In the presence o these types o translation inhibitors the mrna is stabilized, presumably by shielding it rom degradation by RNases. 43 Additionally, more recent work using untranslated RNA suggests that the increased transcription o rrna that occurs in the presence o translation inhibitors creates a drain on RNA degradative capacity, resulting in an additional indirect protection o mrna pools. 44 Based on this knowledge we can summarize the action o ATc as having two eects: it increases mrna stability and it causes the ribosome to stall on the transcript. Increased mrna stability is easily handled by decreasing the mrna decay rate m. A closer look at the translation process is needed to understand the eect o a stalled ribosome. Translation is initiated upon binding o a ribosome to a ribo- Downloaded 30 Jun 006 to Redistribution subject to AIP license or copyright, see