A genetic mammalian proportional-integral feedback control circuit for robust and precise gene regulation

Abstract

The processes that keep a cell alive are constantly challenged by unpredictable changes in its environment. Cells manage to counteract these changes by employing sophisticated regulatory strategies that maintain a steady internal milieu. Recently, the antithetic integral feedback motif has been demonstrated to be a minimal and universal biological regulatory strategy that can guarantee robust perfect adaptation for noisy gene regulatory networks in Escherichia coli. Here, we present a realization of the antithetic integral feedback motif in a synthetic gene circuit in mammalian cells. We show that the motif robustly maintains the expression of a synthetic transcription factor at tunable levels even when it is perturbed by increased degradation or its interaction network structure is perturbed by a negative feedback loop with an RNA-binding protein. We further demonstrate an improved regulatory strategy by augmenting the antithetic integral motif with additional negative feedback to realize antithetic proportional-integral control. We show that this motif produces robust perfect adaptation while also reducing the variance of the regulated synthetic transcription factor. We demonstrate that the integral and proportional-integral feedback motifs can mitigate the impact of gene expression burden, and we computationally explore their use in cell therapy. We believe that the engineering of precise and robust perfect adaptation will enable substantial advances in industrial biotechnology and cell-based therapeutics.

Keywords: proportional–integral feedback; robust perfect adaptation; synthetic gene circuits; variance reduction.

Fig. 1.

The antithetic PI feedback motif. (A) Network topology of an arbitrary molecular network interacting with an antithetic PI feedback motif. The nodes labeled with Z₁ and Z₂ together compose the antithetic motif responsible for realizing integral feedback. Species Z₁ is produced at a rate μ and is functionally annihilated when it interacts with species Z₂ at a rate η. Furthermore, Z₁ interacts with the controlled network by promoting the production of species X₁. To close the feedback loop, species Z₂ is produced at a reaction rate that is proportional to θ and the regulated output species X${}_{\text{L}}$. An additional negative feedback from the output to the production reaction extends the motif to PI feedback. (B) Dynamics of the antithetic integral controller. Subtracting the differential equations of Z₁ and Z₂ reveals the integral action of the controller that ensures that the steady state of the output converges to a value that is independent of the controlled network parameters. Additionally, through linearization (36), the individual integral and the proportional control actions of the antithetic PI motif can be expressed separately. (C) The elements of PI feedback. Without any feedback control, the output of the controlled network may be highly variable and will likely respond drastically to a disturbance in the network. By adding integral feedback, it can be assured that the output will adapt perfectly to disturbances. Conversely, by adding proportional feedback, the variability in the output can be reduced. Combining the two types of feedback reduces the variability of the output while also ensuring perfect adaptation. (D) Graphical demonstration of integral and proportional control. Integral control accounts for error history by mathematically integrating it in time. Consequently, integral controllers have memory and “remember” the past. However, proportional controllers act instantaneously by only accounting for the present error. Consequently, proportional controllers are memoryless and “forget” the past.

Fig. 2.

Perfect adaptation of a synthetic antithetic integral feedback circuit in mammalian cells. (A) Genetic implementation of open- and closed-loop circuits. Both circuits consist of two genes, realized on separate plasmids. The gene in the activator plasmid encodes the synthetic transcription factor tTA (tetracycline transactivator) tagged with the fluorescent protein mCitrine and a chemically inducible degradation tag (SMASh). Its expression is driven by a strong constitutive promoter (P${}_{\text{EF}-1}\alpha$). The gene in the antisense plasmid expresses the antisense RNA under the control of a tTA responsive promoter (P${}_{\text{TRE}}$). In the open-loop configuration, the TRE promoter was exchanged for a noncognate promoter. In this setting, the controlled species is the tTA protein, which can be perturbed externally by addition of ASV, the chemical inducer of the SMASh degradation tag. Another type of (internal) perturbation is introduced by adding a negative feedback in the controlled network. In particular, a negative feedback loop from tTA-mCitrine to its own production was added by expressing the RNA-binding protein L7Ae under the control of a tTA-responsive TRE promoter. This protein binds to the kink-turn hairpin on the sense mRNA to inhibit the translation of tTA. (B) Steady-state levels of the output (mCitrine) for increasing plasmid ratios. The genetic implementation of the closed-loop circuit as shown in A was transiently transfected at different molar ratios (setpoint $≔$ activator/antisense) by varying the concentration of the activator plasmid while keeping the concentration of the antisense plasmid constant. The data are normalized to the lowest setpoint (1/16). This shows that increasing the plasmid ratio increases the steady-state output level. Unpaired two-sided T test with regard to lowest setpoint (1/16). P value: **** < 0.0001, *** < 0.0005, ** < 0.005, * < 0.05. (C) Steady-state response of the open- and closed-loop implementations to induced degradation by ASV. The genetic implementation of the open- and closed-loop circuit as shown in A was transiently transfected at different molar ratios and perturbed with 30 nm of ASV. The data were normalized to the unperturbed conditions for each setpoint separately. Unpaired two-sided T test with regard to No ASV Disturbance condition for all setpoints shown, for open and closed loop. P value: **** < 0.0001, *** < 0.0005, ** < 0.005, * < 0.05. (D) The closed-loop circuit is not affected by the topology of the regulated network. The closed- and open-loop circuits were perturbed by cotransfecting the network perturbation plasmid and by adding 30 nm of ASV. This was done at two setpoints: 1/4 and 1/2 (setpoint $≔$ activator/antisense). The data are normalized to the unperturbed network and no ASV condition. Unpaired two-sided T test with regard to No ASV Disturbance and No Network Perturbation condition for all setpoints shown, for open and closed loop. P value: **** < 0.0001, *** < 0.0005, ** < 0.005, * < 0.05. For all the data, the HEK293T cells were measured using flow cytometry 48 h after transfection, and the normalized data are shown as mean ± SE for n $=$ 3 technical replicates. The unnormalized data are shown in SI Appendix, Figs. S8 and S9 and provided in separate files.

Fig. 3.

A PI controller. (A) Genetic implementation of a PI controller. A negative feedback loop from the RNA-binding protein L7Ae (which is proxy to tTA-mCitrine since it is simultaneously produced from the same mRNA) is added to the antithetic motif. This protein binds in the 5$\prime$ untranslated region of the sense mRNA species to inhibit the translation of tTA and itself simultaneously. Stronger proportional feedback is realized by adding additional L7Ae-binding hairpins. (B) A PI controller does not break the adaptation property. The P and PI circuits were implemented by adding a negative feedback loop from L7Ae to the open- and closed-loop circuits. All circuits were perturbed by adding 30 nm of ASV. The HEK293T cells were measured using flow cytometry 48 h after transfection, and the data are shown as mean per condition normalized to the unperturbed (no ASV) condition ± SE for n $=$ 3 technical replicates. Unpaired two-sided T test with regard to No ASV and No P-Control condition for all setpoints shown, for open and closed loops. P value: ** < 0.005, * < 0.05. (C) PI control reduces the steady-state variance. Computing the normalized coefficient of variation squared on the steady-state flow cytometry distributions reveals a reduction in variation in the presence of proportional feedback. The coefficients of variation squared were normalized to the No P-Control condition for both setpoints and are shown ± SE for n $=$ 3 technical replicates. Unpaired two-sided T test with regard to No P-Control condition for all setpoints shown. The unnormalized data are shown in SI Appendix, Figs. S10 and S11 and provided in a separate file.

Fig. 4.

Mathematical modeling of the various circuits. (A) A chemical reaction network compactly modeling the various circuits presented in Figs. 2 and 3. The sense mRNA, Z1, is constitutively produced at a rate $\mu (G_1)$ that depends on the gene (plasmid) concentration, G₁. Then, Z1 is translated into a fusion of a synthetic transcription factor, fluorescent protein, and inducible-degradation tag, referred to as X1, at a rate k. X1 is either actively degraded by the ASV disturbance D at a rate $\lambda (X_1;D)$ or converted to X2 at a rate c by releasing the SMASh tag. The protein X2 dimerizes to form A, which activates the transcription of the antisense RNA, Z2. The transcription rate, denoted by θ, is a function of A and the gene concentration G₂. The antithetic integral control, shown in the blue box, is modeled by the sequestration of Z1 and Z2 at a rate η. Note that the open-loop circuit is obtained by removing the feedback from the regulated output A. The proportional controller (orange box) is modeled by producing the protein $\mathbf{X}{\prime }_{\mathbf{1}}$, also at a rate k, in parallel with X1 to serve as its proxy. A negative feedback is then achieved by the (un)binding reaction between the proxy $\mathbf{X}{\prime }_{\mathbf{1}}$ and Z1. Finally, the network perturbation (purple box) is modeled by introducing an additional gene $\mathbf{G}{\prime }_{\mathbf{2}}$. This gene is activated by A to transcribe the mRNA $\mathbf{Z}{\prime }_{\mathbf{2}}$ at a rate θ_p which is a function of A and $G_2\prime$. $\mathbf{Z}{\prime }_{\mathbf{2}}$ is then translated into the protein $\mathbf{X}{\prime }_{\mathbf{1}}$ that has, once again, a negative feedback on the production of X1 by binding to Z1. See SI Appendix, Figs. S1, S3, and S4 for a detailed mathematical explanation for each separate circuit. (B and C) Model calibrations to experimental data. (Left) The model fits for the open-loop circuits with/without disturbance (B) and with/without network perturbation (C). (Right) Similarly, the model fits for the closed-loop circuits. The model fits for proportional control are reported in SI Appendix, Fig. S4C. The solid lines denote model fits, while dashed lines denote model predictions. The model fits and predictions show a very good agreement with the experiments over a wide range of plasmid ratios (setpoints) $G_1/G_2$, for all scenarios. (D) Stochastic simulations demonstrating the variance reduction property of the proportional controller. The calibrated steady-state parameter groups of the PI closed-loop circuit, given in SI Appendix, Eq. S42, are fixed, while the time-related parameters are set as follows: $\gamma =\gamma \prime ,k=c=d=1{ \text{min}}^{-1}$ to demonstrate the variance reduction property that is achieved when a proportional controller is appended to the antithetic integral motif. Note that $G_1=0.002 \text{pmol}$, and $G_2=0.004 \text{pmol}$.

Fig. 5.

Mitigating competition for shared limited resources with antithetic integral and PI feedback. (A) A genetic implementation of an antithetic integral and PI feedback circuit for mitigating the effects of limited shared resources. The antithetic integral and PI feedback circuit characterized in Figs. 2 and 3 are repurposed to mitigate the coupling of gene expression induced by shared pools of finite resources. Varying the amounts of an additional disturbance plasmid that constitutively expresses the fluorescent protein miRFP670 introduces a disturbance to the amount of available resources, which indirectly affects the expression levels of tTA-mCitrine-SMASh. (B) Steady-state rejection of disturbances to available limited shared resources. The activator plasmid and antisense plasmid for all conditions were transiently transfected at a setpoint ratio of 1/2 together with disturbance strengths varying from 0.6 to 3.5. The Top and Bottom rows, respectively, show the fluorescence of the miRFP670 disturbance and mCitrine output normalized to the lowest disturbance strength. The disturbance strength describes the amount of disturbance plasmid relative to the activator plasmid. Unpaired two-sided T test with regard to lowest disturbance strength (0.6) for all controllers, for open and closed loop. P value: **** < 0.0001, *** < 0.0005, ** < 0.005, * < 0.05. (C) Reduction in cell-to-cell variability as a result of PI feedback control. The coefficient of variation squared was computed for the first two disturbance strengths and normalized two the I-Control condition. Unpaired two-sided T test with regard to I-Control for both setpoints shown. P value: **** < 0.0001, *** < 0.0005, ** < 0.005, * < 0.05. The data are shown as the mean ± SE for n = 3 technical replicates per condition. The unnormalized data are shown in SI Appendix, Fig. S13 and provided in a separate file.

Fig. 6.

Simulation of glucose regulation in the blood with antithetic PI control. (A) A schematic representation describing the mathematical model of the closed-loop network. The diagram to the right provides a high-level description of the modeled glucose and insulin dynamics based on ref. . This diagram represents the controlled network, where the output of interest (to be controlled) is the glucose concentration (milligrams per dL) in the plasma; whereas, the input that actuates this network is the insulin concentration (picomoles per liter) in the plasma. Note that, unlike the controlled network in the previous figures, this network has a negative gain: Increasing the input (insulin) decreases the output (glucose). Hence, to ensure an overall negative feedback, a P-type controller (with positive gain) is adopted here and shown in the schematic to the left, which models a genetically embedded antithetic PI controller. The P-type property of the integrator is achieved by switching Z1 with Z2; that is, the antisense RNA is now constitutively produced while the sense mRNA “senses” the output (glucose) and actuates the input (insulin). The P-type property of the proportional controller is achieved by using an activation reaction (instead of an inhibition reaction as in Fig. 3A) where glucose activates a gene (in orange) to produce insulin. (B) Robustness to interpatient variability. To demonstrate the robustness of our PI controllers, three parameters $k_{p_1}\in [2.4,3],V_{mx}\in [0.024,0.071]$, and $k_{e_1}\in [0.0003,0.0008]$ (see ref. 49) in the controlled network are varied, while the controller parameters are fixed. Changes of $k_{p_1}$ depict alterations in endogenous glucose production [e.g., in various catabolic or stress states (53)], and V_mx is used to simulate variations in the insulin-dependent glucose utilization (U_id in ref. 49) in the peripheral tissues (e.g., by physiological or pathological changes in GLUT4 translocation), while $k_{e_1}$ is the glomerular filtration rate. The responses are shown for a meal of 40 g of glucose at t = 0. Adaptation is achieved for all these parameters and for both type I and II diabetic subjects. (C) Response to $40 \text{g}$ of glucose at time t = 0 and a disturbance in endogenous glucose production (EGP) rate at t $=$ 24 h. A single meal comprising $40 \text{g}$ of glucose and an increase of endogenous glucose production rate from $k_{p_1}=2.7 \text{mg/min}\to 3 \text{mg/min}$ (see ref. 49) is applied to the models of healthy and diabetic subjects at t = 0 h and t $=$ 24 h, respectively. Top (Bottom) depicts the response of glucose (insulin) concentration, whereas Left (Right) plots correspond to a type I (type II) diabetic subject. The black curves correspond to a healthy subject whose glucose levels quickly return back to the glycemic target range (for adults with diabetes) $[80,130] \text{mg/dl}$ (54) after the meal, due to naturally secreted insulin. In contrast, the red curves correspond to uncontrolled diabetic patients whose glucose levels are incapable of returning back to the healthy range, due to lack of insulin (type I) or low insulin sensitivity (type II). Finally, the solid gray, dashed gray, and green curves correspond to diabetic patients whose glucose levels are controlled by our integral, proportional, and PI controllers, respectively. Both integral and PI controllers are capable of restoring a healthy level of glucose concentration by tuning the setpoint to a desired value (100 mg/dL), whereas the proportional controller alone is capable of neither returning to the desired setpoint nor rejecting the disturbance. Furthermore, the PI controller outperforms the standalone integral controller by speeding up the convergence to the setpoint, especially for type I diabetes.