Systems and Control Theoretic Approaches to Engineer Robust

Systems and Control Theoretic Approaches to Engineer Robust Biological Systems by Yili Qian Submitted to the Department of Mechanical Engineering in partial fulfillment of the requirements for the degree of Doctor of Philosophy at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY September 2020 ○c Massachusetts Institute of Technology 2020. All rights reserved.

Author...... Department of Mechanical Engineering August 7, 2020

Certified by...... Domitilla Del Vecchio Professor of Mechanical Engineering Thesis Supervisor

Accepted by ...... Nicolas G. Hadjiconstantinou Chairman, Department Committee on Graduate Theses 2 Systems and Control Theoretic Approaches to Engineer Robust Biological Systems by Yili Qian

Submitted to the Department of Mechanical Engineering on August 7, 2020, in partial fulfillment of the requirements for the degree of Doctor of Philosophy

Abstract Synthetic biology is an emerging field of research aimed to engineer biological systems by inserting programmed DNA molecules into living cells. These DNAs encode the production and subsequent interactions of biomolecules that allow the cells to have novel sensing, computing, and actuation capabilities. However, most success stories to date rely heavily on trial and error. This is mainly because genetic systems are context-dependent: the expression level of a synthetic gene often depends not only on its own regulatory inputs, but also on the expression of other supposedly unconnected genes. This lack of modularity leads to unexpected behaviors when multiple genetic subsystems are composed together, making it difficult to engineer complex systems that function predictably and robustly in practice. This thesis characterizes resource competition as a form of context dependence, and presents control theoretic approaches to engineer robust, context-independent gene networks. We first present a systems framework to model resource competition, which results in a hidden layer of unintended interactions among genetic subsystems. These unintended interactions lead to failure of the composed network in experiment. We then introduce a set of biomolecular controllers - designed to solve an output regulation problem in vivo - that can decouple a genetic subsystem’s output from its context. We describe challenges applying classical control theory to engineer such controllers due to the physical constraints in living cells, and then present novel theory-guided engineering solutions. Finally, we point to additional design considerations when regulating multiple subsystems using multiple controllers in a single cell. These works have the potential to enhance the robustness of future synthetic biological systems and to fully unleash their power to address pressing societal needs in environment, energy, and health.

Thesis Supervisor: Domitilla Del Vecchio Title: Professor of Mechanical Engineering

3 4 Acknowledgments

First and foremost, I am extremely grateful to my advisor, Prof. Domitilla Del Vecchio, for her relentless support over the past seven years. Her commitment to and enthusiasm for research is second to none. She is such a role model. I’d like to thank my thesis committee members, Profs. Harry Asada, Richard Braatz, and Eduardo Sontag for their feedback on my work. I enjoy working with Prof. Ron Weiss on a number of projects and I want to thank him for his guidance. My deep appreciation goes to my friends and colleagues in the Del Vecchio lab. Many breakthroughs in my research were inspired by conversations with you. I especially want to thank Hsin-Ho and Ross for their patience in teaching me cell culture and cloning, and answering my naïve biology questions. I cannot imagine finishing this thesis without you. I am deeply indebted to Ted, for all the theoretical inspirations I gained through our discussions. I want to thank José and Abdullah for bringing my attention to the difficult research questions that lingered in mymind for the last seven years. Andras, Cameron, Carlos, Fiona, Francesca, Heejin, M.Ali., Max, M.N., Narmada, Nithin, Penny, Rushina, Shaoshuai, Simone, and Ukjin, it is my great fortune to be a colleague with you and thank you for all the support, fun, and intellectual stimulations. My undergraduate mentees Hussein and Anh have provided me many inspirations, and I enjoy working with them. I want to thank Joe for providing all the administrative supports behind the curtain. I am thankful for my friends at MIT and in the greater Boston area, especially my long-time roommates Rujian, Fangchang, and Qingkai, for all the joyful and unforgettable moments. I am heartfully indebted to my dear parents, Wei Qian and Lei Zhou. They have provided me endless support and encouragement over the years, and made many sacrifices for me to finish this thesis. I love you and I missyou. The work reported in this thesis was supported in part by AFOSR grant FA9550- 14-1-0060 and FA9550-12-1-0129, ONR award N000141310074, NSF-CMMI award 1727189, and NIH NIGMS grant P50 GMO98792.

5 6 Contents

1 Introduction 25 1.1 Overview of synthetic biology ...... 25 1.2 Robustness problems in synthetic biology ...... 29 1.3 Statement of contributions ...... 30 1.4 Thesis organization ...... 32

2 Characterization of unintended interactions in genetic circuits 35 2.1 Introduction ...... 35 2.2 General modeling framework ...... 37 2.3 Effective interaction graphs ...... 41 2.4 Experiment: activation cascade ...... 48 2.4.1 Genetic circuit ...... 48 2.4.2 Model guided design to mitigate unintended interactions . . . 50 2.5 Summary ...... 52

3 Robustness of genetic subsystems to disturbances 53 3.1 Introduction ...... 53 3.2 Robustness to constant disturbances ...... 55 3.2.1 Physical constraints arising from cell growth ...... 56 3.2.2 Quasi-integral control (QIC) for set-point regulation ...... 58 3.2.3 Type I and II QICs to regulate genetic subsystems ...... 63 3.2.4 Type I QIC realization: phosphorylation cycle ...... 69 3.2.5 Type II QIC realization: small RNA (sRNA) interference . . . 72

7 3.3 Robust tracking performance ...... 74 3.3.1 A singular singular perturbation (SSP) problem ...... 75 3.3.2 Model reduction of SSP systems ...... 77 3.3.3 QIC tracking performance ...... 88 3.4 Experiment: sRNA-mediated QIC ...... 90 3.4.1 Genetic circuit ...... 92 3.4.2 Model guided controller tuning for set-point regulation . . . . 95 3.5 Summary ...... 101

4 Robustness of networked systems to unintended interactions 103 4.1 Introduction ...... 103 4.2 Motivating example ...... 106 4.3 Problem formulation ...... 110 4.4 Technical background: monotone systems ...... 112 4.5 Network disturbance decoupling with monotone subsystems . . . . . 115 4.6 Network disturbance decoupling with near-monotone subsystems . . . 122 4.7 Application to decentralized QIC-regulated genetic circuits ...... 128 4.8 Summary ...... 136

5 Conclusions and future work 139 5.1 Conclusions ...... 139 5.2 Extensions and future directions ...... 141

A Appendix for Chapter 2 145 A.1 Derivation of graphical rules ...... 145

B Appendix for Chapter 3 149 B.1 Modeling dilution in mass action kinetics ...... 149 B.2 Proof of Theorem 3.1 ...... 151 B.3 Counterexample: increasing only part of the controller reaction rates 154 B.4 Proof of Lemma 3.4 ...... 156 B.5 Analysis of phosphorylation-mediated type I QIC ...... 158

8 B.6 Analysis of sRNA-mediated type II QIC ...... 165 B.7 Additional simulations ...... 170 B.8 Simulation parameters ...... 172 B.9 Experimental methods ...... 173

C Appendix for Chapter 4 175 C.1 Proof of Lemma 4.3 ...... 175 C.2 Proof of Lemma 4.4 ...... 179 C.3 Proof of Lemma 4.5 ...... 180 C.4 Small-gain theorem for convergent-input-convergent-output system . . 182 C.5 Disturbance attenuation of feedback-regulated subsystems ...... 184 C.6 Lipschitz properties of subsystem characteristics ...... 186

9 10 List of Figures

1-1 Architecture of synthetic biology systems. Genetic parts (a) can be assembled to create genetic subsystems (b) that have inputs and outputs. Interconnection of genetic subsystems creates intracellular systems (e.g., genetic circuits) that has sensing, computing, and actuation capabilities. Mul- tiple populations of cells with different encoded programs can create complex synthetic ecology. The focus of this thesis is to provide systems and control theoretic underpinnings to design robust intracellular system from genetic subsystems...... 26

11 1-2 Applications of synthetic biology. (a) A cell type classifier circuit used for cancer diagnostic ex vivo [172]. A reference profile of miRNAs that are expressed in cancer cells is used to construct a genetic logic circuit realized through RNA interactions. When transfected into a cancer cell, the output of the logic circuit triggers expression of a fluorescence protein. (b) Bacteria can be engineered to be smart drug delivery vehicles [44]. A consortium of engineered bacteria is delivered to the target tumor site. Each cell contains a genetic clock, a cell lysis gene, a therapeutic protein production gene and a cell-cell communication module. The synchronized clocks control cell lysis to release the therapeutic proteins periodically. (c) Synthetic genetic circuits increase the specificity and safety of cancer immunotherapy [30]. Receptors can be engineered to trigger T cell activity when cancer cells are detected. Feedback loops can be used to regulate T cell activity to avoid side effects. (d) A synthetic lineage control circuit. By regulating the expression of three transcription factors according to a temporal pattern, hIPSCs can be reprogrammed into insulin-secreting beta-like cell for treating diabetes [136]. . 27

1-3 Thesis organization...... 33

2-1 Setup of a genetic circuit with limited resources. (a) Schematic of gene expression process in a subsystem 푖. (b) In a genetic circuit, all subsystems are connected through prescribed interactions (i.e., transcriptional regulation) and are competing for a conserved amount of resources in the host cell...... 38

2-2 Rules to draw effective interactions in any genetic circuit with resource limitation. Black solid edges represent prescribed regulatory interactions, red dashed edges represent unintended interactions due to resource limitation. If a black and a red edge have the same head and tail, we indicate their combined effects with a gray edge...... 43

12 2-3 Effective interactions arising from resource limitation in twoge- netic circuits. (a-b) Prescribed (i.e., regulatory) and effective interactions

in a single input motif. The output molecule from subsystem 1 (p1) represses subsystems 2 and 3. (c) Static I/O response of the single input

motif. The response curve is monotonically decreasing with 푦1 when there

is no resource limitation (i.e., 퐽푖 = 0). The response curve of 푦2 becomes

biphasic when resource limitation is taken into account (i.e., 퐽푖 ̸= 0). (d-e) Prescribed and effective interactions in an activation cascade. The strength of the unintended repression from subsystem 1 on subsystem 3 is determined

by the resource demand coefficient of subsystem 2(퐽2). (f) The DNA copy

number and RBS strength determines 퐽2, which subsequently leads to three

qualitatively different static I/O response from 푦1 to 푦3. The grey region represent simulation parameters that give rise to monotonically decrease response curves, the dotted region give rise to biphasic response curves, and the grid shaded region represent simulation parameters that give rise to monotonically increasing response curves...... 45

13 2-4 Failure of modular composition in a simple two-stage activation cascade. (a) The first activation stage consists of a subsystem that takes as input the transcription activator LuxR to promote production of RFP as output in the presence of AHL, resulting in a monotonically increasing static I/O response curve. Upward arrows with leftward/rightward tips represent promoters, semicircles represent RBS, and double hairpins represent terminators. The illustrative diagram composed of subsystems and edges at the upper-right corner represents regulatory interactions among species. (b) The second activation stage consists of a subsystem that takes as input the transcription activator NahR to promote production of GFP as output in the presence of SAL, resulting in a monotonically increasing static I/O response curve. (c) The two-stage activation cascade CAS 1/30 was built by connecting the subsystems in a cascade topology. Biphasic static I/O response curve (solid line) of the cascade was observed instead of the expected monotonically increasing static I/O response curve (dashed line). All experimental data represent mean values and standard deviations of populations in the steady state analyzed by flow cytometry in three independent experiments. Each plot is normalized to its maximum fluorescence value...... 49

14 2-5 Model-guided design restores the monotonically increasing static I/O response curve of the cascade. The static I/O response curves of circuits CAS 1/30 (a) and CAS 1/60 (c) were biphasic and monotonically decreasing, respectively. By reducing the RBS strength of NahR, the static I/O response curve of CAS 0.3/30 (b) became monotonically increasing, and that of CAS 0.3/60 (d) became biphasic. Further decreasing the copy number of CAS 0.3/60 to CAS 0.3/30 restored the monotonically increasing static I/O response curve. Experimental results are presented on top of the parameter space created by simulations in Figure 2-3f. Blue and green arrows represent design actions to restore the monotonically increasing static I/O response curves starting from failed cascades CAS 1/30 and CAS 1/60, respectively. Mean values and standard deviations of fluorescence intensities at the steady state are calculated from three independent experiments analyzed by flow cytometry and normalized to the maximum value ineach plot...... 51

3-1 Ideal, leaky, and quasi-integral controller motifs for set-point regulation. (a-c) Two types of ideal, leaky, and quasi-integral control motifs. The controller reactions are boxed in pink, and the rest of the circuit belongs to the process to be regulated. Dilution of the controller species are neglected in IICs. The rates of all controller reactions are increased by a factor of 1/휖 in the quasi-integral controllers. (d) Simulation results for type I and II controller motifs subject to a step disturbance...... 59

3-2 System architecture of a QIC-regulated system. (a) The QIC-regulated system. (b) The auxiliary IIC-regulated system...... 62

15 3-3 Two physical realizations of QICs. (a) Genetic circuit diagram of the phosphorylation-based quasi-integral controller. Chemical reactions realizing the controller are boxed in pink. (b) Simulation of the circuit’s response according to (3.29). A set-point input 푟 = 20 nM is applied at time 0 and a disturbance input 푤 = 0.5 is applied at 20 hr. The vertical axis represents the ratio between output 푦, defined to be proportional to 푝 (푦 = 휎푝), and set- point 푟. The dashed black line is the response of the phosphorylation-based control system assuming no dilution of the active substrate b*. The dotted blue line, the thin green line with square markers and the solid red line represent circuit’s response in the presence of nonzero substrate dilution (훾 = 1

−1 hr ) and decreasing 휖, which corresponds to increasing catalytic rates (푘푖, 푖 = 1, 2). (c) Genetic circuit diagram of the sRNA-based quasi-integral controller. (d) Simulation of the circuit’s response according to (3.32). A set-point input 푟 = 1 is applied at time 0 and a disturbance input 푤 = 0.5 is applied at 30 hr. The vertical axis represents the ratio between output 푦, defined in (3.34), and set-point input 푟. The dashed black line represents response of an ideal integral control system, where RNA decay rate 훿 is set to 0. The dotted blue line, the thin green line with square markers and the solid red line represent circuit’s responses in the presence of nonzero RNA decay rate (훿 = 3 hr−1, corresponding to half-life of about 13 mins) and decreasing 휖. Parameter 휖 is decreased by increasing the mRNA-sRNA removal rate (휃/훽). The DNA copy numbers of the regulated gene and the sRNA are increased simultaneously by a factor of 1/휖 as 휖 decreases. Simulation parameters are listed in Appendix B.8...... 70

3-4 A subsystem regulated by a type II QIC subject to a time-varying disturbance 푤(푡). The regulated process is denoted by 푃 . Other subsystems in the genetic circuit may create time-varying ribosome demand 푑(푡) that becomes a time-varying disturbance input 푤(푡) to the regulated subsystem...... 74

16 3-5 A type II QIC motif and the pole map of its linearized model. (a) Interactions among molecular species. (b) Poles of a linearized type II QIC as 휖 changes. The linearized model is (3.60) with 훼 = 훽 = 휃 = 훿 =푢 ¯ = 1. . 77

3-6 Decomposition of candidate reduced system in SSP problem. The candidate reduced system can be decomposed as two subsystems interconnected through high-gain negative feedback...... 81

3-7 Model reduction and tracking performance of a type II QIC. (a) Output tracking error and (b) model reduction error for the output variable. The dashed lines in panel (a) are for the nonlinear system (3.37) and the solid lines are for the linearized system (3.60). Simulation parameters: 훼 = 휃 = 훿 = 1, 푟(푡) =푟 ¯ = 1, 푤¯ = 0, 푤˜(푡) is a band-limited white noise disturbance input with maximum magnitude of 0.8...... 91

3-8 sRNA-mediated post-transcriptional controllers. (a) Circuit diagram of a regulated subsystem 1 with inducible subsystem 2 functioning as a resource competitor. Upwards arrow with tip rightwards indicate promoters, semicircles represent RBSs, orange box with letter B stands for sRNA-A targeting sequence, and “⊤” symbols represent terminators. The feedback controller consists of ECF32, co-transcribed with GFP, which is used to sense GFP translation rate and to actuate transcription of sRNA-A. sRNA- A antisenses its targeting sequence on the mRNA for degradation of both RNA molecules. (b) Circuit diagram of an unregulated subsystem 1 with inducible subsystem 2 functioning as a resource competitor. The pECF32 promoter and sRNA-A in panel a are removed...... 93

17 3-9 Performance of a regulated subsystem is determined by its feedback gain. (a) Increasing feedback gain enhances the robustness of a regulated subsystem driven by either the stronger Ec-TTL-P109 promoter (a square symbol) or the weaker BBa_J23116 promoter (a circle symbol). A symbol’s color filling in gray scale from light to dark represents a feedback gain from low to high. Simulation results for model (3.63a)-(3.63c) are in the insert box. (b) GFP outputs, growth rates, and RFP outputs for low-gain and high-gain regulated and unregulated subsystems using the Ec-TTL-P109 promoter. GFP per OD values in arbitrary unit (A.U.) were normalized to their respective nominal outputs. All experimental data were obtained with a microplate photometer. Data with error bars represent mean values ± standard deviations. The regulated subsystem using Ec-TLL-P109 promoter and high feedback gain has six replicates (three biological replicates each with two technical replicates). Other subsystems have three biological replicates. Specific value of an independent experiment presents as a black dot. Two-tailed, unpaired 푡-test is used to compute 푝-values...... 99

18 3-10 A regulated subsystem is more robust than an unregulated subsystem with comparable output level. (a) AHL dose responses of the regulated (black square in top panel) and unregulated (white circle in top panel) subsystems. The regulated subsystem uses a stronger Ec-TTL-P109 promoter and the unregulated subsystem uses a weaker BBa_J23116 promoter. For both circuits, the RBS TIRs of ECF32 are 6474, corresponding to high feedback gain. Error bars represent standard deviation from six replicates, including three biological replicates each having two technical replicates. Specific value of an independent experiment presents as a black dot. Two-tailed, unpaired 푡-test is used to compute 푝-values. (b) The mRNA level of mRFP1 gene was quantified by RT-qPCR and normalized to the reference gene cysG. Data represent mean values (± standard deviation). (c) Temporal responses of the unregulated and the regulated subsystems were monitored in parallel by microplate photometer. Cells were first grown in multiple batches in the absence of AHL for 8 hours till they reach steady state GFP expression. Each batch was diluted every 2 hours to maintain exponential cell growth. The resource competitor was then induced at 푡 = 0 with the indicated AHL concentrations. Sampling time interval of microplate photometer was 2 min. Mean GFP per OD values in arbitrary unit (A.U.) are normalized to those of samples without AHL induction to reflect relative change in GFP expression. Error bars represent standard deviation from three biological replicates. (d) GFP fluorescence histograms (normalized to the statistical mode) of three biological replicates measured by flow cytometry 6 hours after AHL induction...... 100

4-1 Schematic of a perturbed network 풩 . It is composed of 푁 subsystems interconnected via prescribed interaction map 퐺 and unintended interaction map Δ...... 105

19 4-2 Network disturbance decoupling for sRNA-regulated subsystems with independent inputs. (a) A genetic circuit consists of three feedback- regulated subsystems (4.3), each taking an independent but identical refer-

ence input 푟푖 = 푟0. The subsystems are coupled through unintended interactions arising from resource conservation. (b) Steady state output error (vector ∞-norm) between perturbed and nominal networks as 휖 varies. For

every 휖푖 and 푟푖, the trajectory converged to an asymptotic stable equilibrium.

−1 Subsystems have identical parameters: 훼푖 = 100 nM/hr, 휆푖 = 1 (nM · hr) , −1 −1 훿 = 1 hr , 훽푖 = 1 hr , and 휅푖 = 1 nM for all 푖...... 109

4-3 A graphical representation of the I/O gain function 휓 for system (4.11). If the input 푤(푡) ultimately enters the box set [푤−, 푤+], the output 푦(푡) ultimately converges to the box set [휓(푤−, 푤+), 휓(푤+, 푤−)]. This

푤 푥 schematic assumes system (4.11) is cooperative, that is, (휎 ; 휎 ) = (1; 1푛), and the output function 휕휌/휕푥 ≥ 0 for all 푥...... 115

4-4 Network disturbance decoupling for a genetic activation cascade. (a) Schematic of a genetic circuit composed of five feedback-regulated subsystems connected in a cascade topology. (b) Simulation results for the ∑︀ * * * ¯ network when 푖̸=푗 휂푗 (푟푗 ) < 1 and thus 푟푖 ∈ ℛ풩2 . (c) Simulation results ∑︀ * * * ¯ for the network when 푖̸=푗 휂푗 (푟푗 ) > 1 and thus 푟푖 ∈/ ℛ풩2 . Simulation pa- −1 rameters are identical for all subsystems: 훼푖 = 70 nM/hr, 휆푖 = 5 (nM·hr) , −1 −1 훿 = 0.5 hr , 훽푖 = 1 hr , 휅푖 = 10 nM, and 휖푖 = 휖 for all 푖. The prescribed

interactions follow equation (4.49) with parameters: 푛푖 = 4, 푘푖 = 6 nM, and

* and 푟1 = 10 nM/hr. For panel (b) 퐵푖 = 10 nM/hr for all 푖 ≥ 1 and for panel

20 B-1 Increasing the integral gain 푘/휖 alone may not be sufficient to diminish leaky integration effect. (a) While the integral gain 푘/휖 in (B.18) is increased, the steady state of the memory variable 푧¯ also increases. (b) The increase in 푧¯ counteract the decrease in 휖, resulting in the system’s inability to quench leaky integration effect (i.e., decrease adaptation error 푒) by increasing the integral gain. An additive external disturbance is applied at 50 hr...... 156 B-2 Adaptation error of the phosphorylation-based QIC with different

magnitudes of times-scale separation (푘푖/훾, 푖 = 1, 2) and binding affinities between the active substrate and the regulated gene (휆). Simulation parameters are listed in Table B.2...... 171 B-3 Adaptation error of the sRNA-based QIC with different gene/sRNA copy numbers and RNA complex removal rates. Simulation parameters are listed in Table B.2...... 172

21 22 List of Tables

B.1 Characteristic parameter values in E. coli bacteria ...... 173 B.2 Simulation parameters ...... 174

23 24 Chapter 1

Introduction

1.1 Overview of synthetic biology

Synthetic biology is the application of engineering principles to the fundamental components of biology, with the aim to create living systems that can be used to address pressing societal needs in health, energy, and environment [25]. This is accomplished by inserting engineered DNAs that encode the production and subsequent interactions of biomolecules into living cells, enabling them to have novel sensing, computation, and actuation capabilities. Ideally, likely other engineering systems, a synthetic biological system can be designed and constructed using a modular and bottom-up approach. Specifically, genetic parts can be assembled to create genetic subsystems that have inputs and outputs (Figure 1-1a-b). The genetic parts are DNA sequences that have specified functionalities, such as recruitment of native cellular machinery to initiate and terminate expression of an engineered gene. Different genetic parts can allow, for example, different rates at which these biochemical reactions occur [14]. Each genetic subsystem encapsulates the dynamical processes of producing a biomolecule from DNA, utilizing the cell’s built-in machinery. Typically, with reference to Figure 1-1b, the input to a subsystem is a regulatory protein r푖 that can bind with the promoter region on the DNA of subsystem 푖 to regulate the expression of output protein p푖. More recently, other forms of biomolecules such as RNAs have been engineered as inputs and outputs of genetic subsystems as well [31]. These

25 (a) genetic (b) genetic (c) intracellular (d) multicellular parts subsystems system system (e.g., genetic circuit)

cell

disturbance cellular context (wi) protein output (p ) i d w DNA mRNA input

(ri) gene

this thesis

Figure 1-1: Architecture of synthetic biology systems. Genetic parts (a) can be assembled to create genetic subsystems (b) that have inputs and outputs. Interconnection of genetic subsystems creates intracellular systems (e.g., genetic circuits) that has sensing, computing, and actuation capabilities. Multiple populations of cells with different encoded programs can create complex synthetic ecology. The focus of this thesis is to provide systems and control theoretic underpinnings to design robust intracellular system from genetic subsystems.

genetic subsystems can then be connected to create a genetic circuit (Figure 1-1c). Interconnection among subsystems is enabled through, for example, transcriptional regulation, where the output protein of one subsystem 푖 serves as an input to another subsystem 푗 to change the rate at which gene 푗 is expressed, thereby changing the abundance of protein p푗. Small scale genetic circuits composed of 2-3 subsystems first appeared in early 2000s. By choosing genetic parts to design the subsystem dynamics as well as their prescribed regulatory interactions, genetic circuits can be engineered to function as toggle switches [53], oscillators [48], or logic computers [106, 111]. By engineering cell-cell communication circuits in different cell strains, multiple populations of cells with different encoded programs can create multi-cellular systems for distributed computing or division of labor [15, 175] (Figure 1-1d). From a systems architecture standpoint, this thesis focuses on analyzing and designing subsystem dynamics and their interconnection to create genetic circuits with improved robustness, predictability, and accuracy for application.

26 (a) ex vivo cancer diagnostic

genetic logic circuit fluorescence classifier circuit output

normal cancer cell cell miRNA inputs

(b) periodic drug release periodic drug release

lysis tumor drug

delivery of engineered bacteria population control & concentration Time drug production circuits

cancer effector patient engineered cell activity T antigen activity regulator T cell T cell

(d) diabetes treatment/cell fate decision diabetes Pdx1 Ngn3 MafA patient synthetic lineage control circuits expression Time insulin

induced pancreatic beta-like cell pluripotent stem cell progenitor cell

transplantation

Figure 1-2: Applications of synthetic biology. (a) A cell type classifier circuit used for cancer diagnostic ex vivo [172]. A reference profile of miRNAs that are expressed in cancer cells is used to construct a genetic logic circuit realized through RNA interactions. When transfected into a cancer cell, the output of the logic circuit triggers expression of a fluorescence protein. (b) Bacteria can be engineered to be smart drug delivery vehicles [44]. A consortium of engineered bacteria is delivered to the target tumor site. Each cell contains a genetic clock, a cell lysis gene, a therapeutic protein production gene and a cell-cell communication module. The synchronized clocks control cell lysis to release the therapeutic proteins periodically. (c) Synthetic genetic circuits increase the specificity and safety of cancer immunotherapy [30]. Receptors can be engineered to trigger T cell activity when cancer cells are detected. Feedback loops can be used to regulate T cell activity to avoid side effects. (d) A synthetic lineage control circuit. By regulating the expression of three transcription factors according to a temporal pattern, hIPSCs can be reprogrammed into insulin-secreting beta-like cell for treating diabetes [136].

27 Applications of synthetic biology

Health. Synthetic biology can revolutionize disease diagnosis and treatment. Syn- thetic genetic circuits can sense the intracellular concentrations of multiple molecular species, carry out logic computations through biomolecular reactions, and output a visible signal (e.g., a fluorescent reporter protein) when a set of logical conditions are met. For example, these logical conditions can be specified to recognize the chemical signature of cancerous cells to trigger a number of actions [172] (Figure 1-2a). Similar circuits provide a promising approach to reduce invasive tests for diagnosis and health monitoring [35, 87]. Programmed bacteria can also serve as smart vehicles for drug delivery by lysing at the tumor site and periodically releasing therapeutic proteins to reduce tumor activity [44] (see Figure 1-2b).

Synthetic biology also provides powerful tools to program T cells, a type of body immune cells, to specifically attack cancer cells. This type of treatment, known as immunotherapy, has recently been demonstrated successful in clinical trials [30]. As shown in Figure 1-2c, synthetic receptors engineered on T cells, possibly combined with biomolecular logic gates, can identify cancer cells with high specificity. Synthetic genetic controllers may then interact with the cellular chemotaxis pathway to migrate T cells to tumor cites and regulate the duration and strength of T cell activity to protect non-cancerous cells [170].

Both theoretical and experimental study in synthetic biology enhance our understanding of natural systems, including cell differentiation and cancer biology [37, 154, 171]. For instance, such understanding can provide unprecedented tools to reprogram cell fate for regenerative medicine [136, 163]. Saxena et al. designed a reprogramming circuit that converts pancreatic progenitor cells derived from human induced pluripotent stem cells (hIPSCs) into insulin-secreting beta-like cells by strictly regulating the timing and expression of three key transcription factors in vivo [136] (Figure 1- 2d). Consequently, it has become possible to implant functional beta-cells in diabetes patients that are derived from the patient’s own tissue cells.

Environment and energy. Programming microbes to detect and report toxi-

28 cants in water, air, soil and food is one of the earliest applications of synthetic biology. To create an environmental biosensor, genes encoding the reporter proteins and proteins that carry out logic computation are artificially brought under the control of the sensory-regulatory system of the host cell [162]. This design technique has been utilized to detect TNT, heavy metals and antibiotics (see [162] for a comprehensive review). More recently, sensors that produce a dynamic output have been developed. Bacterial biosensor can be programmed to produce oscillatory fluorescence output, whose magnitude and frequency reflects the concentration of arsenic inthe environment [116]. In addition, microbes can be programmed to remove contami- nants, including heavy metals and organic pollutants for bioremediation [92, 144]. Microbes may also be programmed to convert biomass feedstock into biofuels [115], and synthetic controllers have been implemented to improve productivity [47, 176].

1.2 Robustness problems in synthetic biology

While these success stories, among many others, demonstrate the great impact that synthetic biology can have on society, they also currently remain mostly at the laboratory stage. In fact, most genetic circuits constructed nowadays rely on lengthy and ad hoc design processes that do not yet give predictable outcomes in less controlled conditions [23, 28]. This is largely because genetic circuits are context-dependent: the expression level of a gene, for instance, not only depends on its own regulatory input, but is also often affected by other unconnected genes. The origins of these problems can to some extent be traced back to molecular biology issues, such as the reliability and orthogonality of genetic parts, and intense research efforts are underway in this direction (see [101, 111], for example). Toa large extent, however, issues of robustness, reliability, and predictability are due to the complex dynamic interactions among system components and can be classified as “system-level” problems that fall beyond the scope of molecular biology. Examples include, but are not limited to, loading effects resulting from direct connectivity [72, 73, 166], for which a mathematical analysis framework was developed [42]; loading

29 effects on the chassis (cell), which in turn impacts the functionality of circuits and/or leads to mutations [29, 160]; dependence of a circuit’s function on the way DNA parts are assembled together [173]; and competition for shared resources, which creates subtle coupling among otherwise unconnected subsystems [63, 120]. Overall, due to these unintended and context-dependent effects, existing genetic subsystems are not robust enough to serve as building blocks for predictive construction of larger systems and we lack a systematic approach to manage uncertainty and complexity to scale up genetic circuits. While systems and control theory is well suited to educate the design of complex and robust dynamical systems, the physics of biomolecular processes pose many challenges unseen in classical engineering settings. The work in this thesis is mainly motivated by the practical context-dependence problem arising from resource competition and provides general theory-grounded rules for control design and system composition in genetic circuits to improve robustness.

1.3 Statement of contributions

The novelties and contributions of this thesis are summarized as follows.

∙ In Section 2.2, we develop a mechanistic model to describe the unintended interactions in genetic circuits due to limited gene expression machinery (i.e., resources). This model has the same dimension as standard textbook models for genetic circuits, where resource limitation is neglected, and is therefore simple enough to guide design. A lumped parameter called resource demand coefficient emerges from the model and can be tuned experimentally to change the strength of unintended interactions due to resource limitation.

∙ In Section 2.3, we provide simple graphical rules to draw effective interactions in a genetic circuit with limited resources. The effective interactions account for both prescribed regulatory interactions and the unintended interactions due to resource limitation. These graphical rules help understand and predict the qualitative behavior of a resource-limited genetic circuit.

30 ∙ In Section 2.4, we use the resource-limited genetic circuit model to design and test a library of genetic activation cascades in E. coli bacteria. We provide model-based rules to tune physical parameters in the cascade to change the resource demand coefficients, thereby varying the static I/O response ofthe cascade from monotonically increasing, to biphscic, to monotonically decreasing.

∙ In Section 3.2, we develop a theory-guided control design framework, based on timescale separation, for feedback controllers in living cells to achieve set-point regulation. This design task is challenged by the physical constraint that all biomolecules are diluting in growing cells, making it physically impossible to create a perfect integral controller. We call the proposed biomolecular controllers quasi-integral controllers (QICs). The set-point regulation performance of a QIC-regulated system improves as the timescale separation between controller dynamics and cell growth rate increases. We propose two biomolecular realizations based on phosphorylation cycle and small RNA (sRNA) interference.

∙ In Section 3.4, we construct and test the sRNA-mediated QIC in bacteria E.coli and demonstrate that it can asymptotically reject a disturbance induced by a step decrease in the availability of resources for gene expression.

∙ In Section 3.3, we address a novel model reduction problem, called singular singular perturbation (SSP), which arises from a class of biomolecular QICs. Such systems have a peculiar model structure, making standard singular perturbation theories inapplicable. We provide conditions on the slow and the fast dynamics for model reduction, and quantify the model reduction error in terms of the small parameter characterizing the timescale separation.

∙ In Section 4.3, we establish a novel theoretical framework to study the effect of unintended interactions on networked dynamical systems, including genetic circuits. Although unintended interactions can be modeled as disturbances for each subsystem, these disturbances are often created by other subsystems

31 in the network. We therefore formulate a new network robustness property called network disturbance decoupling (NDD). While existing works on networked systems focus on robustness to state-independent disturbances, in an NDD problem, the disturbances are state-dependent. The interconnection of disturbance signals can create unexpected network-level behaviors. We seek conditions on the subsystems, the prescribed and the unintended interactions such that the composed network’s long-term behavior is essentially independent of the presence/absence of unintended interactions.

∙ In Section 4.5-4.6, we provide sufficient conditions for NDD. These conditions are centered on the monotonicity properties of the subsystems and their static disturbance attenuation properties. These conditions guarantee robustness of the network to unintended interactions given robustness of the composed subsystems.

∙ In Section 4.7, the sufficient conditions derived for NDD is applied to guide the design of genetic circuits composed of QIC-regulated genetic subsystems. We provide mathematical conditions in terms of experimentlaly tunable parameters for a genetic circuit composed of sRNA-regulated subsystems to achieve NDD.

The works reported in this thesis provides systems and control theoretic underpinnings to analyze, design, and construct genetic circuits to meet the robustness, predictability, reliability, and accuracy requirements in future synthetic biology applications.

1.4 Thesis organization

The organization of this thesis is shown in Figure 1-3 and described as follows. In Chapter 2, we establish a modeling framework to describe unintended interactions in genetic circuits arising from resource competition, which is validated through experiments (Figure 1-3a). Chapter 3 describes a theory-guided framework to design and implement biomolecular feedback controllers in living cells for set-point regulation and tracking (Figure 1-3b). In Chapter 4, we consider the problem of composing

32 Figure 1-3: Thesis organization. feedback-regulated subsystems together to create networked dynamical systems that are robust to unintended interactions (Figure 1-3c). In Chapter 5, we summarize the results in this thesis and outline future research directions.

Related publications

Some materials in this thesis have been published.

∙ Some materials presented in this Chapter have been published in review articles [121, 164] and perspective article [165].

∙ The work presented in Chapter 2 has been published in [76, 120, 122, 124].

∙ The work presented in Chapter 3 has been published in [68, 76, 119, 124–126].

∙ Solutions to a simplified version of the problem in Chapter 4 have been published in [123, 124].

33 34 Chapter 2

Characterization of unintended interactions in genetic circuits

In this chapter, we develop and experimentally validate a general modeling framework to account for the prescribed regulatory interactions and the unintended interactions among subsystems in genetic circuits. Specifically, the unintended interactions arise from competition for limited gene expression resources provided by the host cell.

2.1 Introduction

Predicting the behavior of genetic circuits in living cells is a recurring challenge in synthetic biology [23]. Genetic circuits are often viewed as interconnections of gene expression cassettes, which we call genetic subsystems. Each subsystem is composed of core gene expression processes, chiefly transcription and translation. Here, we view each subsystem as an input/output (I/O) system that takes transcription factors (TFs) as input and gives a TF as output. The input TFs regulate the production of the output TF. Although in an ideal scenario we would like to predict the behavior of a circuit from that of its composing subsystems characterized in isolation, in reality, a subsystem’s behavior often depends on its context, including other subsystems in the same circuit and the host cell environment [28]. This fact significantly limits our current ability to design genetic circuits that behave as intended. There are a number

35 of causes to context dependence, including unknown structural interactions between adjacent genetic sequences [40], loading of TFs by target DNA sites (retroactivity) [42, 72, 104], unintended coupling between synthetic genes and host cell growth (host- circuit interaction) [29, 59, 84], and competition among synthetic genes with each other for common transcriptional and translational resources [27, 38, 63, 98, 127]. Context dependence due to structural interactions and retroactivity has been addressed by engineering insulation parts and devices [17, 93, 104, 109, 112] and that due to host-circuit interaction may be mitigated to some extent by orthogonal RNA polymerase and ribosomes [5, 26, 36, 89, 138]. By contrast, the characterization and mitigation of competition for shared resources among synthetic genes remain largely unexplored. Expression of all genes in a genetic circuit relies on a common pool of transcriptional and translational resources. In particular, the availability of ribosomes has been identified as a major bottleneck for gene expression in bacteria [20, 157,167]. When gene expression in a subsystem is activated, it depletes the pool of free ribosomes, reducing their availability to other subsystems in the network. This can potentially affect the behavior of a network altogether. Recent experimental results have demonstrated that competition for resources can couple the expression of two synthetic genes that are otherwise unconnected [27, 63]. In particular, limitation in ribosome availability has been identified as the key player in this coupling phenomenon [63]. These works further demonstrate that upon induction of a synthetic gene, the expression level of a supposedly unconnected gene on the same DNA plasmid can be reduced by more than 60%. Similar trade-offs have been observed in cell-free systems [142] and in computational models [38, 98, 127, 130].

Chapter overview

In this chapter, we seek to determine how competition for ribosomes by the subsystems constituting a genetic circuit changes the intended network’s behavior. To address this question, we perform a combined modeling and experimental study. In particular, we develop a general mathematical model that explicitly includes competi-

36 tion for ribosomes in Hill-function models of gene expression. In our models, resource demand coefficients quantify the demand for resources by each subsystem andshape the emergent static I/O response curve of a genetic circuit (Section 2.2). For general genetic circuits in bacteria, our model reveals that due to non-zero resource demand coefficients, resource competition gives rise to unintended interactions among subsystems. We give a general rule for drawing the effective interaction graph of any genetic circuit that combines both prescribed and unintended interactions (Section 2.3). To exemplify the applicability of our model and graphical rules to guide the design of genetic circuits, we construct a library of synthetic genetic activation cascades in which we tune the resource demand coefficients by changing the ribosome binding site (RBS) strength of the cascade’s genes and DNA copy number. When the resource demand coefficients are large, the static I/O response curve of thecascade can either be biphasic or monotonically decreasing. When we decrease the resource demand coefficients, we restore the intended cascade’s monotonically increasing static I/O response curve (Section 2.4).

2.2 General modeling framework

A genetic circuit is composed of 푁 genetic subsystems. Each genetic subsystem contains a series of biochemical reactions that express gene 푖 to produce a protein

p푖 as the output molecule (Figure 2-1a). In particular, the gene is first transcribed

to produce mRNA m푖 at rate 푟푖, which is then translated to produce p푖 at rate 푇푖.

Using 푚푖 and 푝푖 (italic) to represent the concentrations of species m푖 and p푖 (roman), ⊤ respectively, the state of a genetic subsystem is 푥푖 = [푚푖, 푝푖] . Based on mass-action kinetics, the dynamics of subsystem 푖 can be written as [41]:

푚˙ 푖 = 훽푖푟푖 − 훿푖푚푖, 푝˙푖 = 푇푖(푚푖) − 훾푖푝푖, (2.1)

where 훿푖 and 훾푖 are decay rate constants of the mRNA and the protein, respectively;

훽푖 is a transcription rate constant and 푇푖(푚푖) is the translation rate increasing with

37 (a) a genetic subsytem (b) a genetic circuit with limited resources and transcriptional regulation

conserved ribosomes protein (pi) yi change in ribosome ribosome availability demand input output mRNA (m ) (r ) r y i 1 (y1) i i subsystem 1 ... subsystem i

reference gene i input (ri) prescribed interactions r =G(y) subsystem i (i.e., transcriptional regulation)

Figure 2-1: Setup of a genetic circuit with limited resources. (a) Schematic of gene expression process in a subsystem 푖. (b) In a genetic circuit, all subsystems are connected through prescribed interactions (i.e., transcriptional regulation) and are competing for a conserved amount of resources in the host cell.

mRNA concentration 푚푖. The transcription rate constant 훽푖 increases with DNA copy number and the promoter strength of subsystem 푖, both of which can be tuned experimentally [14]. The output 푦푖 from a subsystem is the concentration of the active form of protein p푖. The activity of a TF p푖 may be modulated by an external small molecule I푖 that diffuses through cell membrane (see [120] for derivation):

푦푖 = 푝푖 · 퐻푖(퐼푖), (2.2)

where 퐻푖(·): R≥0 → [0, 1] is a monotonic function of 퐼푖. Hence, 퐻푖(퐼푖) determines the fraction of TF p푖 in its active form. If the activity of TF p푖 does not rely on an external inducer or if the concentration of I푖 is constant, we will set 퐻푖(퐼푖) = 1.

Transcriptional regulation

The transcription rate of a gene 푖, 푟푖, can be regulated by the concentration of active TFs in the network. This process is called transcriptional regulation [41]. When a TF p푗 binds with the promoter region on the DNA of subsystem 푖, it can either increase or decrease the transcription rate of m푖 by facilitating or inhibiting the recruitment

38 of RNA polymerase for transcription. In the former case, p푗 is an activator, in the

latter case, p푗 is a repressor. These prescribed interactions are often modeled by

푟푖 = 퐺푖(푦), (2.3)

⊤ where 푦 := [푦1, ··· , 푦푁 ] is the concentration of the active form of protein p := ⊤ 푁 [p1, ··· , p푁 ] . The function 퐺푖(·): R → [0, 1] is a nonlinear function called Hill function [41]. Specifically, based on mass-action kinetics, when subsystem 푖 takes a

single TF input whose concentration is 푦푗, the Hill function 퐺푖(푦푗) takes the following form [41]:

⎧ 푛푖푗 휋푖+(푦푗 /푘푖푗 ) ⎪ 푛푖푗 , if p푗 is an activator ⎨ 1+(푦푗 /푘푖푗 ) 퐺푖(푦푗) = (2.4) ⎪ 1 ⎩ 푛푖푗 , if p푗 is a repressor, (푦푗 /푘푖푗 )

where 푘푖푗 is the dissociation constant of TF p푗 binding with the promoter site of subsystem 푖. The stronger the binding affinity, the smaller the dissociation constant.

Parameter 푛푖푗 is the binding Hill coefficient, and 휋푖 ∈ [0, 1) characterizes basal ex-

pression (i.e. expression when 푦푗 = 0). If the transcription activity of subsystem 푖

is not responsive to any TF in the network, we set 퐺푖(푦푖) ≡ 1. For this scenario,

equivalently, we say gene 푖 is constitutive or protein p푖 is produced constitutively. The above descriptive framework has become standard practice to design 퐺 to obtain prescribed circuit behavior, such as genetic oscillators, toggle switches, and logic gates [48, 53, 111].

Translation dynamics

Translation of mRNA m푖 occurs with a ribosome molecule R binding with m푖 to form

a complex M푖, which is then translated at rate constant 훼¯푖 to produce protein p푖. These reactions can be written as the following chemical reactions:

휅+ i 훼¯i 훾i mi + R Mi, Mi −→ mi + R + pi. pi −→ . (2.5) − ∅ 휅i

39 With the chemical reactions in (2.5), the protein p푖 dynamics in (2.1) can be written more specifically as:

푚푖 푚˙ 푖 = 훽푖푟푖 − 훿푖푚푖, 푝˙푖 =훼 ¯푖푅 − 훾푖푝푖, (2.6) 휅푖

− + where 휅푖 := 휅푖 /휅푖 is the dissociation constant of ribosome binding with the ribosome binding site (RBS) on m푖. A smaller 휅푖 indicates stronger binding of ribosome with m푖. Computational tools exist to design genetic sequences to change the binding strength, and hence 휅푖 [134]. In a standard gene expression model [4, 41], the free amount of ribosomes 푅 available for translation is assumed to be constant. However, in a genetic circuit where multiple subsystems all require the same pool of ribosomes for translation, ribosomes become limited, and the constant free ribosomes assumption used in standard textbook models fails. Since the host cell produces a limited amount of ribosomes [20, 167], resource availability depends on the extent to which different subsystems in the network demand them (see Figure 2-1b). We use 푅푇 , 푅 and 푅푖 to denote the concentrations of total ribosomes, free ribosomes, and that of ribosomes bound to mRNA transcripts in subsystem 푖, respectively. In particular, according to (2.5), the ribosomes bound to subsystem 푖 is

푚푖푅 푅푖 = 푀푖 = , (2.7) 휅푖

which depends on the free ribosome concentration (푅), the concentration of mRNA transcripts in the subsystem (푚푖), and the binding affinity between them휅 ( 푖). On the other hand, in each cell, the concentration of ribosomes follows the conservation law [20]

푁 푁 ∑︁ ∑︁ 푅푇 = 푅 + 푅푖 = 푅 · (1 + 푚푖/휅푖), (2.8) 푖=1 푖=1 indicating that the total ribosome concentration is the summation of its free concentration (푅) and its concentration bound in the subsystems (푅푖). From equation (2.8),

40 the free concentration of ribosomes 푅 can be determined as

푅 푅 = 푇 . (2.9) ∑︀푁 1 + 푖=1 푚푖/휅푖

Replacing the constant free ribosome concentration in (2.6) with the state-dependent amount derived in (2.9) and substituting in (2.3) for transcriptional regulation, the subsystem dynamics in (2.6) can be modified as

(푚푖/휅푖) · 퐻푖(푖푖) 푚˙ 푖 = 훽푖퐺푖(푦) − 훿푖푚푖, 푦˙푖 = 훼푖 ∑︀ − 훾푖푦푖, (2.10) 1 + 푚푖/휅푖 + 푗̸=푖 푚푗/휅푗 where 훼푖 :=훼 ¯푖푅푇 . In the next section, we study the practical implications of model (2.10). In particular, we demonstrate that the model implies that, in addition to prescribed regulatory interactions (2.3), unintended interactions arise due to ribosome competition in genetic circuits.

2.3 Effective interaction graphs

Synthetic biologists often analyze and design genetic circuits based on interaction graphs, which use directed edges to represent regulatory interactions among subsystems. In a standard interaction graph, each node represents a subsystem, and we draw

푗 → 푖 if TF pj, the output protein of subsystem 푗, activates subsystem 푖 by increasing

the rate of production of pi. We draw 푗 − 푖 if TF pj represses production of pi. In this [ section, we expand the concept of interaction graph to incorporate unintended interactions due to resource limitation. We call the resultant graph the effective interaction graph. We then provide simple rules to draw the effective interaction graphs based on prescribed regulatory interactions, and illustrate their applications to predict the behavior of two simple genetic circuits.

Since the half-life of a mRNA in bacteria is ∼ 5−20 minutes and that of a protein is ∼ 1 hr [103], the mRNA dynamics are much faster than those of the proteins. Hence, we draw the interaction graph by setting the mRNA dynamics in (2.10) to

41 quasi-steady state [4, 41] to obtain the following scalar subsystem dynamics:

¯ 푇푖퐺푖(푦) · 퐻푖(퐼푖) 푦˙푖 = ∑︀ −훾푖푦푖, (2.11) 1 + 퐽푖퐺푖(푦) + 푘̸=푖 퐽푘퐺푘(푦) ⏟ ⏞ 퐹푖(푦,퐼):subsystem 푖 effective production rate

¯ where parameter 푇푖 := 훼푖훽푖/훿푖휅푖 represents the maximum protein production rate and parameter

퐽푖 := 훽푖/훿푖휅푖 (2.12)

represents the resource sequestration capability in subsystem 푖, which we call its resource demand coefficient. This resource demand coefficient increases with the transcription rate constant 훽푖, the RBS strength 1/휅푖, and decreases with the mRNA decay rate constant 훿푖. Hence, 퐽푖 can be experimentally decreased by decreasing the DNA copy number encoding gene 푖, reducing its promoter, RBS strength, or enhancing its mRNA decay [14]. Compare (2.11) with a standard textbook model that does not account for resource competition:

¯ 푦˙푖 = 푇푖퐺푖(푦) · 퐻푖(퐼푖) − 훾푖푦푖, (2.13)

the effective production rate of subsystem 푖, 퐹푖(푦, 퐼), now encapsulates the joint effects of regulatory interactions and unintended interactions due to resource competition. While regulatory interaction on subsystem 푖 is characterized by 퐺푖(푦), resource competition effects on the subsystem is described by a common denominator ∑︀푛 퐷(푦) := 1 + 푖=1 퐽푖퐺푖(푦), which is not present in the standard model (2.13).

In particular, the prescribed interaction from subsystem 푗 to 푖 is determined by sign(휕퐺푖/휕푦푗). If sign(휕퐺푖/휕푦푗) > 0, then the regulatory interaction is an activation

(푗 → 푖); if sign(휕퐺푖/휕푦푗) < 0, then the regulatory interaction is a repression (푗 − 푖). [ We define unintended interaction due to resource competition from subsystem 푗 to any subsystem 푖 to be determined by sign(휕퐷/휕푦푗). Specifically, since 퐷(푦) appears in the denominator of 퐹푖(푦, 퐼), it is an unintended activation (푗 → 푖) if sign(휕퐷/휕푦푗) < 0,

42 i ̸= 0 i 푗 /휕푦 (effective ) 푖 푦 [ ( 푖 − 푗

휕퐺 activation is undetermined, if . In particular, we is not its target ) i 푗 i that j to any from

푗 combined/weakened p j j is determined based on

/휕푦 undetermined rule 3: effective interactions rule 3: effective 푖 푖 휕퐹 (c) ( . 0 , and we draw 0 > ) of subsystem if sign > 푗 i i k 푖 k ) 푗 to subsystem

( hidden activation hidden 푗 target

/휕푦 repression 푗 푖 = 휕퐷/휕푦 ( j 43 j 휕퐹 , then the nature of effectiveinteraction from is a ( 푖 , based on prescribed interactions originating 푖 푖 affects the production rate of ) if sign rule 2: effective interactions rule 2: effective to its multiple targets j to its multiple from 푖 j [ p . We draw − 0 (b) 푗 to affect from subsystem < 푗 )

푗 activation (i.e. activation vs. repression) is identical to the regulatory /휕푦 푖 i 푖 i i i Black solid edges represent prescribed regulatory interactions, red has multiple targets, then the effectiveinteraction from subsys-

has only one target regulatory activation regulatory 푗 to 휕퐹 푗 ( . We say subsystem 푗 = = 푗 , representing how j ) j j j Rules to draw effectiveinteractions any in genetic circuit with re- (effective activation) if sign 푗 푖 . These graphical rules are summarized in Figure 2-2, and stated as follows. 푦 effective interaction

/휕푦

to its only target target only its to j from 푖

→ rule 1:effective interactions 1:effective rule 푗 interaction. However, the strength of such interactionthat is predicted weakened by compared a to standard model (see Figure 2-2a). subsystem 휕퐹 ( The 2. If subsystem 1. If subsystem (a) sign Figure 2-2: source limitation. dashed edges represent unintendedred interactions due edge to have the resource same limitation. head If and a tail, black we and indicate a and their a combined unintended effects repression with ( gray a edge. for some draw repression) if sign that is, it depends on parameters and/orBased the on steady (2.11), state we derive the a network set is of operating graphicaloriginating at. rules from to subsystem determine the effectiveinteraction from subsystem tem 푗 to its targets are undetermined (see Figure 2-2b).

3. If subsystem 푖 is not a target of 푗, and pj is an activator (repressor), then pj is

effectively repressing (activating) pi (see Figure 2-2c).

Derivations of these rules can be found in Appendix A.1.

Application examples

As application examples of the graphical rules and the model in (2.11), we consider two example genetic circuits shown in Figure 2-3.

Single input motif

Figure 2-3a shows a simple single-input motif [4], where a TF p1 produced in subsystem 1 represses two downstream subsystems 2 and 3. The output of subsystem 1 is the active form of a TF p1, whose concentration is 푦1 = 푝1퐻1(퐼1). Since we have control over the value of 푦1 via 퐼1, we do not consider unintended interactions acting on subsystem 1.

If we assume there is no resource limitation, the steady state outputs 푦2 = 푝2 and

푦3 = 푝3 should decrease with 푦1. However, in resource-limited cells, according to rule 2, the effective interactions from subsystem 1 to its targets are undetermined (Figure

2-3b). In particular, according to (2.11), the dynamics of 푦2 and 푦3 can be written as: ¯ 푇2퐺2(푦1) 푦˙2 = −훾2푦2, 1 + 퐽1 + 퐽2퐺2(푦1) + 퐽3퐺3(푦1) ⏟ ⏞ 퐹2(푦1) ¯ (2.14) 푇3퐺3(푦1) 푦˙3 = −훾3푦3, 1 + 퐽1 + 퐽2퐺2(푦1) + 퐽3퐺3(푦1) ⏟ ⏞ 퐹3(푦1)

where 퐽푖 are the resource demand coefficient of subsystem 푖. For 푖 = 2, 3, while

the production rates of 푦푖, 퐹푖, are both increasing functions of 푦1 when 퐽푖 = 0, the

sign of 휕퐹푖/휕푦1 is undetermined for 퐽푖 ̸= 0. Simulation of (2.14) confirms this in

Figure 2-3c, where we show that steady state 푦2 indeed increases with 푦1 for low

44 (a) prescribed interactions (d) prescribed interactions (e) effective interactions 1 2 1 2 3 1 2 3 3 (b) effective interactions 1 2 simulation: different parameters lead to (f) qualitatively different static I/O responses 1 3 10 (c) static I/O response

[%] 100 2 ) 2 0 0 normalized y / κ

y1 [nM] 101

100 ( κ 2 [%] 3

Larger J

0 -1 3

normalized y 10 10 y1 [nM] w/o resource limitation (J =0) subsystem 2 RBS strength -1 i 10 10 20 30 40 50 60 70 80

w/ resource limitation (Ji≠0) DNA copy number (D1,T = D2,T = D3,T)

Figure 2-3: Effective interactions arising from resource limitation in two genetic circuits. (a-b) Prescribed (i.e., regulatory) and effective interactions in a single input motif. The output molecule from subsystem 1 (p1) represses subsystems 2 and 3. (c) Static I/O response of the single input motif. The response curve is monotonically decreasing with 푦1 when there is no resource limitation (i.e., 퐽푖 = 0). The response curve of 푦2 becomes biphasic when resource limitation is taken into account (i.e., 퐽푖 ̸= 0). (d-e) Prescribed and effective interactions in an activation cascade. The strength of the unintended repression from subsystem 1 on subsystem 3 is determined by the resource demand coefficient of subsystem 2 (퐽2). (f) The DNA copy number and RBS strength determines 퐽2, which subsequently leads to three qualitatively different static I/O response from 푦1 to 푦3. The grey region represent simulation parameters that give rise to monotonically decrease response curves, the dotted region give rise to biphasic response curves, and the grid shaded region represent simulation parameters that give rise to monotonically increasing response curves.

45 concentrations of 푦1. Physically, such unexpected behavior is due to the fact that as

푦1 increases to repress expression of p3, ribosomes bound to subsystem 3 are released to effectively facilitate production of p2, increasing its concentration. Hence, this unexpected behavior in Figure 2-3c is most prominent when 퐽3 is large.

Activation cascade

Figure 2-3d-e show the prescribed regulatory interactions and the effective interactions in an activation cascade in the presence of resource competition. Similarly, we assume that p1 is produced constitutively and an external inducer I1 is used to modulate 푦1, the concentration of active p1. We do not draw interactions for subsystem 1 because the level of 푦1 can be externally regulated.

According to Rule 3, the unintended interactions create a feed-forward edge 1 − 3 [ and a negative auto-regulation edge 2 − 2. Since subsystems 2 and 3 are the only [ targets of p1, by Rule 1, the effective interactions from subsystem 1 to 2 and from subsystem 2 to 3 are both activation. In the presence of unintended interactions, the topology of this activation cascade effectively becomes a type 3 incoherent feedforward loop (IFFL) [4], where p3 production is jointly affected by regulatory activation from p2 and unintended repression from p1. It is well-known that, depending on parameters, the static I/O response curve of an IFFL can be monotonically increasing, decreasing or biphasic [78, 82]. As we increase increase 푦1, if transcriptional activation 1 → 2 → 3 is stronger than unintended repression 1 − 3, then the static I/O response [ curve is monotonically increasing. Conversely, if the unintended repression is stronger than transcriptional activation, the static I/O response curve becomes monotonically decreasing. Biphasic responses can be expected when transcriptional activation domi- nates at lower inducer levels, and resource-competition-induced unintended repression becomes more significant at higher inducer levels.

More specifically, based on (2.10), in the presence of ribosome limitation, the

46 model of this activation cascade can be written as:

¯ 푇2퐺2(푦1) 푦˙2 = − 훾2푦2, 1 + 퐽1 + 퐽2퐺2(푦1) + 퐽3퐺3(푦2) ¯ 푇3퐺3(푦2) (2.15) 푦˙3 = −훾3푦3. 1 + 퐽1 + 퐽2퐺2(푦1) + 퐽3퐺3(푦2) ⏟ ⏞ 퐹3(푦)

The strength of the unintended interaction from subsystem 1 on subsystem 3 can be computed by:

⃒ ⃒ ¯ ⃒휕퐹3 ⃒ ¯ 퐺3 휕퐺2 푇3퐽2 휕퐺2 ⃒ ⃒ = 푇3퐽2 · 2 · ≤ 2 · , (2.16) ⃒ 휕푦1 ⃒ (1 + 퐽1 + 퐽2퐺3 + 퐽3퐺3) 휕푦1 (1 + 퐽1) 휕푦1

where we used the fact that 퐺2 and 퐺3 are Hill functions of the form (2.4) and hence

takes values in [0, 1]. For the same reason, 휕퐺2/휕푦1 is bounded above, that is, there exists 푀 > 0 such that 휕퐺2/휕푦1 < 푀 for all 푦1 ≥ 0. Hence, for (2.16) to be small

for all 푦1, it is sufficient to reduce the magnitude of 퐽2. This would reduced the strength of the unintended repression 1 − 3, enabling the static I/O response curve [ of an activation cascade to be monotonically increasing. By the same argument,

we expect the static I/O response curve to be monotonically decreasing when 퐽2 is

large, and to be biphasic for intermediate values of 퐽2. Based on the definition of

resource demand coefficient in (2.12), we can decrease 퐽2 by choosing weak subsystem 2 RBS strength and low DNA copy number. We simulated the static I/O response curves of activation cascades with different subsystem 2 RBS strengths and DNA copy numbers, presented in the parameter space in Figure 2-3f. The lower left corner of

the parameter space corresponds to the cascade with the smallest 퐽2, and the upper

right corner corresponds to the largest 퐽2. In accordance with these predictions,

simulations in Figure 2-3f confirms that smaller 퐽2 (weak subsystem 2 RBS and low DNA copy number) results in monotonically increasing response (grid shaded region),

while larger 퐽2 (strong subsystem 2 RBS and high DNA copy number) results in monotonically decreasing response (gray region). The dotted region corresponds to

intermediate values of 퐽2 which result in biphasic response.

47 2.4 Experiment: activation cascade

We built a genetic activation cascade composed of three subsystems to support our analysis in the previous section and in Figure 2-3d-f.

2.4.1 Genetic circuit

The genetic implementation of the activation cascade is described as follows. Sub- system 1 uses the lac promoter to constitutively express LuxR in a LacI-deficient host strain E. coli JTK160. By increasing inducer N-hexanoyl-L-homoserine lactone (AHL) concentration, the active form of LuxR increases, and it can transcriptionally activate the following subsystem. We consider active LuxR as the output of subsystem 1. Subsystem 2 uses transcriptional activation by the active LuxR through the lux promoter (Figure 2-4a). To characterize the static I/O response curve of this subsystem, we placed red fluorescent protein (RFP) under the control ofthe lux promoter. An increase in AHL concentration increases the active LuxR to promote the production of RFP (Figure 2-4a). Subsystem 3 employs transcriptional activation by active NahR and the sal promoter to express green fluorescent protein (GFP) as fluorescence output. Inactive NahR is first produced under the control of the lux promoter. We applied a saturating amount of AHL (100nM) and expressed LuxR constitutively to produce a saturating amount of inactive NahR. By increasing the amount of inducer salicylate (SAL), active NahR concentration increases, activating production of GFP (Figure 2-4b). To build a two-stage activation cascade (CAS 1/30), we connected the three subsystems by replacing the RFP in subsystem 2 by NahR (Figure 2-4c). Active NahR can be regarded as the output of subsystem 2 and the input to subsystem 3. With a constant amount of SAL (1휇M), increased AHL concentration leads to increased active LuxR, and hence to increased concentration of active NahR, resulting in increased production of GFP (cascade output). The expected behavior of this cascade, based on these prescribed regulatory interactions, is therefore a monotonically increasing

48 AHL (a) first activation stage Active RFP LuxR Experiment Model fitting

RFP

rfp

subsystem 2 subsystem 1 SAL

(b) second activation stage Active GFP NahR Experiment Model fitting

subsystem 2 subsystem 3 AHL (c) two-stage activation cascade Active NahR GFP LuxR Expected Experiment

subsystem 2 subsystem 1 subsystem 3

Figure 2-4: Failure of modular composition in a simple two-stage activation cascade. (a) The first activation stage consists of a subsystem that takes as input thetran- scription activator LuxR to promote production of RFP as output in the presence of AHL, resulting in a monotonically increasing static I/O response curve. Upward arrows with leftward/rightward tips represent promoters, semicircles represent RBS, and double hairpins represent terminators. The illustrative diagram composed of subsystems and edges at the upper-right corner represents regulatory interactions among species. (b) The second activation stage consists of a subsystem that takes as input the transcription activator NahR to promote production of GFP as output in the presence of SAL, resulting in a monotonically increasing static I/O response curve. (c) The two-stage activation cascade CAS 1/30 was built by connecting the subsystems in a cascade topology. Biphasic static I/O response curve (solid line) of the cascade was observed instead of the expected monotonically increasing static I/O response curve (dashed line). All experimental data represent mean values and standard deviations of populations in the steady state analyzed by flow cytometry in three independent experiments. Each plot is normalized to its maximum fluorescence value.

49 GFP fluorescence as AHL is increased. Surprisingly, the experimental results contradict this prediction. In fact, although the static I/O response curves of both subsystems 2 and 3 are monotonically increasing (Figure 2-4a-b), their cascade shows a biphasic response curve, in which the GFP fluorescence decreases with increased concentrations of AHL for high AHL concentrations (Figure 2-4c). This fact clearly demonstrates that while the standard model well represents the activation behavior of each individual node, its predictive ability is lost when the subsystems are connected.

2.4.2 Model guided design to mitigate unintended interactions

Based on the simulation map in Figure 2-3f and the mathematical analysis of model (2.15) described in the previous section, we created a library of activation cascades in which each cascade should result into one of the three different behaviors shown in Figure 2-3f. This library is composed of cascades that differ in the value of the resource demand coefficient of NahR퐽 ( 2), with the rationale that we can mitigate the strength of the key unintended interaction 1 − 3 to recover the intended monotonically [ increasing static I/O response curve of the cascade. In particular, starting from CAS 1/30, whose static I/O response curve is biphasic (Figure 2-5a), we designed circuit CAS 0.3/30 with about 30% RBS strength [63] of NahR compared to CAS 1/30, theoretically resulting in a reduction of 퐽2. We therefore expect a reduction of the 1 − 3 interaction strength, leading to a monotonically increasing static I/O response [ curve, which is confirmed by the experiment (Figure 2-5b). Similarly, we constructed another cascade circuit CAS 1/60 in which the DNA copy number is about twice as that of CAS 1/30 (about 60 vs 30). According to our model, resource demand coefficient of NahR 퐽2 in CAS 1/60 should double compared to that of circuit CAS 1/30. Therefore, we expect a possibly monotonically decreasing static I/O response curve. Experiments confirm this prediction (Figure 2-5c). A local increase in GFP fluorescence at about 10 nM AHL is due to the two-step mul- timerization of NahR proteins [114]. To obtain a monotonically increasing static I/O response curve from this circuit, we first reduced NahR resource demand coefficient

50 AHL (a) CAS 1/30 (c) CAS 1/60 Active LuxR NahR GFP

1 10 ) 2 Normalized Fluorescence Normalized Fluorescence /κ 1

log [AHL] (nM) κ 10 log10[AHL] (nM) (b) CAS 0.3/30 (d) CAS 0.3/60

0 10

NahR RBS strength ( -1 10

Normalized Fluorescence 10 20 30 40 50 60 70 80 DNA copy number (D = D = D ) Normalized Fluorescence log [AHL] (nM) 1,T 2,T 3,T 10 log10[AHL] (nM)

Figure 2-5: Model-guided design restores the monotonically increasing static I/O response curve of the cascade. The static I/O response curves of circuits CAS 1/30 (a) and CAS 1/60 (c) were biphasic and monotonically decreasing, respectively. By reducing the RBS strength of NahR, the static I/O response curve of CAS 0.3/30 (b) became monotonically increasing, and that of CAS 0.3/60 (d) became biphasic. Further decreasing the copy number of CAS 0.3/60 to CAS 0.3/30 restored the monotonically increasing static I/O response curve. Experimental results are presented on top of the parameter space created by simulations in Figure 2-3f. Blue and green arrows represent design actions to restore the monotonically increasing static I/O response curves starting from failed cascades CAS 1/30 and CAS 1/60, respectively. Mean values and standard deviations of fluorescence intensities at the steady state are calculated from three independent experiments analyzed by flow cytometry and normalized to the maximum value in each plot.

51 퐽2 by designing a circuit CAS 0.3/60, whose NahR RBS strength is 30% compared to that of CAS 1/60. Theoretically, depending on parameters, reduced 퐽2 can lead to either monotonically increasing or biphasic static I/O response curves (see Fig- ure 2-3f). Our experiment show that the response of CAS 0.3/60 is indeed biphasic (Figure 2-5d). To restore a monotonically increasing static I/O response curve, we can further decrease 퐽2 by reducing DNA copy number to create circuit CAS 0.3/30, whose static I/O response curve is monotonically increasing (Figure 2-5b).

2.5 Summary

The proper function of a genetic circuit requires cellular resources for gene expression. Inevitably, all subsystems in the circuit are forced to compete for a limited amount of resources, introducing unintended interactions in addition to the prescribed regulatory interactions. In this chapter, we have developed a general modeling framework to describe the dynamics of genetic circuits in a resource-limited environment. The model reveals a hidden layer of unintended interactions among subsystems, which have been largely neglected so far but will become more relevant when size of the circuit grows and when resources become limited. We provide simple rules to draw the effective interaction graph of a resource-limited genetic circuit based on prescribed regulatory interactions. This modeling framework can be used to guide the design of genetic circuits to mitigate the effect of unintended interactions. Specifically, we demonstrate this in an example of activation cascade in E. coli bacteria. We show that, as the model predicts, the static I/O response curve of the activation cascade can be monotonically increasing, decreasing, or biphasic depending on the resource demand coefficient of the constituent subsystems. The modeling framework developed in this chapter improves predictability of genetic circuits and provide tools for rationale design of synthetic gene networks in resource-limited cells.

52 Chapter 3

Robustness of genetic subsystems to disturbances

To improve the robustness and predictability of genetic circuits in the presence of unintended interactions, in this chapter, we aim to create genetic subsystems that are context-independent. In particular, we treat variations in the cellular context, including variations in resource availability, as disturbances and aim to design control systems, using biomolecular reactions, to reject the effects of these disturbances on the output of a genetic subsystem.

3.1 Introduction

A long-standing challenge in synthetic biology is the creation of genetic circuits (i.e., synthetic gene network) that perform predictably and reliably in living cells [28, 118]. Variations in the environment [57], unforeseen interactions between a circuit and the host machinery, as we discussed in Chapter 2, and unknown interference between circuit components [173] can all significantly alter the intended behavior of a genetic circuit. Integral control, which is ubiquitous in traditional engineering systems, is a promising approach to mitigate the steady state impact of such unknowns on genetic circuits. At the most basic level, an integral controller ensures that the output of a system can reach a desired set-point while perfectly rejecting (i.e., adapting to)

53 constant disturbances [21, 52, 152]. Due to this unique ability, synthetic biology research has witnessed increasing efforts toward establishing procedures to realize integral control in living cells [7, 8, 22, 45, 83, 128, 161, 164]. At the core of an integral controller is an indispensable “memory" element that accumulates (i.e., integrates) the error between the desired set-point and the measured output over time. While computing this integral is almost trivial using electronic components, realizing it through biomolecular reactions in living cells is problematic. This is because information about the error is usually stored in the form of molecular concentrations, and as host cell grows and divides, biomolecules in every single cell dilute [4, 41], leading to a “leaky" memory that fades away in time. Since cell growth is unavoidable and even beneficial in a number of applications, leaky integration is a fundamental physical limitation for the implementation of integral control through genetic circuits in living cells. Prior theoretical studies have proposed ideal integral controllers (IICs), but neglected the presence of molecule dilution in the analysis [7, 22, 46, 83]. Consequently, while these motifs have theoretically guaranteed performance in cell-free systems, where dilution is non-existent, their performance in the presence of molecule dilution is not preserved. In fact, the deteriorating effect of leaky integration can be significant, as shown here and in other recent studies (e.g., [113, 135]). Therefore, possible solutions to leaky integration have been proposed [7, 135]. Specifically, the approach proposed in [7] hinges on “canceling" the effect of dilution by producing the “memory species” at the same rate as its dilution. This approach, however, relies on exact parameter matching and is hence difficult to realize in practice. A different approach is to use exhaustive numerical simulation to extract circuit parameters that minimize the effects of leaky integration [135], leading to a lengthy and ad hoc design process.

Chapter overview

In this chapter, we propose general design principles for biomolecular set-point controllers, based on existing IICs, that perform well despite the presence of dilution. We then implement one of the designs in vivo. In Section 3.2, we mathematically

54 demonstrate that the undesirable effect of dilution can be arbitrarily suppressed by increasing the rate of all controller reactions. This is true under mild conditions that are largely independent of specific parameter values. This design principle guides the choice of core biomolecular processes and circuit parameters that are most suitable for realizing a biomolecular set-point controller in living cells. We further demonstrate the ability of the proposed biomolecullar controllers to reject time-varying disturbances and track time-varying references in Section 3.3. Analyzing these properties for the system is confounded by the presence of a singular singular perturbation (SSP) structure in the system model. We provide new theories to perform model reduction for SSP systems, resulting in a reduced order whose tracking performance has been analyze previously [64]. Finally, in Section 3.4, we apply the design principles to create a biomolecular feedback controller, mediated through small RNA (sRNA) interference, and demonstrate that its regulation performance improves as a controller parameter is asymptotically reduced in E. coli bacteria.

3.2 Robustness to constant disturbances

To design biomolecular controllers for set-point regulation, we first describe two IIC motifs in Section 3.2.1, which were previously proposed abstract circuit motifs for adapting to constant disturbances in the absence of dilution. We then introduce leaky integral controllers (LICs), which add dilution to these IICs, and demonstrate that the adaptation property is lost. Finally, we describe quasi-integral controllers (QICs) in Sections 3.2.2-3.2.3, the main novelty of this work, in which the controller reactions are engineered to be much faster than dilution, enabling the circuit to restore almost perfect adaptation to constant disturbances. In Section 3.2.4-3.2.5, we illustrate this guided choice on two circuits that are designed to mitigate the effects of transcription/translation resource fluctuations on gene expression, a problem that has gained significant attention recently [63, 120, 127, 141].

55 3.2.1 Physical constraints arising from cell growth

We illustrate in Figure 3-1a two different types of IIC motifs that abstract the two main motifs for biomolecular integral control proposed in the literature. In both types of motifs, we denote by x the species whose concentration needs to be kept at a set point 푟, despite that its production rate is affected by a constant disturbance 푤. Therefore, the output of the process to be regulated is the concentration of species x, which is denoted by 푦 = 푥 (italic). The purpose of implementing an integral controller is thus to have the steady state of 푦, denoted by 푦¯, to satisfy 푦¯ = 푟 regardless of 푤 (i.e., 푦 adapts perfectly to 푤). In these IIC motifs, selected controller species realize the memory function es- sential for integral control. Specifically, a type I IIC (Figure 3-1a-i) regulates the expression of x using a single controller species z. This motif arises from saturating certain Hill-type or Michaelis-Menten-type kinetics such that the production and the removal rates of z become roughly proportional to 푟 and 푦 = 푥, respectively, resulting in the following approximate mass action kinetics model for the controller species z [7, 46, 83]:

푧˙ = 푘(푟 − 푦), (3.1)

where 푘 is a positive constant called integral gain. Since 푧 = ∫︀ 푘(푟−푦)d푡, 푧 is called a memory variable that represents the integral of the error (푟−푦) between the set-point and the output over time. Independent of the exact reactants implementing this IIC, it is immediate from (3.1) that by setting the time derivative to 0, the equilibrium output satisfies 푦¯ = 푟 regardless of 푤. With reference to Figure 3-1a-ii, a type II IIC motif, arising from what was called the antithetic integral control motif in [22], realizes integral action using two controller species z1 and z2. Their production rates are engineered to be proportional to the set-point (푟) and to the concentration of the output species (푦 = 푥), respectively. The two controller species can bind together to form a complex that is removed with 휃 rate constant 휃: z1 + z2 −→ ∅. These biomolecular processes can be described by the

56 following mass action kinetics model:

푧˙1 = 푘푟 − 휃푧1푧2, 푧˙2 = 푘푦 − 휃푧1푧2. (3.2)

The memory function is carried out by the hidden memory variable 푧 := 푧1 − 푧2,

which satisfies 푧˙ =푧 ˙1 − 푧˙2 = 푘(푟 − 푦). As a consequence, independent of the exact choice of reactants, parameters and the magnitude of disturbance 푤, the equilibrium output must satisfy 푦¯ = 푟. With reference to the simulation results in Figure 3-1d (black dashed lines), the output 푦 of both types of IIC motifs reaches the set point 푟 and adapts to disturbance 푤 perfectly at steady state.

When dilution of the controller species due to host cell growth is taken into account, the key integral structure of the memory variables is disrupted (Figure 3-1b). In fact, in a standard mass action kinetics model describing reactions in exponentially growing cells, the average effect of cell growth and division on the dynamics ofany species z can be modeled by a first-order additive term −훾푧, where 훾 is the specific growth rate of the host cell [4, 41, 105, 132] (see Appendix Section B.1). Hence, with reference to Figure 3-1b-i, the dynamics of the memory variable 푧 in a type I IIC now become

푧˙ = 푘(푟 − 푦) − 훾푧. (3.3)

Similarly, as shown in Figure 3-1b-ii, the dynamics of the two controller species z1 and z2 in the type II IIC now become

푧˙1 = 푘푢 − 휃푧1푧2 − 훾푧1, 푧˙2 = 푘푦 − 휃푧1푧2 − 훾푧2, (3.4)

resulting in the dynamics of the hidden memory variable to become

푧˙ =푧 ˙1 − 푧˙2 = 푘(푟 − 푦) − 훾푧. (3.5)

From equations (3.3) and (3.5), we observe that in both types of motifs, the memory

57 variable 푧 is no longer integrating the error between the set-point and the output, but rather carries out leaky integration. We therefore call the motifs in Figure 3-1b leaky integral controllers (LICs). In simulations of Figure 3-1d (blue dash-dot lines), we demonstrate that including dilution significantly hinders the ability of LIC-regulated systems to achieve adaptation to disturbance 푤, resulting in nearly 100% relative adaptation error.

3.2.2 Quasi-integral control (QIC) for set-point regulation

In this section, we introduce QICs, the types of controller we propose to regulate the expression from a genetic subsystem. While LICs cannot achieve perfect adaptation to disturbances, and their performance can be arbitrarily deteriorated by the presence of dilution, we propose that one can achieve almost perfect adaptation to disturbances despite dilution by increasing the rate of all controller reactions in the LICs. Therefore, we call a LIC whose controller reactions are much faster than the slower dilution process a quasi-integral controller (QIC). These controller motifs are shown in Figures 3-1c-i and c-ii respectively for the two types of IIC motifs. In particular, with reference to Figure 3-1c-i, we use a small dimensionless positive parameter 0 < 휖 ≪ 1 to capture the fact that the controller reaction rates in a type I QIC are 1/휖 times faster than those in a type I LIC (Figure 3-1b-i), resulting in the memory variable dynamics to become

푘 푧˙ = (푟 − 푦) − 훾푧. (3.6) 휖

Similarly, when all the controller reaction rates in a type II LIC are increased by a factor of 1/휖 (see Figure 3-1c-ii), the controller species dynamics become

푘 휃 푘 휃 푧˙ = 푟 − 푧 푧 − 훾푧 , 푧˙ = 푦 − 푧 푧 − 훾푧 , (3.7) 1 휖 휖 1 2 1 2 휖 휖 1 2 2

58 (a-i) Ideal integral controllers (b-i) Leaky integral controllers (c-i) є-quasi-integral controlllers (IICs) (LICs) (є-QICs)

w Ø w Ø w

r z x Ø r z x Ø r z x Ø Type I Type

Ø w (a-ii) w (b-ii) Ø w (c-ii)

Ø r z x Ø r z x Ø r z1 x 1 1 Ø Ø Ø z z z2 2 2

Ø Ø Type II Type

(d-i) Type I Type II 1.5 (d-ii)1.5 w w

1 1

0.5 0.5

0 0 0 time (hr) 30 0 time (hr) 30 IICs LICs є-QICs (є=0.02)

Figure 3-1: Ideal, leaky, and quasi-integral controller motifs for set-point regulation. (a-c) Two types of ideal, leaky, and quasi-integral control motifs. The controller reactions are boxed in pink, and the rest of the circuit belongs to the process to be regulated. Dilution of the controller species are neglected in IICs. The rates of all controller reactions are increased by a factor of 1/휖 in the quasi-integral controllers. (d) Simulation results for type I and II controller motifs subject to a step disturbance.

59 resulting in the following hidden memory variable dynamics:

푘 푧˙ =푧 ˙ − 푧˙ = (푟 − 푦) − 훾푧. (3.8) 1 2 휖

As shown in Figure 3-1d (red solid lines), for small 휖, both types of QIC motifs restore the ability to drive the output 푦 to the set point 푟, and to adapt to 푤 almost perfectly despite dilution. These qualitative observations are reflected mathematically in equations (3.6) and (3.8), where for both motifs the steady state adaptation error 푒 can be computed as 푒 = 푟 − 푦¯ = 휖훾푧/푘¯ , whose magnitude can be arbitrarily decreased by detuning 휖 (i.e., increasing controller reaction rates). Although this reasoning is intuitive, it is based on the implicit assumption that the steady state concentration of controller species (푧¯) stays roughly unchanged as 휖 decreases, which is, unfortunately, not always true and hard to verify in general (see Appendix B.3 for such an example). Therefore, in the rest of this section, we provide precise and general mathematical conditions under which performance deterioration of a LIC can be arbitrarily suppressed by faster controller reactions. These results establish a general circuit design principle applicable to any biomolecular process to be regulated and to any leaky integral controller, including, but not limited to those arising from the type I and type II IIC motifs.

We consider a biomolecular process to be regulated, whose dynamics can be written as:

푥˙ = 푓(푥, 푢, 푤), 푦 = ℎ(푥), (3.9) where 푥 represents process states (i.e., the concentrations of all species x forming the biomoelcular process). The process takes two inputs: 푢 is the control input produced by the controller, and 푤 is a disturbance input. The output of the process 푦 is determined by a function ℎ(푥). The process (3.9) is connected to an 휖-parameterized biomolecular controller, which contains a LIC. We study the case where all controller reactions have an 휖-parameterized timescale separation from dilution. To this end,

60 we write the controller in the general form:

1 푘 푧˙ = 푔(푧, 푟, 푦) − 퐿 (푧 ), 푧˙ = (푟 − 푦) − 퐿 (푧 ), 푢 = Θ(푧), (3.10) 1 휖 1 1 2 휖 2 2

⊤ ⊤ where 푧 := [푧1 , 푧2] represents the controller states (e.g., concentrations of controller species) and 휖 is a small positive parameter. Specifically, 푧2 is the intended memory variable that carries out the leaky integration with gain 푘, and 푧1 represents concentrations of additional controller species, if any. The functions 퐿푖(·) for 푖 = 1, 2 accounts for a slightly more general characterization of “integration leakiness” in the controller dynamics. They may arise from a number of biomolecular processes, including dilution and enzymatic degradation. When only dilution is present, we have

퐿푖(푧) = 훾푖푧. The controller takes the output of the process 푦 as input, and compares it with a set-point 푟. The QIC-regulated system is therefore a feedback interconnection of process (3.9) and controller (3.10) (see Figure 3-2a). The two inputs to the QIC-regulated system are constant set-point 푟 and constant disturbance 푤. We assume that they take values on a bounded admissible input set ℛ × 풲. The result we derive is concerned with the equilibrium location of the QIC-regulated system as 휖 decreases, and its stability needs to be verified separately. We therefore make the following assumption.

Assumption 3.1. There is an 휖* > 0 such that for all constant (푟, 푤) ∈ ℛ × 풲 and 0 < 휖 ≤ 휖*, the QIC-regulated system (3.9)-(3.10) has a unique locally asymptotically stable steady state (¯푥, 푧¯). O

We define our control objective as follows.

Definition 3.1. Under Assumption 3.1, system (3.9)-(3.10) achieves 휖-set-point regulation in ℛ × 풲 if for all constant (푟, 푤) ∈ ℛ × 풲, the system’s equilibrium output 푦¯ satisfies

lim 푦¯(푟, 푤, 휖) = 푟. (3.11) 휖→0+

61 (a) w (b) w u u process dynamics process dynamics

r r

є-quasi-integral controller ideal integral controller

Figure 3-2: System architecture of a QIC-regulated system. (a) The QIC-regulated system. (b) The auxiliary IIC-regulated system.

Equation (3.11) implies that, by increasing the rate of controller reactions (i.e., decreasing 휖), the equilibrium output 푦¯ of an 휖-quasi-integral control system can be made arbitrarily close to the set point 푟 in the presence of disturbance 푤. We assume that when the process is regulated by an IIC, shown in Figure 3-2b, it has a locally exponentially stable equilibrium.

Assumption 3.2. For any constant input (푟, 푤) ∈ ℛ × 풲, the closed loop system composed of the regulated process (3.9) and the ideal integral control system

푧˙1 = 푔(푧, 푟, 푦), 푧˙2 = 푘(푟 − 푦), 푢 = Θ(푧). (3.12) has a unique locally exponentially stable steady state (¯푥, 푧¯). O

Assumption 3.3. The functions 푓(푥, 푧, 푤), 푔(푥, 푧, 푟),Θ(푧), 퐿1(푧), 퐿2(푧) and ℎ(푥) are all continuously differentiable in 풳 × 풵 × ℛ × 풲 and have bounded derivatives.

Theorem 3.1. Under Assumptions 3.1-3.3, the QIC-regulated system (3.9)-(3.10) realizes 휖-set-point regulation O

The proof of this Theorem can be found in Appendix Section B.2. This Theorem establishes that if one can use the ideal integral controller (3.12) to obtain perfect adaptation in the absence of dilution, then when dilution is taken into account, one

62 can always speed up all controller reactions to restore adaptation. From an engineering perspective, one should therefore select controller biomolecular processes whose rates are significantly faster with respect to dilution rates. Such processes include, for example, enzymatic reactions (i.e., phosphorylation, methylation, and ubiqui- tination), protein-protein interactions, and RNA-based interactions (i.e., sRNA or microRNA-enabled degradation or sequestration). With these processes, the designer can then realize type I or type II IICs as well as additional IICs that are in the form of system (3.12). Theorem 3.1 provides the theoretical underpinning to the type I and type II QIC designs of Figure 3-1c.

3.2.3 Type I and II QICs to regulate genetic subsystems

We aim to use biomolecular realizations of the type I and II QICs to regulate the output of a genetic subsystem to be independent of disturbances, that is, to allow protein p produced by the regulated gene to become independent of a constant disturbance 푤. For this application, the models in Figure 3-1 for the type I and II QICs are highly simplified and cannot capture the nonlinearities due to biomolecular reaction dynamics. These nonlinearities may appear in the way in which (i) disturbance 푤 affects process dynamics, (ii) controller “senses” process output, and (iii) controller “actuates” the process. Here, we take these potential forms of nonlinearities into account and provide conditions for a class of generalized type I and II QICs with certain structures to satisfy Assumptions 3.1-3.3 and, therefore, to achieve 휖-set-point regulation. Since we aim to regulate a genetic subsystem to produce protein p, in this section, we will specialize the regulated process in (3.9) to be scalar 푥 = 푝 and take the form 푥˙ = 푓(푧, 푤) − 훿푥, where 훿 is the decay rate constant of p. This model is valid if the mRNA dynamics in a genetic subsystem is set to quasi-steady state, or if the regulated process is either the transcription or translation process only. We will assume that the state variable 푥, 푧 evolves in the non-negative orthant.

63 Generalized type I QIC-regulated system

We consider the following model for a generalized type I QIC-regulated system:

푘 푥˙ = 푓(푧, 푤) − 훿푥, 푧˙ = (푟 − 푦) − 퐿(푧), 푦 = ℎ(푥), (3.13) 휖 and its corresponding generalized type I IIC-regulated system:

푥˙ = 푓(푧, 푤) − 훿푥, 푧˙ = 푘(푟 − 푦), 푦 = ℎ(푥). (3.14)

The nonlinear scalar functions 푓(·) and ℎ(·) can model, for example, Hill-type functions. In particular, we make the following assumptions on these nonlinearities:

Assumption 3.4. Functions ℎ(·) and 퐿(·) are continuously differentiable and monotonically increasing. Function 푓(푧, 푤) is continuously differentiable and monotonically increasing in 푧 given a fixed 푤 (i.e., 휕푓/휕푧 > 0 for all 푤). O

Now we demonstrate that systems (3.13) and (3.14) satisfy Assumptions 3.1-3.3, and therefore is guaranteed to achieve 휖-QIC by applying Theorem 3.1. We restrict the inputs (푟, 푤) to the admissible input set ℛ × 풲 defined as any compact subset of:

ℛ˜ × 풲˜ := {(푟, 푤) : 0 < 푟 < ℎ(+∞), 푓(0, 푤) < 훿 · ℎ−1(푟) < 푓(+∞, 푤)}. (3.15)

Lemma 3.1. If Assumption 3.4 is satisfied, then for any given pair (푟, 푤) ∈ ℛ × 풲, the generalized type I IIC-regulated system (3.14) has a unique locally exponentially

* * stable steady state (푥 , 푧 ). O

Proof. When (푟, 푤) ∈ ℛ × 풲, there exists (푧*, 푥*) such that

ℎ(푥*) = 푟, 푓(푧*, 푤) = 훿ℎ−1(푟).

Given Assumptions 3.4, the steady state (푧*, 푥*) must be unique if it exists, since functions 푓 and ℎ are monotonically increasing. Consequently, the existence and

64 uniqueness of the steady state is trivially guaranteed. We focus on studying its stability. Linearizing (3.14) around (푥*, 푧*) gives

⎡ ⎤ ⎡ ⎤ ⎡ ⎤ 푥˙ −훿 푓 ′(푧*) 푥 ⎣ ⎦ = ⎣ ⎦ ⎣ ⎦ , (3.16) 푧˙ −푘ℎ′(푥*) 0 푧 ⏟ ⏞ 퐴0

′ ′ * where, with slight abuse of notation, we denoted ℎ (푥) = dℎ/d푥|푥* and 푓 (푧 ) =

휕푓/휕푧|푧* for a fixed 푑. Due to Assumption 3.4, tr퐴0 = −훿 < 0 and det퐴0 = ′ * ′ * * * 푘ℎ (푥 )푓 (푧 ) > 0, and therefore, 퐴0 is Hurwitz and (푥 , 푧 ) is a locally exponentially stable steady state of system (3.14).

Next, we demonstrate that the generalized type I QIC-regulated system (3.13) has a unique locally asymptotically stable steady state (¯푥, 푧¯).

Lemma 3.2. If Assumption 3.4 is satisfied, then for any given pair (푟, 푤) ∈ ℛ × 풲, the generalized type I QIC-regulated system (3.13) has a unique locally exponentially

stable steady state (¯푥, 푧¯). O

Proof. The steady state (¯푥, 푧¯) of (3.13) can be determined by the following equations:

[︂푓(¯푧, 푤)]︂ 푓(¯푧, 푤) − 훿푥¯ = 0, 푘[푟 − ℎ(¯푥)] − 휖퐿(¯푧) = 0, ⇒ 푘ℎ + 휖퐿(¯푧) = 푘푟. 훿 (3.17)

Since the left hand side of (3.17) increases with 푧¯ and the right hand side of (3.17) is a constant. The equilibrium is unique. Stability of (¯푥, 푧¯) can be determined by studying the linearized system

⎡ ⎤ ⎡ ⎤ ⎡ ⎤ 푥˙ −훿 푓 ′(¯푧) 푥 ⎣ ⎦ = ⎣ ⎦ ⎣ ⎦ (3.18) 푧˙ −푘ℎ′(¯푥)/휖 −퐿′(¯푧) 푧 ⏟ ⏞ 퐴휖

′ ′ The matrix 퐴휖 is always Hurwitz since tr퐴휖 = −훿 − 퐿 (푧) < 0 and det퐴휖 = 훿퐿 (¯푧) + 푘푓 ′(¯푧)ℎ′(¯푥)/휖 > 0. Therefore, the steady state (¯푥, 푧¯) is locally exponentially stable.

Since Lemma 3.1 and 3.2 verifies Assumptions 3.1-3.3, we can apply Theorem 3.1 to a generalized type I QIC-regulated system.

Proposition 3.1. System (3.13) can achieve 휖-set-point regulation in admissible input set ℛ × 풲. O

Generalized type II QIC-regulated system

We consider the following generalized type II QIC-regulated system:

푘 휃 푘 휃 푥˙ = 푓(푤)푧 − 훿푥, 푧˙ = 푟 − 푧 푧 − 훾푧 , 푧˙ = 푦 − 푧 푧 − 훾푧 , 푦 = ℎ(푥) 1 1 휖 휖 1 2 1 2 휖 휖 1 2 2 (3.19) and its corresponding generalized type II IIC-regulated system:

푥˙ = 푓(푤)푧1 − 훿푥, 푧˙1 = 푘푟 − 휃푧1푧2 푧˙2 = 푘푦 − 휃푧1푧2, 푦 = ℎ(푥). (3.20)

The nonlinear scalar functions 푓(푤) describes how disturbance 푤 affects the control action through variable 푧1 and the scalar function ℎ(푥) describes how the controller “senses” the output of the process 푥. We make the following assumptions on functions 푓(·) and ℎ(·).

Assumption 3.5. The scalar function ℎ(푥) is continuously differentiable, monotonically increasing, and 푟 is in the range of ℎ(푥). Furthermore, ℎ(0) = 0, and,

푥 · ℎ′(푥) ≤ 2ℎ(푥), ∀푥 ≥ 0. (3.21)

Since ℎ(푥) functions as a “sensor”, it is natural to expect it to be monotonically increasing with 푥 and vanishing at 0. Input 푟 is in the range of ℎ(푥) if we restrict the

66 inputs to an admissible input set ℛ × 풲. It is defined as any compact subset of

ℛ˜ × 풲˜ := {(푟, 푤) : 0 < 푟 < ℎ(+∞)}. (3.22)

If one has a concave ℎ(푥), such as a Hill function without cooperativity, then one is guaranteed to have 푥 · ℎ′(푥) ≤ ℎ(푥), which satisfies (3.21). Conversely, functions that satisfy Assumption 3.5 need not be concave. We use 푓(푤) to capture the way in which disturbance 푤 affects gene expression. This model can capture a wide variety of sources of disturbances, for example: competition for transcriptional and translational machinery, and uncertainty in gene expression rate constants.

Assumption 3.6. The scalar function 푓(푤) is strictly positive for all 푤 ∈ 풲. O Lemma 3.3. If Assumptions 3.5 and 3.6 are satisfied, then for any fixed pair (푟, 푤) ∈ ℛ × 풲, the generalized type II IIC-regulated system (3.20) has a unique locally

* * * exponentially stable steady state (푥 , 푧1, 푧2). O

* * * Proof. The steady state (푥 , 푧1, 푧2) can be found by setting the time derivatives in (3.20) to zero, from which we have,

* * * * * 푓(푤)푧1 − 훿푥 = 0, 휃푧1푧2 = 푘ℎ(푥 ) = 푘푟. (3.23)

Due to Assumption 3.5, the solution to (3.23) can be uniquely determined:

* * −1 * 훿푥 * 푘푢 푥 = ℎ (푟), 푧1 = , 푧2 = * . (3.24) 푓(푤) 휃푧1

* * * To study the stability of this steady state, we linearize (3.20) around (푥 , 푧1, 푧2) to obtain:

⎡ ⎤ ⎡ ⎤ ⎡ ⎤ 푥˙ −훿 푓(푤) 0 푥 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ = ⎢ * *⎥ · ⎢ ⎥ . (3.25) ⎢푧˙1⎥ ⎢ 0 −휃푧2 −휃푧1⎥ ⎢푧1⎥ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ′ * * * 푧˙2 푘ℎ (푥 ) −휃푧2 −휃푧1 푧2 ⏟ ⏞ 퐴0

67 The linear system (3.25) is exponentially stable if matrix 퐴0 is Hurwitz. To show this, we compute the characteristic polynomial of 퐴0 as

* * * ′ * 2 * * 푃0(푠) = det(푠퐼 − 퐴0) = (푠 + 훿)(푠 + 휃푧2)(푠 + 휃푧1) + 휃푘푧1푓(푤)ℎ (푥 ) − 휃 푧1푧2(푠 + 훿)

3 * * 2 * * * ′ * = 푠 + [휃(푧1 + 푧2) + 훿]푠 + 훿휃(푧1 + 푧2)푠 + 휃푘푧1푓(푤)ℎ (푥 ).

Based on the characteristic polynomial 푃0(푠), according to Routh-Hurwitz condition, (3.25) is stable if the following inequality is satisfied

* * 2 2 * * * ′ * 훿휃(푧1 + 푧2) + 훿 (푧1 + 푧2) > 푘푧1푓(푤)ℎ (푥 ). (3.26)

To verify (3.26), on the one hand, we have

* * 2 2 * * * * 훿휃(푧1 + 푧2) + 훿 (푧1 + 푧2) > 2훿휃푧1푧2. (3.27)

On the other hand, due to (3.24) and Assumption 3.5, we have

* ′ * * ′ * * * * 푘푧1푓(푤)ℎ (푥 ) = 푘훿푥 ℎ (푥 ) ≤ 2푘훿ℎ(푥 ) = 훿휃푧1푧2. (3.28)

* * Since 푧1, 푧2 > 0, from (3.27) and (3.28) we have (3.26) verified, the steady state * * * (푥 , 푧1, 푧2) is therefore locally exponentially stable.

Next, we show that the generalized type II QIC-regulated system has a unique locally asymptotically stable steady state.

Lemma 3.4. If Assumptions 3.5 and 3.6 are satisfied, then for any fixed pair (푟, 푤) ∈ ℛ × 풲 and any 휖 > 0, the generalized type II QIC-regulated system (3.19) has a

unique locally asymptotically stable steady state (¯푥, 푧¯1, 푧¯2).

Proof. See Appendix Section B.4

Since the above two lemmas (Lemma 3.3 and 3.4) verify Assumptions 3.1 and 3.2, we can apply Theorem 3.1 and state the following result.

68 Proposition 3.2. If Assumptions 3.5 and 3.6 are satisfied, then (3.19) can achieve 휖-set-point regulation in admissible input set ℛ × 풲.

In the next Sections, we propose two biomolecular realizations of the generalized type I and II QIC motifs, respectively. For illustration purposes, we use these controllers to mitigate the effect of transcriptional and translational resource fluctuations on gene expression, a problem that was discussed in detail in Chapter 2 and has received considerable attention in synthetic biology in recent years.

3.2.4 Type I QIC realization: phosphorylation cycle

A diagram of the phosphorylation-based quasi-integral control system is shown in Fig- ure 3-3a. The controller is intended to regulate the production of protein p to adapt to a disturbance 푤, which models a reduction in protein production rate due to, for example, depletion of transcriptional and/or translational resources in the host cell [63, 120]. In this system, the intended set-point regulation is accomplished by a phosphorylation cycle. The regulated protein p is co-expressed with a phosphatase and the concentration of the kinase acts as the set point 푟. The substrate b is expressed constitutively. When b is phosphorylated by kinase u to become active substrate b*, it transcriptionally activates production of protein p. A simplified mathematical model of this system is (see Appendix Section B.5 for detailed derivation from chemical reactions):

* * ˙* 푟푏 푝푏 * 푏 푏 = 푘1 − 푘2 * − 훾푏 , 푝˙ = 푅(1 − 푤) * − 훾푝, (3.29) 푏 + 퐾1 푏 + 퐾2 휆 + 푏

where 푘1 and 푘2 are the catalytic rate constants of the phosphorylation and dephosphorylation reactions, respectively, 퐾1 and 퐾2 are the Michaelis-Menten constants, 푅 is the protein production rate constant, and 휆 is the dissociation constant between b* and the promoter of the regulated gene. System (3.29) can be taken to the form of a type I LIC (3.3) if both the kinase and

* the phosphatase are saturated (i.e., 푏 ≫ 퐾1 and 푏 ≫ 퐾2). These design constraints are present in any type I integral control motif, which relies on saturation of Hill-

69 w 1.8 (a) p γ (b) k γ Ø γ b* b w Ø Ø 1

regulated gene substrate є- quasi-integral controller 0 r 0 20 40 time (hr) w (c) p (d) 1.8 Ø Ø w Ø 1 m s Ø r regulated

gene sRNA є- quasi-integral controller 0 0 30 60 time (hr)

Figure 3-3: Two physical realizations of QICs. (a) Genetic circuit diagram of the phosphorylation-based quasi-integral controller. Chemical reactions realizing the controller are boxed in pink. (b) Simulation of the circuit’s response according to (3.29). A set-point input 푟 = 20 nM is applied at time 0 and a disturbance input 푤 = 0.5 is applied at 20 hr. The vertical axis represents the ratio between output 푦, defined to be proportional to 푝 (푦 = 휎푝), and set-point 푟. The dashed black line is the response of the phosphorylation- based control system assuming no dilution of the active substrate b*. The dotted blue line, the thin green line with square markers and the solid red line represent circuit’s response in the presence of nonzero substrate dilution (훾 = 1 hr−1) and decreasing 휖, which corresponds to increasing catalytic rates (푘푖, 푖 = 1, 2). (c) Genetic circuit diagram of the sRNA-based quasi-integral controller. (d) Simulation of the circuit’s response according to (3.32). A set-point input 푟 = 1 is applied at time 0 and a disturbance input 푤 = 0.5 is applied at 30 hr. The vertical axis represents the ratio between output 푦, defined in (3.34), and set-point input 푟. The dashed black line represents response of an ideal integral control system, where RNA decay rate 훿 is set to 0. The dotted blue line, the thin green line with square markers and the solid red line represent circuit’s responses in the presence of nonzero RNA decay rate (훿 = 3 hr−1, corresponding to half-life of about 13 mins) and decreasing 휖. Parameter 휖 is decreased by increasing the mRNA-sRNA removal rate (휃/훽). The DNA copy numbers of the regulated gene and the sRNA are increased simultaneously by a factor of 1/휖 as 휖 decreases. Simulation parameters are listed in Appendix B.8.

70 type or Michaelis-Menten kinetics [7, 46, 83], and can be satisfied in practice when the substrate is over-expressed and the kinase concentration is not too small (see Appendix Section B.5). Under these assumptions and by setting 푥 := 푝, 푧 := 푏* and

휎 := 푘2/푘1, we can approximate (3.29) by

푧 푥˙ = 푅(1 − 푤) − 훾푥, 푧˙ = 푘 (푟 − 푦) − 훾푧, 푦 = 휎푥, (3.30) 휆 + 푧 1 which is in the form of a type I LIC-regulated system. Since phosphorylation and dephosphorayltion reactions are typically much faster than dilution, by choosing a phosphorylation cycle to realize the type I motif, the effect of leaky integration is

−3 naturally mitigated. In fact, by setting 휖 := 훾/푘1 (휖 ∼ 10 in bacteria [41]), system (3.30) is in the form of a type I QIC:

푧 훾 푥˙ = 푅(1 − 푤) − 훾푥, 푧˙ = (푢 − 푦) − 훾푧, 푦 = 휎푥. (3.31) 휆 + 푧 휖

As demonstrated in the simulation results in Figure 3-3b and Appendix Section B.7 , the steady state adaptation error due to leaky integration becomes smaller as 휖 decreases (i.e., larger timescale separation between phosphorylation/dephosphorylation and dilution). In fact, equation (3.31) is in the form of the generalized type I QIC model in (3.13). Let 푓(푧, 푤) := 푅(1 − 푤)푧/(휆 + 푧) and ℎ(푥) = 휎푥, it is easy to verify that 푓(푧, 푤) and ℎ(푥) satisfy the conditions in Assumption 3.4. Furthermore, since 푓(0, 푤) = 0 and 푓(+∞, 푤) = 푅(1 − 푤), we have

ℛ˜ × 풲˜ = {(푟, 푤) : 0 < 푟 < 휎푅(1 − 푤)/훿, 0 ≤ 푤 < 1}, and the admissible input set ℛ × 풲 can be taken as any compact subset of ℛ˜ × 풲˜ . Hence, we can apply Proposition 3.1 to claim the 휖-set-point regulation can be achieved for the phosphorylation-cycle based controller in ℛ × 풲.

71 3.2.5 Type II QIC realization: small RNA (sRNA) interference

The sRNA-based quasi-integral controller is a realization of the generalized type II QIC. It is intended to regulate translation of protein p to adapt to a disturbance 푤 affecting the translation process. This disturbance models uncertainty in translation rate constant due to fluctuation in the amount of available ribosomes [63, 120, 127]. With reference to Figure 3-3c, the controller consists of protein p transcriptionally activating production of an sRNA (s) that is complementary to the mRNA (m) of the regulated protein p. The sRNA and mRNA can bind and degrade together rapidly [90, 97, 174]. The mRNA concentration 푚 is the control input to the translation process. A constant upstream transcription factor regulates transcription rate as a set-point input 푟 to the controller. Based on the chemical reactions in Appendix Section B.6, the ODE model of this system is:

푝/푘푠 푚˙ = 푇 푟 − 휃푚푠/훽 − 훿푚, 푠˙ = 푇푠 − 휃푚푠/훽 − 훿푠, 푝˙ = 푅(1 − 푤)푚 − 훾푝, 1 + 푝/푘푠 (3.32)

where 푇 and 푇푠 characterize the production rates of mRNA and sRNA, respectively. They are proportional to the copy numbers of the regulated gene and the sRNA-

expression DNA. Parameter 푘푠 is the dissociation constant of the binding between protein p and the sRNA promoter, 훽 is the dissociation constant of mRNA-sRNA binding, 휃 is the degradation rate constant of the mRNA-sRNA complexes, and 푅 is the translation rate constant. In addition to dilution due to cell growth, characterized by rate constant 훾, uncoupled mRNA and sRNA are degraded by RNase [69, 96]. Therefore, we model decay (i.e., dilution and degradation) of uncoupled RNAs by a lumped rate constant 훿 such that 훿 ≥ 훾, and assume that this rate constant is the same for mRNA and sRNA without loss of generality. Our model (3.32) is similar to established sRNA-based translation regulation models [90, 100, 140], except that expression of sRNA is activated by the regulated protein (p) to constitute a feedback loop. These additional reactions are modeled using a standard Hill-type function in

72 (3.32), and we refer the readers to Appendix Section B.6 for its derivation.

Setting 푧1 := 푚, 푧2 := 푠 and 푥 := 푝, system (3.32) can be taken to the form of a type II LIC-regulated system:

푧˙1 = 푇 푟 − 휃푧1푧2/훽 − 훿푧1, 푧˙2 = 푇 푦 − 휃푧1푧2/훽 − 훿푧2, (3.33) 푇푠 푥/푘푠 푥˙ = 푅(1 − 푤)푧1 − 훾푥, 푦 = . 푇 1 + 푥/푘푠

In order to achieve negligible adaptation error, we need to ensure that the controller

reactions (i.e., 푧1 and 푧2 dynamics) are much faster than uncoupled RNA decay. On the one hand, coupled degradation of mRNA-sRNA complexes are by nature much more rapid than uncoupled RNA decay [97]. We therefore use 휖 := 훿/휃 ≪ 1 to characterize this timescale separation. Additionally, we can simultaneously increase DNA copy numbers of the controller species m and s by a factor of 1/휖 to increase their production rates. Consequently, the production rate constants of mRNA and

sRNA become 푇/휖 and 푇푠/휖, respectively. Under these assumptions, system (3.33) is in the form of a type II QIC-regulated system:

푇 휃 푇 휃 푧˙ = 푟 − 푧 푧 − 훿푧 , 푧˙ = 푦 − 푧 푧 − 훿푧 , 1 휖 휖훽 1 2 1 2 휖 휖훽 1 2 2 (3.34) 푇푠 푥/푘푠 푥˙ = 푅(1 − 푤)푧1 − 훾푥, 푦 = . 푇 1 + 푥/푘푠

The adaptation error can be arbitrarily decreased by detuning 휖. In practice, this can be accomplished by simultaneously (i) increasing the mRNA-sRNA complex degradation rate constant 휃, and (ii) increasing the copy numbers of the regulated gene and the sRNA-expression DNA. While directly increasing 휃 may be difficult to implement in practice, since the parameters 휃 and 훽 are clustered together in model (3.32), we can achieve the same effect by increasing the affinity between sRNAand mRNA (1/훽) [110]. The above results are confirmed by simulations in Figure 3-3d and Appendix Section B.7 using biologically relevant parameters from bacteria E. coli. The sRNA-regulated system model (3.34) fits the description of generalized type II QIC-regulated system in (3.19). Specifically, let ℎ(푥) = 푇푠 · 푥/푘푠 , Assumption 3.5 푇 1+푥/푘푠

73 Figure 3-4: A subsystem regulated by a type II QIC subject to a time-varying disturbance 푤(푡). The regulated process is denoted by 푃 . Other subsystems in the genetic circuit may create time-varying ribosome demand 푑(푡) that becomes a time-varying disturbance input 푤(푡) to the regulated subsystem. is satisfied because ℎ(0) = 0 and ℎ(푥) is concave. Assumption 3.6 is satisfied for

푤 ∈ [0, 1) because 푓(푤) = 푅(1 − 푤) in (3.34). Additionally, since ℎ(+∞) = 푇푠/푇 , the corresponding admissible input set ℛ × 풲 is any compact subset of

˜ ˜ ℛ × 풲 = {(푟, 푤) : 0 < 푟 < 푇푠/푇, 0 ≤ 푤 < 1}. (3.35)

Due to Proposition 3.2, since Assumptions Assumptions 3.5 and 3.6 are all satisfied in ℛ × 풲, the sRNA-based controller can achieve 휖-set-point regulation in ℛ × 풲.

3.3 Robust tracking performance

As shown in Figure 3-4, in practice, the disturbance 푤(푡) can be created by, for example, other subsystems generating a time-varying ribosome demand 푑(푡), and therefore 푤(푡) is time-varying. On the other hand, in certain applications such as biosensing, it is also desirable for the QIC-regulated subsystem to track a time-varying reference signal 푟(푡) [67, 162]. Hence, we aim to evaluate the performance of the type II QIC in the presence of a time-varying 푢(푡).

In Figure 3-5, we show a type II QIC system, where two controller species c1 and c2 regulate the concentration of a single protein p. Production rate of c1 is proportional to a reference 푟, which often reflects the concentration of a molecular stimulus, and

74 c1 activates the production of protein p. Species c2 is a “sensor”, whose production

rate is proportional to the concentration of the output (푝). All species (i.e., c1, c2, p) are diluted at rate constant 훿 due to cell growth. These processes follow the chemical reactions:

r(t)/휖 훽/휖 휃/휖 ∅ −→ c1, p −→ p + c2, c1 + c2 −→ ∅, (3.36a) 훼 훿 c1 −→ c1 + p, c1, c2, p −→ ∅. (3.36b)

All rate constants are positive, and the small parameter 0 < 휖 ≪ 1 models the fact that the rates of production and annihilation of c1 and c2 in (3.36a) are much larger than the reaction rates in (3.36b). A mass-action kinetic model of (3.36) takes the following form:

휖푐˙1 = 푟 − 휖훿푐1 − 휃푐1푐2,

휖푐˙2 = 훽푝 − 휖훿푐2 − 휃푐1푐2, (3.37)

푝˙ = 훼푐1 − 훿푝 + 푤,

where we use 푤 to represent a time-varying disturbance on the production rate of p. If 푢 := [푟, 푤]⊤ is a constant and 휖 is small enough, the equilibrium output 푝¯ =푝 ¯(휖)

satisfies lim휖→0+ 푝¯ = 푟/훼, as we show in Section 3.2, making type II QIC ideal for synthetic biology applications where protein concentration needs to be robustly and tightly regulated at a constant level. Therefore, we study the tracking performance of (3.37) when 휖 is small.

3.3.1 A singular singular perturbation (SSP) problem

To study the response of system (3.37) to time-varying inputs, we find that it appears

to have two timescales, with 푐1 and 푐2 being the fast variables and 푝 being the slow variable. This hints on the direction to perform first model reduction for (3.37) utilizing this timescale separation property and then study the tracking performance of a reduced order model. We follow the standard singular perturbation (SP) procedure

75 [85] to first write (3.37) in the fast timescale 휏 = 푡/휖:

d푐1/d휏 = 푟(푡) − 휖훿푐1 − 휃푐1푐2, (3.38a)

d푐2/d휏 = 훼푝 − 휖훿푐2 − 휃푐1푐2, (3.38b)

d푝/d휏 = 휖훽푐1 − 휖훿푝 + 푤(푡), (3.38c)

and then set 휖 = 0 to “freeze” the slow variable 푝 to obtain its boundary layer (BL) dynamics:

d푐1/d휏 = 푟 − 휃푐1푐2, d푐2/d휏 = 훽푝 − 휃푐1푐2, (3.39)

whose Jacobian is singular in the entire BL state space. Thus, standard SP is inapplicable to system (3.37) and such a system is called a singular singularly perturbed (SSP)

system [62, 139]. This singularity arises from the annihilation reaction, 푐1 + 푐2 → ∅,

which affects both fast variables푐 ( 1 and 푐2) in an identical fashion (−휃푐1푐2/휖). SSP problems have been studied in [62, 139]. For these results to be applicable, it is necessary for the Jacobian of system (3.38a) evaluated at 휖 = 0:

⎡ ⎤ −휃푐2 −휃푐1 0 ⎢ ⎥ ⎢ ⎥ 퐽(0) := ⎢−휃푐2 −휃푐1 훼⎥ ⎣ ⎦ 0 0 0

to have a zero eigenvalue with same algebraic and geometric multiplicities.

However, the zero eigenvalue of 퐽(0) has algebraic multiplicity 2 and geometric

multiplicity 1 for any positive 푐1, 푐2 and positive parameters 훽 and 휃. Given that the BL Jacobian is singular everywhere in the state space, linearizing (3.37) about any steady state leads to a similar SSP problem. In fact, with reference to Figure3-5b, when we numerically evaluate the poles of the linearized type II QIC, we find that it has a fast mode (that behaves like 푒−푡/휖) and a high frequency damped oscilla- √ tory mode (that behaves like 푒−푡 sin(푡/ 휖)). One would therefore expect the reduced system of a linearized type II QIC to contain the parameter 휖 to capture this high

76 (a) (b) system poles 50

c1 Ø

Ø imag

p c2 Ø output type II QIC -50 Ø -104 -102 real -100 -10-2

Figure 3-5: A type II QIC motif and the pole map of its linearized model. (a) Interactions among molecular species. (b) Poles of a linearized type II QIC as 휖 changes. The linearized model is (3.60) with 훼 = 훽 = 휃 = 훿 =푢 ¯ = 1. frequency damped oscillatory mode. This observation further reinforces our conclu- sion that existing SP and SSP tools are inapplicable to the type II QIC, because existing tools always lead to an 휖-independent reduced system. Therefore, to analyze the tracking property of the type II QIC, we first obtain a model reduction result for general linearized SSP systems and then use the reduced model to study the tracking performance.

3.3.2 Model reduction of SSP systems

In this section, we describe the general linear SSP problem we consider and introduce a transformation to classify the system state variables into three categories. We then construct a candidate reduced system by setting state variables in one of the categories to quasi-steady state.

SSP System Setup

We consider an 휖-parameterized linear SP system subject to a scalar time-varying input 푢(푡). This is for simplicity of presentation and since we consider linear systems,

77 the result is trivially extendable to vector inputs. We use bold face u for the deriva-

(푛) tives of 푢(푡) (i.e., u = [푢, 푢,˙ 푢,¨ ··· , 푢 ]) and write u ∈ ℒ∞ to indicate that 푢(푡) has bounded, 휖-independent derivatives. We write the system as:

˙ 휖 휖 휖 휉1 = 퐴11휉1 + 퐴12휉2 + 퐹1 푢(푡), (3.40) ˙ 휖 휖 휖 휖휉2 = 퐴21휉1 + 퐴22휉2 + 퐹2 푢(푡),

푞+푝 휖 in which (휉1, 휉2) ∈ R . We use 퐴푖푗 to denote that the 휖-dependent matrix 퐴푖푗(휖) is 휖 ∑︀∞ 푘 푘 a power series of 휖 such that 퐴푖푗 := 푘=0 퐴푖푗휖 . In the fast timescale, (3.40) can be equally represented by

휖 휖 휖 d휉1/d휏 = 휖퐴11휉1 + 휖퐴12휉2 + 휖퐹1 푢(푡), (3.41) 휖 휖 휖 d휉2/d휏 = 퐴21휉1 + 퐴22휉2 + 퐹2 푢(푡).

By setting 휖 = 0, we “freeze” the slow 휉1 dynamics to obtain the BL dynamics:

0 0 0 d휉2/d휏 = 퐴21휉1 + 퐴22휉2 + 퐹2 푢(푡). (3.42)

0 We study the SSP problem in which 퐴22 is singular. More specifically, motivated by the properties of matrix 퐽(0) in the FSF, we make the following assumption on the ⎡ ⎤ 0 0 system matrix of (3.41) evaluated at 휖 = 0: 퐴0 := ⎣ ⎦. 0 0 퐴21 퐴22

Assumption 3.7. The zero eigenvalue of 퐴0 has algebraic multiplicity 휇 = 푞 +1 and

0 geometric multiplicity 휆 = 푞. All other eigenvalues of 퐴 have negative real parts. O

0 Remark 3.1. In a standard SP problem, 퐴22 is Hurwitz. As a result, the zero eigenvalue of 퐴0 must have multiplicities 휇 = 휆 = 푞, which is consistent with the

0 dimension of the slow variable 휉1. When 퐴22 has a zero eigenvalue, we are faced with an SSP problem. For a subclass of SSP problems, where the multiplicities of the zero eigenvalue satisfy 휇 = 휆, results in [62, 139] can be applied to transform them into standard SP forms. However, the type of singularity in Assumption 3.7, where 휇 = 푞 + 1 ̸= 푞 = 휆, cannot benefit from these results.

78 Transformation to Normal SSP Form

In this section, we utilize a transform of (3.40) to classify the state variables in an SSP problem into three categories. The following Lemma describes the transformed system, which we say is in normal SSP form.

Lemma 3.5. There exists a non-singular real matrix 푃 , independent of 휖, such that with 푧 = 푃 휉, system (3.40) can be transformed into the following normal SSP form:

휖 휖 휖 휖 푧˙1 = 퐸11푧1 + 퐸12푧2 + 퐸13푧3 + 퐵1푢(푡), (3.43a)

휖 휖 휖 휖 휖푧˙2 = 푅 푧1 + 휖퐸22푧2 + 휖퐸23푧3 + 퐵2푢(푡), (3.43b)

휖 휖 휖 휖 휖푧˙3 = 휖퐸31푧1 + 휖퐸32푧2 + 푆 푧3 + 퐵3푢(푡). (3.43c)

푞 푝−1 0 1×푞 where 푧1 = 휉1 ∈ R , 푧2 is a scalar, and 푧3 ∈ R . The matrix 푅 ∈ R is nonzero, 0 (푝−1)×(푝−1) and 푆 ∈ R is Hurwitz. O

Proof. We derive this result using the fast timescale system (3.41), which can be re-written as

⎛ ⎞ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎜ 휖 휖 ⎟ 휖 d 휉1 ⎜ 0 0 퐴 퐴 ⎟ 휉1 휖퐹 ⎜ 11 12 ⎟ 1 ⎣ ⎦ = ⎜⎣ ⎦ +휖 ⎣ ⎦⎟ ⎣ ⎦ + ⎣ ⎦ 푢, d휏 휉 ⎜ 퐴0 퐴0 퐴˜휖 퐴˜휖 ⎟ 휉 퐹 휖 2 ⎝ 21 22 21 22 ⎠ 2 2 ⏟ ⏞ ⏟ ⏞ 퐴0 퐴휖

˜휖 ∑︀∞ 푘 푘−1 0 where 퐴푖푗 := 푘=1 퐴푖푗휖 = 풪(1). By Assumption 3.7, 퐴22 must have a non- repeated zero eigenvalue. Therefore, there exists a unique invertible matrix 푉 ∈ R푝×푝 ⎡ ⎤ 0 0 0 −1 0 to take 퐴22 to real Jordan form: 푉 퐴22푉 = ⎣ ⎦ , where the (푝 − 1) × (푝 − 1) 0 푆0 ⎡ ⎤ 푅0 0 −1 0 0 1×푞 0 (푝−1)×푞 matrix 푆 is Hurwitz. Let 푉 퐴21 = ⎣ ⎦, where 푅 ∈ R and 푀 ∈ R , 푀 0 the 휖-independent transformation is carried out by matrix 푃 , which is a product of

79 two invertible matrices:

⎡ ⎤ 퐼 0 0 ⎡ ⎤ ⎢ ⎥ 퐼 0 ⎢ ⎥ 푃 := ⎢ 0 1 0 ⎥ ⎣ ⎦ . ⎣ ⎦ 0 푉 −1 푀 0 0 푆0

The resultant 푧 dynamics can be computed as:

⎛⎡ ⎤ ⎞ ⎡ ⎤ 휖 0 0 0 휖퐵1 d ⎜⎢ ⎥ ⎟ ⎢ ⎥ 푧 = ⎜⎢ 0 ⎥ + 휖퐸휖⎟ 푧 + ⎢ 휖 ⎥ 푢, (3.44) ⎜⎢푅 0 0 ⎥ ⎟ ⎢ 퐵2 ⎥ d휏 ⎝⎣ ⎦ ⎠ ⎣ ⎦ 0 휖 0 0 푆 퐵3 where 퐸휖 := 푃 퐴휖푃 −1 and 퐵휖 := 푃 퐹 휖 are all 풪(1). Note that since the upper-left 푞×푞

푞 휖 휖 휖 0 휖 block of 푃 is identity, we have 푧1 = 휉1 ∈ R and 퐵1 = 퐹1 . By denoting 푅 := 푅 +휖퐸21 휖 0 휖 and 푆 := 푆 + 휖퐸33, system (3.44) is equivalent to (3.43) in slow timescale. Matrix 푅0 must not be 0, for if otherwise, one would have 푃 퐴0푃 −1 = diag{0, 푆0}, whose zero eigenvalue has multiplicities 휇 = 휆 = 푞 + 1, violating Assumption 3.7.

Candidate Reduced System

At a high level, the transformation to normal SSP form separates out three sets of

푝−1 state variables. (A) The dynamics of 푧3 ∈ R becomes faster as 휖 decreases, we therefore call 푧3 the fast variable. According to (3.43c), the 풪(1) 푧3 dynamics are decoupled from that of 푧1 and 푧2. Roughly speaking, this decoupling guarantees that fast convergence of 푧3 to its quasi-steady state is minimally affected by the slow dynamics, which may contain a high frequency damped oscillatory mode, as we have

푞 seen in Figure 3-5B. (B) The 풪(1) dynamics of 푧1 ∈ R are unaffected by 휖, we therefore call 푧1 the slow variable. (C) As 휖 decreases, 푧1 has a larger effect on the 휖 scalar 푧2 dynamics (through 푅 /휖). Yet, it does not make 푧2 dynamics faster. We call

푧2 the pseudo-fast variable. Based on this reasoning, we investigate whether we can obtain a reduced model of

(3.43) by setting 푧3 to quasi-steady state. In particular, by setting 휖 = 0 in (3.43c),

80 Figure 3-6: Decomposition of candidate reduced system in SSP problem. The candidate reduced system can be decomposed as two subsystems interconnected through high-gain negative feedback.

we have

0 0 0 −1 0 0 = 푆 푧¯3 + 퐵3 푢(푡) ⇒ 푧¯3 = −(푆 ) 퐵3 푢(푡), (3.45)

We construct a candidate reduced system whose states 푥푖 are intended to approximate 푧푖 in the full system in equation (3.43). The reduced system is obtained by 1) substituting 푧¯3 in (3.45) into the 푧1 and 푧2 dynamics in (3.43), and 2) setting all 풪(휖) terms in 푧1 and 푧2 dynamics to 0. This procedure results in the following candidate reduced system:

⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ 0 0 0 푥˙ 1 퐸11 퐸12 푥1 퐵1푟 ⎣ ⎦ = ⎣ ⎦ ⎣ ⎦ + ⎣ ⎦ 푢(푡), (3.46a) * 0 * 푥˙ 2 푅 /휖 퐸22 푥2 퐵2푟/휖

0 −1 0 푥3 =푧 ¯3 = −(푆 ) 퐵3 푢(푡), (3.46b) where, * 0 1 0 0 0 0 −1 0 푅 = 푅 + 휖푅 , 퐵1푟 = 퐵1 − 퐸13(푆 ) 퐵3 (3.47) * 0 1 0 0 −1 0 퐵2푟 = 퐵2 + 휖퐵2 − 휖퐸23(푆 ) 퐵3 .

As shown in Figure 3-6, the candidate reduced system (3.46a) can be decomposed into the feedback interconnection of two subsystems. Specifically, the two subsystems

81 are: ⎧ 0 0 0 ⎪푥˙ 1 = 퐸 푥1 + 퐸 푣1 + 퐵 푢, 휖 ⎨ 11 12 1푟 Σ1 := * 0 1 ⎩⎪푦1 = −푅 푥1 = −(푅 + 휖푅 )푥1, ⎧ (3.48) 0 ⎨⎪푥˙ 2 = 퐸22푥2 + 푣2, Σ2 := ⎩⎪푦2 = 푥2.

푞 where 푥1 ∈ R and 푦1, 푥2, 푦2 are scalars. The two subsystems are interconnected * according to the rule 푣1 = 푦2 and 푣2 = (퐵2푟푢 − 푦1)/휖. We place the following 0 assumptions on subsystems Σ2 and Σ1.

0 0 0 Assumption 3.8. (i) The scalar 퐸22 < 0. (ii) The pair (퐸11, 퐸12) is controllable, 0 0 0 and the pair (퐸11, 푅 ) is observable. (iii) The transfer function from 푣1 to 푦1 in Σ1:

0 0 0 −1 0 퐻1 (푠) := −푅 (푠퐼 − 퐸11) 퐸12 (3.49)

0 is strictly proper, strictly positive real (SPR) and does not contain a zero at 퐸22. O

0 According to Assumption 3.8, the transfer function of Σ2, 퐻2(푠) = 1/(푠 + 퐸22), is also SPR. Hence, the candidate reduced system can be regarded as a high-gain (1/휖)

휖 feedback interconnection of two SPR subsystems Σ1 and Σ2 (Figure 3-6).

Error Dynamics

Now we are ready to analyze the error dynamics between the full system (3.43) and the candidate reduced system (3.46) to demonstrate that their trajectories are close to each other. We define the error between the full system and the candidate reduced system as 푒푖 := 푧푖 − 푥푖. The resultant error dynamics consist of feedback interconnection of a slow error system (푒1, 푒2) and a fast error system (푒3):

⎡ ⎤ ⎡ ⎤ ⎡ ⎤ 푒˙1 푒1 휖 휖 푥1 휖 ⎣ ⎦ = 퐴푠(휖) ⎣ ⎦ + 퐵푠푒푒3 + 휖퐵푠푥 ⎣ ⎦ + 휖퐵푠푢푢, (3.50) 푒˙2 푒2 푥2 ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ 휖 휖 푒1 휖 푥1 휖 푢 휖푒˙3 = 퐴푓 푒3 + 휖퐵푓푒 ⎣ ⎦ + 휖퐵푓푥 ⎣ ⎦ + 휖퐵푓푢 ⎣ ⎦ . (3.51) 푒2 푥2 푢˙

82 The matrices in the slow error system (3.50) are defined as

⎡ ⎤ ⎡ ⎤ 휖 휖 휖 퐸11 퐸12 휖 퐸13 퐴푠(휖) := ⎣ ⎦ , 퐵푠푒 := ⎣ ⎦ , (3.52) 휖 휖 휖 푅 /휖 퐸22 퐸23 ⎡ ⎤ ⎡ ⎤ 휖 휖 휖 휖 퐻11 퐻12 휖 퐻1푢 퐵푠푥 := ⎣ ⎦ , 퐵푠푢 := ⎣ ⎦ , (3.53) 휖 휖 휖 퐻21 퐻22 퐻2푢

휖 ∑︀∞ 푘 푘−1 휖 ∑︀∞ 푘 푘−2 휖 ∑︀∞ 푘 where 퐻푞 := 푘=1 퐸푞 휖 (푞 ∈ {11, 12, 22}), 퐻21 := 푘=2 푅 휖 , 퐻1푢 := 푘=1[퐵1 − 푘 0 −1 0 푘−1 휖 ∑︀∞ 푘 푘−2 ∑︀∞ 푘 0 −1 0 푘−1 퐸13(푆 ) 퐵3 ]휖 , and 퐻2푢 := 푘=2 퐵2 휖 − 푘=1 퐸23(푆 ) 퐵3 휖 are all 풪(1). 휖 Note that we write 퐴푠(휖) instead of 퐴푠 because it is not a power series of 휖. In the 휖 휖 휖 휖 휖 휖 휖 ∑︀∞ 푘 fast error system (3.51), 퐴푓 := 푆 , 퐵푓푒 = 퐵푓푥 := [퐸31, 퐸32], and 퐵푓푢 := [ 푘=1(퐵3 + 푘 0 −1 0 푘−1 0 −1 0 푆 (푆 ) 퐵3 )휖 , (푆 ) 퐵3 ] are all 풪(1). To determine the magnitude of the approximation errors, we treat the error dynamics as a feedback interconnection of the slow error system (3.50) and the fast error system (3.51). Our subsequent analysis is aimed to obtain their respective input-to-state stability (ISS) properties (see Propo- sition 3.3-3.4) and then apply an ISS small-gain theorem to provide an upper bound for the approximation errors (Theorem 3.2).

ISS Property of the Slow Error System

Here, we first analyze the slow error system (3.50). Weuse | · |푄 to stand for the ⊤ 2 ⊤ 2 vector 2-norm weighted by matrix 푄 = 푄 > 0, so |푥|푄 = 푥 푄 푥 [43]. When 푄 is identity, we omit the subscript 푄 for simplicity. This convention is followed for all norms defined subsequently. For a matrix 퐴, we define |퐴| := sup |퐴푥| = 푄 |푥|퐼 =1 푄 [휆¯(퐴⊤푄2퐴)]1/2, where 휆¯(·) stands for the largest eigenvalue. For a square matrix 퐴, we use 휇푄(퐴) to represent the matrix measure associated with the 푄-weighted 2-norm −1 −1 ⊤ ¯ [43]. In particular, let Π(퐴, 푄) := 푄퐴푄 +푄 퐴 푄, we have 휇푄(퐴) = 휆[Π(퐴, 푄)]/2.

For a signal 푣(푡), ‖푣‖∞ := sup푡≥0 |푣(푡)| and ‖푣‖푎,푄 is the asymptotic signal ℒ∞ norm weighted by 푄: ‖푣‖푎,푄 := lim sup푡→∞ |푣(푡)|푄.

Definition 3.2. A system 푥˙ = 푓(푥, 푢) is said to be ISS if there exists a class 풦ℒ function 훽(·) and a class 풦 function 훾(·), such that for any 푢(푡) ∈ ℒ∞ and any initial

83 condition 푥(0), its trajectory satisfies |푥(푡)| ≤ 훽(|푥(0)|, 푡) + 훾(‖푢‖∞). O

For an LTI system, the following lemma connects its system matrix measure to its ISS property.

Lemma 3.6. Consider an LTI system 푥˙ = 퐴푥 + 퐵푢(푡) with input 푢(푡) ∈ ℒ∞, and suppose that for a positive definite 푄, there exists 푐 > 0 such that 휇푄(퐴) ≤ −푐, then the system is ISS and its trajectory satisfies ‖푥‖푎,푄 ≤ |퐵|푄‖푢‖푎/푐. O

2 ⊤ 2 ˙ ⊤ ⊤ 2 Proof. Consider 푉 (푥) = |푥|푄 = 푥 푄 푥, we have 푉 = 푥 푄[Π(퐴, 푄)]푄푥+2푥 푄 퐵푢 ≤ √ √ 2 2휇푄(퐴)|푥|푄 + 2|푥|푄|퐵|푄‖푢‖∞ ≤ −2푐푉 + 2 푉 |퐵|푄‖푢‖∞. Take 푊 = 푉 = |푥|푄, ˙ we obtain 푊 ≤ −푐푊 + |퐵|푄‖푢‖∞. By comparison lemma [81], 푊 (푡) = |푥(푡)|푄 ≤ −푐푡 −푐푡 |푥(0)|푄푒 + (1 − 푒 )|퐵|푄‖푢‖∞/푐 and thus, the system is ISS. Furthermore, we

have lim푡→∞ |푥|푄 ≤ |퐵|푄‖푢‖∞/푐, and by Lemma 10.4.4 in [71], it is equivalent to

‖푥‖푎,푄 ≤ |퐵|푄‖푢‖푎/푐.

To study the ISS property of the slow error system (3.50), we compute the matrix

measure of 퐴푠(휖) weighted by some 푄 matrix. The following eigenvalue perturbation result facilitates such computation when 휖 is small.

Lemma 3.7. (Corollary 4.13 in [156], Sec. IV.) Suppose matrices 퐴 and 퐸 are ¯ ¯ symmetric, and denote the largest eigenvalues of 퐴 and 퐴+퐸 by 휆 and 휆푝, respectively, ¯ ¯ then |휆 − 휆푝| ≤ |퐸|퐹 , where | · |퐹 stands for the Frobenius norm. O

Using Lemma 3.7, we provide an upper bound on the matrix measure of 퐴푠(휖) when 휖 is sufficiently small.

Lemma 3.8. Under Assumption 3.8, there exists an 휖0 > 0 such that for all 휖 ∈ (0, 휖0],

the matrix measure of 퐴푠(휖) weighted by

⎡ ⎤ 푄1 0 푄 = 푄(휖) = ⎣ √ ⎦ (3.54) 0 휖

satisfies 휇푄[퐴푠(휖)] ≤ −훼, where the positive definite matrix 푄1 and positive constant 훼 are independent of 휖. O

84 Proof. We split 퐴푠(휖) into two parts and write it as:

⎡ ⎤ ⎡ ⎤ 0 0 휖 휖 0 휖 퐸11 퐸12 휖퐷11 휖퐷12 퐴푠(휖) = 퐴푠(휖) + 퐷 := ⎣ ⎦ + ⎣ ⎦ . 0 0 휖 휖 푅 /휖 퐸22 퐷21 휖퐷22

* 0 * We first show that there exists an 휖-independent 훼 such that 휇푄[퐴푠(휖)] ≤ −훼 , 0 휖 and then provide an upper bound on 휇푄[퐴푠(휖)] = 휇푄[퐴푠(휖) + 퐷 ] using Lemma 3.7. Due to (ii)-(iii) in Assumption 3.8 and the KYP lemma [81], there exists a positive definite matrix 푃 and a constant 훼¯ > 0, both 휖-independent, such that

0 0 ⊤ 0 0 ⊤ 푃 퐸11 + (퐸11) 푃 ≤ −2¯훼푃 and 휓 := 푃 퐸12 + (푅 ) = 0. Using these results and by 1/2 taking 푄1 = 푃 , we have,

⎡ ⎤ 0 −1 √ 0 Π(퐸11, 푄1) 푄1 휓/ 휖 Π(퐴푠, 푄) = ⎣ ⎦ 푇 −1 ⊤ √ 0 휓 (푄1 ) / 휖 2퐸22 ⎡ ⎤ ⎡ ⎤ −1 0 0 ⊤ −1 푄1 [푃 퐸11 + (퐸11) 푃 ]푄1 0 −2¯훼퐼 0 = ⎣ ⎦ ≤ ⎣ ⎦ , 0 0 0 2퐸22 0 2퐸22

¯ 0 * 0 and thus, 휆[Π(퐴푠, 푄)]/2 ≤ −훼 := − min(¯훼, −퐸22) < 0. Applying the weight 푄 to 퐷휖, we have

⎡ ⎤ √ 휖 −1 ⊤ ⊤ 휖 √ 휖Π(퐷11, 푄1) 푄1퐷12 + (푄1 ) 퐷21 Π(퐷 , 푄) 휖 ⎣ ⎦ . ⊤ ⊤ −1 √ 퐷12푄1 + 퐷21푄1 2 휖퐷22

휖 √ Therefore, |Π(퐷 , 푄)|퐹 = 풪( 휖). By applying Lemma 3.7, for 휖 sufficiently small, we √ √ ¯ ¯ 0 * have 휇푄[퐴푠(휖)] = 휆[Π(퐴푠(휖), 푄)]/2 ≤ 휆[Π(퐴푠, 푄)]/2 + 풪( 휖) ≤ −훼 + 풪( 휖) ≤ −훼 for some 휖-independent 훼 > 0.

⊤ ⊤ With Lemma 3.6 and 3.8, and by treating 푒3(푡), 푥12(푡) := [푥1 , 푥2] and 푢(푡) as external inputs to the slow error system (3.50), we derive its ISS property.

Proposition 3.3. Under Assumption 3.8, for 푒3(푡), 푥12(푡), 푢(푡) ∈ ℒ∞, there exists an

85 휖1 > 0 such that for all 휖 ∈ (0, 휖1], system (3.50) is ISS and satisfies

‖푒1‖푎 ≤ 푘1푒‖푒3‖푎 + 휖푘1푥‖푥12‖푎 + 휖푘1푢‖푢‖푎, (3.55a) 푘 √ √ ‖푒 ‖ ≤ √2푒 ‖푒 ‖ + 휖푘 ‖푥 ‖ + 휖푘 ‖푢‖ , (3.55b) 2 푎 휖 3 푎 2푥 12 푎 2푢 푎

where the positive constants 푘푖푒, 푘푖푥 and 푘푖푢 (푖 = 1, 2) are 휖-independent. O

⊤ ⊤ Proof. Let 푒12 := [푒1 , 푒2] and take 휖1 < 휖0, we apply 푄 and 훼 in Lemma 3.8 to Lemma 3.6 to obtain:

|퐵휖 | |퐵휖 | |퐵휖 | ‖푒 ‖ ≤ 푠푒 푄 ‖푒 ‖ + 휖 푠푥 푄 ‖푥 ‖ + 휖 푠푢 푄 ‖푢‖ . 12 푎,푄 훼 3 푎 훼 12 푎 훼 푎

√︀ 2 Let 푑 := 휆(푄1) > 0 (휖-independent), where 휆(·) stands for the smallest eigenvalue,

since 푑|푒1(푡)| ≤ |푒12(푡)|푄, we have 푑‖푒1‖푎 ≤ ‖푒12‖푎,푄, and subsequently, ‖푒1‖푎 ≤ 휖 휖 휖 √ |퐵푠푒|푄 |퐵푠푥|푄 |퐵푠푢|푄 훼푑 ‖푒3‖푎 + 휖 훼푑 ‖푥12‖푎 + 휖 훼푑 ‖푢‖푎. Similarly, since 휖‖푒2‖푎 ≤ ‖푒12‖푎, we have |퐵휖 | √ |퐵휖 | √ |퐵휖 | 푠푒√ 푄 푠푥 푄 푠푢 푄 ‖푒2‖푎 ≤ 훼 휖 ‖푒3‖푎 + 휖 훼 ‖푥12‖푎 + 휖 훼 ‖푢‖푎. By (3.52) and Lemma 3.7, 휖 2 ¯ 휖 ⊤ 2 휖 휖 ⊤ 휖 ¯ 0 ⊤ 2 0 2 |퐵푠푒|푄 = 휆[(퐸13) 푄1퐸13 + 휖(퐸23) 퐸23] ≤ 휆[(퐸13) 푄1퐸13] + 풪(휖 ). Hence, for small 휖 enough 휖, there exists an 휖-independent 퐿푠푒 > 0 such that |퐵푠푒|푄 ≤ 퐿푠푒. Similarly, 휖 휖 we can find 휖-independent 퐿푠푥 and 퐿푠푢 such that |퐵푠푥|푄 ≤ 퐿푠푥 and |퐵푠푢|푄 ≤ 퐿푠푢.

Substituting these results into the inequalities for ‖푒푖‖푎 proves Proposition 3.3.

ISS Property of the Fast Error System

⊤ ⊤ Now we study the fast error system (3.51), treating 푒12(푡) = [푒1 , 푒2] , 푢(푡), 푢˙(푡) and

푥12(푡) as its inputs.

Proposition 3.4. If Assumption 3.7 is satisfied, then for 푒12(푡), 푥12(푡), 푢(푡), 푢˙(푡) ∈

ℒ∞, there exists an 휖2 > 0 such that for all 휖 ∈ (0, 휖2], system (3.51) is ISS and satisfies

‖푒3‖푎 ≤ 휖푘3푒‖푒12‖푎 + 휖푘3푥‖푥12‖푎 + 휖푘3푢 ‖[푢, 푢˙]‖푎 , (3.56)

where positive constants 푘3푒, 푘3푥 and 푘3푢 are 휖-independent. O

86 0 0 Proof. Since 퐴푓 = 푆 is Hurwitz, there exists an 휖-independent Θ > 0 such that 0 ⊤ 2 2 0 0 0 −1 2 (퐴푓 ) Θ + Θ 퐴푓 = ΘΠ(퐴푓 , Θ)Θ = −퐼. Thus, Π(퐴푓 , Θ) = −(Θ ) and there exists 0 ¯ 0 0 0 an 휖-independent 훽0 > 0 such that 휇Θ(퐴푓 ) = 휆[Π(퐴푓 , Θ)]/2 ≤ −훽0. Since 퐵푓푒, 퐵푓푥 0 0 0 0 and 퐵푓푢 are all 휖-independent, we can find 퐿푖 such that |퐵푖 |Θ ≤ 퐿푖 . By Lemma 휖 휖 0 0 3.7, since matrices 퐴푓 and 퐵푖 are 휖 perturbations of 퐴푓 and 퐵푖 , respectively, for 휖 휖 small enough, we can find positive constants 훽 and 퐿푖 such that |퐴푓 |Θ ≤ −훽 휖 휖 휖 and |퐵푖 |Θ ≤ 퐿푖. Note that since 휇Θ(퐴푓 /휖) = 휇Θ(퐴푓 )/휖, by Lemma 3.6, we have 퐿푓푒 퐿푓푥 퐿푓푢 √︀ 2 ‖푒3‖푎,Θ ≤ 휖 훽 ‖푒12‖푎 + 휖 훽 ‖푥12‖푎 + 휖 훽 ‖[푢, 푢˙]‖푎 . Since ‖푒3‖푎 ≤ ‖푒3‖푎,Θ/ 휆(Θ ), and that Θ is independent of 휖, we have (3.56) verified.

Error between the Full and the Reduced Systems

In Proposition 3.3 and 3.4, the error bounds depend on the trajectory of the candidate reduced system (3.46). Such a system has been studied in [64] and we bound its trajectory in the next proposition.

Proposition 3.5. Assume that 푢(푡) has bounded derivatives (i.e., u ∈ ℒ∞) and that

Assumption 3.8 is satisfied, then there exists an 휖3 > 0 such that for all 휖 ∈ (0, 휖3], √ the trajectory of (3.46) satisfies ‖푥12‖푎 ≤ 푘푟/ 휖, where the positive constant 푘푟 =

푘푟(‖u‖∞) is independent of 휖. O

Proof. (Sketch.) Theorem 2 in [64] establishes that tracking error of a high-gain (1/휖) √ negative feedback interconnection of two SPR systems is 풪( 휖). With reference to √ Figure 3-6, in the context of (3.48), this implies that 푣2 = 풪(1/ 휖). Regarding 푣2 as 휖 an input to the series interconnection of Σ2 and Σ1, since all matrices representing √ the two systems are 풪(1), all state variables are bounded above by 풪(1/ 휖).

Finally, based on Proposition 3.3-3.5, we state our main result, which provides ultimate upper bounds on the approximation errors. The proof relies on the ISS small-gain theorem.

Lemma 3.9. (Thm. 10.6.1 in [71]) Consider two ISS systems 푥˙ 푖 = 푓푖(푥푖, 푣푖, 푢) (푖 = 푖 1, 2) that each satisfies the asymptotic ISS property: ‖푥푖‖푎 ≤ 훾푖‖푣푖‖푎 + 훾푢‖푢‖푎 in

87 isolation, their feedback interconnection, which is obtained by setting 푣1 = 푥2 and

푣2 = 푥1 and viewed as a system with state (푥1, 푥2) and input 푢, is ISS if 훾1 · 훾2 < 1. O

Theorem 3.2. Suppose u ∈ ℒ∞ and that Assumptions 3.7-3.8 are satisfied, then

there exists 0 < 휖¯ < min푖 휖푖 (푖 = 0, 1, 2, 3), such that for all 휖 ∈ (0, 휖¯], the error

system (3.50)-(3.51) is ISS with states 푒푖 and input 푢. In addition, there exists 휖- √ independent positive constants 퐾푖 = 퐾푖(‖u‖∞), such that ‖푒푗‖푎 ≤ 퐾푗 휖 for 푗 = 1, 3 and ‖푒2‖푎 ≤ 퐾2. O

Proof. We treat the error dynamics as a feedback interconnection of the slow and fast

error dynamics, with states 푒12 and 푒3 and external inputs 푥12(푡), 푢(푡) and 푢˙(푡). By

inequalities (3.55), for all 휖 < 휖¯, there exists constants 푘푠푒, 푘푠푥 and 푘푠푢 such that the

slow error system states 푒12 are bounded by:

푘 √ √ ‖푒 ‖ ≤ √푠푒 ‖푒 ‖ + 휖푘 ‖푥 ‖ + 휖푘 ‖푢‖ . (3.57) 12 푎 휖 3 푎 푠푥 12 푎 푠푢 푎 √ Therefore, the gain of the slow error system (3.50) is 푘푠푒/ 휖. Similarly, by (3.56), √ √ the gain of the fast error system (3.51) is 휖푘3푒. Since 휖푘3푒 · 푘푠푒/ 휖 = 휖푘푠푒푘3푒 < 1 for 휖 small enough, the error system is ISS by Lemma 3.9. Substituting (3.57) into

(3.56), for 휖 small enough, we can find 휖-independent positive constants 휅푥 and 휅푢 √ such that ‖푒3‖푎 ≤ 휖휅푥‖푥12‖푎 + 휖휅푢‖[푢, 푢˙]‖푎. Using the fact that ‖푥12‖푎 ≤ 푘푟/ 휖

(Proposition 3.5), for 휖 small enough, we can find a 퐾3 = 퐾3(‖u‖∞) > 0 such that √ √ √ ‖푒3‖푎 ≤ 휖(휅푥푘푟 + 휖휅푢‖[푢, 푢˙]‖푎) ≤ 퐾3 휖, and subsequently from (3.55), there exists √ a 퐾푗 = 퐾푗(‖u‖∞) > 0 (푗 = 1, 2) such that ‖푒1‖푎 ≤ 퐾1 휖 and ‖푒2‖푎 ≤ 퐾2.

3.3.3 QIC tracking performance

Here, we apply Theorem 3.2 to perform model reduction a linearized model of the type II QIC-regulated system in (3.37) and then use the reduced model to analyze its tracking performance. Suppose the input can be written as 푢(푡) = [¯푟+˜푟(푡), 푤¯+푤 ˜(푡)]⊤, for a sufficiently small positive 휖, as we have shown in Section 3.2, the equilibrium 휁¯

88 corresponding to the constant input 푢¯ satisfies

[︂ 훿푟¯ 푤¯ 훼훽푟¯ 푟¯]︂⊤ 휁¯(¯푢, 휖) := [¯푐휖 , 푐¯휖 , 푝¯휖]⊤ = − , , + 풪(휖) (3.58) 1 2 훼훽 훼 휃훿푟¯ − 휃훽푤¯ 훽

Hence, we linearize (3.37) about (¯푢, 휁¯) to obtain

휖 휖 휖푐˙1 =푟 ˜(푡) − (휃푐¯2 + 휖훿)푐1 − 휃푐¯1푐2, (3.59a)

휖 휖 휖푐˙2 = 훼푝 − 휃푐¯2푐1 − (휃푐¯1 + 휖훿)푐2, (3.59b)

푝˙ = 훽푐1 − 훿푝 +푤 ˜(푡). (3.59c)

Following Lemma 3.5, we find that the 휖-independent transformation 푧1 = 푝, 푧2 = 0 0 0 0 0 0 푐1 −푐2, and 푧3 = 휃푐¯2(¯푐1 +푐 ¯2)푐1/(훼푐¯1)+휃(¯푐1 +푐 ¯2)푐2/훼−푝 takes the system into normal SSP form: 0 0 0 푧˙1 = 퐸11푧1 + 퐸12푧2 + 퐸13푧3 +푤 ˜(푡),

0 0 0 휖푧˙2 = 푅 푧1 + 휖퐸22푧2 + 퐵2 푟˜(푡), (3.60)

휖 휖 휖 0 휖푧˙3 = 휖퐸31푧1 + 휖퐸32푧2 + 푆 푧3 + 퐵3 푟˜(푡),

0 0 0 0 0 0 0 0 훼훽푐¯1 0 where 푅 = −훼, 퐸22 = −훿, 퐵2 = 1, 푆 = −휃(¯푐1 +푐 ¯2), 퐸11 = 0 0 2 − 훿, 퐸12 = 휃(¯푐1+¯푐2) 0 0 0 0 0 훽푐¯1 0 휃푐¯2(¯푐1+¯푐2) 0 훼훽푐¯1 0 0 0 0 , 퐵3 = 0 , and 퐸13 = 0 0 2 . By substituting 푐¯1 and 푐¯2 into the above 푐¯1+¯푐2 훼푐¯1 휃(¯푐1+¯푐2) expressions, it is easy to verify that for any positive rate constants 훼, 훽, 휃, 훿 and input

0 0 푢¯, we have 퐸11 < 0 and 퐸12 > 0. Hence, independent of exact parameter values, 0 0 0 −1 0 0 0 퐻1 (푠) = −푅 (푠퐼 − 퐸11) 퐸12 = 훼퐸12/(푠 − 퐸11) is SPR and Assumption 3.8 can be verified. By (3.46)-(3.47), the reduced system of (3.60) is

0 0 0 푥˙ 1 = 퐸11푥1 + 퐸12푥2 + 퐵1푟푟˜(푡) +푤 ˜(푡), (3.61)

0 −1 0 푥˙ 2 = [˜푟(푡) − 훼푥1]/휖 − 훿푥2, 푥3 = −(푆 ) 퐵3 푟˜(푡),

0 0 0 0 2 where 퐵1푟 = 훽푐¯2/(¯푐1 +푐 ¯2) . By Theorem 3.2, given u˜(푡) ∈ ℒ∞, we have ‖푧푖 − 푥푖‖푎 = √ 풪( 휖) for 푖 = 1, 3. Since 푧1 = 푝 is the concentration of the regulated protein, we

treat 푧1 (푥1) as the output of the full (reduced) system. The reduced system (3.61) is a high-gain negative feedback interconnection of two SPR systems and tracking

89 performance of such systems has been evaluated in [64].

Lemma 3.10. ([64].) Consider (3.61) under Assumption 3.8. Suppose that ˜r(푡), w˜ (푡) ∈ √ ℒ∞, then there exists 퐾 > 0, independent of 휖, such that ‖푥1 − 푟˜(푡)/훼‖푎 ≤ 퐾 휖. O

Thus, by triangle inequality, tracking performance of the full system (3.60) can be certified.

Proposition 3.6. Consider (3.60) under Assumption 3.8. Suppose that ˜r(푡), w˜ (푡) ∈ √ ℒ∞, then ‖푧1 − 푟˜(푡)/훼‖푎 = 풪( 휖). O

In Figure 3-7, we plot the tracking error |푧1 − 푟˜(푡)/훼| (Figure 3-7a) and model reduction error 푒1 = |푧1 − 푥1| (Figure 3-7b) for the linearized type II QIC-regulated system (3.59). In particular, in these simulations, the reference input is set as a constant, hence, 푟(푡) ≡ 푟¯ and the disturbance 푤(푡) =푤 ˜(푡) is a band-limited white noise. In Figure 3-7a, we find the tracking performance of (3.59) improves as 휖 is decreased, which is consistent with the analytical results in Proposition 3.6. As shown in Figure 3-7b, the model reduction error for the slow variable also decreases as 휖 is decreased. In Figure 3-7a, we also use dashed lines to plot the output tracking error for the nonlinear system (3.37) and find that the tracking error is also small. Tracking performance analysis for the nonlinear QIC system is one of our future

0 research directions. Since 퐻1 (푠) is SPR for all positive rate constants, this result is also robust in the sense that arbitrarily small tracking error can be achieved by increasing all controller reaction rates (decreasing 휖) even in the presence of parameter uncertainties.

3.4 Experiment: sRNA-mediated QIC

Here we construct in E. coli bacteria an sRNA-mediated type-II QIC based on our analysis in Section 3.2.5. The controller is constructed and the experiments are designed to allow the expression level of a constitutive gene to reach a set-point regardless of a disturbance arising from step induction of a “resource competitor” gene.

90 (a) 1 w(t)

0.5

tracking error -0.5 є=0.01 є=0.1 є=1 -1 0102030405060708090 100 (b) time

100

10-2

10-4

10-6

10-8 model reduction error

10-10 0102030405060708090 100 time

Figure 3-7: Model reduction and tracking performance of a type II QIC. (a) Output tracking error and (b) model reduction error for the output variable. The dashed lines in panel (a) are for the nonlinear system (3.37) and the solid lines are for the linearized system (3.60). Simulation parameters: 훼 = 휃 = 훿 = 1, 푟(푡) =푟 ¯ = 1, 푤¯ = 0, 푤˜(푡) is a band-limited white noise disturbance input with maximum magnitude of 0.8.

91 3.4.1 Genetic circuit

Figure 3-8a illustrates the genetic circuit diagram of the specific sRNA-mediated QIC we design. In order to evaluate the ability of the controller to make the output of subsystem 1 adapt to changing ribosome demand by subsystem 2, we assembled a library of test-bed genetic circuits with two subsystems shown in Figure 3-8a. Specif- ically, subsystem 2 applies a variable demand for ribosomes. It is externally inducible by AHL and RFP is its output to assess ribosome demand. We embedded a type II QIC enabled by sRNA interference in subsystem 1. For this subsystem, we chose constitutive promoters since our design needs to demonstrate that the subsystem’s output (GFP level) stays unchanged when its input is kept constant, despite a change in ribosome availability. This is a model system for studying competition for ribosomes as employed in earlier works [27, 36, 63, 141]. Specifically, referring to Figure 3-8a, a pLacIQ promoter is used to constitutively express LuxR. In the presence of

LuxR’s effector AHL, AHL-bound LuxR (푟2) is formed to activate subsystem 2 to produce disturbance output RFP (푦2 = 푝2). These two genes constitute a resource competitor that demands more ribosomes when AHL concentration increases, leading to a decrease in the amount of available ribosomes to translate mRNAs in subsystem 1.

Subsystem 1 embeds the sRNA-mediated type II QIC. The controller consists of four key biological parts: sRNA-A and its targeting sequence [55], ECF sigma factor 32_1122 (referred to as ECF32 hereafter), and its cognate promoter pECF32 [129]. The choice of ECF32 and pECF32 is based on their minimal impact on host cell growth and their dose response’s wide dynamic range. Specifically, ECF32 gene is introduced downstream of the output gfp gene to form a bi-cistronic operon. Co- transcription of gfp and ECF32 genes is driven by a constitutive promoter (Figure 3-8a). We constructed systems that either employ the stronger constitutive promoter Ec-TTL-P109 [86] or the weaker constitutive BioBrick promoter BBa_J23116. This provides a means to adjust the transcription rate constant of subsystem 1 as a reference input. The mRNA co-transcript of gfp and ECF32 genes has the sRNA-A’s

92 feedback gain: κ (a) p GFP ECF32 RBS strength regulated subsystem 1 1 k = κ = GFP ECF32 GFP RBS strength post-TX controller pECF32 AHL ColE2 sRNA-A Sp s 1 p LuxR subsystem 2 2 ECF32 pc,1 r2 RFP

r1 m1

Amp p15A B gfp ECF32 luxR rfp pLacIQ pLux Ec-TTL-P109 (stronger) BBa_J23116 (weaker) resource competitor

(b) unregulated subsystem1 ColE2 Sp p terminator 1 AHL GFP

p ECF32 LuxR subsystem 2 2 u2 RFP

Amp p15A B gfp ECF32 luxR rfp pLacIQ pLux Ec-TTL-P109 (stronger) BBa_J23116 (weaker) resource competitor

Figure 3-8: sRNA-mediated post-transcriptional controllers. (a) Circuit diagram of a regulated subsystem 1 with inducible subsystem 2 functioning as a resource competitor. Upwards arrow with tip rightwards indicate promoters, semicircles represent RBSs, orange box with letter B stands for sRNA-A targeting sequence, and “⊤” symbols represent terminators. The feedback controller consists of ECF32, co-transcribed with GFP, which is used to sense GFP translation rate and to actuate transcription of sRNA-A. sRNA-A antisenses its targeting sequence on the mRNA for degradation of both RNA molecules. (b) Circuit diagram of an unregulated subsystem 1 with inducible subsystem 2 functioning as a resource competitor. The pECF32 promoter and sRNA-A in panel a are removed.

93 targeting sequence immediately upstream of the gfp gene’s RBS. The sRNA can complementarily pair with its target mRNA, forming an inert RNA complex that is rapidly cleaved by RNase E, leading to coupled degradation of both mRNA and sRNA [3, 90, 97, 149, 158].

Since GFP and ECF32 proteins are translated from the same mRNA co-transcript

푚1 using the same pool of ribosomes, this bi-cistronic operon design allows ECF32 TL rate to be proportional to GFP TL rate by a factor 푘, which is the ratio between GFP and ECF32 RBS dissociation constants:

휅 푘 := GFP . (3.62) 휅ECF32

This relationship allows the sRNA-mediated post-transcriptional controller to sense a change in GFP TL rate through ECF32 TL rate and respond to this change by adjusting subsequent ECF32 protein and sRNA concentrations, which subsequently

determine mRNA co-transcript concentration 푚1 as the controller’s output. More specifically, when the resource competitor is activated, the amount of ribosome available to translate GFP decreases, causing a reduction ∆ in its TL rate. Due to the bi-cistronic operon design, a decrease in GFP TL rate is always accompanied by a ∆/휖 decrease in ECF32 TL rate. Reduction in ECF32 TL rate, in turn, leads to a decreased ECF32 concentration 푝푐,1, which decreases sRNA-A’s transcription, leading

to an increase in the concentration of GFP/ECF32 mRNA co-transcripts 푚1. This allows GFP TL rate to recover, closing the feedback loop. Based on the above reasoning, a regulated subsystem with a higher 푘 responds to the same decrease ∆ in GFP TL rate with a larger decrease in sRNA transcription, leading to increased control action (sRNA-A transcription). Therefore, we call 푘, as defined in equation (3.62), the feedback gain of the sRNA-mediated embedded post-transcriptional controller. Increasing 푘 also matches the design requirement for type II QIC, as we discuss in more detail in the next section.

To experimentally evaluate the benefit of the sRNA-mediated QIC, we constructed another library of circuits with unregulated subsystem 1 and same resource competitor

94 (Figure 3-8b). In particular, the only difference is the absence of pECF32 promoter and sRNA-A message in the unregulated subsystem. As a result, the mRNA level in the unregulated subsystem 1 (푚1) is not responsive to the output 푦1 = 푝1 TL rate, breaking the feedback loop.

3.4.2 Model guided controller tuning for set-point regulation

Parametric conditions to reach QIC

If we assume that (A) GFP and ECF32 proteins decay with the same rate constant 훾 and that (B) we start our experiment from steady state gene expression, then our circuit design allows ECF32 protein concentration to be theoretically proportional to

GFP concentration by the feedback gain 푘: 푝푐,1(푡) = 푘푝1(푡). As a result, following the model derivation of sRNA-based type II QIC in Appendix Section B.4, the regulated subsystem 1 (Figure 3-8a) can be described by:

푚˙ 1 = 푇 퐷퐻(푟1) − 휆푚1푠1/훽 − 훿푚1, (3.63a)

푠˙1 = 푘푇s푝1 − 휆푚1푠1/훽 − 훿푠1, (3.63b)

푝˙1 = 푅(1 − 푤)푚1/휅GFP − 훾푝1, (3.63c)

where 푚1, 푠1 and 푝1 stand for the concentrations of the GFP/ECF32 mRNA co- transcript, sRNA, and GFP protein, respectively. In this model, function 퐻(푟1) ∈

[0, 1] describes transcriptional regulation by TF input 푟1 and 퐻(푟1) ≡ 1 when the subsystem’s promoter is constitutive. Parameter 퐷 is the plasmid copy number of the regulated gene; 푇 is the transcriptional rate constant per DNA copy, which is primarily dictated by the subsystem’s promoter strength; 푇s is a lumped transcription rate constant for sRNA, which is proportional to its plasmid copy number and pECF32’s promoter strength; 푘 is the feedback gain defined previously in equation (3.62); 훿 is the decay rate constant of uncoupled mRNA and sRNA, which we assume to be identical for both species for simplicity; 휆 is the mRNA-sRNA decay rate constant; 훽 is the dissociation constant of mRNA-sRNA binding, and 푅 is the maximum

95 translation rate constant proportional to the total amount of ribosomes.

We first demonstrate that system (3.63) is a type II QIC-regulated system when the parameter conditions (I) 훿/휆 ≪ 1, (II) 훿/푇 ≪ 1, and (III) 훿/(푘푇s) ≪ 1 are all satisfied. In particular, when (I)-(III) are satisfied, we canwrite 휆 = 훿/휖, 푇 = 휈1훿/휖, and 푘푇s = 휈2훿/휖, with dimensionless parameters 휈1, 휈2 > 0 and 0 < 휖 ≪ 1. Using this re-parameterization, model (3.63) becomes

훿 푚˙ = (휈 퐷퐻(푟 ) − 휖푚 − 푚 푠 /훽), (3.64a) 1 휖 1 1 1 1 1 훿 푠˙ = (휈 푝 − 휇휖푠 − 푚 푠 /훽), (3.64b) 1 휖 2 1 1 1 1

푝˙1 = 푅(1 − 푤)푚1/휅GFP − 훾푝1. (3.64c)

System (3.64) is in the form of the type II QIC we study in Section 3.2.5 and can therefore reach 휖-set-point regulation for sufficiently small 휖.

Now we consider if parameter conditions (I)-(III) can be satisfied in practice. Condition (I) is readily satisfied for the sRNA-mediated feedback because decay rate constant of the sRNA-mRNA complex (휆) is much larger than that of the uncoupled mRNA and/or sRNA (훿) [69, 90]. Condition (II) can be satisfied for a regulated subsystem with a reasonably strong promoter. In particular, the mRNA/DNA ratio for the Ec-TTL-P109 promoter is close to 26 [86]. For constitutive transcription of a gene with copy number 퐷, we have mRNA concentration dynamics 푚˙ = 푇 퐷 − 훿푚. Using this simple model and the mRNA/DNA ratio provided in [86], at steady state, we have 훿/푇 = 퐷/푚 ≈ 2−6 = 0.015, and therefore condition (II) is satisfied. The weaker BBa_J23116 promoter we choose is ∼ 20x weaker than the Ec-TTL-P109 promoter according to our experiment. This implies that for the weaker BBa_J23116 promoter, 훿m/푇 ≈ 0.3. Parameter condition (III) can be reached by either having a sufficiently large 푇s, which is dictated by the pECF32 promoter activity and its plasmid copy number, or by employing a strong feedback gain 푘.

96 Experimental validation

We experimentally demonstrate that increasing the feedback gain 푘 through the ECF32 RBS strength is an effective way to achieve 휖-set-point regulation. We constructed and tested the circuits in Figure 3-8a (b) consisting of a resource competitor and a regulated (unregulated) subsystem. To assess the performance of each regulated subsystem from experimental data, we use its robustness as our main performance metric. In particular, robustness of a subsystem is defined as the percentage of GFP expression when the resource competitor is fully activated (i.e., AHL=1000 nM) relative to its nominal output, which is the GFP expression when the resource competitor is inactive (i.e., AHL=0):

GFP expression when AHL=1000 nM robustness := × 100%. (3.65) GFP expression when AHL=0

Based on this performance metric, robustness of a subsystem that can adapt perfectly to AHL induction as a disturbance is 100%. From our analysis, the ability of the post- transcriptional controller to adapt to variable ribosome availability is dictated by a high feedback gain 푘, which can be increased in our circuit by increasing the RBS strength of ECF32. Therefore, we constructed a library of 6 regulated subsystems with 3 different ECF32 RBS’s and 2 different promoters. In particular, ECF32 RBS’s have translation initiation rates (TIRs) of 565, 1127 and 6474, as calculated by the RBS calculator 2.0 [134], allowing the regulated subsystems to be equipped with low, medium and high gains, respectively. The RBS of GFP is kept unchanged with a TIR of 974 in all experiments. The promoter of a regulated subsystem is either Ec-TTL-P109 (stronger) [86] or BBa_J23116 (weaker). Assessing the performance improvement with increased feedback gain under different promoter strengths allows us to validate that the design principle is independent of GFP level.

We first evaluated the robustness and nominal output for all 6 regulated subsystems (Figure 3-9a). We find that, independent of the choice of promoter, increasing feedback gain 푘 promotes robustness. Specifically, for the circuits with the stronger (weaker) promoter, increasing feedback gain from low to high improved robustness

97 from about 60% (50%) to about 90% (75%). As predicted by our model, the regulated subsystems employing high feedback gain are most robust to resource competitor activation (Figure 3-9a). By contrast, the low-gain regulated subsystems are the least robust. We then compared the performance of the high-gain and the low-gain regulated subsystems for the same promoter (Figure 3-9b). While both the regulated and unregulated subsystems with low gain suffered more than 40% decrease in GFP level when the resource competitor was maximally activated, the high-gain regulated subsystem was practically unaffected by activation of the resource competitor (Figure 3-9b). Since both the regulated and the unregulated subsystems contain the ECF32 gene (Figure 3-8a,c), these results confirm that the robustness of the high-gain regulated subsystem is not due to the mere presence of ECF32. Furthermore, RFP expression levels are comparable in all experiments (Figure 3-9b and Figure S10), indicating comparable disturbances created by the competitor on all circuits. Growth rates of cells containing unregulated high-gain subsystem were appreciably slower than those of cells bearing high-gain regulated subsystems. This is because nominal output from the unregulated subsystem was higher than the output of its regulated counterpart, imposing a larger load on cell growth (Figure 3-9b).

Regulated and unregulated subsystems with similar output levels

In order to further confirm that the controller performance is independent ofGFP level, we performed a detailed comparison between an unregulated subsystem and a regulated subsystem with high-gain and with comparable nominal output. These two subsystems are represented by the white circle and black square, respectively, in Figure 3-10a. The regulated subsystem uses the stronger Ec-TTL-P109 promoter and is embedded with an sRNA-mediated post-transcriptional controller with ECF32 whose RBS TIR is 6474 (i.e., high feedback gain), while the unregulated subsystem is only different in that it uses the weaker BBa_J23116 promoter and removes pECF32 promoter and sRNA-A message. The steady state AHL dose responses of the two circuits are shown in Figure 3-10a. While the nominal output values were comparable in

98 (a) regulated subsystem

6 10 k Model increasing 106

104 40 100 robustness (%) nominal output (A.U.) 105 increasing k nominal output (A.U.)

104 40 60 80 100 robustness (%)

(b) unregulated subsystem regulated subsystem 4 k ×10 × 0.1

1.2 ) p=0.51 7 6 -1 1

0.8 p=0.34 3 5 0.6

RFP/OD (A.U.)

0.4 0 growth rate (hr 3 normalized GFP/OD low feedback gain 0 6 1000 0 6 1000 0 6 1000 AHL (nM) AHL (nM) AHL (nM)

k ×104 × 0.1

1.2 ) 7 p=0.26 p=7.3E-7 6 -1 1

Ec-TTL-P109 promoter 0.8 5 3 0.6

RFP/OD (A.U.)

0.4 growth rate (hr

normalized GFP/OD 0 3 0 6 1000 0 6 1000 0 6 1000 high feedback gain AHL (nM) AHL (nM) AHL (nM)

Figure 3-9: Performance of a regulated subsystem is determined by its feedback gain. (a) Increasing feedback gain enhances the robustness of a regulated subsystem driven by either the stronger Ec-TTL-P109 promoter (a square symbol) or the weaker BBa_J23116 promoter (a circle symbol). A symbol’s color filling in gray scale from light to dark represents a feedback gain from low to high. Simulation results for model (3.63a)-(3.63c) are in the insert box. (b) GFP outputs, growth rates, and RFP outputs for low-gain and high- gain regulated and unregulated subsystems using the Ec-TTL-P109 promoter. GFP per OD values in arbitrary unit (A.U.) were normalized to their respective nominal outputs. All experimental data were obtained with a microplate photometer. Data with error bars represent mean values ± standard deviations. The regulated subsystem using Ec-TLL-P109 promoter and high feedback gain has six replicates (three biological replicates each with two technical replicates). Other subsystems have three biological replicates. Specific value of an independent experiment presents as a black dot. Two-tailed, unpaired 푡-test is used to compute 푝-values.

99 population-level measurements by mircoplate photometer (a) 106 (c) unregulated subsystem regulated subsystem AHL induction AHL induction

1 1 GFP/OD GFP/OD Normalized Normalized 0.4 0.4 × 4 × 104 4 10 nominal output (A.U.) 10 40 robustness (%) 100 6 AHL = 0 6 AHL = 1000 nM 4 4 regulated subsyst. unregulated subsyst. 2 2 (weaker promoter) (stronger promoter)

RFP/OD (A.U.) 0 RFP/OD (A.U.) 0 ×104 ×104 6 6 p=0.016 p=0.024 ) ) p=8.2E-8 p=0.81 -3 -3 600 600 4 4 p=2.2E-4 -4 -4 ln (OD ln (OD GFP/OD (A.U.) p=3.8E-8 GFP/OD (A.U.) -2.5 -0.5 0 1.5 3.5 5.5 7.5 -2.5 -0.5 0 1.5 3.5 5.5 7.5 2 2 time (hr) time (hr) 0 6 1000 0 6 1000 AHL (nM) AHL (nM) normalized fluorescence histograms ×104 ×104 6 6 (b) (d) from single-cell measurements by cytometery RT-qPCR of RFP mRNA AHL = 0 AHL = 1000 nM AHL = 0 3 unregulated subsystem regulated subsystem 3 AHL = 1000 nM 20 biological RFP/OD (A.U.) RFP/OD (A.U.) replicate #1 0 0 6 1000 0 0 6 1000 AHL (nM) AHL (nM) ×0.1 ×0.1 ) ) 7 7 10 -1 -1 biological replicate #2 relative RFP mRNA 5 5

0 biological unregulated regulated growth rate (hr

growth rate (hr replicate #3 3 3 device device 0 6 1000 0 6 1000 AHL (nM) AHL (nM) GFP fluorescence GFP fluorescence

Figure 3-10: A regulated subsystem is more robust than an unregulated subsystem with comparable output level. (a) AHL dose responses of the regulated (black square in top panel) and unregulated (white circle in top panel) subsystems. The regulated subsystem uses a stronger Ec-TTL-P109 promoter and the unregulated subsystem uses a weaker BBa_J23116 promoter. For both circuits, the RBS TIRs of ECF32 are 6474, corresponding to high feedback gain. Error bars represent standard deviation from six replicates, including three biological replicates each having two technical replicates. Specific value of an independent experiment presents as a black dot. Two-tailed, unpaired 푡-test is used to compute 푝-values. (b) The mRNA level of mRFP1 gene was quantified by RT-qPCR and normalized to the reference gene cysG. Data represent mean values (± standard deviation). (c) Temporal responses of the unregulated and the regulated subsystems were monitored in parallel by microplate photometer. Cells were first grown in multiple batches in the absence of AHL for 8 hours till they reach steady state GFP expression. Each batch was diluted every 2 hours to maintain exponential cell growth. The resource competitor was then induced at 푡 = 0 with the indicated AHL concentrations. Sampling time interval of microplate photometer was 2 min. Mean GFP per OD values in arbitrary unit (A.U.) are normalized to those of samples without AHL induction to reflect relative change in GFP expression. Error bars represent standard deviation from three biological replicates. (d) GFP fluorescence histograms (normalized to the statistical mode) of three biological replicates measured by flow cytometry 6 hours after AHL induction.

100 the regulated and unregulated subsystems (∼ 4.5 × 104 for the regulated subsystem and ∼ 5.5 × 104 for the unregulated subsystem), the robustness of the unregulated subsystem was only 50% in contrast to a high robustness of nearly 95% for the regulated subsystem (Figure 3-10a). The resource competitors in both circuits produced similar levels of RFP and imparted comparable growth retardation, indicating that comparable disturbances were applied to the regulated and unregulated subsystems (Figure 3-10a). This is further supported by RT-qPCR measurements, which confirm that when the resource competitor was fully activated, RFP mRNA levels in cells bearing the regulated subsystems were no less than the RFP mRNA levels found in cells bearing the unregulated subsystems (Figure 3-10b). In Figure 3-10c, we report the dynamic responses of the two circuits when subject to a step disturbance input (i.e., AHL induction). Consistent with our analysis, the unregulated subsystem’s output dropped after AHL induction and did not recover. By contrast, the output of the regulated subsystem was nearly unaffected by the applied disturbance. The similar growth profiles of the regulated and unregulated subsystems, reflected in the values of OD at 600 nm in Figure 3-10c, show thatthe sRNA-mediated feedback did not unfavorably affect cell growth. With reference to Figure 3-10d, we did not observe any appreciable difference in gene expression noise between the regulated and unregulated subsystems, consistent with experimental results by others [80]. Altogether, the results in Figure 3-10 demonstrate that the sRNA-mediated quasi-integral control is an effective approach to enable a genetic subsystem to become robust to variations in ribosome availability.

3.5 Summary

Recent trends in synthetic biology to move from prototypes to applications have trig- gered higher expectations on the robustness of genetic circuit. In this chapter, we propose a systematic approach to designing biomolecular controllers for regulating gene expression. Although a number of ideal integral controllers (IICs) have been proposed for set-point regulation, their performance in vivo is challenged by integra-

101 tion leakiness due to dilution, which cannot be neglected in fast growing cells such as bacteria. We propose two types of quasi-integral controller (QIC) motifs designed based on existing IIC motifs and a timescale separation principle. We prove that by engineering all controller reaction rates to be much faster than dilution, set-point regulation can be achieved even in the presence of a leaky integrator. We provide two physical realizations of QICs using phosphorylation cycle and small RNA (sRNA) interference, respectively. When implemented in E.coli bacteria, expression of a QIC- regulated gene show almost perfect adaptation to a step disturbance while expression of an unregulated gene is decreased by more than 50% in the presence of the same disturbance. The proposed QIC designs can also track a time-varying reference in the presence of time-varying disturbances. We prove this by performing model order reduction for the linearized QIC-regulated systems, and then show robust tracking of the reduced order model. Model reduction for the QIC-regulated systems involves a singular singular perturbation (SSP) problem, where the boundary layer dynamics of a two timescale system is singular, making standard singular perturbation inapplicable. We provide I/O conditions on the boundary layer and the reduced system to carry out model reduction for SSP system.

102 Chapter 4

Robustness of networked systems to unintended interactions

Equipped with biomolecular controllers that allow a regulated genetic subsystem to be context-independent, it is tempting to create robust genetic circuits by composing these regulated subsystems together. While appealing, this approach does not in general guarantee that the behavior of the resultant circuit, which we treat as a networked dynamical system here, to be robust to unintended interactions. In fact, while we assume in Chapter 3 that the disturbances to subsystems are external and state-independent, the unintended interactions in the network interconnects the disturbance inputs, making them state-dependent to potentially deteriorating network robust performance. This chapter takes a general networked systems approach to provide sufficient conditions such that the robustness of a networked system tounin- tended interactions can be guaranteed by the robustness of the composed subsystems to state-independent disturbances.

4.1 Introduction

A networked system is the interconnection of input/output (I/O) subsystems through a prescribed interaction map. Many properties of networked systems can be determined using I/O properties of the constituent subsystems and the prescribed inter-

103 action map [13, 49, 74, 77, 108]. Here, we consider the case where a networked system, which we refer to as “nominal network”, is perturbed by unintended interactions among subsystems (Figure4-1). These unintended interactions can arise from, one subsystem physically perturbing the environment that comprises all other subsystems, thereby affecting other subsystems in the network. For example, inclose formation control of aerial vehicles, the vortex created by the propulsion force of the leading vehicle can severely affect the dynamics of its neighbors, creating instabil- ity [51, 117, 137, 145]; in a wind farm with multiple turbines, the wake effect of one turbine alters the surrounding air flow, which, in turn, affects adjacent turbines, reducing efficiency [18, 24]; in building temperature control, the temperature difference between neighboring rooms induces thermal conduction, which results in deviation of each room’s temperature from its set point [95]; in genetic circuits, increased expression of one gene decreases the amount of resources available to express other genes, unintentionally reducing these genes’ expression levels [63, 120].

To retain the prescribed function of a network despite unintended interactions, one approach is to co-design all subsystems and their interactions monolithically [18, 24, 117, 120]. A different approach, taken in networked systems research, is to allow each subsystem to be designed independently of the others, thus allowing scalable network analysis and design [6, 13, 49, 77, 107, 108, 143]. Specifically, works in this direction focused on deriving conditions on subsystem dynamics and interaction map for network stability and performance. Here, we take the networked systems research approach to obtain conditions for robust performance with respect to an unintended interaction maps, as opposed to state independent disturbances as considered in previous works [6, 54, 70, 75, 177]. While these works enable scalable network analysis and design, robust performance with respect to unintended interaction maps has not been considered.

Here, we consider a network composed of heterogeneous nonlinear subsystems interconnected through static prescribed and unintended interaction maps (Figure4-1).

Each subsystem 푖 outputs a disturbance 푑푖 that affects subsystem 푗 through disturbance input 푤푗. We provide mathematical conditions on the subsystems and the

104 Figure 4-1: Schematic of a perturbed network 풩 . It is composed of 푁 subsystems interconnected via prescribed interaction map 퐺 and unintended interaction map Δ. interactions under which the behavior of the perturbed network (with unintended interactions) is arbitrarily close to that of the nominal network (without unintended interactions). Specifically, we are interested in the network’s steady state behavior and thus define a network disturbance decoupling (NDD) property, by which the steady state outputs from all subsystems become essentially independent of the unintended interactions. We prove that if (i) each constituent subsystem is monotone or near-monotone and its output can asymptotically attenuate the effect of a constant external disturbance, (ii) the prescribed interactions do not contain a feedback loop, and (iii) the unintended interaction map is cooperative, then the NDD property of a network can be entirely determined by the static I/O characteristics of the subsystems. This result makes it particularly convenient to analyze and design network behaviors in the presence of unintended interactions. We apply our theoretical results to guide the design of robust genetic circuits in living cells, where unintended interactions arising from resource competition can disrupt network behavior [120]. While solutions have appeared recently to make a single genetic subsystem robust to constant disturbances [1, 2, 11, 68, 80, 113], it remains unclear the extent to which such solutions can be scaled up to enable robustness of a network of genetic subsystems to unintended interactions.

105 Chapter overview

In Section 4.2, we present resource competition in engineered genetic circuits as a motivating example. In Section 4.3, we provide a general system setup to describe unintended interactions and formulate the NDD problem. Section 4.5 studies networks composed of monotone subsystems and states conditions on the interaction maps and subsystem static characteristics for NDD. Section 4.6 extends the result to situations where the subsystems are non-monotone but can be reduced to a monotone system through timescale separation. Finally, in Section 4.7, we revisit the motivating example and provide sufficient conditions to obtain NDD in genetic circuits.

4.2 Motivating example

This work is motivated by the problem of engineering robust genetic circuits in living cells that we described in the previous chapters. In this example, we illustrate how the resource competition problem described in Chapter 2 can be cast within the formulation of Figure 4-1. A genetic circuit is composed of 푁 genetic subsystems. Each genetic subsystem

contains a series of biochemical reactions that express gene 푖 to produce a protein p푖

as output. In particular, the gene is first transcribed to produce mRNA m푖 at rate

푟푖, which is then translated to produce p푖 at rate 푇푖. Using 푚푖 and 푝푖 to represent the concentrations of m푖 and p푖, respectively, the state of a genetic subsystem is ⊤ 푥푖 = [푚푖, 푝푖] and its output is 푦푖 = 푝푖. Based on mass-action kinetics, the dynamics of subsystem 푖 can be written as [41]:

푚˙ 푖 = 푟푖 − 훿0푚푖, 푝˙푖 = 푇푖(푚푖) − 훿푝푖, (4.1)

where 훿0 and 훿 are decay rate constants of the mRNA and the protein, respectively, and 푇푖(푚푖) is the translation rate increasing with mRNA concentration 푚푖. The transcription rate of a gene 푖, 푟푖, can be modulated by the concentration of other proteins in the network, a process called transcriptional regulation [41]. These prescribed in-

106 ⊤ teractions are often modeled by 푟푖 = 퐺푖(푦), where 푦 := [푦1, ··· , 푦푁 ] and 퐺푖(·) is a nonlinear function called Hill function [41].

A major challenge in engineering genetic circuits is the omnipresence of unintended interactions, which severely hamper circuit’s function [61]. One contributor to unintended interactions is resource competition [99]. In particular, translation of mRNA relies on the cellular resource ribosomes, which are demanded by all mRNAs in the cell for translation. When mRNA m푗 is transcribed in genetic subsystem 푗, it binds with free ribosome, reducing its availability to translate m푖, thus decreasing the output of subsystem 푖 unintentionally. Accounting for 푁 subsystems competing for a conserved pool of ribosomes, the translation rate of each gene becomes (recall Section 2.2):

훼푖 · (푚푖/휅푖) ∑︁ 푚푗 푇푖 = 푇푖(푚푖, 푤푖) = , 푤푖 = , (4.2) 1 + 푚푖/휅푖 + 푤푖 휅푗 푗̸=푖

where 훼푖 is the translation rate constant that increases with total ribosome concen-

tration, 휅푖 is the dissociation constant that decreases with the affinity of m푖 with

the ribosome, and 푤푖 is the ribosome demand by all other subsystems in the circuit.

Because translation rate 푇푖 decreases with 푤푖, substituting (4.2) into (4.1), the output

푦푖 = 푝푖 now decreases with 푚푗, creating unintended interactions that give rise to unexpected circuit behavior [120]. Hence, a genetic circuit with ribosome competition can be regarded as a perturbed network with subsystem dynamics (4.1), prescribed interaction map (i.e., transcriptional regulation) 퐺(·), and unintended interaction map ∑︀ 푤푖 = 푗̸=푖 푑푗, where 푑푖 = 푚푖/휅푖 is the subsystem disturbance output. The above descriptive framework is consistent with our modeling framework in Chapter 2.

To reduce the dependence of each subsystem’s output 푦푖 on disturbance 푤푖, an sRNA-based type II QIC, which we describe in Section 3.2.5, can be introduced into each genetic subsystem to create a biomolecular feedback control mechanism [68]. The dynamics in such a feedback-regulated subsystem can be described by the following

107 mass-action kinetic model:

1 1 푚˙ 푖 = 푟푖 − 휆푖푚푖푠푖 − 훿0푚푖, 휖푖 휖푖 1 1 푠˙푖 = 훽푖푝푖 − 휆푖푚푖푠푖 − 훿0푠푖, (4.3) 휖푖 휖푖

푝˙푖 = 푇푖(푚푖, 푤푖) − 훿푝푖,

where 푠푖 is the concentration of sRNA, 휆푖, 훽푖 are constant parameters, and 휖푖 is a positive small parameter that can be decreased experimentally. When 푤푖 is a constant, state-independent disturbance, we have show in Sections 3.2.5 and 3.4 that

+ the steady state output 푦¯푖 of (4.3) satisfies lim휖푖→0 푦¯푖 = 푟푖/훽푖, which is independent of 푤푖 if 0 ≤ 푟푖 < 훼푖/훽푖. Thus, a feedback-regulated subsystem can achieve asymptotic static disturbance attenuation.

Given that each subsystem can asymptotically reject disturbance 푤푖 to reach set- point 푟푖/훽푖, it is tempting to use multiple such feedback controllers, one in each genetic subsystem to ensure that the output of multiple feedback-regulated subsystems become independent of 푤푖, that is, of ribosome usage. This approach, however, can fail depending on the value of reference input 푟푖 to each subsystem. Specifically, we simulated the network in Figure4-2a, which is composed of 3 feedback-regulated subsystems with the dynamics in (4.3) but no prescribed interactions among them

(i.e., 푟푖 = 푟0 is a constant for all 푖). As shown in Figure4-2b, we found that decreasing

휖푖 for all subsystems fails to decrease the tracking error when the reference input value is too large.

These simulation results demonstrate that even if all constituent subsystems of a network are robust to constant, state-independent disturbances, this robustness property may be lost when disturbances are state-dependent through an unintended interaction map 푤 = ∆(푑). Specifically, in this case, the problem occurs because 푑푖 reflects the “control effort” of the feedback mechanism in subsystem 푖. Hence, when

+ 휖푖 → 0 to improve disturbance attenuation of subsystem 푖, depending on 푟푖 level, disturbance output 푑푖 may grow unbounded, leading to 푤푗 → ∞, which cannot be compensated by the control effort in subsystem 푗. The result we present here allows

108 Figure 4-2: Network disturbance decoupling for sRNA-regulated subsystems with independent inputs. (a) A genetic circuit consists of three feedback-regulated subsystems (4.3), each taking an independent but identical reference input 푟푖 = 푟0. The subsystems are coupled through unintended interactions arising from resource conservation. (b) Steady state output error (vector ∞-norm) between perturbed and nominal networks as 휖 varies. For every 휖푖 and 푟푖, the trajectory converged to an asymptotic stable equilibrium. −1 −1 Subsystems have identical parameters: 훼푖 = 100 nM/hr, 휆푖 = 1 (nM · hr) , 훿 = 1 hr , −1 훽푖 = 1 hr , and 휅푖 = 1 nM for all 푖.

109 us to place sufficient conditions on subsystem dynamics, ∆(·), and 퐺(·) such that this does not occur.

4.3 Problem formulation

With reference to Figure4-1, a perturbed network 풩 is a 3-tuple (Σ, 퐺, ∆), where

Σ := (Σ1, ··· , Σ푁 ) is a set of 푁 subsystems, and 퐺 and ∆ describe the prescribed

and unintended interaction maps, respectively. Each subsystem Σ푖 = Σ푖(휖푖) is parameterized by a positive parameter 휖푖 and follows the dynamics:

푥˙ 푖 = 푓푖(푥푖, 푟푖, 푤푖; 휖푖), 푦푖 = 푙푖(푥푖), 푑푖 = 휌푖(푥푖), (4.4)

푛 where 푥푖 is the state variable evolving in 풳푖 ⊆ R . Signals 푟푖 and 푤푖 are scalar reference and disturbance inputs, respectively, taking values on sets ℛ푖 and 풲푖 that contain the origin; 푦푖 and 푑푖 are scalar prescribed and disturbance outputs, respectively, taking values on 풴푖 and 풟푖. We assume that 풳푖, ℛ푖, 풲푖, 풴푖, and 풟푖 are all box sets, which are Cartesian products of (possibly unbounded) closed real intervals. In addition, for all (푟푖(푡), 푤푖(푡)) taking values on ℛ푖 × 풲푖 and for all 휖푖, we assume the set 풳푖 is positively invariant under the dynamics of (4.4). For example, in biological systems, the state variables and I/O signals are all non-negative, hence 풳푖, ℛ푖, 풲푖, 풴푖, and 풟푖 can be chosen as the non-negative orthant. For each fixed 휖푖, we assume the function 푓푖 is differentiable and locally Lipschitz on 풳푖 × ℛ푖 × 풲푖. The output functions 푙푖, 휌푖 are assumed to be differentiable and locally Lipschitz on 풳푖. We assume that each subsystem has the following stability property.

* Assumption 4.1. (Subsystem stability). There exists 휖푖 > 0 such that for each * fixed (푟푖, 푤푖) ∈ ℛ푖 × 풲푖 and 휖푖 ∈ (0, 휖푖 ], system (4.4) has a globally asymptotically 0 stable (GAS) equilibrium 휙푖(푟푖, 푤푖; 휖푖), that is, for all initial conditions 푥푖 ∈ 풳푖, lim푡→∞ 푥푖(푡, 푟푖, 푤푖; 휖푖) = 휙푖(푟푖, 푤푖; 휖푖). O

If Assumption 4.1 is satisfied, 휙푖(· ; 휖푖) is the static I/S characteristic of Σ푖. The

110 corresponding static I/O characteristics are:

푦푖 = ℎ푖(푟푖, 푤푖; 휖푖) := 푙푖 ∘ 휙푖(푟푖, 푤푖; 휖푖), (4.5)

Assumption 4.2. (Subsystem disturbance attenuation). There exists class 풦 func- 0 ¯ tions 훼푖(·) and 훼푖 (·), a non-empty compact set ℛ푖 ⊆ ℛ푖, and a bounded function

퐻푖(푟푖) such that

0 |ℎ푖(푟푖, 푤푖; 휖푖) − 퐻푖(푟푖)| ≤ 훼푖(휖푖)|푤푖| + 훼푖 (휖푖) (4.6)

¯ * for every fixed (푟푖, 푤푖) ∈ ℛ푖 × 풲푖 and 휖푖 ∈ (0, 휖푖 ]. O

We call 퐻푖(푟푖) as the nominal static I/O characteristic. This is because according to

Assumption 4.2, for any bounded and fixed disturbance input 푤푖, the steady state reference output 푦푖 = ℎ푖(푟푖, 푤푖; 휖푖) deviates at most 풪(휖푖) from 퐻푖(푟푖), which is inde- ¯ pendent of disturbance 푤푖. The set ℛ푖 is the admissible reference input set, where this property holds. The subsystems are connected through a static intended interaction map

푟 = 퐺(푦). (4.7)

We assume that 퐺(·) is globally Lipschitz, that is, there exists 퐿퐺 > 0 such that

|퐺(푦1) − 퐺(푦2)| ≤ 퐿퐺|푦1 − 푦2| for all 푦1, 푦2 ∈ 풴. In a perturbed network, the disturbance output of subsystem 푖, 푑푖, perturbs subsystem 푗 through a disturbance input 푤푗.

The dependence of 푤푗 on 푑푖 gives rise to unintended interactions among subsystems, which we model using a static unintended interaction map

푤 = ∆(푑). (4.8)

We assume that ∆ is globally Lipschitz with a Lipschitz constant 퐿Δ. We use 푦 = 푦(푡; 휖; ∆) to represent the output of the perturbed network consisting of (4.4), (4.7), and (4.8), and write 푦(푡; 휖, 0) for the output of a nominal network 풩0 = (Σ, 퐺, ∆ ≡ 0)

111 consisting of (4.4), (4.7), but without disturbance input (i.e., 푤 ≡ 0).

Definition 4.1. (NDD). Given 휇 > 0, the perturbed network 풩 (휖) = (Σ(휖), 퐺, ∆) is said to have the 휇-network disturbance decoupling (휇-NDD) property if

lim sup |푦(푡; 휖, ∆) − 푦(푡; 휖, 0)| ≤ 휇 푡→∞

0 for all initial conditions 푥 ∈ 풳 . O

For small 휇, the output of 풩 becomes close to that of the nominal network 풩0. The 휇-NDD property is therefore a robust performance measure with respect to the unintended interaction map ∆. In general, static disturbance attenuation of the subsystems is insufficient to guarantee NDD. For example, the unintended interactions may result in lim휖→0+ |푤(푡; 휖)| → ∞, as shown in the motivating example of Section 4.2, or they may de-stabilize the network.

Problem Statement. (NDD problem). Given a perturbed network 풩 (휖) = (Σ(휖), 퐺, ∆) consisting of subsystems with the asymptotic static disturbance attenuation property

(4.6), determine conditions on Σ푖(휖푖), 퐺, and ∆ such that given any 휇 > 0, 휇-NDD can be achieved for 휖푖 sufficiently small for all 푖. O

Solution to the NDD problem identifies a class of perturbed networks that are robust to unintended interactions, in the sense that any effect arising from unintended interactions can be mitigated by improving disturbance attenuation of the constituent subsystems (i.e., decreasing 휖푖). As we demonstrate next, one class of such networks are those with certain monotonicity properties.

4.4 Technical background: monotone systems

We present some basic concepts on monotone and mixed-monotone systems theory that are useful here. A more complete and in-depth treatment of these topics can be found in [9, 10, 34, 146, 147].

112 Definition 4.2. ([34]). A function 푓 : 풳 → 풴 is mixed-monotone if there exists a function 푓ˆ : 풳 2 → 풴, called a decomposition function of 푓(·), such that for ˆ all 푥, 푥1, 푥2, 푧 ∈ 풳 the following are satisfied: (i) 푓(푥) = 푓(푥, 푥), (ii) 푥1 ≤ 푥2 ⇒ ˆ ˆ ˆ ˆ 푓(푥1, 푧) ≤ 푓(푥2, 푧), and (iii) 푥1 ≤ 푥2 ⇒ 푓(푧, 푥2) ≤ 푓(푧, 푥1). O

According to the above definition, take any 푥− ≤ 푥 ≤ 푥+, we have 푓ˆ(푥−, 푥+) ≤ 푓(푥) ≤ 푓ˆ(푥+, 푥−). The decomposition function of a mixed-monotone function is not unique. A differentiable function 푓 : R푚 → R푛 has sign-stable partial derivatives if 푛×푚 there exists a matrix Λ ∈ R , whose elements Λ푖푗 take values in {1, −1} and satisfy

Λ푖푗(휕푓푖/휕푥푗) ≥ 0 for all 푖, 푗 and 푥. If 푓 has sign-stable partial derivatives, then one decomposition function of 푓 can be found through Λ. In particular, let

− + − Λ = − min(0, Λ) Λ = 1푚×푛 − Λ , (4.9)

define a vector function 푓ˆ(푥+, 푥−): R2푚 → R푛 whose 푖-th element is:

ˆ + − (︀ + + − −)︀ 푓푖(푥 , 푥 ) := 푓푖 diag(Λ푖 ) · 푥 + diag(Λ푖 ) · 푥 , (4.10)

± ± ˆ where Λ푖 is the 푖-th row of Λ . Then, 푓 is a decomposition function of 푓. In particular, we call 푓ˆ the canonical decomposition function of 푓.

Example 4.1. Given a constant matrix 퐴, the function 푓(푥) = 퐴푥 is mixed-monotone. Its canonical decomposition function is 푓ˆ(푥−, 푥+) = 퐴+푥− + 퐴−푥+, where

⎧ ⎪퐴푖푗, if 퐴푖푗 < 0, − ⎨ + − 퐴푖푗 := 퐴 := 퐴 − 퐴 . ⎩⎪0, otherwise,

Lemma 4.1. Let 푓 and 푔 be two mixed-monotone functions with decomposition functions 푓ˆ and 푔ˆ, respectively. Then ℎ := 푓 ∘ 푔 is also mixed-monotone and ˆ ˆ ℎ(푥1, 푥2) := 푓(ˆ푔(푥1, 푥2), 푔ˆ(푥2, 푥1)) is a decomposition function of ℎ. O

113 Now we consider a system with input 푤 and output 푦:

푥˙ = 퐹 (푥, 푤), 푦 = 휌(푥), (4.11)

where 퐹 : R푛 × R푚 → R푛 and 휌 : R푛 → R푞 are differentiable and their partial derivatives with respect to 푥 and 푤 are sign-stable. We review the notion of input/state (I/S) monotone systems [9].

Definition 4.3. System (4.11) is I/S monotone if there exists 휎푤 ∈ R푚 and 휎푥 ∈ R푛, whose elements are in {1, −1}, such that

푥 푥 휕퐹푖 푥 푤 휕퐹푖 휎푖 휎푗 (푥, 푤) ≥ 0, 휎푖 휎푘 (푥, 푤) ≥ 0, 휕푥푗 휕푤푘 for all indices 푖 ̸= 푗 and 푘, and for all 푥, 푤. Specifically, the system is said to be I/S

푤 푥 monotone with respect to the partial order pair (휎 ; 휎 ). O

If for each fixed 푤, the I/S monotone system (4.11) has a GAS equilibrium 푥¯ = 휙(푤), then 휙(·) is the static I/S characteristic of (4.11). The static I/S characteristic of an I/S monotone system has sign-stable partial derivatives [9]. In particular, the sign pattern of (휕휙/휕푤¯) is Λ = 휎푥(휎푤)⊤, and the canonical decomposition function of 휙 can then be computed according to (4.10). An important property of I/S monotone systems is the following convergent-input-convergent-state/output property.

Lemma 4.2. Suppose that (4.11) is monotone with a static I/S characteristic 푥 = 휙(푤), let 휙ˆ(푤+, 푤−) be the canonical decomposition function of 휙. If 푤(푡) → [푤+, 푤−], then 푥(푡) → [휙 ˆ(푤−, 푤+), 휙ˆ(푤+, 푤−)]. Additionally, if the output function 푦 = 휌(푥) is mixed-monotone with a decomposition function 휌ˆ(푥+, 푥−), then

− + + − + − + − − + 푦(푡) → [휓(푤 , 푤 ), 휓(푤 , 푤 )], where 휓(푤 , 푤 ) :=휌 ˆ(휙 ˆ(푤 , 푤 ), 휙ˆ(푤 , 푤 )). O

Proof. Convergence of 푥(푡) can be found in [10] (Lemma 2). Convergence of 푦(푡) is a consequence of Lemma 4.1.

Here, we will call 휓 the I/O gain function of (4.11), this is because if 푤(푡) eventually enters the box [푤−, 푤+], 푦(푡) will eventually converge to the box [휓(푤−, 푤+), 휓(푤+, 푤−)],

114 Figure 4-3: A graphical representation of the I/O gain function 휓 for system (4.11). If the input 푤(푡) ultimately enters the box set [푤−, 푤+], the output 푦(푡) ultimately converges to the box set [휓(푤−, 푤+), 휓(푤+, 푤−)]. This schematic assumes system (4.11) is 푤 푥 cooperative, that is, (휎 ; 휎 ) = (1; 1푛), and the output function 휕휌/휕푥 ≥ 0 for all 푥.

as shown in Figure4-3.

4.5 Network disturbance decoupling with monotone subsystems

Here we provide a set of sufficient conditions on the subsystem dynamics andpre- scribed/unintended interaction maps for NDD. These conditions are centered around the subsystems having the following monotonicity property. This is satisfied in many biological systems, which can be decomposed into monotone subsystems [12, 153].

* Assumption 4.3. (Subsystem monotonicity). For every 휖푖 ∈ (0, 휖푖 ], each subsystem 푟 푤 푥 Σ푖 in (4.4) is I/S monotone with respect to the partial orders (휎 ; 휎 ; 휎 ). The partial

derivatives of output functions 푙푖 and 휌푖 are sign-stable. O

+ + − − Due to Assumptions 4.1 and 4.3, 휙푖 is mixed-monotone. Let 휙ˆ푖(푟푖 , 푤푖 , 푟푖 , 푤푖 ; 휖푖) + − and 휌ˆ푖(푥푖 , 푥푖 ) be the canonical decomposition function of 휙푖(푟푖, 푤푖; 휖푖) and 휌푖(푥), respectively. We follow Lemma 4.2 and define the disturbance I/O gain of Σ푖 as:

+ + − − + + − − − − + + 휓푖(푟푖 ,푤푖 , 푟푖 , 푤푖 ; 휖푖) :=휌 ˆ푖[휙 ˆ푖(푟푖 , 푤푖 , 푟푖 , 푤푖 ; 휖푖), 휙ˆ푖(푟푖 , 푤푖 , 푟푖 , 푤푖 ; 휖푖)]. (4.12)

We assume that increasing disturbance output from Σ푖 does not decrease disturbance

input to Σ푗. This is a mild assumption satisfied in many scenarios, including our motivating example in Section 4.2, as we will show formally in Section 4.7.

115 Assumption 4.4. (Unintended interactions). The unintended interaction map ∆(·)

− + − + is cooperative, that is, ∆푖(푑푗 ) ≤ ∆푖(푑푗 ) for all 푖, 푗 and 푑푗 ≤ 푑푗 . O

The prescribed interaction map is assumed to have a simple structure.

Assumption 4.5. (Intended interaction). The intended interaction map 푟 = 퐺(푦) does not contain any feedback loop, that is, 퐺푖 is independent of 푦푗 for all 푗 ≥ 푖. O

Given Assumptions 4.1 and 4.5, because there is no feedback loop interconnecting the

* * * ⊤ subsystems through 퐺, equation 푟 = 퐺∘퐻(푟) has a unique solution 푟 = [푟1, ··· , 푟푁 ] . We call 푟* the nominal reference input of the network, since 푟* is computed using

퐺 and the subsystem nominal static I/O characteristic 푦푖 = 퐻푖(푟푖), which is derived assuming 푤 ≡ 0. For simplicity of notation, we use

* + − * * + * − 휓푖 (푤푖 , 푤푖 ; 푟푖 , 휖푖) := 휓푖(푟푖 , 푤푖 , 푟푖 , 푤푖 ; 휖푖) (4.13)

* − + to represent a subsystem’s disturbance I/O gain for a fixed 푟푖 . If 푤푖 → [푤푖 , 푤푖 ] * and 푟푖 ≡ 푟푖 , then the disturbance output 푑푖 is ultimately bounded in the box * − + * * + − * * [휓푖 (푤푖 , 푤푖 ; 푟푖 , 휖푖), 휓푖 (푤푖 , 푤푖 ; 푟푖 , 휖푖)]. We will use 휓푖 to elicit conditions for NDD. Finally, we impose the following technical assumption on each subsystem’s static characteristics.

Assumption 4.6. (Subsystem Lipschitz conditions). The static I/O characteristic ¯ * ℎ푖(푟푖, 푤푖; 휀푖) is Lipschitz continuous in 푟푖 ∈ ℛ푖 uniformly in (푤푖, 휖푖) ∈ 풲푖 × (0, 휖푖 ]. + + − − The disturbance I/O gain function 휓푖(푟푖 , 푤푖 , 푟푖 , 푤푖 ; 휖푖) is Lipschitz continuous in + − ¯ 2 − + 2 * (푟푖 , 푟푖 ) ∈ (ℛ푖) uniformly in 푤푖 , 푤푖 ∈ 풲푖 . In addition, 휓푖 is sub-linear in that there * ± − * * * exists a non-negative function 푎푖(푟푖) such that |휓푖 (푤푖 , 푤푖 ; 푟푖 , 휖푖) − 휓푖 (0, 0; 푟푖 , 휀푖)| ≤ * ± + − ¯ 2 * 푎푖(푟푖 )|푤푖 | uniformly (푟푖 , 푟푖 , 휖푖) ∈ (ℛ푖) × (0, 휖푖 ]. O

NDD for networks composed of monotone subsystems

With reference to Figure 4-1, the perturbed network 풩 can be regarded as a feed-

back interconnection of 풩0 and ∆, where 풩0 is the feedback interconnection of

116 Σ = (Σ1, ··· , Σ푁 ) with 퐺 through the prescribed I/O signals. Although 풩0 contains the prescribed interaction map 퐺, for small 휖푖, it has a convergent-input-convergent- output property, with input 푤 and output 푑, whose I/O gain can be approximated

* * * ⊤ * by 휓 := [휓1, ··· , 휓푁 ] , that is, its I/O gain can be approximated as if 퐺 ≡ 푟 .

Lemma 4.3. Consider 풩0 under Assumptions 4.1-4.6, and suppose that the nominal * * ¯ reference input 푟 satisfies 푟푖 ∈ int(ℛ푖) for all 푖. Then, there exists functions 푃, 푄 : − + R≥0 × R>0 → R>0, such that 푤(푡) → [푤 , 푤 ] implies

푑(푡) → [휓*(푤−, 푤+; 푟*, 휖) − 푃 (|푤±|; 휖), 휓*(푤+, 푤−; 푟*, 휖) + 푃 (|푤±|; 휖)], (4.14a)

푦(푡) → [퐻(푟*) − 푄(|푤±|; 휖), 퐻(푟*) + 푄(|푤±|; 휖)], (4.14b) where 푤± := [(푤−)⊤, (푤+)⊤]⊤. Particularly, the functions 푃, 푄 can be decomposed as

± ± ± ± 푃 (|푤 |; 휖) := 푝1(휖)|푤 | + 푝0(휖), 푄(|푤 |; 휖) := 푞1(휖)|푤 | + 푞0(휖), (4.15)

where 푝1, 푝0, 푞1, 푞0 are positive scalar functions with the following property: for each 푖, ** ** there exists 휖푖 = 휖푖 (휇, 휖푖+1, ··· , 휖푁 ) > 0, such that given any 휇 > 0, 푝1(휖), 푝0(휖), 푞1(휖), 푞0(휖) ≤ ** 휇 if 0 < 휖푖 ≤ 휖푖 (휇, 휖푖+1, ··· , 휖푁 ) for all 푖.

The proof of Lemma 4.3 is in Appendix Section C.1. This property holds because each subsystem has the disturbance attenuation property (Assumption 4.2) and is monotone (Assumption 4.3). Equation (4.14a) allows us to approximate the disturbance

I/O behavior of 풩0 using the disturbance I/O gains of Σ푖. In addition, by (4.14b) and (4.15), for a constant |푤±|, the effect of disturbance input 푤 on the prescribed output 푦 can be arbitrarily diminished by decreasing each 휖푖. Yet, for the perturbed network 풩 , we need to prove that 푤 = 푤(휖) does not grow as 휖 is decreased. To this end, our result involves a boundedness condition for discrete time (DT) systems. Consider

푥(푘 + 1) = 퐹 (푥(푘)), (4.16)

117 where 푥 ∈ R푛, and without loss of generality, we assume that 퐹 (0) = 0. System − + (4.16) is said to be ultimately bounded [81] in a box set [푥* , 푥* ] if, for any initial − + condition 푥(0), there exists a 푘* > 0 such that 푥(푘) ∈ [푥* , 푥* ] for all 푘 ≥ 푘*. We − + − + use 푥(푘) → [푥* , 푥* ] to denote that 푥(푘) is ultimately bounded in [푥* , 푥* ]. We next introduce a Lyapunov characterization of the ultimate boundedness property that is robust to perturbations.

Definition 4.4. System (4.16) is said to be exponentially ultimately bounded if there

푛 exists positive constants 푐1, 푐2, 푐3, 푟0 and a 푉 (·): R → R such that

2 2 푐1|푥| ≤ 푉 (푥) ≤ 푐2|푥| , (4.17a)

|푉 (푥1) − 푉 (푥2)| ≤ 푐4|푥1 − 푥2| · (|푥1| + |푥2|), (4.17b)

2 푉 (퐹 (푥)) − 푉 (푥) ≤ −푐3|푥| , for all |푥| ≥ 푟0. (4.17c)

Specifically, if (4.17) is satisfied, (4.16) is exponentially ultimately bounded in [−푟*, 푟*],

where 푟* := 푐1푟0/푐2. O

If 푟0 = 0, then (4.16) has an exponentially stable equilibrium. Conversely, the existence of a Lyapunov function 푉 (푥) satisfying (4.17) is guaranteed if (4.16) has an exponentially stable equilibrium (see Excercise 4.68 in [81]). The boundedness property of an exponentially ultimately bounded system is robust to perturbations. In fact, consider a perturbation of the nominal system (4.16):

푥(푘 + 1) = 퐹 (푥(푘)) + 푝 · 훿(푥(푘)), (4.18)

where 푝 is a constant parameter and |훿(푥)| ≤ 퐿1|푥| + 퐿2 for all 푥, then we can prove:

Lemma 4.4. Suppose the nominal system (4.16) is exponentially ultimately bounded

in [−푟*, 푟*] and that 퐹 (푥) is globally Lipschitz, then there exists 푝*, 휅 > 0, such that

for all 푝 ∈ [0, 푝*], system (4.18) satisfies 푥(푘) → [−푟* − 휅푝*, 푟* + 휅푝*]. O

Proof of Lemma 4.4 can be found in Appendix Section C.2. Now we are ready to

state our first main result. It takes advantage of the subsystem monotonicity of 풩0

118 and monotonicity of ∆ to provide an 휖-independent bound on 푤(푡; 휖), which allows

each subsystem to decrease 휖푖 for disturbance attenuation. Conditions for this 휖- independent bound only involve I/O properties of Σ푖 and 퐺 (instead of 풩0).

Theorem 4.1. Consider the perturbed network (4.4), (4.7), and (4.8) under Assump- * ¯ tions 4.1-4.6. Suppose that 푟푖 ∈ int(ℛ푖) for all 푖 and that 푤(푡; 휖) is bounded for all 푡 * * * ⊤ for any fixed 휖. If there exists 0 < 휖¯0 ≤ 휖 := [휖1, ··· , 휖푁 ] such that

푤−(푘 + 1) = ∆ ∘ 휓*(푤−(푘), 푤+(푘); 푟*, 휖), (4.19) 푤+(푘 + 1) = ∆ ∘ 휓*(푤+(푘), 푤−(푘); 푟*, 휖)

− + is exponentially ultimately bounded in an 휖-independent set [푤* , 푤* ] for all 0 < 휖 ≤

휖¯0, then given any 휇 > 0, 풩 has the 휇-NDD property if each 휖푖 is sufficiently small, ** that is, if 0 < 휖푖 ≤ 휖푖 (휇, 휖푖+1, ··· , 휖푁 ) for all 푖. O

Proof. We treat 풩 as the feedback interconnection of 풩0 and ∆. By Lemma 4.2,

풩0 has the convergent-input-convergent-output property. Since ∆ is cooperative (As- sumption 4.4) and 푤(푡) is bounded for all 푡, a small-gain theorem for convergent-input-

− + convergent-output systems (Appendix Section C.4) shows that 푤(푡) → [푤**, 푤**] if the DT system

푤+(푘 + 1) = ∆ ∘ [휓*(푤+, 푤−; 푟*, 휖) + 푃 (|푤±(푘)|; 휖)], (4.20) 푤−(푘 + 1) = ∆ ∘ [휓*(푤−, 푤+; 푟*, 휖) − 푃 (|푤±(푘)|; 휖)],

− + is ultimately bounded in [푤**, 푤**]. The small-gain theorem we use is a slight generalization of the small-gain theorem for orthant monotone systems in [10] (Theorem 1) in that we consider convergence of (4.19) to a set instead of to an equilibrium. To

− + show that 푤** and 푤** can be chosen independent of 휖, we treat (4.20) as a perturbation of the nominal DT system (4.19). By Lemma 4.4 and with reference to (4.15), there exists a 푝* > 0, such that if (4.19) is exponentially ultimately bounded in an

− + * − + 휖-independent set [푤* , 푤* ] and |푝1(휖)|, |푝0(휖)| ≤ 푝 , then [푤**, 푤**] is 휖-independent. ** * Therefore, if 0 < 휖푖 ≤ min{휖¯0; 휖푖 (푝 ; 휖푖+1, ··· , 휖푁 )} for every 푖, we can apply Lemma

119 4.3 to find

* ± * ± 푦(푡; 휖, ∆) → [퐻(푟 ) − 푞1(휖)|푤**| − 푞0(휖), 퐻(푟 ) + 푞1(휖)|푤**| + 푞0(휖)],

* * 푦(푡; 휖, 0) → [퐻(푟 ) − 푞0(휖), 퐻(푟 ) + 푞0(휖)].

± ± Hence, lim sup푡→∞ |푦(푡; 휖, 0)−푦(푡; 휖, ∆)| ≤ 푞1(휖)|푤**|+2푞0(휖), where 푤** is 휖-independent.

This implies that, given any 휇 > 0, 휇-NDD can be achieved if, additionally, each 휖푖 ± is taken sufficiently small for 푞1(휖)|푤**| + 2푞0(휖) ≤ 휇.

Under the conditions of Theorem 4.1, NDD of the (푛푁)-dimensional continuous time system 풩 can be certified if the (2푁)-dimensional DT system (4.19) is ultimately bounded in an 휖-independent set. This DT system can be constructed using the static disturbance I/O characteristics of the constituent subsystems and the unintended interaction ∆. It provides an upper bound for the “DC amplification” of disturbance signals in the perturbed network. If the trajectory of (4.19) is ultimately bounded in an 휖-independent set, then NDD can be achieved if the subsystem static disturbance attenuation performance is sufficiently good (small 휖푖). This result is robust in the sense that as long as Σ푖, 퐺, and ∆ satisfy the structural conditions specified in Assumptions 4.3,4.4,4.5, NDD can be achieved by decreasing 휖푖 for all ∆, possibly unknown, that allow the trajectory of (4.19) to be bounded by 휖-independent constants.

* Remark 4.1. Note that 휖푖 is a function of 휖푖+1, ··· ; 휖푁 . This implies that the requirement on disturbance attenuation for an upstream subsystem 푖 is generally stricter than its downstream subsystems 푗 ≥ 푖 + 1 to diminish propagation of the regulation error via prescribed interactions. In the special case where 퐺(푦) ≡ 푟* (i.e., no prescribed

* interactions), 휖푖 can be chosen independent of 휖푖+1, ··· ; 휖푁 . O

* ¯ Remark 4.2. While we require 푟 ∈ int(ℛ푖), there is no need to verify 푟(푡) stays in ¯ ℛ푖 for all 푡. One only needs to verify the nominal reference input to each subsystem is within its admissible reference input set. O

Remark 4.3. The requirement for 푤(푡) = 푤(푡; 휖) to be bounded for all 푡 for a fixed 휖

120 can often be satisfied in many physical systems with nonlinear dynamics. For example, in biomolecular systems, the state variables represent molecular concentrations, which are often bounded above by conservation laws. Note that this assumption does not

imply lim휖→0+ |푤(푡; 휖)| < ∞ for all 푡. However, if an 휖-independent bound for 푤(푡; 휖) can be easily found for 풩 , then there is no need to check the boundedness of the DT system (4.19). This boundedness assumption on 푤(푡) can be relaxed if ∆ and the I/S characteristics of all subsystems are unbounded (see Proposition 4.3 in [9]). O

While many engineering subsystems have I/S monotone dynamics, the presence of controllers is often required for them to achieve asymptotic static disturbance attenuation. When a dynamic controller is used to regulate a subsystem, the resultant dynamics of the regulated subsystem is often not monotone.

Example 4.2. Suppose that a plant has I/S monotone dynamics 푥˙ 푖 = −푥푖 + 푢푖 + 푤푖,

푦푖 = 푥푖, where 푢푖 = 푧푖 is a control input arising from a dynamic feedback controller

푧˙푖 = −푧푖 + (푟푖 − 푥푖)/휖푖. It is easy to show that the regulated subsystem

푥˙ 푖 = −푥푖 + 푧푖 + 푤푖, 푧˙푖 = −푧푖 + (푟푖 − 푥푖)/휖푖 (4.21) has the asymptotic static disturbance attenuation property with a nominal static

I/O characteristic 퐻푖(푟푖) = 푟푖. However, the regulated subsystem is non-monotone. This can be easily checked using the graphical conditions for monotonicity [88]. In particular, a system 푥˙ 푖 = 퐹푖(푥푖) is monotone if and only if the incidence graph induced by 퐹푖 does not contain any (undirected) negative cycle. The incidence graph is drawn according to the following rule: there is a positive (negative) edge “→” (“−”) from [ 푥푖 to 푥푗 if 휕퐹푗/휕푥푖 ≥ 0 (≤ 0) for all 푥. For the regulated subsystem (4.21), where ⊤ 퐹푖 := [−푥푖 + 푧푖 + 푤푖, −푧푖 + (푟푖 − 푥푖)/휖푖] , the incidence graph induced by 퐹푖 contains a negative cycle 푥푖 − 푧푖 → 푥푖, indicating that it is non-monotone. O [ Motivated by this fact, in the next section, we seek conditions for networks composed of non-monotone subsystems to achieve NDD. In the context of Example 4.2, we show that if the dynamics of the feedback controller 푧푖 is sufficiently fast, then (4.21)

121 behaves like an I/S monotone system, thus, the results developed in this section hold with similar conditions.

4.6 Network disturbance decoupling with near-monotone subsystems

Certain non-monotone systems can have dynamic properties similar to monotone systems [155, 169]. In particular, for autonomous systems, if the “non-monotone dynamics” in a two-timescale non-monotone system evolve at a sufficiently fast rate, certain convergence properties for monotone systems are preserved [169]. Based on similar reasonings, we provide conditions for NDD of networks composed of non- monotone subsystems.

Two-timescale subsystem setup

We consider subsystem Σ푖 parameterized by an additional small positive parameter 휈, which induces a timescale separation in the subsystems. For simplicity, we use the same 휈 for all subsystems, although the results are not restricted to this case. We

now write Σ푖 = Σ푖(휖푖, 휈) as:

푥˙ 푖 = 푓푖(푥푖, 푧푖, 푟푖, 푤푖; 휖푖), 푦푖 = 푙푖(푥푖) (4.22) 휈푧˙푖 = 푔푖(푥푖, 푧푖, 푟푖, 푤푖; 휖푖), 푑푖 = 휌푖(푥푖, 푧푖).

푛 푚 where 푥푖 ∈ 풳푖 ⊆ R , 푧푖 ∈ 풵푖 ⊆ R and the I/O signals 푟푖, 푤푖, 푦푖, 푑푖 are defined as before in Section 4.3. We assume that the prescribed output 푦푖 is a function of the slow state 푥푖 only, but the disturbance output 푑푖 depends on both 푥푖 and 푧푖. Subsystem (4.22) is singularly perturbed by 휈. In particular, in the fast time scale 휏 = 푡/휈, the boundary layer dynamics [81] of (4.22) are:

d푧푖/d휏 = 푔푖(푥푖, 푧푖, 푟푖, 푤푖; 휖푖), (4.23)

122 where 푥푖, 푟푖, 푤푖 are treated as fixed parameters.

Assumption 4.7. (Subsystem boundary layer). For every fixed (푥푖, 푟푖, 푤푖) ∈ 풳푖 × * ℛ푖 × 풲푖 and 휖푖 ∈ (0, 휖푖 ], the algebraic equation 푔푖(푥푖, 푧푖, 푟푖, 푤푖; 휖푖) = 0 has a GAS equilibrium 훾푖(푥푖, 푟푖, 푤푖; 휖푖) ∈ 풵푖. O

Substituting 푧푖 = 훾푖(푥푖, 푟푖, 푤푖; 휖푖) into (4.22), a candidate reduced model of (4.22) is:

푟 푟 푟 푟 푟 푟 푥˙ 푖 = 푓푖 (푥푖 , 푟푖, 푤푖; 휖푖), 푑푖 = 휌푖 (푥푖 , 푟푖, 푤푖; 휖푖), (4.24)

푟 푟 and 푦푖 = 푙푖(푥푖 ), where

푟 푟 푟 푟 푓푖 (푥푖 , 푟푖, 푤푖; 휖푖) := 푓푖(푥푖 , 훾푖(푥푖 , 푟푖, 푤푖; 휖푖), 푟푖, 푤푖; 휖푖)

푟 푟 푟 푟 휌푖 (푥푖 , 푟푖, 푤푖; 휖푖) := 휌푖(푥푖 , 훾푖(푥푖 , 푟푖, 푤푖; 휖푖)).

푟 We denote system (4.24) by Σ푖 and require it to have similar stability, disturbance attenuation, monotonicity, and Lipschitz continuity properties as specified for the subsystems in Section 4.5. These conditions are summarized in the assumption below.

푟 Assumption 4.8. (Reduced subsystem). Each reduced subsystem Σ푖 satisfies the following conditions:

푟 푟 푤 푥 (i) System Σ푖 is I/S monotone with respect to the partial orders (휎 ; 휎 ; 휎 ) for all * 푟 휖푖 ∈ (0, 휖푖 ]. The output functions 휌푖 and 푙푖 have sign-stable partial derivatives.

푟 푟 푟 (ii) System Σ푖 is endowed with a well-defined I/S characteristic 푥푖 = 휙푖 (푟푖, 푤푖; 휖푖). 푟 The static I/O characteristics 푦푖 = ℎ푖(푟푖, 푤푖; 휖푖) satisfies Assumption 4.2. O

푟 Since Σ푖 is I/S monotone and its output functions have sign-stable partial derivatives, 푟 푟 푟 the functions 휙푖 (푟푖, 푤푖; 휖푖) and 휌푖 (푥푖 , 푟푖, 푤푖; 휖푖) have canonical decomposition functions 푟 + − 푟 푟+ + 푟− − 휙ˆ푖 (푣푖 , 푣푖 ; 휖푖) and 휌ˆ푖 (푥푖 , 푣푖 , 푥푖 , 푣푖 ; 휖푖), respectively, where for simplicity of notation ⋆ ⋆ ⊤ ⋆ ⊤ ⊤ we write 푣푖 = [(푟푖 ) , (푤푖 ) ] and ⋆ is either + or −. The decomposition functions can be composed according to Lemma 4.1 to obtain the disturbance I/O gain

+ + − − 푟 푟 + − + 푟 − + − 휓푖(푟푖 , 푤푖 ,푟푖 , 푤푖 ; 휖푖) :=휌 ˆ푖 (휙 ˆ푖 (푣푖 , 푣푖 ; 휖푖), 푣푖 , 휙ˆ푖 (푣푖 , 푣푖 ; 휖푖), 푣푖 ; 휖푖).

123 Similar to (4.13), we define

* + − * * + * − 휓푖 (푤푖 , 푤푖 ; 푟푖 , 휖푖) := 휓푖(푟푖 , 푤푖 , 푟푖 , 푤푖 ; 휖푖) (4.25)

* as the subsystem static disturbance I/O gain for a fixed reference input 푟푖 . If several mild technical conditions are satisfied, the conditions to guarantee NDD in Theorem 4.1 for networks with monotone subsystems are also sufficient for networks composed of two-timescale subsystems (4.22). To show this, we provide a convergent-input- convergent-output result for singularly perturbed systems with monotone reduced systems in the next section. It extends the convergent-input-convergent-output result for monotone systems in Lemma 4.2.

A convergent-input-convergent-output property for singularly perturbed monotone systems

We consider two-timescale systems of the form:

휉˙ = 푓¯(휉, 휁, 푢(푡)), 휈휁˙ =푔 ¯(휉, 휁, 푢(푡)), 휒 = 휚(휉, 휁), (4.26)

푛+푚 where (휉, 휁) ∈ R are the state variables, 휒 is the output, and 푢(푡): R≥0 → 풰 is a continuous input signal taking values in a box set 풰 that contains the origin.

The functions 푓,¯ 푔¯ are locally Lipschitz on R푛 × R푚 × 풰 and 휚 is locally Lipschitz on R푛 × R푚. System (4.26) is singularly perturbed by 휈 and we write its boundary layer dynamics as:

d휁/d휏 =푔 ¯(휉, 휁, 푢), (4.27) where 휉 and 푢 are treated as fixed parameters. We assume that (4.27) has aGAS equilibrium 휁¯ = Γ(휉,¯ 푢¯) such that 푔¯(휉,¯ Γ(휉,¯ 푢¯), 푢¯) = 0 for each fixed (휉,¯ 푢¯) ∈ 풳 × 풰.A candidate reduced system of (4.26) is obtained by substituting 휁 = Γ(휉, 푢(푡)) into 휉

124 dynamics: 휉˙푟 = 푓¯푟(휉푟, 푢(푡)) := 푓¯(휉푟, Γ(휉푟, 푢(푡)), 푢(푡)), (4.28) 휒푟(푡) = 휚푟(휉푟, 푢(푡)) := 휚(휉푟, Γ(휉푟, 푢(푡))).

Under certain technical conditions, the singularly perturbed system (4.26) has approximate convergent-input-convergent-state/output properties, as we show next. The 휇 shorthand notation 푥(푡) −→풜 stands for lim sup푡→∞ dist{푥(푡), 풜} ≤ 휇.

Lemma 4.5. (Convergent-input-convergent-state for singularly perturbed monotone systems). Consider system (4.26), suppose that the boundary layer dynamics (4.27) has a GAS equilibrium 휁¯ = Γ(휉,¯ 푢¯) for each fixed (휉,¯ 푢¯), and the reduced system ¯ 푟 (4.28) is I/S monotone with a static I/S characteristic 휉 = 휙휉(¯푢) for every constant 푟 + − input 푢(푡) ≡ 푢¯ ∈ 풰. Let 휙ˆ휉(푢 , 푢 ) be the canonical decomposition function of 푟 휙휉(푢). Assume there exists 푀 > 0, independent of 휈, such that ‖푢‖, ‖푢˙‖ ≤ 푀. Then, given any 휇 > 0, there exists 휈*(휇), also dependent on 푀, such that for all 휇 휇 0 < 휈 ≤ 휈*, 휁(푡) −→ Γ(휉(푡), 푢(푡)). In addition, if 푢(푡) → [푢−, 푢+], then 휉(푡) −→

푟 − + 푟 + − [휙휉(푢 , 푢 ), 휙휉(푢 , 푢 )]. O

푟 Proof. See Section C.3 in the Appendix for details. Note that after translating 휙휉(0) to the origin, an I/S monotone system with a well-defined static I/S characteristic is input-to-state stable (ISS). The proof then uses a singular perturbation theorem for

ISS systems [33].

Remark 4.4. If 푔¯ is not a function of 푢(푡), the requirement ‖푢˙‖ ≤ 푀 can be removed.

As an immediate consequence of Lemma 4.5, we have the following approximate convergent-input-convergent-output result.

Corollary 4.1. Under the assumptions of Lemma 4.5, and suppose that 휚푟(휉) is sign-stable with a canonical decomposition function 휚ˆ푟(휉+, 푢+, 휉−, 푢−). Then, for any 휇 > 0, there exists 휈*(휇) such that for any 0 < 휈 ≤ 휈*(휇), 푢(푡) → [푢−, 푢+] implies

휇 − + + − 휒(푡) −→ [휓휒(푢 , 푢 ), 휓휒(푢 , 푢 )], (4.29)

125 + − 푟 푟 + − + 푟 − + − where 휓휒(푢 , 푢 ) :=휚 ˆ (휙휉(푢 , 푢 ), 푢 , 휙휉(푢 , 푢 ), 푢 ). O Proof. Due to the convergent-input-convergent-state result in Lemma 4.2 and the fact that 휚ˆ푟 is the canonical decomposition function 휚푟, we have

푟 휇 − + + − 휚 (휉(푡)) −→ [휓휒(푢 , 푢 ), 휓휒(푢 , 푢 )].

휇 푟 Since 휚 is Lipschitz and 휁 −→ Γ(휉(푡), 푢(푡)) for sufficiently small 휈, we have lim sup푡→∞ |휚 (휉(푡), 푢(푡))−

휒(푡)| = lim sup푡→∞ |휚(휉(푡), Γ(휉(푡), 푢(푡)))−휚(휉(푡), 휁(푡))| ≤ 퐿휚휇, where 퐿휚 is a Lipschitz constant.

NDD for networks composed of two-timescale subsystems

Using Corollary 4.1, we can determine conditions for NDD of a perturbed network composed of two-timescale non-monotone subsystems (4.7), (4.8), and (4.22). A technical assumption is that the input signal and its derivative need to be bounded in the network to carry out singular perturbation according to Lemma 4.5 and Corollary 4.1.

Assumption 4.9. Consider the perturbed network (4.7), (4.8), and (4.22). There exists 푀1(휖) > 0, independent of 휈, such that |푟(푡)|, |푤(푡)| ≤ 푀1(휖) for all 푡. In addi-

tion, either one of the following is satisfied: (a) There exists 푀2(휖) > 0, independent

of 휈, such that ‖푟˙‖, ‖푤˙ ‖ ≤ 푀2(휖), or (b) the function 푔푖 in (4.22) is independent of

푟푖, 푤푖. O Theorem 4.2. Consider the perturbed network (4.7),(4.8), and (4.22) under As- * ¯ * sumptions 4.4-4.9. Suppose that 푟푖 ∈ int(ℛ푖) for every 푖 and there exists 0 < 휖0 ≤ 휖

such that for all 0 < 휖 ≤ 휖0 the DT system (4.19) is exponentially ultimately bounded − + in an 휖-independent set [푤* , 푤* ]. Then, given any 휇 > 0, 휇-NDD can be achieved if for all 푖

**** * 0 < 휖푖 ≤ 휖푖 (휇, 휖푖+1, ··· , 휖푁 ), 0 < 휈 ≤ 휈 (휇, 휖), (4.30)

**** * where 휖푖 and 휈 are both positive functions non-increasing with 휇. O

126 Proof. (Sketch). The proof is similar to that of Theorem 4.1 but we need to keep track of the model reduction error arising from applying Corollary 4.1 to the subsystems. In particular, for a perturbed network composed of singularly perturbed monotone

** ** subsystems, there exists 휈 (휇1, 휖) such that for all 0 < 휈 ≤ 휈 , the convergence result in (4.14) can be replaced by

휇 푑(푡) −→1 [휓*(푤−, 푤+; 푟*, 휖) − 푃 (|푤±|; 휖), 휓*(푤+, 푤−; 푟*, 휖) + 푃 (|푤±|; 휖)],

휇 푦(푡) −→1 [퐻(푟*) − 푄(|푤±|; 휖), 퐻(푟*) + 푄(|푤±|; 휖)], where 푃 and 푄 are identical to those in (4.15). If the DT system (4.19) converges

− + to [푤* , 푤* ], the small-gain theorem for approximate convergent-input-convergent- 훼(휇 ) output systems (Lemma C.4 in Appendix Section C.4) allows us to claim 푤(푡) −−−→1

− + + − [푤**, 푤**], where 훼(·) is a class 풦0 function and 푤** and 푤** are 휖-independent. The **** rest of the proof is similar to that of Theorem 4.1. For example, one can take 휖푖 := *** ** ** 휖푖 (휇/2, 휖푖+1, ··· , 휖푁 ) and 휈 = 휈 (휇/2, 휖).

In summary, in addition to the conditions of Theorem 4.1, to achieve NDD for networks composed of non-monotone subsystems, Theorem 4.2 requires that the timescale separation in each subsystem is sufficiently large휈 ( is sufficiently small).

This ensures that the behavior of Σ푖, which may be non-monotone, are sufficiently 푟 * close to that of Σ푖 , which is monotone. Since 휈 depends on 휖, when decreasing 휖 to achieve 휇-NDD for a fixed 휇, it is important to ensure that 휈 ≤ 휈*(휇, 휖) remains satisfied.

Remark 4.5. The value of parameter 휈 does not affect the equilibrium location of (4.22). Hence, a small 휖 ensures that the equilibrium location of 풩 is close to that of 풩0. The role of a small 휈 is to guarantee that 풩 is dynamically “well-behaved”. This is a consequence of the convergent-input-convergent-state property for singularly perturbed monotone subsystems in Lemma 4.5. O

127 4.7 Application to decentralized QIC-regulated genetic circuits

In this section, we apply Theorem 4.2 to genetic circuits composed of feedback- regulated subsystems that are described in Section 4.2.

Analytical results

We consider genetic subsystems in (4.3) interconnected through transcriptional regulation (i.e., prescribed interactions). Specifically, the prescribed interactions take the form of (4.7), where 퐺 is a Hill function [41] and we only consider 퐺 that satisfies As- sumption 4.5, that is, 퐺 does not contain a feedback loop. A Hill function has globally bounded derivatives, hence it is globally Lipschitz. The unintended interaction

∑︁ 푤푖 = ∆푖(푑) = 푑푗 (4.32) 푗̸=푖

satisfies Assumption 4.4. The feedback-regulated subsystem dynamics in (4.3) isnot monotone, because the incidence graph induced by the dynamics in (4.3) contains

a negative cycle: 푠푖 − 푚푖 → 푝푖 − 푠푖. However, if the decay rate constant 훿0 of [ [ the RNA species m푖 and s푖 can be made much larger than the decay rate constant

훿 of protein p푖, then the subsystem dynamics can be regarded as a two-timescale system. In particular, let 휈 := 훿/훿0, the feedback-regulated subsystem dynamics can be re-written as:

1 1 휈푚˙ 푖 = 푟푖 − 휆푖푚푖푠푖 − 훿푚푖, (4.33a) 휖푖 휖푖 1 1 휈푠˙푖 = 훽푖푝푖 − 휆푖푚푖푠푖 − 훿푠푖, (4.33b) 휖푖 휖푖 푚푖/휅푖 푝˙푖 = 훼푖 − 훿푝푖. (4.33c) 1 + 푚푖/휅푖 + 푤푖

System (4.33) is in the form of (4.22) if 휈 = 휈/휈0 is sufficiently small, with fast state ⊤ variables 푧푖 = [푚푖, 푠푖] , slow state variable 푥푖 = 푝푖, prescribed output 푦푖 = 푝푖, and

128 disturbance output 푑푖 = 휌푖(푥푖, 푧푖) = 푚푖/휅푖. In practice, the decay rate constant (훿0) of mRNA and sRNA is often faster than that of protein (훿) [90]. To further increase 훿0 to reduce 휈, one can (a) engineer the mRNA sequence to recruit additional RNase for

′ its degradation and (b) introduce an additional mRNA species m푖 that can bind and sequester sRNA s푖 to effectively enhance its removal rate [60, 163]. The parameter 휖푖 can be decreased experimentally by increasing the amount of DNA that encodes m푖 and s푖 [68] and rational design of the sRNA sequence [110].

The state variables represent molecular concentrations, and it is easy to verify that, under the network dynamics (4.7),(4.32),(4.33), the non-negative orthant is positively

2 invariant. Hence, we consider: 풳푖, ℛ푖, 풲푖 = R≥0 and 풵푖 = R≥0. To apply Theorem 4.2, we verify that each feedback-regulated subsystem (4.33) satisfies Assumptions 4.6,4.7,4.8. To verify Assumption 4.7, the boundary layer dynamics are:

d 1 푚 = (푟 − 휆 푚 푠 ) − 훿푚 , d휏 푖 휖 푖 푖 푖 푖 푖 푖 (4.34) d 1 푠푖 = (훽푖푝푖 − 휆푖푚푖푠푖) − 훿푠푖. d휏 휖푖

For each fixed pair (¯푟푖, 푝¯푖) ∈ ℛ푖 × 풳푖 and positive 휖푖, system (4.34) has a unique ⊤ non-negative equilibrium 푧¯푖 = 훾푖(¯푟푖, 푝¯푖; 휖푖) := [푚 ¯ 푖, 푠¯푖] ∈ 풵푖, where

√︀ 2 2 2 1 퐴푖 + 퐴푖 + 4휖푖 훿 휆푖푟¯푖 푚¯ 푖 := 훾푖 (¯푝푖, 푟¯푖; 휖푖) = , (4.35) 2휖푖훿휆푖

and

2 2 퐴푖(¯푟푖, 푝¯푖) :=푟 ¯푖휆푖 − 훽푖휆푖푝¯푖 − 훿 휖푖 . (4.36)

GAS of this equilibrium can be shown using a Lyapunov function [19, 60]. Note that by (4.35), unless 푝¯푖 =푟 ¯푖/훽푖 + 풪(휖푖), the subsystem disturbance output is 풪(1/휖푖).

Hence, as we discussed in Section 4.2, increasing robustness in Σ푖 by decreasing 휖푖 may create larger disturbance to the other subsystems in the network.

1 푟 Substituting 푚푖 = 훾푖 (푝푖, 푟푖; 휖푖) into (4.33c), the reduced subsystem dynamics Σ푖

129 follow:

푟 푟 푟 푟 푟 푟 푝˙푖 = 푓푖 (푝푖 , 푟푖, 푤푖; 휖푖), 푑푖 = 휌푖 (푝푖 , 푟푖; 휖푖), (4.37) where

1 푟 푟 푟 훾푖 (푝푖 , 푟푖; 휖푖)/휅푖 푟 푓푖 (푝푖 , 푟푖, 푤푖; 휖푖) : = 훼푖 1 푟 − 훿푝푖 , 1 + 훾푖 (푝푖 , 푟푖; 휖푖)/휅푖 + 푤푖 푟 푟 1 푟 휌푖 (푝푖 , 푟푖; 휖푖) : = 훾푖 (푝푖 , 푟푖; 휖푖)/휅푖,

1 푟 and 훾푖 (푝푖 , 푟푖; 휖푖) is defined in (4.35), according to which:

(︃ )︃ 휕훾1 1 퐴 + 2휖2훿2 푖 = 1 + 푖 푖 > 0, 휕푟 2휖 훿 √︀ 2 2 2 푖 푖 퐴푖 + 4휖푖 훿 푟푖휆푖 (︃ )︃ (4.38) 휕훾1 훽 퐴 푖 = − 푖 1 − 푖 < 0, 휕푝푟 2휖 훿 √︀ 2 2 2 푖 푖 퐴푖 + 4휖푖 훿 푟푖휆푖

푟 푟 푟 1 1 푟 for all 푝푖 , 푟푖, 푤푖; 휖푖. Hence, 휕푓푖 /휕푟푖 = (휕푓푖 /휕훾푖 ) · (휕훾푖 /휕푟푖) > 0 and 휕푓푖 /휕푤푖 < 0. 푟 The reduced subsystem Σ푖 is thus I/S monotone with respect to the partial orders 푟 푤 푥 푟 1 (휎 , 휎 ; 휎 ) = (1, −1; 1), and the output function 휌푖 = 훾푖 /휅푖 has sign-stable partial derivatives by (4.38). Therefore, Assumption 4.8-(i) is satisfied.

To show that the scalar reduced dynamics (4.37) has a GAS equilibrium for every

푟 푟 푟 fixed (푟푖, 푤푖) ∈ ℛ푖 × 풲푖, note that for all 휖푖 > 0, 휕푓푖 /휕푝푖 < 0, 푓푖 (0, 푟푖, 푤푖; 휖푖) ≥ 0, 푟 푟 and lim 푟 푓 (푝 , 푟 , 푤 ; 휖 ) = −∞. Hence, the reduced system (4.37) has a GAS 푝푖 →+∞ 푖 푖 푖 푖 푖 푟 equilibrium 푝¯푖 in 풳푖:

푟 푟 푝¯푖 = 휙푖 (푟푖, 푤푖; 휖푖) = ℎ푖(푟푖, 푤푖; 휖푖), (4.39)

where the last equality is because 푦푖 = 푝푖 and thus the static I/O characteristic ℎ푖 co- 푟 incides with the static I/S characteristic 휙푖 . Given (4.37) and (4.39), we can compute * 푟 the static disturbance I/O gain 휓푖 using the canonical decomposition functions of 휌푖 푟 푟 푟 and ℎ푖. Specifically, since 휕휌푖 /휕푝푖 < 0, 휕휌푖 /휕푟푖 > 0, the canonical decomposition

130 푟 푟 function 휌ˆ푖 of 휌푖 is:

푟 + + − − 푟 − + 휌ˆ푖 (푝푖 , 푟푖 , 푝푖 , 푟푖 ; 휖푖) = 휌푖 (푝푖 , 푟푖 ; 휖푖). (4.40)

푟 푟 푤 푥 Similarly, since Σ푖 is I/S monotone with respect to the partial orders (휎 , 휎 ; 휎 ) =

(1, −1; 1), we have 휕ℎ푖/휕푟푖 > 0, 휕ℎ푖/휕푤푖 < 0, and the canonical decomposition 푟 function 휙ˆ푖 of 휙푖 = ℎ푖 is

+ + − − * + − 휙ˆ푖(푟푖 , 푤푖 , 푟푖 , 푤푖 ; 푟푖 , 휖푖) = ℎ푖(푟푖 , 푤푖 ; 휖푖), (4.41)

where ℎ푖 is the static I/O characteristic defined in (4.39) and we do not need to explicitly compute its expression. Composing (4.40) and (4.41) according to Lemma 4.1,

+ + − − 푟 − + + the static disturbance I/O gain is 휓푖(푟푖 , 푤푖 , 푟푖 , 푤푖 ; 휖푖) := 휌푖 (ℎ푖(푟푖 , 푤푖 ; 휖푖), 푟푖 ; 휖푖) = 1 − + + 훾푖 (ℎ푖(푟푖 , 푤푖 ; 휖푖), 푟푖 ; 휖푖)/휅푖. Because of this and by equation (4.25), for a fixed input * 푟푖 , we have:

* + − * 1 * + * 휓푖 (푤푖 , 푤푖 ; 푟푖 , 휖푖) = 훾푖 (ℎ푖(푟푖 , 푤푖 ; 휖푖), 푟푖 , 휖푖)/휅푖. (4.42)

1 On the other hand, when (4.37) is set to steady state, 훾푖 necessarily satisfies

1 * + * 훾푖 (ℎ푖(푟푖 , 푤푖 ; 휖푖), 푟푖 , 휖푖)/휅푖 * + 훼푖 1 * + * + = 훿ℎ푖(푟푖 , 푤푖 ; 휖푖). 1 + 훾푖 (ℎ푖(푟푖 , 푤푖 ; 휖푖), 푟푖 ; 휖푖)/휅푖 + 푤푖

Substituting into (4.42), the disturbance I/O gain of (4.37) can be written as:

* + + * + − 훿ℎ푖(푟푖 , 푤푖 ; 휖푖)(1 + 푤푖 ) 휓푖 (푤푖 , 푤푖 ; 휖푖) = * + . (4.43) 훼푖 − 훿ℎ푖(푟푖 , 푤푖 ; 휖푖)

To verify the static disturbance attenuation property in Assumption 4.8-(ii) is

1 2 satisfied, we show in Appendix Section C.5 that there exists constants 퐾푖 , 퐾푖 > 0 such that

1 2 |ℎ푖(푟푖, 푤푖; 휖푖) − 퐻푖(푟푖)| ≤ 휖푖(퐾푖 |푤푖| + 퐾푖 ) (4.44)

131 ¯ for all (푟푖, 푤푖) ∈ ℛ푖 × 풲푖 and for 휖푖 sufficiently small, where the nominal static I/O characteristic is

퐻푖(푟푖) = 푟푖/훽푖, (4.45)

¯ and the admissible reference input set ℛ푖 is any 휖-independent compact subset of

(0, 훼푖훽푖/훿). The proof uses a Lyapunov function to show that the trajectory of (4.37) ultimately converges to an 풪(휖푖) neighborhood of 퐻푖(푟푖). Finally, Appendix Section

C.6 proves that ℎ푖 and 휓푖 possess the required Lipschitz conditions in Assumption 4.6.

We now verify Assumption 4.9, which requires 푟(푡) and 푤(푡) and their derivatives to be bounded. By (4.33c) and the comparison lemma, for any initial condition,

푝푖(푡) is globally attracted to the set [0, 훼푖/훿]. Hence, 푝˙푖(푡) is also bounded. Because

푟푖 = 퐺푖(푦) = 퐺푖(푝) and 퐺(·) is a Hill function, 푟˙푖(푡) and 푟푖(푡) are both bounded.

Similarly, it is easy to verify from (4.33) that [0, 푟푖/휖푖] is a globally attractive set ∑︀ ∑︀ for 푚푖(푡) and hence 푤푖(푡) = 푗̸=푖 푑푗 = 푗̸=푖 푚푗/휅푗 is bounded by an 휈-independent constant. With Assumptions 4.4,4.5,4.6,4.7,4.8,4.9 satisfied, we can apply Theorem 4.2 to determine if NDD can be achieved for genetic circuits composed of subsystems (4.33). Specifically, we find that the DT dynamical system (4.19) is exponentially ultimately bounded in an 휖-independent set for some 푟* values, and hence Theorem 4.2 can be applied to ensure NDD if the nominal reference input 푟* falls within a network admissible reference input set, as we state below.

Proposition 4.1. Given any 휇 > 0, the perturbed network (4.7), (4.32), and (4.33) * ¯ ¯ has the 휇-NDD property if 푟 ∈ ℛ풩 , where ℛ풩 is any 휖-independent compact subset of

{︃ }︃ * ¯ ∑︁ * * ℛ풩 := 푟푖 ∈ ℛ푖 : 휂푖 (푟푖 ) < 1, ∀푖 , (4.46) 푗̸=푖

132 where

* 푟푖훿 휂푖 (푟푖) := , 훼푖훽푖 − 훿푟푖

and additionally, 휈 and each 휖푖 are sufficiently small. O

Proof. With all assumptions in Theorem 4.2 satisfied, we only need to verify that the DT system (4.19) is exponentially ultimately bounded in an 휖-independent set.

* + − Given the form of 휓푖 in (4.43), we find that the dynamics of 푤 and 푤 in (4.19) will be completely decoupled. Hence, it is sufficient to show that the trajectory of the following 푁-dimensional DT system is exponentially ulitmately bounded in an 휖-independent set:

* * * ∑︁ 훿ℎ푗(푟푗 , 푤푗(푘); 휖푗)(1 + 푤푗(푘)) 푤푖(푘 + 1) = ∆푖 ∘ 휓 (푤(푘); 푟 , 휖) = * . (4.47) 훼푗 − 훿ℎ푗(푟 , 푤푗(푘); 휖푗) 푗̸=푖 푗

To this end, we define

훿ℎ푖(푟푖, 푤푖; 휖푖) * 푟푖훿 휂푖(푟푖, 푤푖; 휖푖) := , 휂푖 (푟푖) := , 훼푖 − 훿ℎ푖(푟푖, 푤푖; 휖푖) 훼푖훽푖 − 훿푟푖

¯ * and note that for all (푟푖, 푤푖; 휖푖) ∈ ℛ푖 × 풲푖 × (0, 휖푖 ], the followings are satisfied: (i)

휂푖(푟푖, 푤푖; 휖푖) > 0, (ii) 휂푖(푟푖, 푤푖; 휖푖) < 휂푖(푟푖, 0; 휖푖), and (iii) 훼푖 − 훿ℎ푖(푟푖, 푤푖; 휖푖) is bounded away from 0, and thus by (4.44), there exists constant 푘푖 > 0 such that |휂푖(푟푖, 0; 휖푖) − * 2 휂푖 (푟푖)| ≤ 푘푖휖푖. Using these properties, we consider 푉 (푘) := |푤(푘)| as a candidate Lyapunov function, which satisfies

⃒ ⃒2 ⃒ ⃒ ⃒∑︁ * ⃒ 푉 (푘 + 1) = ⃒ 휂푗(푟푗 , 푤푗(푘); 휖푗)(1 + 푤푗(푘))⃒ ⃒ 푗̸=푖 ⃒ [︃ ]︃2 2 ∑︁ * * ≤ (1 + |푤(푘)|) 휂푗 (푟푗 ) + 푘푗휖푗 . (4.48) 푗̸=푖

* ¯ * ∑︀ * * Suppose (4.46) is satisfied such that for all 푟 ∈ ℛ풩 and 휖 ∈ (0, 휖푖 ], 푗̸=푖(휂푗 (푟푗 ) +

푘푗휖푗) ≤ 1 − 휗 for all 푖, where 0 < 휗 < 1 is an 휖-independent constant. We have

133 푉 (푘+1)−푉 (푘) ≤ (1−휗)(1+|푤|)2 −|푤|2 ≤ −휗|푤|2 +2(1−휗)|푤|+(1−휗) ≤ −휗|푤|2/2 if |푤| ≥ 푤* := max{1, 6(1−휗)/휗}, where 푤* is 휖-independent. This proves that (4.47) is exponentially ultimately bounded in [0, 푤*].

* ¯ Therefore, if 퐺 and Σ푖 are designed such that 푟 ∈ ℛ풩 , then the network behavior can be made independent of ∆ (i.e., NDD is achieved) by making 휖푖 sufficiently small in each subsystem. Our result thus provides an analytical robustness performance limit for genetic circuits composed of feedback-regulated genetic subsystems.

Remark 4.6. The set ℛ풩 is a network admissible reference input set. It is a subset ∏︀푁 ¯ of 푖=1 ℛ푖, which is the cartesian product of admissible reference input sets of iso- lated subsystems. According to (4.46), as the number of subsystems increases, the reference input each subsystem can take for the network to maintain NDD property also decreases.

Feedback-regulated subsystems with independent inputs

We apply Proposition 4.1 to circuit 풩1, which consists of three identical feedback- * * regulated subsystems with identical, independent reference input 푟푖 = 푟 (see Figure

4-2a). Recall that in Figure4-2b, our simulations show that NDD of 풩1 can only be achieved for certain reference input levels. To explain this, we apply Proposition

4.1 to compute the network admissible reference input set ℛ풩 = {0 < 푟0 < 100/3}, where 푟0 is the identical reference input to all subsystems. In accordance with the * ¯ simulation in Figure4-2b, NDD can be achieved by decreasing 휖푖 if 푟 ∈ ℛ풩1 . On the other hand, decreasing 휖푖 does not improve the network’s robustness to unintended * ¯ interactions if 푟 ∈/ ℛ풩 , indicating that ℛ풩 is not conservative. The value of 휈 does not affect the NDD performance of 풩1. In fact, stability of 풩1, which consists of identical subsystems with identical, independent reference inputs, can be shown for any 휈 > 0 through linearization [124]. Thus, with reference to Remark 4.5, there is no need to decrease 휈 to ensure stability of 풩1 and the requirement for 휈 to be sufficiently small in Proposition 4.1 is conservative in this special case.

134 Figure 4-4: Network disturbance decoupling for a genetic activation cascade. (a) Schematic of a genetic circuit composed of five feedback-regulated subsystems connected in ∑︀ * * a cascade topology. (b) Simulation results for the network when 푖̸=푗 휂푗 (푟푗 ) < 1 and thus * ¯ ∑︀ * * * ¯ 푟푖 ∈ ℛ풩2 . (c) Simulation results for the network when 푖̸=푗 휂푗 (푟푗 ) > 1 and thus 푟푖 ∈/ ℛ풩2 . −1 Simulation parameters are identical for all subsystems: 훼푖 = 70 nM/hr, 휆푖 = 5 (nM · hr) , −1 −1 훿 = 0.5 hr , 훽푖 = 1 hr , 휅푖 = 10 nM, and 휖푖 = 휖 for all 푖. The prescribed interactions * follow equation (4.49) with parameters: 푛푖 = 4, 푘푖 = 6 nM, and and 푟1 = 10 nM/hr. For panel (b) 퐵푖 = 10 nM/hr for all 푖 ≥ 1 and for panel (c) 퐵푖 = 10 nM/hr for 푖 = 1, 2 and 퐵푖 = 50 nM/hr for 푖 = 3, 4, 5.

135 Feedback-regulated subsystems in a cascade

We study another circuit 풩2 composed of five feedback-regulated subsystems connected in a cascade topology through prescribed interactions, that is, through transcriptional regulation (see Figure 4-4a). Simulation results for network 풩2 with different (휈; 휖) pairs are shown in Figure4-4b. In particular, we model prescribed interactions as Hill functions [41]:

⎧ 푛 (푦푖−1/푘푖) 푖 ⎪퐵푖 푛 , if 푖 ̸= 1, ⎨ 1+(푦푖−1/푘푖) 푖 푟푖 = 퐺푖(푦푖−1) = (4.49) ⎪ * ⎩푟1, if 푖 = 1,

where 퐵푖 quantifies the maximum transcription rate from gene 푖, 푘푖 is a dissociation constant whose values decreases with the binding affinity between protein p푖−1 and the promoter of gene 푖, and 푛푖 is describes the binding cooperativity. Based on (4.45) * * ¯ and (4.49), we can compute the nominal reference input 푟 and confirm that 푟 ∈ ℛ풩2 . We therefore apply Proposition 4.1 to claim that for arbitrarily small 휇, 휇-NDD of

풩2 can be achieved by decreasing both 휖 and 휈. This is consistent with simulations in Figure 4-4b.

4.8 Summary

In this chapter we study networked dynamical systems perturbed by unintended interactions among subsystems. These unintended interactions make it difficult to predict and design network-level behaviors. We therefore seek conditions on subsystem dynamics, the prescribed interaction map, and the unintended interaction map to achieve network disturbance decoupling (NDD), where the steady state outputs from all subsystems become essentially independent of the unintended interactions. While NDD may be addressed by designing the entire network monolithically, we find that, under certain conditions, NDD can be obtained by simply improving each subsystem’ robustness to a constant, state-independent disturbance. Specifically, these conditions require that (i) all subsystems are I/S monotone, (ii) the prescribed interactions

136 among subsystems do not contain a feedback loop, and (iii) the unintended interactions are cooperative. When the subsystem dynamics are non-monotone, the same result holds with similar conditions if the subsystem dynamics have a timescale separation property, such that the reduced subsystem dynamics are monotone. We apply our theoretical result to engineer robust genetic circuits. We show that the sRNA- mediated QIC in Section 3.4, which enable a single genetic subsystem to asymptotically attenuate a constant disturbance, can be used to regulate multiple genes in a network to reach NDD under additional technical conditions.

137 138 Chapter 5

Conclusions and future work

5.1 Conclusions

The lack of robustness and predictability is a major bottleneck in engineering larger and more complex synthetic biomolecular systems for promising applications in health, energy, and environment. In this thesis, we take a systems and control theoretic approach to address a series of fundamental problems related to the robustness of genetic circuits, which are interconnection of genetic subsystems. These problems are centered on the practical biological problem of eliminating resource competition as a major physical constraint that couples otherwise unconnected genetic subsystems. The majority of the results in this thesis are based on theoretical analysis of nonlinear deterministic models of genetic circuits and most of them have been verified experimentally in living cells.

Characterization of unintended interactions due to resource competition

Expression of synthetic genes relies on a common pool of resources provided by the host cell, which fundamentally couples the expression of all genes. This coupling creates a hidden layer of unintended interactions among genetic subsystems, which can create hard-to-predict network-level behaviors when combined with prescribe regulatory interactions among genetic subsystems. In Chapter 2, we build mechanistic mathematical models to describe these phenomena in bacteria. The model has the

139 same dimension as the standard textbook models for gene regulation, where resource limitations are neglected, and are therefore simple enough to guide design. The models also elicited simple graphical rules to describe resource competition effects as unintended interactions, whose strength can be characterized by a lumped parameter, called resource demand coefficient, for each genetic subsystem. To show the effectiveness of these models to guide network design, we construct a library ofge- netic activation cascades, whose expected static I/O response curve is monotonically increase. However, we demonstrate that, by tuning the resource demand coefficients to vary the strength of the unintended interactions, its static I/O response curve can be either monotonically increasing, decreasing, or biphasic.

Robustness of genetic subsystems to disturbances

Although reducing the expression of all synthetic genes is a trivial solution to resource competition, it is undesired in most application scenarios. To solve this problem, in Chapter 3, we formulated a set-point regulation problem and aimed to design a biomolecular feedback controller in vivo so a genetic subsystem’s output (e.g., protein concentration) can reject a change in gene expression resource availability as a disturbance. While synthesizing a feedback controller for set-point regulation is a well- studied problem in control theory, its biomolecular solution is still largely missing. In fact, engineering a biomolecular feedback controller is confounded by the fact that most biomolecules face decay, making it impossible to precisely implement integral control, which is the main workhorse to achieve perfect set-point regulation in control design. We proposed quasi-integral controllers (QICs) as a general solution to handle this physical constraint. Specifically, QICs require a timescale separation property: if one can find reactions 1/휖 faster than decay of reactants to implement feedback, then the steady state regulation error is guaranteed to be 풪(휖) under mild technical conditions. Based on this principle, we design and implement a QIC through small RNA (sRNA) interference in E. coli bacteria. The regulated genetic subsystem achieved remarkable robustness performance. In particular, while an unregulated subsystem’s output was reduced by more than 50% upon activation of a “resource competitor”

140 gene, a subsystem regulated by the sRNA-based QIC with similar output level was essentially unchanged. When analyzing the dynamics of the sRNA-based QIC, we en- countered a peculiar theoretical problem called singular singular perturbation (SSP), where the boundary layer (i.e., fast dynamics) of a two-timescale system does not have a well-defined steady state. This problem arises from the structure of thechem- ical reactions of the sRNA-based QIC and disallows existing mathematical tools to perform model reduction. We provide a procedure to reduce SSP systems with theoretical bounds on model reduction error, which helps analyze the dynamics of the sRNA-based QIC to time-varying inputs as well as a general class of biomolecular feedback systems with similar architecture.

Robustness of networks to unintended interactions

Equipped with the QICs to achieve robustness of a single genetic subsystem, we questioned whether a network composed of 푁 QIC-regulated subsystems is robust to unintended interactions among them. This problem is nontrivial because control effort for subsystem 푖 becomes a disturbance to all the other (푁 − 1) devices in the network through resource competition. Inspired by this, in Chapter 4, we formulate a system-theoretic problem, which we call network disturbance decoupling (NDD). In an NDD problem, the output from the network is required to be independent of the presence/absence of the unintended interaction. This is distinctive from standard disturbance attenuation problems in control theory, where the disturbances are external and state-independent. We find that NDD can be achieved if the subsystem dynamics are monotone or near-monotone, the prescribed interactions do not have feedback loops, and the unintended interactions are monotone. These conditions are used to guide biomolecular controller design to shape genetic subsystem dynamics for NDD.

5.2 Extensions and future directions

While the results in this thesis are promising, they can be extended in many directions.

141 Resource competition model

One important limitation of our model is the over-simplification of cell metabolism by assuming constant total amount of resources. A cell system has a number of additional complications. Firstly, recent evidence suggests that resources are not distributed evenly in cells [16]. How spatial distribution of resources changes our current models need to be investigated. Secondly, expression of a synthetic gene may retard cell growth. This, in turn, can affect the expression level of synthetic genes [29]. These circuit-host interactions involve some endogenous regulation mechanisms that are still unknown and not accounted for in this resource conservation model. New data-driven and/or computational tools may be needed to account for the combined loading effect from resource competition and growth inhibition [66, 159].

Biomolecular control system design

Our results on tracking performance of QICs is derived for their linearized models. Although simulations suggest satisfactory predictive power of the linearized models even for large inputs, to fully address this robust tracking problem for the nonlinear model, one direction is to extend the SSP model reduction framework to nonlinear systems. In this case, identifying a slow manifold, or equivalently, identifying the pseudo-fast variable, is a non-trivial task, and new data-driven approaches may be helpful [94]. It is also interesting to study situations where the process to be regulated is not passive. In this case, the closed loop system may not remain stable for arbitrarily small 휖, and multiple controllers may need to be employed to regulate such processes. From an implementation standpoint, there is the continuous need to find biomolecules that implement the control actions with faster dynamics and reduced control effort. There is a diverse set of underutilized biomolecular processes, suchas post-translational modification, that are promising to reach this goal. Additionally, since our QIC design essentially requires a high-gain feedback action, its applicability to genome-integrated systems is questionable. This is because DNAs all have single copy on chromosome, making it difficult to achieve large control action. Novel control

142 design strategies may be required in this case.

Robustness of networked systems to unintended interactions

The unintended interactions we consider here are described by a static map. In the future, we plan to consider NDD problems for a larger class of unintended interactions ∆, including, for example, ∆ that are uncertain and contain dynamics. In these cases, it would also be interesting to explore how other I/O properties of ∆, such as passivity, implicate new conditions on subsystem dynamics and on their interactions for NDD. We also plan to extend this study to multi-stable networks and to consider intended interaction maps that contain feedback loops. These studies may provide a road map to engineer networked systems that can function robustly in different contexts and configurations. In the application setting of genetic circuits, experimental implementation of multiple regulated subsystems in vivo has not appeared. It is important to evaluate their performance to verify the design considerations specified here and/or to come up with new controller designs that meet the conditions for NDD. This control system engineering problem will also likely require careful design of control system architecture. In natural biological systems, a combination of feedback, feedforward, buffering, and redundancy are used to achieve robustness, and learning from natural systems will elucidate roles for rational and efficient organization of regulated biomolecular subsystems [65]. The NDD problem setup can also be extended to other engineering systems, such as close formation control of autonomous aerial or under water vehicles, building temperature control, and control of wind turbines in close proximity in a wind farm.

143 144 Appendix A

Appendix for Chapter 2

A.1 Derivation of graphical rules

Directed edges, such as those in Figure 2-2, have been used to represent prescribed regulatory interactions among subsystems, where output of one subsystem (e.g., a

TF p푖) binds with the promoter in another subsystem to regulate the its protein production [4]. Here, we mathematically define the standard to draw interaction graphs and illustrate that resource limitations lead to effective interactions in genetic circuits that do not rely on regulatory interactions.

Definition A.1. Let the dynamics of each subsystem be scalar and represented by:

푦˙푖 = 퐹푖(푦) − 훾푖푦푖. We draw the interaction graph from subsystem 푗 to 푖 based on the following rules:

휕퐹푖 ∙ If ≡ 0 for all 푦푗 ∈ ≥0, then there is no interaction from 푗 to 푖; 휕푦푗 R

휕퐹푖 휕퐹푖 ∙ If ≥ 0 for all 푦푗 ∈ ≥0 and ̸= 0 for some 푦푗, then p푗 activates production 휕푦푗 R 휕푦푗

of p푖 and we draw 푗 → 푖;

휕퐹푖 휕퐹푖 ∙ If ≤ 0 for all 푦푗 ∈ ≥0 and ̸= 0 for some 푦푗, then p푗 represses p푖 and we 휕푦푗 R 휕푦푗 draw 푗 − 푖. [

휕퐹푖 휕퐹푖 ∙ If 0 for some 푦푗 ∈ ≥0 and < 0 for some other 푦푗, then the regulation 휕푦푗 > R 휕푦푗

of p푗 on p푖 is undetermined and we draw 푗 ( 푖.

145 For simplicity of notation, we define 풰푖 as the set of subsystems whose TF output affects subsystem 푖:

풰푖 = {푘 = 1, ··· , 푁 : 휕퐹푖/휕푦푘 ̸= 0, for some 푦푘 ≥ 0}. (A.1)

¯ Based on Definition A.1, for the standard model in equation (2.13), 퐹푖(푦) = 푇푖퐺푖(푦), and therefore there is a link from 푗 to 푖 if and only if 푗 ∈ 풰푖. In our modified model in equation (2.11), instead we have

푇¯ 퐺 (푦) 퐹 (푦) = 푖 푖 , 푖 푁 ∑︀ 1 + 퐽푘퐺푘(푦) 푘=1

which implies that the dynamics of 푦푖 may be belong to the set 풰푖. In what follows, we discuss the effective interactions from subsystem 푗 to sub-

system 푖 when (i) p푖 is the only target of p푗, (ii) p푖 is one of the multiple targets

of p푗, and (iii) p푖 is not a target of p푗. We assume that a TF (e.g., p푗) cannot be

both an activator and a repressor. When p푖 is the only target of p푗, the following Proposition shows that resource limitations do not alter the qualitative effect (i.e., activation/repression) of p푗 on p푖 in the interaction graph.

Proposition A.1. If 푗 ∈ 풰푖 and 푗∈ / 풰푞 for all (푞 ̸= 푖). Then we have sign[휕퐹푖(푦)/휕푦푗] =

sign[휕퐺푖(푦)/휕푦푗]. O

Proof. According to equation (2.11),

positive ⏞ ⏟ 휕퐹 (푦) 휕퐹 휕퐺 (푦) (︂휕퐹 )︂ (︂휕퐺 )︂ 푖 = 푖 · 푖 ⇒ sign 푖 = sign 푖 . 휕푦푗 휕퐺푖 휕푦푗 휕푦푗 휕푦푗

Remark A.1. In the case where 푗 ∈ 풰1, ··· , 풰푘 (푘 ≥ 2), the effective interactions

from subsystem 푗 to its targets are undetermined. For example, if p푗 represses p1 and

p2 simultaneously, the effective interaction from subsystem 푗 to subsystem 1 is given

146 by

transcriptional repression unintended activation ⏞ ⏟ ⏞ ⏟ 휕퐹 (푦) 휕퐹 휕퐺 (푦) 휕퐹 휕퐺 (푦) 1 = 1 · 1 + 1 · 2 . 휕푦푗 휕퐺1 휕푦푗 휕퐺2 휕푦푗 ⏟ ⏞ ⏟ ⏞ ⏟ ⏞ ⏟ ⏞ positive negative negative negative

As sign(휕퐺1/휕푦푗) cannot be determined, the effective interaction from subsystem 푗 to subsystem 1 is undetermined. O

When subsystem 푗 is not a parent of subsystem 푖, the following Proposition shows

p)푗 is an effective repressor for p푖 if p푗 is an activator. Conversely, p푗 is an effective activator for p푖 if it is a repressor.

Proposition A.2. If 푗∈ / 풰푖 but 푗 ∈ 풰푘 for some 푘 ̸= 푖, then we have sign[휕퐹푖(푦)/휕푦푗] =

−sign[휕퐺푘(푦)/휕푦푗]. O

Proof. Since 푗∈ / 풰푖, 휕퐹푖/휕퐺푘 < 0 for all 푘.

휕퐹푖(푦) ∑︁ 휕퐹푖 휕퐺푘(푦) = · . 휕푦푗 휕퐺푘 휕푦푗 푘 ⏟ ⏞ negative

Therefore, sign(휕퐹푖/휕푦푗) = −sign(휕퐺푘/휕푦푗).

The results from Propositions A.1-A.2 and Remark A.1 are summarized in Figure 2-2.

147 148 Appendix B

Appendix for Chapter 3

B.1 Modeling dilution in mass action kinetics

Although modeling dilution of molecule species x due to cell growth and division as −훾푥 has been prevalently used by systems and synthetic biologists, the mechanistic link between cell growth and this mathematical representation may not be immediate. Since the integration leakiness problem arise from this model, we briefly review its derivation in this section. Before looking into detailed model derivation, we want to first emphasize that such models have been extensively validated in experiments since the early 1950s [58, 105], and more recently by Alon et al. in the early 2000s [131, 132]. Moreover, as we shall demonstrate in Theorem 3.1 in Section B.2, we use a more general leakiness model (i.e.,퐿(푥)), which can also take into account various other sources of unmodeled controller dynamics, such as enzymatic degradation. Therefore, whether 퐿(푥) is exactly in the form of 훾푥 does not affect the validity of our main result. With these remarks in mind, we are ready to give a quick review of the derivation of this textbook dilution model. To understand how dilution due to cell growth and division affects the single- cell concentration dynamics of a protein species, we consider a protein x, which is not regulated and not proteolytically tagged (i.e., constitutive) in a cell with initial volume 푉 (0) = 푉0. We assume that the cell grows exponentially in size [79, 168], with 훾푡 growth rate constant 훾 > 0. Therefore, at time 푡, the cell volume is 푉 (푡) = 푉0푒 . We

149 further assume that at 푡 = 푇 , when 푉 (푇 ) = 2푉0, the cell divides into two daughter

cells with identical volume 푉0. After division, we track only one of the daughter cells, neglecting any heterogeneity between the two for simplicity. We use 푁(푡) to denote the number of protein x in a single cell, and use 푥(푡) := 푁(푡)/푉 (푡) to represent the concentration of protein x at time 푡. We assume that after division, the number of protein x is evenly divided into the two daughter cells. Based on these assumptions, when a mother cell divides, the concentrations of x in the daughter cells do not change. This is because

− − − + − 푁(푇 ) 푁(푇 ) 0.5푁(푇 ) 푁(푇 ) + 푥(푇 ) = − = − = = + = 푥(푇 ), (B.1) 푉 (푇 ) 2푉0(푇 ) 푉0 푉 (푇 )

where 푇 − and 푇 + represent the time instant right before and after division, respectively. Therefore, we only need to characterize the concentration dynamics in each cell cycle. Within each cell cycle, the dynamics of 푥 can be derived from:

d (︂푁 )︂ 1 푁 푥˙ = = 푁˙ − 푉˙ . (B.2) d푡 푉 푉 푉 2

˙ Since cell volume grows exponentially, we have 푉 = 훾푉0 exp(훾푡) = 훾푉 (푡). Let 푓(푡) := 푁/푉˙ (푡), equation (B.2) becomes

푥˙ = 푓(푡) − 훾푥 (B.3)

We further assume that the number of proteins produced from a single DNA copy in unit time is a constant 훽. Therefore

푁˙ = 훽퐷(푡), (B.4) where 퐷(푡) is the DNA copy number in the cell, leading to

푁˙ 퐷(푡) 푓(푡) = = 훽 . (B.5) 푉 (푡) 푉 (푡)

150 Here, 퐷(푡)/푉 (푡) represents the copy number of DNA in unit cell volume. Since DNAs are replicated while the cells grow in size, both 퐷(푡) and 푉 (푡) are growing within each cell cycle. In fact, experimental measurements [39, 91] have indicated that the total number of plasmid DNA in a cell population grows (exponentially)

at the same rate as the total cell volume (i.e., 퐷population(푡) ∝ 푉population(푡)). If we make the simplifying assumption that all cells are identical, then the number of plasmid DNA in unit cell volume 퐷(푡)/푉 (푡) is constant over time in a single cell (i.e., 푓(푡) = 훼 := 훽퐷(푡)/푉 (푡) = const.). We therefore arrive at the standard text book model:

푥˙ = 훼 − 훾푥. (B.6)

We remark that the above derivation neglects the stochastic nature of many events in cell growth and division. However, it has been supported by countless experiments to accurately reflect the average concentration dynamics in a cell population. More detailed studies on cell volume control, DNA replication and noise in cell growth and division are deep research topics by their own sake, and are far beyond the scope of this work. References for these topics can be found in, for example [39, 56, 168].

B.2 Proof of Theorem 3.1

The proof is an application of the implicit function theorem, which we state here for completeness.

Lemma B.1. ([133]) Let f be a 풞1-mapping of an open set 퐸 ⊆ R푛+푚 into R푛, such that f(a, b) = 0 for some point (a, b) ∈ 퐸. Let

⃒ ⃒ 휕f(x, y)⃒ 휕f(x, y)⃒ 퐴푥 = ⃒ and 퐴푦 = ⃒ 휕x ⃒x=a,y=b 휕y ⃒x=a,y=b

푛+푚 푚 and assume that 퐴푥 is invertible. Then there exist open sets 푈 ⊆ R and 푊 ⊆ R with (a, b) ∈ 푈 and b ∈ 푊 , having the following property: every y ∈ 푊 corresponds

151 to a unique x such that

(x, y) ∈ 푈 and f(x, y) = 0. (B.7)

If this x is defined to be g(y), then g is a 풞1-mapping of 푊 into R푛 such that,

⃒ 휕g(y)⃒ −1 f(g(y), y) = 0, ∀푦 ∈ 푊, and ⃒ = −(퐴푥) 퐴푦. (B.8) 휕y ⃒y=b

Now we are ready to prove Theorem 3.1.

Proof. Define 퐹 (푥, 푧, 푤) := 푓(푥, Θ(푧), 푤) and 퐺(푥, 푧, 푟) := 푔(푧, ℎ(푥), 푟), and let

(¯푥푖, 푧¯푖) be the exponentially stable equilibrium of the IIC-regulated system (3.9),(3.12). The equilibrium of the closed loop system (3.9)-(3.10) can be solved from the following algebraic equations

⎡ ⎤ 퐹 (푥, 푧, 푤) ⎢ ⎥ ⎢ ⎥ ℱ(푥, 푧, 푤, 푟, 휖) := ⎢ 퐺(푥, 푧, 푟) − 휖퐿1(푧) ⎥ = 0. (B.9) ⎣ ⎦ 푘(푟 − ℎ(푥)) − 휖퐿2(푧)

Since 푟 and 푤 are constants in our problem, for simplicity, we write (B.9) as ℱ(푥, 푧, 휖) = 0 with slight abuse of notation. Due to Assumption 3.3, ℱ is 풞1 for all (푥, 푧) ∈ 풳 ×풵

푖 and all 휖. Meanwhile, based on Assumption 3.2, the steady state (¯푥푖, 푧¯푖) of Σ must satisfy

퐹 (¯푥푖, 푧¯푖, 푤) = 0, 퐺(¯푥푖, 푧¯푖, 푟) = 0, 푘(푢 − ℎ(¯푥푖)) = 0. (B.10)

Since (¯푥푖, 푧¯푖) is locally exponentially stable, its Jacobian matrix evaluated at (¯푥푖, 푧¯푖)

152 must be Hurwitz [81], that is,

⎡ ⎤⃒ ⃒ 휕퐹/휕푥 휕퐹/휕푧1 휕퐹/휕푧2 ⃒ ⎢ ⎥⃒ ⎢ ⎥⃒ 퐷푖 = ⎢ 휕퐺/휕푥 휕퐺/휕푧1 휕퐺/휕푧2⎥⃒ is Hurwitz. (B.11) ⎣ ⎦⃒ ⃒ −푘 · dℎ/d푥 0 0 ⃒ 푥¯푖,푧¯푖

On the other hand, note that when 휖 = 0, any solution (푥*, 푧*) that satisfies ℱ(푥*, 푧*, 0) = 0 can be found from

퐹 (푥*, 푧*, 푤) = 0, 퐺(푥*, 푧*, 푟) = 0, 푘(푟 − ℎ(푥*)) = 0. (B.12)

Comparing (B.10) and (B.12), since (B.10) has a unique solution (Assumption 3.2),

* * * * (푥 , 푧 ) = (¯푥푖, 푧¯푖) must be the unique solution to ℱ(푥 , 푧 , 0) = 0. To apply implicit function theorem, note that

⎡ ⎤⃒ ⃒ 휕퐹/휕푥 휕퐹/휕푧1 휕퐹/휕푧2 ⃒ [︁ ]︁⃒ ⎢ ⎥⃒ 휕ℱ 휕ℱ 퐷휖 = ⃒ = ⎢ 휕퐺/휕푥 휕퐺/휕푧 휕퐺/휕푧 ⎥⃒ 휕푥 휕푧 ⃒ * * ⎢ 1 2⎥⃒ 푥=푥 ,푧=푧 ,휖=0 ⎣ ⎦⃒ −푘 · dℎ/d푥 0 0 ⃒ ⃒ * * 푥=푥 =¯푥푖,푧=푧 =¯푧푖,휖=0 (B.13)

is Hurwitz according to (B.11), and therefore, 퐷휖 is invertible. According to implicit function theorem, there exists a positive constant 휖ˆand an open set 풳˜ ×풵×˜ (−휖,ˆ +ˆ휖) ⊆ * * ˜ ˜ 풳 × 풵 × R such that (a) (푥 , 푧 ) = (¯푥푖, 푧¯푖) ∈ 풳 × 풵, and (b) for 휖 ∈ (−휖,ˆ +ˆ휖), the solution (¯푥, 푧¯) = (¯푥(휖), 푧¯(휖)) to ℱ(¯푥, 푧,¯ 휖) = 0 is continuously differentiable in 휖, that is,

푥¯(휖) =푥 ¯(0) + 풪(휖) = 푥* + 풪(휖), 푧¯(휖) =푧 ¯(0) + 풪(휖) = 푧* + 풪(휖). (B.14)

Since the steady state output satisfies 푦¯ = ℎ(¯푥) and ℎ(·) is continuously differentiable with bounded derivatives, we have

푦¯(휖) = ℎ(¯푥(휖)) = ℎ(푥* + 풪(휖)) = ℎ(푥*) + 풪(휖) = 푟 + 풪(휖).

153 According to Assumption 3.1, since the steady state is locally asymptotically stable for all 0 < 휖 < 휖*, then

푦¯(푢, 푤, 휖) = 푟 + 풪(휖), ∀ 0 < 휖 < min{휖*, 휖ˆ}, (B.15)

where 푦¯ is the steady state of 푦. Note that if (B.15) is satisfied, then trivially

lim휖→0 푦¯(푟, 푤, 휖) = 푟. Based on Definition 3.1, this means that the system can achieve 휖-quasi-integral control.

B.3 Counterexample: increasing only part of the controller reaction rates

Assuming that the integral gain of a LIC can be increased by detuning a small parameter 휖, for any LIC with “leaky” memory dynamics

푘 푧˙ = (푟 − 푦) − 훾푧, 휖 its equilibrium adaptation error can always be computed as:

휖 푒 = 푟 − 푦¯ = 훾푧¯ , 푠푠 푘

where 푧¯ is the equilibrium of the leaky memory variable. Assuming that 푧¯ stays roughly constant as 휖 is reduced, it may seem that arbitrarily small adaptation error can always be achieved by decreasing 휖 (i.e., increasing the integral gain), and therefore our main result (Theorem 3.1) is trivially true. In fact, this intuition leads to previous hypothesis that the effect of leaky integration becomes small when the integral gain 푘/휖 is increased [7, 8, 148]. In this section, we use the type II LIC motif to exemplify that this argument is flawed: increasing the integral gain 푘/휖 alone may not be sufficient to mitigate the effects of leaky integration. This isbecause 푧¯, in general, depends on 휖, and consequently, its magnitude can increase as we detune 휖. This example highlight the need to carry out a precise and systematic mathematical

154 analysis on how to systematically reduce leaky integration error, as we have developed in Theorem 3.1.

Recall that a type II LIC takes the following form (see main text Figure 3-1b-(ii))

푥˙ = 훼푧1 − 훾푥 + 푤, 푧˙1 = 푘푟 − 휃푧1푧2 − 훾푧1, 푧˙2 = 푘푦 − 휃푧1푧2 − 훾푧2, 푦 = 푥. (B.16)

Let 푧 = 푧1 − 푧2, (B.16) can be transformed to:

푥˙ = 훼푧1 − 훾푥 + 푤, 푧˙1 = 푘푟 − 휃푧1(푧1 − 푧) − 훾푧1, 푧˙ = 푘(푟 − 푦) − 훾푧, 푦 = 푥, (B.17)

where 푧 is the “leaky memory variable”. Following the hypothesis that high integral gain mitigates the effect of leaky integration, we can increase 푘 by a factor of 1/휖, where 0 < 휖 ≪ 1, resulting in the following system:

푘 푘 푥˙ = 훼푧 − 훾푥 + 푤, 푧˙ = 푟 − 휃푧 (푧 − 푧) − 훾푧 , 푧˙ = (푟 − 푦) − 훾푧, 푦 = 푥. 1 1 휖 1 1 1 휖 (B.18)

Comparing (B.18) to the type II QIC in Figure 3-1c-ii, the removal rate (휃) of the controller species in (B.18) need not to be large (i.e. removal rate 휃 instead of 휃/휖). Therefore, if the hypothesis that increasing 푘 to 푘/휖 decreases adaptation error were true, system (B.18) seems more favourable than the type II QIC in Figure 3-1c-ii in term of ease of implementation.

However, in fact, the steady state of the memory variable 푧¯ grows as 풪(1/휖) in (B.18) (see simulation in Figure B-1a). As a consequence, adaptation error does not decrease as 휖 decreases (Figure B-1b). System (B.18) therefore cannot achieve 휖- quasi-integral control. This problem can be avoided if one follows the design procedure outlined in Theorem 3.1: increase the rate of all controller species, rather than only part of the controller reactions, as we did in (B.18).

155 (a) (b) w 2 10 5 w/o dilution w/ dilution w/ dilution w/ dilution 10 4

10 3

10 2

10 1

10 0 10 1 10 2 10 3 10 4 10 5 10 6 0 0 100 time (hr)

Figure B-1: Increasing the integral gain 푘/휖 alone may not be sufficient to diminish leaky integration effect. (a) While the integral gain 푘/휖 in (B.18) is increased, the steady state of the memory variable 푧¯ also increases. (b) The increase in 푧¯ counteract the decrease in 휖, resulting in the system’s inability to quench leaky integration effect (i.e., decrease adaptation error 푒) by increasing the integral gain. An additive external disturbance is applied at 50 hr.

B.4 Proof of Lemma 3.4

Proof. The steady state (¯푥, 푧¯1, 푧¯2) can be found by setting the time derivatives in (3.19) to zero, from which we have,

푓(푤)¯푧1 = 훿푥,¯ 0 = 푘푟 − 휃푧¯1푧¯2 − 휖훾푧¯1, 0 = 푘ℎ(¯푥) − 휃푧¯1푧¯2 − 휖훾푧¯2. (B.19)

Solution to (B.19) can be determined by the following equations

푓(푤)¯푧1 푘푟 − 휖훾푧¯1 푥¯ = , 푧¯2 = , (B.20a) 훿 휃푧¯1 [︂ ]︂ 2 2 푓(푤)¯푧1 휖훾푘푢 휖 훾 푘ℎ + 휖훾푧¯1 − = 푘푟 − . (B.20b) 훿 휃푧¯1 휃

Since the left hand side of (B.20b) is a monotonically increasing function of 푧¯1 that

ranges (−∞, ∞) for 푧¯1 > 0, and the right hand side of (B.20b) is a constant, equation

(B.20b) uniquely determines a positive 푧¯1. Equilibrium 푧¯2 and 푥¯ can then be subsequently determined uniquely according to (B.20a). Next, we show that the equilibrium (¯푥, 푧¯1, 푧¯2) determined by (B.20) is locally asymptotically stable. The linearized

156 system of (3.19) is

⎡ ⎤ ⎡ ⎤ ⎡ ⎤ 푥˙ −훿 푓(푤) 0 푥 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ = ⎢ 휃 * 휃 * ⎥ · ⎢ ⎥ . (B.21) ⎢푧˙1⎥ ⎢ 0 − 휖 푧2 − 훾 − 휖 푧1 ⎥ ⎢푧1⎥ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ 푘 ′ * 휃 * 휃 * 푧˙2 휖 ℎ (푥 ) − 휖 푧2 − 휖 푧1 − 훾 푧2 ⏟ ⏞ 퐴휖

Linear system (B.21) is stable if matrix 퐴휖 is Hurwitz. The characteristic polynomial of 퐴휖 is

(︂ 휃 )︂ (︂ 휃 )︂ 휃푘 휃2 푃 (푠) = det(푠퐼 − 퐴 ) = (푠 + 훿) 푠 + 푧¯ + 훾 푠 + 푧¯ + 훾 + 푧¯ 푓(푤)ℎ′(¯푥) − 푧¯ 푧¯ (푠 + 훿) 휖 휖 휖 2 휖 1 휖2 1 휖2 1 2 {︂ [︂ 휃 ]︂}︂ {︂ [︂휃 ]︂ }︂ = 푠3 + 훾 + 훿 + 훾 + (¯푧푧 +푧 ¯ ) 푠2 + (훾 + 훿) (¯푧 +푧 ¯ ) + 훾 + 훿훾 푠 휖 1 2 휖 1 2 [︂휃 ]︂ 휃푘 + 훿훾 (¯푧 +푧 ¯ ) + 훾 + 푓(푤)¯푧 ℎ′(¯푥). 휖 1 2 휖2 1

Based on the characteristic polynomial 푃휖(푠), and according to Routh-Hurwitz condition, (B.21) is stable if the following inequality is satisfied

{︂ [︂ 휃 ]︂}︂ {︂ [︂휃 ]︂ }︂ 훾 + 훿 + 훾 + (¯푧 +푧 ¯ ) · (훾 + 훿) (¯푧 +푧 ¯ ) + 훾 + 훿훾 > 휖 1 2 휖 1 2 [︂휃 ]︂ 휃푘 훿훾 (¯푧 +푧 ¯ ) + 훾 + 푓(푤)¯푧 ℎ′(¯푥) (B.22) 휖 1 2 휖2 1

To verify (B.22), note that on the one hand,

{︂ [︂ 휃 ]︂}︂ {︂ [︂휃 ]︂ }︂ 훾 + 훿 + 훾 + (¯푧 +푧 ¯ ) · (훾 + 훿) (¯푧 +푧 ¯ ) + 훾 + 훿훾 휖 1 2 휖 1 2 {︂ [︂ 휃 ]︂}︂ [︂휃 ]︂ [︂ 휃 ]︂ > 훾 + 훿 + 훾 + (¯푧 +푧 ¯ ) · (훾 + 훿) · (¯푧 +푧 ¯ ) + 훾 + 훿훾 훾 + (¯푧 +푧 ¯ ) 휖 1 2 휖 1 2 휖 1 2 휃2 휃 [︂ 휃 ]︂ > (훾 + 훿) (¯푧 +푧 ¯ )2 + 훿훾 푧¯ + 훿훾 훾 + (¯푧 +푧 ¯ ) 휖2 1 2 휖 2 휖 1 2 2휃2훿 휃 [︂ 휃 ]︂ > 푧¯ 푧¯ + 2훿훾 푧¯ + 훿훾 훾 + (¯푧 +푧 ¯ ) . (B.23) 휖2 1 2 휖 2 휖 1 2

157 On the other hand, given Assumption 3.5 and the equality in (B.20a), we have

휃푘 휃푘 휃푘 휃 휃2훿 휃 푓(푤)¯푧 ℎ′(¯푥) = 훿푥ℎ¯ ′(¯푥) < 2 훿ℎ(¯푥) = 2 훿(휃푧¯ 푧¯ + 휖훾푧¯ ) = 2 푧¯ 푧¯ + 2훿훾 푧¯ . 휖2 1 휖2 휖2 휖2 1 2 2 휖2 1 2 휖 2 (B.24)

Therefore, given inequalities (B.23) and (B.24), condition (B.22) is satisfied for all positive 휖, implying that the steady state (¯푥, 푧¯1, 푧¯2) is locally (exponentially) stable.

B.5 Analysis of phosphorylation-mediated type I QIC

Model derivation

We consider the phosphorylation-based feedback system shown in Figure 3-3a. The objective is to utilize the feedback control to regulate the concentration of protein p to be robust to fluctuations in its protein production rate, arising from, for instance, sequestration of RNA polymerase and ribosomes by other genes in the host cell. In this system, the regulated protein is co-translated with a phosphatase p that * ˜ can de-phosphorylate the active substrate b with catalytic rate constant 푘2. De- phosphorylation can be described by the following chemical reaction:

+ K ˜ * 2 k2 p + b c2 −→ p + b. − K2

The reference input to the circuit is the (total) concentration of a kinase r. We

denote its total concentration by 푟푡 and its free concentration of 푟. The kinase can phosphorylate the substrate b, turning it into active form b* with catalytic rate constant 푘1. The substrate b is produced constitutively with rate constant 휈. These processes can be described by the following chemical reaction:

K+ 1 k1 * 휈 r + b c1 −→ b + r, −→ b. − ∅ K1

158 The active substrate b* is a TF that can activate production of p. We model protein production as a one-step process, which is subject to disturbances/parameter uncertainties. Therefore, the protein production rate constant is 푅˜(1−푤) with disturbance input 푤 ∈ 풲, which is a compact subset of [0, 1). Let 퐷 be the concentration of free promoters of protein p, these processes can be described by the following chemical reactions:

+ ˜ * 휆 R(1−w) b + D C, C −→ p + C. 휆−

All species except DNAs dilute with rate constant 훾:

* 훾 훾 b, b , p, c1, c2 −→ ∅, C −→ D.

These chemical reactions yield the following reaction rate equations:

퐶˙ = 휆+푏*퐷 − 휆−퐶 − 훾퐶, (B.25a)

+ − 푐˙1 = 퐾1 푟푏 − 퐾1 푐1 − 푘1푐1 − 훾푐1, (B.25b) + * − ˜ 푐˙2 = 퐾2 푝푏 − 퐾2 푐2 − 푘2푐2 − 훾푐2, (B.25c) ˙ + − ˜ 푏 = 휈 − 훾푏 − 퐾1 푟푏 + 퐾1 푐1 + 푘2푐2, (B.25d) ˙* − + * * + * − 푏 = 휆 퐶 − 휆 푏 퐷 − 훾푏 − 퐾2 푝푏 + 퐾2 푐2 + 푘1푐1, (B.25e) ˜ ˜ + * − 푝˙ = 푅(1 − 푤)퐶 − 훾푝 + 푘2푐2 − 퐾2 푝푏 + 퐾2 푐2. (B.25f)

Assuming that the binding reactions are much faster than protein dynamics [4, 41], we set (B.25a)-(B.25c) to quasi-steady state (QSS) to obtain:

푏*퐷 푟푏 푝푏* 퐶 = , 푐1 = , 푐2 = , (B.26) 휆 퐾1 퐾2

159 where we define the dissociation constant 휆, and Michaelis-Menten constants 퐾1, 퐾2 as

− − − ˜ 휆 + 훾 퐾1 + 푘1 + 훾 퐾2 + 푘2 + 훾 휆 := + , 퐾1 := + , 퐾2 := + . 휆 퐾1 퐾2

The concentration of the regulated protein 푝푡 is equal to the sum of free phosphatase

푝 and that bound to the substrate 푐2: 푝푡 = 푝 + 푐2. Combine equations (B.25c) and (B.25f), and then use (B.26), we obtain:

푏*퐷 푝˙ = 푅˜(1 − 푤) − 훾푝 . 푡 휆 푡

We assume that the DNA copy number of p is 휋푡, so that 휋푡 = 퐶 + 퐷. We therefore

휋푡 have 퐷 = 1+푏*/휆 and

푏*휋푡 푝˙ = 푅˜(1 − 푤) − 훾푝 . (B.27) 푡 휆 + 푏* 푡

Substituting (B.26) into (B.25e), we obtain the dynamics of 푏*:

˜ (︂ 푡 )︂ ˙* 푘2 + 훾 * 푟푏 * 휋 푏 = − 푝푏 + 푘1 − 훾푏 * + 1 . (B.28) 퐾2 푘1 휆 + 푏

* 휋푡 The 훾푏 휆+푏* term is due to the steady state effect of retroactivity [72], which we assumed to be much less than 1 and neglected when we present the model in Section 3.2.4. However, as we reason in what follows, 휖-quasi-integral control can be achieved without this assumption. To this end, we define

(︂ 휋푡 )︂ 퐿(푏*) := 훾푏* + 1 , 휆 + 푏*

and note that, for any positive 푏*, 퐿(푏*), 퐿′(푏*) > 0. We further use the fact that

푝 = 푝푡 − 푐2 and 푟 = 푟푡 − 푐1 to obtain

푝푡 푟푡 푝 = * , 푟 = , 퐾2 + 푏 퐾1 + 푏

160 where 푟푡 is the total concentration of kinase input (i.e. both free and bound). We substitute these into (B.28) to obtain

* ˙* 푝푡푏 푟푡푏 * 푏 = −푘2 * + 푘1 − 퐿(푏 ), (B.29) 퐾2 + 푏 퐾1 + 푏

˜ where we defined 푘2 := 푘2 + 훾. Therefore, the phosphorylation-based control system can be described the following dynamics:

푝 푏* 푟 푏 푏˙* = −푘 푡 + 푘 푡 − 퐿(푏*), 2 퐾 + 푏* 1 퐾 + 푏 2 1 (B.30) 푏* 푝˙ = 푅(1 − 푤) − 훾푝 , 푡 휆 + 푏* 푡 where 푅 := 푅휋˜ 푡. Note that (B.30) is a slight generalization of model (3.29) in the main text, where 퐿(푏*) = 훾푏*. In the next section, we verify that (B.30) can achieve 휖-quasi-integral control when the kinase and the phosphatase are saturated by their substrates. Since (3.29) in the main text is a special case of (B.30), results in the next section applies to (3.29) as well.

Equilibrium location and stability

When the kinase and the phosphatase are saturated by their substrates, we have

* 푏 ≫ 퐾1 and 푏 ≫ 퐾2, model (B.30) can be approximated by

˙* * 푏 = −푘2푝푡 + 푘1푟푡 − 퐿(푏 ), 푏* (B.31) 푝˙ = 푅(1 − 푤) − 훾푝 . 푡 휆 + 푏* 푡

Here, we first focus on understanding whether the effect of leaky integration onthe closed loop equilibrium location can be mitigated by decreasing 휖.

Since the catalytic rate constants of phosphorylation and de-phosphorylation (푘1 and 푘2) are much larger than dilution rate constant (훾), we define 휖 := 훾/푘1 ≪ 1 and −3 * 휎 := 푘2/푘1. Particularly, in bacteria, 휖 = 풪(10 ) [41]. By letting 푥 = 푝푡 and 푧 = 푏 ,

161 system (B.31) can be re-written as:

푧 훾 푥˙ = 푅(1 − 푤) − 훾푥, 푧˙ = (푟 − 푦) − 퐿(푧), (B.32) 휆 + 푧 휖 푡

where 푦 := 휎푥. System (B.32) fits into the generalized type I QIC in (3.13). In particular, we have

(︂ 휋푡 )︂ 푧 ℎ(푥) = 휎푥, 퐿(푧) = 훾푧 + 1 , 푓(푧, 푤) = 푅(1 − 푤) . 휆 + 푧 휆 + 푧

Since ℎ and 퐿 are monotonically increasing, and 휕푓/휕푧 > 0 for all 푤 ∈ (0, 1), As- sumption 3.4 is satisfied. According to (3.15), the admissible input set ℛ × 풲 can be defined as any compact subset of

˜ ˜ ℛ × 풲 = {(푟푡, 푤) : 0 < 푟푡 < 휎푅(1 − 푤)/훿, 0 ≤ 푤 < 1}. (B.33)

Therefore, based on Proposition 3.1, the phosphorylation-based control system (B.32) can achieve 휖-set-point regulation in the admissible input set (B.33). Simulation results can be found in Section B.7. Next, we give parameter conditions under which both the kinase and phophatase are saturated.

Parameter conditions to saturate the kinase and the phosphatase

In this section, we give parameter conditions under which the steady state of (B.30) ¯ ¯* is such that 푏 ≫ 퐾1 and 푏 ≫ 퐾2. In this situation, the controller dynamics in (B.30) can be approximated as the quasi-integral controller in (B.31) near the steady state. ¯ The condition 푏 ≫ 퐾1 is easy to satisfy since one can produce a sufficient amount of 푏 by increasing its production rate 휈. Production rate 휈 can be increased, for example, ¯ by increasing its DNA copy number. We therefore assume that 푏 ≫ 퐾1 is satisfied ¯* * and seek to find conditions to satisfy 푏 ≫ 퐾2. To this end, let 푥 = 푝푡 and 푧 = 푏 , we first find the steady state (¯푥, 푧¯) of (B.30) by solving:

푧¯ 푥¯푧¯ 푅(1 − 푤) − 훾푥¯ = 0, 푟푡 − 휎 − 휖퐿(¯푧) = 0. (B.34) 휆 +푧 ¯ 퐾2 +푧 ¯

162 When 휖 = 0 and 푟푡 ∈ ℛ = (0, 휎푅(1 − 푤)/훾), equation (B.34) has a steady state (푥*, 푧*) that can be solved from

* * * 푧 * 푥 푧 푅(1 − 푤) * − 훾푥 = 0, 푟푡 − 휎 * = 0, (B.35) 휆 + 푧 퐾2 + 푧 leading to: 훾푟 휆[1 + 휇 + √︀(1 − 휇)2 + 휇훼] 푧* = 푡 , 2[휎푅(1 − 푤) − 훾푟푡] (B.36) 푟 [︁ ]︁ 푥* = 푡 1 − 휇 + √︀(1 − 휇)2 + 휇훼 , 2휎 where

퐾 4푅(1 − 푤)휎 휇 := 2 , and 훼 := . 휆 훾푟푡

By engineering the circuit’s parameters and take the reference input so that 휇 ≪ 1 and 휇훼 ≪ 1, we can write (B.36) as

훾푟 휆 푟 푧* = 푡 + 풪(휇), 푥* = 푡 + 풪(휇). (B.37) 휎푅(1 − 푤) − 훾푟푡 휎

On the other hand, when 휖 is nonzero but sufficiently small, from the Implicit Function Theorem, one can write

푧¯ = 푧* + 풪(휖), and 푥¯ = 푥* + 풪(휖). (B.38)

¯* Therefore, for sufficiently small 휖, 휇 and 휇훼, we have 푏 =푧 ¯ ≫ 퐾2 if

* 훾푟푡휆 푅(1 − 푤)휎퐾2 푧¯ = 푧 + 풪(휖) ≈ ≫ 퐾2 ⇒ ≪ 1. (B.39) 휎푅(1 − 푤) − 훾푟푡 훾푟푡(휆 + 퐾2)

Since

푅(1 − 푤)휎퐾 푅(1 − 푤)휎퐾 4푅(1 − 푤)휎퐾 2 < 2 < 2 = 휇훼, 훾푟푡(휆 + 퐾2) 훾푟푡휆 훾푟푡휆

163 and that we have already assumed 휇훼 ≪ 1, (B.39) is automatically satisfied as long as the assumption 휇훼 ≪ 1 stands. Therefore, there exists 휖 and 휇 sufficiently small ¯* such that 푏 ≫ 퐾2. Now we summarize all the conditions that allow the phosphorylation-based controller to achieve quasi-integral control. Firstly, the following conditions are sufficient to saturate the phosphatase (p) by its substrate (b*).

* 1. 휇 = 퐾2/휆 ≪ 1: the dissociation constant between the substrate b and the promoter of the regulated protein (휆) should be much larger than the Michaelis-

Menten constant between the phosphatase and the active substrate (퐾2). Since it is generaly hard to engineer Michaelis-Menten constants, the promoter needs to be engineered to be weaker.

2. 훼휇 = 4푅(1−푤)휎퐾2 ≪ 1: the input 푟 should be large enough so that a sufficient 훾푟푡휆 푡 amount of substrate b* is produced to saturate the phosphatase p.

The following condition is required to saturate kinase u by the substrate b.

3. 휈 is sufficiently large: this ensures that a large amount of substrate b is produced to saturate the kinase. In practice, 휈 can be increased by, for example, increasing the DNA copy number of the substrate.

Finally, from (B.33), the following condition is needed to guarantee that the reference input is not too large to exceed the maximum protein production capability in the cell.

4. 푟푡 < 휎푅(1 − 푤).

Note that none of the conditions 1-4 require exact parameter tuning. Roughly speaking, if the promoter strength is relatively weak, as long as a sufficient amount of substrate is produced, integral control can be approximated for a range of reference inputs.

164 B.6 Analysis of sRNA-mediated type II QIC

We first derive model (3.32) in the main text from reaction rate equations andthen apply the result in Proposition 3.2 to the sRNA system to verify that it can achieve 휖-quasi-integral control.

We consider the sRNA-mediated feedback system shown in Fig. 3-3c. The feedback circuit is built with the objective to regulate the translation process of the target gene, so that it is robust to disturbances in the translation process, arising from, for example sequestration of ribosomes by other genes in the cell (see Chapter 2). In this controller, the target protein is co-translated with a transcription activator p; a transcription factor input r binds with the DNA promoter site 퐷 of the regulated gene to form a transcription initiation complex C that can be transcribed into mRNAs with rate constant 푇 . The mRNA molecules can be degraded by RNAse [69, 96] with a rate constant 훿˜. The chemical reactions are:

+ 휅 T 훿˜ r + D C −→ m + D + r, m −→ ∅. 휅−

The mRNAs are then translated by ribosomes to produce the target protein and activator p, with translation rate constant 푅. Due to the translation of other ribosome- competing mRNAs in the cell [120], the translation rate constant is reduced by a factor of 푤 ∈ (0, 1), so that the actual translation rate constant is 푅(1 − 푤). We treat 푤 as a disturbance input to the circuit, which could also model uncertainty in parameter 푅. The corresponding chemical reaction is:

R(1−w) m −→ m + p.

The regulated protein p binds with the sRNA promoter site Ds to form a transcription initiation complex Cs, which can then be transcribed into sRNA (s) with rate constant

푇푠. Since the target protein is co-translated with p, its concentration is reflected by

푝푡 := 푝 + 퐶푠, as the target protein does not bind with Ds. We assume that sRNA molecules are degraded by RNAse at the same rate as mRNA (훿˜). The corresponding

165 chemical reactions are:

+ ks Ts 훿˜ p + Ds Cs −→ s + Ds + p, s −→ . − ∅ ks

The sRNA contains complementary sequences to the mRNA, and they can bind together to form a mRNA-sRNA complex 푐, which degrades rapidly with rate constant 휃˜. Experimental results indicate that coupled degradation of mRNA-sRNA complex by RNAse is much faster than uncoupled degradation of RNAs (휃˜ ≫ 훿˜) [90, 97]. These processes can be described by the following chemical reactions:

+ 훽 휃˜ m + s c, c −→ ∅ 훽−

We assume that all species except DNAs dilute with rate constant 훾, which is inversely proportional to the doubling time of the host cell:

훾 훾 훾 m, s, p, c −→ ∅, C −→ D, Cs −→ Ds.

Since the RNAs also degrade with rate constant 훿˜ in addition to dilution, we introduce a lumped decay rate constant 훿 := 훿˜ + 훾. The above chemical reactions yield the following ODEs:

퐶˙ = 휅+푟퐷 − (휅− + 푇 )퐶 − 훾퐶, (B.40a)

˙ + − 퐶푠 = 푘푠 푝퐷푠 − (푘푠 + 푇푠)퐶푠 − 훾퐶푠, (B.40b) 푐˙ = 훽+푚푠 − 훽−푐 − 휃푐˜ − 훾푐, (B.40c)

푚˙ = 푇 퐶 − 훿푚 − 훽+푚푠 + 훽−푐, (B.40d)

+ − 푠˙ = 푇푠퐶푠 − 훿푠 − 훽 푚푠 + 훽 푐, (B.40e)

+ − 푝˙ = 푅(1 − 푑)푚 − 훾푝 − 푘푠 푝퐷푠 + 푘푠 퐶푠 + 푇푠퐶푠. (B.40f)

166 We assume that the concentration of DNAs are conserved such that:

푡 푡 휋 = 퐷 + 퐶, and 휋푠 = 퐷푠 + 퐶푠, (B.41)

푡 푡 where 휋 and 휋푠 are the DNA copy numbers of the regulated gene and the sRNA. We further assume that binding reactions are much faster than mRNA and protein dynamics, such that (B.40a)-(B.40c) can be set to QSS, yielding:

푟퐷 푝퐷푠 푚푠 퐶 = , 퐶푠 = , 푐 = , (B.42) 휅 푘푠 훽 where

− − − ˜ 휅 + 푇 + 훾 푘푠 + 푇푠 + 훾 훽 + 휃 + 훾 휅 := + , 푘푠 := + , 훽 := + . (B.43) 휅 푘푠 훽

Substituting (B.42) into (B.41), we obtain the QSS concentrations of transcription initiation complexes:

푡 푡 푟휋 푝휋푠 퐶 = , 퐶푠 = . (B.44) 휅 + 푢 푘푠 + 푝

Substituting equations (B.44) into (B.40d)-(B.40e), we obtain the following RNA dynamics:

푚˙ = 푇 휋푡푟¯ − 훿푚 − 휃푚푠/훽, (B.45a)

푡 푝 푠˙ = 푇푠휋푠 − 훿푠 − 휃푚푠/훽, (B.45b) 푘푠 + 푝

˜ 푟 where we defined 휃 := 휃 + 훾, and we treated 푟¯ := 휅+푟 as a dimensionless external reference input to the regulated gene. The dynamics of the target protein, whose

concentration is 푝푡 = 푝 + 퐶푠 can be modeled using (B.40b) and (B.40d), resulting in

푝˙푡 = 푅(1 − 푤)푚 − 훾푝푡. (B.46)

The relationship between 푝 and 푝푡 can be found by substituting 퐶푠 in (B.44) into

167 푝 + 퐶푠 = 푝푡, resulting in

푡 푡 √︀ 푡 2 푝휋푠 푝푡 − 푘푠 − 휋푠 + (푘푠 + 휋푠 − 푝푡) + 4푝푡푘푠 푝 + = 푝푡 ⇒ 푝 = 휑(푝푡) := . 푘푠 + 푝 2 (B.47)

Substituting (B.47) into (B.45b), a reduced model of the sRNA-based QIC-regulated system is

푚˙ = 푇 휋푡푟¯ − 훿푚 − 휃푚푠/훽, (B.48a)

푡 휑(푝푡) 푠˙ = 푇푠휋푠 − 훿푠 − 휃푚푠/훽, (B.48b) 푘푠 + 휑(푝푡)

푝˙푡 = 푅(1 − 푤)푚 − 훾푝푡. (B.48c)

푡 When the DNA copy number of sRNA 휋푠 is much less than the dissociation constant 푡 푘푠 (휋푠 ≪ 푘푠), (B.47) can be well-approximated by 푝 ≈ 푝푡 [72]. Under this assumption, (3.32) can be approximated by

푚˙ = 푇 휋푡푟¯ − 훿푚 − 휃푚푠/훽,

푡 푝푡 푠˙ = 푇푠휋푠 − 훿푠 − 휃푚푠/훽, (B.49) 푘푠 + 푝푡

푝˙푡 = 푅(1 − 푤)푚 − 훾푝푡,

which is in the same form as model (3.32) in the main text, and corresponds to a situation where retroactivity due to protein-sRNA promoter binding is small [72]. However, in general, for the circuit to have near perfect adaptation, it is not necessary

푡 to make the additional assumption 휋푠 ≪ 푘푠. Therefore, in the next section, we study the more general model (3.32), and since (B.49) is a special case of (3.32), any result we obtain will be equally applicable to (B.49).

Steady state and stability

Now we show that the sRNA-based control system can be classified as a generalized type II QIC. We then apply the result in Claim 3.2 to verify that it can achieve

168 휖-quasi-integral control. In particular, let 푥 := 푝푡, 푧1 := 푚, 푧2 := 푠, system (3.32) can be re-written as

푥˙ = 푅(1 − 푤)푧1 − 훾푥,

푡 푧˙1 = 푇 휋 푟 − 훿푧1 − 휃푧1푧2/훽, (B.50)

푡 휑(푥) 푧˙2 = 푇푠휋푠 − 훿푧2 − 휃푧1푧2/훽. 푘푠 + 휑(푥) In what follows, we leverage a timescale separation property to put (3.32) into the form of a generalized type II QIC. We exploit the fact that mRNA-sRNA complexes are degraded much more rapidly than uncoupled RNAs [97] to establish a timescale separation between controller reactions and RNA decay, which is the source of leaky integration in this circuit. We use 휖 := 훿/휃 ≪ 1 to characterize the rate difference between mRNA-sRNA complex degradation and uncoupled RNA decay, and simultaneously increase DNA copy numbers of the controller species m and s to 휋푡/휖 and

푡 휋푠/휖 to increase their production rates. System (B.50) becomes:

푥˙ = 푅(1 − 푤)푧1 − 훾푥, 푇 휋푡 훿 푧˙ = 푟 − 훿푧 − 푧 푧 , 1 휖 1 휖훽 1 2 (B.51) 푡 푇푠휋푠 휑(푥) 훿 푧˙2 = − 훿푧2 − 푧1푧2. 휖 푘푠 + 휑(푥) 휖훽

System (B.51) is in the form of a generalized type II QIC in equation (3.19): specifically, by comparing (B.51) and (3.19), we have

푡 푇푠휋푠 휑(푥) 푓(푑) = 푅(1 − 푤), ℎ(푥) = 푡 . 푇 휋 푘푠 + 휑(푥)

According to (B.47), 휑(0) = 0, and 휑(푥) is monotonically increasing for all positive 푥. Therefore, ℎ(0) = 0 and ℎ(푥) is monotonically increasing. One can also verify easily that, based on the definition of 휑(푥) in (B.47), 푥휑′(푥) < 2휑(푥), leading to

′ ′ ′ ′ 푥ℎ (푥) 푥휑 (푥)푘푠 푘푠 + 휑(푥) 푥휑 (푥)푘푠 푥휑 (푥) = 2 · = < < 2. ℎ(푥) (푘푠 + 휑(푥)) 휑(푥) 휑(푥)(푘푠 + 휑(푥)) 휑(푥)

169 In addition, since lim푥→∞ 휑(푥) = +∞,

푇 휋푡 lim ℎ(푥) = 푠 푠 , 푥→∞ 푇 휋푡 and therefore, Assumption 3.5 is satisfied if 푟 is in the range of ℎ(푥) (i.e., 푟 <

푡 푡 푇푠휋푠/(푇 휋 )). Assumption 3.6 can be satisfied if we have 푤 < 1. This is a reasonable assumption as any disturbance on translation rate would not render it to become non-positive. Therefore, both Assumptions 3.5 and 3.6 are satisfied if we restrict the inputs to be within any compact subset of

{︂ 푇 휋푡 }︂ ℛ˜ × 풲˜ := (푟, 푤) : 0 < 푟 < 푠 푠 , 0 ≤ 푤 < 1 . (B.52) 푇 휋푡

Therefore, based on Proposition 3.2, the sRNA-based control system (B.51) can achieve 휖-set-point regulation in the admissible input set (B.52). Simulation results can be found in Section B.7.

B.7 Additional simulations

We show additional simulations to consolidate analytical results discussed in Sections 3.2.4-3.2.5. All simulations were carried out using MATLAB R2014b with variable step ODE solver ode23s. We use relative adaptation error (푒푟) to evaluate the performance of quasi-integral controllers given constant reference input 푢 and disturbance input 푤. Here, relative adaptation error is defined as the relative percentage change of steady state output due to the presence of 푤

|푦¯(푟, 푤) − 푦¯(푟, 0)| 푒 = 푒 (푟, 푤) := × 100%. (B.53) 푟 푟 푦¯(푟, 0)

Simulations of the two circuits were carried out using their full reaction models (B.25) and (B.40).

170 Adaptation error (%) 10 3 40

30 10 2

25 (nM)

λ 20

1 10 15

5 10 0 10 1 10 2 10 3 10 4 10 5 -1 k 1=k 2 (hr )

Figure B-2: Adaptation error of the phosphorylation-based QIC with different magnitudes of times-scale separation (푘푖/훾, 푖 = 1, 2) and binding affinities between the active substrate and the regulated gene (휆). Simulation parameters are listed in Table B.2.

Phosphorylation-based QIC

Figure B-2 shows the relative adaptation error of the phosphorylation-based quasi- integral controller in the presence of 50% reduction in protein production rate (푤 = 0.5). Our analysis in Section B.5 reveals that in order to saturate the phosphatase, we need 휇 = 퐾2/휆 to be sufficiently small. In addition, to mitigate the leaky integration effect due to dilution of the substrate, weneed 휖 = 푘푖/훾 to be sufficiently large

(we use 푘1 = 푘2 in the simulation). The latter condition can be easily satisfied due to the natural time-scale separation between phosphorylation/dephosphorylation and dilution. In fact, in bacteria, 휖 is at most ∼ 10−3 [41]. On the other hand, as decreasing the Michaelis-Menten constant 퐾2 is in general difficult, we decrease 휇 by decreasing the binding affinity between the substrate b* and the regulated promoter. The upper right corner of the color map corresponds to the largest time-scale separation between phosphorylation/de-phosphorylation catalytic rates and dilution, and weakest binding affinity between b* and the regulated promoter, leading to the minimum relative adaptation error of 2%.

171 Adaptation error (%) 10 4 45

40 3 10 35 )

-1 30 hr -1 10 2 25 (nM

β 20 / θ 15 10 1 10

10 0 10 0 10 1 10 2 10 3 πt πt = s (nM)

Figure B-3: Adaptation error of the sRNA-based QIC with different gene/sRNA copy numbers and RNA complex removal rates. Simulation parameters are listed in Table B.2. sRNA-based QIC

Our analytical results suggest that quasi-integral control can be realized by simulta-

푡 푡 neously increasing the gene/sRNA copy numbers (휋 and 휋푠) and the mRNA-sRNA complex removal rate constant (휃/훽). This result is confirmed by the simulation in Figure B-3. The color map represents the relative adaptation error of the regulated protein concentration. The upper right corner of the color map corresponds to the largest copy number and removal rate constant, and therefore the smallest 휖. Accord- ing to Figure B-3, this is where the relative adaptation error is minimum (0.6%). We also confirm that increasing the copy number or the removal rate alone is insufficient to guarantee good adaptation performance, as we discussed in Section B.3. Based on our choice of parameters, a medium copy number (∼ 30 copies) and a physically reasonably removal rate of ∼ 50 nM−1·hr−1 is sufficient to keep the adaptation error to be around 10% in the presence of 50% reduction in translation rate constant.

B.8 Simulation parameters

Simulation parameters were taken from a set of characteristic values from literature for E. coli bacteria. These characteristic values are listed below in Table B.1.

172 Table B.1: Characteristic parameter values in E. coli bacteria phosphorylation-based quasi-integral controller model Parameters Characteristic values Sources Protein production rate constant 푅 = 푅휋˜ 푡 102 ∼ 103 nM·hr−1 [103] * * 3 Dis. const. between b (p) & promoter (b ) 휆, 퐾2 0.1 ∼ 10 nM [41] 3 −1 Catalytic rate constants 푘1, 푘2 & 10 hr [41] sRNA-based quasi-integral controller model Parameters Characteristic values Sources 푡 푡 Gene/sRNA copy number 휋 , 휋푠 1 copy ∼ 1 nM [103] −1 Transcription rate constant 푇 , 푇푠 ∼ 1 hr [69, 103] Translation rate constant 푅 ∼ 100 hr−1 [103] Uncoupled RNA decay rate constant 훿 1 ∼ 15 hr−1 [103] Protein dilution rate constant 훾 0.1 ∼ 1 hr−1 [102] 3 Dis. const. between p and sRNA 푘푠 0.1 ∼ 10 nM [41] 2 −1 −1 mRNA-sRNA removal rate constant 휃/훽 & 10 nM ·hr [50, 69]

B.9 Experimental methods

The experimental data reported in Figures 3-9 and 3-10 are collected using the following protocol. DNA sequences are available on Addgene (www.addgene.org, plasmid # 120890-120901).

Cell culture and microplate photometer

Overnight culture was prepared by inoculating a -80∘C glycerol stock in 800 휇L growth medium per well in a 24-well plate (Falcon, 351147) and grew at 30∘C, 250 rpm in a horizontal orbiting shaker for 7 hours. Overnight culture was first diluted to an initial OD at 600 nm of 0.02 in 200 휇L growth medium per well in a 96-well plate (Falcon, 351172) and grew for 2 to 3 hours to ensure exponential growth before induction. The 96-well plate was incubated at 30∘C in a Synergy MX (Biotek, Winooski, VT) microplate reader in static condition and was shaken for 3 seconds right before OD and fluorescence measurements. Sampling interval was 5 minutes unless stated otherwise in figure captions. Excitation and emission wavelengths to monitor GFP fluorescence were 485 and 530 nm, respectively and those to monitor RFP fluorescence were 584 and 619 nm, respectively. To ensure exponential growth, cell culture was diluted

173 Table B.2: Simulation parameters Figure 2d (model (3.6)) Figure 2b (model (3.3)) 푇 1 nM · hr−1 푅 100 nM · hr−1 −1 푇푠 4 nM · hr 휆 500 nM 푅 50 hr−1 훾 1 hr−1 −1 훾 1 hr 퐾2 5 nM −1 −1 훿 3 hr 푘1 = 푘2 10 hr 푘푠 50 nM Figure S5 (model (S82)) 휃/훽 10 nM−1·hr−1 푇 1 hr−1 −1 Figure S4 (model (S67)) 푇푠 10 hr 휈 5 휇M · hr−1 푅 50 hr−1 퐾1 10 nM 푘푠 100 nM −1 퐾2 10 nM 훾 1 hr 푅˜ 50 hr−1 휃 100 hr−1 훾 1 hr−1 휅 50 nM 휋푡 1 nM every 2 hours to OD of 0.02 as one batch. Multiple batches were used to ensure cell growth remains in exponential phase and gene expression reaches steady state. Growth rates were computed from the last batch of each experiment.

Flow Cytometry

Single-cell fluorescence data of Escherichia coli cells after 6-h AHL induction were measured by the Accuri C6 flow cytometer (Special Order 2B2LYG RUO System, 656035). The optical system equipped 448-nm and 552-nm lasers to excite GFP and RFP, and employed the 525/50-nm and 610/25-nm band-pass filters to detect the emission of GFP and RFP, respectively. The fluid system set the flow rate to66 휇L/min and the core size to 22 휇m. The detection threshold was set as 10,000 on FSC-H channel. Singlet events were gated in a FSC-A vs. FSC-H plot. At least 120,000 singlet events were collected for each sample for data analysis.

174 Appendix C

Appendix for Chapter 4

C.1 Proof of Lemma 4.3

To prove Lemma 4.3, we note that the I/O gain function of each subsystem has the following property.

ˆ + + − − Lemma C.1. Suppose that Assumptions 4.1,4.2,4.6 are satisfied and let ℎ푖(푟푖 , 푤푖 , 푟푖 , 푤푖 ; 휀푖) be the canonical decomposition function of ℎ푖(푟푖, 푤푖; 휀푖), then, for any 푒푖 > 0 such that ¯ ˆ 푟푖 − 푒푖, 푟푖 + 푒푖 ∈ ℛ푖, the function ℎ푖 satisfies:

ˆ + − ± 0 |ℎ푖(푟푖 + 푒푖, 푤푖 ,푟푖 − 푒푖, 푤푖 ; 휀푖) − 퐻푖(푟푖)| ≤ 퐿ℎ|푒푖| + 훼푖(휀푖)|푤푖 | + 훼푖 (휀푖), (C.1)

where 퐿ℎ > 0 is the Lipschitz constant of ℎ푖 . O

Proof. Due to Assumption 4.3, the static I/O characteristic 푦푖 = ℎ푖(푟푖, 푤푖; 휀푖) is also + − sign-stable. For 푠 = 푟, 푤, define Λ푠 := sign(휕ℎ푖/휕푠푖), and let Λ푠 and Λ푠 be defined according to (4.9). By equation (4.10), let Λ푠,푗 be the 푗-th row of matrix Λ푠, the canonical decomposition function

ˆ + + − − ± ± ℎ푖(푟푖 , 푤푖 , 푟푖 , 푤푖 ; 휀푖) := ℎ푖(p푟(푟푖 ), p푤(푤푖 ); 휀푖), (C.2)

175 where ± + + − − p푟,푗(푟 ) := diag(Λ푟,푗)푟 + diag(Λ푟,푗)푟 , (C.3) ± + + − − p푤,푗(푤 ) := diag(Λ푤,푗)푤 + diag(Λ푤,푗)푤 ,

are the 푗-th elements of the vector-valued functions p푟 and p푤, respectively. Note ± ± that (C.3) satisfies |p푠,푗(푠 )| ≤ |푠 |. Therefore,

ˆ + − ± |ℎ푖(푟푖, 푤푖 , 푟푖, 푤푖 ; 휀푖) − 퐻푖(푟푖)| = |ℎ푖(푟푖, p푤(푤푖 ); 휀푖) − 퐻푖(푟푖)|

± 0 ≤ 훼푖(휀푖)|p푤(푤푖 )| + 훼푖 (휀푖)

± 0 ≤ 훼푖(휀푖)|푤푖 | + 훼푖 (휀푖). (C.4)

ˆ On the other hand, by the definition of ℎ푖 in (C.2) and the Lipschitz property of ℎ푖 in ˆ ± ¯ 2 Assumption 4.6, the decomposition function ℎ푖 is Lipschitz continuous in 푟푖 ∈ (ℛ푖) ± uniformly in 푤푖 and 휀푖 with a Lipschitz constant 퐿ℎ. Hence, we have

ˆ + − ˆ + − |ℎ푖(푟푖 + 푒푖, 푤푖 , 푟푖 − 푒푖, 푤푖 ; 휀푖) − ℎ푖(푟푖, 푤푖 , 푟푖, 푤푖 ; 휀푖)| ≤ 퐿ℎ|푒푖|. (C.5)

Combining (C.4) and (C.5), we have (C.1) proven by triangle inequality.

Proof. (Lemma 4.3). We prove Lemma 4.3 through induction. In particular, given 푤(푡) → [푤−, 푤+], we find the ultimate bound for each element of 푑(푡) using the disturbance I/O gain function of each subsystem in (4.12), the subsystem static disturbance attenuation property (4.6), and Assumptions 4.5 and 4.6. For 푖 = 1, according to

* Assumption 4.5, we necessary have 푟1(푡) ≡ 푟1, which is independent of the state of all other subsystems. Since Σ1 is I/S monotone and the prescribed output function 푙푖 has sign-stable Jacobian, the static I/O characteristic ℎ푖 is necessarily equipped with ˆ + + − − a canonical decomposition function ℎ푖(푟푖 , 푤푖 , 푟푖 , 푤푖 ; 휀푖) that serves as the I/O gain − + function for the reference output 푦. Thus, if 푤1(푡) → [푤1 , 푤1 ], then we have

ˆ * − * + ˆ * + * − 푦1(푡) → [ℎ1(푟1, 푤1 , 푟1, 푤1 ; 휀1), ℎ1(푟1, 푤1 , 푟1, 푤1 ; 휀1)], (C.6) * − * + * + * − 푑1(푡) → [휓1(푟1, 푤1 , 푟1, 푤1 ; 휀1), 휓1(푟1, 푤1 , 푟1, 푤1 ; 휀1)].

176 * * Let 푦1 := 퐻1(푟1), by Lemma C.1, we can write

* ± * ± 푦1(푡) → [푦1 − 푄1(푤1 ; 휀1), 푦1 + 푄1(푤1 ; 휀1)], (C.7)

± ± 0 where 푄1(푤1 ; 휀1) := 훼1(휀1)|푤1 | + 훼1(휀1). On the other hand, by the definition of * 휓푖 in (4.13), the convergence result for 푑1(푡) in (C.6) can be re-written as 푑1(푡) → * − + * * + − * [휓1(푤1 , 푤1 ; 푟1, 휀1), 휓1(푤1 , 푤1 ; 푟1, 휀1)]. Due to Assumption 4.5, the reference input * * 푟2 = 퐺2(푦) to Σ2 is only a function of 푦1. Let 푟2 := 퐺2(푦1), let 퐿퐺 be the Lipschitz constant of 퐺(·), we have

* ± * ± 푟2(푡) → [푟2 − 퐿퐺푄1(푤1 ; 휀1), 푟2 + 퐿퐺푄1(푤1 ; 휀1)]. (C.8)

− * ± + * ± We use 푟2 := 푟2 − 퐿퐺푄1(푤1 ; 휀1) and 푟2 := 푟2 + 퐿퐺푄1(푤1 ; 휀1) to denote the ultimate * ¯ ± ¯ bounds for 푟2(푡). Since 푟2 ∈ int(ℛ2), for sufficiently small 휀1, 푟2 ∈ ℛ2. Similar to our treatment in (C.6) for Σ1, we have

ˆ − − + + ˆ + + − − 푦2(푡) → [ℎ2(푟2 , 푤2 , 푟2 , 푤2 ; 휀2), ℎ2(푟2 , 푤2 , 푟2 , 푤2 ; 휀2)], (C.9) − − + + + + − − 푑2(푡) → [휓2(푟2 , 푤2 , 푟2 , 푤2 ; 휀2), 휓2(푟2 , 푤2 , 푟2 , 푤2 ; 휀2)].

* * By the subsystem disturbance attenuation property (4.6), let 푦2 := 퐻2(푟2), we have

* ± * ± 푦2(푡) → [푦2 − 푄2(푤≤2; 휀≤2), 푦2 + 푄2(푤≤2; 휀≤2)] (C.10) where

± ± ± 0 푄2(푤≤2; 휀≤2) := 퐿ℎ퐿퐺푄1(푤1 ; 휀1) + 훼2(휀2)|푤2 | + 훼2(휀2),

according to Lemma C.1. Also due to Assumption 4.6, the convergence of 푑2(푡) in (C.9) can be re-written as:

* − + * * + − * 푑2(푡) → [휓2(푤2 , 푤2 ; 푟2; 휀2) − 푃2, 휓2(푤2 , 푤2 ; 푟2; 휀2) + 푃2],

177 where

± ± ± 0 푃2 = 푃2(푤1 ; 휀2) : = 퐿휓(휀2)퐿퐺푄1(푤1 ; 휀1) = 퐿휓(휀2)[훼1(휀1)|푤1 | + 훼1(휀1)],

− + and 퐿휓(휀) is the Lipschitz constant of 휓푖 for variables 푟푖 and 푟푖 as stated in Assump- tion 4.6. Since we do not assume the Lipschitz property of 휓푖 to hold uniformly in 휀푖, 0 퐿휓 is in general dependent on 휀푖. Note that, for a fixed 휀2, since 훼1 and 훼1 are class 풦 functions, 푃2 can be made arbitrarily small if 휀1 is sufficiently small. Using (C.7) and ± (C.10) to determine 푟3 , we can continue the iteration to find the boxes that bounds ⊤ ⊤ 푟3(푡) and 푑3(푡). After 푘 iterations, let 푤≤푘 := [푤1, ··· , 푤푘] and 휀≤푘 := [휀1, ··· , 휀푘] , we have

* * * − + * * + − * 푦푘(푡) → [푦푘 − 푄푘, 푦푘 + 푄푘], 푑푘(푡) → [휓푘(푤푘 , 푤푘 ; 푟푘, 휀푘) − 푃푘, 휓푘(푤푘 , 푤푘 ; 푟푘, 휀푘) + 푃푘],

* * where 푦푘 = 퐻푘(푟푘)

푘 ± ∑︁ 푘−푖 ± 0 푄푘(푤≤푘; 휀≤푘) := (퐿ℎ퐿퐺) · (훼푖(휀푖)|푤푖 | + 훼푖 (휀푖)), 푖=1 푘−1 ± ∑︁ 푘−1−푖 푘−푖 ± 0 푃푘(푤≤푘; 휀≤푘) :=퐿휓(휀푘) 퐿ℎ 퐿퐺 · (훼푖(휀푖)|푤푖 | + 훼푖 (휀푖)). 푖=1

Note 푄(푤±; 휀) and 푃 (푤±; 휀) can be arranged as in (4.15). Specifically, let

푘−1 ∑︁ 푘−1−푖 푘−푖 푝1,푘(휀≤푘) := 퐿휓(휀푘) 퐿ℎ 퐿퐺 훼푖(휀푖), 푖=1 푘−1 ∑︁ 푘−1−푖 푘−푖 0 푝0,푘(휀≤푘) := 퐿휓(휀푘) 퐿ℎ 퐿퐺 훼푖 (휀푖), 푖=1

⊤ 0 we have 푝푗(휀) = [푝푗,1, ··· , 푝푗,푁 ] for 푗 = 0, 1. Since 훼푖 and 훼푖 are class 풦 functions, for each 푘, given any 휇 > 0, 푝1,푘 ≤ 휇, and hence 푝1 ≤ 휇, can be satisfied if

(︃ −1 )︃ −1 휇퐿휓 (휀푘) ** 휀푖 ≤ 훼푖 푘−1−푖 푘−1 =: 휀푖,푘(휇, 휀푘) (푘 − 1)퐿ℎ 퐿퐺

178 ∀푖 ≤ 푘 − 1, ∀푘. We can then take

** ** 휀푖 (휇, 휀≥푖+1) := min 휀푖,푘(휇, 휀푘). 푘=푖+1,··· ,푁

** A similar upper bound 휀 can be established for 푝0, 푞1, 푞0 ≤ 휇 to be satisfied. This completes the proof.

C.2 Proof of Lemma 4.4

Proof. Consider 푉 (푥) in (4.17) for the nominal system as a candidate Lyapunov function for the perturbed system, then we have

∆푉 := 푉 (퐹 (푥) + 푝훿(푥)) − 푉 (푥)

=푉 (퐹 (푥) + 푝훿(푥)) − 푉 (퐹 (푥)) + 푉 (퐹 (푥)) − 푉 (푥)

2 ≤푐3푝(퐿1|푥| + 퐿2)(|퐹 (푥)| + |퐹 (푥) + 푝훿(푥)|) − 푐4|푥| , ∀|푥| ≥ 푟0

2 ≤푎(푝)|푥| + 푏(푝)|푥| + 푐(푝), ∀|푥| ≥ 푟0 (C.11) where

2 푎(푝) = −푐4 + 푝푐3(2퐿1퐿퐹 + 푝퐿1),

2 2 푏(푝) = 2푐3푝퐿2(푝퐿1 + 퐿퐹 ), 푐(푝) = 푐3푝 퐿2.

(︁ )︁ 퐿퐹 푐4 If we take 푝* := min , , then inequality (C.11) can be re-written as: 퐿1 6푐3퐿퐹 퐿1

2 ∆푉 ≤ − 푐4|푥| /2 + 푏(푝)|푥| + 푐(푝), ∀|푥| ≥ 푟0

2 2 2 ≤ − 푐4|푥| /2 + 4푐3푝퐿2|푥| + 푐3푝 퐿2, ∀|푥| ≥ 푟0 (C.12)

(︁ √︁ )︁ 32푐3퐿2 8푐3 2 2 For |푥| ≥ 푟푝 := max 푝 , 푝퐿2 , one can find that 4푐3푝퐿2|푥|, 푐3푝 퐿 ≤ 푐4 푐4 2 2 2 푐4|푥| /8. Hence, ∆ ≤ −푐4|푥| /4 for all |푥| ≥ 푟0 + 푟푝. Therefore, the perturbed system (4.18) is also exponentially ultimately bounded by Definition 4.4.

179 C.3 Proof of Lemma 4.5

The proof has several parts. We first show that the reduced system satisfying the assumptions in Lemma 4.5 is ISS after a coordinate translation, which then allows us to use a singular perturbation result for ISS systems [33] to compute the model reduction error for the fast variable 휁. This is then combined with a robustness result for convergent-input-convergent-state systems to prove Lemma 4.5. Specifically, recall 푟 ˜푟 푟 푟 휙휉(푢) is the static I/S characteristic of the reduced system. We let 휉 := 휉 − 휙휉(0) and write the translated reduced system as:

˜˙푟 ˜푟 ˜푟 ¯푟 ˜푟 푟 휉 = 푓 (휉 , 푢(푡)) := 푓 (휉 + 휙휉(0), 푢(푡)). (C.13)

Lemma C.2. Under the assumptions of Lemma 4.5, the translated reduced system

(C.13) is ISS. O

Proof. To show that (C.13) is ISS, we first show that it has the asymptotic gain prop- ˜푟 erty (see [151]), that is, there exists a class 풦0 function 훾(·) such that lim sup푡→∞ |휉 | ≤ 훾(‖푢‖) for all initial conditions. This property, combined with the fact that (C.13) is GAS for 푢 ≡ 0, is equivalent to (C.13) being ISS (see Theorem 1 in [151]). Given

푟 Assumption 4.8, system Σ푖 is I/S monotone with a mixed-monotone static I/S char- 푟 ˆ acteristic 휑(푢) := 휙휉(푢). Let 휑(푢1, 푢2) be the canonical decomposition function of 휑(푢) and suppose that 풰 = [푢, 푢], where 푢 and 푢 can possibly be ∞. Let 푢−(‖푢‖) :=

+ max(−1푛‖푢‖, 푢), 푢 (‖푢‖) := min(1푛‖푢‖, 푢), where 1푛 is an 푛-vector with all elements being 1. Therefore, the input 푢(푡) to (C.13) satisfies 푢(푡) → [푢−(‖푢‖), 푢+(‖푢‖)], and by Lemma 4.2, we have 휉˜푟 → [휑˜−(‖푢‖), 휑˜+(‖푢‖)], where 휑˜−(‖푢‖), 휑˜+(‖푢‖): R → R푛 are defined as:

휑˜+(‖푢‖) := 휑ˆ(푢+(‖푢‖), 푢−(‖푢‖)) − 휑ˆ(0, 0),

휑˜−(‖푢‖) := 휑ˆ(푢−(‖푢‖), 푢+(‖푢‖)) − 휑ˆ(0, 0).

˜+ ˜− Define 훾(‖푢‖) = max푣≤‖푢‖ max{|휑 (푣)|, |휑 (푣)|}, since 훾(0) = 0 and it is non- decreasing, it is an asymptotic gain of (C.13). The GAS property of (C.13) when

180 푢 = 0 is a consequence of the existence of the I/S characteristic for all 푢 ∈ 풰.

Since the convergent-input-convergent-state property we aim to prove is translation-

푟 푟 invariant, we will assume in the sequel that 휙휉(0) = 0 and hence Σ푖 is ISS.

Lemma C.3. Under the assumptions of Lemma 4.5, given any 휇 > 0, ∃휈* = 휈*(휇) such that the trajectory of (4.26) is bounded (by an 휇-independent constant) for all 푡 ≥ 0 and

lim sup |휁(푡) − Γ(휉, 푢(푡))| ≤ 휇 (C.14) 푡→∞

* for all 0 < 휈 ≤ 휈 . O

Lemma C.3 is a direct application of Theorem 1 in [33] and the proof for trajectory boundedness can be found in [32]. This proves the convergence result for 휁 in Lemma 4.5. To show the convergent-input-convergent-state property of the slow variable 휉, let 푦푏(푡) := 휁 − Γ(휉, 푢(푡)). The dynamics of 휉 in (4.26) can be written as:

˙ ¯ 휉 = 퐹 (휉, 푦푏(푡), 푢(푡)) := 푓(휉, Γ(휉, 푢) + 푦푏, 푢). (C.15)

We treat (C.15) as a perturbation of the reduced system (4.28), whose dynamics follow

휉˙푟 = 퐹 (휉푟, 0, 푢(푡)) = 푓¯(휉푟, Γ(휉푟, 푢), 푢) = 푓¯푟(휉푟, 푢). (C.16)

We therefore aim to show that the trajectories of (C.15) and (C.16) are ultimately close to each other. Since (C.16) is I/S monotone with respect to 푢(푡), so is (C.15).

푟 + − 푟 + − Suppose that 휙ˆ휉(푢 , 푢 ) := 휙휉(Λ+푢 + Λ−푢 ) is the canonical decomposition func- 푟 + + − − − + tion of 휙휉(푢), and let 푢* := Λ+푢 + Λ−푢 and 푢* := Λ+푢 + Λ−푢 . Suppose − + 푢(푡) ∈ [푢 , 푢 ], the trajectory of the perturbed system (C.15), 휉(푡, 휉(0), 푦푏(푡), 푢(푡))

181 necessarily satisfy

− 휉(푡, 휉(0), 푦푏(푡), 푢* ) ≤ 휉(푡, 휉(0),푦푏(푡), 푢(푡))

+ ≤휉(푡, 휉(0), 푦푏(푡), 푢* ) (C.17) for all 푡 ≥ 0. Under the assumptions for the reduced system in Lemma 4.5, for each fixed 푢, 휉˙ = 퐹 (휉, 0, 푢) has a GAS equilibrium 휉¯(푢, 0). By Lemma C.3,

lim lim sup |푦푏(푡)| = 0 휈→0+ 푡→∞

and the trajectory of (C.15) is bounded. Hence, by Theorem 1 in [150], pick any 휇 > 0

± ± there exists 휈 sufficiently small such that lim sup푡→∞ |휉(푡, 휉(0), 0, 푢* )−휉(푡, 휉(0), 푦푏(푡), 푢* )| ≤ ± 푟 ± 푟 ± ∓ 휇. Since lim푡→∞ 휉(푡, 휉(0), 0, 푢* ) = 휙푖 (푢* ) =휙 ˆ휉(푢 , 푢 ), by triangle inequality, we 휇 푟 − + 푟 + − have 휉(푡) = 휉(푡, 휉(0), 푦푏(푡), 푢(푡)) −→ [휙 ˆ휉(푢 , 푢 ), 휙ˆ휉(푢 , 푢 )]. This completes the proof for Lemma 4.5.

C.4 Small-gain theorem for convergent-input-convergent- output system

We state and prove the small-gain theorem for approximate convergent-input-convergent- output (CICO) systems used in here. Consider system (4.26) interconnected with a

cooperative function 푢 = ∆(휒) with the global Lipshitz property: ∃퐿Δ > 0 such that + − + − |∆(휒 ) − ∆(휒 )| ≤ 퐿Δ|휒 − 휒 |.

Lemma C.4. Suppose that (4.26) has the approximate CICO property: 푤(푡) → [푤−, 푤+] implies

휇 휒(푡) −→ [휓(푢−, 푢+), 휓(푢+, 푢−)]. (C.18)

+ − − + Assume that there exists 푢0 and 푢0 such that 푢(푡) ∈ [푢0 , 푢0 ] for all 푡, if the DT

182 dynamical system 푢−(푘 + 1) = ∆ ∘ 휓(푢−(푘), 푢+(푘)), (C.19) 푢+(푘 + 1) = ∆ ∘ 휓(푢+(푘), 푢−(푘)).

− + is exponentially ultimately bounded in [푢* , 푢* ], then given any 휇 > 0, there exists a

class 풦0 function 훼(·) such that

휇 − + 푢(푡) −→ [푢* , 푢* ] (C.20)

* for all 0 < 휈 ≤ 휈 (훼(휇)). O

− + − + Proof. Since the closed loop 푢(푡) is bounded in [푢 (0), 푢 (0)] := [푢0 , 푢0 ], by Lemma 4.5, for any 휇 > 0, ∃휈*(휇) such that ∀0 < 휈 ≤ 휈*(휇)

휇 휒(푡) −→ [휓(푢−(0), 푢+(0)), 휓(푢+(0), 푢−(0))],

By the cooperativity and Lipschitz property of ∆, we have that 푢(푡) → [푢−(1), 푢+(1)], where

− − + 푢 (1) : = ∆ ∘ 휓(푢 (0), 푢 (0)) − 퐿Δ휇,

+ + − 푢 (1) : = ∆ ∘ 휓(푢 (0), 푢 (0)) + 퐿Δ휇.

After (푘 + 1)-iterations, 푢(푡) → [푢−(푘 + 1), 푢+(푘 + 1)], where

− − + 푢 (푘 + 1) = ∆ ∘ 휓(푢 (푘), 푢 (푘)) − 퐿Δ휇, (C.21) + + − 푢 (푘 + 1) = ∆ ∘ 휓(푢 (푘), 푢 (푘)) + 퐿Δ휇, and 푢(푡) → [푢−(푘), 푢+(푘)] for every integer 푘 ≥ 0. We treat (C.21) as a perturbation of the DT dynamical system (C.19). Since (C.19) is exponentially ultimately

− + * * bounded in [푢* , 푢* ], according to Lemma 4.4, ∃휇 , 휅 > 0 such that if 휇 ∈ (0, 휇 ], − + 휅휇 − + 휅휇 − + [푢 (푘), 푢 (푘)] −→ [푢* , 푢* ]. Therefore, 푢(푡) −→ [푢* , 푢* ]. For (C.20) to be satisfied, * * we can take 훼(휇) := min(휇/휅, 휇 ) and 0 < 휈 ≤ 휈 (훼(휇)).

If the CICO property is exact, as for I/S monotone systems in Lemma 4.2, then we

183 only need to set 휇 = 0 in (C.18) and (C.20) and the rest of the statement in Lemma C.4 still holds.

C.5 Disturbance attenuation of feedback-regulated subsystems

To show (4.44), we take two step: we first show that |ℎ푖(푟푖, 0; 휀푖) − 퐻푖(푟푖)| is 휀푖-small,

we then show that |ℎ푖(푟푖, 푤푖; 휀푖) − ℎ푖(푟푖, 0; 휀푖)| is small. The first part can be shown

using the following property of ℎ푖.

* Claim C.1. There exists 퐾푖 > 0, independent of 푟푖, such that

* |ℎ푖(푟푖, 0; 휀푖) − 푟푖/훽푖| ≤ 퐾푖 휀푖 (C.22)

¯ for all 푟푖 ∈ ℛ푖 and for 휀푖 sufficiently small. O

* The proof for a constant 푟푖 can be found in [125], and 퐾푖 can be chosen independent ¯ of 푟푖 because ℛ푖 is compact. We now show that ℎ푖(푟푖, 푤푖; 휀푖) is close to ℎ푖(푟푖, 0; 휀푖).

* Claim C.2. Consider system (4.37), there exists a positive constant 푘푖 , independent ¯ of 푟푖, such that for any fixed pair (푟푖, 푤푖) ∈ ℛ푖 × 풲푖,

* * |ℎ푖(푟푖, 푤푖; 휀푖) − ℎ푖(푟푖, 0; 휀푖)| ≤ 푘푖 휀푖|푤푖| + 퐾푖 휀푖 (C.23)

* for 휀푖 sufficiently small, where 퐾푖 is as defined in Claim C.1.

푟 푟 Proof. Since Σ푖 has a GAS equilibrium, it is sufficient to show lim sup푡→∞ |푦푖 (푡) − * * 푟 푟 ℎ푖(푟푖, 0; 휀푖)| ≤ 휀푖푘푖 |푤푖| + 퐾푖 휀푖 for all initial condition 푝푖 (0) = 푦푖 (0). We first fix a ¯ * 푟 * 푟푖 ∈ ℛ푖, and let 푦푖 = ℎ푖(푟푖, 0; 휀푖) and 푦˜푖 := 푦푖 − 푦푖 . The dynamics of 푦˜푖 follow:

˙ * 푦˜푖 = 푇푖(˜푦푖, 푟푖, 푤푖) − 훿(푦푖 +푦 ˜푖), (C.24)

184 where

1 * 훾푖 (˜푦푖 + 푦푖 , 푟푖; 휀푖)/휅푖 푇푖(˜푦푖, 푟푖, 푤푖) := 훼푖 1 * . 1 + 훾푖 (˜푦푖 + 푦푖 , 푟푖; 휀푖)/휅푖 + 푤푖

* 1 * * * and because 푇푖(0, 푟푖, 0) − 훿푦푖 = 0, we have 훾푖 (푦푖 , 푟푖; 휀푖) = 휅푖훿푦푖 /(훼푖 − 훿푦푖 ). Let

* * 훿휅푖푦푖 2훿 푘푖(푦푖 ) := * · , (C.25) 훼푖 − 훿푦푖 훽푖

* we show that the trajectory of (C.24) is ultimately bounded in the set 풫푖(푦푖 ) := * 2 {−푘푖휀푖푤푖 − 퐾푖 휀푖 ≤ 푦˜푖 ≤ 0} using the Lyapunov function 푉푖(˜푦푖) =푦 ˜푖 /2. Note that if ¯ 푟푖 ∈ ℛ푖, 훼푖 − 훿퐻푖(푟푖) ≥ 휗푖, where 휗푖 is an 휖푖 independent constant. Due to Claim * C.1, 푘푖(푦 ) > 0 for sufficiently small 휀푖. For 푦˜푖 ≥ 0, since 휕푇푖/휕푤푖, 휕푇푖/휕푦˜푖 < 0, we ˙ * * have 푉푖 =푦 ˜푖[푇푖(˜푦푖, 푤푖, 푟푖) − 훿푥푖 − 훿푦˜푖] ≤ 푦˜푖[푇푖(0, 0, 푟푖) − 훿푦푖 − 훿푦˜푖] = −2훿푉푖. By Claim * * * * C.1, 푦푖 ≤ 퐻푖(푟푖) + 퐾푖 휀푖 = 푟푖/훽푖 + 퐾푖 휀푖, and therefore, for 푦˜푖 ≤ −푘푖휀푖푤푖 − 퐾푖 휀푖 < 0, 푟 * * * we have 푦푖 =푦 ˜푖 + 푦푖 ≤ −푘푖휀푖푤푖 − 퐾푖 휀푖 + 푦푖 ≤ 푟푖/훽푖 − 푘푖휀푖푤푖. Substituting into 푟 * (4.36), 퐴푖(푟푖, 푦푖 ) ≤ 0 for all 푦˜푖 ≤ −푘푖휀푖푤푖 − 퐾푖 휀푖 and for all 푤푖 ≥ 0. This fact can be 1 1 휕훾푖 휕훾푖 훽푖 combined with (4.38) to show 푟 = ≤ − . As a result, 휕푦푖 휕푦˜푖 2휀푖훿

1 * 1 * 훽푖 1 * 훾푖 (푦푖 +푦 ˜푖, 푟푖; 휀푖) ≥ 훾푖 (푦푖 , 푟푖; 휀푖) − 푦˜푖 ≥ 훾푖 (푦푖 , 푟푖; 휀푖)(1 + 푤푖) (C.26) 2휀푖훿

* if 푦˜푖 ≤ −푘푖휀푖푤푖 − 퐾푖 휀푖. Substituting into (C.24), we find

1 * 훾푖 (푦푖 , 푟푖; 휀푖)/휅푖 푇푖(˜푦푖, 푟푖, 푤푖) ≥ 훼푖 1 * = 푇푖(0, 푟푖, 0) 1 + 훾푖 (푦푖 , 푟푖; 휀푖)/휅푖

* ˙ * * if 푦˜푖 ≤ −푘푖휀푖푤푖 − 퐾푖 휀푖. Thus, 푉푖 =푦 ˜푖[푇푖(˜푦푖, 푟푖, 푤푖) − 훿푦푖 − 훿푦˜푖] ≤ 푦˜푖[푇푖(0, 푟푖, 0) − 훿푦푖 −

훿푦˜푖] = −2훿푉푖. Hence, we have shown that 푦˜푖(푡) eventually enters 풫푖 for any constant ¯ ¯ * (푟푖, 푤푖) ∈ ℛ푖 × 풲푖. Since ℛ푖 is compact, due to Claim C.1, 푦푖 is also bounded in a * * * compact set. Thus, there exists 푘푖 ≥ 푘푖(푦푖 ) for all 푦푖 .

With Claim C.1 and C.2, inequality (4.44) can be shown via triangle inequality.

185 C.6 Lipschitz properties of subsystem characteristics

+ + − − 푟 − + + Since 휓푖(푟푖 , 푤푖 , 푟푖 , 푤푖 ; 휀푖) = 휌푖 (ℎ푖(푟푖 , 푤푖 ; 휀푖), 푟푖 ; 휀푖), to show Assumption 4.6 is 푟 satisfied, we prove that 휌푖 and ℎ푖 each has the Lipschitz conditions stated below.

Claim C.3. There are positive functions 푐푥(·) and 푐푟(·) such that:

¯ * ∀(푥푖, 푟푖, 푤푖; 휀푖) ∈ 풳푖×ℛ푖×풲푖×(0, 휀푖 ], where 푐푟(휀푖) = 1/(훿휀푖) and 푐푥(휀푖) = max푖(훽푖)/(훿휀푖). ¯ 1 In addition, ℎ is Lipschitz in 푟푖 ∈ ℛ푖 uniformly in 푤푖, 휀푖 with 퐿ℎ = 1/훽푖 + 훼푖/(2훿휗푖 ). O

Proof. We first show that ℎ푖(푟푖, 푤푖; 휀푖), which is the steady state output of feedback- ¯ regulated subsystem given fixed (푟푖, 푤푖) ∈ ℛ푖 × 풲푖 is Lipschitz in 푟푖 uniformly in ¯ ¯ 푤푖, 휀푖. Since ℛ푖 is any 휀푖-independent compact subset of (0, 훼푖훽푖/훿), we let ℛ푖 := 1 푖 푖 푖 [휗푖 , 훼푖훽푖/훿 − 휗2], where 0 < 휗1, 휗2 < 훼푖훽푖/훿 are 휀푖-independent constants. By setting

the dynamics of (4.33) to steady state, the equilibrium 푚¯ 푖 =푚 ¯ 푖(푟푖, 푤푖) is the solution to

2 훼푖훽푖 푚¯ 푖/휅푖 훿푟푖 휀푖 훿 F푖(푚 ¯ 푖, 푟푖, 푤푖) := − 푟푖 + 휀푖훿푚¯ 푖 − 휀푖 + = 0, 훿 1 +푚 ¯ 푖/휅푖 + 푤푖 휆푖푚¯ 푖 휆푖

and the equilibrium output 푦¯푖 =푝 ¯푖 can subsequently determined via

훼푖 푚¯ 푖/휅푖 푦¯푖 = G푖(푚 ¯ 푖, 푤푖) = . 훿 1 +푚 ¯ 푖/휅푖 + 푤푖

Using chain rule and the implicit function theorem, we have:

휕ℎ 휕G 휕푚¯ 휕G 휕F (︂ 휕F )︂−1 푖 = 푖 · 푖 = − 푖 푖 푖 , 휕푟푖 휕푚¯ 푖 휕푟푖 휕푚¯ 푖 휕푟¯푖 휕푚¯ 푖

186 from which we find

휕ℎ푖 1 훼푖 0 < ≤ + 1 =: 퐿ℎ 휕푟푖 훽푖 2훿휗푖

¯ for all (푟푖, 푤푖) ∈ ℛ푖 × 풲푖. To show (C.27) is satisfied, we use (4.38). Because

퐴푖(푟푖, 푥푖; 휀푖) ∈ (−∞, +∞) for 푟푖, 푥푖 taking positive values. Thus, we have

휕훾1 1 훽 휕훾1 0 < 푖 ≤ , − 푖 ≤ 푖 < 0. (C.28) 휕푟푖 2훿휀푖 2훿휀푖 휕푥푖

푟 1 1 max푖(훽푖) Since 휌 = 훾 /휅푖, we can take 푐푟(휀푖) = and 푐푥(휀푖) = and (C.27) 푖 푖 2훿 min푖(휅푖)휀푖 2 min푖(휅푖)훿휀푖 is satisfied.

+ − ¯ 2 − + Claim C.3 implies that 휓푖 is Lipschitz in (푟푖 , 푟푖 ) ∈ (ℛ푖) uniformly in 푤푖 , 푤푖 ∈ 풲푖 * with Lipschitz constant. 퐿휓(휀푖) = 푐푟(휀푖) + 푐푥(휀푖)퐿ℎ. The I/O gain function 휓푖 is * + − * * + + sub-linear because, according to (4.43), 휓푖 (푤푖 , 푤푖 ; 푟푖 , 휀푖) = 휂푖(푟푖 , 푤푖 ; 휀푖)(1 + 푤푖 )

and, as we have shown in the proof of Proposition 4.1, 휂푖 is positive and bounded for * + ¯ * (푟푖 , 푤푖 ; 휀푖) ∈ ℛ푖 × 풲푖 × (0, 휀푖 ].

187 188 Bibliography

[1] Deepak K. Agrawal, Ryan Marshall, Vincent Noireaux, and Eduardo D Sontag. In vitro implementation of robust gene regulation in a synthetic biomolecular integral controller. Nature Communications, 10(1), 2019.

[2] Deepak K. Agrawal, Xun Tang, Alexandra Westbrook, Ryan Marshall, Colin S. Maxwell, Julius Lucks, Vincent Noireaux, Chase L. Beisel, Mary J. Dunlop, and Elisa Franco. Mathematical modeling of RNA-based architectures for closed loop control of gene expression. ACS Synthetic Biology, 7(5):1219–1228, 2018.

[3] Hiroji Aiba. Mechanism of RNA silencing by Hfq-binding small RNAs. Curr. Opin. Microbiol., 10(2):134–139, Apr 2007.

[4] Uri Alon. An Introduction to Systems Biology: Design Principles of Biological Circuits. Chapman & Hall/CRC Press, 2006.

[5] Wenlin An and Jason W. Chin. Synthesis of orthogonal transcription- translation networks. Proc. Natl. Acad. Sci. U. S. A., 106(21):8477–8482, 2009.

[6] Martin Andreasson, Dimos V. Dimarogonas, Henrik Sandberg, and Karl Henrik Johansson. Distributed control of networked dynamical systems: Static feedback, integral action and consensus. IEEE Transactions on Automatic Control, 59(7):1750–1764, 2014.

[7] Jordan Ang, Sangram Bagh, Brian P. Ingalls, and David R. McMillen. Con- siderations for using integral feedback control to construct a perfectly adapting synthetic gene network. J. Theor. Biol., 266(4):723–738, 2010.

[8] Jordan Ang and David R. McMillen. Physical constraints on biological integral control design for homeostasis and sensory adaptation. Biophys. J., 104(2):505– 515, 2013.

[9] D. Angeli and E.D. Sontag. Monotone control systems. IEEE Transactions on Automatic Control, 48(10):1684–1698, 2003.

[10] David Angeli, Germán A. Enciso, and Eduardo D. Sontag. A small-gain result for orthant-monotone systems under mixed feedback. Systems & Control Letters, 68:9–19, 2014.

189 [11] Stephanie K. Aoki, Gabriele Lillacci, Ankit Gupta, Armin Baumschlager, David Schweingruber, and Mustafa Khammash. A universal biomolecular integral feedback controller for robust perfect adaptation. Nature, 570(7762):533–537, 2019.

[12] Murat Arcak. Pattern formation by lateral inhibition in large-scale networks of cells. IEEE Transactions on Automatic Control, 58(5):1250–1262, 2013.

[13] Murat Arcak and Eduardo D. Sontag. Diagonal stability of a class of cyclic systems and its connection with the secant criterion. Automatica, 42(9):1531– 1537, 2006.

[14] James A. J. Arpino, Edward J. Hancock, James Anderson, Mauricio Barahona, Guy-Bart V. Stan, Antonis Papachristodoulou, and Karen Polizzi. Tuning the dials of synthetic biology. Microbiology, 159(Pt_7):1236–1253, 2013.

[15] F. K. Balagaddé, H. Song, J. Ozaki, C. H. Collins, M. Barnet, F. H. Arnold, S. R. Quake, and L. You. A synthetic Escherichia coli predator-prey ecosystem. Mol. Syst. Biol., 4, 2008.

[16] Carlos Barajas and Domitilla Del Vecchio. Genetic circuit-host ribosome transactions: Diffusion-reaction model. In 2019 American Control Conference (ACC). IEEE, 2019.

[17] Caleb J Bashor and James J Collins. Insulating gene circuits from context by RNA processing. Nat. Biotechnol., 30:1061–1062, 2012.

[18] Eilyan Bitar and Pete Seiler. Coordinated control of a wind turbine array for power maximization. In 2013 American Control Conference. IEEE, 2013.

[19] Franco Blanchini and Elisa Franco. Structurally robust biological networks. BMC Systems Biology, 5(1):74, 2011.

[20] Hans Bremer and Patrick P. Dennis. Modulation of chemical composition and other parameters of the cell by growth rate. In Frederick C. Neidhardt, editor, Escherichia coli and Salmonella: Cellular and Molecular Biology. ASM Press, 1996.

[21] K. J. Astr˘ ¨om and R. M. Murray. Feedback Systems: An Introduction for Sci- entists and Engineers. Princeton University Press, 2008.

[22] Corentin Briat, Ankit Gupta, and Mustafa Khammash. Antithetic integral feedback ensures robust perfect adaptation in noisy bimolecular networks. Cell Systems, 2(1):15–26, 2016.

[23] Jennifer A N Brophy and Christopher A Voigt. Principles of genetic circuit design. Nat. Methods, 11(5):508–520, 2014.

190 [24] Lucas Buccafusca, Joao P. Jansch-Porto, Geir E. Dullerud, and Carolyn L. Beck. An application of nested control synthesis for wind farms. IFAC-PapersOnLine, 52(20):199–204, 2019.

[25] D Ewen Cameron, Caleb J Bashor, and James J Collins. A brief history of synthetic biology. Nature Reviews Microbiology, 12(5):381–390, 2014.

[26] Bartholomew Canton. Engineering the interface between cellular chassis and synthetic biological systems. PhD thesis, Massachusetts Institute of Technology, 2008.

[27] M. Carbonell-Ballestero, E. Garcia-Ramallo, R. Montañez, C. Rodriguez-Caso, and J. Macía. Dealing with the genetic load in bacterial synthetic biology circuits: convergences with the ohm’s law. Nucleic Acids Res., 44(1):496–507, 2015.

[28] Stefano Cardinale and Adam Paul Arkin. Contextualizing context for synthetic biology- identifying causes of failure of synthetic biological systems. Biotechnol. J., 7:856–866, 2012.

[29] Francesca Ceroni, Rhys Algar, Guy-Bart Stan, and Tom Ellis. Quantifying cellular capacity identifies gene expression designs with reduced burden. Nat. Methods, 12(5):415–422, 2015.

[30] Deboki Chakravarti and Wilson W Wong. Synthetic biology in cell-based cancer immunotherapy. Trends Biotechnol., 33(8):449–461, 2015.

[31] James Chappell, Kyle E Watters, Melissa K Takahashi, and Julius B Lucks. A renaissance in RNA synthetic biology: new mechanisms, applications and tools for the future. Current Opinion in Chemical Biology, 28:47–56, 2015.

[32] P.D. Christofides and A.R. Teel. Singular perturbations and input-to-state stability. In Proceedings of 3rd European Control Conference, pages 1845–1850, 1995.

[33] P.D. Christofides and A.R. Teel. Singular perturbations and input-to-state stability. IEEE Transactions on Automatic Control, 41(11):1645–1650, 1996.

[34] Samuel Coogan and Murat Arcak. Efficient finite abstraction of mixed monotone systems. In Proceedings of the 18th International Conference on Hybrid Systems Computation and Control. ACM Press, 2015.

[35] Tal Danino, Arthur Prindle, Gabriel A Kwong, Matthew Skalak, Howard Li, Kaitlin Allen, Jeff Hasty, and Sangeeta N Bhatia. Programmable probiotics for detection of cancer in urine. Sci. Transl. Med., 7(289):289ra84–289ra84, 2015.

[36] Alexander P. S. Darlington, Juhyun Kim, José I. Jiménez, and Declan G. Bates. Dynamic allocation of orthogonal ribosomes facilitates uncoupling of co-expressed genes. Nat. Commun., 9(1), 2018.

191 [37] Jamie Davies. Using synthetic biology to explore principles of development. Development, 144(7):1146–1158, 2017.

[38] Dirk De Vos, Frank J. Bruggeman, Hans V. Westerhoff, and Barbara M. Bakker. How molecular competition influences fluxes in gene expression networks. PLoS One, 6(12):e28494, 2011.

[39] G. del Solar, R. Giraldo, M. J. Ruiz-Echevarria, M. Espinosa, and R. Diaz- Orejas. Replication and control of circular bacterial plasmids. Microbiol. Mol. Biol. Rev., 62(2):434–464, 1998.

[40] Domitilla Del Vecchio. Modularity, context-dependence, and insulation in engineered biological circuits. Trends Biotechnol., 33(2):111–119, 2015.

[41] Domitilla Del Vecchio and Richard M. Murray. Biomolecular Feedback Systems. Princeton University Press, Princeton, 2014.

[42] Domitilla Del Vecchio, Alexander J Ninfa, and Eduardo D Sontag. Modular cell biology: retroactivity and insulation. Mol. Syst. Biol., 4:161, 2008.

[43] C. A. Desoer and M. Vidyasagar. Feedback Systems: Input-Output Properties. Academic Press, 1975.

[44] M Omar Din, Tal Danino, Arthur Prindle, Matt Skalak, Jangir Selimkhanov, Kaitlin Allen, Ellixis Julio, Eta Atolia, Lev S Tsimring, Sangeeta N Bhatia, and Jeff Hasty. Synchronized cycles of bacterial lysis for in vivo delivery. Nature, 536:81–85, 2016.

[45] T. Drengstig, I. W. Jolma, X. Y. Ni, K. Thorsen, X. M. Xu, and P. Ruoff. A Basic Set of Homeostatic Controller Motifs. Biophys. J., 103:2000–2010, 2012.

[46] T. Drengstig, X. Y. Ni, K. Thorsen, I. W. Jolma, and P. Ruoff. Robust adaptation and homeostasis by autocatalysis. J. Phys. Chem. B, 116(18):5355–5363, 2012.

[47] Mary J. Dunlop, Jay D. Keasling, and Aindrila Mukhopadhyay. A model for improving microbial biofuel production using a synthetic feedback loop. Systems and Synthetic Biology, 4(2):95–104, 2010.

[48] Michael B. Elowitz and Stanislas Leibler. A synthetic oscillatory network of transcriptional regulators. Nature, 403(6767):335–338, January 2000.

[49] J.A. Fax and R.M. Murray. Information flow and cooperative control of vehicle formations. IEEE Transactions on Automatic Control, 49(9):1465–1476, 2004.

[50] AurÃ○c lie Fender, Johan Elf, Kornelia Hampel, Bastian Zimmermann, and E. Gerhart H. Wagner. RNAs actively cycle on the Sm-like protein Hfq. Genes. Dev., 24(23):2621–2626, 2010.

192 [51] R. Fierro, C. Belta, J.P. Desai, and V. Kumar. On controlling aircraft formations. In Proceedings of the 40th IEEE Conference on Decision and Control. IEEE, 2001.

[52] B.A. Francis and W.M. Wonham. The Internal Model Principle of Control Theory The Internal Model Principle of Control Theory. Automatica, 12:457– 465, 1976.

[53] Timothy S. Gardner, Charles R. Cantor, and James J. Collins. Construction of a genetic toggle switch in escherichia coli. Nature, 403(6767):339–342, 2000.

[54] D.T. Gavel and D.D. Siljak. Decentralized adaptive control: structural conditions for stability. IEEE Transactions on Automatic Control, 34(4):413–426, April 1989.

[55] Amar Ghodasara and Christopher A. Voigt. Balancing gene expression without library construction via a reusable sRNA pool. Nucleic Acids Research, 45(13):8116–8127, June 2017.

[56] Khem Raj Ghusinga, Cesar A. Vargas-Garcia, and Abhyudai Singh. A mechanistic stochastic framework for regulating bacterial cell division. Sci. Rep., 6(1), 2016.

[57] H. Giladi, D. Goldenberg, S. Koby, and A. B. Oppenheim. Enhanced activity of the bacteriophage lambda PL promoter at low temperature. FEMS Microbiol. Rev., 17(1-2):135–140, 1995.

[58] L. Gorini and W. K. Maas. The potential for the formation of a biosynthetic enzyme in Escherichia coli. Biochim. Biophys. Acta, 25(1):208–209, 1957.

[59] Thomas E Gorochowski, Irem Avcilar-kucukgoze, Roel A L Bovenberg, Jo- hannes A Roubos, and Zoya Ignatova. A Minimal Model of Ribosome Allo- cation Dynamics Captures Trade- offs in Expression between Endogenous and Synthetic Genes. ACS Synth. Biol., 5(7):710–720, 2016.

[60] Theodore W. Grunberg and Domitilla Del Vecchio. Time-scale separation based design of biomolecular feedback controllers. In 2019 IEEE 58th Conference on Decision and Control (CDC). IEEE, 2019.

[61] Theodore W. Grunberg and Domitilla Del Vecchio. Modular analysis and design of biological circuits. Current Opinion in Biotechnology, 63:41–47, 2020.

[62] Zhong-Mei Gu, N. N. Nefedov, and R. E. O’Malley Jr. On singular singularly perturbed initial value problems. SIAM J. Appl. Math., 49(1):1–25, 1989.

[63] Andras Gyorgy, José I. Jiménez, John Yazbek, Hsin-Ho Huang, Hattie Chung, Ron Weiss, and Domitilla Del Vecchio. Isocost lines describe the cellular econ- omy of genetic circuits. Biophys. J., 109(3):639–646, 2015.

193 [64] Abdullah Hamadeh, Eduardo D Sontag, and Domitilla Del Vecchio. A contrac- tion approach to input tracking via high gain feedback. In Proc. 54th IEEE Conf. Decis. Control (CDC), pages 7689–7694, Osaka, 2015. [65] Edward J. Hancock, Jordan Ang, Antonis Papachristodoulou, and Guy-Bart Stan. The interplay between feedback and buffering in cellular homeostasis. Cell Systems, 5(5):498–508.e23, 2017. [66] Aqib Hasnain, Diveena Becker, Atsede Siba, Narendra Maheshri, Ben Gordon, Chris Voigt, Enoch Yeung, Subhrajit Sinha, Yuval Dorfan, Amin Espah Boru- jeni, Yongjin Park, Paul Maschhoff, Uma Saxena, Joshua Urrutia, and Niall Gaffney. A data-driven method for quantifying the impact of a genetic circuit on its host. In 2019 IEEE Biomedical Circuits and Systems Conference (BioCAS). IEEE, 2019. [67] Victoria Hsiao, Emmanuel L. C. de los Santos, Weston R. Whitaker, John E. Dueber, and Richard M. Murray. Design and implementation of a biomolecular concentration tracker. ACS Synthetic Biology, 4(2):150–161, 2014. [68] Hsin-Ho Huang, Yili Qian, and Domitilla Del Vecchio. A quasi-integral controller for adaptation of genetic modules to variable ribosome demand. Nature Communications, 9(1), 2018. [69] Razika Hussein and Han N. Lim. Direct comparison of small RNA and transcription factor signaling. Nucleic. Acids. Res., 40(15):7269–7279, 2012. [70] P. Ioannou. Decentralized adaptive control of interconnected systems. IEEE Transactions on Automatic Control, 31(4):291–298, 1986. [71] Alberto Isidori. Nonlinear Control Systems II. Springer London, 1999. [72] Shridhar Jayanthi, Kayzad Soli Nilgiriwala, and Domitilla Del Vecchio. Retroac- tivity controls the temporal dynamics of gene transcription. ACS Synth. Biol., 2(8):431–441, 2013. [73] P. Jiang, A. C. Ventura, E. D. Sontag, S. D. Merajver, A. J. Ninfa, and D. Del Vecchio. Load-induced modulation of signal transduction networks. Sci. Signal., 4(194):ra67–ra67, 2011. [74] Z. P. Jiang, A. R. Teel, and L. Praly. Small-gain theorem for ISS systems and applications. Mathematics of Control, Signals, and Systems, 7(2):95–120, 1994. [75] Zhong-Ping Jiang, D.W. Repperger, and D.J. Hill. Decentralized nonlinear output-feedback stabilization with disturbance attenuation. IEEE Transactions on Automatic Control, 46(10):1623–1629, 2001. [76] Ross D. Jones, Yili Qian, Velia Siciliano, Breanna DiAndreth, Jin Huh, Ron Weiss, and Domitilla Del Vecchio. An endoribonuclease-based feedforward controller for decoupling resource-limited genetic modules in mammalian cells. bioRxiv, 2020.

194 [77] Ulf T. Jonsson and Chung-Yao Kao. A scalable robust stability criterion for systems with heterogeneous LTI components. IEEE Transactions on Automatic Control, 55(10):2219–2234, 2010.

[78] Shai Kaplan, Anat Bren, Erez Dekel, and Uri Alon. The incoherent feed-forward loop can generate non-monotonic input functions for genes. Mol. Syst. Biol., 4:203, 2008.

[79] J. D. Keasling and B. O. Palsson. On the kinetics of plasmid replication. J. Theor. Biol., 136(4):487–492, 1989.

[80] Ciarán L Kelly, Andreas W K Harris, Harrison Steel, Edward J Hancock, John T Heap, and Antonis Papachristodoulou. Synthetic negative feedback circuits using engineered small RNAs. Nucleic Acids Research, 46(18):9875–9889, 2018.

[81] Hassan K. Khalil. Nonlinear systems. Prentice Hall, Upper Saddle River, New Jersey, 3rd edition, 2002.

[82] Dongsan Kim, Yung-Keun Kwon, and Kwang-Hyun Cho. The biphasic behavior of incoherent feed-forward loops in biomolecular regulatory networks. Bioessays, 30(11-12):1204–1211, 2008.

[83] E. Klavins. Proportional-integral control of stochastic gene regulatory networks. In Proceedings of the 49th IEEE Conference on Decision and Control, pages 2547–2553, Atlanta, GA, 2010.

[84] Stefan Klumpp, Zhongge Zhang, and Terence Hwa. Growth-rate dependent global effect on gene expression in bacteria. Cell, 139:1366–1375, 2009.

[85] Petar Kokotović, Hassan K. Khalil, and John O’Reilly. Singular Perturbation Methods in Control: Analysis and Design. SIAM, 1999.

[86] Sriram Kosuri, Daniel B. Goodman, Guillaume Cambray, Vivek K. Mutalik, Yuan Gao, Adam P. Arkin, Drew Endy, and George M. Church. Composability of regulatory sequences controlling transcription and translation in Escherichia coli. Proc. Natl. Acad. Sci. U. S. A., 110(34):14024–14029, 2013.

[87] Jonathan W Kotula, S Jordan Kerns, Lev A Shaket, Layla Siraj, James J Collins, and Jeffrey C Way. Programmable bacteria detect and record an environmental signal in the mammalian gut. Proceedings of the National Academy of Sciences of the United States of America, 111(13):4838–4843, 2014.

[88] H. Kunze and D. Siegel. A graph theoretical approach to monotonicity with respect to initial conditions II. Nonlinear Analysis: Theory, Methods & Appli- cations, 35(1):1–20, 1999.

[89] Manish Kushwaha and Howard M Salis. A portable expression resource for engineering cross-species genetic circuits and pathways. Nat. Commun., 6:7832, 2015.

195 [90] Erel Levine, Zhongge Zhang, Thomas Kuhlman, and Terence Hwa. Quantitative characteristics of gene regulation by small RNA. PLoS Biology, 5(9):e229, 2007.

[91] S. Lin-Chao and H. Bremer. Effect of the bacterial growth rate on replication control of plasmid pbr322 in escherichia coli. Mol. Gen. Genet., 203(1):143–149, 1986.

[92] Shuang Liu, Fan Zhang, Jian Chen, and Guoxin Sun. Arsenic removal from contaminated soil via biovolatilization by genetically engineered bacteria under laboratory conditions. Journal of Environmental Sciences, 23(9):1544–1550, 2011.

[93] Chunbo Lou, Brynne Stanton, Ying-Ja Chen, Brian Munsky, and Christo- pher A. Voigt. Ribozyme-based insulator parts buffer synthetic circuits from genetic context. Nat. Biotechnol., 30(11):1137–1142, 2012.

[94] Bethany Lusch, J. Nathan Kutz, and Steven L. Brunton. Deep learning for universal linear embeddings of nonlinear dynamics. Nature Communications, 9(1), 2018.

[95] Yudong Ma, Anthony Kelman, Allan Daly, and Francesco Borrelli. Predictive control for energy efficient buildings with thermal storage: Modeling, stimula- tion, and experiments. IEEE Control Systems, 32(1):44–64, February 2012.

[96] George A. Mackie. RNase E: at the interface of bacterial RNA processing and decay. Nat. Rev. Micro., 11(1):45–57, 2013.

[97] Eric Massé, Freddy E. Escorcia, and Susan Gottesman. Coupled degradation of a small regulatory RNA and its mRNA targets in Escherichia coli. Genes Dev., 17(19):2374–2383, 2003.

[98] William H. Mather, Jeff Hasty, Lev S. Tsimring, and Ruth J. Williams. Trans- lation cross talk in gene networks. Biophys. J., 104:2564–2572, 2013.

[99] Cameron McBride, Rushina Shah, and Domitilla Del Vecchio. The effect of loads in molecular communications. Proceedings of the IEEE, 107(7):1369–1386, 2019.

[100] Pankaj Mehta, Sidhartha Goyal, and Ned S. Wingreen. A quantitative comparison of sRNA-based and protein-based gene regulation. Mol. Syst. Biol., 4:221, 2008.

[101] Adam J. Meyer, Thomas H. Segall-Shapiro, Emerson Glassey, Jing Zhang, and Christopher A. Voigt. Escherichia coli “marionette” strains with 12 highly opti- mized small-molecule sensors. Nature Chemical Biology, 15(2):196–204, Novem- ber 2018.

196 [102] Ron Milo, Paul Jorgensen, Uri Moran, Griffin Weber, and Michael Springer. BioNumbers The database of key numbers in molecular and cell biology. Nu- cleic. Acids. Res., 38(SUPPL.1):750–753, 2009.

[103] Ron Milo and Rob Phillips. Cell Biology by the Numbers. Garland Science, 2015.

[104] Deepak Mishra, Phillip M Rivera, Allen Lin, Domitilla Del Vecchio, and Ron Weiss. A load driver device for engineering modularity in biological networks. Nat. Biotechnol., 32:1268–1275, 2014.

[105] J Monod, A.M Pappenheimer, and G Cohen-Bazire. La cinétique de la biosyn- thèse de la 훽-galactosidase chez e. coli considérée comme fonction de la crois- sance. Biochimica et Biophysica Acta, 9:648–660, 1952.

[106] Tae Seok Moon, Chunbo Lou, Alvin Tamsir, Brynne C. Stanton, and Christo- pher A. Voigt. Genetic programs constructed from layered logic gates in single cells. Nature, 491(7423):249–253, October 2012.

[107] P. Moylan. A connective stability result for interconnected passive systems. IEEE Transactions on Automatic Control, 25(4):812–813, 1980.

[108] P. Moylan and D. Hill. Stability criteria for large-scale systems. IEEE Trans- actions on Automatic Control, 23(2):143–149, 1978.

[109] Vivek K Mutalik, Joao C Guimaraes, Guillaume Cambray, Colin Lam, Marc Juul Christoffersen, Quynh-Anh Mai, Andrew B Tran, Morgan Paull, Jay D Keasling, Adam P Arkin, and Drew Endy. Precise and reliable gene expression via standard transcription and translation initiation elements. Nat. Methods, 10:354–360, 2013.

[110] Dokyun Na, Seung Min Yoo, Hannah Chung, Hyegwon Park, Jin Hwan Park, and Sang Yup Lee. Metabolic engineering of escherichia coli using synthetic small regulatory RNAs. Nature Biotechnology, 31(2):170–174, 2013.

[111] A. A. K. Nielsen, B. S. Der, J. Shin, P. Vaidyanathan, V. Paralanov, E. A. Strychalski, D. Ross, D. Densmore, and C. A. Voigt. Genetic circuit design automation. Science, 352(6281):aac7341–aac7341, 2016.

[112] Kayzad Soli Nilgiriwala, José I. Jiménez, Phillip Michael Rivera, and Domitilla Del Vecchio. Synthetic tunable amplifying buffer circuit in E. coli. ACS Synth. Biol., 4(5):577–584, 2015.

[113] Noah Olsman, Ania-Ariadna Baetica, Fangzhou Xiao, Yoke Peng Leong, Richard M. Murray, and John C. Doyle. Hard limits and performance tradeoffs in a class of antithetic integral feedback networks. Cell Systems, 9(1):49–63.e16, 2019.

197 [114] Hoo Hwi Park, Woon Ki Lim, and Hae Ja Shin. In vitro binding of purified NahR regulatory protein with promoter psal. Biochimica et Biophysica Acta (BBA) - General Subjects, 1725(2):247–255, September 2005.

[115] P. P. Peralta-Yahya, F. Zhang, S. B. del Cardayre, and J. D. Keasling. Microbial engineering for the production of advanced biofuels. Nature, 488:320–328, 2012.

[116] Arthur Prindle, Phillip Samayoa, Ivan Razinkov, Tal Danino, Lev S Tsimring, and Jeff Hasty. A sensing array of radically coupled genetic ’biopixels’. Nature, 481:39–44, 2012.

[117] Andrew Proud, Meir Pachter, and John D’Azzo. Close formation flight control. In Guidance, Navigation, and Control Conference and Exhibit. American Institute of Aeronautics and Astronautics, 1999.

[118] Priscilla E. M. Purnick and Ron Weiss. The second wave of synthetic biology: from modules to systems. Nat. Rev. Mol. Cell Biol., 10:410–422, 2009.

[119] Yili Qian, Theodore W. Grunberg, and Domitilla Del Vecchio. Multi-time-scale biomolecular ‘quasi-integral’ controllers for set-point regulation and trajectory tracking. In 2018 Annual American Control Conference (ACC). IEEE, June 2018.

[120] Yili Qian, Hsin-Ho Huang, José I. Jiménez, and Domitilla Del Vecchio. Re- source competition shapes the response of genetic circuits. ACS Synth. Biol., 6(7):1263–1272, 2017.

[121] Yili Qian, Cameron McBride, and Domitilla Del Vecchio. Programming cells to work for us. Annual Review of Control, Robotics, and Autonomous Systems, 1(1):411–440, 2018.

[122] Yili Qian and Domitilla Del Vecchio. Effective interaction graphs arising from resource limitations in gene networks. In 2015 American Control Conference (ACC). IEEE, July 2015.

[123] Yili Qian and Domitilla Del Vecchio. Mitigation of ribosome competition through distributed sRNA feedback. In 2016 IEEE 55th Conference on De- cision and Control (CDC). IEEE, 2016.

[124] Yili Qian and Domitilla Del Vecchio. The “power network” of genetic circuits. In Lecture Notes in Control and Information Sciences - Proceedings, pages 109– 121. Springer International Publishing, 2018.

[125] Yili Qian and Domitilla Del Vecchio. Realizing ‘integral control’ in living cells: how to overcome leaky integration due to dilution? Journal of The Royal Society Interface, 15(139):20170902, February 2018.

198 [126] Yili Qian and Domitilla Del Vecchio. A singular singular perturbation problem arising from a class of biomolecular feedback controllers. IEEE Control Systems Letters, 3(2):236–241, 2019.

[127] Alon Raveh, Michael Margaliot, Eduardo D Sontag, and Tamir Tuller. A model for competition for ribosomes in the cell. J. R. Soc. Interface, 13:20151062, 2015.

[128] Xinying Ren, Ania-Ariadna Baetica, Anandh Swaminathan, and Richard M Murray. Population regulation in microbial consortia using dual feedback control. In Proceedings of the 56th IEEE Conference on Decision and Control, Melbourne, Australia, 2017.

[129] Virgil A. Rhodius, Thomas H. Segall-Shapiro, Brain D. Sharon, Amar Gho- dasara, Ekaterina Orlova, Hannah Tabakh, David H. Burkhardt, Kevin Clancy, Todd C. Peterson, Carol A. Gross, and Christopher A. Voigt. Design of orthogonal genetic switches based on a crosstalk map of 휎s, anti-휎s, and promoters. Mol. Syst. Biol., 9(1):702–702, 2013.

[130] Yannick Rondelez. Competition for catalytic resources alters biological network dynamics. Phys. Rev. Lett., 108(1):018102, 2012.

[131] Nitzan Rosenfeld and Uri Alon. Response delays and the structure of transcription networks. J. Mol. Biol., 329(4):645–654, 2003.

[132] Nitzan Rosenfeld, Michael B Elowitz, and Uri Alon. Negative autoregulation speeds the response times of transcription networks. J. Mol. Biol., 323(5):785– 793, 2002.

[133] Walter Rudin. Principles of Mathematical Analysis. McGraw-Hill, 3rd edition, 1976.

[134] Howard M. Salis, Ethan A. Mirsky, and Christopher A. Voigt. Automated design of synthetic ribosome binding sites to control protein expression. Nat. Biotechnol., 27(10):946–50, 2009.

[135] Christian Cuba Samaniego and Elisa Franco. An ultrasensitive biomolecular network for robust feedback control. In Proceedings of the 20th World Congress of International Federation of Automatic Control (IFAC), pages 11438–11443, 2017.

[136] Pratik Saxena, Boon Chin Heng, Peng Bai, Marc Folcher, Henryk Zulewski, and Martin Fussenegger. A programmable synthetic lineage-control network that differentiates human IPSCs into glucose-sensitive insulin-secreting beta- like cells. Nature Communications, 7:11247, 2016.

[137] C.J. Schumacher and Rajeeva Kumar. Adaptive control of UAVs in close- coupled formation flight. In Proceedings of the 2000 American Control Con- ference. IEEE, 2000.

199 [138] Thomas H Segall-Shapiro, Adam J Meyer, Andrew D Ellington, Eduardo D Sontag, and Christopher A Voigt. A “resource allocator” for transcription based on a highly fragmented T7 RNA polymerase. Mol. Syst. Biol., 10:742, 2014.

[139] Elena Shchepakina, Vladimir Sobolev, and Michael P. Mortell. Singular singularly perturbed systems. In Singular Perturbations. Springer, 2014.

[140] Yishai Shimoni, Gilgi Friedlander, Guy Hetzroni, Gali Niv, Shoshy Altuvia, Ofer Biham, and Hanah Margalit. Regulation of gene expression by small non- coding RNAs: a quantitative view. Mol. Syst. Biol., 3, 2007.

[141] Tatenda Shopera, Lian He, Tolutola Oyetunde, Yinjie J. Tang, and Tae Seok Moon. Decoupling resource-coupled gene expression in living cells. ACS Synth. Biol., 6(8):1596–1604, 2017.

[142] Dan Siegal-gaskins, Zoltan A Tuza, Jongmin Kim, Vincent Noireaux, and Richard M Murray. Gene circuit performance characterization and resource usage in a cell- free ‘breadboard’. ACS Synth. Biol., 3:416–425, 2014.

[143] Dragoslav D. Siljak. Stability of large-scale systems under structural perturbations. IEEE Transactions on Systems, Man, and Cybernetics, SMC-2(5):657– 663, 1972.

[144] Jay Shankar Singh, P. C. Abhilash, H. B. Singh, Rana P. Singh, and D. P. Singh. Genetically engineered bacteria: An emerging tool for environmental remediation and future research perspectives. Gene, 480(1-2):1–9, 2011.

[145] Sahjendra N. Singh, Phil Chandler, Corey Schumacher, Siva Banda, and Meir Pachter. Nonlinear adaptive close formation control of unmanned aerial vehicles. Dynamics and Control, 10(2):179–194, 2000.

[146] Hal L. Smith. Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems. American Mathematical Society, 1995.

[147] H.L. Smith. Global stability for mixed monotone systems. Journal of Difference Equations and Applications, 14(10-11):1159–1164, 2008.

[148] Pramod R. Somvanshi, Anilkumar K. Patel, Sharad Bhartiya, and K. V. Venkatesh. Implementation of integral feedback control in biological systems. Wiley Interdiscip. Rev. Syst. Biol. Med., 7(5):301–316, 2015.

[149] Elisabeth Sonnleitner, Johanna Napetschnig, Taras Afonyushkin, Karin Ecker, Branislav Večerek, Isabella Moll, Vladimir R. Kaberdin, and Udo Bläsi. Func- tional effects of variants of the RNA chaperone Hfq. Biochem. Biophys. Res. Commun., 323(3):1017–1023, 2004.

[150] E.D. Sontag. A remark on the converging-input converging-state property. IEEE Transactions on Automatic Control, 48(2):313–314, 2003.

200 [151] E.D. Sontag and Yuan Wang. New characterizations of input-to-state stability. IEEE Transactions on Automatic Control, 41(9):1283–1294, 1996. [152] Eduardo D. Sontag. Adaptation and regulation with signal detection implies internal model. Syst. Control Lett., 50(2):119–126, 2003. [153] Eduardo D. Sontag. Monotone and near-monotone biochemical networks. Sys- tems and Synthetic Biology, 1(2):59–87, 2007. [154] Eduardo D. Sontag. Some remarks on a model for immune signal detection and feedback. In 2016 IEEE 55th Conference on Decision and Control (CDC). IEEE, 2016. [155] Aivar Sootla and Alexandre Mauroy. Operator-theoretic characterization of eventually monotone systems. IEEE Control Systems Letters, 2018. [156] G. W. Stewart and Ji-Guang Sun. Matrix Perturbation Theory. Academic Press, 1990. [157] Daniel M. Stoebel, Antony M. Dean, and Daniel E. Dykhuizen. The cost of expression of escherichia coli lac operon proteins is in the process, not in the products. Genetics, 178(3):1653–1660, 2008. [158] Gisela Storz, Jörg Vogel, and Karen M. Wassarman. Regulation by small RNAs in bacteria: Expanding frontiers. Mol. Cell, 43(6):880–891, 2011. [159] Stefan Streif, Kwang-Ki K. Kim, Philipp Rumschinski, Masako Kishida, Dongy- ing Erin Shen, Rolf Findeisen, and Richard D. Braatz. Robustness analysis, prediction, and estimation for uncertain biochemical networks: An overview. Journal of Process Control, 42:14–34, 2016. [160] Cheemeng Tan, Philippe Marguet, and Lingchong You. Emergent bistability by a growth-modulating positive feedback circuit. Nature Chemical Biology, 5(11):842–848, 2009. [161] Zhe F. Tang and David R. McMillen. Design principles for the analysis and construction of robustly homeostatic biological networks. J. Theor. Biol., 408:274– 289, 2016. [162] Jan Roelof van der Meer and Shimshon Belkin. Where microbiology meets microengineering: design and applications of reporter bacteria. Nature Reviews Microbiology, 8:511–522, 2010. [163] Domitilla Del Vecchio, Hussein Abdallah, Yili Qian, and James J. Collins. A blueprint for a synthetic genetic feedback controller to reprogram cell fate. Cell Systems, 4(1):109–120.e11, 2017. [164] Domitilla Del Vecchio, Aaron J. Dy, and Yili Qian. Control theory meets synthetic biology. Journal of The Royal Society Interface, 13(120):20160380, 2016.

201 [165] Domitilla Del Vecchio, Yili Qian, Richard M. Murray, and Eduardo D. Sontag. Future systems and control research in synthetic biology. Annual Reviews in Control, 45:5–17, 2018.

[166] A. C. Ventura, P. Jiang, L. Van Wassenhove, D. Del Vecchio, S. D. Merajver, and A. J. Ninfa. Signaling properties of a covalent modification cycle are altered by a downstream target. Proc. Natl. Acad. Sci. U.S.A., 107(22):10032–10037, 2010.

[167] Jesper Vind, Michael A. Sørensen, Michael D. Rasmussen, and Steen Pedersen. Synthesis of proteins in Escherichia coli is limited by the concentration of free ribosomes: Expression from reporter gene does not always reflect functional mRNA levels. J. Mol. Biol., 231:678–688, 1993.

[168] Mats Wallden, David Fange, Ebba Gregorsson Lundius, Özden Baltekin, and Johan Elf. The synchronization of replication and division cycles in individual e. coli cells. Cell, 166(3):729–739, 2016.

[169] Liming Wang and Eduardo D. Sontag. Singularly perturbed monotone systems and an application to double phosphorylation cycles. Journal of Nonlinear Science, 2008.

[170] Ping Wei, Wilson W. Wong, Jason S. Park, Ethan E. Corcoran, Sergio G. Peisajovich, James J. Onuffer, Arthur Weiss, and Wendell A. Lim. Bacterial virulence proteins as tools to rewire kinase pathways in yeast and immune cells. Nature, 488(7411):384–388, 2012.

[171] Min Wu, Ri-Qi Su, Xiaohui Li, Tom Ellisc, Ying-Cheng Lai, and Xiao Wang. Engineering of regulated stochastic cell fate determination. Proceedings of the National Academy of Sciences of the United States of America, 110(26):10610– 10615, 2013.

[172] Zhen Xie, Liliana Wroblewska, Laura Prochazka, Ron Weiss, and Yaakov Be- nenson. Multi-input RNAi-based logic circuit. Science, 333:1307–1312, 2011.

[173] Enoch Yeung, Aaron J. Dy, Kyle B. Martin, Andrew H. Ng, Domitilla Del Vecchio, James L. Beck, James J. Collins, and Richard M. Murray. Biophysical constraints arising from compositional context in synthetic gene networks. Cell Systems, 5(1):11–24.e12, 2017.

[174] Seung Min Yoo, Dokyun Na, and Sang Yup Lee. Design and use of synthetic regulatory small RNAs to control gene expression in Escherichia coli. Nat. Protocols, 8(9):1694–1707, 2013.

[175] L. You, R. S. Cox, R. Weiss, and F. H. Arnold. Programmed population control by cell-cell communication and regulated killing. Nature, 428(6985):868–871, 2004.

202 [176] Fuzhong Zhang, James M Carothers, and Jay D Keasling. Design of a dynamic sensor-regulator system for production of chemicals and fuels derived from fatty acids. Nature Biotechnology, 30(4):354–9, 2012.

[177] Xu Zhang and Yan Lin. Nonlinear decentralized control of large-scale systems with strong interconnections. Automatica, 50(9):2419–2423, September 2014.

203