Math j Bio Computational Re-design of Synthetic Genetic Oscillators for Independent Amplitude and Frequency Modulation Graphical Abstract Authors Marios Tomazou, Mauricio Barahona, Karen M. Polizzi, Guy-Bart Stan Correspondence g.stan@imperial.ac.uk In Brief Tomazou et al. re-design synthetic genetic oscillators to enable independent tuning of their core characteristics (amplitude and period) over extended ranges. Highlights d Design of genetic oscillators with independent control of amplitude and period d d d Orthogonal degradation mechanisms allow decoupling of amplitude and period Dual-input gene control elements allow for extended amplitude and period ranges Applications such as multi-input biosensors and periodic signal generators Tomazou et al., 2018, Cell Systems 6, 508 520 April 25, 2018 ª 2018 The Author(s). Published by Elsevier Inc. https://doi.org/10.1016/j.cels.2018.03.013
Cell Systems Math j Bio Computational Re-design of Synthetic Genetic Oscillators for Independent Amplitude and Frequency Modulation Marios Tomazou, 1,4 Mauricio Barahona, 2,4 Karen M. Polizzi, 3,4 and Guy-Bart Stan 1,4,5, * 1 Department of Bioengineering, Imperial College London, London SW7 2AZ, UK 2 Department of Mathematics, Imperial College London, London SW7 2AZ, UK 3 Department of Life Sciences, Imperial College London, London SW7 2AZ, UK 4 Imperial College Centre for Synthetic Biology, Imperial College London, London SW7 2AZ, UK 5 Lead Contact *Correspondence: g.stan@imperial.ac.uk https://doi.org/10.1016/j.cels.2018.03.013 SUMMARY To perform well in biotechnology applications, synthetic genetic oscillators must be engineered to allow independent modulation of amplitude and period. This need is currently unmet. Here, we demonstrate computationally how two classic genetic oscillators, the dual-feedback oscillator and the repressilator, can be re-designed to provide independent control of amplitude and period and improve tunability that is, a broad dynamic range of periods and amplitudes accessible through the input dials. Our approach decouples frequency and amplitude modulation by incorporating an orthogonal sink module where the key molecular species are channeled for enzymatic degradation. This sink module maintains fast oscillation cycles while alleviating the translational coupling between the oscillator s transcription factors and output. We characterize the behavior of our re-designed oscillators over a broad range of physiologically reasonable parameters, explain why this facilitates broader function and control, and provide general design principles for building synthetic genetic oscillators that are more precisely controllable. INTRODUCTION Accurate temporal control of biological processes is of fundamental importance across all kingdoms of life and is often realized through genetic clocks, i.e., transcriptional networks with periodic gene expression. Heart beats, cell cycles, circadian rhythms, developmental processes, and energy metabolism are all, in one way or another, driven by robust genetic clocks (Hess, 2000; Bozek et al., 2009; Uriu, 2016). Such oscillators can also form the building blocks to engineer complex genetic networks that can be used to synchronize cellular activity or to optimize the efficiency of metabolic pathways. Understanding the design principles of genetic oscillators and finding ways to engineer and precisely control them is therefore of crucial importance, as conveyed by the numerous synthetic oscillators proposed to date (Goodwin, 1963; Elowitz and Leibler, 2000; Atkinson et al., 2003; Fung et al., 2005; Stricker et al., 2008; Danino et al., 2010; Tigges et al., 2010; Toettcher et al., 2010; Purcell et al., 2011; Chuang and Lin, 2014). Genetic transcriptional oscillators are typically based on the same principle introduced by Goodwin (1963): a negative feedback loop with a time delay. Perhaps the best-studied example of such a system is the repressilator (RLT) (Elowitz and Leibler, 2000), a ring of transcriptional repressors acting in sequence, which naturally introduces an intrinsic lag or delay. More recently, it was demonstrated that the addition of a positive feedback loop (as is commonly encountered in nature) increases the robustness of the oscillations and adds some tunability to their amplitude and period (Hasty et al., 2002; Tsai et al., 2008; Purcell et al., 2010). A prime example of this approach is the dualfeedback oscillator (DFO) (Stricker et al., 2008), which comprises a relatively slow negative feedback loop and a faster positive feedback loop. The DFO has been shown to function robustly when implemented using different components (Danino et al., 2010), and also as part of large networks in different organisms and conditions (Danino et al., 2012; Prindle et al., 2012), as well as in populations of cells (Danino et al., 2010; Prindle et al., 2011, 2014). However, an important feature yet to be implemented in either the DFO or other genetic oscillators is the ability to control the period and amplitude of oscillations independently. Such control would allow a wide range of applications, including frequency analysis of downstream networks (Davidson et al., 2013; Olson et al., 2014), frequency encoding and pulse-based signal processing (Lu et al., 2009), development of two-dimensional biosensors where period and amplitude give distinct responses (proposed below), designs for burden relief and metabolic pathway optimization (Sowa et al., 2014), or even periodic administration of therapeutic molecules (Tigges et al., 2010; Ritika et al., 2012). Frequency modulation with constant amplitude, or vice versa, is common in natural oscillatory processes such as circadian rhythms, cell cycles, and heart beats (Tsai et al., 2008). Typically, independent frequency and period modulation is realized through complex networks of interlinked positive and negative feedback loops, and 508 Cell Systems 6, 508 520, April 25, 2018 ª 2018 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).
Dual Feedback Oscillator (DFO) Repressilator (RLT) Repressor (R) Output Oscillator Gene of Interest (G) Oscillator Repressor 1 (R1) I R3 I R1 Repressor 3 (R3) I R2 Output Original networks Gene of Interest (G) I a Activator (A) Repressor 2 (R2) Sink Protease (C) Sink Protease (C) Output Gain Dial 1 (I 1 ) Activator (U) Output Gain Re-Design I. Output and protease dials A DFO B DFO Rd I C DFO Rd II Positive Regulation I r Oscillator Repressor 3 (R3) Dial 1 (I 1 ) Activator (U) Repressor (R) Oscillator Gene of Interest (G) Repressor 1 (R1) Gene of Interest (G) Activator (A) Output Repressor 2 (R2) Output Protease (C) D RLT E RLT Rd I F I R3 I r Negative Regulation I r I a I a Enzymatic Degradation h(u) h(u) Protease Gain Protease Gain Dial 2 (I 2 ) Activator (Y) j(y) Sink Dial 2 (I 2 ) Sink Activator (Y) j(y) Protease (C) Re-Design II. Output with orthogonal degradation Output Gain Dial 1 (I 1 ) Activator (U) RLT Rd II Oscillator Dial 2 (I 2 ) Negative Feedback Gain Output Gain Repressor (R) h(u) Repressor 3 (R3) Oscillator Gene of Interest (G) Repressor 1 (R1) Activator (Y) j(y) Dial 1 (I 1 ) Activator (U) Gene of Interest (G) h(u) Activator (A) Repressor 2 (R2) Protease Gain Output Output Dial 2 (I 2 ) Activator (Y) j(y) Protease (C) Dual Sink Protease (L) Dual Sink Protease (C) Protease (L) Figure 1. Diagrams of the Original and Re-designed Oscillators Considered architectures of the dual-feedback oscillator (DFO; A C) and repressilator (RLT; D F). For each architecture, positive (green arrow lines) and negative (black solid lines) regulations are indicated. Enzymatic degradation reactions are indicated by black dashed arrows. Each network consists of a core oscillator module, a degradation (also known as sink) module, and an output module. The inputs considered for the original DFO and RLT designs are inducers titrating the effect of the transcription factors (TFs) in the core oscillator module. For all the re-designed networks (labeled Rd), an input I 1 (Dial 1) controls the affinity of an orthogonal TF U, which in turn controls the expression rate of the gene of interest (G) in the output module. For the re-designs DFO Rd I and RLT Rd I, a second external input I 2 (Dial 2) is used to modulate the levels of the protease C via an activator Y. For re-designs DFO Rd II and RLT Rd II, a second orthogonal protease L is used to decouple the degradation of G from the degradation of the TFs in the oscillator module. In RLT Rd II, instead of modulating the amount of C like in DFO Rd II, we use the second input I 2 to modulate the expression rate of the repressor R2. is particularly challenging for networks like the RLT that have only negative feedback (Tsai et al., 2008). Achieving independent amplitude and period modulation by design in synthetic gene regulatory networks is a challenging problem that has, so far, received little attention. In this work, we use mathematical modeling and numerical simulation to propose simple, implementable re-designs of the DFO (Stricker et al., 2008) and RTL (Elowitz and Leibler, 2000) that enable independent control of amplitude and period over increased dynamic ranges. Through simulations of a parsimonious model, we demonstrate how to introduce orthogonal inputs or dials (Arpino et al., 2013) based on dual-input promoters (Lutz and Bujard, 1997; Tomazou et al., 2014). The introduction of these new dials does not require modifications of the core topologies of the DFO and RLT, yet it allows oscillations to be tuned in (a) amplitude only, (b) period and amplitude simultaneously, or (c) period with the amplitude maintained at nearconstant levels. In addition, some of our re-designs allow the system to transition from a stable OFF state, to a tunable oscillatory regime, and, finally, to a stable ON state as a function of the input dials. Hence, the same architecture can be used to generate controllable oscillations and/or switch between high and low steady states (Tyson et al., 2003). While our models are parameterized based on a typical Escherichia coli host, we anticipate the proposed structures to be applicable to other organisms if the required genetic parts are available for those organisms. RESULTS Our main objective is to re-design synthetic genetic oscillators to achieve independent amplitude and frequency modulation over increased ranges. We consider the two most popular oscillators in synthetic biology (DFO and RTL) and study key modifications of their network architecture that extend the range of achievable amplitudes and periods, while at the same time decoupling these characteristics. To keep our results general, we study parsimonious models that adequately reproduce their behaviors. The models comprise three modules (Figure 1): (1) a core oscillator module composed of a negative feedback loop with delay (with an additional positive feedback loop for the DFO); (2) an output module with the gene of interest as a readout; (3) a sink module where the molecular species of both the oscillator and output modules are channeled for enzymatic degradation. Tuning the amplitude without affecting the period has proved difficult for both the DFO and RLT. Although a previous study (Tsai et al., 2008) showed that different amplitudes with identical periods (or vice versa) are possible when a number of parameters are modulated, the parameters (e.g., cooperativity coefficients, transcription factor [TF]-DNA binding affinities) are typically not easily accessible or dynamically tunable. Instead, and akin to the design of electronic oscillators, our proposed redesigns introduce an amplification stage to the genetic oscillator output as a simple means of independently tuning the amplitude. Furthermore, our analysis identified the post-translational Cell Systems 6, 508 520, April 25, 2018 509
coupling introduced by enzymatic degradation (also known as the protease queuing effect [Cookson et al., 2014]) as a major obstacle for independent amplitude modulation: this coupling affects the degradation rate of the TFs and consequently the period of the oscillations (Cookson et al., 2014; Moriya et al., 2014). To mitigate this effect, we introduce an orthogonal enzymatic degradation pathway (Cameron and Collins, 2014), which enables modulation of the amplitude with reduced impact on the period. To achieve independent period modulation, we followed two approaches. Firstly, we considered a re-design that modulates the expression levels of the protease, which resulted in a significant increase in the achievable range of periods. Secondly, we considered re-designs that allow the tuning of the negative feedback gain by modulation of the expression rate of one of the repressors in the RTL. With such re-designs, our simulations showed that the period can be tuned within a greater range compared with chemical induction alone, while the amplitude is kept nearly constant. Herein we give a description of the original and modified networks considered in this work. Our coarse-grained phenomenological models employ Hill-type functions (Gjuvsland et al., 2007) to capture the regulatory action of single- and dual-input promoters as well as of the inputs of DFO and RLT. Although these models are too simplistic to reproduce the full range of dynamics exhibited by detailed mechanistic models corresponding to characterized biological parts (Stricker et al., 2008), they capture the core responses of the oscillators and produce stable oscillations with periods within the range observed in experimentally implemented systems (Elowitz and Leibler, 2000; Stricker et al., 2008). The Dual-Feedback Oscillator and Its Re-designs The Original Dual-Feedback Oscillator The DFO basic structure (Figure 1A) comprises a positive and a negative feedback loop mediated by a transcriptional activator (A) and repressor (R), respectively. The output is an oscillating protein of interest denoted by G. All three proteins (A, R, and G) are tagged for fast enzymatic degradation via the same protease (C). A difference between our model and the original DFO model (Stricker et al., 2008) is that we take into account the output protein and its queuing effect (Cookson et al., 2014) on the degradation rates. The rate of change of the mrna concentrations of the three genes m i associated with the three proteins (A, R and G) is given by 0 m_ X = p X B @ b 0 + b 1 Degradation zfflfflfflfflfflffl} fflfflfflfflfflffl{ m $ _ {z} ðd + d mþ m X Activating HIll Function zfflfflfflfflfflffl} fflfflfflfflfflffl{ n fðaþ k n A + fðaþn ; X fa; R; Gg; $ Repressing HIll Function zfflfflfflfflfflffl} fflfflfflfflfflffl{ kr n kr v + gðrþn 1 C A (Equation 1.1) where p X is the gene copy number, and b 0 and b 1 are the basal and maximum transcription rate constants, respectively. Here, k A and k R are Hill constants describing the amounts of TFs required for half-maximal activation and repression rates, respectively, whereas n and v are Hill cooperativities reflecting whether the TFs are active as monomers or multimers. As per the original published model for the DFO (Stricker et al., 2008), we used higher cooperativity degree for the repressor (v = 4) compared with the activator (n = 2). The removal rate per mrna is given by m, which is equal to the sum of the dilution rate due to cell division (d) and the mrna decay rate (d m ). The inducer-dependent functions f(a) and g(r) are defined as I2 a fðaþ = A ka 2 + ; gðrþ = R k2 r I2 a kr 2 + ; (Equation 1.2) I2 r where I a and I r are inducer concentrations that increase or decrease the effect of the activator (A) or repressor (R) with half-maximal constants k a and k r, respectively. In the original DFO, the external inputs that are used to tune the oscillations are the chemical inducers, I a and I r, which modulate the affinity of the TFs to their cognate promoter binding sites and, consequently, their transcriptional regulation effectiveness. The Hill cooperativity coefficients in Equation 1.2 are set to 2 to represent a typical dimeric-inducer-tf complex. Additional simulations (not shown) indicate that similar behaviors are obtained for other Hill cooperativities (e.g., Hill coefficients of 1 and 4 in Equation 1.2). In what follows, we only present simulations for Hill cooperativities of 2. For the dynamics of the proteins we have also considered a folding step with respect to previously published models (Stricker et al., 2008). Preliminary results (not shown) of such phenomenological models of the DFO and RLT without the folding step generated very narrow oscillatory regimes that deviated from experimentally observed periods. This is highly likely because the time delay inherent in both oscillators is not adequately captured without the considering the protein folding dynamics. The dynamics of the proteins (folded and unfolded) are described, therefore, by Translation zffl} ffl{ X_ u = t X m X f X X u Folding zffl} ffl{ 0 1 Enzymatic Degradation zfflfflfflfflffl} fflfflfflfflffl{ C B @ d + n m $C k c + P P C A X u; X fa; R; Gg (Equation 1.3) X _ = f X X u d + n m$c k c + P X; X fa; R; Gg; (Equation 1.4) P where X u and X are the concentrations of unfolded and folded proteins, respectively. Here t X and f X are the translation and folding rate constants, respectively, and d is the dilution rate due to cell division. The enzymatic degradation term obeys Michaelis-Menten (MM) kinetics (Cookson et al., 2014) with v m representing the catalysis rate, k c its MM constant, and C the total concentration of protease in the cell. P P represents the total concentration of degradation tags competing for the same protease sites (Cookson et al., 2014), denoted hereafter as the load. Assuming TFs with identical degradation tags can bind to the protease binding site with equal affinities, we have 510 Cell Systems 6, 508 520, April 25, 2018
X P = Au + R u + G u + A + R + G: (Equation 1.5) The above model was parameterized to reproduce the oscillatory behavior observed in experiments (Elowitz and Leibler, 2000; Stricker et al., 2008). The full set of equations, kinetic scheme, assumptions, and parameter values are given in STAR Methods and Tables S1 S8. DFO Re-designs The DFO features an output (G) whose genetic expression is controlled by both a repressor (R) and an activator (A), so that G is periodically expressed when A and R levels oscillate. We explore computationally a number of re-designs of the basic DFO architecture (Figure 1A) governed by the ordinary differential equation model (Equations 1.1 1.5). The re-designs introduce additional biological dials inserted at different locations in the DFO gene regulation network (Figures 1B and 1C). Re-design 1: DFO Rd I. The first re-design DFO Rd I (Figure 1B) introduces an activation input (U) that controls the expression of G independently from the TF, A. This can be achieved by a dualinput promoter (Lutz and Bujard, 1997; Tomazou et al., 2014) that responds to two different TFs (U and R). The total amount of U is assumed to be unregulated (constitutive gene expression) and constant at steady state. The activation effect of U on the expression of G is modulated by an external input signal I 1.In this re-design, the expression rate of the protease (C) is also tunable via an orthogonal TF, denoted Y. The effect of Y on C is modulated by an external input signal I 2. The corresponding model is thus updated as follows: 0 1 Auxiliary Activation zfflfflfflfflfflfflfflfflffl} fflfflfflfflfflfflfflfflffl{ 0 Output Gain 0 2 m_ G = p G B @ b hðuþ k n R 0 + b g ku 2 + $ hðuþ2 kr n + C gðrþn A m$m G; where the protease concentration C depends on Y, itself modulated by the second input, I 2 : lðcþ = C jðyþ2 ky 2 + jðyþ2; jðyþ = Y I2 2 k2 2 + : (Equation 1.9) I2 2 Re-design 2: DFO Rd II. The second re-design DFO Rd II (Figure 1C) differs from DFO Rd I in the degradation mechanism. In DFO Rd II, an additional orthogonal protease (L) is specifically dedicated to the degradation of the output (G). This second enzymatic degradation pathway eliminates the post-translational coupling between G and the TFs of the core oscillator module, i.e., TFs A and R are both targeted for degradation by C, whereas G is specifically targeted for degradation by the orthogonal protease L (Cameron and Collins, 2014). This gives rise to the following model: x_ u = t x m x f x x u d + n m$c k c + X x u ; x fa; Rg P x _ = f x x u d + n m$c k c + X x; x fa; Rg P G_ n ml L u = t G m G f G G u d + G u k L + G + G u G _ n ml L = f G G u d + G; k L + G + G u (Equation 1.10) where v ml and k L are the MM parameters for the secondary enzymatic degradation pathway. The load for the primary degradation pathway is now independent of the output (G): X P = A + Au + R + R u : (Equation 1.11) (Equation 1.6) where the affinity of the activating TF U is modulated by the input (inducer) I 1, which is orthogonal to I a and I r. The function h(u) in (1.6) captures the influence of I 1 on the steady-state concentration of U. This is modeled using a Hill function with cooperativity 2 and half-activation constant k 1 : hðuþ = U I2 1 k1 2 + : (Equation 1.7) I2 1 The dynamics of G allow the output to oscillate following oscillations in R and the rate of expression of G, when R is not fully repressing, is titrated by I 1.AsinEquation 1.2, using cooperativity exponents other than 2 gives similar results, which are omitted here for the sake of brevity. In addition, the steady-state level of C is a function of a second input I 2, as follows: X_ u = t X m X f X X u d + n m$lðcþ X u ; X fa; R; Gg; X _ = f i X u d + n m$lðcþ k c + X P k c + X P X; X fa; R; Gg (Equation 1.8) Computational Characterization of the DFO Re-designs In its original experimental implementation, the DFO (Figure 1A) employed two of the most commonly used TFs, LacI and AraC, which can be controlled by the level of the chemical inducers, IPTG (isopropyl b-d-1-thiogalactopyranoside) and arabinose, respectively. These inducers are modeled in Equation 1.2 by the parameters I a and I r in the functions f(a) and g(r), respectively. Figures 2A and 2D show the change in the amplitude (h) and period (t) of the oscillations of the original DFO as a result of varying I a and I r from low to saturating levels. Our computations show that h and t are strongly correlated: the inputs I a and I r affect both amplitude and period, hence these oscillation characteristics cannot be independently tuned. To quantify the dependence between amplitude and period, we used the mutual information (MI) between two variables x and y, which is defined as (Kraskov et al., 2004; Kinney and Atwal, 2014) MIðx; yþ = X X rðx; yþ rðx; yþlog ; (Equation 1.12) rðxþrðyþ x y where r(x,y), r(x), and r(y) are the sample distributions of the variables obtained from the observations. When two variables x and Cell Systems 6, 508 520, April 25, 2018 511
A B C D E F G H Figure 2. Simulation Results for the DFO and Its Re-designs (A C) Time-course simulations for the DFO and its re-designs showing the oscillatory output G over time under four different induction conditions (+ = 20 mm, +++ = 90 mm). In DFO (A) and DFO Rd I (B), when an input results in an increase of the amplitude, the period is increased. (C) In DFO Rd II, the amplitude can be increased with no effect on the period (t = 40 min for black and green and t = 25 min for red and orange trajectories). I 2 primarily affects the period but with some impact on the amplitude. (D F) Amplitude versus period dot plots for DFO, DFO Rd I, and DFO Rd II, respectively. The color of the dot indicates the amount of input with the green component indicating I 1 and the red component indicating I 2. Yellow indicates that both inputs are high while black corresponds to their absence. The range of amplitude is denoted as Rh, while the range of period is denoted as Rt. Both ranges are visualized by dashed boxes where Rh and Rt are the height and width of the boxes, respectively. (D) The original DFO exhibited a narrower range for both amplitude and period, and changes in either affect both characteristics in a highly coupled manner. (E) The DFO Rd I exhibits a wider tunable range with I 1 primarily increasing the amplitude and I 2 primarily modulating the period. However, none of these characteristics were independently tunable. (F) In the DFO Rd II, the amplitude can be tuned in response to I 1 independently of the period. The opposite was not feasible in this case. The effect of each dial in isolation is further demonstrated by the arrows. The green arrow in (E) and (F) shows the shift in amplitude and period as I 1 increases with a fixed amount of I 2. The red arrow shows the shift in amplitude and period as I 2 increases while I 1 is fixed. (G) The input function when either I 1 or I 2 are varied. Gray areas indicate the range of values of I 2 when the system converges to a stable equilibrium point instead of oscillating. (H) Bar chart showing mutual information (MI) between calculated amplitude and period, MI(h,t), where high values correspond to highly coupled period and amplitude values. The heatmaps demonstrate the orthogonality of inputs using the MI value calculated between h and t against the inputs. The optimal case is a diagonal map where MI(I 1,h) and MI(I 2,t) are high while MI(I 2,h) and MI(I 1,t) are zero. y are independent, it implies that r(x,y) = r(x)r(y) and therefore MI(x,y) = 0. By definition, the input dials I a and I r are independent orthogonal inputs, and therefore MI(I a, I r ) = 0. Under independent variation of the inputs I a and I r, we evaluated the MI between the amplitude h(i a,i r ) and the period t(i a,i r ). In an ideal situation, we aim for: (1) maximum orthogonality: MI(h,t) = 0 (i.e., amplitude and period vary independently under independent variation of the inputs I a and I r ); and (2) maximum specificity: e.g., MI(I r,h) =MI(I a,t) = 0, while MI(I a,h)andmi(i r,t) are maximized (i.e., each external dial I a or I r induces a change in only one of the characteristics, either amplitude or period). However, as seen in Figure 2H, the original DFO exhibits low orthogonality (shown by the large value of MI(h,t) = 5.5) and low specificity (as shown by the fact that MI(I a,h) z MI(I a,t) z 0andMI(I r,h) z MI(I r,t) >> 0). Our two re-designs introduce two additional inputs, I 1 and I 2, which affect the functions h(u) and l(c) given by Equations 1.7 and 1.9, respectively. Our aim is to improve the orthogonality (MI(h,t) = 0) and specificity of the DFO under the variation of these inputs (MI(I 2,h) =MI(I 1,t) = 0 and MI(I 1,h) >> 0, MI(I 2,t) >> 0), as well as to enlarge the dynamic range of the oscillatory responses in both amplitude and period. In addition to the MI metrics, we have calculated the local sensitivities of amplitude to period (S ht ) and period to amplitude (S th ) for our various re-designs as defined in STAR Methods and shown in Figures S1A and S1B. 512 Cell Systems 6, 508 520, April 25, 2018
DFO Rd I Exhibits an Increased Range of Achievable Amplitudes and Periods Our simulations show that the range of achievable period and amplitude is increased for DFO Rd I in comparison with the original DFO (Figures 2B and 2E). However, although the amplitude can be tuned over a wider range, the increase in amplitude is accompanied by a nonlinear shift in the period. This is explained by the post-translational coupling between the output (G) and the TFs (R and A) due to the protease queuing effect, which results from the sharing of the protease (C) among them. Hence, as the total amount of proteins tagged for enzymatic degradation increases their degradation rates become smaller, which in turn increases the period. To investigate whether this effect can be attenuated by increasing the abundance of the protease, we consider the effect of the second external input I 2 that is used to control the expression rate of the protease C (via the additional TF, Y). Figure 2E shows that even at the highest levels of C (red points) the period is still affected by the increase in amplitude when I 1 is increased from 0 (red dots) to saturating amounts (yellow dots). This is likely because the protease degradation activity cycles between the underloaded and overloaded queuing regime as the system transitions from low protein levels to peak levels during an oscillation. Modulating the amount of the protease increases the range of achievable periods by about 70% compared with the original DFO. MI(h,t) for the DFO Rd I is significantly lower than the original DFO tuned by I a and I r, but greater than DFO Rd II. At moderate to high amounts of protease, the local sensitivity (STAR Methods and Figure S1A) of the amplitude is relatively small but increases as the protease concentration drops to low levels. DFO Rd II Allows Tuning of Amplitude with No Effect on Period As stated above, our objective is to find re-designs for which the mutual information metrics MI(I 1,t) z MI(I 2,h) z 0, whereas MI(I 1,h) andmi(i 2,t) are maximized. In DFO Rd II, we aimed at eliminating the observed post-translational coupling by introducing an additional, orthogonal protease dedicated to the degradation of the output G. Indeed, the results in Figures 2C and 2F show that the period is no longer affected when I 1 is varied (MI(I 1,t) = 0) over the full range of accessible amplitudes. The amplitude is still sensitive to I 2 (and, therefore, MI(I 2,h) > 0); however, when compared with DFO Rd I, MI(I 2,h) is approximately 50% lower for DFO Rd II. A disadvantage of DFO Rd II is that the range of achievable periods and amplitudes is comparable with, but not larger than, that of the original DFO, although, for any given period, an extended range of amplitudes is accessible by varying I 1. While Figure 2F shows the DFO Rd II behavior in the oscillatory regime, our simulation results also predict that when the input function f(i 2 ) becomes too high, i.e., for I 2 >100mM, or too low, i.e. for I 2 <15mM, the system stops oscillating and reaches a stable equilibrium (Figure 1G). For high values of I 2, the abundance of protease C is high enough to degrade the TFs faster than they can accumulate, thereby disrupting the oscillatory mechanism. When this happens, R becomes low and therefore output G reaches a high steady-state level. On the other hand, at values of I 2 that are too low, protease C abundance is too low to effectively degrade the repressor and ultimately the system reaches a stable equilibrium where the output G is repressed (Figure 1G). Repressilator Design and Re-design Original RLT The original repressilator model comprises a ring of three TFs, each repressing the following gene in the ring (Figure 1D). Compared with the original RLT (Elowitz and Leibler, 2000), we model the enzymatic degradation mechanism catalyzed by the protease C, and we include the output gene module G explicitly. In addition, the repression is modulated by external inputs I Ri, with i {1,2,3}. As above, we derived a model for the RLT, as follows. The mrna rate equations are modeled as: where! m_ Ri = p i b i + a i ki 1 2 ki 1 2 + fðr i 1Þ 2 m$m Ri ; i f1; 2; 3g! k m_ 3 2 G = p G b g + a g k3 2 + fðr m$m 3Þ 2 G ; (Equation 1.13) k 0 bk 3 ; R 0 br 3 and the induction function for each inducer is given as I 2 Ri fðr i Þ = R i kri 2 + ; i f1; 2; 3g: (Equation 1.14) I2 Ri The unfolded and folded protein dynamics are given by: X_ n m C u = t X m X k fx X u d + k c + X X u ; X fg; R1; R2; R3g P X _ n m C = k fx X u d + k c + X X; X fg; R1; R2; R3g P (Equation 1.15) X P = Gu + G + X3 i = 1 ðri u + RiÞ; i f1; 2; 3g: (Equation 1.16) Repressilator Re-designs The re-designs of RLT follow similar trends as those presented for DFO, yet the additional flexibility supported by the RLT architecture leads to more independent amplitude and period tuning. This flexibility is due to the fact that the negative feedback loop is mediated by three repressors, hence offering a larger number of biological dials (Arpino et al., 2013) that can be tuned. Re-design 1: RLT Rd I. Two orthogonal dials (TFs U and Y) are introduced in RLT Rd I (Figure 1E): U modulates the expression of G, as a function of the external input I 1, whereas Y modulates the expression of the degradation protease C, as a function of the external input I 2. Cell Systems 6, 508 520, April 25, 2018 513
Following the same structural modifications discussed for DFO Rd I, the output mrna rate equation is given by! hðuþ 2 k m_ 3 2 G = p g b g + a u ku 2 + hðuþ2$ k3 2 + fðr m$m 3Þ 2 G (Equation 1.17) with h(u) given in Equation 1.7, and the second input I 2 regulating the amount of protease in the system: X_ u = t X m X k fx X u d + X _ = k fx X u X P = Gu + G + X3 where l(c) is n mlðcþ k c + X P X u ; X fg; R1; R2; R3g d + n mlðcþ k c + X P X; X fg; R1; R2; R3g (Equation 1.18) i = 1 ðri u + RiÞ; i f1; 2; 3g; (Equation 1.19) lðcþ = C jðyþ2 ky 2 + jðyþ2; jðyþ = Y I2 2 k2 2 + : (Equation 1.20) I2 2 Re-design 2: RLT Rd II. In the second re-design (Figure 1F), R1 and R3 obey the same regulation scheme as in RLT Rd I, while R2 is now regulated by a dual-input promoter that is repressed by R1 and activated by Y:! m_ Ri = p i b i + a i ki 1 2 ki 1 2 + fðr i 1Þ 2 m$m Ri ; i f1; 3g! m_ R2 = p y b 2 + a y jðyþ 2 k1 2 ky 2 + jðyþ2 k1 2 + fðr1þ2 m$m R2 k 0 bk 3 ; R 0 br 3 jðyþ = Y I2 2 k2 2 + : (Equation 1.21) I2 2 Furthermore, a secondary protease L is specifically dedicated to the enzymatic degradation of the output gene of interest (G), which leads to X_ n m C u = t X m X k fx X u d + X u ; X fr1; R2; R3g X _ = k fx X u k c + X P n m C d + k c + X X; X fr1; R2; R3g P X P = X 3 i = 1 ðri u + RiÞ; i f1; 2; 3g! m_ G = p G b 0 + b 1 k3 2 k3 2 + fðr 3Þ 2 m$m G G_ u = t G m G k fg G u d + G _ n ml L = k fg G u d + k L + G + G u n ml L k L + G + G u G: G u (Equation 1.22) Computational Characterization of the RLT Re-designs As in the DFO case, we have simulated and characterized the results in terms of the MI metrics defined above. For the original RLT we have varied the chemical inducers for the three repressors (I R1, I R2, and I R3 ). Numerical simulations of the original RLT (Figures 3A and 3D) demonstrate the effect of the inputs I R2 and I R3 corresponding to the chemical inducers used to modulate the affinity of repressors. Other combinations of titratable inputs (I R1 -I R2 and I R1 -I R3 ) produced similar results (see Figure S3A). The period is predicted to be tunable over a range of approximately 200 min, whereas the amplitude range is relatively narrow. As in the original DFO, the amplitude and period in the RLT are highly correlated, with a high value of MI(h,t)(Figure 3H). Hence independent tuning of the amplitude and the period of the RLT using chemical inducers I R1, I R2, and I R3 is not possible. RLT Rd I Exhibits a Wider Amplitude Range but a Strongly Nonlinear Amplitude and Period Dependency Numerical simulations of the RLT Rd I (Figures 3B and 3E) indicate that the range of achievable amplitudes is increased by approximately 3.5-fold compared with the original RLT, but with large period shifts. Specifically, MI(h,t) values are significantly larger to those of the DFO Rd I (Figure 2E). The shape of the tunable region also indicates strong and highly nonlinear coupling between period and amplitude. Similarly to DFO Rd 1, the protease queuing effect cannot be eliminated through the use of increased amounts of the protease. The values of the MI and local cross-sensitivities S th and S th (STAR Methods; Figures S1A and S1B) metrics we have defined previously indicate that this re-design is significantly far from the orthogonality and specificity objectives defined in Equation 2.2. RLT Rd II Exhibits a Wide Range of Periods and Amplitudes, Tunable in a Nearly Independent Manner Simulations of RLT Rd II show that the amplitude is tunable independently of the period (Figures 3C and 3F). Here we exploit the architecture of the RLT to tune the period by introducing a dual-input promoter at the second position of the ring because it neither directly regulates nor is influenced by the output (G). Such a re-design exhibits the lowest MI(h,t) (Figure S3B). Furthermore, simulation results indicate a wider range of tunable periods than for RLT or RLT Rd I. Tuning the period by varying I 2 results in the lowest observed MI(I 2,h) across all oscillators tested. This is likely because the duration of the cycle of each repressor affects the overall period, but not the amplitude of oscillations of the other repressors. This effect becomes even more apparent when more genes are added to the ring, as seen in simulation results of the generalized repressilator (Strelkowa and Barahona, 2010a, 2010b, 2012) with n = 5 and n = 7 repressors (STAR Methods and Figure S3). 514 Cell Systems 6, 508 520, April 25, 2018
A B C D E F G H Figure 3. Simulation Results for the RLT and Its Re-designs (A C) Time-course simulations for the RLT and its re-designs showing the output G over time under four different input conditions (+ = 20 mm, +++ = 90 mm). (D F) As with DFO and DFO Rd I, numerical simulation results show that the period of RLT in (D) and RLT Rd I in (E) is significantly sensitive to input I 1 and, therefore, that the amplitude cannot be modulated independently of the period. On the other hand, numerical simulations presented in (F) indicate that the redesign corresponding to RLT Rd II allows for the tuning of the amplitude of the oscillations (green trajectory) over a wider range than either RLT or RLT Rd I, and with no effect on the period. Additionally, tuning the period by modulating I 2 in RLT Rd II has a minimal effect on the amplitude. The mutual information MI(h,t) was the lowest for RLT Rd II across all designs. In each amplitude versus period plot the range for amplitude is denoted as Rh and the range for period as Rt. Both ranges are visualized by the dashed box where Rh and Rt are the height and width of the box, respectively. Green color indicates the amount of I 1 and red color indicates the amount of I 2 according to the input key insets. The green arrows show the isolated effect of increasing input I R2 with a fixed amount of I R3 in (D) and of increasing I 1 for a fixed amount of I 2 in the re-designs (E and F). The red arrows similarly indicate the effect of the second input when the first is fixed. (G) Input function applied for the RLT Rd I and RLT Rd II re-designs. Unlike the re-design DFO Rd II, in the RLT Rd II only a small range of low I 2 values resulted in non-oscillating behavior. (H) Bar chart comparing all the MI metrics used to assess the performance of the RLT oscillators. Here the RLT Rd II is the closest to the optimality since MI(t,h), MI(I 2,h), and MI(I 1,t) are close to zero while MI(I 1,h) and MI(I 2,t) are high. This re-design maintained oscillations throughout the input range of I 2 except for I 2 <5mM where the expression rate of R2 is so low that the systems reaches a stable equilibrium point. Overall, RLT Rd II satisfies our objective of high orthogonality and specificity since MI(t,h), MI(I 2,h), MI(I 1,t) approach zero while MI(I 1,h) and MI(I 2,t) are high (Figure 3H). The range of achievable amplitude and period is also greater than that of all other oscillators discussed in this work. Application of the Re-designed RLT Rd II: An Orthogonally Tunable Oscillator as a Multiple-Input Single-Output Biosensor Orthogonally tunable oscillators can be used to regulate precisely the magnitude and temporal characteristics of downstream networks or metabolic pathways (Hess, 2000; Fung et al., 2005; Bozek et al., 2009). It could also be used as a periodic signal generator employed to better understand the dynamics of cellular and biological systems (Lux et al., 2012; Davidson et al., 2013; Olson et al., 2014; Nielsen et al., 2016) or even pulsatile (e.g., circadian) release and administration of drugs and therapeutic molecules (Bakken and Heruth, 1991; Arora et al., 2006; Ritika et al., 2012). An alternative application is to reverse their use: instead of applying specific inputs to generate periodic oscillations of desired amplitude and period, the user can read the periodic signal and accurately determine the concentrations of the external inputs, hence using the oscillator as a dual-input single-output biosensor (Goers et al., 2013). According to our computational analyses, the RLT Rd II redesign, which is the most orthogonal (i.e., the oscillator with the lowest MI(h,t)), would be the most appropriate to build such a biosensor. The inputs can be mediated by any transcriptional regulatory network, including TFs, two-component systems, or even CRISPR-derived circuits (Qi et al., 2013; Didovyk et al., 2016) that respond to ligands of interest (e.g., arsenic [Voigt, 2012] or aspartate [Utsumi et al., 1989]) or environmental Cell Systems 6, 508 520, April 25, 2018 515
A B C D Figure 4. Diagram of a Conceptual Two-Input, Single-Output Oscillator-Based Biosensor (A) A hypothetical biosensor where the concentration of two small molecules of interest I 1 and I 2 determine the characteristics of the oscillator in the processing unit that generates an oscillatory fluorescence signal, which is read by the output module. (B) The readout of the biosensor is analyzed over a period to determine the amplitude and period of the oscillation. (C) Heatmap and contour map showing the period as a function of both inputs. The isolines are vertical as I 1 has no effect on the period. Therefore, the concentration of I 2 can be determined by mapping the period to a standard curve of period versus I 2. (D) Contour plot of the amplitude versus both inputs. Once the I 2 concentration is determined from (B), the I 1 concentration is simply found at the intersection point of the isolines bounding the measured amplitude value (blue isoclines) and the isoline of I 2. The error for I 1 and I 2 is determined by the width of the blue and red isolines, respectively. signals (e.g., light [Levskaya et al., 2005; Tabor et al., 2011; Schmidl et al., 2014] or temperature [Galkin et al., 2009]). Figure 4 illustrates the concept with the RLT Rd II used as a biosensor. The network is exposed to ligand 1 (I 1 ), which modulates the amplitude of the oscillations, and to ligand 2 (I 2 ), which modulates their period. The output reporter (e.g., GFP) can be recorded with a basic setup comprising a light excitation source and a sensor that records the periodic fluorescent output over time. The analysis of the signal would map the period to the level of I 2 using a precomputed function (red line), while, for a given I 2, the amplitude versus the I 1 -precomputed function would return the concentration of I 1. This approach could have significant advantages over coexpression of multiple single-input-single-output biosensors. A multi-input-single-output biosensor requires a single excitation source and sensor, and a simpler electronic interface for implementing a low-cost biosensor. From a biological perspective, a single reporter imposes less metabolic burden on the host, and periodic expression of a single fluorescent reporter would reduce the toxic effects, e.g., due to production of free radicals upon excitation (Chalfie and Kain, 2006). Finally, the error due to stochastic gene expression effects and measurement method can be reduced in periodic signals by increasing the duration of the readings, in contrast to the single readings from traditional steadystate output biosensors. Naturally, any attempt to build such a device will have to overcome potential caveats and practical difficulties, which are discussed in detail in the Biology Box section. DISCUSSION Using computational modeling, we have investigated network re-designs of the DFO and RLT (the two oscillators most widely used in synthetic biology) with the goal of proposing re-designs with extended ranges and independent tuning of amplitude and period. In particular, the re-designs of the RLT outperform the DFO in achieving independent tuning of amplitude and period and extending their achievable ranges. Our DFO re-design showed fast oscillations (period below 1 hr) but within a limited range of achievable amplitudes compared with the RLT. This is because the activating and repressing TFs in the DFO oscillate in phase, while in the RLT each repressor is expressed in succession. This allows the modulation of the expression rate of one of the RLT TFs with less effect on the others. Positive feedback loops are known to increase the tunability range (Tsai et al., 2008). While superficially the RLT and its re-designs do not appear to include any positive feedback loops, deeper analysis of the protease queuing effect suggests the opposite. The abundance of every TF in the RLT decreases its degradation rate and therefore increases its overall accumulation rate, acting as a positive feedback loop. This might explain why the RLT showed a broader tunable range in our re-designs. In vivo implementations of the system would be affected by other couplings, such as competition for shared cellular resources (e.g., energy, amino acids, ribosomes, and polymerases [Ceroni et al., 2015; Weisse et al., 2015]). Taking these into account is beyond the scope of this study, but will likely result in narrower ranges over which amplitude and period can be tuned independently. To attenuate host context effects, the oscillator TFs should be expressed at minimal levels where burden and resource competition effects are not dominant. Recently, a TX- TL implementation of these oscillators has been successfully demonstrated in cell-free microfluidic platforms (Niederholtmeyer et al., 2015), where a continuous flow of TX-TL medium rich in nutrients and translational and transcriptional resources was used (Sun et al., 2013). Here the competition for shared resources is minimized, and we anticipate this setup to constitute the best platform for a successful implementation of our re-designs. Engineering Design Principles of Orthogonally Tunable Oscillators with Increased Ranges of Achievable Amplitude and Period Through our computational simulation and analysis, we identify the following core principles for the implementation of orthogonally tunable oscillators: (1) Orthogonality of the degradation of the output and the TFs enables decoupling of amplitude and period. In our proposed re-designs (DFO Rd II and RLT Rd II) 516 Cell Systems 6, 508 520, April 25, 2018
Box 1. Biology Box This study aimed to identify minimal structural modifications to the DFO and RLT networks that decouple amplitude and period tuning and increase their achievable ranges. Our goal was to introduce these modifications without disrupting the core structure of the original gene regulatory networks. Below we highlight the most important considerations for a possible experimental implementation of our proposed re-designs. HOST CONTEXT AND BURDEN Host-related parameters such as growth rates and mrna degradation rates were chosen within the physiological range for E. coli since this was the host context in which both the RLT and DFO were originally implemented. Nevertheless, since our re-designs are based on structural rather than parametric modifications, if the genetic regulatory parts and proteases are available these redesigns should be applicable to other bacterial hosts (e.g., Salmonella [Prindle et al., 2012]) or even yeast. While other synthetic oscillators were reported previously in mammalian cells (Tigges et al., 2009), these did not have the same architecture as the RLT or DFO and so would need to be analyzed in this context. In addition, mammalian cells have different gene expression, dilution, and degradation time scales, and their capacity for heterologous protein expression is vastly different from bacteria and yeast. Another consideration is the competition for shared cellular resources or gene expression burden. Our models do not account for shared cellular resource competition, which could be another potential coupling mechanism between period and amplitude. The expression of proteases at low numbers (100 300 molecules per cell volume) is not anticipated to have a significant impact. However, expression levels of the TFs and output proteins are typically 1 2 orders of magnitude higher. At high expression levels burden effects can potentially result in period drifts and limit the achievable amplitude range (Weisse et al., 2015). Inevitably an optimization process would be needed to mitigate burden-related effects in our proposed re-designs. We anticipate that a cellfree in vitro approach (Niederholtmeyer et al., 2015) should provide enough latitude for modulating the nutrient supply (sugar, amino acids, energy) and the translational machinery to minimize any significant burden-mediated coupling effects. At the genetic network level, a number of studies have highlighted burden alleviation strategies such as feedback control (Ceroni et al., 2015; Borkowski et al., 2016; Qian and Del Vecchio, 2016), or trading lower translational rates for higher transcription rates to reach the same output levels while imposing less heterologous gene expression burden on the host (Ceroni et al., 2015). ROBUSTNESS TO STOCHASTIC FLUCTUATIONS The original implementation of the RLT suffered from a lack of robustness as only a small percentage of cells exhibited oscillations. Furthermore, cells that did oscillate were not synchronized. However, recent studies have offered significant insights into structural features that extend the robustness and reliability of oscillatory networks and allow for synchronous periodic oscillations of cells over extended periods of time without any active quorum coupling mechanism (Niederholtmeyer et al., 2015; Potvin-Trottier et al., 2016). Potential implementation of our proposed re-designs should consider the points raised by these studies. According to Potvin-Trottier et al. (2016), plasmid copy number fluctuations can be reduced by combining the reporter gene and the core oscillator genes on the same backbone, preferably one with a stable medium-to-low copy origin of replication (e.g., p15a or psc101). While the same study shows that removing the SsrA degradation tags will improve the consistency of the oscillations, another work (Niederholtmeyer et al., 2015) demonstrated that synchronous and consistent periods and amplitudes can be achieved in the presence of enzymatic degradation when the proteins are strongly expressed. It is worth noting that RLT Rd II does not necessarily require enzymatic degradation to enable amplitude and period decoupling. However, if the oscillator s sink relies only on dilution, our model predicts a reduced period range, with higher amplitude to period sensitivity (S ht ), and reduced robustness against growth rate fluctuations. The decrease in tunability was well documented previously by a study that showed that interlinking positive and negative feedback loops in oscillatory networks is necessary for increasing the robustness and bandwidth of the oscillations under different parameter sets (Tsai et al., 2008). In fact, in the RLT case the enzymatic queuing effect acts as a positive feedback loop at each node of the network, since the abundance of a TF or of the output reduces its own degradation by congesting the shared protease pool. In an experimental implementation, it is expected that some parameters might vary depending on the bacterial growth phase or environmental conditions. To assess the effect of such variations, we investigated in STAR Methods and Figure S1 the deviations of the amplitude, period, and period-to-amplitude cross-sensitivity S ht under large parameter perturbations (±90% around their nominal values). When the entire parameter set was scanned for the best performing re-designs (DFO Rd II and RLT Rd II) the parameters that appeared to affect the DFO Rd II most were the mrna degradation rate and k R3 (the half-activation constant of the repressor). On the other hand, RLT Rd II oscillation characteristics exhibited the largest changes when v m (the protease catalysis rate) and t 3 (R3 translation rate) were perturbed. For both re-designs the effect of dilution rate due to cell division was relatively low. Finally, stochastic effects were assessed by performing stochastic simulations of mechanistic mass-action models that share the same structure as the phenomenological models we used in the main text for DFO/RLT Rd II. The results and their analysis is provided in STAR Methods and Figure S4. The analysis suggests that although the DFO Rd II exhibited smaller SDs for amplitude and (Continued on next page) Cell Systems 6, 508 520, April 25, 2018 517