A universal biomolecular integral feedback controller for robust perfect adaptation

Abstract

Homeostasis is a recurring theme in biology that ensures that regulated variables robustly—and in some systems, completely—adapt to environmental perturbations. This robust perfect adaptation feature is achieved in natural circuits by using integral control, a negative feedback strategy that performs mathematical integration to achieve structurally robust regulation1,2. Despite its benefits, the synthetic realization of integral feedback in living cells has remained elusive owing to the complexity of the required biological computations. Here we prove mathematically that there is a single fundamental biomolecular controller topology3 that realizes integral feedback and achieves robust perfect adaptation in arbitrary intracellular networks with noisy dynamics. This adaptation property is guaranteed both for the population-average and for the time-average of single cells. On the basis of this concept, we genetically engineer a synthetic integral feedback controller in living cells4 and demonstrate its tunability and adaptation properties. A growth-rate control application in Escherichia coli shows the intrinsic capacity of our integral controller to deliver robustness and highlights its potential use as a versatile controller for regulation of biological variables in uncertain networks. Our results provide conceptual and practical tools in the area of cybergenetics3,5, for engineering synthetic controllers that steer the dynamics of living systems3,4,5,6,7,8,9.

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Fig. 1: Integral feedback enables robust perfect adaptation.
Fig. 2: Synthetic antithetic integral feedback control circuit.
Fig. 3: Growth-rate control.

Data availability

All relevant data are included as Source Data and/or are available from the corresponding author on reasonable request. Plasmid sequences are deposited in GenBank under the accession codes MK775703MK775710. Strains and plasmids used in this study are available from the corresponding author on reasonable request.

Code availability

Code used for simulations is available on reasonable request from the corresponding author.

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Acknowledgements

We thank T. Frei, C. Briat, D. Meyer and G. Schmidt for assistance with the project. This project has received funding from the Swiss National Science Foundation (31003A-149802), 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 (COSY-BIO; grant agreement 766840).

Reviewer information

Nature thanks Jeff Hasty and the other anonymous reviewer(s) for their contribution to the peer review of this work.

Author information

Affiliations

Authors

Contributions

M.K., G.L. and S.K.A. conceived the project and designed the circuit. S.K.A., G.L. and D.S. constructed the circuits. G.L. performed the computational modelling. A.G. proved the theoretical results. S.K.A., G.L. and A.B. planned and performed the experiments and data analysis. M.K., S.K.A., A.G. and G.L. wrote the manuscript. M.K. secured funding and supervised the project.

Corresponding author

Correspondence to Mustafa Khammash.

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The authors declare no competing interests.

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Extended data figures and tables

Extended Data Fig. 1 Deterministic modelling of the non-ideal antithetic integral feedback circuit.

a, Deterministic model of the non-ideal antithetic integral feedback (AIF) circuit. b, c, Output steady state and steady-state error for the non-ideal AIF circuit with dilution and negative disturbance. The black lines indicate curves of constant output steady state (in nM) for a dilution rate corresponding to a typical bacterial growth rate. The green, blue and yellow shadings denote the regions of the parameter space in which the relative output steady-state error is less than 5% for dilution rates corresponding to typical values for bacteria, yeast and mammalian cells, respectively. k = 0.037 min−1 in b and 3.85 min−1 in c. d, Output steady state and steady-state error for the non-ideal AIF circuit with dilution, negative disturbance and actuator saturation. The black lines indicate curves of constant output steady state for a dilution rate corresponding to a typical bacterial growth rate. The green, blue and yellow shadings denote the regions of the parameter space in which the relative output steady-state error is less than 5% for dilution rates corresponding to typical values for bacteria, yeast and mammalian cells, respectively. In this plot k = 1 min−1.

Extended Data Fig. 2 Dependency of the adaptation region of the non-ideal AIF circuit with dilution, negative disturbance and saturation on the actuator gain k.

The black curve has constant steady-state of 10,000 nM for a dilution rate corresponding to a typical bacterial growth rate. The green shading shows the region of the parameter space in which the relative output steady-state error is less than 5%.

Extended Data Fig. 3 Fluorescence distributions of dynamic closed-loop circuit response.

a, Example of gating strategy for flow cytometry fluorescence measurements. Cells are first gated manually from background noise using a FSC-H–SSC-H plot (left). The gated cells are then selected for singlets using a SSC-width–SSC-H plot (right). The percentage of events within each gate is indicated. b, sfGFP fluorescence distributions for the dynamic step-response experiment in Fig. 2d as a function of time. The vertical orange line represents the mean. The vertical dashed grey line represents the mean of the background distribution, which was obtained by measuring non-fluorescent SKA703 cells containing pSKA570 and pSKA417. Data are representative of three biological replicates.

Extended Data Fig. 4 AIF circuit with sfGFP output does not affect cell growth rate.

a, Dynamic growth rates for closed-loop step responses presented in Fig. 2d. Circles indicate the independent biological replicates (n = 3) and the average of the points is fit with a cubic spline. Shaded regions indicate the s.d. b, Steady-state growth rates for closed- and open-loop circuits in Fig. 2e. Data show mean ± s.d. (n = 3). Circles indicate the independent biological replicates. c, Dynamic growth rates for closed- and open-loop circuits in Fig. 2f. Circles indicate the independent biological replicates (n = 3) and the average of the points is fit with a cubic spline. Shaded regions indicate the s.d. Source Data

Extended Data Fig. 5 MfLon protease-mediated degradation of AraC and sfGFP is closely matched.

Protein levels of V5-tagged AraC with a Pdt#3c degradation tag with and without aTc-induction of Mflon were compared to fluorescence levels of sfGFP with a Pdt#1 tag. a, Mean sfGFP fluorescence of the cultures used for lysate preparation. sfGFP fluorescence normalized to the predisturbance level. Error bars denote standard deviation (n = 3). Circles denote the individual replicates. b, Means of V5-tagged AraC immunoblot band intensities normalized to total protein. Band intensity normalized to the pre-disturbance level. Error bars denote standard deviation (n = 3). Circles denote the individual replicates. c, Immunoblot. Left, total protein on membrane post-transfer. Right, V5-AraC-Pdt#3c immunoblot. V5-AraC-Pdt#3c is indicated with an arrow. SKA703 was included as a no-plasmid negative control (neg ctrl). For immunoblot source data, see Supplementary Fig. 1. Data are representative of three independent experiments. Source Data

Extended Data Fig. 6 Output steady-state error of the AIF circuit.

a, Steady-state error simulations of the non-ideal AIF circuit with dilution, negative disturbance and actuator saturation from Extended Data Fig. 1d. The black lines indicate curves of constant output steady state (in nM) for a dilution rate corresponding to a typical bacterial growth rate. The green, blue and yellow shading denotes the regions of the parameter space in which the relative output steady-state error is less than 5% for dilution rates corresponding to bacteria, yeast and mammalian cells, respectively. In this plot k = 1 min−1. b, Non-normalized output steady states of closed- and open-loop circuits in vivo in the presence of MfLon protease perturbation from Fig. 2e and four additional HSL and ARA conditions. Data show mean ± s.d. for n = 3 independent biological replicates (grey circles). Two-tailed, unpaired, unequal variance t-test. c, Heat map of the steady-state error for the closed-loop and open-loop data presented in b. d, Matched output steady states of the closed-loop circuit with 0.2% ARA and 7 nM HSL and the open-loop circuit with 0.2% ARA and 0 nM HSL. Data show mean ± s.d. for n = 3 independent biological replicates (grey circles). Two-tailed, unpaired, unequal variance t-test. e, Fluorescence distributions for the closed- and open-loop circuits in b. Data are representative of three independent experiments. f, Non-normalized output for the closed- and open-loop dynamic disturbance rejection experiment presented in Fig. 2f. Mean sfGFP is plotted as a function of time and fit with a cubic spline for n = 3 independent biological replicates. Shaded regions indicate the s.d. Source Data

Extended Data Fig. 7 Application of a negative perturbation reduces the output variance of closed-loop and open-loop circuits.

sfGFP fluorescence variance for the closed-loop and open-loop circuits in Fig. 2e. Data show mean ± s.d. for n = 3 independent biological replicates (grey circles). Source Data

Extended Data Fig. 8 Additional growth-rate control data.

a, Non-normalized growth-rate data for the closed- and open-loop growth-rate control circuits in Fig. 3d induced with 0.2% ARA and 10 nM HSL, an open-loop growth-rate control circuit with reduced-plasmid-copy number (pSKA571-p15A, open loop (p15A)) induced with 0.2% ARA and 0 nM HSL, and a metE+ wild-type strain (SKA703, wild type) containing an empty plasmid vector induced with 0.2% ARA and 10 nM HSL. Closed- and open-loop circuits were placed in a ΔmetE host strain for testing. All strains were grown in methionine-free medium at 37 °C and 30 °C and steady-state growth rates were determined. Data show mean ± s.d. for n = 3 independent biological replicates (grey circles). Two-tailed, unpaired, unequal variance t-test. b, Cell concentration plotted over time for the closed-loop circuit (closed circles), the open-loop circuit (open circles) and the reduced-plasmid-copy open-loop circuit (pSKA571-p15A, open triangles) in a and Fig. 3d. Grey circles indicate the independent biological replicates (n = 3) and the average of the points is fit with a cubic spline. Error bars indicate the s.d. c, Cell concentration plotted over time for the wild-type strain from a. Grey circles indicate n = 3 independent biological replicates and the average of the points is fit with a cubic spline. Error bars indicate the s.d. d, Example of gating strategy for absolute cell counts. Cells and beads are gated manually from background noise using a FSC-H/SSC-H plot (top). The bead population can be seen in a magnified view (bottom). The percentage of events within each gate is indicated. Source Data

Extended Data Fig. 9 Plasmids used in this study to construct host strain SKA703.

a, Suicide plasmids used to delete the ARA, lactose and rhamnose metabolic pathways from MG1655. The disrupted gene region and flanking upstream and downstream genomic sequences were inserted into suicide vector pRE112. The remaining deletion scar is indicated in red. b, λ-Integration plasmid used for the chromosomal integration of constitutively expressed lacYA177C. c, Tn7-integration plasmid used for the chromosomal integration of constitutively expressed tetR.

Extended Data Fig. 10 Circuit plasmids used in this study.

a, Mflon perturbation plasmid. b, Synthetic-circuit precursor plasmids. c, Closed- and open-loop plasmids. d, Closed- and open-loop growth-rate control plasmids.

Supplementary information

Supplementary Information

Supplementary Notes for Modeling (Section S1) and The Universality of antithetic feedback controllers for robust perfect adaptation of noisy biomolecular networks (Section S2); Supplementary Methods (Section S3); Supplementary Table S1; Supplementary Table S2.

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Uncropped Western blot scans with size markers indications.

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Aoki, S.K., Lillacci, G., Gupta, A. et al. A universal biomolecular integral feedback controller for robust perfect adaptation. Nature 570, 533–537 (2019). https://doi.org/10.1038/s41586-019-1321-1

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