Theoretical and experimental analysis of the forced LacI-AraC oscillator with a minimal gene regulatory model

Transcription

1 Theoretical and experimental analysis of the forced LacI-AraC oscillator with a minimal gene regulatory model Guillermo Rodrigo, Boris Kirov, Shensi Shen, and Alfonso Jaramillo Citation: Chaos 23, (2013); doi: / View online: View Table of Contents: Published by the American Institute of Physics. Additional information on Chaos Journal Homepage: Journal Information: Top downloads: Information for Authors:

2 CHAOS 23, (2013) Theoretical and experimental analysis of the forced LacI-AraC oscillator with a minimal gene regulatory model Guillermo Rodrigo, 1,2 Boris Kirov, 1,2 Shensi Shen, 1,2 and Alfonso Jaramillo 1,2 1 Institute of Systems and Synthetic Biology (issb), CNRS, F Evry, France 2 Universited Evry Val d Essonne, issb, F Evry, France (Received 24 January 2013; accepted 24 May 2013; published online 12 June 2013) Oscillatory dynamics have been observed in multiple cellular functions and synthetic constructs; and here, we study the behavior of a synthetic oscillator under temporal perturbations. We use a minimal model, involving a single transcription factor with delayed self-repression and enzymatic degradation, together with a first-order perturbative approach, to derive an analytical expression for the power spectrum of the system, which characterizes its response to external forces and molecular noise. Experimentally, we force and monitor the dynamics of the LacI-AraC oscillator in single cells during long time intervals by constructing a microfluidics device. Pulse dynamics of IPTG with different periods serve to perturb this system. Due to the resonance of the system, we predict theoretically and confirm experimentally the dependence on the forcing frequency of the variability in gene expression with time and the synchronization of the population to the input signal. The reported results show that the engineering of gene circuits can provide test cases for dynamical models, which could be further exploited in synthetic biology. VC 2013 AIP Publishing LLC. [ The genetic motif consisting in a delayed self-repression is particularly interesting in molecular biology because of its structural simplicity and the wide range of dynamical behaviors that displays. When combined with an external force changing with time, intriguing dynamical features can emerge. In this work, we develop a theoretical framework to study this motif in response to an external, periodic force. A simple mathematical model based on a stochastic delay differential equation results fruitful to tackle analytically the amplitude amplification and phase synchronization of the system to the external force. Experimental results of forcing a genetic oscillator support this work. I. INTRODUCTION Genetic oscillations are the result of nonlinear interactions between several elements of a regulatory network in the cell. 1,2 However, cells are usually subjected to perturbations that can change the dynamical behavior of the network. Together with variations in certain intracellular factors (i.e., molecular noise), 3 temporal fluctuations of external signals that modulate regulatory elements of the system need to be considered for a proper understanding of a genetic oscillator. Details of its dynamical behavior that can emerge as a consequence of such coupling are difficult to predict without a quantitative model. Exploiting a minimal gene regulatory model is of particular interest to be able to derive analytical expressions, which can be taken into account to understand more complex systems. 4 In this work, we aim at studying mathematically and experimentally how the dynamical behavior of a genetic clock is perturbed when the environment changes deterministically or stochastically with time. In essence, oscillatory circuits, both natural and synthetic, 5 8 could be reduced to a minimal model of delayed self-repression, where the expression of a given gene entails a negative signal that affects its own expression after few or multiple steps. Proteins in such systems are usually short-lived by interactions with proteases to better respond dynamically. 9 It is also important to consider enzymatic degradation kinetics because it enhances the oscillatory region in the phase space. 10 Here, we consider a self-repressed gene circuit with external forcing to develop a theoretical framework. A mathematical model based on a couple of sigmoidal functions, one with delay for protein synthesis and another for protein degradation, serve to account for the small set of molecular interactions that are required to predict the resulting dynamics. Previous theoretical works have disentangled the basic principles of genetic oscillators, 2 and more precisely have characterized the stochastic dynamics of that circuit, but multiple details remain uncertain related to the coupling of stochastic and periodic signals with a system having a natural frequency. In this work, we will show how a minimal oscillator model is already able to provide a wide range of complex behaviors, from stationary to complex oscillatory dynamics when periodically forced. Moreover, we will show that this simple model can describe the dynamics of an experimental system. Using the LacI-AraC oscillator 8 and a microfluidics platform, we will explore experimentally the effect on the dynamical behavior of an external force, constant or periodic to see whether the system can enter in resonance or get entrained due to that perturbation All together, this /2013/23(2)/025109/9/$ , VC 2013 AIP Publishing LLC

3 Rodrigo et al. Chaos 23, (2013) study will reveal important aspects about forced, or even coupled, genetic clocks. II. METHODS A. Model We construct an effective model for a self-repressed gene (Fig. 1(a)) using a delay differential equation (DDE) involving the concentrations of the protein (X) and an external molecule that is able to inhibit the repressive action of the protein (U), which reads dx dt ¼ A 0 1 X s 1 K 1 þ UðtÞ A K U n bex lx; (1) K E þ E þ X where we neglect the transcription leakage at full repression and consider an enzymatic degradation of the protein. 18,19 The model for the regulation is based on an effective Hilllike formulation. 12,13 For enzymatic degradation, an effective term accounting for a limited amount of proteins and proteases in the cell is considered. 19 Here, A is the maximal protein synthesis rate, K is the effective protein-dna dissociation constant, n is the Hill coefficient, and K U is the effective protein-molecule dissociation constant. U(t) is an arbitrary function that we use to model an external, temporal signal. In addition, s is the delay of the regulation (produced by the different processes of transcription, translation, multimerization, and binding to DNA), with X s ¼ Xðt sþ. Finally, b is the enzymatic degradation rate (mediated by a host protease, such as ClpXP in bacteria 18 ), E is the protease (ClpXP) concentration, K E is the effective enzyme-protein dissociation constant, and l is the specific cell growth rate (assuming exponential phase). It is useful to normalize the mathematical model for subsequent analyses. To this end, we define x ¼ X/K, u ¼ U=K U ; a ¼ A=K; b ¼ be=k, and r ¼ðK E þ EÞ=K (see Ref. 30 for typical parameter values). Experimentally, we can have a control of the system with parameter a by playing with promoters of different strengths. Then, we can write the DDE of the system (from Eq. (1)) as dx dt ¼ Pðx s; uðtþþ QðxÞ; (2) where we define P and Q, production and degradation terms, as a Pðx; uðtþþ ¼ x n ; 1 þ 1 þ uðtþ (3) QðxÞ ¼ bx r þ x þ lx: To account for the inherent stochasticity of biological systems, we exploit a Langevin formulation 20 (see also Ref. 30). Assuming that the process of translation acts as a noise amplifier, we only consider the noise at the mrna level and discard the noise at the protein one. 21 We also assume that most of the noise is extrinsic, which has been observed in bacteria. 3 Note that extrinsic noise depends on the variability of the cellular machinery (i.e., number of RNA polymerases or ribosomes), the growth rate, the plasmid copy number (if the system is expressed heterologously), and even the environmental conditions. Therefore, the resulting stochastic DDE (SDDE), in terms of P and Q, reads dx dt ¼ Pðx s; uðtþþ QðxÞþqn 0m ðtþ; (4) FIG. 1. (a) Scheme of the self-repressed gene circuit mathematically analyzed in this work. Gene X is assumed to have a degradation tag. 18 The different processes of transcription, translation, multimerization, and binding to DNA are assumed to introduce the delay of the regulation (X m denotes mrna, and X protein). (b) Simplified scheme of the LacI-AraC oscillator (the actual system involves two plasmids). 8 Transcription factors AraC and LacI, as well as GFP as reporter, were under the control of the same promoter, P lac=ara. 23 All proteins were ssra tagged (LAA). 18 The circuit consists of a self-repression mediated by LacI together with a self-activation by AraC. where n 0m ðtþ is a stochastic process that models the extrinsic noise of the mrna change rate, and q is the corresponding magnitude. We take hn 0m ðtþi ¼ 0. According to previous experimental results, 22 the autocorrelation time for the extrinsic noise is similar to the cell growth rate, which leads us to take hn 0m ðt 0 Þn 0m ðt 0 þ tþi ¼ l 2 e ljtj. h:i represents a statistical ensemble average. B. Single cell experiments The genetic circuit that we studied experimentally in the bacterium Escherichia coli was the LacI-AraC oscillator engineered by Hasty and coworkers (Fig. 1(b)). 8 Transcription factors AraC (activator/repressor of the arabinose operon) and

4 Rodrigo et al. Chaos 23, (2013) LacI (repressor of the lactose operon), as well as green fluorescent protein (GFP) as reporter, were under the control of the same promoter, P lac=ara. 23 All proteins had been engineered to have enhanced degradation rates by fusing at the C-terminus of them a ssra tag (LAA). 18 The E. coli K-12 strain used in all experiments was the JS006 (MG1655 DaraC DlacI). Microfluidics experiments were performed in microfabricated devices comprising supply channels and rectangular growth chambers. Isopropyl b-d-1-thiogalactopyranoside (IPTG) and L-arabinose were used at different concentrations. Images were processed by first segmenting the individual bacterial cells, and then extracting the fluorescence levels. 30 III. RESULTS A. Delayed self-repression as a model for genetic oscillators To understand qualitatively the behavior of our onecomponent genetic oscillator with effective delay (s), we use a minimal regulatory model. In our DDE, the intersection of the production (P) and degradation (Q) terms could be assumed to yield a single stable point (Fig. S1). 30 However, the effect of a delay in the production term creates a hysteresis that prevents reaching a stable steady state in favor of oscillations. We can obtain analytical approximations when n!1; r! 0, and l! 0, provided a > b. Under this regime, we can use a piecewise equation Pðx; 0Þ ¼ahð1 xþ and QðxÞ ¼b, where h is the Heaviside step function (in absence of forcing). This way, the gene expression dynamics is governed by a piecewise linear DDE, which helps to analyze the dynamics and gives rise to degrade-and-fire oscillations. 12 The corresponding dynamics could be qualitatively understood as follows. According to the value of x s, the system will be in a region dominated by the production, dx dt ¼ a b (when x s > 1), or by the degradation term, dx dt ¼ b (when x s < 1). When being at low expression levels (x 0) and assuming a close history where the gene is not expressed (x s 0), the system enters into a production regime, which increases gene expression. This regime elapses in s, the time that needs the repressor to be active, which gives a maximum concentration about ða bþs. After that, the system is at high expression levels (x > x s > 1) and the promoter is fully repressed. Hence, the system goes into the degradation regime (relaxation). This makes that gene expression to decrease, taking a much longer time to recover the initial state. Following the early work by Glass and Mackey, 15 we perform a stability analysis to determine the parameter conditions to get oscillations with this genetic system. At the steady state ( dx dt ¼ 0) in the deterministic regime and without introducing any forcing, we have Pðx 0 ; 0Þ ¼Qðx 0 Þ, where x 0 is the stationary value. We can solve this algebraic equation in the limit r! 0 (zeroth-order enzymatic degradation 10 ), provided a > b l. This gives x 0 ¼ a 1=n b 1 : (5) This is also a good approximation when r ¼ 1. For small amplitude oscillations around x 0, we can linearize Eq. (2), with u ¼ To simplify the notation, we define the following partial derivates: / x Pðx 0 ; 0Þ; w x Qðx 0 Þ, and f u Pðx 0 ; 0Þ. Then, we apply the Laplace transform to obtain the characteristic quasi-polynomial dðsþ ¼s þ /e ss þ w; (6) which can be exploited to look for bifurcations (d(s) ¼ 0). 24 The Pitchfork bifurcation (bistability) implies s ¼ 0, giving / þ w ¼ 0, which is absurd in this case (/ > 0 and w > 0). The Hopf bifurcation (oscillations) implies s ¼ ix, giving w ¼ / cosðxsþ and x ¼ / sinðxsþ. Because we are considering a perturbative approach, this is still a classical result. The first value of the arccos defines the critical delay (s 0 ) from which oscillations occur arccos w / s 0 ¼ q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; (7) / 2 w 2 and the oscillations for systems closed to the Hopf bifurcation will have a period of 2p T ¼ q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : (8) / 2 w 2 Accordingly, as enzymatic degradation and nonlinearity in the system increase, the region for sustained oscillations gets larger (Fig. S2). 30 Subsequently, by fixing s ¼ 2:2 min, n ¼ 4, b ¼ 1min 1 ; r ¼ 1, and l ¼ 0:02 min 1, we can analyze the oscillations shape in function of a, which can be tuned by varying, for example, the gene copy number. We obtain damped oscillations for a ¼ 1000 min 1 (Fig. 2(a)), reaching steady state after a transient period of oscillations, lim t!1 xðtþ ¼x 0, and the dynamics can be written as xðtþ x 0 þ he lt cos 2pt ; (9) T where h is a constant. The period of those oscillations is measured from numerical simulation to be T 8:7 min, which agrees with our theoretical estimate (Eq. (8)). This is understandable because it is a damped oscillator close to the bifurcation. In another regime, we obtain quasi-harmonic oscillations for a ¼ 100 min 1 (Fig. 2(b)), resulting in estimated dynamics given by xðtþ h 0 þ h 1 cos 2pt T ; (10) where h 0 and h 1 are two constants, and the period is measured from numerical simulation to be T 9:4min. The value disagrees with the estimate (Eq. (8)) because the system is far from the bifurcation. Moreover, for a ¼ 10 min 1, we obtain degrade-and-fire oscillations (Fig. 2(c)), with period T 10:8 min, and the dynamics can be approximated by

5 Rodrigo et al. Chaos 23, (2013) FIG. 2. Dynamics of the system for different values of a to modulate the oscillation shape, obtained by numerical simulation of Eq. (2) with u ¼ 0. (a) a ¼ 1000 min 1 (damped oscillations), (b) a ¼ 100 min 1 (quasi-harmonic oscillations), and (c) a ¼ 10 min 1 (degrade-and-fire oscillations). The rest of parameter values are s ¼ 2:2 min, n ¼ 4, b ¼ 1 min 1 ; r ¼ 1, and l ¼ 0:02 min 1. 8 >< x fire t=s; t 0 t < t 0 þ s xðtþ x fire bt 0 ; t 0 þ s t < t 0 þ s þ t deg >: h 0 ; t 0 þ s þ t deg t < T; (11) where x fire ¼ð a 1þh 0n bþs; t0 ¼ t t 0 s; t deg ¼ðx fire h 0 Þ=b, and h 0 is a constant. Following this model, we attempt to study the experimental dynamics at the single cell level of the LacI-AraC oscillator, 8 which has been characterized in bacteria showing fast and robust oscillations. The LacI-AraC oscillator consists of a self-repression mediated by LacI together with a self-activation by AraC (Fig. 1(b)). But can a single gene with delayed negative feedback be a relevant model to study the dynamical properties of a given oscillatory circuit? If so, this could be generalized to many other systems, with two or more components, that in an effective way might be reduced to that minimal motif. Certainly, in presence of a constant amount of arabinose, the system could be reduced to a selfrepressed motif by assuming that AraC is stronger than LacI in the regulation. 25 There, the time to accumulate a critical amount of AraC together with the multimerization and binding to DNA of LacI introduce the required delay to sustain oscillations. 4,8 Using our microfluidics platform, we show that this oscillator exhibits different dynamics according to different concentrations of IPTG and arabinose. In absence of inducers, the system does not oscillate, and this is because the promoter P lac=ara is not active (Fig. 3(a)). RNA polymerase needs to be enhanced by the complex formed by AraC and arabinose; hence in that regime, we only observe some eventual spikes due to molecular noise. However, when arabinose is present in the medium, the system shows oscillations (monitored with GFP) due to a higher transcription rate of the promoter P lac=ara. When arabinose is the only inducer present, we observe different modes of oscillations, which may reflect the effect of molecular noise (Fig. 3(b)). Several cells oscillate with a period of 60 min and a small amplitude, while some others do with higher period and amplitude. By simulating the stochastic model, we recover a dynamical pattern where the amplitude can change with time, and thus also from cell to cell (Fig. S3). 30 When both arabinose and IPTG are present (the latter at a concentration that does not fully inhibit the repressive action of LacI), the system shows the more regular behavior (Fig. 3(c)). The addition of IPTG in the medium, which influences the regulatory coefficient of LacI (K in Eq. (1)), can be simulated by decreasing the value of a. According to our model, the tunable parameter a modulates the oscillatory behavior. This way, for a fixed small delay, the system falls more likely within the region of sustained oscillations with lower values of this parameter, up to a limit (Fig. S2). 30 B. Dependence of variability in gene expression on forcing frequency It is important to study how dynamical perturbations over a system that shows oscillations alter its behavior. 2 This is of relevance for understanding important aspects about the functions of natural clocks, as well as for implementing de novo genetic circuits. 1 For instance, the nuclear factor j-light-chain-enhancer of activated B cells (NF-jB) induces oscillations thanks to a delayed negative feedback loop, 6 and because it is a sensor of external stimuli, gene expression levels in different environments could be predicted. Here, we account for the combination of a deterministic perturbation that is introduced manually with the inherent stochasticity due to extrinsic noise. We study how variability in gene expression can be modulated by the frequency of an external, periodic signal. To study analytically this question, it is useful to obtain the power spectrum of the system, which relates, in the frequency domain, the internal variables of the system and the perturbations. 30 In case of a damped oscillator, where the system reaches a steady state, we can apply linear expansions and easily solve the problem.

6 Rodrigo et al. Chaos 23, (2013) loss of generality and for simplicity, we can take uðtþ ¼u 0 cosðx 1 tþ, even if u(t) takes negative values, with u 0 1. Therefore, and assuming that u(t) and n 0m ðtþ are uncorrelated, we calculate the power spectrum (S Dx ) by squaring and averaging Eq. (12), which results in l2 l 2 þx 2 S Dx ðxþ ¼ pf2 u 2 0 2jdðixÞj 2 ðdðx x 1Þþdðxþx 1 ÞÞ þ q2 jdðixþj 2 : (13) It is well-known that noise can enhance oscillations, because a perturbation over a system close to the Hopf bifurcation leads to a transition. 26 However, the system with a natural frequency will respond with different amplitudes under periodic perturbations (resonance), which can be characterized by exploiting Eq. (13). The term 1 represents jdðixþj 2 the power spectrum of the system under an external pulse. The algebraic solution x jdðixþj ¼ 0 gives the value of the natural frequency of the system, x In our case qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (damped oscillator), x 0 is very close to x 0 ¼ / 2 w 2, the frequency of the bifurcation. We indeed observe the resonance at x 0 x 0, and also that its distribution is very narrow (Fig. 4(a)). This could be approached by a Lorentzian function to model a generic excitable system. Hence, by forcing with x 1 ¼ x 0, the amplitude of the oscillations will be enhanced. In presence of colored noise, the resonance is FIG. 3. Experimental GFP dynamics in a single cell of the LacI-AraC oscillator. We show two different dynamics (continuous and dashed lines), corresponding to different cells. In (a), [IPTG] ¼ 0 and [arabinose] ¼ 0; in (b), [IPTG] ¼ 0 and [arabinose] ¼ 0.3%; in (c), [IPTG] ¼ 2 mm and [arabinose] ¼ 0.3%. In the insets, we show the distribution of fluorescence for all cells and all time points. AFU, arbitrary fluorescence units. We apply the Fourier transform over the linearized SDDE (from Eq. (4)), giving D^x ¼ f^uðxþþq^n 0m ðxþ ; (12) dðixþ where dðixþ is the characteristic quasi-polynomial in the frequency domain (Eq. (6)). If the external signal to force the system is periodic, we denote x 1 its frequency. Without FIG. 4. Power spectrum showing the resonance. (a) Natural power spectrum, as a function of the period. (b) Power spectrum of the fluctuation (S Dx ) for a damped oscillator without forcing (Eq. (13) with u 0 ¼ 0). Parameter values of Fig. 2(a), and q ¼ 1.7.

7 Rodrigo et al. Chaos 23, (2013) still maintained at x 0 with l 1 (Fig. 4(b)). Interestingly, the analytical derivation of the resonance features (frequency, bandwidth or maximum value) could be exploited to infer kinetic parameters of the system from experimental data. 27 Furthermore, the power spectrum S Dx can be inverse Fourier transformed (Wiener-Khinchin theorem) with t ¼ 0 to calculate the variance of the fluctuation hdx 2 i¼ f2 u 2 0 2jdðix 1 Þj 2 þ q2 v; (14) where q 2 v is the contribution to the variance due to the noise (see definition of v in Ref. 30). Thus, the variability in normalized protein expression is given by pffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi hdx g x ¼ 2 i ¼ 1 f 2 u 2 0 x 0 x 0 2jdðix 1 Þj 2 þ q2 v: (15) Thus, by forcing with x 1 < x 0, we can modulate g x by taking advantage of the resonance (Fig. 5(a)). We obtain a good agreement between the numerical simulations and the theoretical predictions. Nevertheless, the system may behave unpredictably when forced at much higher amplitudes (Fig. S4). 30 Indeed, in nonlinear systems, complex dynamical behaviors may emerge due to a coupling between different frequencies in the system, 15 and complete theoretical analyses of the frequency-amplitude forcing maps require nonperturbative approaches. We then analyze the single cell response of the forced LacI-AraC oscillator. Experimentally, it is possible to introduce a periodic forcing with small amplitude over a system by using a microfluidics platform. In particular, we consider a square wave of IPTG maintaining a constant concentration of arabinose. In this regime, the LacI-AraC oscillator can still be analyzed following our theoretical results. In absence of IPTG, we estimate the natural frequency of the system in x 0 0:105 rad=min. We can calculate the variability in GFP expression for different forcing frequencies to see whether the resulting trend is similar to the one shown in Fig. 5(a). Thus, from N single cell dynamics, we can calculate vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u1 X N hdx 2 j g x ¼ t i N hx j i 2 ; (16) j¼1 where averages are done over time. We also observe a dependence of GFP variability on x 1, noting that fluctuations are higher in the experimental system (Fig. 5(b)). This difference could be explained, at least in part, thanks to the positive feedback in the LacI-AraC oscillator, 4 which is not considered in our theoretical model, and a higher magnitude of the extrinsic noise (q). In addition, we find that g x is also large when x 1 is high (x 1 3x 0 ), which might be due to an increase on extrinsic noise. Moreover, Fig. S5 30 shows the distribution of GFP expression from cell to cell and for all time points, which could be approached by a log-normal, to illustrate the higher variability (due to higher oscillations amplitude) when forcing with x 1 x 0. C. Synchronization of gene expression to periodic forcing In absence of external forcing, even though noise makes the system oscillate, each cell will do so at a different phase, which also changes with time. This will mask the single cell behavior if we look at the population level. One way to quantify the synchrony between two cells (assuming ergodicity) is by the normalized autocorrelation function (C Dx ), defined by C Dx ðtþ ¼ hdxðt 0ÞDxðt 0 þtþi hdx 2 i. In our case, we can assume that the effective correlation time of Dx will be the same as the one for the extrinsic noise, and that it will oscillate at the natural frequency. This way, it can be approached by FIG. 5. Variability in protein expression in function of the forcing frequency (x 1 ). (a) Theoretical calculations. In circles and triangles, values calculated from the dynamics obtained by numerical simulation, with u 0 ¼ 0:1. The lines show the theoretical predictions from Eq. (15). All numerical simulations are performed with Eq. (4). Parameter values of Fig. 2(a), which give v ¼ 0:033. (b) Experimental data. Values calculated with the GFP dynamics obtained by single cell experiments, with constant [arabinose] ¼ 0.3% and a square wave of IPTG (between 0 and 20 lm) for different periods. C Dx ðtþ e ljtj cosðx 0 tþ: (17) In Fig. S6, 30 we show a good agreement between the numerical simulation and the theoretical prediction. With the experimental system, we also show that the phase of GFP dynamics of a single cell changes with time.

8 Rodrigo et al. Chaos 23, (2013) Using our theoretical model and experimental system, it is possible to explore the degree of synchronization to the input of a population of genetic oscillators. An entrainment by periodic signals of arabinose and IPTG has been already observed for the LacI-AraC oscillator, 17 where the positive feedback is believed to enlarge the entrainment region. We here study how a periodic perturbation can be transmitted in phase by our regulatory motif, and how this depends on the signal frequency. 15,28 We can use our model to simulate the dynamics of the LacI-AraC oscillator in presence of a square wave of IPTG, with constant arabinose. For that, we take a ¼ 10 min 1 ; s ¼ 10 min, and n ¼ 2. We consider a higher delay due to the effect of the positive feedback. Furthermore, as the circuit has several tagged proteins and the host protease is limited, we consider b ¼ 0:25 min 1 and r ¼ 0:25 as the parameter values characterizing the enzymatic degradation. And from our own experimental data, we estimate l ¼ 0:01 min 1. With this parameterization, the system is close to the Hopf bifurcation and the period of the oscillations is 60 min (using Eq. (8)). Then, we show both theoretically and experimentally forced oscillations in phase with x 1 x 0 (Fig. 6), but we expect different degrees of synchronization for different values of x 1. To tackle analytically the input/output phase synchronization, we define the synchronization coefficient (q) as the correlation between the input and the fluctuation of the output hdxuðtþi q ¼ q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; (18) hdx 2 ihuðtþ 2 i which we use to characterize the system quantitatively. Higher values of q will indicate better synchronization. To perform this calculation, we consider as before a harmonic perturbative approach, where the amplitude of external forcing is small and then linear expansions can be exploited. 30 We start from Eq. (12) to multiply by ^u ðxþ (complex conjugate of ^uðxþ). Then, by averaging and assuming that u(t) and n 0m ðtþ are uncorrelated, we get hd^x^u ðxþi ¼ pfu2 0 2dðixÞ ðdðx x 1Þþdðx þ x 1 ÞÞ: (19) Therefore, we can apply the generalized Wiener-Khinchin theorem to obtain the cross-correlation function hdxuðtþi ¼ fu 2 Reðdðix 1 ÞÞ 0 2jdðix 1 Þj 2 ; (20) where Reðdðix 1 ÞÞ ¼ / cosðx 1 sþþw. Then, we can combine this with Eq. (14), knowing that huðtþ 2 i¼ u2 0 2, to obtain an expression for the synchronization coefficient Reðdðix 1 ÞÞ q ¼ r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : (21) jdðix 1 Þj 1 þ 2q2 vjdðix 1 Þj 2 f 2 u 2 0 FIG. 6. (a) Dynamics obtained by numerical simulation of Eq. (4) (continuous line), with a ¼ 10 min 1 ; s ¼ 10 min; n ¼ 2; b ¼ 0:25 min 1 ; r ¼ 0:25; l ¼ 0:01 min 1,andq ¼ 1.7. The dashed line corresponds to u(t) (between0 and 0.5). (b) Experimental GFP dynamics in a single cell of the forced LacI- AraC oscillator (continuous line), with constant [arabinose] ¼ 0.3%. The dashed line corresponds to the square wave of IPTG (between 0 and 20 lm). AFU, arbitrary fluorescence units. Forcing period 2p=x 1 ¼ 60 min in both cases. In Fig. 7(a), we indeed show the existence of an optimal synchronization when forcing with the natural frequency (x 1 ¼ x 0 ). However, when x 1 ¼ x 0, a very close point to the optimum, Reðdðix 1 ÞÞ ¼ 0 and then q ¼ 0. This indicates that the oscillations of the forcing signal (u(t)) and the ones of the output (x) are phase-shifted p=2. Numerical simulations confirm this theoretical result. We also show that the parameter q=u 0, the ratio of amplitudes between noise and forcing, modulates the shape of the function. By assuming that molecular noise is a major cause of fluctuation in gene expression (q=u 0 1), it turns out q / Reðdðix 1ÞÞ jdðix 1 Þj 2. This synchronization coefficient, however, does not take into account the amplitude of the output oscillations. It only quantifies the phase. For instance, with q=u 0 ¼ 10, the oscillator is very well synchronized by the external force with x 1 < x 0, but it has a very low amplitude (according to Eq. (14) and for a given separation from the optimum). A way to balance these two terms would be, for example, with the variable g x q. We then analyze the population of single cell responses of the forced LacI-AraC oscillator, using a square wave of IPTG. We expect an input/output synchronization with reasonable signal-to-noise ratio when forcing with periods close

9 Rodrigo et al. Chaos 23, (2013) FIG. 8. Experimental GFP dynamics in a single cell of the forced LacI-AraC oscillator, with constant [arabinose] ¼ 0.3% and a square wave of IPTG (between 0 and 20 lm). Forcing period 2p=x 1 ¼ 60 min. Each single cell trajectory (about 100 in total) is shown in horizontal. Arbitrary fluorescence units in log scale. FIG. 7. Synchronization coefficient of the system in function of the forcing frequency (x 1 ). (a) Theoretical calculation with Eq. (21) for different values of q=u 0 (ratio of amplitudes between noise and forcing). Parameter values of Fig. 2(a). (b) Experimental calculation for the LacI-AraC oscillator. It is calculated with the GFP dynamics obtained by single cell experiments using Eq. (22). to the one of GFP oscillations in absence of IPTG. 17 From N single cell dynamics, we can calculate vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u1 X N hdx j uðtþi 2 q ¼ t ; (22) N hdx 2 j ihuðtþ2 i j¼1 where averages are done over time. We calculate q for different values of x 1 (Fig. 7(b)), showing a maximal synchronization when forcing with x 1 x 0. The synchronization to the 60 min-period wave of IPTG when forcing with a period of 20 min (x 1 3x 0 ) is also high (q ¼ 0:245), which indicates that the system is excited at this frequency (see also Fig. 5(b)). Additionally, we illustrate the moderate correlation between GFP and IPTG when x 1 x 0 for a significant number of cell trajectories, although not all (Fig. 8; see also the movie 30 ). Certainly, noise induces variability in the period of the oscillations, 29 which prevents full synchronization between cells with a small amplitude signal. IV. DISCUSSION Using an effective model for a genetic oscillator based on delayed self-repression, we have been able to study analytically the effects of stochasticity and periodic forcing on the dynamics of the system. Oscillations in gene expression are enhanced for forcing frequencies close to the natural one, due to a resonant behavior. The analytical derivation of the power spectrum has resulted instrumental to explicit our theoretical results. Although these results have been developed for a particular model (self-repressed transcription factor with enzymatic degradation), any oscillatory system would show similar dynamical behaviors. We have tested our theoretical predictions using an experimental system consisting in a gene circuit exhibiting robust oscillations and a microfluidics platform to monitor gene dynamics in single cells. Our results, both theoretical and experimental, indicate that it could be possible to use periodic forcing to modulate gene expression variability and synchrony in a bacterial population. And from a synthetic biology perspective, such a regulatory motif could be exploited as a frequency filter and amplifier of weak environmental signals. ACKNOWLEDGMENTS We thank J. Hasty for the gift of the strain JS006, the plasmids pjs167 and pjs169, and his support with the microfluidics manipulation, the chip design, and image processing technology. We also thank O. Mondragon-Palomino and I. Razinkov for their help with the microfluidics technology, C. Giuraniuc for useful conversations, and W. Rostain for a proofreading of the manuscript. This work was supported by the FP7-ICT (BACTOCOM), PRES Paris Sud (MICROSCILA), and the Fondation pour la Recherche Medicale Grants (to A.J.). G.R. acknowledges the ALTF-

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