Computational Re-Design of Synthetic Genetic Oscillators for Independent

bioRxiv preprint doi: https://doi.org/10.1101/179945; this version posted August 23, 2017. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY-NC-ND 4.0 International license.

Computational Re-Design of Synthetic Genetic Oscillators for Independent Amplitude and Frequency Modulation Tomazou M.*†, Barahona M.‡, Polizzi K.§*, Stan G.-B.†*¶ † Department of Bioengineering, Imperial College London ‡ Department of Mathematics, Imperial College London § Department of Life Sciences, Imperial College London * Centre for Synthetic Biology and Innovation, Imperial College London ¶ Corresponding author: [email protected] (Guy-Bart Stan)

Abstract Engineering robust and tuneable genetic clocks is a topic of current interest in Systems and Synthetic Biology with wide applications in biotechnology. Synthetic genetic oscillators share a common structure based on a negative feedback loop with a time delay, and generally display only limited tuneability. Recently, the dual-feedback oscillator was demonstrated to be robust and tuneable, to some extent, by the use of chemical inducers. Yet no engineered genetic oscillator currently allows for the independent modulation of amplitude and period. In this work, we demonstrate computationally how recent advances in tuneable synthetic degradation can be used to decouple the frequency and amplitude modulation in synthetic genetic oscillators. We show how the range of tuneability can be increased by connecting additional input dials, e.g. orthogonal transcription factors that respond to chemical, temperature or even light signals. Modelling and numerical simulations predict that our proposed re-designs enable amplitude tuning without period modulation, coupled modulation of both period and amplitude, or period adjustment with near-constant amplitude. We illustrate our work through computational re-designs of both the dual- feedback oscillator and the repressilator, and show that the repressilator is more flexible and can allow for independent amplitude and near-independent period modulation.

1. Introduction nature) increases the robustness and adds tuneability to Accurate temporal control of biological processes is of the oscillator’s amplitude and period14–16. A prime fundamental importance across all kingdoms of life and is example in this category is the dual-feedback oscillator often realised through genetic ‘clocks’, i.e., transcriptional (DFO)10, which comprises a relatively slow negative networks with periodic gene expression. Heart beats, cell feedback loop and a faster positive feedback loop. The cycles, circadian rhythms, developmental processes and DFO has been shown to function robustly when energy metabolism are all, in one way or another, driven implemented using different components13, and also as by robust genetic clocks1–3. Such oscillators can also form part of large networks in different organisms and the building blocks to engineer complex genetic networks conditions17,18 as well as in populations of cells13,19,20. that can be used to synchronise cellular activity or to However, an important feature yet to be implemented in optimise the efficiency of metabolic pathways. either the DFO or other genetic oscillators is the ability to Understanding the design principles of genetic oscillators control the period and amplitude of oscillations and finding ways to engineer and precisely control them is independently. Such control would allow a wide range of therefore of crucial importance, as conveyed by the applications, including frequency analysis of downstream numerous synthetic oscillators proposed to date4–13. networks21,22, frequency encoding and pulse-based signal All genetic transcriptional oscillators are based on the processing23, development of two-dimensional biosensors same principle introduced in 1963 by Goodwin4: a where period and amplitude give distinct responses, negative feedback loop with a time-delay. Perhaps the designs for burden relief and metabolic pathway best studied example of such a system is the repressilator optimisation or even periodic administration of (RLT)5, a ring of transcriptional repressors acting in therapeutic molecules. sequence, thus providing an intrinsic lag or delay. More In this work, we use mathematical modelling and recently, it was demonstrated that the addition of a numerical simulation to show how simple, implementable positive feedback loop (as is commonly encountered in modifications of the DFO10 and RTL5 allow the user to bioRxiv preprint doi: https://doi.org/10.1101/179945; this version posted August 23, 2017. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY-NC-ND 4.0 International license.

Figure 1 Considered architectures of the dual-feedback oscillator (A,B,C) and repressilator (D,E,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 (sink) module, and an output module. The inputs considered for the original DFO and RLT designs are inducers titrating the effect of the TFs in the oscillator module. For all the

redesigned networks (labelled Rd), an input I1 (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 redesigns DFO Rd I and the RLT Rd I, a second external input I2 (Dial 2) is used to modulate the levels of the protease C via TF Y. For redesigns 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 TF 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 I2 to modulate the expression rate of the repressor R2.

control their amplitude and period independently and over achievable amplitudes and periods, at the same time an increased dynamic range. Through deterministic decoupling these characteristics. To keep our results numerical simulations of a parsimonious model, we general, we study parsimonious models that adequately demonstrate how to introduce orthogonal inputs or reproduce their behaviours. The models comprise three ‘’dials’’24 based on dual-input promoters25,26, that do not modules: (i) a core oscillator module composed of a affect the core topology of the DFO and RLT, yet allow negative feedback loop with delay mediated by repressors oscillations to be tuned in (a) amplitude only, (b) period (with an additional positive feedback loop for the DFO); (ii) and amplitude simultaneously, or (c) period with the an output module, with the gene of interest as a readout; amplitude maintained at near-constant levels. In addition, (iii) a sink module where the species of the oscillator and some of our re-designs allow the system to transition from output modules are channelled for enzymatic degradation. a stable OFF state, to a tuneable oscillatory regime, and Our analysis identified the post-translational coupling finally to a stable ON state as a function of the input dials introduced by enzymatic degradation (a.k.a. protease (See SI, S6). Hence, the same architecture can be used to ‘queuing effect’28) as a major challenge for independent generate controllable oscillations and/or switching amplitude modulation. This coupling affects the between high and low steady-states.27 degradation rate of the transcription factors (TF) and consequently the period28,29. To mitigate this effect, we 2. Results introduce an orthogonal enzymatic degradation pathway30 Our main objective is to re-design genetic oscillators with that enables modulation of amplitude with no impact on enhanced amplitude and frequency tuneability. We the period. consider the two most popular oscillators in synthetic To achieve independent period modulation, we followed biology (DFO and RTL) and discuss key modifications of two approaches. First, a re-design that modulates the their network architecture that extend the range of expression levels of the protease showed that the bioRxiv preprint doi: https://doi.org/10.1101/179945; this version posted August 23, 2017. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY-NC-ND 4.0 International license.

oscillators exhibit a significant increase in the achievable I 2 k2 a r range of periods. Second, we considered tuning of the f (A)= A , g(R) = R k2 + I 2 k2 + I 2 negative feedback loop by modulating the expression rate a a r r (2.2)

of one of the repressors in the RTL. Our simulations where Ia and Ir are the inducer concentrations that showed that the period can be tuned within a greater increase or decrease the effect of the activator (A) or

range compared to chemical induction alone, while the repressor (R) with half-maximal constants ka and kr, amplitude is kept nearly constant. respectively. For simplicity, their Hill cooperativities are set Below we give a description of the original and modified to 2 to represent a typical dimeric transcription factor. networks considered in this work. Our coarse-grained Additional simulation results (not shown here) indicate 31 phenomenological models employ Hill-type functions to that similar qualitative behaviours are obtained when capture the regulatory action of single and dual-input considering other Hill cooperativities, e.g. cooperativities 1 promoters as well as the inputs of DFO and RLT. Although and 4 in (2.2). In what follows, we therefore only present these models are too simplistic to reproduce the full range simulation results for Hill cooperativities set to 2. of dynamics captured by detailed mechanistic models The dynamics of the associated proteins (folded and using characterised biological parts10, they capture the unfolded) are described as follows: core responses of the oscillators and produce stable ⎛ Enzy"m$atic# Deg$r%adation⎞ Translation Folding oscillations with periods within the range observed in ! ! ⎜ v ⋅C ⎟ X! = t m − f X − ⎜δ + m ⎟ X , X ∈ A,R,G 5,10 u X x X u k P u { } experimentally implemented systems . ⎜ c + ∑ ⎟ ⎜ ⎟ ⎝ ⎠ (2.3) 2.1. The dual-feedback oscillator and its redesigns ⎛ v ⋅C ⎞ X! f X m X , X A,R,G 2.1.1 The original DFO = X u − ⎜δ + ⎟ ∈{ } ⎝ k + P ⎠ The DFO basic structure (Fig. 1A) comprises a positive and c ∑ (2.5)

a negative feedback loop mediated by a transcriptional where Χu and Χ are the concentrations of unfolded and

activator (A) and repressor (R), respectively. The output is folded proteins, respectively. Here tΧ and fΧ are the defined as the oscillating protein of interest denoted by G. translation and folding rate constants, respectively, and δ All three proteins (A, R, and G) are tagged for fast is the dilution rate due to cell division. The enzymatic enzymatic degradation via the same protease (C). Note degradation term obeys Michaelis-Menten (MM) kinetics28

that a basic difference in our modelled network from the with vm representing the catalysis rate, kc its MM constant original published DFO model10, is that we take in account and C the total concentration of protease in the cell. � the output protein and its queuing effect28 on the represents the total concentration of degradation tags degradation rates. competing for the same protease sites28, denoted The rate of change of the mRNA concentrations of the hereafter as ‘the load’. Assuming TFs with identical

three genes mi is given by: degradation tags can bind on the protease binding site ⎛ Activating HIll Function Repressing HIll Function⎞ "$#$% "$#$% Degradation with equal affinities then n v "$#$% ⎜ f (A) k ⎟ m! = p ⎜ b + b ⋅ R ⎟ − µ ⋅m! , X ∈ A,R,G X X 0 1 kn + f (A)n k v + g(R)v ! X { } ⎜ A R ⎟ (δ +d ) P = A + R +G + A + R +G ⎜ ⎟ m ∑ u u u ⎝ ⎠ (2.1) (2.5) The above model was parametrised to reproduce the where pX is the gene copy number, and b0 and b1 the basal oscillatory behaviour observed in experiments5,10. The full and maximum transcription rate constant, respectively. kA set of equations, kinetic scheme, assumptions, and and kR are Hill constants describing the amounts of transcription factors (TFs) required for half-maximal parameter values is given in SI (S7). activation and repression rates, respectively, whereas n and v are Hill cooperativities reflecting whether the TFs are 2.1.2 Dual-feedback oscillator re-designs active as monomers or multimers. As per the original DFO features an output (G) whose genetic expression is published model for DFO10 we used higher cooperativity controlled by a repressor (R), so that G is periodically degree for the repressor (v=4) compared to the activator expressed when R levels periodically transition from high (n=2). The removal rate constant per mRNA is given by μ to low during an oscillation. Through numerical simulations and captures the sum of the dilution rate due to cell and analysis of the Ordinary Differential Equation (ODE) model (2.1)-(2.5), we explore computationally a number of division (δ) and the mRNA decay rate (dm). The inducer- dependent functions f(A) and g(R) are defined as: re-designs of the basic DFO architecture. These redesigns bioRxiv preprint doi: https://doi.org/10.1101/179945; this version posted August 23, 2017. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY-NC-ND 4.0 International license.

involve the introduction of additional biological dials 2 2 j(Y ) I2 inserted at different locations in the DFO gene regulation l(C)= C , j(Y )= Y k2 + j(Y )2 k2 + I 2 network (Fig. 1B,C). The external inputs for tuning the y 2 2 (2.9)

oscillation are chemical inducers (Ia and Ir) that modulate Re-design 2: DFO Rd II the affinity of the TFs to their cognate promoter binding The second re-design DFO Rd II (Fig. 1C) differs from Rd I sites, and, therefore, their transcriptional regulation in the degradation mechanism. In DFO Rd II, an additional effectiveness. orthogonal protease (L) is dedicated to the specific degradation of the output (G). This second enzymatic Re-design 1: DFO Rd I degradation pathway eliminates the post-translational The first re-design DFO Rd I (Fig. 1B) introduces an coupling between G and the TFs of the core oscillator activation input (U) that controls the expression of G module, i.e., TFs A,R are targeted for degradation by C, orthogonally to the TF (R) of the core oscillator module. whereas G is specifically targeted for degradation by the This can be achieved by a dual-input promoter25,26 that orthogonal protease L30: responds to two different TFs (U and R). The total amount ⎛ v ⋅C ⎞ x! = t m − f x − δ + m x , x ∈ A,R of U is assumed to be unregulated (constitutive gene u x x x u ⎜ k P ⎟ u { } ⎝ c + ∑ ⎠ expression) and constant at steady state, but the ⎛ v ⋅C ⎞ activation effect of U on G is modulated by an external x f x m x, x A,R ! = x u − ⎜δ + ⎟ ∈ k + P { } input signal I1. In this re-design, the protease expression ⎝ c ∑ ⎠ rate is also tuneable via an orthogonal TF (Y). The effect of ⎛ v L ⎞ ! mL Gu = tGmG − fGGu − δ + Gu Y on C is modulated by an external input signal I2. ⎜ ⎟ ⎝ kL +G +Gu ⎠ The corresponding model is thus updated as follows: ⎛ v L ⎞ Auxiliary Activation ' Output Gain' G! f G mL G ⎛ "$$#$$% ⎞ = G u − ⎜δ + ⎟ 2 v k G G ⎜ h(U) k ⎟ ⎝ L + + u ⎠ m! = p ⎜ b + b ⋅ R ⎟ − µ ⋅m (2.10) G G 0 g k2 + h(U)2 kv + g(R)v G ⎜ U R ⎟ where vmL and kL are the Michaelis-Menten parameters for ⎝⎜ ⎠⎟ (2.6) the secondary enzymatic degradation pathway. The load where the affinity of the activating TF U is modulated by for the primary degradation pathway is now independent the input (inducer) I1 which is orthogonal to Ia and Ir. The of the output (G) function h(U) in (2.6) captures the influence of I1 on the P A A R R ∑ = + u + + u steady state concentration of U, captured with a Hill (2.11) function with cooperativity 2 and half-activation constant 2.2 Computational characterisation of the DFO re-designs The experimental implementation of the original DFO k1: (Fig. 1A) employed two of the most commonly used TFs, I 2 h(U)= U 1 LacI and AraC, which can be controlled by the chemical k2 + I 2 inducers IPTG and arabinose, respectively. We model the 1 1 (2.7) effect of these inducers by varying the parameters I and The dynamics of G allows the output to oscillate following a I . Fig. 2A,B shows model predictions of how the oscillations in R, and the rate of expression of G when R is r amplitude and period change as a result of varying the not fully repressing is titrated by I . As in (2.2) using 1 concentrations of the inputs I , I , I and I from 0 to cooperativity exponents other than 2 gives similar results. a r 1 2 saturating levels with respect to their corresponding Hill In addition, the steady state levels of C vary as a function functions (2.2), (2.7) and (2.9). of a second input I2 , as follows: The simulation results show that period and amplitude ⎛ v ⋅l(C) ⎞ X! t m f X m X , X A,R,G are correlated with Ia and Ir, in the original DFO (Fig 2A,D). = X X − X u − ⎜δ + ⎟ u ∈ k + P { } ⎝ c ∑ ⎠ Hence both inputs simultaneously affect amplitude and ⎛ v ⋅l(C) ⎞ period so that these characteristics cannot be X! f X m X , X A,R,G = X u − ⎜δ + ⎟ ∈ k + P { } independently tuned. This is quantified through the local ⎝ c ∑ ⎠ (2.8) cross-sensitivities for a given amplitude and period value where the protease concentration C is a function of Y (η*, τ*) defined as: modulated by the second input, I2: bioRxiv preprint doi: https://doi.org/10.1101/179945; this version posted August 23, 2017. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY-NC-ND 4.0 International license.

Figure 2 (A,B,C) Time course simulations showing the oscillatory output G over time under three different induction conditions (+ = 20 uM, +++ = 100 uM). In DFO (A) and DFO Rd I (B) when an input results in an increase in the amplitude, the period is elongated. In DFO Rd II the amplitude can be increased with no effect on the period (τ=40 min for black and green and τ=25 min for red and

orange trajectories). I2 primarily affects the period but with some impact on the amplitude. (D,E,F) Amplitude versus period dot plots for DFO, DFO Rd I and DFO Rd II, respectively. The colour of the dot indicates the amount of input with the green component

indicating I1 and the red component indicating I2. Yellow indicates that both inputs are high while black corresponds to their absence. (D) The original DFO exhibited a narrower range for both amplitude and period and either affects both characteristics in a

highly-coupled manner. (E) The DFO Rd I shows a wider tuneable range with I1 primarily increasing the amplitude and I2 modulating the period. However, none of these characteristics were independently tuneable. (F) Finally, in the DFO Rd II the amplitude can be

tuned in response to I1 independently of the period. The opposite was not feasible in this case.

∂η even with the highest levels of C (red points) the period S = is still affected by the increase in amplitude when I1 is ητ ∂τ (η ,τ )=(η*,τ *) (2.12) increased from 0 (red dots) to saturating amounts (yellow dots). ∂τ S = Interestingly, modulating the amount of the protease τη ∂η increases the range of achievable periods by about 70% (η ,τ )=(η*,τ *) (2.13) compared to the original DFO. At moderate to high The cross-sensitivities for the entire range of amplitudes amounts of protease, the sensitivity of the amplitude is and periods are presented in SI (Fig. S1). relatively small but increases as the protease drops to

low levels. 2.2.1 DFO Rd I exhibits an increased range of

achievable amplitudes and periods 2.2.2 DFO Rd II exhibits independent tuneability of Our simulations show that the range of period and period and amplitude amplitude tuneability of DFO Rd I is increased (Fig. 2B,E). Our objective is to find oscillatory re-designs and However, although the amplitude can be tuned over a associated conditions for which the cross-sensitivities wider range, the increase in amplitude is accompanied (2.12-2.13) are significantly reduced. In DFO Rd II we by a nonlinear shift in the period. This is explained by the aimed to eliminate the observed post-translational post-translational coupling between the output G and coupling by introducing an additional, orthogonal the TFs in the core oscillatory module introduced by the protease dedicated to the degradation of the output G. protease queuing effect, which results from the sharing The results in Fig. 2C,F show a negligible sensitivity of of the protease among the total amount of degradation the period to changes in the amplitude (Sτη=0) over the tags in the system. Hence, the degradation for G and all full range of accessible amplitudes, i.e., the amplitude is TFs becomes slower, which in turn increases the period. tuneable over the full range with no effect on the To investigate whether this effect can be attenuated by period. However, the amplitude to period sensitivity Sητ increasing the abundance of protease we use a second was increased compared to DFO Rd I. A downside of DFO external input I to modulate the expression rate of the 2 Rd II is a range of achievable periods comparable to that protease C (via the additional TF, Y). Fig. 2E shows that bioRxiv preprint doi: https://doi.org/10.1101/179945; this version posted August 23, 2017. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY-NC-ND 4.0 International license.

of the original DFO, although, for a given period, an ⎛ 2 2 ⎞ h(U) k3 extended range of amplitudes is accessible by varying I1. m! G = pg ⎜ bg + au 2 ⋅ 2 2 ⎟ − µ⋅mG k + h(U)2 k + f (R ) ⎝ U 3 3 ⎠ (2.18)

2.3. The Repressilator re-designs with h(U) given in (2.7), and the second input I2 regulating 2.3.1. The original RLT the amount of protease in the system: The original repressilator model comprises a ring of three ⎛ v l(C) ⎞ X! = t m − k X − δ + m X , X ∈ G,R1,R2,R3 TFs, each repressing the following gene in the ring (Fig. u X X fX u ⎜ k P ⎟ u { } ⎝ c + ∑ ⎠ 5 1D). Compared to the original RLT , we model the ⎛ v l(C) ⎞ X! k X m X , X G,R1,R2,R3 = fX u − ⎜δ + ⎟ ∈ enzymatic degradation mechanism catalysed by the k + P { } ⎝ c ∑ ⎠ (2.19) protease C, and we include explicitly the output gene 3 P G G Ri Ri , i 1,2,3 module (G). In addition, the repression is modulated by ∑ = u + + ∑( u + ) ∈{ } i=1 (2.20) external inputs IRi, with iÎ{1,2,3}. where l(C) is Similarly to above, we derived a model for RLT, as 2 2 follows. The mRNA rate equation is modelled as: j(Y ) I2 l(C)= C 2 2 , j(Y )= Y 2 2 2 ⎛ k ⎞ k y + j(Y ) k2 + I2 m! = p b + a i−1 − µ⋅m , i ∈ 1,2,3 (2.21) Ri i ⎜ i i 2 2 ⎟ Ri { } ⎝ k + f (R ) ⎠ i−1 i−1 ⎛ k2 ⎞ Re-design 2: RLT Rd II m p b a 3 m ! G = G ⎜ g + g 2 2 ⎟ − µ⋅ G k + f (R ) In the second redesign (Fig. 1F), R1 and R3 obey the ⎝ 3 3 ⎠ (2.14) same regulation scheme as in RLT Rd I, while R2 is now where, regulated by a dual-input promoter that is repressed k ! k ,R ! R by R1 and activated by Y: 0 3 0 3 and the induction function for each inducer is given as: ⎛ k2 ⎞ m p b a i−1 m , i 1,3 ! Ri = i ⎜ i + i 2 2 ⎟ − µ⋅ Ri ∈ 2 k f (R ) { } I ⎝ i−1 + i−1 ⎠ f (R )= R Ri , i ∈ 1,2,3 i i 2 2 { } 2 2 kRi + IRi ⎛ j(Y ) k ⎞ (2.15) m! = p ⎜ b + a 1 ⎟ − µ⋅m R2 y ⎜ 2 y k2 + j(Y )2 k2 + f (R1)2 ⎟ R2 The unfolded and folded protein dynamics are given by: ⎝ y 1 ⎠ ⎛ v C ⎞ I 2 X! t m k X m X , X G,R1,R2,R3 2 u = X X − fX u − ⎜δ + ⎟ u ∈ k k ,R R j(Y )= Y k + P { } 0 ! 3 0 ! 3 2 2 ⎝ c ∑ ⎠ k + I 2 2 (2.22) ⎛ v C ⎞ X! = k X − δ + m X , X ∈ G,R1,R2,R3 fX u ⎜ k P ⎟ { } Furthermore, a secondary protease L is specifically ⎝ c + ∑ ⎠ (2.16) dedicated to the enzymatic degradation of the output 3 P G G Ri Ri , i 1,2,3 gene of interest (G) yielding: ∑ = u + + ∑( u + ) ∈{ } i 1 = (2.17) ⎛ ⎛ v vC C ⎞ ⎞ X! t mm fkX X m m X X, X, X R1R,R12,,RR23,R3 2.3.2. Repressilator re-designs u = xX xX−− x fXu −u ⎜−δ δ+ + ⎟ u u ∈ ∈ ⎜ k k+ P P ⎟ { { } } ⎝ ⎝ c c ∑+ ∑⎠ ⎠ The redesigns of RLT follow similar lines to the ones ⎛⎛ vvCC ⎞ ⎞ presented for DFO, yet the additional flexibility supported X! = kf XX − δ + m m X X, ,X X∈ R1R,R12,R,R23,R3 = xfX u u −⎜⎜δ + ⎟ ⎟ {∈ } ⎝ kk++ PP⎠ { } by the RLT architecture leads to more independent ⎝ c c ∑∑ ⎠ 3 amplitude and period tuning. P Ri Ri , i 1,2,3 ∑ = ∑( u + ) ∈{ } i=1 Re-design 1: RLT Rd I ⎛ k2 ⎞ m! = p b + b 3 − µ⋅m , i ∈ 1,2,3 Two orthogonal “dials” (TFs U and Y) are introduced in G G ⎜ 0 1 2 2 ⎟ G { } ⎝ k3 + f (R3 ) ⎠ RLT Rd I (Fig. 1E): U modulates the expression of G as a function of the external input I whereas Y modulates the ⎛ v L ⎞ 1, G! t m k G mL G u = G G − fG u − ⎜δ + ⎟ u expression of the degradation protease C as a function of ⎝ kL +G +Gu ⎠ the external input I2. ⎛ v L ⎞ G! k G mL G Following the same structural modifications discussed for = fG u − ⎜δ + ⎟ ⎝ k +G +G ⎠ DFO Rd I, the output mRNA rate equation is given by: L u (2.23)

Figure 3 (A,B,C) Time course simulations of the output G over time under four different input conditions (+ = 20 uM, +++ = 100 uM). Similar to 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 I1 such that the amplitude cannot be modulated independently of the period. On the other hand, numerical simulations presented in (F) indicate that the re-design corresponding to RLT Rd II allows to tune the amplitude of the oscillations (green trajectory) over a wider range than both RLT and RLT Rd I and with no effect on the period. More surprising, tuning the

period by modulating I2, had a minimal effect on the amplitude. The sensitivity of both Sητ and Sτη was the lowest for RLT Rd II across all designs.

2.4 Computational characterisation of the RLT re-designs 2.4.2. RLT Rd II exhibits a wide range of periods and The numerical simulations of the original RLT (Fig. 3A,D) amplitudes tuneable in a near-independent manner

demonstrate the effect of the inputs IR2 and IR3 The simulations of RLT Rd II show that the amplitude is corresponding to chemical inducers used to modulate tuneable independently of the period as shown in Fig. the affinity of repressors. Other combinations of 3C, F. Note that in this case, we exploit the architecture

titratable inputs (IR1-IR2 and IR1-IR3) shown in SI (Fig. S2) of RLT to tune the period by introducing a dual-input produced similar results. The period is predicted to be promoter at the second position of the ring. This tuneable over a range of approximately 185 min, position was chosen as it is not directly regulating or whereas the amplitude range is narrower. As for the being regulated by the position of the output (G) and is original DFO, the amplitude and period are highly expected to have the least possible period to amplitude

correlated; hence tuning the amplitude and the period sensitivity (Sτη). The latter is further supported by of the RLT independently is not possible. simulation results shown in SI (S5). Simulation results indicate a significantly wider range of tuneable periods. 2.4.1. RLT Rd I exhibits a wider amplitude range yet More importantly, tuning the period results in low

with a strong non-linear amplitude and period cross-sensitivity Sητ (period to amplitude variations). dependency This follows from the fact that the duration of the cycle Numerical simulations of the RLT Rd I (Fig. 3B,E) indicate of each repressor affects the overall period, but not the that the range of achievable amplitudes is increased by amplitude of oscillations of the other repressors. In fact, approximately 3.5-fold compared to RLT, yet with large this effect becomes more apparent if more genes are period shifts. Specifically, the cross-sensitivity of period added in the ring, as seen in simulation results of the 32–34 to amplitude Sτη is significantly larger compared to DFO generalised repressilator (with n=5 and n=7 Rd I (Fig. 2E). The shape of the tuneable region also repressors) in SI (Fig. S5). indicates strong and highly non-linear coupling of the period with the amplitude. Increasing the amount of the protease C results in shorter periods but with a strong coupling between amplitude and period. Indeed, the range of achievable periods was narrower than in the original RLT but with an extended range for amplitude. bioRxiv preprint doi: https://doi.org/10.1101/179945; this version posted August 23, 2017. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY-NC-ND 4.0 International license.

Figure 4 Diagram of a conceptual two input, single output oscillator based biosensor. (A) A diagram of a hypothetical biosensor

where the concentration of two small molecules of interest I1 and I2 determine the characteristic of the oscillator in the processing unit which generates an oscillatory fluorescence signal, read in the output module. (B) The readout of the biosensor is analysed over a period to determine the amplitude and period of the oscillation. (C) Contour map showing the period as a function of both

inputs. The isoclines are vertical as I1 has no effect on the period, therefore at this stage the concentration of I2 is determined by

mapping the period to a standard curve of period versus I2. (D) Contour plot of the amplitude versus both inputs. Once the I2

concentration is determined from (B) the I1 concentration is simply found at the intersection point of the isoclines bounding the

measured amplitude value (blue isoclines) and the isocline of I2. The error for I1 and I2 is determined by the width of the blue and red isoclines, respectively.

38 39 2.5 Application of the re-designed RLT Rd II: An arsenic or aspartate ) or environmental signals (e.g. orthogonally tuneable oscillator as a multiple-input light40–42 or temperature43). single-output biosensor Fig. 4 illustrates the concept with RLT Rd II used as a

Orthogonally tuneable oscillators can be used to biosensor. The network is exposed to ligand 1 (I1), which precisely regulate the magnitude and temporal leads to modulation of the amplitude, and to ligand 2

characteristics of downstream networks or metabolic (I2), which modulates the period of the oscillations. The pathways1,3,11. An alternative application is to reverse output reporter (e.g., green fluorescent protein (GFP)) their use: instead of applying specific inputs to generate can be recorded with a basic setup comprising a light periodic oscillations of desired amplitude and period, excitation source and a sensor that records the periodic the user can read the periodic signal and accurately fluorescent output over time. The analysis of the signal

determine the concentrations of the inputs; hence using would map the period to the level of I2 using a 35 the oscillator as a dual-input single-output biosensor . precomputed function (red line); for a given I2, the

According to our computational analyses, the RLT Rd II amplitude versus I1 precomputed function then returns

re-design is most orthogonal in terms of its cross- the concentration of I1. sensitivities, and would thus be most appropriate to This approach could have significant advantages over co- build such a biosensor. The inputs can be mediated by expressing multiple single-input-single-output any transcriptional regulatory network, including TFs, biosensors. A multi-input-single-output biosensor two component systems, or even CRISPR derived requires a single excitation source and sensor, and a circuits36,37, that respond to ligands of interest (e.g. simpler electronic interface for implementing a low-cost bioRxiv preprint doi: https://doi.org/10.1101/179945; this version posted August 23, 2017. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY-NC-ND 4.0 International license.

biosensor. From a biological perspective, a single would offer a powerful basis for several applications reporter imposes less metabolic burden on the host, and including our proposed dual-input single-output periodic expression of a single fluorescent reporter biosensor. It could also be used to generate periodic would reduce the toxic effects, e.g. due to production of inputs to drive downstream networks, which can be free radicals upon excitation44. Finally, the error due to useful to better understand the dynamics of cellular stochastic gene expression effects and measurement and biological systems21,22,49,50. Furthermore this can method, can be reduced in periodic signals by increasing be utilised to optimise temporal enzyme expression in the duration of the readings, in contrast to the single metabolic pathways22 or even pulsatile (e.g. circadian) readings from traditional biosensors. release and administration of drugs and therapeutic molecules51–53. 3. Discussion Using computational modelling, this work has Engineering design principles of orthogonally identified network re-designs that expand the range of tuneable oscillators with increased ranges of achievable amplitudes and periods of the two achievable amplitude and period oscillators most widely used in Synthetic Biology, the Through the computational simulation and analysis DFO and RLT. Additionally, we have identified those that we have carried out above and further data re-designs that enable orthogonal tuning of amplitude provided in the Supplemental Material, we identify and period. In particular, we have shown that the RLT the following core principles for the implementation outperforms the DFO in achieving independent tuning of an orthogonally tuneable oscillator: of amplitude and period and extending the range of a) Orthogonality of the degradation of the output their achievable values. For instance, our DFO re- with respect to the degradation of the oscillator’s TFs design showed fast oscillations (period below 1 hour) enables decoupling of the amplitude and period but within a limited range of achievable amplitudes tuning. In our proposed re-designs (DFO Rd II and RLT compared to the RLT. This is due to the fact that the Rd II) this was realised by considering a second activating and repressing TFs in the DFO oscillate in orthogonal protease pathway dedicated to the output. phase, while in the RLT each repressor is expressed in Alternatively, this decoupling can be achieved by succession such that the effects of tuning a repressor removing the enzymatic degradation altogether like it are temporarily separated to some extent. was recently demonstrated experimentally for the In vivo implementations of the system would be repressilator54. The trade-off for using the latter affected by other couplings, such as competition for approach is that the achievable periods will inevitably shared cellular resources (e.g., energy, amino acids, become longer and dependent on the growth rate of ribosomes and polymerases45,46). This means that the the host. range over which amplitude and period can be tuned b) Modulation of the output expression rate independently should become narrower for independently of the oscillator’s TF levels is required experimental conditions where the TFs or output gene for setting the amplitude levels. We simulated the are overexpressed. To attenuate host context effects, above by considering a dual-input promoter that the oscillator TFs should be expressed at low enough oscillates due to the periodic expression of the levels, i.e. levels at which burden and resource oscillator’s repressors. The rate of transcription competition effects are not dominant. Recently, TX-TL obtained from this dual-input promoter is tuneable implementation of these oscillators has been through the use of its second input, which in our case successfully demonstrated in cell-free microfluidic was an activator orthogonal to the oscillator’s based platforms47 where a continuous flow of TX-TL48 repressors. Nevertheless, the same effect can be rich in nutrients, translational and transcriptional obtained by modulating other gene expression resources was used. In this setup, the competition for elements like the promoter, RBS, gene copy number shared resources is minimised and thus we anticipate or the degradation tags. The downside is that this this setup to constitute the best platform for a modulation will require the creation of different successful implementation of our re-designs. genetic constructs for a required amplitude level and A successful implementation of an orthogonally is not easily or externally tuneable in a dynamic tuneable oscillator, such as our proposed RLT Rd II, manner. bioRxiv preprint doi: https://doi.org/10.1101/179945; this version posted August 23, 2017. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY-NC-ND 4.0 International license.

c) Control over the degradation rate of the effect on the amplitude when the repressor was not oscillator’s TFs enables period modulation. This was directly regulating or regulated by the output (SI, S5). demonstrated in DFO Rd II where the protease abundance is controlled by an orthogonal input which, 4. Computational Methods in turns, regulates the amount of activating TF (Y). The models given in section 3 were computationally Again, alternatively the same effect can be obtained implemented in MATLAB’s Simbiology V3 as a set of by varying the protease levels through the use of reactions, species, parameters and rules (Tables given promoters, RBSs and degradation tags of different in SI Tables 7.1-7.8). The models were simulated using strengths or by varying the copy number of the the ODE15s (Stiff/NDF) solver. The simulated time for protease gene. However, when doing so, the range of individual runs was set to 1000 min in the case of the achievable amplitudes was observed to be much more DFO and its redesigned models and 4000 min in the dependent on protease abundance (Fig. 2F, Fig. S3D). case of the RLT and its corresponding redesigns. The d) Varying the strength of the negative feedback loop RLT required longer simulations time to be able to enables control of the period of the oscillations. As capture a sufficient number of oscillations, which is we showed in our re-designs, controlling the important for downstream analysis as the RLT expression rate of one of the repressilator’s repressors produced longer periods (150-400 min) compared to in RLT Rd II by the use a dual-input promoter enables the DFO (15-55 min) as shown in Fig 3. Each input was period modulation. This promoter retains the original parameter scanned from 0 to saturating amounts of regulation from the oscillator’s TF but in addition is input (100 uM) in 50 steps. Each simulation dataset also responsive to an orthogonal activating TF (Y). As was obtained through parallel computations of the mentioned above, modifying the gene expression 50X50 dual-input titration matrix associated with elements (promoter, RBS, etc.) of the affected these parameter scans. repressor will generate the same effects. However, The simulation results were analysed using custom- when doing so for the DFO case, the range of written scripts in MATLAB. The analysis was achievable periods was significantly narrower (Fig 3F, performed by, first, extracting the trajectories of the Fig S3B.) output (G) and the transcription factors. This was then e) Increasing the network distance between the followed by identification of the peaks of the signal modulated repressor and the RLT allows for nearly using the ‘findpeaks’ function of MATLAB. The period completely orthogonal period to amplitude was then calculated by determining the time modulation. When we considered repressilator rings difference for consecutive peaks. The amplitude was with 5 and 7 repressors our models predicted that, determined by computing the difference between the when the rate of expression of one of the repressors peak and minimum levels of G for a given cycle. was tuneable as per point d), the period had the least

8. Tigges, M., Dénervaud, N., Greber, D., Stelling, J. & Fussenegger, M. A synthetic low-frequency mammalian oscillator. Nucleic Acids 5. References Res. 38, 2702–2711 (2010). 1. Hess, B. Periodic patterns in biology. Naturwissenschaften 87, 9. Toettcher, J. E., Mock, C., Batchelor, E., Loewer, A. & Lahav, G. A 199–211 (2000). synthetic-natural hybrid oscillator in human cells. Proc. Natl. 2. Uriu, K. Genetic oscillators in development. Development Growth Acad. Sci. U. S. A. 107, 17047–17052 (2010). and Differentiation 58, 16–30 (2016). 10. Stricker, J. et al. A fast, robust and tunable synthetic gene 3. Bozek, K. et al. Regulation of clock-controlled genes in mammals. oscillator. Nature 456, 516–9 (2008). PLoS One 4, (2009). 11. Fung, E. et al. A synthetic gene-metabolic oscillator. Nature 435, 4. Goodwin, B. C. Temporal organization in cells. A dynamic theory 118–22 (2005). of cellular control processes. Temporal Organ. cells. A Dyn. theory 12. Chuang, C.-H. & Lin, C.-L. Synthesizing genetic sequential logic Cell. Control Process. (1963). circuit with clock pulse generator. BMC Syst. Biol. 8, 63 (2014). 5. Elowitz, M. B. & Leibler, S. A synthetic oscillatory network of 13. Danino, T., Mondragón-Palomino, O., Tsimring, L. & Hasty, J. A transcriptional regulators. Nature 403, 335–8 (2000). synchronized quorum of genetic clocks. Nature 463, 326–330 6. Atkinson, M. R., Savageau, M. A., Myers, J. T. & Ninfa, A. J. (2010). Development of genetic circuitry exhibiting toggle switch or 14. Hasty, J., Dolnik, M., Rottschäfer, V. & Collins, J. Synthetic Gene oscillatory behavior in Escherichia coli. Cell 113, 597–607 (2003). Network for Entraining and Amplifying Cellular Oscillations. Phys. 7. Purcell, O., di Bernardo, M., Grierson, C. S. & Savery, N. J. A multi- Rev. Lett. 88, 148101 (2002). functional synthetic gene network: A frequency multiplier, 15. Tsai, T. Y.-C. et al. Robust, Tunable Biological Oscillations from oscillator and switch. PLoS One 6, (2011). bioRxiv preprint doi: https://doi.org/10.1101/179945; this version posted August 23, 2017. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY-NC-ND 4.0 International license.

Interlinked Positive and Negative Feedback Loops. Sci. 321, 126– 35. Goers, L. et al. Engineering Microbial Biosensors. Methods 129 (2008). Microbiol. 40, 119–156 (2013). 16. Purcell, O., Savery, N. J., Grierson, C. S. & di Bernardo, M. A 36. Qi, L. S. et al. Repurposing CRISPR as an RNA-guided platform for comparative analysis of synthetic genetic oscillators. J. R. Soc. sequence-specific control of gene expression. Cell 152, 1173–83 Interface 7, 1503–1524 (2010). (2013). 17. Prindle, A. et al. Genetic Circuits in Salmonella typhimurium. ACS 37. Didovyk, A., Borek, B., Hasty, J. & Tsimring, L. Orthogonal Modular Synth. Biol. 1, 458–464 (2012). Gene Repression in Escherichia coli Using Engineered 18. Danino, T., Lo, J., Prindle, A., Hasty, J. & Bhatia, S. N. In Vivo Gene CRISPR/Cas9. ACS Synth. Biol. 5, 81–88 (2016). Expression Dynamics of Tumor-Targeted Bacteria. ACS Synth. Biol. 38. Voigt, C. A. Synthetic biology: Bacteria collaborate to sense 1, 465–470 (2012). arsenic. Nature 481, 33–34 (2012). 19. Prindle, A. et al. A sensing array of radically coupled genetic 39. Utsumi, R. et al. Activation of bacterial porin gene expression by a ‘biopixels’. Nature 481, 39–44 (2011). chimeric signal transducer in response to aspartate. Science 245, 20. Prindle, A. et al. Rapid and tunable post-translational coupling of 1246–1249 (1989). genetic circuits. Nature 508, 387–91 (2014). 40. Schmidl, S. R., Sheth, R. U., Wu, A. & Tabor, J. J. Refactoring and 21. Olson, E. J., Hartsough, L. A., Landry, B. P., Shroff, R. & Tabor, J. J. Optimization of Light-Switchable Escherichia coli Two-Component Characterizing bacterial gene circuit dynamics with optically Systems. ACS Synth. Biol. 3, 820–831 (2014). programmed gene expression signals. Nat. Methods 11, 449–55 41. Levskaya, A. et al. Synthetic biology: engineering Escherichia coli (2014). to see light. Nature 438, 441–2 (2005). 22. Davidson, E. A., Basu, A. S. & Bayer, T. S. Programming microbes 42. Tabor, J. J., Levskaya, A. & Voigt, C. A. Multichromatic control of using pulse width modulation of optical signals. J. Mol. Biol. 425, gene expression in Escherichia coli. J. Mol. Biol. 405, 315–24 4161–6 (2013). (2011). 23. Lu, T. K., Khalil, A. S. & Collins, J. J. Next-generation synthetic gene 43. Galkin, V. E. et al. Cleavage of Bacteriophage λ cI Repressor networks. Nat. Biotechnol. 27, 1139–1150 (2009). Involves the RecA C-Terminal Domain. J. Mol. Biol. 385, 779–787 24. Arpino, J. A. J. et al. Tuning the dials of Synthetic Biology. (2009). Microbiology 159, 1236–53 (2013). 44. Chalfie, M. & Kain, S. Green Fluorescent Protein: Properties, 25. Lutz, R. & Bujard, H. Independent and tight regulation of Applications, and Protocols. Wiley (2006). doi:10.1021/np068247i transcriptional units in escherichia coli via the LacR/O, the TetR/O 45. Weisse, A. Y., Oyarzun, D. A., Danos, V. & Swain, P. S. A and AraC/I1-I2 regulatory elements. Nucleic Acids Res. 25, 1203– mechanistic link between cellular trade-offs, gene expression and 1210 (1997). growth. bioRxiv (2015). doi:10.1101/014787 26. Tomazou, M., Stan, G.-B., Polizzi, K. & Barahona, M. Towards light 46. Ceroni, F., Algar, R., Stan, G.-B. & Ellis, T. Quantifying cellular based dynamic control of synthetic biological systems. (Imperial capacity identifies gene expression designs with reduced burden. College London, 2014). Nat. Methods 1–8 (2015). doi:10.1038/nmeth.3339 27. Tyson, J. J., Chen, K. C. & Novak, B. Sniffers, buzzers, toggles and 47. Niederholtmeyer, H. et al. Rapid cell-free forward engineering of blinkers: Dynamics of regulatory and signaling pathways in the novel genetic ring oscillators. Elife 4, (2015). cell. Current Opinion in Cell Biology 15, 221–231 (2003). 48. Sun, Z. Z. et al. Protocols for implementing an Escherichia coli 28. Cookson, N. A. et al. Queueing up for enzymatic processing: based TX-TL cell-free expression system for synthetic biology. J. correlated signaling through coupled degradation. Mol. Syst. Biol. Vis. Exp. e50762 (2013). doi:10.3791/50762 7, 561–561 (2014). 49. Lux, M. W., Bramlett, B. W., Ball, D. A. & Peccoud, J. Genetic 29. Moriya, T., Yamamura, M. & Kiga, D. Effects of downstream genes design automation: Engineering fantasy or scientific renewal? on synthetic genetic circuits. BMC Syst. Biol. 8, S4 (2014). Trends in Biotechnology 30, 120–126 (2012). 30. Cameron, D. E. & Collins, J. J. Tunable protein degradation in 50. Nielsen, A. A. K. et al. Genetic circuit design automation. Science bacteria. Nat. Biotechnol. 32, 1276–1281 (2014). (80-. ). 352, aac7341-aac7341 (2016). 31. Gjuvsland, A. B., Plahte, E. & Omholt, S. W. Threshold-dominated 51. Bakken, E. E. & Heruth, K. Temporal control of drugs. Ann. N. Y. regulation hides genetic variation in gene expression networks. Acad. Sci. 618, 422–427 (1991). BMC Syst. Biol. 1, 57 (2007). 52. Ritika, S., Arjun, S., Sunil, K. & Faraz, J. Pulsatile drug delivery 32. Strelkowa, N. & Barahona, M. Transient dynamics around system: a review. Int. Res. J. Pharm. 3, 934–41 ST–Pulsatile Drug unstable periodic orbits in the generalized repressilator model. Delivery System for Co (2012). Chaos 21, (2010). 53. Arora, S., Ali, J., Ahuja, A., Baboota, S. & Qureshi, J. Pulsatile drug 33. Strelkowa, N. & Barahona, M. Switchable genetic oscillator delivery systems: An approach for controlled drug delivery. Indian operating in quasi-stable mode. J. R. Soc. Interface 7, 1071–1082 J. Pharm. Sci. 68, 295 (2006). (2010). 54. Potvin-Trottier, L., Lord, N. D., Vinnicombe, G. & Paulsson, J. 34. Strelkowa, N. & Barahona, M. Stochastic oscillatory dynamics of Synchronous long-term oscillations in a synthetic gene circuit. generalized repressilators. in AIP Conference Proceedings 1479, Nature 538, 514–517 (2016). 662–666 (AIP, 2012).