Research Article Cite This: ACS Synth. Biol. XXXX, XXX, XXX−XXX
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A Robust Molecular Network Motif for Period-Doubling Devices Christian Cuba Samaniego and Elisa Franco* Mechanical Engineering, University of California at Riverside, Riverside, California 92521, United States S Supporting Information *
ABSTRACT: Life is sustained by a variety of cyclic processes such as cell division, muscle contraction, and neuron firing. The periodic signals powering these processes often direct a variety of other downstream systems, which operate at different time scales and must have the capacity to divide or multiply the period of the master clock. Period modulation is also an important challenge in synthetic molecular systems, where slow and fast components may have to be coordinated simultaneously by a single oscillator whose frequency is often difficult to tune. Circuits that can multiply the period of a clock signal (frequency dividers), such as binary counters and flip-flops, are commonly encountered in electronic systems, but design principles to obtain similar devices in biological systems are still unclear. We take inspiration from the architecture of electronic flip-flops, and we propose to build biomolecular period-doubling networks by combining a bistable switch with negative feedback modules that preprocess the circuit inputs. We identify a network motif and we show it can be “realized” using different biomolecular components; two of the realizations we propose rely on transcriptional gene networks and one on nucleic acid strand displacement systems. We examine the capacity of each realization to perform period-doubling by studying how bistability of the motif is affected by the presence of the input; for this purpose, we employ mathematical tools from algebraic geometry that provide us with valuable insights on the input/output behavior as a function of the realization parameters. We show that transcriptional network realizations operate correctly also in a stochastic regime when processing oscillations from the repressilator, a canonical synthetic in vivo oscillator. Finally, we compare the performance of different realizations in a range of realistic parameters via numerical sensitivity analysis of the period-doubling region, computed with respect to the input period and amplitude. Our mathematical and computational analysis suggests that the motif we propose is generally robust with respect to specific implementation details: functionally equivalent circuits can be built as long as the species-interaction topology is respected. This indicates that experimental construction of the circuit is possible with a variety of components within the rapidly expanding libraries available in synthetic biology. KEYWORDS: synthetic biology, oscillations, period-doubling, bistability, network motif, frequency divider the number of flip-flops employed determines the frequency division factor. (Thus, a device performing a single frequency division operation, or period-doubling, relies on a single flipflop.) On the basis of this well-known design principle in electronics, we ask if molecular period-doubling devices could be built from a resettable bistable component. An affirmative answer to this question has been recently suggested with coarse numerical simulations of a few molecular networks built around toggle switches, that could be reset by periodic inputs.7−10 However, a minimal network motif capable of period-doubling has not been identified; in addition, operational trade-offs and limitations of this class of devices have not been systematically explored. These steps are essential to support experimental implementations that could take advantage of the wealth of bistable networks and molecular networking strategies available in synthetic biology, both in vitro and in vivo.11−15 Here we describe and analyze a general molecular network motif to achieve period-doubling of an oscillatory input signal.
M
olecular oscillators time many processes of life, from cell division to differentiation.1,2 Evidence of the existence of a master clock regulating multiple downstream processes is found in many organisms from single cells to mammals.3 Yet, molecular events with different time scales must operate simultaneously to sustain life, suggesting the existence of robust mechanisms that process and modulate the period of the master clock without perturbing it. In synthetic biology, network motifs for period (frequency) modulation of oscillations could enable the coordination of multiple, temporally distinct parallel processes.4,5 These devices would also bypass the challenges encountered when tuning the period of synthetic oscillators, and could expand their range of operation. The use of frequency dividers is widespread in analog and digital computing devices and communication systems, which are typically driven by a single, fast and robust clock. Design principles for fequency dividers have been well-known for many decades:6 the most common approach to achieve integer frequency division is to interconnect multiple flip-flop elements. A flip-flop circuit is in essence a bistable component that is toggled between its two stable states by the input (Figure 1A1); © XXXX American Chemical Society
Received: June 20, 2017
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DOI: 10.1021/acssynbio.7b00222 ACS Synth. Biol. XXXX, XXX, XXX−XXX
Research Article
ACS Synthetic Biology
and numerical simulations. In particular, we rely on algebraic tools to determine how the input affects bistability of the core of the network motif.24 First, we use Sturm’s theorem to obtain exact parametric expressions that describe the bistability region; then, we use these exact expressions to identify how inputs can alter the bistability region and the capacity of the circuit to perform correctly, because the input baseline and amplitude can be viewed as time-varying perturbations of some of the bistable circuit parameters. In parallel, we use numerical simulations to compute the temporal response of each network to a variety of periodic inputs to evaluate their input-output rating. Perioddoubling of in vivo transcriptional realizations is also tested in a stochastic regime where the input is provided by a stochastic simulation of Elowitz’s repressilator.25 For each realization we adopt parameters that are in a realistic range for either in vivo artificial genetic circuits11 or in vitro nucleic acid systems.20,21
1. METHODS 1.1. Design of the Network Motif. Taking inspiration from the structure of electronic frequency dividers, as the circuit shown in Figure 1A1, we propose the network motif in Figure 1A2 to build molecular frequency dividers: the core of the architecture is a bistable switch, whose states are expected to toggle periodically in response to a periodic stimulus. The outputs of the network are the components of the bistable switch that exhibit a bistable behavior (in isolation); when toggling periodically, the outputs are expected to have opposite phase because their stable equilibria reach either a “low” or “high” state while never being simultaneously low or high, as highlighted in the qualitative phase diagram in Figure 1A2 (purple dots represent stable steady states). Toggling of the bistable switch in response to a periodic input is mediated by circuit components we name “push modules”. The network topology of each component (bistable switch and push modules) is in Figure 1B. Here, pointed arrows connecting two nodes mean that an increase in the concentration of the species associated with the start node causes an increase in the concentration of the species at the end node; conversely, hammer-head arrows indicate that an increase in the concentration of the start species causes a decrease in concentration of the end species. Feedback loops are highlighted in blue, and are akin to those shown in the electronic circuit in Figure 1A1. The outputs w 1 and w 2 of the push modules are synchronously driven by the oscillatory input u and passed to the bistable circuit; the push modules receive the states of the bistable switch as inputs, generating two negative feedback loops. We illustrate the desired toggling process of the network elements with the support of Figure 1C, where we consider an example oscillatory input u having period T. When the input signal u increases during a cycle, the output wi of each push module should increase. However, each push module is also controlled by the switch-driven negative feedback loop: if the ith node of the switch is in its high state, the output wi of the push module is inhibited by the intermediate variable yi. The presence of the intermediate species yi introduces a time lag in the push reactions to allow toggling.9 Suppose, for instance, that initially the bistable switch is in a state we indicate as S1, where x1 is at its low steady state value, and x2 is at its high steady state value. As u increases in its first cycle, w1 increases (while w2 remains low due to the high value of x2); thus, w1 pushes x1 to become high, and forces the bistable switch to toggle to equilibrium S2, where x1 is high and x2 is low; this
Figure 1. Network motif for period-doubling of oscillatory inputs. (A1) Frequency division is commonly achieved in electronic circuits using bistable switches that are toggled by the input signal. The D flipflop shown here includes a bistable switch built with two NAND gates, as well as NAND-based negative feedback loops for toggling. (A2) We propose a molecular network motif that resembles the architecture of a flip-flop, and is the interconnection of two push modules and a bistable subsystem. The bistable switch is forced to toggle between its stable steady states in the presence of periodic inputs. (B) The push modules are designed to include two intermediate species that generate a negative feedback loop when connected to the bistable switch, achieving an architecture similar to the flip-flop example in panel A1. Dashed lines indicate alternative regulatory interactions within the push modules, which would still guarantee an overall negative feedback loop (the output wi of each push module should be high only when xi is low and u is high). (C) Expected “state transitions” for each variable in the network when the input is a sinusoid with period T.
The motif is defined by the interconnection of three elements: the first is a two-node bistable switch, and the other two are upstream negative feedback loops that preprocess the oscillatory input (Figure 1A2). The outputs of the motif are given by the bistable species of the molecular switch. We claim that the motif is robust with respect to implementation, as long as the required feedback loops are present; we support this claim by demonstrating that three distinct realizations of the circuit have the capacity to operate period doubling. The first two realizations rely on Gardner and Collins’s bistable switch and on negative feedback loops implemented with cooperative transcriptional interactions,11,16 which are typically used to model synthetic gene networks in vivo. The third realization is based on a bistable switch and feedback loops built with noncooperative interactions (uni- and bimolecular chemical reactions),17,18 which are commonly adopted when modeling in vitro DNA and RNA-based circuits.4,19−23 We show that, as their electronic counterpart, molecular frequency dividers can process inputs having frequency, amplitude, and baseline within a given range. To characterize this “input-output rating” we combine mathematical analysis B
DOI: 10.1021/acssynbio.7b00222 ACS Synth. Biol. XXXX, XXX, XXX−XXX
Research Article
ACS Synthetic Biology
Figure 2. A transcriptional realization of the frequency divider motif that relies on competitive interactions. (A) Illustration of the gene network corresponding to model (2)−(3); the bistable switch genes are controlled by a single promoter, for which species xi and wi compete. (B) Top: Example integrated solutions x1 and x2 of model (2)−(3) when the input is the gray sinusoid u. Bottom: Equilibrium conditions of the bistable switch (SI Section 2.2.2) change over time, and their number of intersection changes; variables wi are taken as time-varying parameters; at the peak value of the input (t = 5 h and t = 15 h), the system transiently loses bistability and becomes monostable. (C) The frequency divider adapts to changes in input period, and maintains the period-doubling property. (D) Left, 500 stochastic (Gillespie) simulations showing the circuit response to the repressilator as input signal u; right, spectral analysis shows that the period of the input is doubled (solid line: mean, shaded regions: standard deviation). (E) The circuit doubles the period of the input in a limited range of input amplitude and frequency, shown as the dark orange region. The area of the period-doubling region shrinks as the baseline of the input increases, as a consequence of permanent loss of bistability.
state is maintained until u decreases at the end of the cycle. The same process repeats at the next cycle of u, which will bring the bistable switch back to its initial stable state S1. Overall, the period of each toggling output of the bistable switch is 2T. In the next sections we consider three different realizations of this network motif, and show that each of them has the capacity to work as a period-doubling device. Thus, we claim that this network motif is robust with respect to its biomolecular implementation. 1.2. Modeling and Analysis Approach. The perioddoubling network motif at Figure 1B describes the desired dynamic interactions between components that should be satisfied by a biomolecular implementation. Because we consider ordinary differential equation (ODE) models, we associate the (signed) interaction pattern to the sign pattern of the Jacobian matrix of the model.26 The distinct realizations considered in the next sections have a sign pattern consistent with the topology of the motif, but the specific interactions among species have different dynamics depending on the implementation. In particular we consider two kinds of dynamic interactions: cooperative, which can be modeled using Hill functions, and stoichiometric (noncooperative). Cooperative interactions are often used to model transcriptional gene networks in cells,11,25,27 while stoichiometric interactions are well suited to model chemical reaction networks such as in vitro nucleic acid systems.4,21−23 A bistable switch is at the core of the network motif: bistability is a global property of a dynamical system, and it is defined by the coexistence of three equilibria, two stable and one unstable (Figure 1A2). (In the following, we use the words “equilibrium” and “steady state” as synonyms.) In general, numerical simulations are required to evaluate the bistable
region of a system in parameter space. However, under assumptions that include dissipativity and positivity, certain systems can be classified as bistable as long as they present three (positive) equilibria, as we show in detail in the Supporting Information (SI) Section 1.24,28,29 Exploiting this fact, we recast the problem of characterizing bistability regions of our systems to the problem of studying their number of steady states. In turn, this task can be reduced to studying the roots of equilibrium polynomial conditions in a given interval. The bistable subsystem of all the realizations considered here satisfies the assumptions required to identify their bistability regions by simply counting the number of positive equilibria. Equilibria can be found as the zeros of polynomial expressions that depend on the system parameters and inputs. In turn, analytical parametric conditions for a polynomial to have exactly three zeros in an appropriate region (positive orthant) can be found using Sturm’s theorem (SI Section 1),24 a well-known algebraic geometry tool. We use these conditions to identify analytically the bistable regions of our realizations as a function of relevant combinations of the parameters. The bistable switch in our motif is connected to additional components forming negative feedback loops that preprocess the input signal u. Regardless of the chosen biomolecular realization, the push modules have an influence on the dynamics of the bistable core, and broadly have two effects: the (desired) effect of toggling the stable state of the circuit, and the (undesired) effect of driving the circuit outside of its bistable regime. To quantify these two effects, we treat the interconnections with the push modules as time-varying parametric perturbations on the bistable system, and evaluate the influence of these perturbations on the bistable region. For this purpose, we use the analytical expressions describing the C
DOI: 10.1021/acssynbio.7b00222 ACS Synth. Biol. XXXX, XXX, XXX−XXX
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ACS Synthetic Biology
interactions can be realized using an AND logic gate for the production of wi, whose input are yi and u.16,10 (Because yi is repressed by xi, we expect that a high concentration of xi causes a decrease in the concentration of yi, and therefore a decrease in wi.) The ODEs describing the push modules are ρ − δy1 , y1̇ = 1 + (x1/κ )m (y1 /κ y)n ψ (u/κu)r · − δw1 , w1̇ = 1 + (u/κu)r 1 + (y1 /κ y)n
bistability regions that were obtained using Sturm’s theorem, and we derive insights on the range of input period and amplitude allowing the circuit to perform frequency division. In addition to the mathematical analysis outlined above, we rely on numerical simulations to characterize the input-output behavior of each realization (due to size and nonlinearity of the models). We integrate the ODEs using custom MATLAB scripts to obtain the temporal response of each realization to periodic inputs. We also test the temporal response of some of the realizations in a stochastic regime, using Gillespie’s algorithm.30 We quantify the period and amplitude of the network output as a function of the period and amplitude of the input using MATLAB’s fft routine (Fast Fourier Transform): because the system is nonlinear, the output frequency spectrum is expected to include a range of frequencies distinct from the input frequency. We classify the circuit as doubling the period of the input only if the dominant component of the output frequency spectrum is half the input frequency. In the SI we examine how the period-doubling regions vary with respect to changes in relevant parameters of the circuits such as reaction rates and component concentrations.
y2̇ =
where x1 and x2 are protein concentrations, α is their maximal expression rate, δ is the degradation rate, κ is the apparent dissociation constant and m is their cooperativity coefficient for repression. (For simplicity we assume that reaction rates and cooperativity are the same for the dynamics of each gene, so the circuit is symmetric.) To connect the push module outputs w1 and w2 to the toggle switch we consider a competitive interaction, where xi and wi, i = 1, 2 bind to the same promoter region. Competition yields a model with additive terms (highlighted with boxes below) introduced by the push signals:
1 + (x 2/κ )m + w1/κw
x 2̇ =
1 + (x1/κ )m + w2 /κw
(y2 /κ y)n ψ (u/κu)r · − δw2 1 + (u/κu)r 1 + (y2 /κ y)n
(3)
Table 1. Simulation Parameters for the Realizations Relying on Gardner’s Toggle Switch
− δx1 ,
α + θw2 /κw
− δy2 ,
Here we assume that x1 and x2 are repressors for y1 and y2 with the same cooperativity coefficient m characterizing their mutual interactions within the bistable subsystem. Terms modeling the interconnection between the push modules and the bistable switch are highlighted with boxes. Both u and yi are activators for wi, i = 1, 2. While there are many ways to build a genetic AND gate,32 it is generally convenient to model this interaction by taking the product of the two corresponding activator Hill functions.16 If we abstract each chemical species as a binary variable, this product has the same sign pattern found in the truth table of an AND gate.33,34 Parameters κy and κu are apparent dissociation constants, ρ and ψ are maximal expression rates, and r and n are cooperativity coefficients. If the maximal expression rate of wi is at least as large as the maximal expression rate of xi (θ ≥ α), the Jacobian matrix (Section 2.3 of the SI file), is a sign definite matrix having a sign pattern consistent with the regulatory interactions of the network motif in Figure 1B. Figure 2B shows the outputs x1 and x2 of this realization, where the model was simulated using parameters listed in Table 1 and a sinusoidal input signal with period T = 10 h (gray trace
(1)
α + θw1/κw
1 + (x 2/κ )m
w2̇ =
2. RESULTS 2.1. A Transcriptional Network with Competitive Push Signals. Our first realization of the period-doubling motif relies on the well-known transcriptional toggle switch by Gardner and Collins.11 Many natural network motifs operate with transcriptional regulators, which have been exploited in synthetic biology to build a multitude of artificial circuits.16,31 A schematic of the overall realization is in Figure 2A. Gardner’s switch in isolation consists of two mutually repressing genes. If we neglect the dynamics of RNA translation, the switch is modeled by two ODEs: α α − δx1 , x 2̇ = − δx 2 x1̇ = 1 + (x 2/κ )m 1 + (x1/κ )m
x1̇ =
ρ
− δx 2
rate
description
value
other studies
θ (μM/h) α (μM/h) ρ = ψ (μM/h) δ (/h) κ (μM) κy = κu (μM) κw (μM) n=r m
Production rate Maximal production rate Reactivation Degradation Dissociation constant
3 1 2 1 0.2 1 0.1 1 2
0.1−100, refs 35, 36
Hill coefficient
0.4−1, ref 37 10−5−1, refs 38, 39
1−5, refs 36, 40−42
in the figure). As the input goes through two cycles, each variable of the bistable subsystem undergoes a single cycle. The overall circuit is able to handle fluctuations in the period: Figure 2C shows that if the input period changes, the output period adapts to maintain period-doubling. We note that, depending on the input characteristics, the output waveform may significantly differ from a perfect sinusoid: this is to be expected due to the nonlinearity of the model. We classify the output as correctly period-doubling as long as its principal
(2)
Here kw is the apparent dissociation constant of wi to the gene expressing xi, and θ is the maximal expression rate of xi induced by wi. The realization of the push modules is chosen based on the following rationale: the push module output wi should be low when u and xi are both high; in contrast, the push module output wi should be high if u is high but xi is low. These D
DOI: 10.1021/acssynbio.7b00222 ACS Synth. Biol. XXXX, XXX, XXX−XXX
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ACS Synthetic Biology frequency component is half the input frequency (spectra are computed using MATLAB’s fft routine). The circuit can also operate in a stochastic regime: Figure 2D shows an ensemble of 500 trajectories of output x1 obtained using Gillespie’s algorithm30 to simulate the reactions corresponding to the deterministic model (2)−(3). A stochastic simulation of the repressilator (gray traces) was used as input to the system25 (SI Section 5). Because in each stochastic simulation the phase of the input and the phase of the output may vary, the mean behavior is difficult to interpret. Phase variations should explain the change in the mean of the stochastic input and the variation of waveform of the mean of the output; phase variability may also explain the increase in output standard deviation, and the apparent similarity of input and output periodicity. To clarify whether period doubling occurs, we plot the frequency spectrum of input and output signals in Figure 2D (right), which shows that on average the dominant frequency of the output is half the dominant frequency of the input. 2.1.1. Baseline and Amplitude of the Input Signal Reduce the Bistability Region of the Core Toggle Switch and Reduce the Period-Doubling Regime. In the absence of input (u = 0) the push variables w1 and w2 converge to zero and the bistable circuit reverts to its model in isolation. To characterize the bistability region of the switch in isolation, we find equilibrium conditions by setting eq 1 to zero. The positive roots of the resulting polynomial are the admissible equilibria of the system; the circuit is bistable as long as three equilibria are found (SI Section 2.1). Using Sturm’s theorem (SI Section 2.2), we identify the region in parameter space where the circuit presents three equilibria; for m = 2 (dimerizing repressors), this region is characterized by the following inequality: α >2 (4) δκ
αi ⎛α⎞ =⎜ ⎟ ⎝ δκ ⎠ δκi
1
κi = κ m 1 + wj /κw ,
wj
wi κw
κw
i = 1, 2,
(6)
(
)
(
)
the amplitude uA and the period T are varied. Figure 2E shows the range of input period and amplitude in which perioddoubling is achieved, assuming the nominal parameters in Table 1. Simulations indicate that the circuit doubles the period of the input in a bounded region of input amplitude and period. In this region, a small input period admits a range of amplitudes that can also be very large; however, a large input period correlates with a small amplitude. The qualitative inverse relationship (large period, small amplitude) can be analytically predicted, as shown in Figure S6: we find that the duration of a step input (during which u is constant) required to toggle the bistable subsystem is proportional to the distance between the stable equilibria, and inversely proportional to the amplitude of the step; this result is obtained with sensible approximations described in SI Section 2.2.4. As suggested by our bistability conditions, the area of the period-doubling region decreases as the constant baseline uB of the input increases. Yet, the area appears robust with respect to variations of most circuit parameters as shown by our sensitivity analysis, reported in the SI Section 2.4. Notably, we find that the period-doubling region expands for large values of the transcription rate α, and low values of the degradation rate δ and of the dissociation coefficient κ. 2.2. A Transcriptional Network with Noncompetitive Push Signals. The realization based on Gardner’s toggle
(5)
i
m
These expressions are similar to expression (4) derived earlier, but depend on the time-varying variables w1 and w2, which in turn depend on u(t). While inequalities (6) cannot be analytically related to the input u, they clearly highlight that a large input may cause loss of bistability even though the system taken in isolation is fully within its bistable region in parameter space. This observation allows us to conclude that the circuit will not operate correctly whenever the input presents a large, constant baseline, which would permanently push the bistable network outside of its bistable region. However, a transient loss of bistability due to a large-amplitude oscillation may not have the same detrimental effect. 2.1.2. Period-Doubling Occurs with Transient Loss of Bistability. Numerical simulations suggest that a transient loss of bistability does not impact the correct operation of the circuit. In Figure 2B, bottom, we plot the equilibrium conditions of the bistable circuit as they change over time as a function of the integrated variables w1(t) and w2(t), taken as time-varying parameters in the equilibrium conditions. When the input signal u is at its lowest value, the circuit admits two stable equilibria, however when u reaches its highest value, the circuit is in a monostable regime; we conjecture that this temporary transition to a monostable regime promotes faster toggling. In contrast, the circuit performance could be compromised by permanent loss of bistability, i.e., a violation of conditions (6) at all times. This case would occur if the input has a large constant bias, or baseline. The observations above are validated by systematic numerical exploration of the period-doubling region. We compute the output period and amplitude in response to a sinusoidal input u u π t u(t ) = uB + 2A + 2A sin 2π T − 2 , where the baseline uB,
where θ wi α κw w + κi w
θ wi α κw
j = 2, 1
If this condition is satisfied, then the system admits three equilibria; this implies bistability because the system is dissipative and positive, and stability of its equilibria depends exclusively on the sign of the constant term of the characteristic polynomial (SI Section 2.1).24,28 In the presence of an input (u > 0), bistability conditions can be derived by rewriting eq 2 as follows: α1 α2 x 2̇ = x1̇ = − δx 2 m − δx1 , 1 + (x 2/κ2) 1 + (x1/κ1)m
(1 + ) , α =α
(1 + ) < 2, ( 1 + )(1 + )
i = 1, 2,
j = 2, 1
With this notation, and assuming m = 2, we can derive polynomial equilibrium conditions and use again Sturm’s theorem to identify the parameter region in which three equilibria are present for u(t) ≥ 0. It is possible to find tight analytical expressions describing the region of bistability as a function of the aggregated parameters αi/κiδ, i = 1, 2, but these expressions can only be evaluated numerically due to their complexity (SI Section 2.2.2). In contrast, the simple (but conservative) inequalities below describe a region where bistability is certainly lost: E
DOI: 10.1021/acssynbio.7b00222 ACS Synth. Biol. XXXX, XXX, XXX−XXX
Research Article
ACS Synthetic Biology
Figure 3. Realization of the frequency divider motif that relies on transcriptional noncompetitive interactions. (A) Illustration of the gene regulatory network, where two distinct promoters control the bistable switch, eliminating competition of species xi and wi like in the previous realization. (B) Top: Example integrated solutions x1 and x2 in model (7)−(3) when the input is the gray sinusoid; because this realization exhibits long transient dynamics (see also panel C), we show a portion of the stationary solution (when periodic behavior is reached). Bottom: Evolution of the equilibrium conditions of the bistable switch, where species wi are taken as time-varying parameters. (C) The circuit adapts to variations in the input period, maintaining period-doubling. (D) Left, ensemble of 500 stochastic (Gillespie) simulations of the circuit responding to the repressilator as input oscillatory signal; right, spectral analysis shows that the input period is doubled (solid line: mean, shaded regions: standard deviation). (E) Deterministic period-doubling region (dark orange) as a function of amplitude and period of the input. The input baseline generally reduces the area of the region.
Finally, stochastic Gillespie simulations in Figure 3D suggest that the system can operate in conditions where random fluctuations affect the input and the expression levels of the circuit components. A stochastic simulation of the repressilator25 was used as input to the system, and spectral analysis of the output x1 (Figure 3D, right) show that period doubling occurs as desired. 2.2.1. The Period-Doubling Region Is Robust with Respect to Input Amplitude, Period, and Baseline. Unlike the previous realization, the bistable subsystem (7) in isolation (u = 0, wi = 0) collapses to a system whose unique equilibrium is the origin; this feature may be an advantage as it implies that in the absence of stimuli the device is fully ”off”. If u is a positive constant, and therefore wi > 0, the system becomes bistable. If the input is a periodic signal with no baseline, the system can become transiently bistable. The example simulation at Figure 3 B, bottom, shows how the equilibrium conditions of the bistable subsystem vary as a function of variables wi taken as time-varying parameters: the subsystem is bistable only when u(t) becomes large. The bistability region in parameter space can be studied with the same approach used in the previous realization. In this case, the multiplicative terms in model (7) (boxed terms) can be treated as time-varying factors altering the transcription rate α. The positive term in each equation could therefore be rewritten as αi(wi)/(1 + (xi/κ)m), for i, j = 1,2, i ≠ j, where αi(wi) = αθ(wi/κw)/(1 + (wi/κw)). Coefficients αi(wi) vary in a bounded interval: their value ranges between zero (when u = 0 and wi = 0) and αθ (when wi → ∞). Parametric conditions for loss of bistability are similar to expressions (6):
switch can be modified to present noncompetitive regulatory interactions between the bistable subsystem and the push modules: w
x1̇ =
x 2̇
θ κ1
α
w
x2 m κ
− δx1 ,
(1 + ) (θ ) α = 1 + ( ) (1 + ) 1+
( )
w1 κw
w2 κw
x1 m κ
w2 κw
− δx 2 (7)
Here all the parameters are defined as in the previous section; boxed terms highlight the influence of the push modules on the bistable switch. This realization could be implemented in a synthetic circuit by using distinct promoter regions (binding sites) for xi and wi, i = 1, 2. A scheme of the interactions among expressed proteins and promoters is shown in Figure 3A. The model of the push subsystems is unchanged with respect to eq 3. The Jacobian matrix (SI Section 3.2), is a sign definite matrix having a sign pattern consistent with the regulatory interactions of the network motif in Figure 1B, regardless of the parameters adopted. An example numerical simulation of this realization is shown in Figure 3B, top, where we used the nominal parameters listed in Table 1; the system responds to the gray input sinusoid with a periodic signal having twice the input period. As the previous realization, but with a notably longer transient, this network can handle fluctuations in the input period: the output response adapts to maintain period-doubling, as shown in Figure 3C. F
DOI: 10.1021/acssynbio.7b00222 ACS Synth. Biol. XXXX, XXX, XXX−XXX
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ACS Synthetic Biology
Figure 4. Monomeric regulators can be used to implement the period-doubling motif. (A) Illustration of an example implementation network, where two enzymes (represented by blue and red circles, species X1 and X2) are inhibited and activated by RNA species. (B) Top: Example integrated trajectories of species x1 and x2 in model (8)−(9) when the input is the gray sinusoid. Bottom: Evolution of the equilibrium conditions of the bistable subsystems where variables wi are taken as time-varying inputs. Bistability is transiently lost at the peaks of the input signal (t = 5 h and t = 15 h). (C) The circuit outputs adapt to fluctuations of the input period, maintaining period-doubling. (D) The area of the period-doubling region (dark orange) decreases as the baseline of the input signal increases, as observed for the other realizations (Figure 2D and Figure 3D).
⎛ wi ⎛ α ⎞ ⎜ κw ⎜ ⎟θ ⎝ δκ ⎠ ⎜ 1 + wi ⎝ κw
⎞ ⎟ δ /γ ,
Protein/Protein: 104− 106, refs 60, 61 RNA: 10−5−10−3, ref 62 Proteins: 10−4−10−3, ref 56
150
β
