Retroactivity Attenuation in Bio-Molecular Systems Based on Timescale Separation

748 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 56, NO. 4, APRIL 2011

Retroactivity Attenuation in Bio-MolecularSystems Based on Timescale SeparationShridhar Jayanthi, Student Member, IEEE, and Domitilla Del Vecchio, Member, IEEE

Abstract—As with several engineering systems, bio-molecularsystems display impedance-like effects at interconnections, calledretroactivity. In this paper, we propose a mechanism that exploitsthe natural timescale separation present in bio-molecular systemsto attenuate retroactivity. Retroactivity enters the dynamics of abio-molecular system as a state dependent disturbance multipliedby gains that can be very large. By virtue of the system structure,retroactivity can be arbitrarily attenuated by internal systemgains even when these are much smaller than the gains multi-plying retroactivity terms. This result is obtained by employinga suitable change of coordinates and a nested application of thesingular perturbation theorem on the finite time interval. As anapplication example, we show that two modules extracted fromnatural signal transduction pathways have a remarkable capa-bility of attenuating retroactivity, which is certainly desirable inany (engineered or natural) signal transmission system.

Index Terms—Bio-molecular system, retroactivity.

I. INTRODUCTION

M ODULARITY is a fundamental property that allows theprediction of the behavior of a system from the behavior

of its components, guaranteeing that the input/output behaviorof a component does not change upon interconnection. Thisproperty is often taken for granted and tacitly exploited in sev-eral engineering areas, such as electrical engineering. Modu-larity is usually a fair assumption because mechanisms such asoperational amplifiers in suitable feedback configurations areemployed so that impedance effects at interconnections can beneglected [1]. As a result, systems can be conveniently com-posed by simple static output-to-input assignments. Modularityhas been more recently advocated also in systems biology andin synthetic biology, in which networks of bio-molecular in-teractions between species, such as proteins, enzymes, DNAsites, and signaling molecules take place. In particular, in sys-tems biology one seeks to understand the behavior of a nat-ural bio-molecular network from the behavior of the composingmodules or motifs [2]–[4]. Complementary to systems biology,researchers in synthetic biology aim at constructing complexnetworks of interactions between genes and proteins in living

Manuscript received June 29, 2009; revised February 13, 2010; accepted June29, 2010. Date of publication August 23, 2010; date of current version April 06,2011. This work was supported in part by AFOSR Award FA9550-09-1-0211.Recommended by Associate Editor D. Angeli.

S. Jayanthi is with the Electrical Engineering and Computer ScienceDepartment, University of Michigan, Ann Arbor MI 48109 USA, (e-mail:[email protected]).

D. Del Vecchio is with the Department of Mechanical Engineering, Massa-chusetts Institute of Technology, Cambridge MA 02139 USA (e-mail: [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TAC.2010.2069631

cells with the ultimate goal of controlling cell behavior. A keyapproach in doing so is the design and construction of simplebio-molecular systems, such as oscillators [5], [6] and toggles[7], which are then interconnected in a modular fashion to de-sign bio-molecular circuits with more complex functionalities[8], [9].

A fundamental systems engineering issue that arises wheninterconnecting systems with each other is how the dynamicstate of the sending system (upstream system) is affected by thedynamic state of the receiving system (downstream system).The effect of downstream loads has been well characterizedand accounted for in electrical, mechanical, and hydraulicsystems. It has been recently argued that similar problemsappear in bio-molecular systems. In particular, Alon statesthat bio-molecular modules, just like engineering modules,should have special features that make them easily embeddedin almost any system. For example, output connections shouldhave “low impedance” so that connecting additional down-stream clients should not change the output to existing clientsup to some limit [10]. A recent theoretical study has shown,however, that output connections in bio-molecular systems donot always have low impedance. Instead, they can be affectedby large impedance-like effects that dramatically distort thedynamics of a system in the face of downstream loads [11].These impedance-like effects have been called retroactivity toextend the notion of impedance to non-electrical systems andin particular to bio-molecular systems.

From a systems biology point of view, one method to dealwith retroactivity is to partition large networks into “modules”for which retroactivity effects are minimal, by employing graphand information theoretic approaches [12]–[16]. By contrast,the studies in [11], [17] consider fixed modules, which is morealigned with the synthetic biology perspective. Specifically,[11] characterizes retroactivity, and investigates interconnectionmechanisms that provide arbitrary retroactivity attenuation.To this end, it proposes an alternative model to the standardinput/output model employed in virtually every systems engi-neering book [18] (a notable exception to the standard systeminput/output model is Willem’s work [19], which blurs thedistinction between inputs, states, and outputs). Within thisalternative modeling framework, an input/output system modelis augmented with two additional signals: the retroactivity tothe input and the retroactivity to the output (Fig. 1).

In this formalism, achieving low output impedance becomesthe problem of attenuating retroactivity to the output. Theproblem of arbitrarily attenuating the retroactivity to the outputis in turn conceptually similar to problems of disturbanceattenuation and decoupling [20], [21]. Insulation devices arethen designed in such a way to (a) arbitrarily attenuate theretroactivity to the output (thus they can keep the same output

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JAYANTHI AND DEL VECCHIO: RETROACTIVITY ATTENUATION IN BIO-MOLECULAR SYSTEMS 749

Fig. 1. System� with input and output signals, along with the interconnectionstructure with its upstream and its downstream systems. The retroactivity to theoutput � accounts for the change in the system� dynamics when it is connectedto downstream systems. The retroactivity to the input � accounts for changes that� causes on upstream systems when it connects to receive the information �.

independently of the downstream systems connected to suchan output) and (b) have low retroactivity to the input (thus theydo not affect the upstream system from which they receive thesignal). Insulation devices can be placed between the upstreamsystem sending the signal and the downstream one receivingthe signal to insulate these systems from retroactivity. Onedesign method for bio-molecular insulation devices has beenillustrated in [11]. It employs a large amplification gain in anegative feedback loop (in analogy to the design of non-in-verting amplifiers in electronics) to attenuate the retroactivityto the output.

In this paper, we show that a special interconnection structurefound in bio-molecular systems enables a different mechanismfor retroactivity attenuation. Retroactivity to the output entersthe system dynamics as a state dependent disturbance, which isoften multiplied by very large gains. These gains are large dueto the fact that bio-molecular system interconnection often oc-curs through processes that can be among the fastest processesin bio-molecular systems [10], [22]. We show that, for a classof systems with this interconnection structure, whenever the dy-namics of a system evolves on a timescale faster than that ofits upstream system, the retroactivity to the output of can bearbitrarily attenuated. We also show that this attenuation prop-erty is independent of the gains multiplying retroactivity andthat the faster the timescale of system with respect to its up-stream system, the better the retroactivity to the output atten-uation achieved. As a consequence, one can arbitrarily atten-uate state dependent disturbances even when these enter the dy-namics of system through gains that are orders of magnitudehigher than the gains internal to itself. In order to show thisretroactivity attenuation capability enabled by timescale sepa-ration, we employ singular perturbation techniques for systemswith one and multiple small parameters [23]–[25].

Singular perturbation arguments have been used in biochem-ical applications to show the validity of the quasi-steady stateapproximation for enzyme kinetics [26]–[31]. In these studies,the timescale separation stems from large initial conditionsof either substrate or enzyme, or due to large values for theMichaelis-Menten constant. The separation of timescales inthe systems studied in this paper are also due to differences inthe order of magnitude of the reaction rates of the processesconsidered. In bacterial systems, for example, the timescale ofgene expression and protein dilution is of the order of minutes[10], [32], the one of post-translational modification processesrange from the order of milliseconds to seconds [33], [34], thatof proteins binding to small signaling molecules can be in thesub-second timescale [10], and the timescale of transcriptionfactor-DNA interactions can be as fast as few milliseconds[22], [35], [36].

Despite several timescales being present in the processes hereconsidered, the resulting models are not in standard singularperturbation form. This issue arises because the states involvedin the interconnection are shared by systems with dynamics indifferent timescales. This problem is often encountered also inchemical reaction systems [37], [38] and in biochemical sys-tems [27], [29]. A common solution is to employ a change ofvariables for which the system is in standard singular pertur-bation form. In this study, we provide sufficient conditions forthe existence of a linear coordinate transformation that takes theoriginal system to standard singular perturbation form. Then, weperform a nested application of Tikhonov singular perturbationtheorem on the finite time interval as it appears in standard [25].Finally, by taking the reduced system back to the original coor-dinates, we find that the dynamics of the original system on theslow manifold is independent of the retroactivity to the output.

As an application example, we show how modules extractedfrom natural signal transduction systems can attenuate theretroactivity to the output based on the separation of time scalemechanism illustrated in the paper. It is also shown that thecapacity of attenuating retroactivity holds independently of thetimescale of the downstream interconnection. The examplesin this paper employ a phosphorylation cycle and a phospho-transfer module, both of which are ubiquitous in natural signaltransduction systems [39], [40].

This paper is organized as follows. In Section II, we intro-duce the bio-molecular system model and the retroactivity tothe output attenuation problem. The main result is provided inSection III, in which a change of coordinates and a nested appli-cation of Tikhonov singular perturbation theorem is performed.Section IV shows the application of the theory to two motifsextracted from natural signal transduction systems: a phospho-rylation system and a phosphotransfer system.

II. SYSTEM MODELAND PROBLEM FORMULATION

In this paper, we consider the system model depicted in Fig. 1.In addition to the usual input and output signals, we add twoadditional signals traveling from downstream to upstream: aretroactivity to the output and a retroactivity to the input .The retroactivity to the output is a signal (which may dependon and on the internal variables of the downstream system)that appears in the dynamics of whenever is connected tothe downstream system. The retroactivity to the input (whichmay depend on and on ) is a signal that system applies toits upstream system as an input whenever connects to the up-stream system to receive the information . The system is saidisolated when it is not connected to the downstream system. Insuch a case, .

From an engineering point of view, signals and do notnecessarily carry information. They are present only because ofthe physics of the interconnection between system components.For example, if is a voltage generator with voltage andinternal resistance , the value of its output when is iso-lated is exactly equal to . However, when is connected toa downstream load, a voltage drop is caused by current flowingthrough the internal resistance so that the new value ofits output voltage will be smaller than what we had in the iso-lated configuration. In this case, is due to the non-zero cur-rent flowing through upon interconnection with the down-


stream load. A similar situation is found in bio-molecular sys-tems. When a synthetic bio-molecular oscillator, such as thoseof [5], [6], is employed as a signal generator and connected todownstream clients to, for example, synchronize them, the os-cillator dynamics can be dramatically affected [11].

A. Bio-Molecular System Model

In this section, we specialize the general interconnectionstructure of Fig. 1 to the case of bio-molecular systems sothat , , and arevectors whose components denote concentrations of chemicalspecies, such as proteins, enzymes, DNA sites, etc. We employa model similar from a formal point of view to that of metabolicnetworks [41]. Let and be reactionrate vectors modeling the interaction of species in the vector

with species in the vector and of species in the vectorwith species in the vector , respectively. Let ,

, , and be constant matrices.Let , , and be vector fieldsand be positive constants. The model that we considerfor in the interconnection of Fig. 1 is thus the following:

(1)

with initial conditions . The model ofwhen it is isolated from the downstream system becomes( )

(2)

with initial conditions , .System (1) is a general model for a bio-molecular system.

Interconnections always occur through reactions, whose rates( and , in this case) appear in both the upstream and the down-stream systems with different coefficients (captured by matrices

, , , and ). Constants and explicitly model the factthat some of the reactions may be several orders of magnitudefaster than others. Constant models the timescale of system

. In this paper, we are interested in those cases in whichevolves on a faster timescale than its input, that is, .This situation is encountered, for example, when modelsprotein modification processes (such as phosphorylation, al-losteric modification, dimerization, etc.), while its upstreamsystem models slower processes such as protein productionand decay or signaling from outside the cell (here modeledby ) [10], [33], [34]. Constant models the timescaleof the interconnection mechanism of with its downstreamsystem. For example, when this downstream system modelsgene expression, models the binding and unbinding processof transcription factors to DNA binding sites. This reaction isfaster than expression and degradation of proteins and therefore,

[10], [22]. Additionally, it is possible for the proteinmodification processes to be in the same range as, much fasterthan, or much slower than DNA-transcription factor bindingand unbinding [33]–[36]. Therefore, it is important to considerthe cases in which , , and .

B. Retroactivity Attenuation Problem

In this paper, we are interested in determining conditionsthat allow to attenuate the retroactivity to the output andin quantifying the retroactivity to the input. To this end, wedefine the retroactivity to the output attenuation property ofsystem in the interconnection structure of Fig. 1 as follows.Let , ,and , be the unique solutions for

with to systems (1) and (2), respectively.Definition 1: System has the retroactivity to the output at-

tenuation property provided there are constants ,, , and a compact set

such that the following properties hold for and:

(i)when as ;

(ii)when as and .

If a system enjoys the retroactivity to the output attenua-tion property, its dynamics are not affected by the retroactivityto the output as grows, independent of the value of . Inparticular, independently of whether is smaller than, muchlarger than, or of the same order as , state dependent distur-bances can be arbitrarily attenuated by a sufficientlylarge . Furthermore, one can achieve arbitrary retroactivityattenuation by properly adjusting the system parameter .

The remainder of this paper focuses on providing sufficientconditions under which system enjoys the retroactivity tothe output attenuation property. Furthermore, we quantify theretroactivity to the input of by determining the impact ofon the dynamics of .

III. PROBLEM SOLUTION

System (1) models processes occurring at multipletimescales. Specifically, since , there are at leasttwo timescales and when there are three timescales.However, there may not be a separation of timescales. Thesystem is however not in standard singular perturbation form[25]. This situation is typical of bio-molecular and chemicalsystems. Such systems often display multiple timescales butthere is no explicit separation between fast and slow variables[29], [37]. However, when the interconnection occurs troughbinding processes, faster reaction rates appear in the dynamicsof both upstream and downstream systems multiplied by inte-gers related to the stoichiometric coefficients [41]. Thereforeit is possible to extract the slow variables of a system througha linear combination of the states of the upstream and down-stream systems. This motivates an approach that employs alinear coordinate transformation to take the original system tostandard singular perturbation form.

In what follows, we first determine conditions for the exis-tence of a linear coordinate transformation independent ofand that transforms systems (1) and (2) to standard singularperturbation form. Then, we employ Tikhonov singular pertur-bation theorem to study the dynamics of the system on the slowmanifold. To this end, we restrict the class of systems (1) to thosewith the two following central properties:


P1 There is an invertible matrix and a matrixsuch that (i) ; (ii)

for all ; and (iii) .P2 There is an invertible matrix , and a matrix

such that (i) ; (ii) forall .

Let , , and

(3)

(4)

(5)

Then, we prove the following proposition.Proposition 1: Under properties P1 and P2, the linear

change of coordinates

(6)

takes systems (1) and (2) respectively to the standard singularperturbation forms

(7)

and

(8)

Proof: From the linear coordinate transformation (6), wehave that and . By substitutingin these relations the expressions of , and from system(1) (from system (2)), writing , and

, one obtains system (7) (system (8)).Conditions and from prop-

erty P1 and the conditions from property P2 ensure the exis-tence of a linear coordinate transformation that takes the systemto standard singular perturbation form. Additionally, condition

from property P1 is necessary to ensure that once, the dynamics of do not depend on , and thus

that the retroactivity to the output does not directly propagate tothe input. Properties P1 and P2 give sufficient conditions on theinterconnection structure that allows for insulation employingseparation of timescales. For low dimensional systems, matrices

, , and that satisfy properties P1 and P2 can be easilydetermined by inspection of matrices , , and . This isillustrated in Section IV with two five-dimensional applicationexamples. For more general cases, prior work has focused onthe existence and construction of non-linear coordinate trans-formations that bring a system to standard singular perturbationform [37].

For , define the domains andto be the images of through transformations

(6). We also define the map forall as . Note thatthis map is continuous and invertible. Similarly, define the map

for allas . Note

that this map is also continuous and invertible.

A. Technical Assumptions

In the following sections, a nested application of Tikhonovsingular perturbation theorem, as found in standard references,is employed in systems (7) and (8). To assure validity of thetheorems, we pose technical assumptions which are consideredvalid on the domains , and . In what follows,we say that a square matrix depending on isHurwitz uniformly for if there is a real such that

for all .A1 The functions are smooth;A2 The functions are Lipschitz continuous for all

;A3 The function is the unique solution of

, it is Lipschitz continuous and smooth;A4 The function is the unique solution of

and it is Lipschitz continuous;A5 The function is the unique solution of

and it is Lipschitz continuous;A6 We have that is Hurwitz uni-formly for ;A7 We have that is Hurwitzuniformly for ;A8 We have that

is

Hurwitz uniformly for ;A9 We have that is Hurwitz uni-

formly for .Assumptions A1 and A2 guarantee existence and uniqueness ofthe solutions of systems (1) and (2). As a consequence, assump-tions A1 and A2 also guarantee the existence and uniquenessof the solutions of systems (7) and (8). Assumptions A3, A4,A6 and A7 guarantee the stability of the boundary layer systemsobtained when employing a nested application of Tikhonov the-orem to system (7) for the case in which . Along withassumption A8, assumptions A3 and A4 are also employed toguarantee the stability of the boundary layer system in the appli-cation of Tikhonov theorem to system (7) for the case in which

and are of the same order of magnitude. AssumptionsA5 and A9 guarantee the stability of the boundary layer systemwhen employing Tikhonov theorem to system (8) and to system(7) when .

Proposition 2: Let be a solutionto . Then, such a solution is unique.Furthermore and .

Proof: Since is a solution to equa-tion , we have that

. By A3, this implies that . This along withimply that .

This along with A4 imply that . As a conse-quence, we have that .


B. Main Result

The main result of the paper is based on the two followinglemmas, which employ Tikhonov singular perturbation theoremin the form presented in [25]. Specifically, Lemmas 1 and 2 pro-vide approximations of the isolated and connected system tra-jectories, respectively, when we consider as small parameters

and . These approximations are thencompared with each other to obtain the retroactivity to the outputattenuation property, which is the main result of the paper.

Before giving the first lemma, we define the two followingsets. For any , define the set by

(9)

and let be any compact subset of . For ,define the set by

(10)

and let be any compact subset of . The nextproposition shows the relationship between the sets and

.Proposition 3: Consider the sets defined in (9) and (10).

Then, for all there is such thatimplies that .

Proof: Since is the unique solution ofand , it follows from the definition

of ((4)) that . Sinceis invertible, for all there is such that

implies. By applying the triangular inequality, one

can show that for all there is such thatand

imply .Finally, the continuity of along with the triangular inequalityimply that for all there is such that

and imply. Let .

As a consequence of this proposition, if iscompact, then the set is a compactsubset of .

Under properties P1-P2 and assumptions A1-A9, we give thetwo following lemmas.

Lemma 1: Let be the unique solu-tion of system (2) for with initial condition

and . Let be the unique solution of system

(11)

for with initial conditionwhere and

is the locally unique solution of. Then, there is such that for all there

exists such that andhold uniformly for

provided and .

Proof: For convenience, defineand denote the solution of

system (8) by for with. Let be the unique

solution of the algebraic equation and denote bythe unique solution of the reduced system

(12)

for and (the uniqueness of the so-lution follows from the fact that is Lipschitz continuous on itsdomain by Assumptions A2 and A5). Assumption A9 furtherguarantees that the boundary layer system is locally exponen-tially stable. The region of attraction thus contains the set ofsuch that for some sufficientlysmall. Define the set .Let be any compact subset of . By Tikhonovtheorem, for all , there exists such that

(13)

provided and .To obtain these approximations in the original coordinate

system, define

(14)

We seek to show that satisfies the differential (11). Sinceis the locally unique solution of , we

must have that

(15)

Since, (15) implies that

(16)

From the assumptions of the lemma, we have thatis the locally unique solution of

. This along with (16) imply that. As a consequence,

we can re-write (14) as . Taking thetime derivative of both sides of this expression, we obtain

. Employing (12) on theleft-hand side and re-arranging the terms, we obtain that

, in which (by (8))we have that

, leading to satisfying the differential (11) forwith .

Since and (13)hold, we have that

, which, by employing (14) and the factthat , leads to

(17)

provided that and . Sinceis the image of under the continuous map ,


we have that for any compact set , thereis a compact subset such that

. As a consequence, (17) hold provided andwith .

Set and .The following lemma provides approximations to the solution

of system (1) in a way similar to what was performed in Lemma1 for system (2). The main technical difference between Lemma1 and Lemma 2 is that system (1) has two small parameters, thatis, and , which can take different relativevalues. The proof of the lemma thus considers three differentcases: as (i.e., and are ofthe same order of magnitude); as (i.e.,

is orders of magnitude larger than ); as(i.e., is orders of magnitude larger than ). In par-

ticular, in the latter case the system has three different timescalesand therefore it is treated by performing a nested application ofTikhonov singular perturbation theorem.

Lemma 2: Let , ,be the unique solution of system (1) for

with initial conditions. Let be the unique solution of system

(18)

for with initial conditionwith and

the locally unique solution of . Then,there is such that for all there areand such that the following properties hold for

and :(i) and

uniformly forwhen as ;

(ii) anduniformly for

when as and.

Proof: Define for convenience the functionand

let be the uniquesolution of system (7) for with initial conditions

, ,and . There are three cases: as ,

as , and as .Case 1: as . We perform a nested ap-

plication of Tikhonov singular perturbation theorem. Define thenew small parameters and and re-writesystem (7) as

(19)

Set and let be the locally unique solution of. Let also and be the unique solution

of the reduced system obtained once

(20)

for , , and (unique-ness of the solution follows from Assumptions A2 and A3). As-sumption A6 further guarantees that the boundary layer systemis locally exponentially stable. For some sufficientlysmall, the region of attraction contains the set of such that

is sufficiently small. Define the setand let

be compact. Then, by Tikhonov theorem, for allthere is such that

(21)

hold provided and .System (20) is also in standard singular perturbation form

with small parameter . Set and let be thelocally unique solution of . Let be theunique solution of the resulting reduced system when

(22)

for with (uniqueness of the solutionfollows from Assumptions A2, A3, and A4). Furthermore, As-sumption A7 guarantees that the boundary layer system is lo-cally exponentially stable. For some sufficiently small,the region of attraction contains the set of such that

. Define the setand let be compact.

Then, from Tikhonov theorem, for all , there issuch that

(23)

hold provided and . As a con-sequence of relations (23), for the solution of system(20) is uniquely defined for . We can thus letso that for , with sufficiently small, also the so-lution of system (19) is uniquely defined for . Let

and define

Let be any compact set. Combining ex-pression (21) with and expression (23), the solution ofsystem (1) satisfies

(24)

in which we have used thatsince is smooth. In order to return to the original co-

ordinate system, define

(25)


We seek to show that satisfies (11). Since is thelocally unique solution of , by the definitionof ((4)), we have that

This equation along with the fact that is the locallyunique solution of lead to

(26)

Substituting this into (25) and re-arranging the terms, we obtainthe equation . Taking the time derivative bothsides, we obtain that . Employing(22) on the left-hand side and re-arranging the terms, we ob-tain , inwhich we have that

withfrom (25). Therefore, is the unique so-lution of (18) for and

,in which . Since is the unique solutionof and , it follows fromthe definition of ((4)) that .Thus, .

From the coordinate transformation (6), we have.

Employing the relations for and from (24), we obtain

By employing (25) and (26), one obtains that. Similarly, from the change of vari-

able

Equations (24), and (25), we obtain that. Hence, we have that

(27)

uniformly for provided , ,and . Since isthe image of under the continuous map , wehave that for any compact set ,there is a compact set such that

. As a consequence, (27) hold pro-vided , , andwith . Define and

.

Case 2: as . Letting ,system (7) becomes

(28)

Denote the solution of system (28) by , , andfor . By Proposition 2,is the locally unique solution of

. Define to simplify notation. Let bethe unique solution of the reduced system

(29)

for and (uniqueness of the solutionfollows from Assumptions A2, A3, and A4). Furthermore, As-sumption A8 guarantees that the boundary layer system is lo-cally exponentially stable. For some sufficiently small,the region of attraction contains the set of all such that

. Define the set

and let be compact. Then, by Tikhonovtheorem, for all there is such that

(30)

provided and .Define

(31)

We seek to determine the differential equation that obeys.Since , we have by the definition of((4)) that

Given that by assumption is the locally unique solu-tion of , we must have that

Substituting this in (31), we obtain that .Computing the time derivative both sides of this equa-tion, employing (29) and (31), one obtains thatis the solution of (18) for with


and

. Since, as for Case 1, we have that, then

.Finally, employing the change of coordinates (6) and approx-

imations (30), we obtain that

(32)

hold uniformly for providedand . Define the new region

. Bythe continuity of and the triangular inequality, it follows thatfor all there is such that .Since is an arbitrary compact subset of ,it can be chosen such that for

a suitable compact subset of . Sinceand is a continuous mapping,

we have that for all compact setsthere is a compact set such that

. As a consequence, (32) hold pro-vided and with

. Define .Case 3: as . In this case, only the

change of coordinates is applied to system (1),leading to the system in the new coordinates

(33)

Let be the locally unique solution to the equation(in which we have that

) and let be the unique solution of thereduced system

(34)

for with and (unique-ness of the solution follows from Assumptions A2 and A5). As-sumption A9 further guarantees that the boundary layer systemis locally exponentially stable. For some sufficientlysmall, the region of attraction contains the set of such that

. Define the set. Let be any compact set

contained in . From Tikhonov theorem, there issuch that for all , there exist such that

(35)

provided and . In orderto obtain the approximations in the original coordinate system,define . Since is the locallyunique solution of and is thelocally unique solution of

, we have that. Then, we can write and conclude

that is the unique solution to system (11) forwith . By employing thecoordinate change as performed in Case 1, wefinally obtain that

(36)

uniformly for provided and. Since is the image of

under the continuous map , for any compact set, there is a compact set

such that . As a consequence, (36)hold provided and .By Proposition 3, for all there is suchthat implies .Let be any compact set. Then,

implies for somecompact set . As a consequence, (36) holdprovided . Let .

By combining Case 1, Case 2, and Case 3, the result of thetheorem follows with ,

with , and.

By combining the results of the lemmas, we can obtain themain result of the paper.

Theorem 1: Under Properties P1-P2 and AssumptionsA1-A9, system has the retroactivity to the output attenuationproperty.

Proof: By virtue of Lemma 1, we have that there issuch that for all there exists such that

hold uniformly forwhenever and .

Similarly, Lemma 2 shows that there is such that forall there are and such that (i)and (ii) hold uniformly for in wheneverand . By Proposition 3, forall there is such thatimplies . Let beany compact set. Then, implies

for some compact set . Lettingand , we obtain the de-

sired result.Remark 1: (Retroactivity to the input) An immediate con-

sequence of Lemma 2 is the quantification of the retroactivityto the input of system , that is, the impact of on the dy-namics of the upstream system. Specifically, one can make theretroactivity to the input small by choosing the parameters ofin such a way to make small. Therefore,can be considered as a measure of the retroactivity to the inputof system .

IV. APPLICATION EXAMPLES

In this section, we show how the interconnection structureof system (1) is found in bio-molecular systems extracted fromnatural signal transduction pathways and how it can be used tobuild insulation devices. In particular, we consider as system


two post-translational modification systems which are recur-rent motifs in signal transduction: phosphorylation cycles andphosphotransfer systems. In both examples, the system outputis connected to the downstream system through the binding oftranscription factors to DNA. Studies show that this type of in-teraction can be much faster than, much slower than or of thesame order as the post-translational modification processes an-alyzed here. For example, [35] gives a first-order reaction rateof 40 for DNA-protein interaction, while [33], [34] givefirst-order reaction rates ranging from tofor phosphorylation cycles and phosphotransfer systems. There-fore, in this application, it is important to show that the retroac-tivity to the output attenuation property holds when the down-stream interconnection dynamics are of the same order as, muchfaster or much slower than the dynamics of system . Phospho-rylation cycles are among the most common intracellular signaltransduction mechanisms. They have been observed in virtuallyevery organism, carrying signals that regulate processes suchas cell motility, nutrition, interaction with environment and celldeath[42]. In this paper, we describe a phosphorylation systemextracted from the MAPK cascade [33], similar to the deviceproposed in [11]. While in [11] the timescales of the down-stream interconnection and that of the phosphorylation cycle arethe same, here we consider the situation in which the timescaleof the downstream interconnection is much faster than that ofthe phosphorylation cycle reactions. Phosphotransfer systemsare also a common motif in cellular signal transduction [43],[44]. These structures are composed of proteins that can phos-phorylate each other. By contrast to kinase-mediated phospho-rylation, in which the phosphate donor is usually ATP, in phos-photransfer the phosphate group comes from the donor proteinitself. Each protein carrying a phosphate group can donate itto the next protein in the system through a reversible reaction.In this paper, we describe a module extracted from the phos-photransferase system [45]. In this example, we consider all thethree possible relationships between the timescale of the down-stream interconnection and that of the phosphotranfer device.

A. Example 1: Phosphorylation

In this section, we analyze the dynamics of a system mod-eling a phosphorylation cycle as shown in Fig. 2. This systemtakes as input a kinase that phosphorylates a protein . Thephosphorylated form of , denoted , is a transcription factor,which binds to downstream DNA promoter binding sites .Therefore, the downstream system in terms of Fig. 1 is thebinding and unbinding process to DNA sites. The phosphory-lated protein is converted to the original dephosphorylatedform by phosphatase . A standard two-step reaction modelfor the phosphorylation and dephosphorylation reactions is

given by ,

respectively, in which and are the complexes of proteinwith substrate and of protein with protein , respec-

tively [46]. The binding reactions of transcription factor

with downstream binding sites p are given by , in

which is the complex of bound to site . In this system,the total amounts of proteins and and the total amountof promoter p are conserved. Their total amounts are denoted

, , and , respectively, so that the conservation laws are

Fig. 2. System� is a phosphorylation cycle. Its product� activates transcrip-tion through the reversible binding of� to downstream DNA promoter sites �.

given by , ,and . Assuming is expressed at time-varyingrate and decays at rate , the differential equations forthe concentrations of the various species of system whenconnected to the downstream system are given by

(37)

A common approach to take a system to the standard singularperturbation form is to rewrite it in terms of non-dimensionalvariables [25], [30]. To this end, let and de-fine the non-dimensional input . Define alsothe new variables

For a variable , denote . The system (37) in thesenew variables becomes

(38)

In this example, we assume the parameter to be muchlarger than , , which are in turn


much larger than [10], [32], [33], [35]. This timescale dif-ferences can be made explicit by defining the large parameters

and , in which .Define also the non-dimensional constants

Define also the dissociation constant . By em-ploying these constants, system (38) can be rewritten as

(39)

The domains for the variables of this system are givenby , , and

. Compare system (39) with the struc-ture of model (1). The retroactivity to the input term

is a function of the downstream system state . This im-plies that the retroactivity to the output of impacts directlythe retroactivity to the input. In order to remove this effect,and therefore, match the structure of system (1), in which

does not depend on , we require the ratio to besmall enough so that the term becomes negli-gible with respect to one, since . This assumptiongives a limit to the amount of load that can be added to thesystem for any fixed value of . Under this assumption,the system fits the structure (1) with ,

,

,

, , ,, , and

. By inspection of the matrices , , and , wecan choose matrices , , (3by 3 identity matrix) and that satisfyproperties P1 and P2. This can be verified by checking that in-deed , , ,and, trivially, . The linear coordinate transformationthat takes this system to the standard singular perturbationform is, thus, given by and

.Since we are considering the case in which , it

is necessary to show that technical assumptions A1–A7 and A9

are satisfied. For brevity, we show the properties A3, A6 and A7only. Expression leads to

which leads to the unique isolated solution

in the domain . This function is Lipschitzcontinuous as the argument of the square root is boundedaway from zero and thus A3 is satisfied. The Jacobianmatrix evaluated at is given by

,

in which the argument of the square root is always boundedaway from zero. Therefore, A6 is satisfied. The Jacobian

gives , in which

,, ,

, .We show that this Jacobian matrix is Hurwitz by employingthe Routh-Hurwitz criterion. Note first that , , and areall positive terms. The characteristic equation of the Jacobianis given by

Employing Routh-Hurwitz method, the terms in the first columnof the Routh-Hurwitz table are given by ,

and . Provided that is largeenough, all the coefficients are positive and, therefore, the realpart of all eigenvalues of is negative and propertyA7 is satisfied. Similarly, it is possible to show that assumptionsA4, A5 and A9 are satisfied.

Fig. 3 shows that, for low values of , the system does notattenuate the retroactivity to the output as the permanent be-havior of the isolated and connected systems are different. Bycontrast, and in accordance to the theory, large values oflead to retroactivity to the output attenuation. Note also thatthis property is achieved even if the gain multiplying thestate-dependent disturbance is much larger than .

In practice, while reactions rates , , and are oftenmuch larger than , constants and may not achieve suchhigh values [33]. It is, however, possible to compensate for thisand obtain the desired timescale separation by having largeramounts of and . Large values of and are alsoinstrumental in removing the direct effect of retroactivity to theouput on the retroactivity to the input. Finally, large values of

and are also necessary to guarantee the stability of theboundary layer system, as concluded when showing that prop-erty A7 holds. In a synthetic bio-molecular system, expression


Fig. 3. Output response to a sinusoidal signal �� ofthe phosphorylation system �. The parameter values are given by � � ��,� � ��, � � ��, � � ��, � � � � � �� , and � � � � � � � �� , in which � �� (left-side panel), and �� (right-side panel). The downstream system parameters are � � ��,� � �� and, thus, � ��. Simulations for the connected system(� �� ) correspond to � � �� while simulations for the isolated system(� � �) correspond to � � �.

Fig. 4. System � is a phosphotransfer system. The output activates tran-scription through the reversible binding of to downstream DNA promotersites �.

level of proteins X and Y can be tuned by having their respec-tive genes under the control of inducible promoters. It is there-fore possible to tune this system so that the retroactivity to theoutput attenuation property holds.

B. Example 2: Phosphotransfer

In this section, we model the phosphotransfer module shownin Fig. 4. Let be a transcription factor in its inactive formand let be the same transcription factor once it has beenactivated by the addition of a phosphate group. Let be aphosphate donor, that is, a protein that can transfer its phos-phate group to the acceptor . The standard phosphotransfer

reactions [34] can be modeled according to the two-step reac-

tion model , in which is the complex

of bound to bound to the phosphate group. Additionally,protein can be phosphorylated and protein dephospho-rylated by other phosphotransfer interactions. These reactionsare modeled as one step reactions depending only on the con-centrations of and , that is, , . Proteinis assumed to be conserved in the system, that is,

. We assume that protein is producedwith time-varying production rate and decays with rate .The active transcription factor binds to downstream DNAbinding sites p with total concentration to activate tran-

scription through the reversible reaction . Since the

total amount of is conserved, we also have that .The ODE model corresponding to this system is thus given bythe equations

(40)

As performed in Example 1, we introduce non-dimensionalvariables for this system. Let and define thenon-dimensional input . Define also the non-di-mensional variables , , ,

, and . For a variable , denote. System (40) in these new variables becomes

(41)

Phosphotranferase reactions are much faster than gene expres-sion and protein decay rates [34]. To make this timescale sepa-ration explicit, we define the large parameterand define the non-dimensional constants ,

, , and .The fact that the process of protein binding and unbinding topromoter sites is much faster than protein production and decay[10], [32] is made explicit by the ratio . Inthis example we do not make any assumption on the relation-ship between and . Let also the dissociation constant be


. By using these constants, system (41) can bewritten as

(42)

The domain for the states of this system are givenby , and

. Compare system (42) with system (1).In system (42), the internal dynamics term is given by

and it depends on output term . Therefore, in orderfor system (42) to fit the structure of system (1), werequire that the ratio to be small enough so that

becomes negligible with respect to 1 in the term, as . This assumption,

in practice, limits the amount of load this insulation devicecan accommodate for a given amount of . Under thisassumption, system (42) fits the structure of model (1) with

, ,

,

, ,

, , ,

and . By inspecting matrices , , and itis possible to choose matrices , ,

and , which satisfyproperties P1 and P2. This can be verified by checkingthat indeed , , ,

and, trivially, . By applying thelinear coordinate transformation given by and

, we obtain the system

(43)

In this example, we do not claim any relationship betweenand . In this the situation it is necessary to show that all

assumptions A1-A9 are satisfied to prove that the retroactivityto the output property holds. For brevity we restrict to show thatassumptions A3, A6 and A7 hold.

As in the phosphorylation system, we have that. Therefore, A3 and A6 are satisfied

as it was for the phosphorylation system.Since the function

issufficiently smooth (the argument of the square root isbounded away from zero) we define the diffeomorphism

. Define. Since under a

diffeomorphism the linearization of a nonlinearsystem is invariant [47], it is sufficient to show that

is Hurwitz. We have that

, in

which , , ,, , and

. The characteristic equation of thisJacobian is given by

Write the characteristic equation aswhere are implicitly defined. The terms on

the first column of the Routh-Hurwitz table are given byand . Since all are positive,

we are guaranteed to have only positive terms on the firstcolumn of the Routh-Hurwitz table if .In particular, the term can be reduced to

,in which the term . It remains to show that

on themanifold . From the system of equa-tions , one can obtain the identity

. Sub-stituting in this identity, we obtain that

. As a result,


Fig. 5. Output response of the phosphotransfer system with a periodic signal�� . The parameters are given by � � ��,� � ��,� � � � � � � � � � � � �� in which � � � (left-sidepanel), and � � �� (right-side panel). The downstream system parametersare given by � � � and � � �� , in which � assumes the valuesindicated on the legend. The isolated system ( � �) corresponds to � �while the connected system ( �� ) corresponds to � ��.

and thus, the Jacobian matrix is Hur-witz satisfying condition A7.

We illustrate the retroactivity to the output attenuation prop-erty of this system using simulations for the cases in which

, , and . Fig. 5 shows that, for aperiodic input , the system with low value for suffers theimpact of retroactivity to the output. However, for a large valueof , the permanent behavior of the connected system be-comes similar to that of the isolated system, whether ,

or . Notice that, in the bottom panel ofFig. 5, when , the impact of the retroactivity to theoutput is not as dramatic as it is when or .This is due to the fact that is scaled by and it is not relatedto the retroactivity to the output attenuation property. This con-firms the theoretical result that, independently of the order ofmagnitude of , the system can arbitrarily attenuate retroac-tivity for large enough .

V. CONCLUSION

In this paper, we have proposed a mechanism for attenuatingthe retroactivity to the output of a bio-molecular system whichis based on timescale separation. A special structure found inbio-molecular systems allows the attenuation of state dependentdisturbances that enter the dynamics through arbitrarily largegains. This attenuation can be achieved even when the internalsystem gains are several orders of magnitude smaller than thegains that multiply the disturbance. One structural assumption

at the basis of our result is that the retroactivity to the inputof the system and the vector field do not explicitly depend onthe variables of the downstream system. In future work, wewill investigate how the retroactivity to the output attenuationproperty may be relaxed when both the retroactivity and thefunction depend on .

We illustrated this mechanism by presenting two instancesof bio-molecular systems that have the capability of attenuatingthe retroactivity to the output based on timescale separation.These are a phosphorylation cycle and a phosphotransfersystem, which are ubiquitous in natural signal transductionsystems. Our finding suggests that a reason why these systemsare fundamental building blocks of natural signal transductionsystems is that, in addition to their well recognized signalamplification capability, they can attenuate retroactivity to theoutput and therefore enforce unidirectional signal propagation.This property is certainly desirable in any signal transmissionsystem, natural or engineered. More interestingly, this findingsuggests that phosphorylation and phosphotransfer systems canbe employed in synthetic bio-molecular circuits to attenuateretroactivity and to thus allow modular interconnection ofsynthetic circuit components.

ACKNOWLEDGMENT

The authors would like to thank Dr. E. Sontag andDr. L. Wang for constructive discussions.

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Shridhar Jayanthi (S’07) received the B.S. degreein computer engineering from the Instituto Tec-nológico de Aeronáutica (ITA), Brazil, in December2005 and is currently pursuing the Ph.D. degree inthe Electrical Engineering and Computer ScienceDepartment, University of Michigan, Ann Arbor.

He worked as Research and Development Engi-neer at Dixtal Biomédica from August 2005 to April2006 and was a Research Scholar with the MedicalImaging Processing Group, University of Pennsyl-vania from May 2006 to August 2007. His research

interests are in the stochastic properties of bio-molecular systems and in the de-sign of synthetic biology devices.

Domitilla Del Vecchio (M’05) received the and theLaurea degree in electrical engineering from the Uni-versity of Rome, Tor Vergata, in 1999 and the Ph.D.degree in control and dynamical systems from theCalifornia Institute of Technology, Pasadena, in 2005.

From January 2006 to May 2010, she has been anAssistant Professor in the Department of ElectricalEngineering and Computer Science and in the Centerfor Computational Medicine and Bioinformatics atUniversity of Michigan, Ann Arbor. She joined the

Department of Mechanical Engineering at the Massachusetts Institute of Tech-nology as an Assistant Professor in June 2010. Her research interests are in thecontrol of hybrid dynamical systems with imperfect information and in the anal-ysis and design of bio-molecular feedback systems.

Dr. Del Vecchio received the Donald P. Eckman Award from the AmericanAutomatic Control Council (2010), the NSF Career Award (2007), the CrosbyAward, University of Michigan (2007), the American Control Conference BestStudent Paper Award (2004), and the Bank of Italy Fellowship (2000).

Retroactivity Attenuation in Bio-Molecular Systems Based on Timescale Separation

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