Abstract
ProportionalIntegralDerivative (PID) feedback controllers are the most widely used controllers in industry. Recently, the design of molecular PIDcontrollers has been identified as an important goal for synthetic biology and the field of cybergenetics. In this paper, we consider the realization of PIDcontrollers via biomolecular reactions. We propose an array of topologies offering a compromise between simplicity and high performance. We first demonstrate that different biomolecular PIcontrollers exhibit different performanceenhancing capabilities. Next, we introduce several derivative controllers based on incoherent feedforward loops acting in a feedback configuration. Alternatively, we show that differentiators can be realized by placing molecular integrators in a negative feedback loop, which can be augmented by PIcomponents to yield PIDcontrollers. We demonstrate that PIDcontrollers can enhance stability and dynamic performance, and can also reduce stochastic noise. Finally, we provide an experimental demonstration using a hybrid setup where in silico PIDcontrollers regulate a genetic circuit in single yeast cells.
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Introduction
One of the most salient features of biological systems is their ability to adapt to their noisy environments. For example, cells often regulate gene expression to counteract intrinsic and extrinsic noise in order to maintain a desirable behavior in a precise and timely fashion. This resilience toward undesired disturbances is often achieved via feedback control that has proved to be ubiquitous in both natural (e.g.^{1,2,3}) and engineered systems (e.g.^{4,5}). In fact, synthetic engineering of biomolecular feedback controllers is gaining wide attention from biologists and engineers (e.g.^{6,7,8,9,10,11,12,13,14}).
A standard general setup for feedback controllers is depicted as a block diagram in Fig. 1a. Refer to Supplementary Box 1 (A Primer on Block Diagrams). The “Plant" block represents the process to be controlled. It can be actuated through its input, denoted here by u, to dynamically manipulate its output of interest, denoted here by y. The objective of such control systems is to design a feedback controller that automatically actuates the plant in a smart autonomous fashion which guarantees that the output y meets certain performance goals despite the presence of disturbances in the plant. These performance goals, described in Fig. 1b, include (but are not limited to) robust steadystate tracking—also known as robust perfect adaptation (RPA) in biology, stability enhancement, desirable transient response and variance reduction. Control theory developed a wide set of tools to design feedback controllers that meet certain performance objectives. For instance, it is well known in control theory (Internal Model Principle^{15}) that a controller should include integral (I) action to be able to achieve RPA. Furthermore, proportionalintegralderivative (PID) feedback controllers—first rigorously introduced by Nicolas Minorsky^{16} around a hundred years ago—adds proportional (P) and derivative (D) action to the integrator (I) to be able to tune the transient dynamics and enhance stability while preserving RPA. Interestingly, after almost a century, PID controllers are still the most widely used controllers in industrial applications^{17,18,19}.
Originally, PID feedback controllers were designed to control mechanical (later, electrical and chemical) systems such as automatic ship steering^{20}. Such control systems involve controlling quantities that can take both negative and positive values such as angles, velocities, electric currents, voltages, etc. Furthermore, traditional PID controllers possess linear dynamics since all three operations of a PID are linear. Two classes of linear PID controllers, adopted from Chapter 10 of ref. ^{21}, are shown in Fig. 1c, d. In Fig. 1c, the error signal e(t) ≔ r − y(t) is fed into the three (P, I, and D) components. The outputs of the three components are summed up to yield the control action u that serves as the actuation input to the plant. However, in Fig. 1d, the controller has two degrees of freedom since both the error e and the output y are used separately and simultaneously. Particularly, the error is fed into the integrator, while the output is fed into the proportional and derivative components. Observe that both architectures require that the integrator operates on the error (and not the output). This is necessary to achieve RPA and can be easily seen using a very simple argument explained next. Let u_{I}(t) denote the output of the integrator, that is,
Assuming that the dynamics are stable, then at steady state we have \(\mathop{\lim }\limits_{t\to \infty }{\dot{u}}_{I}(t)=0\). This implies that, at steady state, the error e ≔ r − y has to be zero, and thus \(\mathop{\lim }\limits_{t\to \infty }y(t)=r\), hence achieving the steadystate tracking property. Observe that although this argument requires closedloop stability, it does not depend on the particular structure and/or parameters of the plant, hence achieving the robustness property.
For mechanical and electrical systems, the linearity of the PID controllers is convenient because of the availability of basic physical parts (e.g., dampers, springs, RLC circuits, opamps, etc.) that are capable of realizing these linear dynamics. However, this realization quickly becomes challenging when designing biomolecular controllers. This difficulty arises because (a) biomolecular controllers have to respect the structure of BioChemical Reaction Networks (BCRN) that are inherently nonlinear, and (b) the quantities to be controlled (protein copy numbers or concentrations) cannot be negative (see^{22} for positive integral control). To achieve RPA, BCRN realizations of standalone integral (I) controllers received considerable attention^{23,24,25,26,27,28}. In previous work^{25}, the antithetic integral (aI) feedback controller was introduced to realize integral action that ensures RPA. More recently, it was shown in^{9} that the antithetic motif is necessary to achieve RPA in arbitrary intracellular networks with noisy dynamics. A detailed mathematical analysis of the performance tradeoffs that may arise in the aI controller is presented in^{29,30}, and optimal tuning is treated in^{31}. Furthermore, practical design aspects, particularly the dilution effect of controller species, are addressed in^{9,27}. Biological implementations of various biomolecular integral and quasiintegral controllers appeared in bacteria in vivo^{6,8,9} and in vitro^{13}, and more recently in mammalian cells in vivo^{14} and in yeast using the cyberloop in silico^{32}.
In the pursuit of designing highperformance controllers while maintaining the RPA property, BCRN realizations of PI and PID controllers are starting to receive more focused attention^{33,34,35,36,37,38}. Particularly in^{33}, a proportional component is separately appended to the antithetic integral motif via a repressing Hilltype function to tune the transient dynamics and reduce the variance. The resulting PI controller follows the concept of Fig. 1d where error and output feedback are used to build separate (but nonlinear) P and I components. Several successful attempts were carried out to devise BCRN realizations that approximate derivatives^{39,40,41,42,43}. A BCRN realization of a full PID controller was reported in^{35}, where the authors introduced additional controller species to obtain a derivative component. The resulting PID controller uses error feedback (similar to the concept of Fig. 1c) to build separate nonlinear P, I, and D components and successfully improves the dynamic performance in the deterministic setting. Using a different approach, ^{37} and ^{38} exploit the dualrail representation from^{23}, where additional species are introduced to overcome the nonnegativity challenge of the realized PID controller. The authors demonstrate the resulting improvement of the performance in the deterministic setting. On a different note, ^{36} analyzed the effects of separate proportional and derivative controllers on (bursty) gene expression models in the stochastic setting.
Interestingly, previous research in this direction shares two intimately related aspects. Firstly, the P, I, and D components are realized separately such that they enter the dynamics additively. This aspect is motivated by traditional PID controllers where the controller dynamics are constrained to be linear, and thus the three components have to be added up (rather than multiplied for example). However, since feedback mechanisms in BCRNs are inherently nonlinear, there is no reason to restrict the controller to have linear dynamics and/or additive components. Secondly, the proposed designs introduce additional species to mathematically realize the controller, and thus making the biological implementation more difficult. To this end, we consider in this paper (more general) nonlinear PID controllers that do not have to be explicitly separable into their three (P, I, and D) components. This allows controllers to involve P, I, and D architectures in one (inseparable) block as depicted in Fig. 1e where both, error and output, feedback are allowed. The nonlinearity and inseparability features of the proposed PI and PID controllers provide more flexibility in the BCRN design and allow simpler architectures that do not require introducing additional species to the standalone integral controller. Next, we slightly increase the complexity (or order) of the controller designs by introducing up to two additional controller species. This provides more degrees of freedom for the controller and, as a result, offers a higher achievable performance. Furthermore, the higher the order of the controller, the more separable it is which facilitates the tuning of the PID gains by the biomolecular parameters. This hierarchical approach offers the designer a natural compromise between simplicity and performance enhancement. A rich library of biomolecular PID controllers of variable complexity is presented in this paper to offer the biologists a flexible and wide range of designs that can be selected depending on the desired performance and application at hand.
Results
General framework for biomolecular feedback controllers
The framework for feedback control systems is traditionally described through block diagrams (e.g., Fig. 1a). In this section, we lay down a general framework for feedback control systems where both the plant and the controller are represented by BCRNs. With this framework, the controller can either represent an actual biomolecular circuit or it can be implemented as a mathematical algorithm in silico^{44,45,46} to regulate a biological circuit (through light for example^{32,47}).
Consider a general plant, depicted in Fig. 2, comprised of L species X ≔ {X_{1}, ⋯ , X_{L}} that react with each other through K reaction channels labeled as \({{{{{{{\mathcal{R}}}}}}}}:= \{{{{{{{{{\mathcal{R}}}}}}}}}^{1},{{{{{{{{\mathcal{R}}}}}}}}}^{2},\cdots \,,{{{{{{{{\mathcal{R}}}}}}}}}^{K}\}\). Each reaction \({{{{{{{{\mathcal{R}}}}}}}}}^{k}\) has a stoichiometry vector denoted by \({\zeta }_{k}\in {{\mathbb{Z}}}^{L}\) and a propensity function \({\lambda }_{k}:{{\mathbb{R}}}_{+}^{L}\to {{\mathbb{R}}}_{+}\). Let \(S:= \left[{\zeta }_{1}\quad{\zeta}_{2}\quad \cdots \quad {\zeta}_{K}\right]{\in}\,{{\mathbb{Z}}}^{L\times K}\) denote the stoichiometry matrix and let \(\lambda := {\left[{\lambda }_{1}\quad{\lambda }_{2}\quad\cdots \quad{\lambda }_{K}\right]}^{T}\) denote the (vectorvalued) propensity function. Then, the plant constitutes a BCRN that is fully characterized by the triplet \({{{{{{{\mathcal{N}}}}}}}}:= ({{\mbox{X}}}\,,S,\lambda )\), which we shall call the “openloop” system.
The goal of this work is to design a controller network, denoted by \({{{{{{{{\mathcal{N}}}}}}}}}_{c}\), that is connected in feedback with the plant network \({{{{{{{\mathcal{N}}}}}}}}\), as illustrated in Fig. 2a, to meet certain performance objectives such as those mentioned in Fig. 1b. We assume that all the plant species are inaccessible by the controller except for species X_{1} and X_{L}. Particularly, the controller “senses” the plant output species X_{L}, then “processes" the sensed signal via the controller species Z ≔ {Z_{1}, ⋯ , Z_{M}}, and “actuates" the plant input species X_{1}. The controller species are allowed to react with each other and with the plant input/output species through K_{c} reaction channels labeled as \({{{{{{{{\mathcal{R}}}}}}}}}_{c}:= \{{{{{{{{{\mathcal{R}}}}}}}}}_{c}^{1},{{{{{{{{\mathcal{R}}}}}}}}}_{c}^{2},\cdots \,,{{{{{{{{\mathcal{R}}}}}}}}}_{c}^{{K}_{c}}\}\). Let \({\bar{S}}_{c}\in {{\mathbb{Z}}}^{(M+2)\times {K}_{c}}\) and \({\lambda }_{c}:{{\mathbb{R}}}_{+}^{M+2}\to {{\mathbb{R}}}_{+}^{{K}_{c}}\) denote the stoichiometry matrix and propensity function of the controller, respectively. Since the controller reactions \({{{{{{{{\mathcal{R}}}}}}}}}_{c}\) involve the controller species Z and the plant input/output species X_{1}/X_{L}, the stoichiometry matrix \({\bar{S}}_{c}\) can be partitioned as
where S_{1} and \({S}_{L}\in {{\mathbb{Z}}}^{1\times {K}_{c}}\) encrypt the stoichiometry coefficients of the plant input and output species X_{1} and X_{L}, respectively, among the controller reaction channels \({{{{{{{{\mathcal{R}}}}}}}}}_{c}\). Furthermore, \({S}_{c}\in {{\mathbb{Z}}}^{M\times {K}_{c}}\) encrypts the stoichiometry coefficients of the controller species Z_{1}, ⋯ , Z_{M}. Hence, the controller design problem boils down to designing S_{1}, S_{L}, S_{c} and λ_{c}. Note that, for simplicity, we consider plants with singleinputsingleoutput species. However, this can be straightforwardly generalized to multipleinputmultipleoutput species by adding more rows to S_{1} and S_{L}. Finally, the closedloop system constitutes the openloop network augmented with the controller network so that it includes all the plant and controller species X_{cl} ≔ X ∪ Z and reactions \({{{{{{{{\mathcal{R}}}}}}}}}_{cl}:= {{{{{{{\mathcal{R}}}}}}}}\cup {{{{{{{{\mathcal{R}}}}}}}}}_{c}\). Thus, the closedloop network, \({{{{{{{{\mathcal{N}}}}}}}}}_{{{{{{{{\rm{cl}}}}}}}}}:= {{{{{{{\mathcal{N}}}}}}}}\cup {{{{{{{{\mathcal{N}}}}}}}}}_{c}\), can be fully represented by the closedloop stoichiometry matrix S_{cl} and propensity function λ_{cl} described in Fig. 2a. We close this section, by noting that our proposed controllers range from simple designs involving M = 2 controller species, up to more complex designs involving M = 4 controller species.
Antithetic proportionalintegral (aPI) feedback controllers
Equipped with the BCRN framework for feedback control systems, we are now ready to propose several PI feedback controllers that are capable of achieving various performance objectives. All of the proposed controllers involve the antithetic integral motif introduced in^{25} to ensure RPA. However, other additional motifs are appended to mathematically realize a proportional (P) control action.
Consider the closedloop network, depicted in Fig. 3, where an arbitrary plant is connected in feedback with a class of controllers that we shall call aPI controllers. Observe that there are three different inhibition actions that are color coded. Each inhibition action gives rise to a single class of the proposed aPI controllers. Particularly, when no inhibition is present, we obtain the standalone antithetic integral (aI) controller of^{25} whose reactions are summarized in the left table of Fig. 3, whereas aPI of Class 1 (resp. Class 2) involves the inhibition of the input X_{1} by the output X_{L} (resp. controller species Z_{2}), and aPI of Class 3 involves the inhibition of the controller species Z_{1} by the output X_{L}. Furthermore, each aPI class encompasses various types of controllers depending on the inhibition mechanisms that enter the controller network as actuation reactions. We consider three types of biologically relevant inhibition mechanisms detailed in Fig. 3: additive, multiplicative (competitive) and degradation. Considering all three aPI classes with the various inhibition mechanisms, Fig. 3 proposes eight different aPI control architectures. Note that, it can be shown that a degradation inhibition in the case of aPI Class 3 would destroy the RPA property and is thus omitted. All of these controllers are compactly represented by a single general closedloop stoichiometry matrix S_{cl} and propensity function λ_{cl} depicted in Fig. 3. The various architectures can be easily obtained by suitably selecting the functions h ≔ h^{+} − h^{−} and g from the tables of Fig. 3. A theoretical linear perturbation analysis is carried out in Supplementary Information Section 1.1 to verify the proportionalintegral control structure of the proposed controllers. In fact, the analysis applies to any smooth function h which is monotonically increasing (resp. decreasing) in z_{1} (resp. z_{2}, x_{1} and x_{L}), and any smooth function g that is monotonically increasing (resp. decreasing) in μ (resp. x_{L}). For example, linear terms in h such as kz_{1} can be replaced with increasing Hilltype functions to model saturation whenever needed. The genetic implementation of the various control mechanisms presented here (and throughout the paper) can be carried out using basic activators, repressors and proteases which do not require any complicated biophysical mechanisms to implement the control designs. For instance, in the case of Class 1 aPI with degradation (see Fig. 3), the output of interest X_{L} can be fused to a protease capable of degrading the input X_{1} (see below for more details on the genetic implementations).
Deterministic steadystate analysis: robust perfect adaptation (RPA) of aPI controllers
The deterministic dynamics of the closedloop systems, for all the aPI controllers given in Fig. 3 can be compactly written as a set of ordinary differential equations (ODEs) given by
where \({e}_{1}:= {[1\quad 0\quad \cdots \quad 0]}^{T}\in {{\mathbb{Z}}}^{L}\). Note that the total actuation and reference propensities h and g take different forms for different aPI control architectures as depicted in Fig. 3. The fixed point of the closedloop dynamics cannot be calculated explicitly for a general plant; however, the output component (x_{L}) of the fixed point solves the following algebraic equation
where overbars denote steadystate values (if they exist), that is \({\overline{x}}_{L}:= \mathop{\lim }\limits_{t\to \infty }{x}_{L}(t)\). Two observations can be made based on (3). The first observation is that (3) has a unique nonnegative solution since g is a monotonically decreasing function in x_{L}. The second observation is that (3) does not depend on the plant. As a result, if the closedloop system is stable (i.e., the dynamics converge to a fixed point), then the output concentration converges to a unique setpoint that is independent of the plant. This property is valid for any initial condition, and is referred to as RPA. Particularly, for the aI and aPI controllers of Class 1 and 2, the reference propensity is g(μ, x_{L}) = μ, and thus \({\overline{x}}_{L}=\frac{\mu }{\theta }\). Furthermore, for the aPI of Class 3, \({\overline{x}}_{L}\) solves a polynomial equation of degree n + 1, where n denotes the Hill coefficient depicted in Fig. 3 (see Supplementary Information Section 3). In conclusion, all the proposed aPI controllers maintain the RPA property that is obtained by the antithetic integral motif, while introducing additional control knobs as extra degrees of freedom to enhance other performance objectives.
Deterministic stability analysis and performance assessment of aPI controllers
Next, we show that Class 1 aPI controllers with degradation and multiplicative inhibitions yield superior stability and performance properties. To compare the stability properties of the various proposed aPI controllers, we consider a particular plant, depicted in Fig. 4a, that is comprised of two species X_{1} and X_{2} (L = 2). This plant may represent a gene expression network where X_{1} is the mRNA that is translated to a protein X_{2} at a rate k_{1}. The degradation rates of X_{1} and X_{2} are denoted by γ_{1} and γ_{2}, respectively. The closedloop stoichiometry matrix and propensity function are also shown in Fig. 4a. Using the RouthHurwitz stability criterion^{48}, one can establish the exact conditions of local stability of the fixed point (Supplementary Equation (17)) for the various proposed aPI controllers. These conditions, once satisfied, guarantee that the dynamics locally converge to the fixed point.
For the remainder of this section, we consider fast sequestration reactions, that is, η is large. Under this assumption, one can obtain simpler stability conditions that are calculated in Supplementary Information Section 3, and tabulated in Fig. 4b. The stability conditions are given as inequalities that have to be satisfied by the various parameters of the closedloop systems. A particularly significant lumped parameter group is \(\rho := \frac{k{k}_{1}\theta }{{\gamma }_{1}{\gamma }_{2}({\gamma }_{1}+{\gamma }_{2})}\) that depends only on the plant and standalone aI controller parameters. To study the stabilizing effect of the appended proportional (P) component, we fix all the parameters related to the plant and standalone aI controller (hence ρ is fixed), and investigate the effect of the other controller parameters related to the appended proportional component. By examining the table in Fig. 4b, one can see that, compared to the standalone aI, the aPI controller of Class 1 with multiplicative (resp. degradation) inhibition enhances stability regardless of the exact values of κ (resp. δ) and n. This gives rise to a structural stability property: adding these types of proportional components guarantees better stability without having to finetune parameters.
In contrast, although the aPI controller of Class 1 with additive inhibition may enhance stability, special care has to be taken when tuning α. In fact, if α is tuned to be larger than a threshold given by \({\alpha }_{TH}:= \frac{{\gamma }_{1}{\gamma }_{2}}{{k}_{1}}r\left[1+{(r/\kappa )}^{n}\right]\), then stability is lost. Figure 4c elaborates more on this type of aPI controller. Three cases arise here. Firstly, if ρ < 1, that is, the standalone aI already stabilizes the closedloop dynamics, then the (α, κ) parameter space is split into a stable and unstable region. In the latter (α > α_{TH}), z_{2} grows to infinity, and the output x_{2} never reaches the desired setpoint r = μ/θ. Secondly, if 1 < ρ < 2, that is the standalone aI is unstable, then the (α, κ)parameter space is split into three regions: (1) a stable region, (2) an unstable region with a divergent response similar to the previous scenario where ρ < 1, and (3) another unstable region where sustained oscillations emerge as depicted in the bottom plot of Fig. 4c. Note that the closer ρ is to 2, the narrower the stable region is. Thirdly, for ρ > 2, the stable region disappears and thus this aPI controller has no hope of stabilizing the dynamics without retuning the parameters related to the standalone aI controller (e.g., k and/or θ). Clearly, multiplicative and degradation inhibitions outperform additive inhibition if stability is a critical objective. To this end, Fig. 4d shows how the settling time and overshoot can be tuned by the controller parameters α, κ, and δ for additive, multiplicative, and degradation inhibitions, respectively. It is shown that with multiplicative and degradation inhibitions, one can simultaneously suppress oscillations (settling time) and remove overshoots. In contrast, a proportional component with additive inhibition can suppress oscillations but is not capable of removing overshoots as illustrated in the simulations of Fig. 4d right panel. Furthermore, one can lose stability if α is increased above a threshold, as mentioned earlier. Nevertheless, for multiplicative and degradation inhibitions, increasing the controller parameters (κ^{−1}, δ) too much can make the response slower but can never destroy stability.
It can be shown that the other two classes (2 and 3) are undesirable in enhancing stability. For instance, observe that for Class 2, the stability conditions are the same as the standalone aI controller (in the limit as η → ∞) with the exception of the case of additive inhibition when \(\alpha \, > \, \frac{{\gamma }_{1}{\gamma }_{2}}{{k}_{1}}r\). In this case, the inequality is structurally very different from all other stability conditions. In fact, the actuation via Z_{2} dominates Z_{1}, and hence Z_{2} becomes responsible for the integral (I) action instead of Z_{1}. The detailed analysis of this network is not within the scope of this paper, and is left for future work. Finally, aPI controllers of Class 3 deteriorate the stability margin, since the righthand side of the inequalities is strictly less than one. However, this class of controllers can be useful for slow plants if the objective is to speed up the dynamics.
Stochastic analysis of the aPI controllers: RPA and stationary variance
We now investigate the effect of the aPI controllers on the stationary (steadystate) behavior of the output species X_{L} in the stochastic setting. First, we examine the stationary expectation \({\mathbb{E_\pi }}\left[{X}_{L}\right]\). It is shown in Supplementary Information Section 8.1 that for RPA to be achieved in the stochastic setting, the function g has to be affine in x_{L}. Hence only aPI controllers of Class 1 and 2 achieve RPA in the stochastic setting with \({\mathbb{E_\pi }}\left[{X}_{L}\right]=\mu /\theta =:r\). Next, we examine the stationary variance \({{\mbox{Var}}}_\pi \left[{X}_{L}\right]\) to demonstrate that aPI controllers with degradation inhibition excel in reducing celltocell variability. In Supplementary Information Section 8.2, we develop a tailored momentclosure technique based on^{33} to derive an analytic closed formula for the stationary variance \({{\mbox{Var}}}_\pi \left[{X}_{L}\right]\). Unfortunately, a general analysis for an arbitrary plant cannot be done. As a case study, we consider again the particular plant given in Fig. 4a in feedback with the aPI controller of Class 1 with the various inhibition mechanisms. Note, however, that the analysis can be generalized to any (affinelinear) plant with monomolecular reactions (see Supplementary Information Section 8.2). The resulting formula is given in Supplementary Table 2 that shows that the proportional controller decreases \({{\mbox{Var}}}_\pi \left[{X}_{2}\right]\). Figure 4e demonstrates this stationary variance reduction via simulations and the approximate formula. Unlike additive inhibition, multiplicative and degradation inhibitions provide a structural property of decreasing the \({{\mbox{Var}}}_\pi \left[{X}_{2}\right]\) without risking the loss of ergodicity (similar to the deterministic setting).
Antithetic proportionalintegralderivative feedback (aPID) control topologies
In this section, we append a derivative (D) control action to the aPI (Class 1) controller of Fig. 3 to obtain an array of aPID controllers depicted in Fig. 5. The proposed aPID controllers range from simple second order (involving only two controller species Z_{1} and Z_{2}) up to fourth order (involving four controller species Z_{1} to Z_{4}). Furthermore, the various controllers are categorized as two types: Ntype and Ptype. Ntype (negative feedback) controllers are usually suitable for plants with positive gain (increasing the input yields an increase in the output), while Ptype (positive feedback) controllers are usually suitable for plants with negative gains. This ensures that the overall control loops realize negative feedback. Note that one can easily construct hybrid PNtype controllers, where the individual P, I, and D components have different P/Ntypes. This hybrid design is shown to be very useful for certain plants (see Fig. 7d for example).
We begin with the Ntype secondorder design (first row of Fig. 5) whose main advantage is its simplicity. Note that the rationale behind the various Ptype designs is similar. Intuitively, the antithetic integral motif is cascaded with an incoherent feedforward loop (IFFL) to yield a PID architecture whose P, I and D components are inseparable as described in Fig. 1e. More precisely, the output X_{L} directly inhibits the input X_{1} and simultaneously produces it via the intermediate species Z_{1}. As a result, Z_{1} simultaneously plays the role of both an intermediate species for the IFFL and the integral control action. It is shown in Supplementary Information Section 1.2.1 that this simple design embeds a lowpassfiltered PID controller. The Ntype thirdorder design (second row of Fig. 5) involves one additional controller species Z_{3} to realize an IFFL that is disjoint from the antithetic motif. This yields an inseparable PD component appended to the separate I controller. It is shown in Supplementary Information Section 1.2.2 that this design embeds a lowpassfiltered PD + I controller when η is large enough. In contrast, the Ntype fourthorder design (third row of Fig. 5) involves two additional controller species Z_{3} and Z_{4} to realize a completely separable PID control architecture. It is shown in Supplementary Information Sections 1.2.3 and 1.2.4 that this design embeds a PI + lowpassfiltered D controller when η and η_{0} are large enough. The key idea behind mathematically realizing the derivative component here is fundamentally different from the previous two designs. This controller realizes an “antithetic differentiator”, whereby the antithetic motif feeds back into itself: Z_{3} feeds back into Z_{4} via the rate function g(z_{3}, x_{L}). In fact, this idea is inspired by a mathematical trick in control theory (see Supplementary Information Section 7) that basically exploits an integral controller, in feedback with itself to implement a lowpassfiltered derivative controller. For this fourthorder design, the derivative action can be achieved in two ways. One way is by mutually producing Z_{4} and X_{1} at a rate proportional to g(z_{3}, x_{L}) such that g is monotonically increasing (resp. decreasing) in z_{3} (resp. x_{L}). This implementation is treated separately in Supplementary Information Section 1.2.3. The other way is by producing Z_{4} while degrading X_{1} at a mutual rate of g(z_{3}, x_{L}) such that g is monotonically increasing in both z_{3} and x_{L}. This implementation is treated separately in Supplementary Information Section 1.2.4. Both designs have the same underlying PID control structure, but one might be easier to experimentally implement than the other.
Deterministic analysis and properties of the various aPID controllers
It is straightforward to show that the setpoint for the secondorder design is given by \({\bar{x}}_{L}=\frac{\mu }{\theta \beta }\) with the requirement that β < θ; whereas the setpoints for both higherorder designs are given by \({\bar{x}}_{L}=\frac{\mu }{\theta }\). Furthermore, the effective PID gains, denoted by (K_{P}, K_{I}, K_{D}), and cutoff frequency ω of the embedded lowpass filter for each of the proposed aPID controllers can be designed by tuning the various biomolecular parameters: β, η, η_{0}, γ_{0}, μ_{0} and the parameters of the propensity functions h and g. These functions depend on the specific implementation adopted. In particular, they can be picked in a similar fashion to the functions used to realize the three inhibition mechanisms (additive, multiplicative, or degradation) of the aPI controllers shown in Fig. 3. In the subsequent examples, we use degradation inhibitions, but the other mechanisms can also be used.
Next, we demonstrate various properties of the proposed controller designs in the deterministic setting and highlight the benefits of the added complexity. The mappings between the effective PID parameters (K_{P}, K_{I}, K_{D}, ω) and the biomolecular parameters (μ, θ, η, β, γ_{0}, η_{0}, . . . ) are given in Supplementary Information Section 4 for each controller. It is fairly straightforward to go back and forth between the two parameter spaces. For control analysis, these mappings can compute the various PID parameters from the biomolecular parameters, whereas for control design, these mappings can compute the various biomolecular parameters that achieve some desired PID gains and cutoff frequency. As a result, one can use existing methods in the literature (e.g.^{49}) to carry out the controller tuning in the PID parameter space, and then map them to the actual biomolecular parameter space. Nevertheless, it is of critical importance to note that different controllers yield different coverage over the PID parameter space. For instance, for the fourthorder design, there are enough biomolecular degrees of freedom to design any desired positive \(({K}_{P},{K}_{I},{K}_{D},\omega )\in {{\mathbb{R}}}_{+}^{4}\). The lower the order of the controller, the fewer the biomolecular degrees of freedom, and hence the more constrained the coverage in the PID parameter space. For instance, for the thirdorder design, the achievable PID parameters are constrained to satisfy K_{P} ≤ K_{D}ω. For the secondorder design, the constraint becomes even stricter. The details are all rigorously reported in Supplementary Information Section 4.
Flexibility of aPID controllers
We first show the limitation of aPI controllers, and then demonstrate the flexibility that comes with an added derivative component. More precisely, we show that the aPI controller alone is incapable of speeding up the response beyond a certain threshold without incurring oscillations—a limitation that a full aPID controller overcomes. We also show that the higherorder aPID controllers exhibit more flexibility in shaping the transient response. Consider the controlled gene expression network depicted in Fig. 6a where the ODEs of the various controllers are shown to explicitly specify the adopted propensity functions h and g. In this example, we consider both the P and D components acting on the input species X_{1} as negative actuation via degradation reactions. In fact, inhibition with degradation is used whenever possible because it outperforms the other inhibition mechanisms by achieving better stability properties and dramatically reducing the stationary variance (see Fig. 4). We start by highlighting the fundamental limitation of aPI controllers alone (without a D) in Fig. 6b. Using simple rootlocus arguments (see Supplementary Information Section 5.1), it is shown that as the proportional gain K_{P} is increased, two complex eigenvalues of the linearized dynamics around the fixed point approach a vertical asymptote intersecting the real axis at \(\frac{{\gamma }_{1}+{\gamma }_{2}}{2}\), while one real eigenvalue approaches the origin (due to integral control). This is numerically demonstrated in the two rootlocus plots of Fig. 6b, where K_{P} ≈ δ (for a sufficiently small κ_{1}). Clearly, the asymptotic limit is independent of all other parameters, including the integral gain K_{I}. This analysis highlights a fundamental limitation of the aPI controller, because no matter how we tune K_{P} and K_{I}, two of the eigenvalues are constrained to remain close to the imaginary axis when γ_{1} and γ_{2} are small. In the time domain, these constraints impose either a slowly rising response or a fasterrising response but with lightly damped oscillations, as illustrated in the simulation examples of Fig. 6b. These limitations can be mitigated by appending a derivative control action via the various aPID controllers. To demonstrate this, we consider a design problem where the end goal is to achieve a fast response without oscillations and with minimal overshoot. This can be achieved by placing the eigenvalues far to the left on the real axis. Hence the design problem can essentially be translated to the following objective: place the four most dominant eigenvalues (or poles) at s = −a where a > 0 and make a sufficiently large. The design steps start by first (1) deciding where to place the poles s = −a for some desired a, then (2) computing the PID parameters so as to place the poles as desired; this can be achieved using Supplementary Equation (46), and finally (3) mapping the PID parameters to the actual biomolecular parameters using the formulas in Supplementary Information Section 4. This is pictorially demonstrated in Fig. 6c for each aPID controller. However, it is shown in Supplementary Information Section 5.2 that the secondorder aPID imposes a lower bound on the achievable poles given by \((2+\sqrt{2})\frac{{\gamma }_{1}+{\gamma }_{2}}{2}\) as demonstrated in Fig. 6c. As a result, with a secondorder aPID, the performance can be made better than the aPI controller; however, the performance is also limited and cannot be made faster than a threshold, dictated by γ_{1} and γ_{2}, without causing overshoots and/or oscillations. In contrast, it is also shown in Supplementary Information Section 5.2 that the third and fourthorder aPID controllers can make a as large as desired without any theoretical upper bound. This means that the added complexity of the higherorder controllers is capable of shaping the response of the gene expression network freely and as fast as desired with no overshoots nor oscillations. This is also demonstrated in the simulations depicted in Fig. 6c.
Deterministic performance of aPID control of complex networks
We consider a more complex plant to be controlled. The plant, comprised of L = 6 species, is depicted in Fig. 7a where X_{i} degrades at a rate γ_{i} and catalytically produces X_{i+1} at a rate k_{i}. Furthermore, the output species X_{6} feeds back into X_{2} by catalytically degrading it at a rate γ_{F}. This plant is adopted from^{35}; however, to challenge our controllers in a way that demonstrates their features, the feedback degradation rate γ_{F} is chosen to be larger than that reported in^{35}, which yields a plant that is unstable when operating in an open loop, as shown in Fig. 7a. In fact, the root locus in the integral gain K_{I} ≈ k (for large η) demonstrates that this plant cannot be stabilized with a standalone aI controller, that is no matter how we tune k, the response will remain unstable. It was shown in^{35} that, for this plant, the P control is not useful. This is the case because the proportional gain K_{P} was restricted to have a positive value. One of the nice features of our proposed second and thirdorder aPID controllers is their ability to achieve negative proportional gains K_{P} (see Supplementary Equations (27) and (32)) without having to rewire, that is without switching topologically from Ntype to Ptype. This is a consequence of the inseparability of the P component from other components (I and D for the second order, and D for the third order). In Fig. 7b, c we show that, for this plant, tuning the proportional gain K_{P} to be negative is critical to achieving high performance, whereby oscillations and overshoots are almost completely removed while maintaining a fast response. This is demonstrated using the intensity plots of a performance index that quantifies the overshoot, settling time, and rise time of the output response over a range of the relevant biomolecular controller parameters. With the completely separable fourthorder aPID, the gains cannot be tuned to be negative; however, one can always switch between Ntype and Ptype topologies or even resort to hybrid designs where different PID components are of different P/Ntypes. Indeed, Fig. 7d shows that by using a fourthorder hybrid aPID controller, high performance can be attained.
To demonstrate the effectiveness of aPID control of high dimensional plants, we carry out a simulation study of cholesterol control in the plasma using the secondorder aPID controller. The mathematical wholebody model of cholesterol metabolism is adopted from^{50} that involves 34 state variables (species). The simulation description and results can be found in Supplementary Information Section 6 that demonstrates that aPID controllers are also capable of achieving high performance for high dimensional systems.
Effect of derivative control on the stationary variance
In Supplementary Information Section 9, we examine the effect of the derivative component in the various aPID controllers on the celltocell variability (e.g., stationary variance). We consider two plants: the gene expression network of Fig. 6a and the sixspecies network of Fig. 7a. The results in Supplementary Information Section 9 demonstrate that, for both plants, the third and fourthorder aPID controllers are capable of reducing the stationary variance, whereas the secondorder aPID increases it. Unfortunately, momentclosure techniques similar to the one used for the aPI controllers failed to approximate the stationary variance here. Hence, the conclusion here is based on simulations only, but seems to be consistent. Further stochastic analysis similar to^{51} that exploits linear noise approximations is left for future work.
Genetic circuit designs
Here we propose and describe a particular genetic design in Escherichia coli that realizes the thirdorder aPID controller topology presented in Fig. 5 (See Supplementary Information Section 10 for another genetic design of the secondorder aPID controller). We also perform numerical simulations using biologically realistic parameters to demonstrate the effectiveness of the controllers in ameliorating the dynamic performance. The genetic circuit is depicted in Fig. 8a where the controller circuit augments the Icontrol module (in blue), which is based on^{9}, with additional circuitry to implement additional P and D controls (in red and green). The only difference between our Icontrol module and that of^{9} is the choice of the promoter P_{RM} driving the expression of the antiσ factor (rsiW) that is activated by the transcription factor cI acting as a dual activator for P_{RM} and repressor for P_{R} (see^{52}^{,}^{53}). The P and D control modules are implemented via the Mflon protease which is capable of degrading the input species X_{1}. The additional disturbance circuit (in yellow) serves as a source of external perturbation to the closedloop circuit by degrading the regulated output X_{2}. The set of ODEs describing the deterministic dynamics is also shown Fig. 8a and the various parameters are chosen to reflect biologically realistic regimes and account for controller species dilution δ_{c} (see Supplementary Information Section 11). Figure 8b shows the simulation results for I, PI, and PID control. The responses are shown for a step change of setpoint μ/θ, which is tunable with HSL^{9}, at t = 8 h, and for a step change of disturbance Δ, which is tunable with aTc, at t = 16 h. The simulations demonstrate that the full PID controller is capable of dramatically enhancing the stability and performance by not only shaping the transient dynamics but also reducing the steadystate error that can be incurred by the dilution effect (see^{9}^{,}^{25}^{,}^{27}).
Experimental demonstration—Cyberloop implementation
To validate the performance benefits of the proposed aPID controllers, we implemented and tested our fourthorder PID controller (presented in Fig. 5) in a hybrid in vivo–in silico optogenetic platform^{54} using the rapid prototyping “Cyberloop” framework developed in^{32}. This platform provides an interface (at singlecell resolution) between real biological circuits (in vivo) in cells placed under the microscope and stochastic computer simulation (in silico) of controllers via light stimulation and fluorescence measurement. Multiple cells can be targeted and observed individually in parallel on this platform. Under the cyberloop framework, at first, individual cellular outputs are observed and quantified via fluorescence imaging and subsequent image processing. The quantified value for each cell is then fed to a stochastic simulation^{55} of controller reactions (one controller simulation per cell) that computes the light intensity (based on the controller species abundance) which the corresponding cell should be stimulated with. The light intensity data, once computed for every target cell in the microscope field of view, are sent to a specialized custombuilt projection hardware^{54} that then stimulates target cells with their corresponding light intensities in a parallel fashion, thus closing the control loop. This fluorescence measurement and subsequent light stimulation steps are repeated every fixed interval providing us with singlecell timecourse data for the controller performance and output behavior.
In our cyberloop experiments, we used a target biological circuit genetically engineered in Saccharomyces cerevisiae (previously presented and used in^{32,54}). As shown in Fig. 9a, this circuit includes an optogenetic tool for gene expression regulation designed in such a way that the transcription rate in a target cell can be changed by varying blue light intensity which the cell is stimulated with. This provides a light control over nascent RNA abundance in the cell. The nascent RNAs are engineered with multiple stem loops that can bind with available fluorescent proteins in the cell and hence, they can be observed and quantified via fluorescence imaging under the microscope. The reader is referred to^{54} and^{32} for further technical details about this target circuit and the cyberloop framework, respectively.
Following the approach in^{32}, we implemented our fourthorder PID controller network. As shown in Fig. 9b, the PID controller was capable of reducing the oscillations on both the population and singlecell levels. This is demonstrated by plotting the time response of the population average, and the power spectral density (PSD) where sharp peaks indicate singlecell oscillations^{56}. Particularly, the added derivative control action was capable of considerably enhancing the response by getting rid of the overshoot of the mean response across the cells, while simultaneously smoothing out the peak of the PSD and as a result suppressing the stochastic singlesingle oscillations.
Alternative differentiators
In Fig. 5, the antithetic integral motif is exploited to yield an antithetic differentiator; however, other integral motifs such as zeroorder^{57}^{,}^{58} and autocatalytic^{24} integrators can also be similarly exploited as depicted in Supplementary Fig. 9. These differentiators can be carefully appended to the aPI controllers of Class 1 (see Fig. 3) to obtain an alternative set of aPID controllers depicted in Fig. 10. These differentiators act on the concentration x_{L} of the output species to approximate its derivative as a rate u_{D} ≔ g(z_{3}, x_{L}). This is one of the differences between our differentiators and those proposed in^{43} where the computed derivative is encoded as a concentration of another species. Having the computed derivative encoded directly as a rate rather than a concentration is particularly convenient for controllers with a fewer number of species. Another technical difference is that our differentiators realize a derivative with a firstorder lowpass filter, whereas the differentiators in^{43} realize derivatives with a secondorder lowpass filter due to the additional species introduced. We close this section by noting that it is also possible to replace the antithetic integral motif with other integrators to design yet another collection of PID controllers (see Supplementary Fig. 10).
Discussion
This paper proposes a library of PID controllers that can be realized via BCRNs. The proposed PID designs are introduced as a hierarchy of controllers ranging from simple to more complex designs. This hierarchical approach that we adopt offers the designer a rich library of controllers that gives rise to a natural compromise between simplicity and achievable performance. At the lower end of the hierarchy, we introduce simple PID controllers that are mathematically realized with a small number of biomolecular species and reactions making them easier to implement biologically. As we move up in the hierarchy, more biomolecular species and/or reactions are introduced to push the limit on the achievable performance. More precisely, higherorder PID controllers cover a wider range of PID gains that can be tuned to further enhance performance. Of course, this comes at the price of more complex designs making the controllers more difficult to implement biologically.
In this work, we start by introducing a library of PI controllers based on the antithetic integral motif^{25} and an appended feedback control action where the input species is directly actuated by the output species. This is similar in spirit to previous works in^{33} and^{35} where the proportional control action enters the dynamics additively via a separate repressive production reaction. While this mechanism succeeds in enhancing the overall performance, we introduce other biologically relevant mechanisms, for the P component, that are capable of achieving even higher performance without risking instability and further reducing the stationary variance (see Fig. 4). However, it is shown rigorously and through simulations (see Fig. 6) that a PI controller alone is limited, while adding a D component adds more flexibility. Interestingly, it is shown that the performance of a gene expression network can be arbitrarily enhanced with full PID controllers: the PID can be tuned to achieve an arbitrarily fast response without triggering any oscillations or overshoots. This example highlights the power of full PID control. Another nice feature of PID control is the availability of various systematic tuning methods in the literature (see^{49} for example). Wellknown design tools in control theory (such as the pole placement performed in Fig. 6) can be exploited to perform the tuning in the PID parameter space instead of the biomolecular parameter space. Then the obtained PID parameters (PID gains and cutoff frequency) can be mapped by the formulas we derived (see Supplementary Information Section 4) to the actual biomolecular parameters. This approach considerably facilitates the biomolecular tuning process. It is worth mentioning that the biomolecular tuning is the easiest for the fourthorder aPID due to the separability of its components which allows tuning each PID gain separately with a different biomolecular parameter. In contrast, the lower order aPID controllers mix the various P, I, and D components and render them inseparable (see Fig. 1e) that results in each biomolecular parameter tuning multiple gains simultaneously. This is the price one has to pay for obtaining simpler designs. However, this can also be leveraged in some cases. For example, a single biomolecular parameter can tune both the integral and proportional gains simultaneously to enhance the dynamics and variance without risking instability (see the multiplicative aPI in Fig. 4). This inseparability also offers a nice advantage where the proportional gains can be tuned to be negative without having to switch topologies from Ntype to Ptype. For certain plants, achieving negative gains is critical to achieve a high performance (see Fig. 7).
We would like to point out that the proposed control structures are all designed based on linear perturbation analysis (see Supplementary Information Section 1). This is motivated by the rich set of existing tools to design and analyze linear control systems; whereas nonlinear control design and analysis is challenging and is often treated on a casebycase basis. In the linearization, the PID structures are verified and hence the dynamics behave exactly like what is expected from classical PID control. However, full nonlinear simulations are always carried out to back up the theoretical analyses and implications. Of course, the dynamical behavior of the nonlinear PID controllers may deviate from their linear counterparts when the dynamics are (initially) far from the fixed point. This is a limitation that we believe can serve as a good future research direction where small signal analysis should be extended to large signal analysis as well. Another possible future direction is to analyze the effects of dilution on the full aPID controllers in a similar fashion to the analysis carried out for Icontrollers only in^{9} and^{27}. In fact, the simulations in Fig. 8b show promising results on the roles of P and D controls in reducing the steadystate error incurred by dilution. Furthermore, in our work, we lay down a general mathematical framework for biomolecular feedback control systems that can be used to pave the way for other possible controllers in the future. We believe that research along these directions helps building highperformance controllers that are capable of reliably manipulating genetic circuits for various applications in synthetic biology and biomedicine in the same way that PID controllers revolutionized other engineering disciplines such as navigation, telephony, aerospace, etc.
Methods
Yeast strain
No new strains were engineered in this study. Strain DBY96 from^{54} was used for the cyberloop experiments. All plasmids, strains, and related details are summarized in the Key Resources Table in^{54}.
Culture media and initialization
Yeast cell cultures were started from a –80 °C glycerol stock at least 24 h prior to the experiment, and were grown in an incubator (Innova 42R, New Brunswick) at 30 °C in SD dropout medium (2% glucose, low fluorescence yeast nitrogen base (ForMedium), 5 g/L ammonium sulfate, 8 mg/L methionine, pH 5.8). The cell density was maintained at OD_{600} < 0.2 in the incubator (30 °C) for the last 12 h leading to the experiment. Approximately 400 μL of cell culture was centrifuged at 3000 RCF for 6 min, and then sufficient volume of supernatant was removed to get a concentrated culture with OD_{600} ~ 4.
Microfluidic chip loading protocol
The microfluidic chip proposed in^{54} was used in the cyberloop experiments in this study. As mentioned in^{54}, this chip is a single layer poly(dimethylsiloxane) (PDMS, Sylgard 184, Dow Corning, USA) device, attached to a cover glass (thickness: 150 mm, size: 24 mm × 60 mm, MenzelGlaser, Germany). Before loading, the PDMS device and cover glass were rinsed with acetone, isopropanol, deionized water and dried using an air gun. The chip loading protocol in^{54} was followed: using a conventional pipette, 0.4 μL of the concentrated cell solution (as described before) was loaded into each chamber of the clean and dried microfluidic chip. The cover glass was placed on top of the PDMS device and pressed down very gently, creating an electrostatic bond between the glass and the PDMS. The loaded microfluidic chip was placed onto a custombuilt microscope holder. A syringe pump (Model no. 300, New Era Pump Systems, Inc.) was used to maintain 30 μL/min of media flow through the loaded microfluidic chip. Cells were allowed to settle in the new conditions for 2 h prior to the start of any experiment.
Imaging and light delivery system
All image acquisitions were performed as described in^{54}. Briefly, images were taken under an automated Nikon TiEclipse inverted microscope (Nikon Instruments), equipped with a 40× oilimmersion objective (MRH01401, Nikon AG, Egg, Switzerland) and CMOS camera ORCAFlash4.0 (Hamamatsu Photonic, Solothurn, Switzerland). Brightfield imaging was done using LED 100 (Märzhäuser Wetzlar GmbH & Co. KG) with diffuser and green interference filter placed in the light path. Fluorescence (mRuby3) imaging was done using Spectra X Light Engine fluorescence excitation light source (Lumencor, Beaverton, USA) with 550/15 nm LED line from the light source, 561/4 nm excitation filter, HCBS573 beam splitter, 605/40 nm emission filter (filters and beam splitter acquired from AHF Analysetechnik AG, Tubingen, Germany). The microscope sample temperature was maintained at 30 °C by enclosing the microscope with an opaque environment box setup (Life Imaging Systems, Switzerland), which also shielded the cell sample from external light.
To achieve optogenetic stimulation with singlecell resolution under the microscope, a Digital Micromirror Device (DMD)based projection hardware developed in^{54} was used. An additional neutral density filter (ND 1.3, 25 mm absorptive filter from Thorlabs) was placed in the light stimulation pathway to reduce blue light intensity reaching the cells. The microscope and DMD projector was operated using an open source microscope control software YouScope^{59}.
Image analysis
In this study, each of the cyberloop experiments was run for 4 h duration with imaging/sampling done every 2 min. At every imaging step, two brightfield images above and below the focal plane (±5 a.u. Nikon Perfect Focus System) were acquired, with an exposure of 100 ms each. These images were used for cell segmentation and tracking over the course of the experiment. For nascent RNA count quantification, five fluorescence images (Z stacks with step size ~0.5 μm) were also captured, with an exposure of 300 ms each. The software tools developed in^{54} and^{32} were employed for cell segmentation, tracking and (nascent RNA) quantification. These image analysis software routines were run in MATLAB (MathWorks) environment.
Stochastic simulation of proposed controllers
The in silico simulations of the proposed biomolecular controllers were run in MATLAB (MathWorks) environment. Routines developed in^{32} were used in the cyberloop experiments. Briefly, at every sampling time, the following steps were performed:

1.
The quantified cellular readout (nascent RNA count) was used to compute and update controller reaction network propensities for every tracked cell.

2.
Gillespie’s Stochastic Simulation Algorithm^{55} was then employed to obtain the controller species abundance.

3.
These abundance values for individually tracked cells were used to compute blue light intensities (proportional to the X_{0} abundance) which the corresponding cells were stimulated with.
Stimulation of individually tracked cells was done via a light delivery system mentioned previously.
Data analysis and formatting
All data obtained from Cyberloop experiments (Fig. 9b) were analyzed and plotted using MATLAB R2018a (academic use) platform. For these experiments, data from nonresponding cells were manually removed from the analysis and further consideration. These outliers (nonresponding cells) constituted around 5–10% of total cells tracked throughout the experiment. To identify and remove these cells from our data, we first observed the temporal profile of the actuation species X_{0} abundance, which determines the blue light intensity a cell receives, for each tracked cell. Cells that were receiving constantly increasing blue light intensity and showed no appreciable increase in the output response were then removed from our final analysis. All data analyses and simulations in this work were performed on MATLAB R2018a and R2021a (academic use) platforms. Different plots and figures were structured and formatted using Inkscape (v0.92, open source), MATLAB and TexStudio (v3.1.1, open source) software.
Reporting summary
Further information on research design is available in the Nature Research Reporting Summary linked to this article.
Data availability
The raw data (MATLAB .mat files) for the results of Fig. 9b are available in the Source Data file. They are also available at https://doi.org/10.5281/zenodo.6373177^{60}. Source Data are provided with this paper.
Code availability
The MATLAB code for generating all the figures is available at a dedicated GitHub repository: https://github.com/MauriceFilo/BiomolecularPIDControl^{60}. In the same repository, a MATLAB application is also available and is capable of simulating various biomolecular controllers using a graphical user interface. The custom code for the cyberloop experiments presented in Fig. 9b is run on an integrated experimental setup and hardware^{54}, and cannot be executed without the full associated hardware–software suite. Codes for experiments and hardware configuration files are available upon request.
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Acknowledgements
This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (CyberGenetics; grant agreement 743269), and from the European Union’s Horizon 2020 research and innovation programme (COSYBIO; grant agreement 766840).
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M.K. conceived the study; M.F. and M.K. designed the controller topologies; M.F. performed the control analysis and simulations, developed the code, and wrote the initial version of the paper; S.K. performed the cyberloop experiments with input from M.F.; M.K. supervised the project and secured funding; all authors contributed to the final version of the paper.
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ETH Zürich has filed a patent application on behalf of the inventors T.F., C.H.C., M.F. and M.K. that includes the designs described (application no. EP21187316.1). The remaining authors declare no competing interests.
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Filo, M., Kumar, S. & Khammash, M. A hierarchy of biomolecular proportionalintegralderivative feedback controllers for robust perfect adaptation and dynamic performance. Nat Commun 13, 2119 (2022). https://doi.org/10.1038/s41467022296407
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DOI: https://doi.org/10.1038/s41467022296407
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