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An ultrasensitive biomolecular network for robust feedback control
Proceedings of the 20th World Congress Proceedings of 20th The International Federation of Congress Automatic Control Proceedings of the the 20th World World Congress Proceedings of the 20th World Congress Control The of Toulouse, France,Federation July 9-14, 2017 The International International Federation of Automatic Automatic Control Available online at www.sciencedirect.com The International Federation of Automatic Control Toulouse, Toulouse, France, France, July July 9-14, 9-14, 2017 2017 Toulouse, France, July 9-14, 2017
ScienceDirect
An An An
IFAC PapersOnLine 50-1 (2017) 10950–10956
ultrasensitive biomolecular network ultrasensitive biomolecular network ultrasensitive biomolecular network robust feedback control robust feedback control robust feedback ∗control ∗
for for for
Christian Cuba Samaniego ∗ Elisa Franco ∗ ∗ Elisa Franco ∗ Christian Cuba Samaniego Christian Christian Cuba Cuba Samaniego Samaniego ∗ Elisa Elisa Franco Franco ∗ ∗ Mechanical Engineering, University of California, Riverside, ∗ ∗ Mechanical Engineering, University of California, Riverside, Engineering, University of California, Riverside, ∗ Mechanical Riverside, CA 92521 (ccuba002@ ucr.edu, [email protected]). Mechanical Engineering, University of California, Riverside, Riverside, CA 92521 (ccuba002@ ucr.edu, [email protected]). Riverside, CA 92521 (ccuba002@ ucr.edu, [email protected]). Riverside, CA 92521 (ccuba002@ ucr.edu, [email protected]). Abstract: We describe an ultrasensitive reaction network to achieve closed loop control Abstract: We describe an ultrasensitive reaction network to achieve closed loop control Abstract: We describe an reaction network closed control of biomolecular systems. This network is based on non-cooperative interactions of activator Abstract: We systems. describe This an ultrasensitive ultrasensitive reaction network to to achieve achieve closed loop loop control of biomolecular network is based on non-cooperative interactions of activator of biomolecular systems. This network is based on non-cooperative interactions of activator andbiomolecular inhibitor species thatThis regulate the iscontroller output level with interactions a mechanismofsimilar to of systems. network based on non-cooperative activator and inhibitor species that regulate the controller output level with a mechanism similar to and inhibitor species that regulate the controller output level with aa mechanism similar to protein phosphorylation. The controller reactions could potentially be realized using nucleic and inhibitor species that regulate the controller output level with mechanism similar to protein phosphorylation. The controller reactions could potentially be realized using nucleic protein phosphorylation. The controller reactions could potentially be realized using nucleic acids or monomeric proteins. Using numerical analysis we demonstrate that ultrasensitivity of protein phosphorylation. The controller reactions could potentially be realized using nucleic acids or monomeric proteins. Using numerical analysis we demonstrate that ultrasensitivity of acids or monomeric proteins. Using numerical analysis we demonstrate that ultrasensitivity of the controller output renders the closed loop system robust to perturbations in the system acids or monomeric proteins. numerical we demonstrate that ultrasensitivity of the controller output rendersUsing the closed closed loop analysis system robust robust to perturbations perturbations in the the system the controller parameters. the controller output output renders renders the the closed loop loop system system robust to to perturbations in in the system system parameters. parameters. parameters. © 2017, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Biomolecular controller, negative feedback, ultrasensitivity, chemical reactions, Keywords: controller, negative Keywords: Biomolecular Biomolecular controller,tracking. negative feedback, feedback, ultrasensitivity, ultrasensitivity, chemical chemical reactions, reactions, adaptation, disturbance rejection, Keywords: Biomolecular controller, negative feedback, ultrasensitivity, chemical reactions, adaptation, disturbance rejection, tracking. adaptation, disturbance rejection, tracking. adaptation, disturbance rejection, tracking. 1. INTRODUCTION 1. INTRODUCTION INTRODUCTION 1. 1. INTRODUCTION Fulfilling the promises of synthetic biology to be employed Fulfilling the promises and of synthetic synthetic biology to to be employed Fulfilling promises of biology employed for energy,the agriculture health applications will require Fulfilling the promises and of synthetic biology to be be employed for energy, agriculture health applications will require for energy, agriculture and health applications will require a precise control of regulation of many molecular processes forprecise energy, agriculture and health applications will require control of regulation regulation of many many molecular processes aa precise control of processes such as transcription, translation andmolecular post-translational a precise control of of regulation of many molecular processes such as transcription, translation and post-translational such as translation and post-translational post-translational modification. Biologicaltranslation cells have evolved feedback pathsuch as transcription, transcription, and modification. Biological cells have have of evolved feedback pathmodification. Biological cells evolved feedback ways so that the concentration proteins and pathsmall modification. Biological cells have of evolved feedback pathways so that the concentration proteins and small ways so that the concentration of proteins and small molecules remains within a desirable range;and negative ways so that the concentration of proteins small molecules remains within aahasdesirable desirable range; negative molecules remains within range; negative autoregulation, for instance, been found in more than molecules remains within ahasdesirable range; negative autoregulation, for instance, been found in more than autoregulation, for instance, has been found in more than half of genes in E coli (Thieffry et al. [1998]). Although autoregulation, for instance, has been found in more than half of genes in E coli (Thieffry et al. [1998]). Although half genes inaE coli (Thieffry et al. [1998]). Although thereof has beenin great deal of progress in understanding half of genes E coli (Thieffry et al. [1998]). Although there hasharness been aa feedback great deal dealtoof ofbuild progress in understanding understanding there has been great progress in how to various biomolecular there hasharness been a feedback great dealto ofbuild progress in understanding how to various biomolecular how to harness feedback to build various biomolecular circuits, including toggle switches, oscillators, and a varihow to harness feedback to build various biomolecular circuits, including toggle switches, oscillators, and varicircuits, including toggle switches, oscillators, and aaa variety of logic gates, it is still unclear how to design controller circuits, including toggle switches, oscillators, and variety of logic gates, it is still unclear how to design controller ety of logic it is still unclear to design controller circuits in agates, rational manner thathow is comparable to feedety of logic gates, it is still unclear how to design controller circuits in aa rational rational mannerorthat that is comparable comparable to feedfeedcircuits in manner is to back controllers in electrical mechanical systems. circuits in a rational mannerorthat is comparable to feedback controllers controllers in electrical electrical mechanical systems. back in or mechanical systems. back electrical mechanical systems. In thiscontrollers paper we in describe andoranalyze a molecular controlIn this paper we describe and analyze a molecular controlIn this paper we describe and analyze a molecular controller this network that relies on non-cooperative interactions In paper we describe and analyze a molecular controller network that relies relies on non-cooperative non-cooperative interactions ler network that on interactions between components. In other words, the regulatory mechler network that relies on non-cooperative interactions between components. In other words, the regulatory mechbetween components. In other words, the regulatory mechanisms require stoichiometric interactions, rather than between components. In other words, the regulatory mechanisms require stoichiometric interactions, rather than anisms require stoichiometric interactions, rather than multiple copies of any chemical species. The controller anisms require stoichiometric interactions, rather than multiple copies of any chemical species. The controller multiple copies species. controller network is used of to any steerchemical the output of a The target system multiple copies of any chemical species. The controller network is used to steer the output of a target system network is used used into toclosed steer loop, the output output of in target system to be controlled as shown Fig. 1 A. network is steer the of aa target system to be be controlled controlled in closed closed loop, as shown shown in Fig. 11 A. A. to in loop, as in Fig. to controlled in closed loop, as shown in Fig. 1 A. is Thebemost important feature of the controller we present The most state important feature of ofresponse the controller controller we present present is The most important feature the we is its steady ultrasensitive which has a tunable The most state important feature ofresponse the controller we present is its steady ultrasensitive which has a tunable its steady state ultrasensitive response has tunable threshold. The threshold is designed to which “track” theaa controlits steady state ultrasensitive response which has tunable threshold. Theinput, threshold is designed designed to “track” “track” the controlcontrolthreshold. The threshold is to ler reference so that the closed loop the equilibrium threshold. Theinput, threshold is designed to “track” the controller reference so that that the closed loop equilibrium equilibrium ler reference input, so the closed loop tracks the reference robustly. Ultrasensitivity is achieved ler reference input, robustly. so that the closed loop equilibrium tracks the reference reference Ultrasensitivity is achieved achieved tracks the robustly. Ultrasensitivity is by combining two mechanisms: molecular titration (Buchtracks the reference robustly. Ultrasensitivity is achieved by combining two mechanisms: molecular titration (Buchby combining two mechanisms: molecular titration (Buchler and Louis [2008], Cuba Samaniego et al. [2016b]), by combining two mechanisms: molecular titration (Buchler andthan Louis [2008], Cuba Samaniego et al. al. [2016b]), ler and Louis [2008], et rather cooperativity, andSamaniego an activation/deactivation ler andthan Louis [2008], Cuba Cuba Samaniego et al. [2016b]), [2016b]), rather cooperativity, and an activation/deactivation rather than and cycle of thecooperativity, controller variable, similar to a phosphorather than and an an activation/deactivation activation/deactivation cycle of thecooperativity, controller variable, variable, similar(Ferrell to aa phosphophosphocycle of the controller similar to rylation/dephosphorylation monocycle and Ha cycle of the controller variable, similar(Ferrell to a phosphorylation/dephosphorylation monocycle and and Ha rylation/dephosphorylation monocycle (Ferrell and Ha [2014]). Each of these mechanisms, taken in isolation rylation/dephosphorylation monocycle (Ferrell and and Ha [2014]). Each of these mechanisms, taken in isolation [2014]). Each of these mechanisms, taken in isolation and modeledEach withofmass action kinetics,taken are not sufficientand to [2014]). theseaction mechanisms, in isolation modeled with mass kinetics, are not sufficient to modeled with mass action kinetics, are not sufficient to provide high ultrasensitivity. Titration reactions facilitate modeled with mass action kinetics, are not sufficient to provide high ultrasensitivity. ultrasensitivity. Titration reactions facilitate provide high Titration reactions the tunability of the response threshold, and facilitate titration provide high ultrasensitivity. Titration reactions facilitate the tunability of the response threshold, and titration the tunability of the response threshold, and titration the tunability of the response threshold, and titration
r A rr A Ayr y y y
Biomolecular u u u Biomolecular u Biomolecular Biomolecular Controller Controller Controller Controller p
B B Bu u u u
+p +p +p +p
p p p
y y y y
Biomolecular Biomolecular Biomolecular Biomolecular u System u System u System u System
-p -p -p r -p rr r
y y y y
Fig. 1. Closed loop molecular network with an ultraFig. 1. Closed loop molecular network with ultraFig. Closed molecular network with an ultrasensitive controller A) Schematic thean interconFig. 1. 1. Closed loop loop molecular network of with an ultrasensitive controller A) Schematic of the interconsensitive controller A) Schematic of the interconnection of a biomolecular controller and a biomolecsensitive controller A) Schematic of the interconnection of a biomolecular controller and a biomolecnection of and ular system; p represents acontroller given parameter perturnection of aa biomolecular biomolecular and aa biomolecbiomolecular system; p represents aacontroller given parameter perturular system; p represents given parameter perturbation in the system. B) shows an illustration of ular system; p system. represents a shows given parameter perturbation in the B) an illustration of bation in the system. B) shows an illustration of input-output equilibrium conditions of illustration the controller bation in the system. B) shows an of input-output equilibrium conditions of the controller input-output equilibrium conditions of the (in grey) and the system (in orange). The controller input-output equilibrium conditions of the controller (in grey) and response the system (in orange). The controller (in grey) the (in The controller ultrasensitive threshold is tunable setting (in grey) and and response the system system (in orange). orange). The by controller ultrasensitive threshold is tunable by setting ultrasensitive response threshold is tunable by the reference signal r, shown in red. Ultrasensitivity of ultrasensitive response threshold is tunable by setting setting the reference signal r, shown in red. Ultrasensitivity of the reference signal r, shown in red. Ultrasensitivity of controller implies that, inin a certain regime, perturthe reference signal r, shown red. Ultrasensitivity of the controller implies that, in aa certain regime, perturthe controller implies that, in certain regime, perturbations in the input/output curve of the systems have the controller implies that, in a certain regime, perturbations in the input/output curve ofpoint. the systems have bations in curve the minor influence on the equilibrium bations in the the input/output input/output curve of ofpoint. the systems systems have have minor influence on the equilibrium minor influence on the equilibrium point. minor influence on the equilibrium point. together with the activation/deactivation cycle mechanism together the activation/deactivation cycle mechanism together with the activation/deactivation cycle mechanism creates awith highly input/output This together with the ultrasensitive activation/deactivation cyclecurve. mechanism creates a highly ultrasensitive input/output curve. This creates a highly ultrasensitive input/output curve. This ultrasensitive behavior is very similar to zero-order ultracreates a highly ultrasensitive input/output curve.ultraThis ultrasensitive behavior is very similar to zero-order ultrasensitive behavior is very similar to zero-order ultrasensitivity when enzymes are saturable and their activities ultrasensitive behavior is very similar to zero-order ultrasensitivity when enzymes are saturable and their activities sensitivity when enzymes and their are given by Michaelis-Menten equations and Ha sensitivity when enzymes are are saturable saturable and(Ferrell their activities activities are given by Michaelis-Menten equations (Ferrell and Ha are given by Michaelis-Menten equations (Ferrell and Ha [2014]. The controller reaction network is directly impleare given bycontroller Michaelis-Menten equations (Ferrell and Ha [2014]. The reaction network is directly imple[2014]. The controller reaction network is directly implementable using in vitro synthetic transcriptional systems, [2014]. The controller reaction network is directly implementable using in in vitro vitro synthetic synthetic transcriptional systems, mentable using such as negative systemstranscriptional (Franco et al.systems, [2014]) mentable using infeedback vitro synthetic transcriptional systems, such as negative feedback systems (Franco et al. [2014]) such as negative feedback systems (Franco et al. [2014]) to balance production rates. The module, taken as a such as negative feedback systems (Franco et al. [2014]) to balance production rates. The module, taken as a to balance production rates. The module, taken as subsystem in a larger network, can also be used to build to balance inproduction rates. The module, taken as a a subsystem a larger network, can also be used to build subsystem in aa larger network, can also to build oscillatory systems (Cuba Samaniego et be al. used [2016a]). We subsystem in larger network, can also be used to build oscillatory systems (Cuba Samaniego et et al. [2016a]). We oscillatory (Cuba Samaniego [2016a]). We exploit the systems controller ultrasensitivity to al. tune the closed oscillatory systems (Cuba Samaniego et al. [2016a]). We exploit the controller ultrasensitivity to tune the closed exploit the controller ultrasensitivity to tune the closed loop equilibrium point, and to achieve robustness relative exploit the controller ultrasensitivity to tune the closed loop equilibrium point, and to relative loop equilibrium point, achieve robustness relative to parameter perturbations, as achieve sketchedrobustness in Fig. 1 B. loop equilibrium point, and and to to achieve robustness relative to parameter perturbations, as sketched in Fig. 1 B. to parameter perturbations, as sketched in Fig. 1 B. to parameter perturbations, as sketched in Fig. 1 B.
Copyright © 2017, 2017 IFAC 11437 2405-8963 © IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Copyright © 2017 IFAC 11437 Copyright ©under 2017 responsibility IFAC 11437 Peer review of International Federation of Automatic Control. Copyright © 2017 IFAC 11437 10.1016/j.ifacol.2017.08.2466
Proceedings of the 20th IFAC World Congress Toulouse, France, July 9-14, 2017 Christian Cuba Samaniego et al. / IFAC PapersOnLine 50-1 (2017) 10950–10956
Titration reactions in vivo without degradation were used recently by Briat et al. [2016] to propose an antithetic integral feedback controller, whose reactions are analogous to the first stage of our controller. Our network differs from the anthitetic controller for two reasons: first, it includes an additional activation/deactivation cycle for the controller molecule; second, we take into account degradation (or dilution) reactions that are known to be present in most biochemical and cellular processes. Our simulations show that large degradation reduces ultrasensitivity of the controller, and this can therefore disrupt its tracking and adaptation performance (Del Vecchio et al. [2016]). Montefusco et al. [2016] found that an ultrasensitive network can model osmoregulation behaviors better than proportional and/or integral networks; this suggests that natural circuits may have evolved ultrasensitivity to robustify equilibria as shown in this paper. In Section 2 we describe the controller network in isolation, and we characterize its ultrasensitive behavior. In Section 3 we describe the target system to be controlled. Finally, in Section 4 we consider the closed loop system and examine the capacity of the system output to track a reference, as well as its sensitivity with respect to perturbations in the reaction rates and concentration of components. We discuss the reactions chosen to design the controller in relation to a potential in vitro implementation with synthetic transcriptional networks Kim et al. [2006]. Throughout the paper we consider biomolecular systems that are described by chemical reactions. Species are indicated as uppercase letters, concentrations as lowercase letters (e.g. species X has concentration x). Given a set of chemical reactions, we use the law of mass action to derive ordinary differential equation (ODE) models that describe the kinetics of the ensemble of reactions.
2. DESIGN OF AN ULTRASENSITIVE MOLECULAR CONTROLLER
The controller network (Fig. 2A) consists of a single output species U and two input species, an activator A and an inhibitor I. The output species can be in active form U , or inactive form U ∗ , and its total concentration is conserved (utot = u + u∗ ). Active U is assumed to have a regulatory function in downstream reactions, which cannot be performed by U ∗ . The inputs A and I respectively produce species RA and RI , which regulate the active fraction of output U . RI binds to and deactivates U , forming an inert species U ∗ . In contrast, species RA binds to and reactivates U ∗ by displacing RI from U ∗ , yielding a waste complex RA · RI . In addition, free RA and RI bind and mutually titrate forming waste complex RA · RI . The biomolecular reactions constituting our controller network are:
κ
A −−c A + RA θc
I −− I + RI
10951
Production Production
γc
RA + RI −− RA · RI
Titration
RA −− 0
Degradation
φc
φc
RI −− 0 ∗
Degradation αc
RA + U −− U + RA · RI
Reactivation
RI + U −− U
Inhibition
βc
∗
Using the law of mass action, we derive the following ODEs that describe the dynamics of the system: u˙ = αc rA u∗ − βc rI u (1) r˙A = κc a − αc rA u∗ − γc rA rI − φc rA (2) (3) r˙I = θc i − βc rI u − γc rA rI − φc rI Fig. 2 B and C illustrate the desired interactions of species U , RA , and RI , and how they result in an ultrasensitive response. First, consider the case where the system receives a step input in i. Then, active species u is expected to be quickly converted to its inactive form u∗ . However, the presence of a (which produces rA ) has a “buffering” effect: free rA binds to rI and titrates it before it can inhibit u (this of course requires that γc is fast). Due to this buffering effect, we can think of i as the input to the controller, and of the titrating species a as a “reference” signal: the controller responds only if i exceeds a sufficiently. A similar reasoning can be followed to explain the behavior of the system when the system receives a step input a: in this case, free rI acts like a buffer for rA . We can think of a as the input to the controller, and of the titrating species i as the reference; only when a exceeds i sufficiently, the controller responds. In both cases, the transition point (at which the concentration of u switches from low to high and vice-versa) depends on the concentration of the titrating species. This reaction network could be experimentally realized using nucleic acids and proteins. Species U could be an RNA polymerase whose activity can repressed using an RNA aptamers (RI ) as those proposed by Mori et al. [2012], Ohuchi et al. [2012] (species RI ). RNA polymerase activity could be restored via another RNA species (RA ) displacing the aptamer as suggested in Cuba Samaniego et al. [2016a]. Synthetic templates (A, I) could be used to produce the RNA regulators. Numerical integration of equations (1)–(3) is done assuming nominal values of parameters (Table 1) that are realistic in the implementation context just described. In the next sections we examine the behavior of the controller network. We derive its steady state input/output response analytically, and then we test numerically the sensitivity of the controller response to parameter perturbations. 2.1 Boundedness and monotonicity The solutions of the biomolecular controller model are bounded. After a state transformation, the Jacobian of the system is a Metlzer matrix, which is Hurwitz stable, and we can conclude that the input-output equilibrium maps of u(A) and u(I) are monotonically increasing and
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Proceedings of the 20th IFAC World Congress 10952 Toulouse, France, July 9-14, 2017 Christian Cuba Samaniego et al. / IFAC PapersOnLine 50-1 (2017) 10950–10956
A A A
A
RA
o
I
RI
o
C
I
RA
U*
o
II
B A
Biomolecular Controller U
U
U
C
S
A I
U
C
S
u
u
RI
ar
ir
i
a
Fig. 2. Ultrasensitive network. A) Schematic summary of the reactions defining the controller network. B) Controller network within a feedback loop, where signal A is kept constant and represents a reference signal, shown in red. C) Controller in a feedback loop where signal A is kept constant and acts as a reference, shown in red. The thresholds ar and is are functions of the reference a and i (in red) and of the parameters of the system. decreasing. As a consequence, also their inverse maps (which will be used later in the manuscript) are monotonic functions. These claims can be demonstrated following the analysis in Cuba Samaniego et al. [2016a], which are not reported here.
1
1 10 c 1 c 0.1 c
0 0
1
0 0
2.2 Input-output characterization We begin by finding an expression for the concentration of input A as a function of the concentrations of input I and output U . (These expressions are more convenient to find relative to deriving the concentration of U as a function of A and I). a(u, i) =
θc φc i+ κc κc
where
1−
α c u∗ βc u
r¯A (u, i),
(4)
b2A − 4aA cA , 2aA aA = αc γc u∗ , bA = αc u∗ (βc u + φc ) and cA = −uβc θc I. r¯A (u, i) =
−bA +
Similarly, we find an expression relating the equilibrium concentration of I as a function of U and A. i(u, a) =
where
φc κc a− θc θc
1
βc u − 1 r¯I (u, a), αc u∗
(5)
1
2
0
2
1
1 10 c 1 c 0.1 c
0
2
1
c
0 0
1 10 c 1 c 0.1 c
10 c 1 c 0.1
c
0
2
1
1 10 c 1 c 0.1
0
1
2
10 c 1 c 0.1 c 0 0
1
2
1 10i 1i 0.1i
10utot 1utot tot
0.1u
0 0
1
0 2
0
1
2
Fig. 3. Controller ultrasensistive response. Inputoutput equilibrium conditions from equation (4) as parameters are varied. Nominal simulation parameters are listed in Table 1. Normalization of axis are given by an = a(u, i)/ir and un = u/utot . behavior of the input-output expression (4) is evaluated as parameters are individually varied. We note that curves are very sensitive to the inhibition rate (βc ), the activation rate (αc ), the degradation rate (φc ) and total amount of input (utot ). The input/output curve steepness increases when these parameters are large, with exception for the degradation rate (φc ), which largely reduces the ultrasensitivity. The last panel of Fig. 3 shows the influence of variations of i on the input-output response. Because of how the variables are normalized, the curve transition always occurs at an = 1: this indicates that the output equilibrium always perfectly tracks the desired threshold, which can be seen as a “reference” in the closed loop system, as we further explain in the next sections. (A non-normalized plot of the input-output response is in Fig. 6 B, C, D.)
−bI + b2I − 4aI cI , r¯I (u, a) = 2aI aI = βc γc u, bI = βc u(αc u∗ + φc ) and cI = −αc u∗ κc A.
2.3 Ultrasensitivity characterization To assess the ultrasensitivity of the input/output curves with respect to parameter variations, we vary individual parameters of the network. We focus on plotting expression (4): we fix the concentration of i, and then evaluate r¯A (u, i) when u varies between 0 and utot ; this allows us to evaluate the map a(u, i) (x axis) versus u (y axis) as shown in Fig. 3. We note that if the concentration of i is fixed, we can think of term ir = κθcc i as the threshold of the ultrasensitive response. The plots in Fig. 3 were obtained using normalized concentrations of species an = a(u, i)/ir , and un = u/utot . The
It is possible to plot (5) by defining similar threshold parameters and normalized variables, and we obtain similar results (not shown for brevity), illustrated on Fig. 2B. 3. CONTROLLED SYSTEM We consider an example transcription process as a target system to be controlled. Specifically, we focus on a class of synthetic in vitro genetic switches which have been used to build a variety of autonomous circuits including bistable and oscillatory networks Kim et al. [2006], Kim and Winfree [2011], Franco et al. [2011].
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Table 1. Nominal simulation parameters of the controller Description
Value
θc (/s) κc (/s) βc (/M/s) αc (/M/s) γc (/M/s) φc (/s) utot (nM ) i(nM ) a(nM )
Production Production Inhibition Reactivation Titration Degradation Concentration Concentration Concentration
7.5 · 10−4 5 · 10−4 6 · 104 6 · 104 3 · 104 3.85 · 10−4 500 100 100
B
Biomolecular System
A u U
Parameter
Z
u U
Y
Y* W*
W
Transcription: 10−3 − 1 Vogel and Jensen [1994], Chen et al. [2015] 104 − 106 Kim et al. [2006], Zhang et al. [2007] 104 − 106 Kim et al. [2006], Zhang et al. [2007] 104 − 106 Kim et al. [2006], Zhang et al. [2007] 10−4 − 10−3 Kim et al. [2014].
u(y) =
y
W
o
W*
Other studies
Z
yY
ytot
0 w -y tot
tot
y
tot
as =
z
Fig. 4. Biomolecular system or process A) Schematic of a simplified model of a synthetic gene switch with input u and output y. B) An illustration of input/output response with input z and output y. The target system receives a single input species U and produces an output species Y . As in Kim et al. [2006], Y is a synthetic template (genelet) that is activated by a DNA activator molecule W , and deactivated by an RNA inhibitor molecule Z. The input U could be an RNA polymerase transcribing inhibitor Z from a constitutively active template; Z is designed to bind to Y and convert it to inactive Y ∗ (this occurs by displacement of the activator W ). In addition, Z directly binds to and titrates W converting it to inactive species W ∗ . We assume that inhibited activator W ∗ spontaneously reverts to its active form W . The total concentrations of Y and W are assumed to be constant, y tot = y + y ∗ and wtot = w + y + w∗ (activation of y ∗ requires binding of one copy of w). These regulatory interactions are illustrated in Fig. 4A, and the corresponding set of chemical reactions is: κ
U −−s U + Z γs
Z + W −− W
Production ∗
Titration
φs
Z −− 0
(9)
where
αs y ∗ w, z= βs y
α s γs y ∗ βs y ,
bs = αs y ∗ + θs and cs = −θs (wtot − y).
w=
−bs +
b2s − 4as cs 2as
In the next sections, we numerically integrate the system ODEs using the nominal parameters listed in Table 2, which are realistic values relative to the proposed circuit implementation using transcriptional circuits (Kim et al. [2006]). 4. CLOSED LOOP SYSTEM We now analyze the performance of the closed loop system where controller and system are interconnected. The goal is to control the concentration of active template Y as shown in Fig.5, following a reference signal R. Because the input/output behavior of this particular system is monotone decreasing (Fig. 4), we operate our controller so that the feedback loop counteracts the system response. So we connect variable Y in the feedback loop to serve as input A (activator) for the controller. The controller input I is now considered the reference signal R. We note that if the system to be controlled had a monotonically increasing input-output map, we would have operated the controller in the opposite way, so that input A would be the reference. This suggests that this controller architecture can be adapted to work with systems having very different input-output behaviors; a theoretical characterization of this property is left for future work.
Degradation
θs
W ∗ −− W ∗
θ s w ∗ + φs z , κs
αs
Recovery ∗
W + Y −− W + Y βs
Y + Z −− Y ∗
Biomolecular Controller
Activation Y
Inhibition
Using the law of mass action we derive the following ODEs: y˙ = αs wy ∗ − βs yz w˙ = θs w∗ − αs wy ∗ − γs zw z˙ = κs u − βs yz − γs zw − φs z
(6) (7) (8)
In earlier work (Cuba Samaniego et al. [2016b]) we demonstrated analytically that the solutions of ODEs (6)–(8) are bounded, that the system is structurally stable and monotone, and the input-output map is monotonically decreasing. We find the equilibrium values of u and y are related according to the following expression:
Ry
o
R
R
Ry
U*
o
Rr
o
Biomolecular System U U
Rr
Z
o
W
Y*
W* W
W*
Y
Y
Z
Fig. 5. Feedback control system. Closed loop system between the biomolecular controller and biomolecular system. The set point, output and actuator are r, y and u respectively. We combine equations (1)–(3) (controller) and equations (6)–(8) (controlled system) and obtain the closed loop system equations:
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Proceedings of the 20th IFAC World Congress 10954 Toulouse, France, July 9-14, 2017 Christian Cuba Samaniego et al. / IFAC PapersOnLine 50-1 (2017) 10950–10956
Table 2. Nominal simulation parameters of the controlled system
C
Rate
Description
Value
Other studies
κs (/s) θs (/s) βs (/M/s) αs (/M/s) γs (/M/s) φs (/s) y tot (nM ) wtot (nM )
Production Recovery Inhibition Reactivation Titration Degradation Concentration Concentration
5 · 10−4 3 · 10−4 6 · 104 6 · 104 3 · 104 3.85 · 10−4 800 800
Transcription: 10−3 − 1 Vogel and Jensen [1994], Chen et al. [2015] 10−5 − 10−2 Chen et al. [2015], Beelman and Parker [1995] 104 − 106 Kim et al. [2006], Zhang et al. [2007] 104 − 106 Kim et al. [2006], Zhang et al. [2007] 104 − 106 Kim et al. [2006], Zhang et al. [2007] 10−4 − 10−3 Kim et al. [2014]
u˙ = αc ry u∗ − βc rr u controller output r˙y = −αc ry u∗ − γc ry rr − φc ry κc y
The system’s output y, shown in purple, closely tracks each reference input (dark gray line). Fig.6 B, C and D show the input-output equilibrium conditions of the controller and system at the corresponding reference value: the threshold of the ultrasensitive input/output controller map tracks the reference r¯. As noted earlier, if φc is too large and ultrasensitivity is lost, then equilibria computed in Fig. 6 B–D will not satisfy y ≈ r¯.
controller input r ˙ = θc r −βc rr u − γc ry rr − φc rr r external input y˙ = αs wy ∗ − βs yz system output w˙ = θs w∗ − αs wy ∗ − γs zw S κs u −βs yz − γs zw − φs z z˙ = system input
The system output is y, as annotated in the equations above, and its external input or reference input is r. We assume utot = u + u∗ , y tot = y + y ∗ and wtot = w + y + w∗ . The experimental implementation of these two reaction systems in isolation was discussed earlier. Their interconnection would be immediately feasible: transcriptional circuits and aptamer systems have been characterized in compatible in vitro conditions; the RNA polymerase U can be easily used to transcribe RNA species Z from a constitutively active template. 4.1 Reference tracking and disturbance rejection We now define a rescaled reference signal as r¯ = κθcc r; then, we define the error e = r¯ − y to quantify the difference between the output y and the rescaled reference r¯. We also define the controller “internal” error er = rr − ry . We then derive the following expression for e˙ r : e˙ r = r˙r − r˙y = θc r − κc y − φc er + u˙ = κc e − φc er + u. ˙ At steady state, e˙ r = 0 and u˙ = 0, and we find: κc e = φc er . This equation indicates that if φc = 0 (i.e. if there is no degradation in the system), then the steady state error e is exactly zero, therefore the output tracks the reference, y = r¯. If φc is sufficiently small, we expect that y ≈ r¯. If φc is large, then the output cannot track the reference well. This behavior is related to the fact that large φc induces a loss of ultrasensitivity in the controller module (as shown in simulations in Fig. 3). We now test the ability of y to track changes in r¯ using numerical simulations of the closed loop system. Fig. 6A shows the response of the output to step-increases in rescaled reference signal r¯ = 0.15, 0.3 and 0.45µM . The rescaled reference and the reference input are related as: r¯ = κθcc r = 1.5 r, given the nominal parameters in Tables 1 and 2.
Fig. 6 E shows the behavior of the output in the presence of a step-perturbation of transcription rate κs , given a constant reference r¯ = 0.15µM . The system converges to the reference input rejecting the disturbance; Fig.6 F, G and H show input-output equilibrium conditions of the controller and system for different disturbance values on κs . The system is able to reject those disturbance in the ultra sensitivity region. Gillespie simulations (light purple color in Figs. 6 A and E), Gillespie et al. [1977], suggest that the tracking and disturbance rejection of the closed loop system hold also in a stochastic setting. In these single stochastic simulations, initial conditions are set to zero. We consider 100 molecules at 100nM concentration in a 1f L volume. 4.2 Sensitivity analysis 1. Sensitivity to variations of system parameters: When the equilibrium of the closed loop system is in the ultrasensitive regime of the controller response, the system transient and stationary responses are robust with respect to parameter variations of the system to be controlled. Numerical sensitivity analysis is shown in Fig. 7. The closed loop system transient (Fig. 7 A) and equilibria (Fig. 7 B) are robust to most parameter variations spanning 0.5 to 2 times the nominal values listed on Table 2. We note that the controller can “saturate” in a certain range of the production rate of Z (κs ), the recovery rate of W (θs ), the total concentration of template (y tot ) and total concentration of activator (wtot ). Large θs or wtot increase free w, which can bind faster to y ∗ . To balance this process, the system increases the demand of controller species u in order to inhibit w. This can cause the controller to reach a saturation point and fail. If y tot is much lower than the reference value, the controller may fail to supply the system’s demand of u because there is not enough free y available. (Note that Y is not depleted by the controller reactions.) 2. Sensitivity to variations of controller parameters: We test how changes in the parameters of the biomolecular controller affect the closed loop system behavior. Fig. 3
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Fig. 6. Tracking and adaptation A) The output y of the closed loop system (purple) tracks three different references (¯ r = 0.15, 0.3 and 0.45µM , dark gray). B,C and D) show equilibrium conditions as they change as a function of reference changes, where grey corresponds to the controller (4), and orange corresponds to the system (9). E) Adaptation of the system output in the presence of a step perturbation of the production rate (κs ) applied at 15 and 30h, where the final values are two and four times its nominal value listed on Table 2.
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Fig. 7. Rejection of system parametric disturbances A) Output y is shown as parameter values vary from 0.5 to 2 times their nominal values listed on Table 1 (yellow to red). B) Input-output equilibrium conditions of the controller (grey) and system (color scale), as the same parameters are varied.
Fig. 8. Controller robustness Top shows y trajectories for different parameter values that varies from 0.5 to 2 times its nominal values listed on Table 1 (from gray to black). Bottom shows input-output equilibrium condition of the controller (grey scale) and system (orange).
and Fig. 8 show that the system is generally very robust to changes in controller parameters. An exception is parameter utot , which defines the upper bound of the controller’s output. By increasing this parameter we improve the robustness of the ultrasensitive response of the controller. In contrast, if utot is reduced, the controller may saturate and fail to supply enough input U to the system.
To create a highly ultrasensitive behavior we combine two mechanisms: 1) titration and 2) an activation and inhibition cycle. Titration is important for tuning the threshold of the ultrasensitive response and both mechanisms are important to raise the highly ultrasensitive response.
5. DISCUSSION We presented a biomolecular controller that operates via titration reactions, and showed its closed loop performance when the system to be controlled is an RNA transcription process.
We suggest that this ultrasensitive controller overcomes the challenge of designing an integral controller that takes into account degradation and dilution processes Briat et al. [2016]. We believe that our studies could be useful to guide experimental implementation of a variety of molecular controllers, beyond the in vitro implementation that we suggest here. Ultrasensitivity may be obtained using mechanisms other than molecular sequestration; for instance, Goldbeter and Koshland [1984] showed that phosphoryla-
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tion cycles have an ultra sensitive response. Many posttranslational modification pathways such as phosphorylation, glycosylation, acetylation and ubiquitination could be used to design an ultrasensitive feedback controller for tracking and disturbance rejection. ACKNOWLEDGEMENTS This research was partially supported by a ThinkSwiss fellowship to CCS; by the US National Science Foundation under grant CMMI 1266402, which provided funding for CCS and EF. EF was also partially supported by the Department of Energy under grant DE-SC0010595. CCS finalized this research as a visiting scholar at ETH-Zurich: we thank Corentin Briat and Mustafa Khammash for kindly hosting CCS, for helpful discussions and support. REFERENCES Beelman, C.A. and Parker, R. (1995). Degradation of mrna in eukaryotes. Cell, 81(2), 179–183. Briat, C., Gupta, A., and Khammash, M. (2016). Antithetic integral feedback ensures robust perfect adaptation in noisy bimolecular networks. Cell systems, 2(1), 15–26. Buchler, N.E. and Louis, M. (2008). Molecular titration and ultrasensitivity in regulatory networks. Journal of molecular biology, 384(5), 1106–1119. Chen, H., Shiroguchi, K., Ge, H., and Xie, X.S. (2015). Genome-wide study of mRNA degradation and transcript elongation in escherichia coli. Molecular systems biology, 11(1), 781. Cuba Samaniego, C., Giordano, G., Blanchini, F., and Franco, E. (2016a). Stability analysis of an artificial biomolecular oscillator with non-cooperative regulatory interactions. Journal of Biological Dynamics, xxx. Cuba Samaniego, C., Giordano, G., Kim, J., Blanchini, F., and Franco, E. (2016b). Molecular titration promotes oscillations and bistability in minimal network models with monomeric regulators. ACS synthetic biology, 5(4), 321–333. Del Vecchio, D., Dy, A.J., and Qian, Y. (2016). Control theory meets synthetic biology. Journal of The Royal Society Interface, 13(120), 20160380. Ferrell, J.E. and Ha, S.H. (2014). Ultrasensitivity part i: Michaelian responses and zero-order ultrasensitivity. Trends in biochemical sciences, 39(10), 496–503. Franco, E., Friedrichs, E., Kim, J., Jungmann, R., Murray, R., Winfree, E., and Simmel, F.C. (2011). Timing molecular motion and production with a synthetic transcriptional clock. Proceedings of the National Academy of Sciences, 108(40), E784–E793. Franco, E., Giordano, G., Forsberg, P.O., and Murray, R.M. (2014). Negative autoregulation matches production and demand in synthetic transcriptional networks. ACS synthetic biology, 3(8), 589–599. Gillespie, D.T. et al. (1977). Exact stochastic simulation of coupled chemical reactions. J. phys. Chem, 81(25), 2340–2361. Goldbeter, A. and Koshland, D. (1984). Ultrasensitivity in biochemical systems controlled by covalent modification. interplay between zero-order and multistep effects. Journal of Biological Chemistry, 259(23), 14441–14447.
Kim, J., Khetarpal, I., Sen, S., and Murray, R.M. (2014). Synthetic circuit for exact adaptation and fold-change detection. Nucleic acids research, gku233. Kim, J., White, K.S., and Winfree, E. (2006). Construction of an in vitro bistable circuit from synthetic transcriptional switches. Molecular systems biology, 2(1), 68. Kim, J. and Winfree, E. (2011). Synthetic in vitro transcriptional oscillators. Molecular systems biology, 7(1), 465. Montefusco, F., Akman, O.E., Soyer, O.S., and Bates, D.G. (2016). Ultrasensitive negative feedback control: a natural approach for the design of synthetic controllers. PLoS One, 11(8), e0161605. Mori, Y., Nakamura, Y., and Ohuchi, S. (2012). Inhibitory RNA aptamer against SP6 RNA polymerase. Biochemical and Biophysical Research Communications, 420(2), 440–443. Ohuchi, S., Mori, Y., and Nakamura, Y. (2012). Evolution of an inhibitory RNA aptamer against T7 RNA polymerase. FEBS open bio. Thieffry, D., Huerta, A.M., P´erez-Rueda, E., and ColladoVides, J. (1998). From specific gene regulation to genomic networks: a global analysis of transcriptional regulation in Escherichia coli. Bioessays, 20(5), 433– 440. Vogel, U. and Jensen, K.F. (1994). The RNA chain elongation rate in escherichia coli depends on the growth rate. Journal of bacteriology, 176(10), 2807–2813. Zhang, D.Y., Turberfield, A.J., Yurke, B., and Winfree, E. (2007). Engineering entropy-driven reactions and networks catalyzed by dna. Science, 318(5853), 1121– 1125.
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ultrasensitive biomolecular network ultrasensitive biomolecular network ultrasensitive biomolecular network robust feedback control robust feedback control robust feedback ∗control ∗
for for for
Christian Cuba Samaniego ∗ Elisa Franco ∗ ∗ Elisa Franco ∗ Christian Cuba Samaniego Christian Christian Cuba Cuba Samaniego Samaniego ∗ Elisa Elisa Franco Franco ∗ ∗ Mechanical Engineering, University of California, Riverside, ∗ ∗ Mechanical Engineering, University of California, Riverside, Engineering, University of California, Riverside, ∗ Mechanical Riverside, CA 92521 (ccuba002@ ucr.edu, [email protected]). Mechanical Engineering, University of California, Riverside, Riverside, CA 92521 (ccuba002@ ucr.edu, [email protected]). Riverside, CA 92521 (ccuba002@ ucr.edu, [email protected]). Riverside, CA 92521 (ccuba002@ ucr.edu, [email protected]). Abstract: We describe an ultrasensitive reaction network to achieve closed loop control Abstract: We describe an ultrasensitive reaction network to achieve closed loop control Abstract: We describe an reaction network closed control of biomolecular systems. This network is based on non-cooperative interactions of activator Abstract: We systems. describe This an ultrasensitive ultrasensitive reaction network to to achieve achieve closed loop loop control of biomolecular network is based on non-cooperative interactions of activator of biomolecular systems. This network is based on non-cooperative interactions of activator andbiomolecular inhibitor species thatThis regulate the iscontroller output level with interactions a mechanismofsimilar to of systems. network based on non-cooperative activator and inhibitor species that regulate the controller output level with a mechanism similar to and inhibitor species that regulate the controller output level with aa mechanism similar to protein phosphorylation. The controller reactions could potentially be realized using nucleic and inhibitor species that regulate the controller output level with mechanism similar to protein phosphorylation. The controller reactions could potentially be realized using nucleic protein phosphorylation. The controller reactions could potentially be realized using nucleic acids or monomeric proteins. Using numerical analysis we demonstrate that ultrasensitivity of protein phosphorylation. The controller reactions could potentially be realized using nucleic acids or monomeric proteins. Using numerical analysis we demonstrate that ultrasensitivity of acids or monomeric proteins. Using numerical analysis we demonstrate that ultrasensitivity of the controller output renders the closed loop system robust to perturbations in the system acids or monomeric proteins. numerical we demonstrate that ultrasensitivity of the controller output rendersUsing the closed closed loop analysis system robust robust to perturbations perturbations in the the system the controller parameters. the controller output output renders renders the the closed loop loop system system robust to to perturbations in in the system system parameters. parameters. parameters. © 2017, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Biomolecular controller, negative feedback, ultrasensitivity, chemical reactions, Keywords: controller, negative Keywords: Biomolecular Biomolecular controller,tracking. negative feedback, feedback, ultrasensitivity, ultrasensitivity, chemical chemical reactions, reactions, adaptation, disturbance rejection, Keywords: Biomolecular controller, negative feedback, ultrasensitivity, chemical reactions, adaptation, disturbance rejection, tracking. adaptation, disturbance rejection, tracking. adaptation, disturbance rejection, tracking. 1. INTRODUCTION 1. INTRODUCTION INTRODUCTION 1. 1. INTRODUCTION Fulfilling the promises of synthetic biology to be employed Fulfilling the promises and of synthetic synthetic biology to to be employed Fulfilling promises of biology employed for energy,the agriculture health applications will require Fulfilling the promises and of synthetic biology to be be employed for energy, agriculture health applications will require for energy, agriculture and health applications will require a precise control of regulation of many molecular processes forprecise energy, agriculture and health applications will require control of regulation regulation of many many molecular processes aa precise control of processes such as transcription, translation andmolecular post-translational a precise control of of regulation of many molecular processes such as transcription, translation and post-translational such as translation and post-translational post-translational modification. Biologicaltranslation cells have evolved feedback pathsuch as transcription, transcription, and modification. Biological cells have have of evolved feedback pathmodification. Biological cells evolved feedback ways so that the concentration proteins and pathsmall modification. Biological cells have of evolved feedback pathways so that the concentration proteins and small ways so that the concentration of proteins and small molecules remains within a desirable range;and negative ways so that the concentration of proteins small molecules remains within aahasdesirable desirable range; negative molecules remains within range; negative autoregulation, for instance, been found in more than molecules remains within ahasdesirable range; negative autoregulation, for instance, been found in more than autoregulation, for instance, has been found in more than half of genes in E coli (Thieffry et al. [1998]). Although autoregulation, for instance, has been found in more than half of genes in E coli (Thieffry et al. [1998]). Although half genes inaE coli (Thieffry et al. [1998]). Although thereof has beenin great deal of progress in understanding half of genes E coli (Thieffry et al. [1998]). Although there hasharness been aa feedback great deal dealtoof ofbuild progress in understanding understanding there has been great progress in how to various biomolecular there hasharness been a feedback great dealto ofbuild progress in understanding how to various biomolecular how to harness feedback to build various biomolecular circuits, including toggle switches, oscillators, and a varihow to harness feedback to build various biomolecular circuits, including toggle switches, oscillators, and varicircuits, including toggle switches, oscillators, and aaa variety of logic gates, it is still unclear how to design controller circuits, including toggle switches, oscillators, and variety of logic gates, it is still unclear how to design controller ety of logic it is still unclear to design controller circuits in agates, rational manner thathow is comparable to feedety of logic gates, it is still unclear how to design controller circuits in aa rational rational mannerorthat that is comparable comparable to feedfeedcircuits in manner is to back controllers in electrical mechanical systems. circuits in a rational mannerorthat is comparable to feedback controllers controllers in electrical electrical mechanical systems. back in or mechanical systems. back electrical mechanical systems. In thiscontrollers paper we in describe andoranalyze a molecular controlIn this paper we describe and analyze a molecular controlIn this paper we describe and analyze a molecular controller this network that relies on non-cooperative interactions In paper we describe and analyze a molecular controller network that relies relies on non-cooperative non-cooperative interactions ler network that on interactions between components. In other words, the regulatory mechler network that relies on non-cooperative interactions between components. In other words, the regulatory mechbetween components. In other words, the regulatory mechanisms require stoichiometric interactions, rather than between components. In other words, the regulatory mechanisms require stoichiometric interactions, rather than anisms require stoichiometric interactions, rather than multiple copies of any chemical species. The controller anisms require stoichiometric interactions, rather than multiple copies of any chemical species. The controller multiple copies species. controller network is used of to any steerchemical the output of a The target system multiple copies of any chemical species. The controller network is used to steer the output of a target system network is used used into toclosed steer loop, the output output of in target system to be controlled as shown Fig. 1 A. network is steer the of aa target system to be be controlled controlled in closed closed loop, as shown shown in Fig. 11 A. A. to in loop, as in Fig. to controlled in closed loop, as shown in Fig. 1 A. is Thebemost important feature of the controller we present The most state important feature of ofresponse the controller controller we present present is The most important feature the we is its steady ultrasensitive which has a tunable The most state important feature ofresponse the controller we present is its steady ultrasensitive which has a tunable its steady state ultrasensitive response has tunable threshold. The threshold is designed to which “track” theaa controlits steady state ultrasensitive response which has tunable threshold. Theinput, threshold is designed designed to “track” “track” the controlcontrolthreshold. The threshold is to ler reference so that the closed loop the equilibrium threshold. Theinput, threshold is designed to “track” the controller reference so that that the closed loop equilibrium equilibrium ler reference input, so the closed loop tracks the reference robustly. Ultrasensitivity is achieved ler reference input, robustly. so that the closed loop equilibrium tracks the reference reference Ultrasensitivity is achieved achieved tracks the robustly. Ultrasensitivity is by combining two mechanisms: molecular titration (Buchtracks the reference robustly. Ultrasensitivity is achieved by combining two mechanisms: molecular titration (Buchby combining two mechanisms: molecular titration (Buchler and Louis [2008], Cuba Samaniego et al. [2016b]), by combining two mechanisms: molecular titration (Buchler andthan Louis [2008], Cuba Samaniego et al. al. [2016b]), ler and Louis [2008], et rather cooperativity, andSamaniego an activation/deactivation ler andthan Louis [2008], Cuba Cuba Samaniego et al. [2016b]), [2016b]), rather cooperativity, and an activation/deactivation rather than and cycle of thecooperativity, controller variable, similar to a phosphorather than and an an activation/deactivation activation/deactivation cycle of thecooperativity, controller variable, variable, similar(Ferrell to aa phosphophosphocycle of the controller similar to rylation/dephosphorylation monocycle and Ha cycle of the controller variable, similar(Ferrell to a phosphorylation/dephosphorylation monocycle and and Ha rylation/dephosphorylation monocycle (Ferrell and Ha [2014]). Each of these mechanisms, taken in isolation rylation/dephosphorylation monocycle (Ferrell and and Ha [2014]). Each of these mechanisms, taken in isolation [2014]). Each of these mechanisms, taken in isolation and modeledEach withofmass action kinetics,taken are not sufficientand to [2014]). theseaction mechanisms, in isolation modeled with mass kinetics, are not sufficient to modeled with mass action kinetics, are not sufficient to provide high ultrasensitivity. Titration reactions facilitate modeled with mass action kinetics, are not sufficient to provide high ultrasensitivity. ultrasensitivity. Titration reactions facilitate provide high Titration reactions the tunability of the response threshold, and facilitate titration provide high ultrasensitivity. Titration reactions facilitate the tunability of the response threshold, and titration the tunability of the response threshold, and titration the tunability of the response threshold, and titration
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Fig. 1. Closed loop molecular network with an ultraFig. 1. Closed loop molecular network with ultraFig. Closed molecular network with an ultrasensitive controller A) Schematic thean interconFig. 1. 1. Closed loop loop molecular network of with an ultrasensitive controller A) Schematic of the interconsensitive controller A) Schematic of the interconnection of a biomolecular controller and a biomolecsensitive controller A) Schematic of the interconnection of a biomolecular controller and a biomolecnection of and ular system; p represents acontroller given parameter perturnection of aa biomolecular biomolecular and aa biomolecbiomolecular system; p represents aacontroller given parameter perturular system; p represents given parameter perturbation in the system. B) shows an illustration of ular system; p system. represents a shows given parameter perturbation in the B) an illustration of bation in the system. B) shows an illustration of input-output equilibrium conditions of illustration the controller bation in the system. B) shows an of input-output equilibrium conditions of the controller input-output equilibrium conditions of the (in grey) and the system (in orange). The controller input-output equilibrium conditions of the controller (in grey) and response the system (in orange). The controller (in grey) the (in The controller ultrasensitive threshold is tunable setting (in grey) and and response the system system (in orange). orange). The by controller ultrasensitive threshold is tunable by setting ultrasensitive response threshold is tunable by the reference signal r, shown in red. Ultrasensitivity of ultrasensitive response threshold is tunable by setting setting the reference signal r, shown in red. Ultrasensitivity of the reference signal r, shown in red. Ultrasensitivity of controller implies that, inin a certain regime, perturthe reference signal r, shown red. Ultrasensitivity of the controller implies that, in aa certain regime, perturthe controller implies that, in certain regime, perturbations in the input/output curve of the systems have the controller implies that, in a certain regime, perturbations in the input/output curve ofpoint. the systems have bations in curve the minor influence on the equilibrium bations in the the input/output input/output curve of ofpoint. the systems systems have have minor influence on the equilibrium minor influence on the equilibrium point. minor influence on the equilibrium point. together with the activation/deactivation cycle mechanism together the activation/deactivation cycle mechanism together with the activation/deactivation cycle mechanism creates awith highly input/output This together with the ultrasensitive activation/deactivation cyclecurve. mechanism creates a highly ultrasensitive input/output curve. This creates a highly ultrasensitive input/output curve. This ultrasensitive behavior is very similar to zero-order ultracreates a highly ultrasensitive input/output curve.ultraThis ultrasensitive behavior is very similar to zero-order ultrasensitive behavior is very similar to zero-order ultrasensitivity when enzymes are saturable and their activities ultrasensitive behavior is very similar to zero-order ultrasensitivity when enzymes are saturable and their activities sensitivity when enzymes and their are given by Michaelis-Menten equations and Ha sensitivity when enzymes are are saturable saturable and(Ferrell their activities activities are given by Michaelis-Menten equations (Ferrell and Ha are given by Michaelis-Menten equations (Ferrell and Ha [2014]. The controller reaction network is directly impleare given bycontroller Michaelis-Menten equations (Ferrell and Ha [2014]. The reaction network is directly imple[2014]. The controller reaction network is directly implementable using in vitro synthetic transcriptional systems, [2014]. The controller reaction network is directly implementable using in in vitro vitro synthetic synthetic transcriptional systems, mentable using such as negative systemstranscriptional (Franco et al.systems, [2014]) mentable using infeedback vitro synthetic transcriptional systems, such as negative feedback systems (Franco et al. [2014]) such as negative feedback systems (Franco et al. [2014]) to balance production rates. The module, taken as a such as negative feedback systems (Franco et al. [2014]) to balance production rates. The module, taken as a to balance production rates. The module, taken as subsystem in a larger network, can also be used to build to balance inproduction rates. The module, taken as a a subsystem a larger network, can also be used to build subsystem in aa larger network, can also to build oscillatory systems (Cuba Samaniego et be al. used [2016a]). We subsystem in larger network, can also be used to build oscillatory systems (Cuba Samaniego et et al. [2016a]). We oscillatory (Cuba Samaniego [2016a]). We exploit the systems controller ultrasensitivity to al. tune the closed oscillatory systems (Cuba Samaniego et al. [2016a]). We exploit the controller ultrasensitivity to tune the closed exploit the controller ultrasensitivity to tune the closed loop equilibrium point, and to achieve robustness relative exploit the controller ultrasensitivity to tune the closed loop equilibrium point, and to relative loop equilibrium point, achieve robustness relative to parameter perturbations, as achieve sketchedrobustness in Fig. 1 B. loop equilibrium point, and and to to achieve robustness relative to parameter perturbations, as sketched in Fig. 1 B. to parameter perturbations, as sketched in Fig. 1 B. to parameter perturbations, as sketched in Fig. 1 B.
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Proceedings of the 20th IFAC World Congress Toulouse, France, July 9-14, 2017 Christian Cuba Samaniego et al. / IFAC PapersOnLine 50-1 (2017) 10950–10956
Titration reactions in vivo without degradation were used recently by Briat et al. [2016] to propose an antithetic integral feedback controller, whose reactions are analogous to the first stage of our controller. Our network differs from the anthitetic controller for two reasons: first, it includes an additional activation/deactivation cycle for the controller molecule; second, we take into account degradation (or dilution) reactions that are known to be present in most biochemical and cellular processes. Our simulations show that large degradation reduces ultrasensitivity of the controller, and this can therefore disrupt its tracking and adaptation performance (Del Vecchio et al. [2016]). Montefusco et al. [2016] found that an ultrasensitive network can model osmoregulation behaviors better than proportional and/or integral networks; this suggests that natural circuits may have evolved ultrasensitivity to robustify equilibria as shown in this paper. In Section 2 we describe the controller network in isolation, and we characterize its ultrasensitive behavior. In Section 3 we describe the target system to be controlled. Finally, in Section 4 we consider the closed loop system and examine the capacity of the system output to track a reference, as well as its sensitivity with respect to perturbations in the reaction rates and concentration of components. We discuss the reactions chosen to design the controller in relation to a potential in vitro implementation with synthetic transcriptional networks Kim et al. [2006]. Throughout the paper we consider biomolecular systems that are described by chemical reactions. Species are indicated as uppercase letters, concentrations as lowercase letters (e.g. species X has concentration x). Given a set of chemical reactions, we use the law of mass action to derive ordinary differential equation (ODE) models that describe the kinetics of the ensemble of reactions.
2. DESIGN OF AN ULTRASENSITIVE MOLECULAR CONTROLLER
The controller network (Fig. 2A) consists of a single output species U and two input species, an activator A and an inhibitor I. The output species can be in active form U , or inactive form U ∗ , and its total concentration is conserved (utot = u + u∗ ). Active U is assumed to have a regulatory function in downstream reactions, which cannot be performed by U ∗ . The inputs A and I respectively produce species RA and RI , which regulate the active fraction of output U . RI binds to and deactivates U , forming an inert species U ∗ . In contrast, species RA binds to and reactivates U ∗ by displacing RI from U ∗ , yielding a waste complex RA · RI . In addition, free RA and RI bind and mutually titrate forming waste complex RA · RI . The biomolecular reactions constituting our controller network are:
κ
A −−c A + RA θc
I −− I + RI
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Production Production
γc
RA + RI −− RA · RI
Titration
RA −− 0
Degradation
φc
φc
RI −− 0 ∗
Degradation αc
RA + U −− U + RA · RI
Reactivation
RI + U −− U
Inhibition
βc
∗
Using the law of mass action, we derive the following ODEs that describe the dynamics of the system: u˙ = αc rA u∗ − βc rI u (1) r˙A = κc a − αc rA u∗ − γc rA rI − φc rA (2) (3) r˙I = θc i − βc rI u − γc rA rI − φc rI Fig. 2 B and C illustrate the desired interactions of species U , RA , and RI , and how they result in an ultrasensitive response. First, consider the case where the system receives a step input in i. Then, active species u is expected to be quickly converted to its inactive form u∗ . However, the presence of a (which produces rA ) has a “buffering” effect: free rA binds to rI and titrates it before it can inhibit u (this of course requires that γc is fast). Due to this buffering effect, we can think of i as the input to the controller, and of the titrating species a as a “reference” signal: the controller responds only if i exceeds a sufficiently. A similar reasoning can be followed to explain the behavior of the system when the system receives a step input a: in this case, free rI acts like a buffer for rA . We can think of a as the input to the controller, and of the titrating species i as the reference; only when a exceeds i sufficiently, the controller responds. In both cases, the transition point (at which the concentration of u switches from low to high and vice-versa) depends on the concentration of the titrating species. This reaction network could be experimentally realized using nucleic acids and proteins. Species U could be an RNA polymerase whose activity can repressed using an RNA aptamers (RI ) as those proposed by Mori et al. [2012], Ohuchi et al. [2012] (species RI ). RNA polymerase activity could be restored via another RNA species (RA ) displacing the aptamer as suggested in Cuba Samaniego et al. [2016a]. Synthetic templates (A, I) could be used to produce the RNA regulators. Numerical integration of equations (1)–(3) is done assuming nominal values of parameters (Table 1) that are realistic in the implementation context just described. In the next sections we examine the behavior of the controller network. We derive its steady state input/output response analytically, and then we test numerically the sensitivity of the controller response to parameter perturbations. 2.1 Boundedness and monotonicity The solutions of the biomolecular controller model are bounded. After a state transformation, the Jacobian of the system is a Metlzer matrix, which is Hurwitz stable, and we can conclude that the input-output equilibrium maps of u(A) and u(I) are monotonically increasing and
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A A A
A
RA
o
I
RI
o
C
I
RA
U*
o
II
B A
Biomolecular Controller U
U
U
C
S
A I
U
C
S
u
u
RI
ar
ir
i
a
Fig. 2. Ultrasensitive network. A) Schematic summary of the reactions defining the controller network. B) Controller network within a feedback loop, where signal A is kept constant and represents a reference signal, shown in red. C) Controller in a feedback loop where signal A is kept constant and acts as a reference, shown in red. The thresholds ar and is are functions of the reference a and i (in red) and of the parameters of the system. decreasing. As a consequence, also their inverse maps (which will be used later in the manuscript) are monotonic functions. These claims can be demonstrated following the analysis in Cuba Samaniego et al. [2016a], which are not reported here.
1
1 10 c 1 c 0.1 c
0 0
1
0 0
2.2 Input-output characterization We begin by finding an expression for the concentration of input A as a function of the concentrations of input I and output U . (These expressions are more convenient to find relative to deriving the concentration of U as a function of A and I). a(u, i) =
θc φc i+ κc κc
where
1−
α c u∗ βc u
r¯A (u, i),
(4)
b2A − 4aA cA , 2aA aA = αc γc u∗ , bA = αc u∗ (βc u + φc ) and cA = −uβc θc I. r¯A (u, i) =
−bA +
Similarly, we find an expression relating the equilibrium concentration of I as a function of U and A. i(u, a) =
where
φc κc a− θc θc
1
βc u − 1 r¯I (u, a), αc u∗
(5)
1
2
0
2
1
1 10 c 1 c 0.1 c
0
2
1
c
0 0
1 10 c 1 c 0.1 c
10 c 1 c 0.1
c
0
2
1
1 10 c 1 c 0.1
0
1
2
10 c 1 c 0.1 c 0 0
1
2
1 10i 1i 0.1i
10utot 1utot tot
0.1u
0 0
1
0 2
0
1
2
Fig. 3. Controller ultrasensistive response. Inputoutput equilibrium conditions from equation (4) as parameters are varied. Nominal simulation parameters are listed in Table 1. Normalization of axis are given by an = a(u, i)/ir and un = u/utot . behavior of the input-output expression (4) is evaluated as parameters are individually varied. We note that curves are very sensitive to the inhibition rate (βc ), the activation rate (αc ), the degradation rate (φc ) and total amount of input (utot ). The input/output curve steepness increases when these parameters are large, with exception for the degradation rate (φc ), which largely reduces the ultrasensitivity. The last panel of Fig. 3 shows the influence of variations of i on the input-output response. Because of how the variables are normalized, the curve transition always occurs at an = 1: this indicates that the output equilibrium always perfectly tracks the desired threshold, which can be seen as a “reference” in the closed loop system, as we further explain in the next sections. (A non-normalized plot of the input-output response is in Fig. 6 B, C, D.)
−bI + b2I − 4aI cI , r¯I (u, a) = 2aI aI = βc γc u, bI = βc u(αc u∗ + φc ) and cI = −αc u∗ κc A.
2.3 Ultrasensitivity characterization To assess the ultrasensitivity of the input/output curves with respect to parameter variations, we vary individual parameters of the network. We focus on plotting expression (4): we fix the concentration of i, and then evaluate r¯A (u, i) when u varies between 0 and utot ; this allows us to evaluate the map a(u, i) (x axis) versus u (y axis) as shown in Fig. 3. We note that if the concentration of i is fixed, we can think of term ir = κθcc i as the threshold of the ultrasensitive response. The plots in Fig. 3 were obtained using normalized concentrations of species an = a(u, i)/ir , and un = u/utot . The
It is possible to plot (5) by defining similar threshold parameters and normalized variables, and we obtain similar results (not shown for brevity), illustrated on Fig. 2B. 3. CONTROLLED SYSTEM We consider an example transcription process as a target system to be controlled. Specifically, we focus on a class of synthetic in vitro genetic switches which have been used to build a variety of autonomous circuits including bistable and oscillatory networks Kim et al. [2006], Kim and Winfree [2011], Franco et al. [2011].
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Table 1. Nominal simulation parameters of the controller Description
Value
θc (/s) κc (/s) βc (/M/s) αc (/M/s) γc (/M/s) φc (/s) utot (nM ) i(nM ) a(nM )
Production Production Inhibition Reactivation Titration Degradation Concentration Concentration Concentration
7.5 · 10−4 5 · 10−4 6 · 104 6 · 104 3 · 104 3.85 · 10−4 500 100 100
B
Biomolecular System
A u U
Parameter
Z
u U
Y
Y* W*
W
Transcription: 10−3 − 1 Vogel and Jensen [1994], Chen et al. [2015] 104 − 106 Kim et al. [2006], Zhang et al. [2007] 104 − 106 Kim et al. [2006], Zhang et al. [2007] 104 − 106 Kim et al. [2006], Zhang et al. [2007] 10−4 − 10−3 Kim et al. [2014].
u(y) =
y
W
o
W*
Other studies
Z
yY
ytot
0 w -y tot
tot
y
tot
as =
z
Fig. 4. Biomolecular system or process A) Schematic of a simplified model of a synthetic gene switch with input u and output y. B) An illustration of input/output response with input z and output y. The target system receives a single input species U and produces an output species Y . As in Kim et al. [2006], Y is a synthetic template (genelet) that is activated by a DNA activator molecule W , and deactivated by an RNA inhibitor molecule Z. The input U could be an RNA polymerase transcribing inhibitor Z from a constitutively active template; Z is designed to bind to Y and convert it to inactive Y ∗ (this occurs by displacement of the activator W ). In addition, Z directly binds to and titrates W converting it to inactive species W ∗ . We assume that inhibited activator W ∗ spontaneously reverts to its active form W . The total concentrations of Y and W are assumed to be constant, y tot = y + y ∗ and wtot = w + y + w∗ (activation of y ∗ requires binding of one copy of w). These regulatory interactions are illustrated in Fig. 4A, and the corresponding set of chemical reactions is: κ
U −−s U + Z γs
Z + W −− W
Production ∗
Titration
φs
Z −− 0
(9)
where
αs y ∗ w, z= βs y
α s γs y ∗ βs y ,
bs = αs y ∗ + θs and cs = −θs (wtot − y).
w=
−bs +
b2s − 4as cs 2as
In the next sections, we numerically integrate the system ODEs using the nominal parameters listed in Table 2, which are realistic values relative to the proposed circuit implementation using transcriptional circuits (Kim et al. [2006]). 4. CLOSED LOOP SYSTEM We now analyze the performance of the closed loop system where controller and system are interconnected. The goal is to control the concentration of active template Y as shown in Fig.5, following a reference signal R. Because the input/output behavior of this particular system is monotone decreasing (Fig. 4), we operate our controller so that the feedback loop counteracts the system response. So we connect variable Y in the feedback loop to serve as input A (activator) for the controller. The controller input I is now considered the reference signal R. We note that if the system to be controlled had a monotonically increasing input-output map, we would have operated the controller in the opposite way, so that input A would be the reference. This suggests that this controller architecture can be adapted to work with systems having very different input-output behaviors; a theoretical characterization of this property is left for future work.
Degradation
θs
W ∗ −− W ∗
θ s w ∗ + φs z , κs
αs
Recovery ∗
W + Y −− W + Y βs
Y + Z −− Y ∗
Biomolecular Controller
Activation Y
Inhibition
Using the law of mass action we derive the following ODEs: y˙ = αs wy ∗ − βs yz w˙ = θs w∗ − αs wy ∗ − γs zw z˙ = κs u − βs yz − γs zw − φs z
(6) (7) (8)
In earlier work (Cuba Samaniego et al. [2016b]) we demonstrated analytically that the solutions of ODEs (6)–(8) are bounded, that the system is structurally stable and monotone, and the input-output map is monotonically decreasing. We find the equilibrium values of u and y are related according to the following expression:
Ry
o
R
R
Ry
U*
o
Rr
o
Biomolecular System U U
Rr
Z
o
W
Y*
W* W
W*
Y
Y
Z
Fig. 5. Feedback control system. Closed loop system between the biomolecular controller and biomolecular system. The set point, output and actuator are r, y and u respectively. We combine equations (1)–(3) (controller) and equations (6)–(8) (controlled system) and obtain the closed loop system equations:
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Table 2. Nominal simulation parameters of the controlled system
C
Rate
Description
Value
Other studies
κs (/s) θs (/s) βs (/M/s) αs (/M/s) γs (/M/s) φs (/s) y tot (nM ) wtot (nM )
Production Recovery Inhibition Reactivation Titration Degradation Concentration Concentration
5 · 10−4 3 · 10−4 6 · 104 6 · 104 3 · 104 3.85 · 10−4 800 800
Transcription: 10−3 − 1 Vogel and Jensen [1994], Chen et al. [2015] 10−5 − 10−2 Chen et al. [2015], Beelman and Parker [1995] 104 − 106 Kim et al. [2006], Zhang et al. [2007] 104 − 106 Kim et al. [2006], Zhang et al. [2007] 104 − 106 Kim et al. [2006], Zhang et al. [2007] 10−4 − 10−3 Kim et al. [2014]
u˙ = αc ry u∗ − βc rr u controller output r˙y = −αc ry u∗ − γc ry rr − φc ry κc y
The system’s output y, shown in purple, closely tracks each reference input (dark gray line). Fig.6 B, C and D show the input-output equilibrium conditions of the controller and system at the corresponding reference value: the threshold of the ultrasensitive input/output controller map tracks the reference r¯. As noted earlier, if φc is too large and ultrasensitivity is lost, then equilibria computed in Fig. 6 B–D will not satisfy y ≈ r¯.
controller input r ˙ = θc r −βc rr u − γc ry rr − φc rr r external input y˙ = αs wy ∗ − βs yz system output w˙ = θs w∗ − αs wy ∗ − γs zw S κs u −βs yz − γs zw − φs z z˙ = system input
The system output is y, as annotated in the equations above, and its external input or reference input is r. We assume utot = u + u∗ , y tot = y + y ∗ and wtot = w + y + w∗ . The experimental implementation of these two reaction systems in isolation was discussed earlier. Their interconnection would be immediately feasible: transcriptional circuits and aptamer systems have been characterized in compatible in vitro conditions; the RNA polymerase U can be easily used to transcribe RNA species Z from a constitutively active template. 4.1 Reference tracking and disturbance rejection We now define a rescaled reference signal as r¯ = κθcc r; then, we define the error e = r¯ − y to quantify the difference between the output y and the rescaled reference r¯. We also define the controller “internal” error er = rr − ry . We then derive the following expression for e˙ r : e˙ r = r˙r − r˙y = θc r − κc y − φc er + u˙ = κc e − φc er + u. ˙ At steady state, e˙ r = 0 and u˙ = 0, and we find: κc e = φc er . This equation indicates that if φc = 0 (i.e. if there is no degradation in the system), then the steady state error e is exactly zero, therefore the output tracks the reference, y = r¯. If φc is sufficiently small, we expect that y ≈ r¯. If φc is large, then the output cannot track the reference well. This behavior is related to the fact that large φc induces a loss of ultrasensitivity in the controller module (as shown in simulations in Fig. 3). We now test the ability of y to track changes in r¯ using numerical simulations of the closed loop system. Fig. 6A shows the response of the output to step-increases in rescaled reference signal r¯ = 0.15, 0.3 and 0.45µM . The rescaled reference and the reference input are related as: r¯ = κθcc r = 1.5 r, given the nominal parameters in Tables 1 and 2.
Fig. 6 E shows the behavior of the output in the presence of a step-perturbation of transcription rate κs , given a constant reference r¯ = 0.15µM . The system converges to the reference input rejecting the disturbance; Fig.6 F, G and H show input-output equilibrium conditions of the controller and system for different disturbance values on κs . The system is able to reject those disturbance in the ultra sensitivity region. Gillespie simulations (light purple color in Figs. 6 A and E), Gillespie et al. [1977], suggest that the tracking and disturbance rejection of the closed loop system hold also in a stochastic setting. In these single stochastic simulations, initial conditions are set to zero. We consider 100 molecules at 100nM concentration in a 1f L volume. 4.2 Sensitivity analysis 1. Sensitivity to variations of system parameters: When the equilibrium of the closed loop system is in the ultrasensitive regime of the controller response, the system transient and stationary responses are robust with respect to parameter variations of the system to be controlled. Numerical sensitivity analysis is shown in Fig. 7. The closed loop system transient (Fig. 7 A) and equilibria (Fig. 7 B) are robust to most parameter variations spanning 0.5 to 2 times the nominal values listed on Table 2. We note that the controller can “saturate” in a certain range of the production rate of Z (κs ), the recovery rate of W (θs ), the total concentration of template (y tot ) and total concentration of activator (wtot ). Large θs or wtot increase free w, which can bind faster to y ∗ . To balance this process, the system increases the demand of controller species u in order to inhibit w. This can cause the controller to reach a saturation point and fail. If y tot is much lower than the reference value, the controller may fail to supply the system’s demand of u because there is not enough free y available. (Note that Y is not depleted by the controller reactions.) 2. Sensitivity to variations of controller parameters: We test how changes in the parameters of the biomolecular controller affect the closed loop system behavior. Fig. 3
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E 0.8 Stochastic Deterministic
=0.45 =0.3 =0.15 0 0
B
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30
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45
D 0.5
0.15 0 0
F
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H 0.5
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0.3
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0.45
0 0 0.15
0.8
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0.8
0 0.15
0.8
Fig. 6. Tracking and adaptation A) The output y of the closed loop system (purple) tracks three different references (¯ r = 0.15, 0.3 and 0.45µM , dark gray). B,C and D) show equilibrium conditions as they change as a function of reference changes, where grey corresponds to the controller (4), and orange corresponds to the system (9). E) Adaptation of the system output in the presence of a step perturbation of the production rate (κs ) applied at 15 and 30h, where the final values are two and four times its nominal value listed on Table 2.
A
s
0.8
s
0.8
0
0 0
time (h)
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s
y
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c
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s
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0.8
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0.8
Fig. 7. Rejection of system parametric disturbances A) Output y is shown as parameter values vary from 0.5 to 2 times their nominal values listed on Table 1 (yellow to red). B) Input-output equilibrium conditions of the controller (grey) and system (color scale), as the same parameters are varied.
Fig. 8. Controller robustness Top shows y trajectories for different parameter values that varies from 0.5 to 2 times its nominal values listed on Table 1 (from gray to black). Bottom shows input-output equilibrium condition of the controller (grey scale) and system (orange).
and Fig. 8 show that the system is generally very robust to changes in controller parameters. An exception is parameter utot , which defines the upper bound of the controller’s output. By increasing this parameter we improve the robustness of the ultrasensitive response of the controller. In contrast, if utot is reduced, the controller may saturate and fail to supply enough input U to the system.
To create a highly ultrasensitive behavior we combine two mechanisms: 1) titration and 2) an activation and inhibition cycle. Titration is important for tuning the threshold of the ultrasensitive response and both mechanisms are important to raise the highly ultrasensitive response.
5. DISCUSSION We presented a biomolecular controller that operates via titration reactions, and showed its closed loop performance when the system to be controlled is an RNA transcription process.
We suggest that this ultrasensitive controller overcomes the challenge of designing an integral controller that takes into account degradation and dilution processes Briat et al. [2016]. We believe that our studies could be useful to guide experimental implementation of a variety of molecular controllers, beyond the in vitro implementation that we suggest here. Ultrasensitivity may be obtained using mechanisms other than molecular sequestration; for instance, Goldbeter and Koshland [1984] showed that phosphoryla-
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tion cycles have an ultra sensitive response. Many posttranslational modification pathways such as phosphorylation, glycosylation, acetylation and ubiquitination could be used to design an ultrasensitive feedback controller for tracking and disturbance rejection. ACKNOWLEDGEMENTS This research was partially supported by a ThinkSwiss fellowship to CCS; by the US National Science Foundation under grant CMMI 1266402, which provided funding for CCS and EF. EF was also partially supported by the Department of Energy under grant DE-SC0010595. CCS finalized this research as a visiting scholar at ETH-Zurich: we thank Corentin Briat and Mustafa Khammash for kindly hosting CCS, for helpful discussions and support. REFERENCES Beelman, C.A. and Parker, R. (1995). Degradation of mrna in eukaryotes. Cell, 81(2), 179–183. Briat, C., Gupta, A., and Khammash, M. (2016). Antithetic integral feedback ensures robust perfect adaptation in noisy bimolecular networks. Cell systems, 2(1), 15–26. Buchler, N.E. and Louis, M. (2008). Molecular titration and ultrasensitivity in regulatory networks. Journal of molecular biology, 384(5), 1106–1119. Chen, H., Shiroguchi, K., Ge, H., and Xie, X.S. (2015). Genome-wide study of mRNA degradation and transcript elongation in escherichia coli. Molecular systems biology, 11(1), 781. Cuba Samaniego, C., Giordano, G., Blanchini, F., and Franco, E. (2016a). Stability analysis of an artificial biomolecular oscillator with non-cooperative regulatory interactions. Journal of Biological Dynamics, xxx. Cuba Samaniego, C., Giordano, G., Kim, J., Blanchini, F., and Franco, E. (2016b). Molecular titration promotes oscillations and bistability in minimal network models with monomeric regulators. ACS synthetic biology, 5(4), 321–333. Del Vecchio, D., Dy, A.J., and Qian, Y. (2016). Control theory meets synthetic biology. Journal of The Royal Society Interface, 13(120), 20160380. Ferrell, J.E. and Ha, S.H. (2014). Ultrasensitivity part i: Michaelian responses and zero-order ultrasensitivity. Trends in biochemical sciences, 39(10), 496–503. Franco, E., Friedrichs, E., Kim, J., Jungmann, R., Murray, R., Winfree, E., and Simmel, F.C. (2011). Timing molecular motion and production with a synthetic transcriptional clock. Proceedings of the National Academy of Sciences, 108(40), E784–E793. Franco, E., Giordano, G., Forsberg, P.O., and Murray, R.M. (2014). Negative autoregulation matches production and demand in synthetic transcriptional networks. ACS synthetic biology, 3(8), 589–599. Gillespie, D.T. et al. (1977). Exact stochastic simulation of coupled chemical reactions. J. phys. Chem, 81(25), 2340–2361. Goldbeter, A. and Koshland, D. (1984). Ultrasensitivity in biochemical systems controlled by covalent modification. interplay between zero-order and multistep effects. Journal of Biological Chemistry, 259(23), 14441–14447.
Kim, J., Khetarpal, I., Sen, S., and Murray, R.M. (2014). Synthetic circuit for exact adaptation and fold-change detection. Nucleic acids research, gku233. Kim, J., White, K.S., and Winfree, E. (2006). Construction of an in vitro bistable circuit from synthetic transcriptional switches. Molecular systems biology, 2(1), 68. Kim, J. and Winfree, E. (2011). Synthetic in vitro transcriptional oscillators. Molecular systems biology, 7(1), 465. Montefusco, F., Akman, O.E., Soyer, O.S., and Bates, D.G. (2016). Ultrasensitive negative feedback control: a natural approach for the design of synthetic controllers. PLoS One, 11(8), e0161605. Mori, Y., Nakamura, Y., and Ohuchi, S. (2012). Inhibitory RNA aptamer against SP6 RNA polymerase. Biochemical and Biophysical Research Communications, 420(2), 440–443. Ohuchi, S., Mori, Y., and Nakamura, Y. (2012). Evolution of an inhibitory RNA aptamer against T7 RNA polymerase. FEBS open bio. Thieffry, D., Huerta, A.M., P´erez-Rueda, E., and ColladoVides, J. (1998). From specific gene regulation to genomic networks: a global analysis of transcriptional regulation in Escherichia coli. Bioessays, 20(5), 433– 440. Vogel, U. and Jensen, K.F. (1994). The RNA chain elongation rate in escherichia coli depends on the growth rate. Journal of bacteriology, 176(10), 2807–2813. Zhang, D.Y., Turberfield, A.J., Yurke, B., and Winfree, E. (2007). Engineering entropy-driven reactions and networks catalyzed by dna. Science, 318(5853), 1121– 1125.
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