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BMC Bioinformatics Research Stability analysis of the GAL regulatory network in Saccharomyces cerevisiae and Kluyveromyces lactis Vishwesh V Kulkarni* 1 , Venkatesh Kareenhalli 2 , Pushkar Malakar 2 , Lucy Y Pao 3 , Michael G Safonov 4 and Ganesh A Viswanathan 2 Addresses: 1 Department of Electrical Engineering, Indian Institute of Technology Bombay, Mumbai 400 076, India, 2 Department of Chemical Engineering, Indian Institute of Technology Bombay, Mumbai 400 076, India, 3 Department of Electrical, Computer, and Energy Engineering, University of Colorado, Boulder, CO 80302, USA and 4 Department of Electrical Engineering, University of Southern California, Los Angeles, 90089-2563, USA E-mail: Vishwesh V Kulkarni* - [email protected]; Venkatesh Kareenhalli - [email protected]; Pushkar Malakar - [email protected]; Lucy Y Pao - [email protected]; Michael G Safonov - [email protected]; Ganesh A Viswanathan - [email protected] *Corresponding author from The Eighth Asia Pacific Bioinformatics Conference (APBC 2010) Bangalore, India 18-21 January 2010 Published: 18 January 2010 BMC Bioinformatics 2010, 11(Suppl 1):S43 doi: 10.1186/1471-2105-11-S1-S43 This article is available from: http://www.biomedcentral.com/1471-2105/11/S1/S43 © 2010 Kulkarni et al; licensee BioMed Central Ltd. This is an open access article distributed under the terms of the Creative Commons Attribution License ( http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract Background: In the yeast Saccharomyces cerevisiae, interactions between galactose, Gal3p, Gal80p, and Gal4p determine the transcriptional status of the genes required for the galactose utilization. Increase in the cellular galactose concentration causes the galactose molecules to bind onto Gal3p which, via Gal80p, activates Gal4p, which induces the GAL3 and GAL80 gene transcription. Recently, a linear time-invariant multi-input multi-output (MIMO) model of this GAL regulatory network has been proposed; the inputs being galactose and Gal4p, and the outputs being the active Gal4p and galactose utilization. Unfortunately, this model assumes the cell culture to be homogeneous, although it is not so in practice. We overcome this drawback by including more biochemical reactions, and derive a quadratic ordinary differential equation (ODE) based model. Results: We show that the model, referred to above, does not exhibit bistability. We establish sufficiency conditions for the domain of attraction of an equilibrium point of our ODE model for the special case of full-state feedback controller. We observe that the GAL regulatory system of Kluyveromyces lactis exhibits an aberration of monotone nonlinearity and apply the Rantzer multipliers to establish a class of stabilizing controllers for this system. Conclusion: Feedback in a GAL regulatory system can be used to enhance the cellular memory. We show that the system can be modeled as a quadratic nonlinear system for which the effect of feedback on the domain of attraction of the equilibrium point can be characterized using linear matrix inequality (LMI) conditions that are easily implementable in software. The benefit of this result is that a mathematically sound approach to the synthesis of full-state and partial-state feedback controllers to regulate the cellular memory is now possible, irrespective of the number of state-variables or parameters of interest. Page 1 of 8 (page number not for citation purposes) BioMed Central Open Access

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  • BMC Bioinformatics

    ResearchStability analysis of the GAL regulatory networkin Saccharomyces cerevisiae and Kluyveromyces lactisVishwesh V Kulkarni*1, Venkatesh Kareenhalli2, Pushkar Malakar2,Lucy Y Pao3, Michael G Safonov4 and Ganesh A Viswanathan2

    Addresses: 1Department of Electrical Engineering, Indian Institute of Technology Bombay, Mumbai 400 076, India, 2Department of ChemicalEngineering, Indian Institute of Technology Bombay, Mumbai 400 076, India, 3Department of Electrical, Computer, and Energy Engineering,University of Colorado, Boulder, CO 80302, USA and 4Department of Electrical Engineering, University of Southern California, Los Angeles,90089-2563, USA

    E-mail: Vishwesh V Kulkarni* - [email protected]; Venkatesh Kareenhalli - [email protected]; Pushkar Malakar - [email protected];Lucy Y Pao - [email protected]; Michael G Safonov - [email protected]; Ganesh A Viswanathan - [email protected]*Corresponding author

    from The Eighth Asia Pacific Bioinformatics Conference (APBC 2010)Bangalore, India 18-21 January 2010

    Published: 18 January 2010

    BMC Bioinformatics 2010, 11(Suppl 1):S43 doi: 10.1186/1471-2105-11-S1-S43

    This article is available from: http://www.biomedcentral.com/1471-2105/11/S1/S43

    2010 Kulkarni et al; licensee BioMed Central Ltd.This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0),which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

    Abstract

    Background: In the yeast Saccharomyces cerevisiae, interactions between galactose, Gal3p,Gal80p, and Gal4p determine the transcriptional status of the genes required for the galactoseutilization. Increase in the cellular galactose concentration causes the galactose molecules to bindonto Gal3p which, via Gal80p, activates Gal4p, which induces the GAL3 and GAL80 genetranscription. Recently, a linear time-invariant multi-input multi-output (MIMO) model of this GALregulatory network has been proposed; the inputs being galactose and Gal4p, and the outputs beingthe active Gal4p and galactose utilization. Unfortunately, this model assumes the cell culture to behomogeneous, although it is not so in practice. We overcome this drawback by including morebiochemical reactions, and derive a quadratic ordinary differential equation (ODE) based model.

    Results: We show that the model, referred to above, does not exhibit bistability. We establishsufficiency conditions for the domain of attraction of an equilibrium point of our ODE model forthe special case of full-state feedback controller. We observe that the GAL regulatory system ofKluyveromyces lactis exhibits an aberration of monotone nonlinearity and apply the Rantzermultipliers to establish a class of stabilizing controllers for this system.

    Conclusion: Feedback in a GAL regulatory system can be used to enhance the cellular memory.We show that the system can be modeled as a quadratic nonlinear system for which the effect offeedback on the domain of attraction of the equilibrium point can be characterized using linearmatrix inequality (LMI) conditions that are easily implementable in software. The benefit of thisresult is that a mathematically sound approach to the synthesis of full-state and partial-statefeedback controllers to regulate the cellular memory is now possible, irrespective of the number ofstate-variables or parameters of interest.

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  • BackgroundIntroduction to the GAL regulatory systemNaturally occurring networks of genes and proteins,especially in eukaryotic organisms, feature multiple com-plex nested feedback loops. So, although gene expressionscan be affected at many levels including protein-DNAinteractions, protein-protein interactions, and protein-small molecule interactions, it is difficult to characterize,a priori, the systemic effect of these changes. An example ofsuch networks is the galactose signalling pathway in theyeast Saccharomyces cerevisiae. Despite extensive data on itsmolecular interactions, an a priori prediction of its systemicbehavior remains challenging (see [1,2], and [3]). In theGAL regulatory network (see Fig. 1), the galactose signalpropagates through a four-stage signalling cascade. At theuppermost stage is Gal2p, which imports extracellulargalactose into the cell.

    Subsequently, intracellular galactose binds to andactivates Gal3p (see [2] and [3]). At the third stage ofthis cascade, the activated Gal3p binds to and sequestersGal80p in the cytoplasm, depleting Gal80p from thenucleus. The transcriptional activator Gal4p, which isconstitutively bound to promoters of the GAL genes, isthen released from the inhibitory action of Gal80p andactivates expression of genes at the output of the cascade,including GAL1, GAL2, GAL3 and GAL80. Because anincrease in Gal2p and Gal3p concentration results in

    enhanced transcriptional activity, these two proteinseach enforce a positive feedback loop whereas Gal80penforces a negative feedback loop (see [1]).

    Modelling assumptionsWe mostly follow [1] to model the GAL regulatorynetwork. We denote genes in all capital letters, andproteins with only first letter in capital letters. We focuson only early stages of the galactose induction, anddisregard the events that occur after the Gal4p phos-phorylation. We overlook the details of signal transmis-sion from galactose to Gal4p. In other words, Gal4pencountered in our model could be bound to DNA orcould be bound to DNA and Gal80p. Likewise, Gal80pin our model is either bound to DNA and Gal4p orbound to Gal3p or unbound. Gal1p and Gal3p are takento play a similar role, and are together referred to asGal3p.

    Modelling the GAL regulatory systemWe shall first summarize the logic of [1] and describe itsmodel of the GAL regulatory system, and then relaxsome of its assumptions to derive a nonlinear model.The states of interest of the system are shown in Fig. 2. Inthe absence of galactose, Gal4p can bind to Gal80p andhas no transcriptional activity. Following a step increasein galactose, Gal3p rapidly binds galactose, and Gal80pis consumed by being recruited in a complex withGal3p. As the concentration of the unbound Gal80pdecreases, the Gal4p/80p complex is destabilized, whichactivates Gal4p. Activated Gal4p then initiates theslower biosynthetic reactions, viz., transcription of theGAL genes including GAL3 and GAL80, followed bytranslation into their protein products. Following Gal4pactivation, and consequent GAL gene expression, newlysynthesized Gal3p and Gal80p shift the equilibriumback towards Gal4p inactivation. As a result, GALtranscriptional activity decreases. Newly formed proteinscan bind to the incoming galactose molecules, thusrestoring sensitivity to any further galactose input. Thiseffectively closes the feedback loop. We lump Gal3p andGal80p together as the Gal3/80p complex. In theabsence of galactose, Gal3p, Gal80p and Gal4p forman inactive complex Gal4/3/80p called receptor R.A bound receptor BR comprising Gal3p, Gal80p andgalactose remains inactive or may be degraded. The totalGal4p concentration is assumed to be constant duringthe GAL response, as suggested by transcriptomics data(see [4,3], and [5]).

    Biochemical reactions in the GAL regulatory systemWe represent a gene, its encoded mRNA and protein as asingle entity. G4 denotes Gal4p protein and G4

    denotes

    Figure 1The galactose signalling pathway. The external galactosesignal controls the transcriptional activity of the GAL genes.Galactose can shuttle between the cytoplasm and thenucleus. The galactose bound stage of the protein Gal3p isGal3p*. The pointed arrows indicate activation whereas theblunt arrows indicate inhibition.

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  • activated Gal4p protein. The reactions of interest are asfollows:

    G G Gal G Gal R R Gal G BR4 4 3 80 4 3 80 4 + + + + / , / , .

    (1)

    The first reaction is the slow biosynthesis of transcriptionand translation, comprising the binding of Gal4p to theGAL3 and GAL80 gene promoters, and all subsequentactions until Gal3/80p molecules are newly synthesizedone at a time, and Gal4p stays activated. The secondreaction is the inactivation of Gal4p into its inactiveform called the receptor R, and the third reaction is theactivation of Gal4p due to the binding of galactose, Gal,to the receptor, yielding the bound receptor, BR. Forsimplicity, these three reactions are reduced to thefollowing two reactions in [4]:

    R Gal G BR G RK K

    + + 3 1

    4 4, . (2)

    Let S0 denote the initial quantity of galactose and let R0denote the initial quantity of Gal4p. Let us normalizeK1 to unity. Then, it is shown in [4] that the abovemodel of the GAL regulatory system gives rise to asystem of differential equations that can be analyzedusing the phase-plane method to better understandhow the GAL regulatory system is robust to parametervariations and gene transcription time-delays. In deriv-ing this model, [4] makes the following assumptionswhich may not hold in practice: (i) the cell culture hasa homogeneous distribution whence Ki are equal; and(ii) the feedback loops of GAL3 and GAL80 can belumped together. In this paper, we relax the above twoassumptions and derive a less simplified nonlinearmodel. We then apply multiplier theory to betterunderstand stability and robustness of the GALregulatory network.

    MethodsA nonlinear state-space model of the GAL regulatorysystemThe state-space model derived in [4] is as follows. Let Rand Gal be the states x1 and x2 of the system, and let x [x1 x2]

    T. Define a1 = -(K3S0 + 1), a2 = -K3R0, a3 = - K3S0where Ki are the kinetic reaction constants, and thenonlinearity f(1, 2) = (K3 - K2) 12. Then, a state-spacemodel of the GAL system is SL initialx Ax x x: ( )= + + ,where

    A xf x x

    f x xxinitial=

    =

    =

    1 21 2

    1 2

    1 21, ( )

    ( , )

    ( , ), 11 0

    1 01

    R

    R( ).

    Remark 1 In [4], only the initial condition response, i.e.,the response to S0 and R0, is considered. The two inputsof interest are the galactose injected in the cell, and R; thefirst input can be varied using Gal2p, and the secondinput can be varied by transforming Gal4 deleted cellswith a plasmid expressing Gal4.

    Remark 2 Arguing that Ki are all equal, F(x) is set to zeroin [4], and the phase-plane method is applied on thelinearized 2-state system to determine the conditionsunder which the system is stable and robust to the geneexpression delays. In practice, however, the cells are notuniformly distributed whence Ki are not equal so that thenonlinearity F cannot be neglected. Further, as thefollowing lemma shows, SL fails to exhibit bistability, akey property of the GAL regulatory network, even after Fis accounted for.

    Lemma 1 SL has a unique steady state and does not exhibit aHopf bifurcation.

    Proof: See Additional file 1.

    Figure 2The galactose induction loop. This figure is reproducedfrom [4]. (i) In the absence of galactose, the transcriptionalactivity of Gal4p is inhibited by Gal3/80p. (ii) The associationof galactose with Gal3/80p allows Gal4p to be freed fromGal80p inhibition and to activate the transcription of newGal3/80p. (iii) Newly synthesized Gal3/80p inhibits thetranscriptional activity of Gal4p.

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  • Remark 3 Lemma 1 implies that the GAL regulatorysystem model of [4] is not bistable. However, it is wellknown that the GAL regulatory system exhibits bist-ability (see [1]). This anomaly results because, inderiving SL , the nonlinear feedback loops of GAL3and GAL80, one of which is positive whereas the other isnegative, are overly simplified using a single negativefeedback loop in [4]. We propose a correction byincluding more molecular reactions and, hence, morestate variables in our model.

    Let us choose the state variables x1 = G4 , x2 = Gal3/80, x3

    = R, x4 = BR, x5 = Gal, and let x [x1 x2 x3 x4 x5]T . Then(1) can be expressed as x = Ax + F(x) + Bu, where

    A

    b a

    b a

    b

    1 2 1

    2 2 1

    3 2

    4

    5

    0 0

    0 0

    0 0 0 0

    0 0 0 0

    0 0 0 0

    +

    +, ( ) x

    a x x a x x b x x

    a x x

    a x x a x x b x x

    a

    2 1 2 3 3 5 3 1 4

    2 1 2

    2 1 2 3 3 5 3 1 4

    33 3 5 3 1 4

    3 3 5 3 1 4

    1

    1

    0

    0

    0

    x x b x x

    a x x b x x

    B

    a

    a

    +

    ,

    (3)

    where ai and bi are the kinetic reaction constants, i arethe degradation rates, and u is the input galactose. This isour model SG of the GAL regulatory system. Note thatthe nonlinearity F(x) is quadratic and can be expressedas F(x) = xTNx where N [N1 N2... N5]T for some Ni R55. Literature on the stability analysis of such systemsis sparse although sufficiency conditions have beenestablished in [6]. It appears that L2 stability cannot beexpected of multistable models due to the followingreason.

    Lemma 2 A bistable controllable state-space system is not L2-stable.

    Proof: Let u, x denote the input and output of the system.Since the system is bistable, there exists a time andcontrol signals u1, u2 P L2 that drive the system outputto each of two distinct constant equilibrium outputvalues, say x1o and x2o, at time such that x1(t) = x1o andx2(t) = x2o for all t . Hence, u1, u2 L2, but x1 - x2 L2. Therefore, either x1 L2 or x2 L2 or both. QED.

    As a result, we focus only on establishing a domain ofattraction for an equilibrium point of such models.Determination of the domain of attraction is useful sinceit determines the stability region for cellular memorythat can be controlled using a linear feedback of the geneexpression states. An extreme example is that ofpersistent memory, obtained by deleting the GAL80feedback loop; this phenomenon is observed in mutantgenes [1].

    Remark 4 Experimentally, we have observed that theinput-output map of Kluyveromyces lactis with GAL80 as

    the output and galactose as the input is an aberration offriction nonlinearity. Multiplier theoretic stability analy-sis results (see [7-10], and [11]) can be applied todetermine the finite-gain stability of such reduced ordermodels as we demonstrate in the Results section.

    Stability and multipliersWe now formally introduce the notation and the notion ofstability; a detailed description of these notions is availablein [9] and [10]. Let (R+) R denote the set of all(nonnegative) real numbers. Let () (or ()T) denote thetranspose of a vector or a matrix (). Let the inner-product

    x y y t x t dtT, ( ) ( ) and let the norm x x x , .

    The vector space L2 comprises all signals x for which ||x|| 0, asymmetric positive definite matrix P Rnn, and a matrix Ksuch that

    10

    10 1 2

    iT

    iT T

    iT

    i

    Pc

    Pc Pcand

    cPi

    ( ), , , , .

    = ..., ,5

    (4)

    Herm A BK P

    N

    N

    N

    PT

    Ti

    Ti

    Ti

    ( )+ +

    1

    2

    5

    < =0 1 2 5, , ,..., ,i

    (5)

    where Herm() denotes the Hermitian of (). The desiredcontroller is given by u = Kx.

    Proof: Our proof uses the results derived in [17] and[18], and can be sketched as follows. Let us consider thefunction V (x) = xTP-1x as the candidate Lyapunovfunction. Since P is a symmetric positive definite matrix,(x) is positive definite. It needs to be shown that V x( )is negative definite along the system trajectories on P .Observe that the inequality (5) holds not only for thevertices i but for all points x inside the scaled polytopeP P1/ since the function on the left-hand side is anaffine function of x. It can be observed that the left handside of this inequality is V x( ) along the trajectories ofSG so that V x( ) is indeed a Lyapunov function for SG .We next show that the polytope P contains a level curveof the chosen Lyapunov function. It is well known thatthe ellipsoid {x R5 | xTP-1x c}contains thepolytope P (see [[19], pp. 69]). Now, the polytope Pcan be expressed as P = {x R

    5|gaix 1 i = 1, 2, ..., 5}.Now, using the Schur complement, the condition (4) canbe re-written as ( )a cPa ii

    Ti 1 Hence, by [[19], pp.

    70], it follows that P contains . Hence V (x) is aLyapunov function on . Further, the boundary of is alevel curve of V (x) whence is an invariant set. Hence,by Theorem 1, P is an estimate of the domain ofattraction. Hence the proof. QED.

    Remark 6 Theorem 2 establishes a lower bound P onthe domain of attraction of an equlibrium point and alsoyields a full-state feedback controller u = Kx whichasymptotically drives a state within P to the equili-brium point. The result applies only for the special casewherein the equilibrium point xois the origin, and can beextended to cover the case of other equilibrium points.

    Remark 7 The domains of attraction of the equilibriumpoints have been experimentally reported as the regionsof persistent and non-persistent memory in [1]. Theorem2 characterizes the domain of attraction for the specialcase in which a linear time-invariant feedback from theexpressed genes is used to control the input galactose.

    Stabilizing feedback controller for gal4expression in kluyveromyces lactisIf the objective is to control only GAL4 expression, asopposed to controlling all individual gene expressionlevels, the classical multiplier theory might provide awide range of linear and nonlinear stabilizing controllers.We have experimentally observed that the GAL4 expressionexhibits an aberration of monotone nonlinearity when thecell is excited with galactose (see Fig. 4); the expression isfurther inhibited in the presence of glucose. Someexperimental set-ups require that the galactose be injected

    Figure 4Nonmonotone nonlinearity in Kluveromycetes lactis.We experimentally observed that the synthesis of G4

    usinggalactose in Kluveromycetes lactis exhibits the shownnonlinearity. WT culture was grown in different concentrationsof lactose and galactose in liquid culture. The concentrationsused were 0.1%, 0.5%, 1%, and 2%. The beta-galactosidaseactivity was measured in different concentrations of sugar by aprotocol described by Miller method (see [20]). The maximumenzyme activities at different concentrations are plotted; theplots are not to scale.

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  • in a cell such that the GAL4 expression is regulated to adesired value. For these applications, a class of stabilizingcontrollers may be obtained as follows using the frame-work of [15]. Let N denote this nonlinearity, and let denote the dip in the nonlinearity (see Fig. 4). Let C be thecontroller to be designed. Then, feedback system R ofinterest is as follows: y1 = N (u1), u1 = C(e1), e1 = r - y1.Using Theorem 1 of [15], the following result is readilyestablished.

    Lemma 3 Let R denote the class of convolution operators,either continuous-time or discrete-time, such that the impulseresponse of an M R is of the form

    m g h with h g h t t( ) ( ) ( ) ( ) , ( ) , = + + < 1

    0

    where g, h() R. Then R is finite-gain stable if C R.

    Proof: The proof follows as a ready consequence of[[15], Theorem 1].

    Remark 8 This controller can be used to control theexpression of GAL4. However, it cannot control thecellular memory since it cannot regulate the expressionof other genes.

    ConclusionWe have derived an ODE model of the GAL regulatorynetwork of Saccharomyces cerevisiae. We have shown thatalthough the ODE model of [4] gives an elegantexplanation of the transient response of a subset of thisnetwork, it does not exhibit bistability, a key property ofthe GAL regulatory network. By including more chemicalreactions in the approach of [4], we have proposed a 5-state quadratic model of the GAL regulatory network. Forthis model, we have established sufficiency conditions forthe domain of attraction of an equilibrium point for thespecial case of full-state feedback control. This result isuseful in characterizing the persistence of cellular memory.We have experimentally observed that the GAL4 expressionin Kluyveromyces lactis exhibits an aberration of monotonenonlinearity. For a simplified model of this system,wherein the input is galactose and the output is GAL4expression, we have derived a class of stabilizing controllersusing the results of [15]. Unlike the existing literature onGAL regulatory systems, our approach is not limited to 2state-variables or 2 parameters; our LMI conditions scalewell to address more state-variables and parameters, as isthe case in the GAL regulatory system, and can be easilyimplemented in software.

    Competing interestsThe authors declare that they have no competinginterests.

    Authors contributionsThe experimental results for Kluyveromyces lactis wereobtained by Prof. Venkatesh Kareenhalli and PushkarMalakar. Lemma 1 was derived by Prof. Ganesh Viswa-nathan and Prof. Vishwesh Kulkarni. Lemma 2 was derivedby Prof. Michael Safonov. Theorem 2 was derived byProf. Vishwesh Kulkarni. The ODE model was derived byPushkar Malakar and Prof. Vishwesh Kulkarni. Theapplication of [15] is due to Prof. Vishwesh Kulkarni,Prof. Lucy Pao, and Prof. Michael Safonov.

    Proof of Lemma 1: At an equilibrium, the followingequalities hold:

    ( ) ( ) ( )

    ( )

    K K GalR K R Gal K S K R R K S

    K K GalR K S3 2 3 0 3 0 1 0 3 0

    3 2 3

    1 0 + + + = + 00 0 3 0 3 0 0R K S R K R Gal = .

    The steady state solution is given by

    RRK

    GalK S K

    RK

    R K S

    K KRK

    K Sss ss= =

    + +

    01

    3 0 101

    0 3 0 1

    3 201

    3 0

    ,( ) ( )

    ( ).

    So, the Jacobian at the steady state (Rss, Galss) is given by

    JK K R K R K K Gal K S K

    K K R K R Kss ss

    ss=

    +

    ( ) ( ) ( )

    ( ) (3 2 3 0 3 2 3 0 1

    3 3 3 0 3 KK Gal K Sss2 3 0)

    As the Jacobian is non-singular for all values of themodel parameters, it satisfies the implicit functiontheorem. As a result, the system cannot exhibit anysteady state bifurcations and, therefore, does not exhibitbistability. Next, we investigate the presence of a Hopfbifurcation point that may lead to oscillatory solution ordynamic equilibrium. Now, the model exhibits Hopfbifurcation if and only if trace(J) = 0 and det(J) >0.Hence, the given system has a Hopf bifurcation if andonly if

    ( ) ( ) , (( ) )K K R K R K K Gal K S and K K R K R Kss ss ss3 2 3 0 3 2 3 0 3 1 3 0 10 + = >> 0.

    Since K1 is positive, a Hopf bifurcation exists if and onlyif

    ( ) [( ) ] ( ) .K K R K R K K Gal K S K K Gal K Sss ss ss3 2 3 0 3 2 3 0 3 2 3 00 = > . (6)

    Hence, Hopf bifurcation exists if and only if (6) holdstogether with the following equation:

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  • ( ) .K KRK

    K R K K S R3 2 3 0

    2

    2 3 0 001

    0

    + =

    Since all parameters are strictly positive and since R0, theinitial quantity of GAL4p, and S0, the initial quantity ofgalactose, are non-zero quantities (see [4]), the abovecondition cannot be satisfied. Therefore, Hopf bifurca-tion does not exist for SL . QED.

    Additional material

    Additional file 1The single-page PDF file contains the proof of Lemma 1. Name of thefile: APBC2010-ID151-Appendix.pdf.Click here for file[http://www.biomedcentral.com/content/supplementary/1471-2105-11-S1-S43-S1.pdf]

    AcknowledgementsThis research is supported, in parts, by the Wadhwani Foundation at theIndian Institute of Technology Bombay, Mumbai, India. Pushkar Malakarwas supported by a junior research fellowship of the Council of Scientificand Industrial Research, New Delhi. Prof. Lucy Y. Pao was supported bythe US National Science Foundation grant CMMI-0700877.

    This article has been published as part of BMC Bioinformatics Volume 11Supplement 1, 2010: Selected articles from the Eighth Asia-PacificBioinformatics Conference (APBC 2010). The full contents of thesupplement are available online at http://www.biomedcentral.com/1471-2105/11?issue=S1.

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    http://www.biomedcentral.com/1471-2105/11?issue=S1http://www.biomedcentral.com/1471-2105/11?issue=S1http://www.ncbi.nlm.nih.gov/pubmed/15889097?dopt=Abstracthttp://www.ncbi.nlm.nih.gov/pubmed/15889097?dopt=Abstracthttp://www.ncbi.nlm.nih.gov/pubmed/11406593?dopt=Abstracthttp://www.ncbi.nlm.nih.gov/pubmed/11406593?dopt=Abstracthttp://www.ncbi.nlm.nih.gov/pubmed/11340206?dopt=Abstracthttp://www.ncbi.nlm.nih.gov/pubmed/11340206?dopt=Abstracthttp://www.ncbi.nlm.nih.gov/pubmed/11340206?dopt=Abstracthttp://www.ncbi.nlm.nih.gov/pubmed/11125145?dopt=Abstracthttp://www.ncbi.nlm.nih.gov/pubmed/11125145?dopt=Abstracthttp://www.biomedcentral.com/http://www.biomedcentral.com/info/publishing_adv.asphttp://www.biomedcentral.com/

    AbstractBackgroundResultsConclusion

    BackgroundIntroduction to the GAL regulatory systemModelling assumptionsModelling the GAL regulatory systemBiochemical reactions in the GAL regulatory system

    MethodsA nonlinear state-space model of the GAL regulatory systemStability and multipliers

    Results and discussionDetermination of the domain of attraction for equilibrium gene expression statesStabilizing feedback controller for gal4 expression in kluyveromyces lactis

    ConclusionCompeting interestsAuthors contributionsAdditional materialAcknowledgementsReferences