Annual Reviews in Control 37 (2013) 333–345

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Annual Reviews in Control

journal homepage: www.elsevier .com/ locate /arcontrol

A control theoretic framework for modular analysis and designof biomolecular networks q,qq

1367-5788/$ - see front matter � 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.arcontrol.2013.09.011

q This article is a substantial extension of a plenary paper presented at the IFACSymposium on Nonlinear Control Systems – NOLCOS 2013, Toulouse, France,September 4-6, 2013.qq This work was supported in part by the National Science Foundation and the AirForce Office of Scientific Research.

E-mail address: ddv@mit.edu

Domitilla Del VecchioDepartment of Mechanical Engineering, Laboratory for Information and Decision Systems (LIDS), Massachusetts Institute of Technology, Cambridge, MA, United States

a r t i c l e i n f o a b s t r a c t

Article history:Received 18 May 2013Accepted 14 August 2013Available online 19 October 2013

Control theory has been instrumental for the analysis and design of a number of engineering systems,including aerospace and transportation systems, robotics and intelligent machines, manufacturingchains, electrical, power, and information networks. In the past several years, the ability of de novo cre-ating biomolecular networks and of measuring key physical quantities has come to a point in whichquantitative analysis and design of biological systems is possible. While a modular approach to analyzeand design complex systems has proven critical in most control theory applications, it is still subject ofdebate whether a modular approach is viable in biomolecular networks. In fact, biomolecular networksdisplay context-dependent behavior, that is, the input/output dynamical properties of a module changeonce this is part of a network. One cause of context dependence, similar to what found in many engineer-ing systems, is retroactivity, that is, the effect of loads applied on a module by downstream systems. Inthis paper, we focus on retroactivity and review techniques, based on nonlinear control and dynamicalsystems theory, that we have developed to quantify the extent of modularity of biomolecular systemsand to establish modular analysis and design techniques.

� 2013 Elsevier Ltd. All rights reserved.

1. Introduction

Modularity is the property that allows to predict the behavior ofa complicated system from that of its components, guaranteeingthat the salient properties of individual components do not changeafter interconnection. The assumption that is often made whendesigning or analyzing a network modularly is that the input/out-put behavior of a module does not change upon connection toother modules. Researchers in the systems biology communityhave hypothesized that modularity may be a property of biologicalsystems, proposing functional modules as a critical level of biolog-ical organization (Alon, 2007a; Hartwell, Hopfield, Leibler, & Mur-ray, 1999). This view has profound implications on evolution(Kirschner & Gerhart, 2005) and further suggests that biology, justlike engineering, can be understood in a hierarchical fashion(Asthagiri & Lauffenburger, 2000; Lauffenburger, 2000). Lauffen-burger (2000) in fact elaborates that biological systems can be ana-lyzed in a nested manner, similar to what is performed inengineering design, in which individual components are first char-acterized and tested in isolation prior to incorporation into incre-mentally larger and more sophisticated systems. However,

modularity is still subject of intense debate and remains one ofthe most vexing questions in systems biology (Alexander, Kim,Emonet, & Gerstein, 2009; Cardinale & Arkin, 2012; Kaltenbach &Stelling, 2012).

Modularity is also critical for the field of synthetic biology(Purnick & Weiss, 2009), in which researchers are pursuing abottom-up approach to designing biomolecular networks(Andrianantoandro, Basu, Karig, & Weiss, 2006; Baker et al.,2006). The past decade has seen tremendous advances to the pointthat creation of simple biomolecular networks, or ‘‘circuits’’, thatcontrol the behavior of living organisms is now possible (Becskei& Serrano, 2000; Elowitz & Leibler, 2000; Gardner, Cantor, & Collins,2000; Atkinson, Savageau, Meyers, & Ninfa, 2003; Stricker et al.,2008; Danino, Mondrag ón-Palomino, Tsimring, & Hasty, 2009;Bleris et al., 2011; Moon, Lou, Tamsir, Stanton, & Voigt, 2012). Anear future is envisioned in which re-engineered cells will performa number of useful functions from turning feedstock into energy(Peralta-Yahya, Zhang, del Cardayre, & Keasling, 2012; Zhang,Carothers, & Keasling, 2012), to killing cancer cells in ill patients(Xie, Wroblewska, Prochazka, Weiss, & Benenson, 2011), to detect-ing pathogens in the environment (Kobayashi et al., 2004), to regu-late cell differentiation in diseases like diabetes (Miller et al., 2012).To meet this vision, one key challenge must be tackled, namelydesigning biomolecular networks that can realize substantiallymore complex functionalities than those currently available.

A main bottleneck in advancing in this direction is that modulesdisplay context-dependent behavior (Cardinale & Arkin, 2012).

334 D. Del Vecchio / Annual Reviews in Control 37 (2013) 333–345

Hence, accurate prediction of a module’s behavior once part of anetwork is still an open question. As a result, synthetic circuitsneed to be re-tuned/re-designed through a lengthy and ad hoc pro-cess, which takes years, every time they are inserted into a differ-ent context (Slusarczyk, Lin, & Weiss, 2012). Context-dependenceis due to a number of different factors. These include direct in-ter-modular interactions that result in load effects, a phenomenonknown as retroactivity (Saez-Rodriguez, Kremling, Conzelmann,Bettenbrock, & Gilles, 2004; Saez-Rodriguez, Kremling, & Gilles,2005; Del Vecchio, Ninfa, & Sontag, 2008); interactions of syntheticcircuits with the cell ‘‘chassis’’, a phenomenon broadly known as‘‘metabolic burden’’ (Bentley, Mirjalili, Andersen, Davis, & Kompal-a, 1990), which encompasses a number of different effects, such aseffects on cell growth (Scott, Gunderson, Mateescu, Zhang, & Hwa,2010) and competition for shared resources, such as ATP, transcrip-tion/translation machinery and proteases (Mather et al., 2011;Yeung, Kim, & Murray, 2013; Mather, Hasty, Tsimring, & Williams,2013); and perturbations in the cell environment, such as changesin temperature, acidity and nutrients’ level.

In this paper, we address the fundamental question of modular-ity in biology focusing on the problem of retroactivity. Retroactiv-ity extends the notion of impedance or loading to non-electricalsystems and, in particular, to biological systems (Saez-Rodriguezet al., 2004; Saez-Rodriguez et al., 2005; Del Vecchio et al., 2008).Because of retroactivity, the dynamical behavior of a systemchanges upon connection to other systems and, therefore, retroac-tivity effects imply a loss of modularity. We review some of our re-sults towards (a) understanding the extent of modularity ofbiomolecular systems by quantifying the effects of retroactivity,(b) establishing a predictive modeling framework for modularanalysis in the presence of retroactivity, and (c) uncovering designprinciples ensuring that the behavior of modules is, to some extent,robust to retroactivity effects.

This work is complementary to but different from studies focus-ing on partitioning biological networks into modules using graph-theoretic approaches (Saez-Rodriguez et al., 2005; Vivek Sridharan,Hassoun, & Lee, 2011; Anderson, Chang, & Papachristodoulou,2011). Instead, the work presented here describes a theoreticalframework to accurately predict both the quantitative and thequalitative behavior of interconnected modules from their behav-ior in isolation and from key physical properties. In this sense,our approach is closer to that of disciplines in biochemical systemsanalysis, such as metabolic control analysis (MCA) (Fell, 1992;Heinrich & Schuster, 1999; Sauro & Kholodenko, 2004). However,while MCA is primarily focused on steady state and near-equilib-rium behavior, our approach focuses on global nonlinear dynamicsevolving possibly far from equilibrium situations.

To make the retroactivity problem amenable for a mathematicalanalysis, we propose a system concept that explicitly incorporatesretroactivity (Del Vecchio & Sontag, 2009). We then leverage thenatural separation of time scales in biomolecular networks to pro-vide an operative quantification of retroactivity for general net-works as a function of measurable biomolecular parameters andnetwork topology (Gyorgy & Del Vecchio, 2012, 2013). From thisoperative quantification and the dynamic model of a module in iso-lation, we can accurately predict how the behavior of the modulewill change after connection to other systems. The conceptual sim-ilarity of this framework with the electrical circuit theory based onequivalent impedances (Thevenin, 1883) is apparent. The key dif-ference is that electrical circuit theory is largely based on linearsystems, while our framework relies on tools from nonlinear sys-tem theory, due to the inherent nonlinearity of biomolecular net-works, which makes linear systems theory inapplicable.

We then consider the problem of designing larger systems bycomposing smaller modules. We first briefly describe how theframework developed in Gyorgy and Del Vecchio (2013) allows

to determine how the parameters of an upstream system and adownstream system should be tuned such that the effects of retro-activity are minimized. This approach to minimizing retroactivityeffects implies that modules should be co-designed, a procedurethat can become impractical as the number of modules composinga system increases. A different approach is that to design and opti-mize modules in isolation and then connect modules to each otherthrough a special device called insulation device, such that the mod-ules retain their isolated behavior. The insulation device should ap-ply negligible loading to the upstream module while being able tokeep the desired input/output response in the face of significantretroactivity to its output (Jayanthi & Del Vecchio, 2011). This ap-proach has been used in electrical engineering with the introduc-tion of operational amplifiers and their derived circuits such asinverting and non-inverting amplifiers (Schilling & Belove, 1968).We illustrate a design principle for insulation devices, which ex-ploits the distinctive feature of the interconnection structure inbiomolecular networks and the separation of time scales amongkey processes. We illustrate the implementation of this mechanismthrough covalent modification (Heinrich, Neel, & Rapoport, 2002;Goldbeter & Koshland, 1981), such as phosphorylation (Jiang, Ven-tura, Sontag, Ninfa, & Del Vecchio, 2011).

This paper is organized as follows. In Section 2, we introducethe retroactivity problem through a motivating example. In Sec-tion 3, we introduce the system concept to model retroactivity,illustrate analogies with other engineering systems, provide anoperative quantification of retroactivity for a simple system, andthen for general gene transcription networks. Section 4 formulatesthe problem of designing insulation devices as a disturbance atten-uation problem and tackles it by using singular perturbation the-ory. In this section, we also illustrate one specific biomolecularimplementation of an insulation device through phosphorylationand discuss implications for natural systems.

2. Motivating example

As a motivating example, consider the activator-repressor clockof Atkinson et al. (2003) showed in Fig. 1(a). This oscillator is com-posed of an activator A activating itself and a repressor R, which, inturn, represses the activator A. Both activation and repression oc-cur through transcription regulation. Transcription regulation is acommon regulatory mechanism used in synthetic gene circuits.In transcription regulation, a protein, called a transcription factor,binds specific DNA sites, called operator sites, located on a regioncalled promoter, which is upstream of the gene of interest and re-presses or activates the gene’s expression. When the gene is ex-pressed, the protein that the gene encodes is produced, throughthe process of transcription/translation (a process called the cen-tral dogma of molecular biology (Alberts, Johnson, Lewis, Raff, &Roberts, 2007)). This protein can, in turn act as an activator orrepressor for other genes, creating circuits of activation and repres-sion interactions (Alon, 2007b). It is common practice in syntheticbiology to fabricate the desired DNA sequences of promoters andgenes on plasmids (see (Bleris et al., 2011), for example). Plasmidsare circular pieces of DNA that come with a nominal copy number,from low copy (1-5 copies of plasmid per cell) to high copy (hun-dreds copies of plasmid per cell).

There are several different modeling frameworks that can beused to mathematically describe the dynamic evolution of biolog-ical networks, depending on the desired level of granularity inspace and time. At the coarsest level, we have reaction rate equa-tion models in the form of ordinary differential equations (ODE),in which individual states represent concentrations (number ofmolecules divided by the cell volume) of chemical species. Thesemodels provide a good approximation of the system dynamics

0 20 40 60 80 1000

0.1

0.2

0.3

0.4

0.5Activator

time

isolatedconnected

0 20 40 60 80 1000

2

4

6

8

10Repressor

time

(c)

(a) (b)

(d)

Fig. 1. (a) Shows the activator–repressor clock when isolated and (b) when connected to a downstream system. (c) and (d) Shows the trajectories of the activator andrepressor concentrations when the clock is isolated (solid line) and when it is connected (dashed line). In all simulations, we have aA = aR = 100, a0,A = .04, a0,R = .004, dA = 1,dR = 0.5, Ka = Kr = 1, n = 2 and m = 4 (model (1)). In the connected clock configuration (b), we also have kon = koff = 100 and pT = 20 (model where the dynamics of A modify tothose modeled in Eq. (2)).

D. Del Vecchio / Annual Reviews in Control 37 (2013) 333–345 335

when the number of molecules is sufficiently high and the space is‘‘well stirred’’, that is, there are no significant spatial gradients inthe number of molecules. When, instead, the number of moleculesis low, stochastic models need to be employed. These models in-clude the chemical master equation (Van Kampen, 2007; Gillespie,1977), which describes the evolution in time of the probability ofthe number of molecules of each species, and the chemical Lange-vin equation (Gillespie, 2000), which describes the dynamic evolu-tion of the noise under the assumption of sufficiently largevolumes and molecule numbers. Reaction-diffusion equations,which take the form of a parabolic partial differential equation,are instead employed when there is a significant spatial gradientin the concentration of chemical species (Shvartsman & Baker,2012).

In this paper, we focus on ODE models under the assumptionthat the number of molecules in the cell volume is sufficientlyhigh. Letting italics denote the concentration of species, the ODEmodel of the clock can be written as

_A ¼ aAðA=KaÞn þ a0;A

1þ ðA=KaÞn þ ðR=KrÞm� dAA;

_R ¼ aRðA=KaÞn þ a0;R

1þ ðA=KaÞn� dRR;

ð1Þ

which describes the rate of change of the activator A and repressorR concentrations. In this model, dA and dR represent protein decay,

due to dilution and/or degradation. The functions aAðA=KaÞnþa0;A

1þðA=KaÞnþðR=Kr Þm

and aRðA=KaÞnþa0;R

1þðA=KaÞnare Hill functions (Alon, 2007b). The first one in-

creases with A and decreases with R, since A is an activator and Ris a repressor for A, while the second one increases with A as A isan activator for R.

It was shown in the work of Vecchio (2007) that the key mech-anism by which this system displays sustained oscillations is aHopf bifurcation with bifurcation parameter given by the relativetime scale between the activator and the repressor dynamics. Spe-cifically, as the activator dynamics become faster than the repres-sor dynamics, the system goes through a supercritical Hopf

bifurcation and a periodic orbit appears (solid plots of Fig. 1c andd).

Assume now that we would like to use the clock as a signal gen-erator to, for example, provide the timing to or synchronize down-stream systems. To do so, we need to take one of the proteins of theclock, say protein A, as an input for a downstream system(Fig. 1(b)), in which A will activate the expression of another pro-tein D, for example. In this case, one needs to add to the clockdynamics the description of the physical process by which infor-mation is transmitted from the upstream system to the down-stream one. In any biomolecular system, information istransmitted through (reversible) binding reactions. In this case, nmolecules of A will reversibly bind with the promoter p controllingthe expression of protein D to form a transcriptionally active com-plex C (see Jayanthi & Del Vecchio, 2012 for more details on themodel). Letting pT denote the total concentration of this promoter(given by the plasmid copy number divided by the cell volume)and kon and koff the association and dissociation rate constants,we have that the A dynamics modify to

_A ¼ aAðA=KaÞn þ a0;A

1þ ðA=KaÞn þ ðR=KrÞm� dAA � nkonAnðpT � CÞ þ nkoff C;

_C ¼ konAnðpT � CÞ � koff C;

ð2Þ

while the differential equation for R remains the same. Since A is anactivator of D, we will also have that _D ¼ kC � dDD, in which k anddD are the production rate constant and the decay rate constant,respectively.

As a result of the interconnection, the clock stops functioning(dashed plots in Fig. 1(c-d)). This effect has been called retroactivityto extend the notion of loading or impedance to non-electrical sys-tems, and in particular to biomolecular systems (Del Vecchio et al.,2008). It is due to the fact that the communicating species, A in thiscase, when occupied in the reactions of the downstream systemcannot participate in the reactions of the upstream system andhence the clock behavior is affected.

336 D. Del Vecchio / Annual Reviews in Control 37 (2013) 333–345

In the next sections, we illustrate a systems theory frameworkto explicitly model retroactivity so to make the problem of retroac-tivity amenable of theoretical study.

Fig. 3. Interconnection of two systems. (a) The upstream system is sending thesignal y and the downstream system is receiving it. Information travels fromupstream to downstream. The r and s arrows represent retroactivity to the inputand retroactivity to the output, respectively. (b) V and I in the electrical systemrepresent voltage and current, respectively; (c) f and p in the hydraulic systemrepresent flow and pressure, respectively; (d) the upstream system is the activator/repressor clock described in Section 2. Retroactivity takes different forms depend-ing on the specific physical domain, but it always has the physical meaning of aflow.

3. Retroactivity

In order to model retroactivity, we propose to model systems asshown in Fig. 2. Specifically, we explicitly add retroactivity as sig-nals traveling from downstream to upstream, that is, in the direc-tion opposed to that in which information is traveling (fromupstream to downstream). Signal s is the retroactivity to the out-put and models the fact that whenever the output y of R becomesan input to a downstream system, this system affects the upstreamsystem dynamics because of the physics of the interconnectionmechanism. Similarly, r is called the retroactivity to the inputand models the fact that whenever R receives signal u, it changesthe dynamics of the sending system. Related system concepts in-clude that of Paynter (1961) and that of Polderman and Willems(2007). Differently from Paynter (1961), our framework does notrequire that signals r and u (s and y) are generalized effort and flowvariables and hence that their product is the power flowingthrough the port. Differently from Willems (1999), we keep adirectionality to signals as we consider upstream-to-downstreamas the direction in which we think information is being transmit-ted. From a practical point of view, this is useful because a moduleis usually characterized by forcing input signals and measuring theconsequent output signal. So, there is an intrinsic directionality al-ready associated with the information transfer within a module.

The concept of retroactivity is not restricted to biomolecularsystems and it applies in general to many engineering systems(Fig. 3). For illustration, we provide two examples: an electricaland an hydraulic system.

Electrical Systems. Fig. 3(b) depicts a voltage generator withinternal resistor R0 (upstream system) and the downstream systemto which it is connected, a load resistor with value RL. The output ofthe isolated system is y = V0 while the output of the connected sys-tem is given by y = V0 � R0I. The two outputs are the same whenthe current drawn by the resistor is I = 0. Hence, we can takes = I. Note that for the connected system, I can be rendered arbi-trarily small by taking RL very large, corresponding to a down-stream system with high input impedance (a downstreamsystem with low retroactivity to the input).

Hydraulic Systems. Fig. 3(c) depicts the interconnection of twotanks. In the case in which the upstream tank is isolated, we havethat the valve at the output pipe is closed and hence the outlet flowis zero, that is, f = 0. When the tank is connected to the down-stream tank, we have that f ¼ qk

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðp� p1Þ

pfor p > p1, in which q

is the fluid density and k is a parameter that depends on the outputvalve geometry. In this case, the retroactivity to the output can betaken as s = f. This additional flow will cause a change in the pres-sure p of the upstream tank. For the connected system this flow canbe rendered small by decreasing k, which corresponds to decreas-ing the aperture of the valve.

Biomolecular Systems. Fig. 3(d) depicts the biomolecular clockmodule with downstream system with sites p to which A binds.In the case in which the clock is not connected, A is not taken asan input to any downstream system, while when it is connected,A serves as an input to a downstream system. This interconnection

Fig. 2. System concept with retroactivity.

takes place by having A bind to targets p through a reversiblebinding reaction. The corresponding reaction rate is given by�nkon An p + nkoff C. Hence, we can take s = � nkon An p + nkoff C. This‘‘usage’’ of A can change the behavior of the clock and hence thevalue of the A concentration as discussed in Section 2.

Retroactivity physically has the form of a ‘‘flow’’ between sys-tems: it is a current in the case of the electrical example, a flowof fluid in the case of the hydraulic system, and the rate of a chem-ical reaction in the biomolecular system. Also, note that retroactiv-ity is fundamentally different from feedback. In fact, s can beremoved only if the connection from the upstream system to thedownstream system is broken, that is, y is not transmitted to thedownstream system. If s were feedback, it could be removed whilekeeping the transmission of y from upstream to downstream.

3.1. Quantification of retroactivity in a simple biomolecular system

Because of retroactivity, the connected behavior of a modulediffers from the behavior of the same module in isolation (s = 0).We seek to predict how the behavior of a module will change uponinterconnection as a function of measurable biochemical parame-ters. As a simple example, consider the dynamics of a protein Xsubject to production and decay:

_X ¼ KðtÞ � dX; ð3Þ

in which the production rate K(t) varies with the activity of the pro-moter controlling the expression of X. When this protein is used asan activator or repressor for another protein, such as protein A inthe clock is used to activate protein D, we need to modify thedynamics of X to

_X ¼ KðtÞ � dX � konXðpT � CÞ þ koff C;_C ¼ konXðpT � CÞ � koff C;

ð4Þ

in which we have assumed for simplicity that X binds to DNA pro-moter sites p as a monomer (in one copy). In this system, we havethat s = �konX(pT � C) + koffC is the retroactivity to the output. Howthis additional reaction rate affects the X dynamics when comparedto the isolated system (3) is the next question we address.

D. Del Vecchio / Annual Reviews in Control 37 (2013) 333–345 337

To answer this question, we can exploit the natural time-scaleseparation between protein production and decay (d), of the orderof minutes to hours, and binding/unbinding rates (kon/koff), of theorder of seconds/subseconds (Alon, 2007b). By defining the smallparameter � = d/koff and the slow variable z = X + C, system (4)can be taken to the standard singular perturbation form (Khalil,2002)

_z ¼ KðtÞ � dðz� CÞ;

� _C ¼ dKdðz� CÞðpT � CÞ � dC;

ð5Þ

in which Kd = koff/kon is the dissociation constant of the binding be-tween X and p. One can show that the slow manifold of this systemis locally exponentially stable (Del Vecchio et al., 2008), so that sys-tem (5) can be approximated up to order � by the reduced systemwhere � = 0 given by _z ¼ KðtÞ � dX and CðXÞ ¼ pT X

XþKd. Considering that

_z ¼ _X þ _C ¼ _X 1þ dCdX

� �;

and that _z ¼ KðtÞ � dX, we finally obtain

_X ¼ ðKðtÞ � dXÞ 11þRðXÞ

� �; RðXÞ ¼ pT=Kd

ðX=Kd þ 1Þ2; ð6Þ

in whichRðXÞ ¼ dCdX. Comparing this system with the isolated system

(3), we note that the net effect of retroactivity is that of decreasingthe rate of change of the species X by a factor 1=ð1þRðXÞÞ. Theexpression of RðXÞ provides an operative quantification of retroac-tivity as a function of relevant biochemical parameters. Specifically,retroactivity increases when pT increases (the load increases) and/orKd decreases (the affinity of the binding increases), which is physi-cally intuitive. Furthermore, since RðXÞ > 0, we have that Eq. (6)implies that the dynamics of X slow down upon connection to thedownstream system. Further, as Kd becomes smaller, this ‘‘slowdown’’ becomes close to a finite-time delay (Fig. 4).

The effects of retroactivity on the behavior of biomolecular sys-tems have been experimentally demonstrated in the MAPK cascadein eukaryotic cells (Kim et al., 2011), in gene circuits in bacterialcells (Jayanthi, Nilgiriwala, & Del Vecchio, 2013), and in signalingsystems in vitro (Ventura et al., 2010a; Jiang et al., 2011).

0.4

0.5

trat

ion

3.2. Revisiting the effects of retroactivity on the clock

Since retroactivity slows down the dynamics of the output spe-cies, it is natural that the clock in Section 2 stops oscillating if theload is sufficiently high. In fact, the addition of the load causes theactivator dynamics to slow down compared to the repressordynamics and hence the system moves to the ‘‘left’’ of the super-critical Hopf bifurcation so that the equilibrium point becomes sta-ble. Using the results of the previous section, the clock dynamics

0 20 40 60 80 100 120 140 160 180 2000

0.2

0.4

0.6

0.8

1

1.2

time

A c

once

ntra

tion

pT=0

pT=100,K

d=1

pT=100,K

d=0.1

pT=100,K

d=0.001

Fig. 4. Effect of retroactivity on the dynamics of X when subject to constantproduction rate K(t) = 1.

with load on the activator become (see Jayanthi & Del Vecchio,2012 for the full derivation)

_A ¼ aAðA=KaÞn þ a0;A

1þ ðA=KaÞn þ ðR=KrÞm� dAA

� �1

1þR0ðAÞ

� �;

_R ¼ aRðA=KaÞn þ a0;R

1þ ðA=KaÞn� dRR;

ð7Þ

in which we have R0ðAÞ ¼ npTðAðn�1Þ=Kn

aÞ=ð1þ ðA=KaÞnÞ2, so that the

activator dynamics become slower compared to the unloaded case(1). In particular, the effect of retroactivity is that of making the rateof change of A slower compared to the rate of change of R. As a con-sequence, when the activator load pT is increased, the system goesthrough a Hopf bifurcation so that the stable limit cycle disappearsand the system trajectories converge to a stable equilibrium point(Fig. 5).

At this point, one may ask how the clock behavior would be af-fected if the load were applied to the repressor. Since retroactivityslows down the dynamics of the repressor, one should expect thata non-oscillating clock can be turned into an oscillating clock forsufficiently high load as the system moves through the Hopf bifur-cation. This is in fact the case as formally shown in the work ofJayanthi and Del Vecchio (2012).

These results illustrate that downstream targets allow to tunethe dynamics of a module without changing the parts of the mod-ule itself, hence providing an additional degree of freedom in de-sign. Therefore, retroactivity does not always have a negativeconnotation and natural systems may actually use it in advanta-geous ways. It is very well known, for example, that transcriptionfactors in natural systems can have large numbers of DNA bindingsites, several of which do not even have regulatory functions (seeBurger, Walczak, & Wolynes, 2010). Our finding highlights an addi-tional role that these DNA binding sites may have, that is, to tuneeffective kinetic rates in natural gene regulatory networks.

3.3. A Thevenin-like framework to quantify retroactivity in genenetworks

The operative quantification of retroactivity outlined in the pre-vious sections can be extended to the interconnection of any twogeneral gene transcription networks (Fig. 6). In particular, giventhe isolated modules dynamics _xA ¼ f A

0 ðxA;uAÞ and _xB ¼ f B0 ðxB; uBÞ,

in which xA, xB, uA, uB are vectors with dim(yA) = dim(uB), one can

0 1 2 3 4 50

0.1

0.2

0.3

pT

Act

ivat

or C

once

n

Fig. 5. Bifurcation diagram for the system in Eq. (7), in which activator load is pT. AHopf bifurcation occurs in correspondence to the asterisk. Parameters are the sameas in Fig. 1.

Module A

Fig. 6. Network A connects to network B.

338 D. Del Vecchio / Annual Reviews in Control 37 (2013) 333–345

demonstrate (see Gyorgy & Del Vecchio, 2013 for the technical de-tails) that the dynamics of connected module A will have the form

_xA ¼ ðI þ ðI þ RAÞ�1SBÞ

�1ðf A

0 ðxA; uAÞ þ ðI þ RAÞ�1MBf B

0 ðxB; yAÞÞ; ð8Þ

in which RA, SB, and MB are state-dependent matrices. Specifically,RA depends only on parameters (promoter amounts and dissocia-tion constants) of module A, while SB and MB depend only onparameters of module B. Matrix RA is called internal retroactivityand quantifies the effect of intramodular connections on thedynamics of module A in isolation. In fact, the vector fieldf A0 ðxA;uAÞ contains in its expression also matrix RA according to

f A0 ðxA;uAÞ ¼ ðI þ RAÞ�1½ðgAðxA;uAÞ � QA _uA�, in which gA(xA,uA) is a vec-

tor field that models the dynamics of the nodes in module Aneglecting retroactivity and QA is a matrix that determines howthe rate of change of the input affects the state (see Gyorgy & DelVecchio, 2013 for details). Matrix MB is called the mixing retroactiv-ity and quantifies the ‘‘coupling’’ between the isolated dynamics ofmodule A and the isolated dynamics of module B. Specifically, whenMB = 0, the isolated dynamics f B

0 of module B do not appear in thedynamics of module A, so that the two dynamics are not ‘‘mixed’’.From a physical point of view, this mixing occurs when nodes inmodule B have parents both from module B itself and module A,so that transcription factors from A and B interfere with each otherwhile binding to promoter sites in module B. Matrix MB models thephenomenon by which transcription factors in B can force tran-scription factors from A to bind/unbind promoter sites in moduleB, thus effectively changing the free concentration of transcriptionfactors in A. When MB = 0, the dynamics of module A are simply gi-ven by

_xA ¼ ðI þ ðI þ RAÞ�1SBÞ

�1f A0 ðxA;uAÞ; ð9Þ

so that the dynamics of module A are a ‘‘matrix-scaled’’ version ofthe dynamics of A in isolation. This is why matrix SB is called thescaling retroactivity and quantifies the loading effect that moduleB has on module A due to transcription factors in A binding to pro-moter sites in module B. When also SB = 0, the dynamics of moduleA are the same as in isolation.

These retroactivity matrices can be calculated once the measur-able parameters (promoter amounts and dissociation constants)and the interconnection graph are known. Specifically, they canbe calculated as:

RAðxA;uAÞ ¼X

i2 nodes in A

VTi RiðxA;uAÞVi

SBðxA; xBÞ ¼X

i2 input nodes in B

WTi RiðxA;uAÞWi

MBðxA; xBÞ ¼X

i2 input nodes in B

WTi RiðxA;uAÞDi;

in which Ri(xA,uA) is the retroactivity of the node and depends on thepromoter concentration at the node and on the dissociation con-stant of the same promoter. For example, if the node has only oneparent, Ri is a scalar quantity and it has the expression given inEq. (6). If the node has two parents, Ri is a two by two matrix whoseoff-diagonal entries are zero if the parents bind independently tothe promoter, while they are non-zero when the parents bindnon-independently (cooperatively or competitively). The specificexpressions can be found in the work of Gyorgy and Del Vecchio(2012). The matrices Vi, Wi, and Di have entries equal to 0 or 1and encode the interconnection graph among nodes. Specifically,Vi encodes the topology of the interconnection graph internal tomodule A, while Wi and Di encode the topology of the interconnec-tion graph between nodes in module A and nodes in module B.

The conceptual analogy between Eq. (8) and Thevenin’s theo-rem for electrical networks is instructive. Specifically, Thevenin’sresult shows that a linear electrical network between two termi-nals can be reduced, independently of its complexity, to an equiv-alent ideal energy source and an equivalent impedance (Agarwal &Lang, 2005). In particular, an upstream module A has an equivalentvoltage source f A

0 and an equivalent output impedance ZA, while itsdownstream module B can be represented by an equivalent voltagesource f B

0 and an equivalent input impedance ZA (Fig. 7). Referringto Fig. 7, when module A and module B are not connected to eachother the voltage at the output terminals of module A is given byf A ¼ f A

0 . When these modules are connected, the output voltageof module A becomes

f A ¼ 1ðZA=ZBÞ þ 1

f A0 þ

ZA=ZB

ðZA=ZBÞ þ 1f B0 ;

so that when ZA/ZB � 0, that is, the output impedance of module A ismuch smaller than the input impedance of module B, then we havethat f A � f A

0 . Comparing this equation with Eq. (8), it is apparentthat the ratio ZA/ZB plays a conceptually similar role to the ‘‘ratios’’(I + RA)�1SB and (I + RA)�1MB: as SB and MB become ‘‘small’’ com-pared to (I + RA), equivalently, ZA becomes ‘‘small’’ compared to ZB,the output of module A is the same as that in isolation (not con-nected to module B). Here, we have placed quotes on ‘‘small’’ sincewe are dealing with matrices for the case of gene networks and withcomplex numbers for the case of electrical circuits. The notion ofsmall in the case of the gene network can be made precise by calcu-lating the matrix 2-norm (see Gyorgy & Del Vecchio, 2013 for moredetails). Therefore, from a conceptual point of view the internal ret-roactivity RA plays a similar role to the output admittance 1/ZA ofmodule A in the electrical circuit, while SB plays a similar role tothe input admittance 1/ZB of module B in the electrical circuit. Thisanalogy is purely conceptual since the structure of the two systems(gene network and electric network) is fundamentally differentespecially since one (the electrical network) is linear while theother (gene network) is nonlinear.

We can further quantify the difference between trajectories ofconnected and isolated systems as function of measurable bio-chemical parameters. Specifically, let x(t) and x̂ðtÞ denote the tra-jectory of the state of Module A when in isolation and whenconnected to module B, respectively. When MB = 0, it can be shownunder mild assumptions that the difference between the isolatedand connected module trajectories satisfies

kxðtÞ � x̂ðtÞk2 ¼ OðlÞ; l ¼ rMðSBÞ=rmðI þ RAÞ1� rMðSBÞ=rmðI þ RAÞ

;

in which rM(SB)/rm(I + RA) < 1, and rM(A) and rm(A) denote the larg-est and smallest singular values of A, respectively. Parameter l pro-vides a metric of robustness to interconnections. It can be employedto either partition a natural network into modules whose behaviorcan be isolated (for modules with small l), to some extent, from

Fig. 7. Equivalent representation of an upstream electrical network (Module A) and a downstream electrical network (Module B).

UpstreamSystem

DownstreamSystem

D. Del Vecchio / Annual Reviews in Control 37 (2013) 333–345 339

that of the surrounding modules. At the same time, it can also beemployed in the bottom-up design of synthetic circuits by matchingthe output retroactivity of module A to the scaling retroactivity ofmodule B so that l is small, leading to a module A that preservesits isolated behavior after connection to module B.

In summary, this framework can be employed to predict thebehavior of a transcription network once it is connected into a lar-ger system, that is, to understand how the input/output propertiesof a module depend on its context.

4. Insulation

From a design point of view, it is often desirable that the behav-ior of the upstream system does not change when it is connected toa downstream one. However, we have seen in the previous sectionsthat retroactivity causes a potentially dramatic change in thedynamics of the upstream system. If one cannot design the down-stream system to have low scaling and mixing retroactivity com-pared to the internal retroactivity of the upstream system, adifferent approach is required to connect the two systems whilepreserving their isolated input/output response. A viable optionis to design a device that, once placed between module A and mod-ule B, insulates them from retroactivity effects (Fig. 8). We call thisdevice an insulation device and we formally specify its properties byrequiring that (a) r � 0 and (b) the effect of s on y is completelyattenuated. The first requirement can be satisfied by picking theparts of the input nodes of the device so that they have low retro-activity to the input (low scaling and mixing retroactivity). Thesecond requirement can be formulated as a disturbance attenua-tion problem and it is the focus of the next section.

4.1. Retroactivity attenuation through high-gain feedback

One established technique for disturbance attenuation is high-gain negative feedback (Young, Kokotovic, & Utkin, 1977). We illus-trate how this idea applies to our problem by considering again thedynamics of X in the isolated

_X ¼ KðtÞ � dX

and connected

_X ¼ ðKðtÞ � dXÞð1=ð1þRðXÞÞ

Fig. 8. An insulation device is placed between an upstream module A and adownstream module B in order to protect these systems from retroactivity. Aninsulation device should have r � 0 and the dynamic response of y to u should bepractically independent of s.

upstream system configuration. In this case, the idea of high-gainfeedback is to apply a negative feedback gain G and, in order notto attenuate the signal X(t), to apply a similarly large gain G0 = aGfor some a > 0 to the input K(t). In this case, the dynamics of the iso-lated and connected systems become

_X ¼ G0KðtÞ � dX � GX ðisolatedÞ_�X ¼ ðG0KðtÞ � d�X � G�XÞ 1

1þRð�XÞ

� �ðconnectedÞ; ð10Þ

so that when G increases, we have that j XðtÞ � �XðtÞ j! Oð1=GÞ (DelVecchio et al., 2008), which implies that the effect of retroactivityon �XðtÞ is attenuated as G increases.

The next question is how to implement this mechanismthrough a biomolecular system. To this end, consider a phosphor-ylation cycle (Fig. 9), in which protein X is converted to the activeform X⁄ through an enzymatic reaction with a kinase K (the inputof the system) and converted back through another enzymaticreaction with a phosphatase P (Klipp, Herwig, Kowald, Wierling,& Lehrach, 2005). Only when active, protein X can act as a tran-scription factor and activate or repress the expression of D. The ba-sic idea is that amplification of the input K should occur throughthe forward cycle reaction while the negative feedback should oc-cur through the reverse cycle reaction. In order to understand howthis can be explained mathematically, we consider a simple modelof the cycle, in which enzymatic reactions are modeled throughone-step reactions:

K þ X!k1 K þ X�

P þ X� !k2 P þ X;

in which we let XT denote the total amount of cycle protein. Alongwith these equations, we need to model also the binding reaction ofX⁄ with sites in the downstream system so that, after applying sin-gular perturbation as performed before, we obtain

_X� ¼ ðk1XT KðtÞð1� X�=XTÞ � k2PX�Þ 11þRðX�Þ

� �: ð11Þ

X*X

KInput

P

OutputD

Fig. 9. A phosphorylation cycle is a protein modification mechanism in which aninactive protein X is converted by a kinase K to an active form X⁄, which isconverted back to X by a phosphatase P.

340 D. Del Vecchio / Annual Reviews in Control 37 (2013) 333–345

Comparing this equation with Eq. (10), we see that the gain on theinput is given by G0 = k1XT, while the negative feedback gain is givenby G = k2P. As a consequence, we can conclude that as XT and P areincreased, the behavior of X⁄ should be minimally affected by theretroactivity to the output of the phosphorylation cycle. This resultimplies that phosphorylation cycles can function as insulation de-vices, suggesting another reason why these cycles are ubiquitousin natural signal transduction: they can enforce unidirectional sig-nal propagation, which is certainly desirable in any (human-madeor natural) signal transmission system. Related works have pro-posed that adding an explicit negative feedback to the cycle shouldfurther improve its robustness (Sauro & Ingalls, 2007). Here, we fo-cus on the cycle without explicit negative feedback due to simplerexperimental implementation.

The hypothesis that phosphorylation cycles function as insula-tion devices when both the amounts of cycle protein and phospha-tase are sufficiently large is appealing since it can beexperimentally tested. However, model (11), while providing anintuition behind the mechanisms responsible for retroactivityattenuation, is overly simplified and hides many important detailsthat may be relevant to the overall cycle robustness. Specifically,covalent modification cycles can be modeled considering the enzy-matic reactions as two-step processes (Goldbeter & Koshland,1981) and can even include all of the details of the units that ma-keup a protein, some (or all) of which can be modified by the en-zymes (Ventura et al., 2010b). In the latter case, the ODE modelof the upstream system in Fig. 9 can even have several dozens ofstate variables. Hence, a mathematical framework to study insula-tion from retroactivity is required to handle models of arbitrarydimension. This is illustrated in the next section.

4.2. Retroactivity attenuation through time scale separation

We have developed a technique to analyze and design retroac-tivity attenuation, which holds for arbitrarily complex biomolecu-lar network models. The basic idea exploits separation of timescales and the specific structure of the interconnection betweenbiomolecular modules. This can be illustrated through the follow-ing simplified treatment. For the general results, the reader is re-ferred to the work of Jayanthi and Del Vecchio (2011).

For the insulation device of Fig. 8, which we refer to as systemR, we seek to determine conditions under which the dynamic re-sponse of y to u is minimally affected by retroactivity s. In orderto do so, we write the model of the system in its isolated configu-ration (s = 0) and in its connected configuration (s – 0) and quan-tify the difference between the trajectories of the isolated andconnected systems. Specifically, letting u be a vector variable andassuming for simplicity that y 2 Rn is the state of R, we can writethe dynamics of the isolated system as

_u ¼ f0ðt; uÞ þ rðu; yÞ_y ¼ G1f1ðu; yÞ; ð12Þ

in which G1 > 0 is a positive constant, and we can write the dynam-ics of the connected system as

_�u ¼ f0ðt; �uÞ þ rð�u; �yÞ_�y ¼ G1f1ð�u; �yÞ þ G2Msð�y;vÞ_v ¼ �G2Nsð�y;vÞ;

ð13Þ

in which G2 > 0 is also a positive constant and M and N are matricescalled stoichiometry matrices (Klipp et al., 2005). Here, v is a vectorvariable that models the dynamics of the downstream system towhich R is connected. The distinctive structure of the interconnec-tion comes in the fact that matrices M and N are such that there is anon-singular n � n matrix B and a matrix T such that B M � T N = 0.

This is the case because the entries of s physically represent the rateof reversible binding between two species and it always affects withopposite signs the species involved in the binding, one of which be-longs to the upstream system and the other belongs to the down-stream system. The constant G2 models the fact that bindingreactions are among the fastest reactions in biomolecular networks,so that G2� 1.

Now, assume that we can take G1� 1 probably not as large asG2 but still sufficiently larger than 1. This can be achieved, forexample, by letting y be driven by protein modification reactions,such as phosphorylation or allosteric modification, which are usu-ally much faster than protein production and decay processes(Klipp et al., 2005). In this case, we can re-write the dynamics ofthe connected system (13) by using the change of variablesz ¼ B�yþ Tv; �1 ¼ 1=G1, and �2 = 1/G2 as

_�u ¼ f0ðt; �uÞ þ rð�u; �yÞ�1 _z ¼ Bf1ð�u; �yÞ�2 _v ¼ �sð�y;vÞ;

which is in standard singular perturbation form with two smallparameters. Under the assumption that the slow manifold is locallyexponentially stable (the technical conditions can be found inJayanthi & Del Vecchio (2011)) the above system can be wellapproximated by one in which �1 = 0 and �2 = 0. This leads to�y ¼ cð�uÞ (the locally unique solution of f1ð�u; �yÞ ¼ 0), which is thesame solution as found in the isolated system (12) when �1 = 0.As a consequence, we can conclude that kyðtÞ � �yðtÞk ¼ Oð�1Þ for0 < tb 6 t < T for some tb > 0 and T > 0, independently of the valueof G2. This result implies that if the time scale of R is sufficiently fas-ter than the time scale of the input and suitable stability conditionsare satisfied, then R attenuates the effects of retroactivity s on theresponse of y to u.

4.3. Implementation through phosphorylation cycles

In view of the result of the previous section, we can consider amore realistic model of the phosphorylation cycle and exploit thenatural time scale separation between phosphorylation and geneexpression (controlling K(t)) to show retroactivity attenuation.

Consider a two-step reaction model for the phosphorylationreactions, given by

Kþ X �b1

b2

C1!k1 X� þ K

Pþ X� �a1

a2C2!

k2 Xþ P

with conservation laws PT = P + C2,AT = X + X⁄ + C1 + C2 + C, alongwith the binding of X⁄ with downstream sites p, that is,

X� þ p �kon

koff

C:

The resulting ODE model is given by

_K ¼ kðtÞ � dK � b1XT K 1� X�

XT� C1

XT� C2

XT� C

XT

� �

þ ðb2 þ k1ÞC1

_C1 ¼ �ðb2 þ k1ÞC1 þ b1XT K 1� X�

XT� C1

XT� C2

XT

�� C

XT

�

_C2 ¼ �ðk2 þ a2ÞC2 þ a1PT X� 1� C2

PT

� �

_X� ¼ k1C1 þ a2C2 � a1PT X� 1� C2

PT

� �

þ koff C � konX�ðpT � CÞ_C ¼ �koff C þ konX�ðpT � CÞ: ð14Þ

0

1

2

time

High G1

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000

0

1

2Low G1

isolatedconnected

Fig. 10. Simulation of the phosphorylation cycle in (14) with low gain G1 and highgain G1 when K(t) is a periodic signal. Specifically, we have d = 0.01, XT = 5000,PT = 5000, a1 = b1 = 2 � 10�6G1, and a2 = b2 = k1 = k2 = 0.01G1, in which G1 = 10(upper panel), and G1 = 1000 (lower panel). The downstream system parametersare kon = 100,koff = 100 and, thus, G2 = 10000. Simulations for the connected system(s – 0) correspond to pT = 100 while simulations for the isolated system (s = 0)correspond to pT = 0.

D. Del Vecchio / Annual Reviews in Control 37 (2013) 333–345 341

To take this system in the form (13), we can define the gain G1 byconsidering the separation of time scales between gene expressionand protein phosphorylation, so that

G1 :¼ b1XT

d� 1;

b2 :¼ b2G1; a2 :¼ a2

G1; a1 :¼ a1PT

G1, and ji :¼ ki

G1for i = 1, 2. Similarly, we can

define the gain G2 by considering the separation of time scales be-tween gene expression and binding reactions, so that

G2 :¼ koff

d� 1

and Kd :¼ koffkon

. By using the change of variables z = K + C1 and ensur-ing that XT� pT so that C/XT� 1, we can re-write the system as

_z ¼ kðtÞ � dðz� C1Þ

_C1 ¼ G1 �ðb2 þ j1ÞC1 þ dðz� C1Þ 1� X�

XT� C1

XT

��� C2

XT

��

_C2 ¼ G1 �ðj2 þ a2ÞC2 þ a1X� 1� C2

PT

� �� �

_X� ¼ G1 j1C1 þ a2C2 � a1X� 1� C2

PT

� �� �

þ G2 dC � dKd

X�ðpT � CÞ� �

_C ¼ �G2 dC � dKd

X�ðpT � CÞ� �

;

which is in the form of system (13) with

u ¼ z; y ¼ ðC1;C2;X�Þ0; v ¼ C;

s ¼ koff C � konX�ðpT � CÞ;M ¼ ð0;0;1Þ0; N ¼ 1; r ¼ 0:

Therefore, the main result of the previous section applies with B = Iand T = (0,0,1)0, so that as G1 increases, the response of X⁄ to K be-comes insensitive to retroactivity s, as also reflected by the simula-tion results of Fig. 10.

This technique is applicable to large models and was utilized fordesigning experiments on a covalent modification cycle in vitro(Jiang et al., 2011) and for designing a buffer between an in vitrobiomolecular oscillator and a load (Franco, Del Vecchio, & Murray,2009; Franco et al., 2011).

From a design point of view, the gain G1 can be made larger byincreasing the total concentration of substrate XT and of phospha-tase PT in comparable amounts. Note, however, that in doing so theloading applied to the input kinase K increases, so that the retroac-tivity to the input r of the insulation device also increases. As a con-sequence, by increasing G1 this way, the retroactivity attenuationproperty improves to the expense of applying increased retroactiv-ity to the input K. As a result, the output is not the desired one as itmirrors an input K(t) that has been distorted due to retroactivity r.In this case, it becomes necessary to perform an optimal design inwhich the amounts of substrate and phosphatase are optimized tominimize the overall error due to both the contribution of r and ofs. This optimal design can be performed by explicitly computingbounds on the errors due to r and s and my minimizing these,which, in turn, can be obtained by employing robustness resultsfrom contraction theory. For details on these, the reader is referredto the work of Rivera-Ortiz and Del Vecchio (2013).

A further downside in overexpressing proteins is that the cost ofthe circuit in terms of cellular resources increases (Stoebel, Dean, &Dykhuizen, 2008). Of particular concern in a phosphorylation cycleis the consumption of ATP, the cellular energy currency. An exten-sive analysis of how increased gains (increased substrate and phos-phatase concentrations) reflect in an increased energy cost of thecycle can be found in the work of Barton and Sontag (2013). Finally,

it is well known from the Bode’s integral formula for linear systemsthat increased gains can lead to undesired sensitivity increase incertain frequency ranges, ultimately potentially leading to noiseamplification. This limitation is found also in the presented insula-tion design, for which a simplified analysis using the chemicalLangevin equation can be found in the work of Jayanthi and Delvecchio (2009). All these constraints should be taken into accountwhen implementing the insulation device design.

4.4. Connecting the clock to its downstream system through aninsulation device

In this section, we illustrate how the insulation device imple-mented through phosphorylation of Fig. 9 can be employed to con-nect the clock to the desired downstream system such that (a) theclock behavior is not significantly perturbed and (b) the signal fromthe clock is transmitted to the downstream system robustly withrespect to the load that this downstream system applies. To thisend, consider the diagrams depicted in Fig. 11. The top diagramillustrates a simplified genetic layout of the clock. The activator Ais expressed from a gene under the control of a promoter activatedby A and repressed by R, while the repressor is expressed from agene under the control of a promoter activated by A. Protein A,in turn, activates the expression of protein D in the downstreamsystem. In this case, the promoter p controlling the expression ofD contains a sequence that A recognizes, so that A can bind to it.

When the insulation device of Fig. 9 is employed to interconnectthe clock to the downstream system, two modifications need to bemade to enable the connections. Since A is not a kinase, we need toinsert in sequence to the gene expressing A the gene expressing thekinase K (bottom diagram of Fig. 11). Since both A and K are underthe control of the same promoter, they will be produced at thesame rates and hence the concentration of K should mirror thatof A. Note that a solution in which we insert in sequence to thegene expressing A a transcription factor K that directly binds todownstream promoter sites p0 to produce D (without the insulationdevice in between) would not solve the problem. In fact, while theclock behavior would be preserved in this case, the behavior of theconcentration of K would not mirror that of A since protein Kwould be loaded by the downstream promoter sites p0 (Fig. 12).As a consequence, we would still fail to transmit the clock signalto protein D. The second modification that needs to be made isto change the promoter p to a new promoter p0 that has a sequencethat the protein X⁄ recognizes (bottom diagram of Fig. 11).

R A

p

D

R AK

X*X

P p’

D

Fig. 11. Illustration of how one can interconnect the clock to its downstream system through the insulation device of Fig. 9. The top diagram illustrates a simplified geneticlayout of the activator–repressor clock of Fig. 1(a). The boxes represent the genes expressing R, A, and D, while the arrows upstream of the genes represent the promoters thatcontrol these genes. The bottom diagram illustrates how the genetic layout of the clock should be modified such that it can connect to the phosphorylation cycle that takes asinput the kinase K. In this case, the downstream system still expresses protein D, but its expression is controlled by a different promoter that is activated by X⁄ as opposed tobeing activated by A.

342 D. Del Vecchio / Annual Reviews in Control 37 (2013) 333–345

In the case of the bottom diagram of Fig. 11, the dynamics of theclock remain that of model (1). To these, we need to add thedynamics of K(t), which, when the phosphorylation cycle is notpresent, will be given by

_K ¼ aAðA=KaÞn þ a0;A

1þ ðA=KaÞn þ ðR=KrÞm� dK K: ð15Þ

When the phosphorylation cycle is present, this ODE changes to

_K ¼ aAðA=KaÞn þ a0;A

1þ ðA=KaÞn þ ðR=KrÞm� dK K

� b1XT K 1� X�

XT� C1

XT� C2

XT� C

XT

� �;

ð16Þ

in which the term r ¼ �b1XT K 1� X�

XT� C1

XT� C2

XT� C

XT

� �represents the

retroactivity to the input of the insulation device realized by thephosphorylation cycle. The ODE model of the insulation device withthe downstream system remains the same as before and given bythe second to fifth equations of system (14), in which we replacepT by p0T , so that the retroactivity to the output of the insulation de-vice is given by s ¼ koff C � konX�ðp0T � CÞ. Fig. 13 shows the trajecto-ries of A(t),K(t), and X⁄(t) for the system of Fig. 11. As desired, thesignal X⁄(t), which drives the downstream system, closely tracks thatof A(t) despite the retroactivity due to load applied by the down-stream sites p0. Note that because of a non-zero retroactivity to theinput r of the insulation device, the trajectory of K(t) is slightly

0 20 40 60 80 1000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

time

K c

once

ntra

tion

without downstream systemwith downstream system

Fig. 12. Simulation results for the concentration of protein K in Fig. 11 in the case inwhich this were used directly as an input to the downstream system, thus bindingsites p0 (dashed red plot). (For interpretation of the references to colour in thisfigure legend, the reader is referred to the web version of this article.)

different from the same trajectory in the absence of the insulationdevice (Fig. 13(b)). The retroactivity to the output s only slightlyaffects the output of the insulation device (Fig. 13(c)). The plot ofFig. 13(c), showing the signal that drives the downstream system,can be directly compared to the signal that would drive the down-stream system in the case in which the insulation device wouldnot be used (Fig. 12). In the latter case, the downstream systemwould not be properly driven, while with the insulation device it is.

4.5. Natural mechanisms for retroactivity attenuation

In natural signal transduction systems, phosphorylation cyclesoften appear in cascades, such as in the MAPK cascades (Seger &Krebs, 1995; Rubinfeld & Seger, 2005), where signals from outsidethe cell are transmitted down to the gene expression machinery.Cascades often intersect with each other by sharing common sub-strates, so that perturbations at the bottom or at intermediatestages of a cascade are frequently applied by the intersecting cas-cade(s) (Roux & Blenis, 2004; Müller, 2004). Since, we have seenthat a load applied to the output protein of a cycle can affect thedynamic state of the cycle itself, a natural question is how thesedownstream perturbations propagate backward toward the up-stream stages of the cascade. Specifically, the role of the lengthof a cascade has been subject of many studies in the systems biol-ogy community, indicating that it has specific functions in signalamplification, signal duration, and signaling time (Heinrich et al.,2002; Chaves, Sontag, & Dinerstein, 2004). Here, we investigatewhether the length of the cascade has any role in how perturba-tions applied downstream of a cascade propagate backward.

To investigate this question, we refer to the cascade depicted inFig. 14. Here, the downstream substrate D has total availableamount DT, which we assume is subject to perturbations due toother intersecting cascades. Letting dT be a small perturbation ofDT about a steady state value, the ODE system can be linearizedabout the steady state corresponding to DT, and a static gain fromdT to the total phosphorylated ith cycle protein perturbation zi canbe obtained (see the work of Ossareh, Ventura, Merajver, & DelVecchio (2011) for details). Let

Ui :¼ zi

ziþ1and Utot :¼ Pn�1

i¼1 Ui

represent the gain from downstream stage i to upstream stage i + 1and the gain from stage n to stage 1, respectively. Then, it is possibleto show that

j Ui j< 1 and Ui < 0;

0 20 40 60 80 1000

0.1

0.2

0.3

0.4

0.5

time

A c

once

ntra

tion

0 20 40 60 80 1000

0.1

0.2

0.3

0.4

0.5

time

K c

once

ntra

tion without insulation device

with insulation device

0 20 40 60 80 1000

0.1

0.2

0.3

0.4

0.5

time

X*

conc

entr

atio

n

without downstream systemwith downstream system

(a)

(c)

(b)

Fig. 13. Simulation results for the system of Fig. 11. Panel (a) shows theconcentration of the clock activator protein A. Panel (b) shows the concentrationof the kinase K with and without (r = 0 in Eq. (16)) the insulation device. Panel (c)shows the behavior of the output of the insulation device without (s = 0) and withthe downstream system. The clock parameters are the same as those in Fig. 1 anddK = dA. The phosphorylation cycle parameters are as follows: k1 = 5000, k2 = 5000,a1 = 1, b1 = 1, b2 = 1000, a2 = 1000, PT = 1500, and XT = 1500. The load parameters aregiven by kon = koff = 100 and p0T ¼ 100.

Fig. 14. Cascade of phosphorylation cycles. Proteins W�i are kinases that lead to

phosphorylation of the substrates at the next downstream stage i + 1. Proteins Ei arephosphatases for stage i. D represents a downstream substrate, where we assume aperturbation is applied.

D. Del Vecchio / Annual Reviews in Control 37 (2013) 333–345 343

that is, the perturbation from stage i+1 to stage i is attenuated inmagnitude and there is a reversal of the sign as the perturbationtravels from one stage to the next upstream. As a consequence,the total gain Utot is also strictly less than one and it becomes smal-ler as the number of stages increases. That is, the perturbation in thetotal phosphorylated protein is attenuated as it propagates up-stream in the cascade. Hence, longer cascades present enhancedretroactivity attenuation ability. For a more detailed study alsoincluding the free phosphorylated protein and a computationalstudy with parameters from established databases, the reader is re-ferred to the work of Ossareh et al. (2011).

This finding illustrates that natural systems may mitigate retro-activity effects by employing multiple stages of covalent modifica-tion, allowing each stage to have smaller gains. In fact, the same

attenuation of retroactivity can be obtained by employing onestage with high gains or two stages, each with lower gain. The factthat natural systems may prefer using multiple stages with lowergains is not surprising. In fact, high gains imply significant expen-diture both in terms of energy (ATP) and in terms of gene expres-sion machinery, such as ribosomes, RNA polymerase, and aminoacids (Barton & Sontag, 2013).

5. Conclusions and discussion

In this review paper, we have illustrated how problems of load-ing, called retroactivity, are found in biomolecular systems just likethey are found in many engineering systems. These effects alter thebehavior of a module upon interconnection and hinder modularanalysis and design approaches. Differently from electrical circuits,which can be analyzed to a large extent through linear systemstheory, biomolecular network models are highly nonlinear andhence their study requires nonlinear systems theory. We haveillustrated how, using singular perturbation theory, we can analyzeand quantify retroactivity effects by obtaining equivalent systemrepresentations, just like Thevenin’s theorem does for electricalcircuits. Inspired by the approach used in electrical circuit design,we analyzed the question of how to design devices that attenuatethe retroactivity to the output and have low retroactivity to the in-put (insulation devices). We described an approach to answer thisquestion by exploiting the structure of the interconnection foundin biomolecular networks and employing singular perturbationtheory after a suitable change of variables. We illustrated on a sim-ple example, how an insulation device could be used in syntheticbiology circuits to connect systems to each other, while keepingthe isolated systems behaviors.

Many system-level challenges remain to be addressed for theanalysis and design of biological networks. First of all, we havenot touched on the problem of noise in biology, which is a central

344 D. Del Vecchio / Annual Reviews in Control 37 (2013) 333–345

issue since all the reactions that make the circuits are probabilisticin nature. Unfortunately, while many analysis and design tools fornonlinear dynamical systems are available, only a limited pool ofsimilar results are available for stochastic systems (Khammash &EISamad, 2005). Hence, problems such as designing the probabilitydistribution of a system’s trajectories, as opposed to designing theirmean value, remain mostly open, while being very important forengineering biological circuits. We have briefly mentioned theinteractions that arise between biomolecular circuits and the cellmachinery, such as due to loading of cellular resources or to inter-fering directly with important cellular pathways. These interactionsare rarely modeled due to the fact that most of them are unknown.As a result, it is currently very difficult to design circuits that behaverobustly once in the cellular chassis. Designing circuits such thattheir function is robust to external perturbations, parameter uncer-tainty, unwanted interactions, and competition for shared resourcesis a central problem, which is currently open. Robust design ap-proaches have been extensively developed in the control commu-nity, but most of these tools are hardly applicable to biomolecularsystems, which are highly nonlinear and often rely on nonlinearitiesfor proper functioning. Finally, another hard question that remainsopen is the ‘‘realization’’ question, that is, how to implement a givenfunction with the available biological parts. Addressing this ques-tion is especially challenging as it requires, even more than the otherquestions, a synergistic interplay of system-level analytical exper-tise and a deep knowledge of the available biological mechanisms.

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Domitilla Del Vecchio received the Ph.D. degree in Control and Dynamical Systemsfrom the California Institute of Technology, Pasadena, and the Laurea degree inElectrical Engineering from the University of Rome at Tor Vergata in 2005 and 1999,respectively. From 2006 to 2010, she was an Assistant Professor in the Departmentof Electrical Engineering and Computer Science at the University of Michigan, AnnArbor. In 2010, she joined the Department of Mechanical Engineering and theLaboratory for Information and Decision Systems (LIDS) at the MassachusettsInstitute of Technology (MIT), where she is currently an associate professor. She is arecipient of the Donald P. Eckman Award from the American Automatic ControlCouncil (2010), the W.M. Keck Career Development Professorship, MIT (2010), theNSF Career Award (2007), the Crosby Award, University of Michigan (2007), theAmerican Control Conference Best Student Paper Award (2004), and the Bank ofItaly Fellowship (2000). Her research interests include nonlinear control theorywith application to biological and transportation networks.