Qualitative control of periodic solutions in piecewise affine systems; application to genetic networks

apport   de recherche ISSN 0249-6399 ISRN INRIA/RR--7130--FR+ENG INSTITUT N A TION AL DE RECHERCHE EN INFORMA TIQUE ET EN A UTOMA TIQUE Qualitativ e control of periodic solutions in piecewise af fine systems; application to genetic netw orks Etienne Farc ot — Jean-Luc Gouzé N° 7130 Decembre 2009 Centre de recherche INRIA Sophia Antipolis – Méditerr anée 2004, route des Lucioles, BP 93, 06902 Sophia Antipolis Cedex Téléphone : +33 4 92 38 77 77 — Téléco pie : +33 4 92 38 77 65 Qualitativ e on trol of p erio di solutions in pieewise ane systems; appliation to geneti net w orks Etienne F arot ∗ , Jean-Lu Gouzé † Thème : Observ ation, mo délisation et ommande p our le viv an t Équip es-Pro jets Virtual Plan ts et Comore Rapp ort de re her he n ° 7130  Deem bre 2009  20 pages Abstrat: Hybrid systems, and esp eially pieewise ane (PW A) systems, are often used to mo del gene regulatory net w orks. In this pap er w e elab orate on previous w ork ab out on trol problems for this lass of mo dels, using also some reen t results guaran teeing the existene and uniqueness of limit yles, based solely on a disrete abstration of the system and its in teration struture. Our aim is to on trol the transition graph of the PW A system to obtain an osilla- tory b eha viour, whi h is indeed of primary funtional imp ortane in n umerous biologial net w orks; w e sho w ho w it is p ossible to on trol the app earane or disapp earane of a unique stable limit yle b y h ybrid qualitativ e ation on the degradation rates of the PW A system, b oth b y stati and dynami feedba k, i.e. the adequate oupling of a on trolling subnet w ork. This is illustrated on t w o lassial gene net w ork mo dules, ha ving the struture of mixed feedba k lo ops. Key-w ords: Gene Net w orks, F eedba k Con trol, Pieewise Linear, P erio di Solutions ∗ Virtual Plan ts INRIA, UMR D AP , CIRAD, T A A-96/02, 34398 Mon tp ellier Cedex 5, F rane, farotsophia.inria.fr † COMORE INRIA, Unité de re her he Sophia An tip olis Méditerranée , 2004 route des Luioles, BP 93, 06902 Sophia An tip olis, F rane, gouzesophia.inria.fr Commande qualitativ e de solutions p ério diques de systèmes anes par moreaux ; appliation aux réseaux génétiques Résumé : Les systèmes h ybrides, en partiulier anes par moreaux (APM), son t souv en t emplo y és omme mo dèles de réseaux génétiques. Dans e rapp ort nous approfondissons des tra v aux an térieurs sur la ommande de tels systèmes, utilisan t égalemen t des résultats réen ts garan tissan t l'existene et l'uniité de yles limites, sur la seule base d'une abstration disrète du système et de sa struture d'in teration. L'ob jetif est de on trler le graphe de transitions d'états du système APM p our obtenir un omp ortemen t p ério dique, e qui est une propriété très imp ortan te de nom breux systèmes biologiques. Nous mon trons ommen t ommander l'apparition ou la suppression d'un yle limite unique, par une ation qualitativ e sur les taux de dégradation d'un système APM, aussi bien par ommande statique que dynamique, 'est-à-dire par le ouplage adéquat d'un sous-réseau on trleur. Cei est illustré sur deux ré- seaux de gènes lassiques, présen tan t une struture de b oules de retro-ation im briquées. Mots-lés : Réseaux génétiques, Commande en feedba k, Linéaire par mor- eaux, Solutions p ério diques Contr ol of limit yles in PW A gene networks 3 1 In tro dution Gene regulatory net w orks often displa y b oth robustness and steep, almost swit h- lik e, resp onse to transriptional on trol. This motiv ates the use of an appro xi- mation of these resp onse la ws b y pieewise ane dieren tial (PW A) equations, to build h ybrid mo dels of geneti net w orks. PW A systems are ane in ea h retangular domain (or b o x) of the state spae. They ha v e b een in tro dued in the 1970's b y Leon Glass [18 ℄ to mo del geneti net w orks. It has led to a long series of w orks b y dieren t authors, dealing with v arious asp ets of these equa- tions, e.g. [4, 9 , 11 , 18 , 20 ℄. They ha v e b een used also as mo dels of onrete biologial systems [5 ℄. F rom an h ybrid system p oin t of view, the b eha viour of PW A systems an b e desrib ed b y a transition graph, whi h is an abstration (in the h ybrid system sense) of the on tin uous system. This transition graph desrib es the p ossible transition b et w een the b o xes. It is also p ossible to  he k prop erties of the transition graph b y mo del  he king te hniques [2 ℄. No w ada ys, the extraordinary dev elopmen t of biomoleular exp erimen tal te h- niques mak es it p ossible to design and implemen t on trol la ws in the ell system. The authors ha v e reen tly dev elop ed a mathematial framew ork for on trolling gene net w orks with h ybrid on trols [13 ℄; these on trols are dened on ea h b o x. It is easy to see that this amoun ts to  hange the transition graph to obtain the desired one. F rom another p oin t of view, more orien ted to w ards dynamial systems, it is also p ossible to obtain results onerning the limit yles in PW A systems (see [19 ℄ and the reen t generalisation in [12 ℄). F or example, one an sho w that a simple negativ e lo op in dimension greater that t w o pro dues a unique stable limit yle. It is lear that biologial osillations pla y a fundamen tal role in the ell ([10 ℄). Our aim in this pap er is to on trol PW A systems to mak e a single stable limit yle app ear or disapp ear. T o full that goal, after some realls onerning the PW A systems, w e use some results enabling to dedue the existene of a single stable limit yle in the state spae from a p erio di b eha viour in a b o x sequene (setion 3 ), then the results on the on trol of the transition graph in the spae of b o xes (setion 4), to obtain our main results illustrated b y 2 examples (setion 5). Related w orks on on trol asp ets onern the ane or m ulti-ane h ybrid systems ([22, 3 ℄). The authors deriv e suien t onditions for driving all the solutions out of some b o x. Other related w orks study the existene of limit yles in the state spae [19 , 25 ℄. W e are not a w are of w orks linking on trol theory and limit yle for this lass of h ybrid systems. 2 Pieewise ane mo dels 2.1 General form ulation This setion on tains basi denitions and notations for pieewise ane mo dels [18 , 8 , 11 , 6 ℄. The general form of these mo dels an b e written as: dx dt = κ ( x ) − Γ( x ) x (1) RR n ° 7130 4 F ar  ot & Gouzé The v ariables ( x 1 . . . x n ) represen t lev els of expression of n in terating genes, meaning in general onen trations of the mRNA or protein they o de for. W e will simply all genes the n net w ork elemen ts in the follo wing. Sine gene transriptional regulation is often onsidered to follo w a steep sigmoid la w, an appro ximation b y a step funtion has b een prop osed to mo del the resp onse of a gene (i.e. its rate of transription) to the ativit y of its regulators [ 18 ℄. W e use the notation:  s + ( x, θ ) = 0 if x < θ , s + ( x, θ ) = 1 if x > θ , This desrib es an eet of ativ ation, whereas s − ( x, θ ) = 1 − s + ( x, θ ) represen ts inhibition. Unless further preision are giv en, w e lea v e this funtion undened at its threshold v alue θ . The maps κ : R n + → R n + and Γ : R n + → R n × n + in (1 ) are usually m ultiv ari- ate p olynomials (in general m ulti-ane), applied to step funtions of the form s ± ( x i , θ i ) , where for ea h i ∈ { 1 , · · · , n } the threshold v alues b elong to a nite set Θ i = { θ 0 i , . . . , θ q i i } . (2) W e supp ose that the thresholds are ordered (i.e. θ j i < θ j +1 i ), and the extreme v alues θ 0 i = 0 and θ q i i represen t the range of v alues tak en b y x i rather than thresholds. Γ is a diagonal matrix whose diagonal en tries Γ ii = γ i , are degradation rates of v ariables in the system. Ob viously , Γ and the pro dution rate κ are pieewise- onstan t, taking xed v alues in the retangular domains obtained as Cartesian pro duts of in terv als b ounded b y v alues in the threshold sets ( 2). These ret- angles, or b oxes , or r e gular domains [27 , 6℄, are w ell  haraterised b y in teger v etors: w e will often refer to a b o x D a = Q i ( θ a i − 1 i , θ a i i ) b y its lo w er-orner index a = ( a 1 − 1 . . . a n − 1) . The set of b o xes is then isomorphi to A = n Y i =1 { 0 , · · · , q i − 1 } , (3) Also, the follo wing pairs of funtions will b e on v enien t notations: θ ± i : A → Θ i , θ − i ( a ) = θ a i − 1 i and θ + i ( a ) = θ a i i . Let us all singular domains the in tersetions of losure of b o xes with threshold h yp erplanes, where some x i ∈ Θ i \ { θ 0 i , θ q i i } . On these domains, the righ t-hand side of (1) is undened in general. Although the notion of Filipp o v solution pro vides a generi solution to this problem [20℄, in the ase where the normal of the v etor eld has the same sign on b oth side of these singular h yp erplanes, it is more simply p ossible to extend the o w b y on tin uit y . In the remaining of this pap er, w e will only onsider tra jetories whi h do not meet an y singular domain, a fat holding neessarily in absene of auto-regulation, i.e. when no κ i dep ends on x i . This leads to the follo wing h yp othesis: ∀ i ∈ { 1 , · · · , n } , κ i and γ i do not dep end on x i . (H1) On an y regular domain of index a ∈ A , the rates κ = κ ( a ) and Γ = Γ( a ) are onstan t, and th us equation (1) is ane. Its solution is expliitly kno wn, for ea h o ordinate i : ϕ i ( x, t ) = x i ( t ) = φ i ( a ) + e − γ i t ( x i − φ i ( a )) , (4) INRIA Contr ol of limit yles in PW A gene networks 5 where t ∈ R + is su h that x ( t ) ∈ D a , and φ ( a ) = ( φ 1 ( a ) · · · φ n ( a )) =  κ 1 ( a ) γ 1 ( a ) · · · κ n ( a ) γ n ( a )  . It is learly an attrativ e equilibrium of the o w (4). It will b e alled fo  al p oint in the follo wing for reasons w e explain no w. Let us rst mak e the generi assumption that no fo al p oin t lies on a singular domain: ∀ a ∈ A , φ ( a ) ∈ [ a ′ ∈ A D a ′ . (H2) Then, if φ ( a ) ∈ D a , it is an asymptotially stable steady state of system (1). Otherwise, the o w will rea h the (top ologial) b oundary ∂ D a in nite time. A t this p oin t, the v alue of κ (and th us, of φ )  hanges, and the o w  hanges its diretion, ev olving to w ards a new fo al p oin t. The same pro ess arries on rep eatedly . It follo ws that the on tin uous tra jetories are en tirely  haraterised b y their suessiv e in tersetions with the b oundaries of regular domains (ex- tending them b y on tin uit y , as men tioned previously). This sequene dep ends essen tially on the p osition of fo al p oin ts with resp et to thresholds. A tually , { x | x i = θ − i ( a ) } (resp. { x | x i = θ + i ( a ) } ) an b e rossed if and only if φ i ( a ) < θ − i ( a ) (resp. φ i ( a ) > θ + i ( a ) ). Then, let us denote I + out ( a ) = { i ∈ { 1 , · · · , n }| φ i > θ + i ( a ) } , and similarly I − out ( a ) = { i ∈ { 1 , · · · , n }| φ i < θ − i ( a ) } . Then, I out ( a ) = I + out ( a ) ∪ I − out ( a ) is the set of esaping diretions of D a . Also, w e all wal ls the in tersetions of threshold h yp erplanes with the b oundary of a regular domain. When it is unam biguous, w e will omit the dep endene on a in the sequel. No w, in ea h diretion i ∈ I out the time at whi h x ( t ) enoun ters the orresp onding h yp erplane, for x ∈ D a , is readily alulated: τ i ( x ) = − 1 γ i ln  φ i − θ ± i φ i − x i  , i ∈ I ± out . (5) Then, τ ( x ) = min i ∈ I out τ i ( x ) , is the exit time of D a for the tra jetory with initial ondition x . Then, w e dene a tr ansition map T a : ∂ D a → ∂ D a : T a x = ϕ ( x, τ ( x )) = φ + α ( x )( x − φ ) . (6) where α ( x ) = exp( − τ ( x )Γ) . The map ab o v e is dened lo ally , at a domain D a . Ho w ev er, under our assump- tion (H1) , an y w all an b e onsidered as esaping in one of the t w o regular domains it b ounds, and inoming in the other. Hene, on an y p oin t of the in te- rior of a w all, there is no am biguit y on whi h a to  ho ose in expression ( 6), and there is a w ell dened global transition map on the union of w alls, denoted T . On the b oundaries of w alls, at in tersetions b et w een sev eral threshold h yp er- planes, the onept of Filipp o v solution w ould b e required in general [ 20 ℄. This problem will either b e solv ed on a ase b y ase basis, or w e impliitly restrit our atten tion to the (full Leb esgue measure) set of tra jetories whi h nev er in terset more than one threshold h yp erplane. RR n ° 7130 6 F ar  ot & Gouzé T o onlude this setion let us dene the state tr ansition gr aph TG asso iated to a system of the form (1) as the pair ( A , E ) of no des and orien ted edges, where A is dened in (3) and ( a, b ) ∈ E ⊂ A 2 if and only if ∂ D a ∩ ∂ D b 6 = ∅ , and there exists a p ositiv e Leb esgue measure set of tra jetories going from D a to D b . It is not diult to see that this is equiv alen t to b b eing of the form a ± e i , with i ∈ I ± out ( a ) and e i a standard basis v etor. F rom no w on, it will alw a ys b e assumed that (H1) and (H2) hold, at least in some region of state spae (or transition graph) on whi h w e fo us. 2.2 Illustrativ e example Let us no w illustrate the previous notions on a w ell-kno wn example with t w o v ariables, in order to help the reader's in tuition. Consider t w o genes repressing ea h other's transription. In the on text of pieewise-ane mo dels, this w ould b e desrib ed b y the system b elo w: 1 2 ( ˙ x 1 = κ 0 1 + κ 1 1 s − ( x 2 , θ 1 2 ) − γ 1 x 1 ˙ x 2 = κ 0 2 + κ 1 2 s − ( x 1 , θ 1 1 ) − γ 2 x 2 where inhibition is mo deled b y s − ( x, θ ) , as already men tioned. A usual notation for the in teration graph uses to denote inhibition, and to denote ativ ation. The t w o onstan ts κ 0 i represen t the lo w est lev el of pro dution rates of the t w o sp eies in in teration. It will b e zero in general, but ma y also b e a v ery lo w p ositiv e onstan t, in some ases where a gene needs to b e expressed p ermanen tly . In the giv en equation, arbitrary parameters ma y lead to spurious b eha viour, in partiular an inhibition whi h w ould not drop its target v ariable b elo w its threshold. T o a v oid this, it sues to assume the follo wing onditions on fo al p oin ts' o ordinates: κ 0 i γ i < θ 1 i and κ 0 i + κ 1 i γ i > θ 1 i , for i = 1 , 2 . This migh t b e alled strutur al  onstr aints on parameters. The phase spae of this system is s hematised on Figure 1 . Then, the transition graph of the system tak es the form: 01 11 00 10 where irled states are those with no suessor. It app ears in this ase that TG onstitutes a reliable abstration of the system's b eha viour. In general, things are not as on v enien t, and some paths in the transition graph ma y b e spurious. In partiular, yli paths ma y orresp ond to damp ed osillations of the original system, but ev en this annot b e alw a ys asertained without a preise kno wledge of the parameter, see setion 3 for related results. Ho w ev er, one general goal of INRIA Contr ol of limit yles in PW A gene networks 7 θ 1 1 θ 2 1 θ 1 2 θ 2 2 κ 0 1 γ 1 κ 0 1 + κ 1 1 γ 1 κ 0 2 γ 2 κ 0 2 + κ 1 2 γ 2 Figure 1: The dashed lines represen t threshold h yp erplanes, and dene a retan- gular partition of state spae, and dotted lines indiate fo al p oin ts' o ordinates. Arro ws represen t s hemati o w lines, p oin ted to w ard these limit p oin ts. Note that piees of tra jetories are depited as straigh t lines, whi h is the ase when all degradation rates γ i oinide, a fat w e nev er assume in the presen t study . the presen t study will b e to sear h for feedba k on trol la ws ensuring that giv en systems are indeed w ell  haraterised b y their abstration. Su h a prop ert y an b e dedued from the shap e of the abstration TG itself, whene the term 'qualitativ e on trol'. 3 Stabilit y and limit yles P erio di solutions ha v e so on b een a prominen t topi of study for systems of the form (1 ) [19 , 29 , 26 , 8 , 25 ℄. With the notable exeption of [29 ℄, all these studies fo used on the sp eial ase where Γ is a salar matrix, whi h greatly simplies the analysis, sine tra jetories in ea h b o x are then straigh t lines to w ards the fo al p oin t, as in Figure 1 . In a reen t w ork [12 , 15 , 14 ℄, w e ha v e sho wn that the lo al monotoniit y prop erties of transition maps an b e used to pro v e existene and uniqueness of limit yles in systems lik e ( 1). In this setion w e reall without pro of some of these results. In the rest of this setion w e onsider a pieewise-ane system su h that there exists a sequene C = { a 0 . . . a ℓ − 1 } of regular domains whi h is a yle in the transition graph, and study p erio di solutions in this sequene. W e abbreviate the fo al p oin ts of these b o xes as φ i = φ ( a i ) . Let us no w dene a prop ert y of these fo al p oin ts: w e sa y that the p oin ts φ i are aligne d if ∀ i ∈ { 0 , · · · , ℓ − 1 } , ∃ ! j ∈ { 1 , · · · , n } , φ i +1 j − φ i j 6 = 0 , (7) where φ ℓ and φ 0 are iden tied. Sine C is supp osed to b e a yle in TG , for ea h pair ( a i , a i +1 ) of suessiv e b o xes there m ust b e at least one o ordinate at whi h their fo al p oin ts dier, namely the only s i ∈ I out ( a i ) su h that a i +1 = a i ± e s i . W e k eep on denoting s i this swithing o ordinate in the follo wing. Hene ondition (7) means that s i is the only o ordinate in whi h φ i and φ i +1 dier. This implies in partiular that RR n ° 7130 8 F ar  ot & Gouzé a i +1 is the only suessor of a i , i.e. there is no edge in TG from C to A \ C . It migh t seem in tuitiv e in this ase that all orbits in C on v erge either to a unique limit yle, or to a p oin t at the in tersetion of all rossed thresholds. Ho w ev er, this fat has only b een pro v ed for uniform dea y rates (i.e. Γ salar), [ 19 ℄, and its v alidit y with distint dea y remains an op en question. The ondition ( 7 ) is of purely geometri nature. Ho w ev er, it an b e sho wn that it holds neessarily when the in teration graph has degree one or less, see [14℄ for more details. If { s i } 0 6 i<ℓ = { 1 , · · · , n } , i.e. all v ariables are swit hing along C , then the in tersetion of all w alls b et w een b o xes in C is either a single p oin t, whi h w e denote θ C , or it is empt y . The latter holds when t w o distint thresholds are rossed in at least one diretion. When dened, θ C is a xed p oin t for an y on tin uous extension of the o w in C , see [14 ℄. Let us no w rephrase the main result from [14 ℄. Theorem 1. L et C = { a 0 , a 1 · · · a ℓ − 1 } denote a se quen e of r e gular domains whih is p erio di al ly visite d by the ow, and whose fo  al p oints satisfy  ondition (7 ). Supp ose also that al l variables ar e swithing at le ast on e. L et W denote the wal l ∂ D a 0 ∩ ∂ D a 1 , and  onsider the rst r eturn map T : W → W dene d as the  omp osite of lo  al tr ansition maps along C . A ) If a single thr eshold is r osse d in e ah dir e tion, let λ = ρ ( D T ( θ C )) , the sp e tr al r adius of the dier ential D T ( θ C ) . Then, the fol lowing alternative holds: i) if λ 6 1 , then ∀ x ∈ W , T n x → θ C when n → ∞ . ii) if λ > 1 then ther e exists a unique xe d p oint dier ent fr om θ C , say q = T q . Mor e over, for every x ∈ W \ { θ C } , T n x → q as n → ∞ . B ) If ther e ar e two distint r osse d thr esholds in at le ast one dir e tion, then the  onlusion of ii) holds. In [12 , 15 ℄ w e ha v e resolv ed the alternativ e ab o v e for a partiular lass of systems: Theorem 2. Consider a ne gative fe e db ak lo op system of the form ˙ x i = κ 0 i + s ε i ( x i − 1 , θ i − 1 ) − γ i x i , ε i ∈ {− , + } i ∈ { 1 , · · · , n } , with subsripts understo o d mo dulo n , and an o dd numb er of ne gative ε i . It  an b e shown that ther e exists a yle C in TG whose fo  al p oints satisfy (7). Then, in The or em 1 , A. i) holds in dimension n = 2 , and A. ii) holds for al l n > 3 . 4 Pieewise Con trol F eedba k regulation is naturally presen t in man y biologial systems, as the widespread app elation 'regulatory net w ork' suggests. Hene, it seems appro- priate to tak e adv an tage of the imp ortan t b o dy of w ork dev elop ed in feedba k on trol theory for deades, in order to study gene regulatory net w orks and re- lated systems [ 23 , 30 ℄. In partiular, the reen t adv en t of so alled syntheti biolo gy [ 1, 24 ℄ , has led to a situation where gene regulatory pro esses are not only studied, but de- signed to p erform ertain funtions. Hene, autonomous systems of the form (1) should to b e extended, so as to inlude p ossible input v ariables. In [ 13 ℄, INRIA Contr ol of limit yles in PW A gene networks 9 w e ha v e presen ted su h an extension, where b oth pro dution and dea y terms ha v e some additional argumen t u ∈ R p , of whi h they w ere ane funtions. In this on text, w e dened a lass of qualitativ e on trol problems, and sho w ed that w ere equiv alen t to some linear programming problems. As in our previous w ork, w e onsider qualitativ e feedba k la ws, in the sense that they dep end only on the b o x on taining the state v etor, rather than its exat v alue. This  hoie is motiv ated b y robustness purp oses. More pragmatially , it is also due to the fat that reen t te hniques allo w for the observ ation of qual- itativ e  harateristis of biologial systems, for instane b y liv e imaging, using onfo al mirosop y , of GFP mark er lines, where the measured state is loser to an ON/OFF signal than to a real n um b er. Reen t exp erimen tal te hniques allo w furthermore for the rev ersible indution of sp ei genes at a  hosen instan t, for instane using promoters induible b y ethanol [7℄, or ligh t [28 ℄, to name only t w o. Also, degradation rates ma y b e mo died, either diretly b y in tro duing a drug [ 31 ℄, or via a designed geneti iruit [21℄, whi h migh t b e indued using previously men tioned te hniques. T o simplify the presen tation, w e fo us in this pap er on the partiular ase where dea y rates an b e linearly on trolled b y a salar and b ounded input u . F or ea h i ∈ { 1 , · · · , n } , let us denote this as: dx i dt = κ i ( x ) − ( γ 1 i ( x ) u + γ 0 i ( x )) x, u ∈ [0 , U ] ⊂ R + , (8) where γ 0 i and γ 1 i are pieewise onstan t funtions assumed to satisfy γ 0 i > 0 and γ 1 i > − γ 0 i U , in an y b o x. This ensures that dea y rates are p ositiv e, but y et an b e dereasing funtions of u (for γ 1 i < 0 ). No w, a feedba k la w dep ending only on the qualitativ e state of the system is simply a expressed as the omp osite of a map S a D a → A indiating the b o x of the urren t state, with a funtion u : A → [0 , U ] whi h represen ts the on trol la w itself. In other w ords, in ea h b o x a onstan t input v alue is  hosen. F or a xed la w of this form, it is lear that the dynamis of ( 8 ) is en tirely determined, and in partiular w e denote its transition graph b y TG ( u ) . Let us no w reall our denition of on trol problem. Global Con trol Problem : Let TG ⋆ = ( A , E ⋆ ) b e a transition graph. Find a feedba k la w u : A → [0 , U ] su h that TG ( u ) = TG ⋆ . Clearly , E ⋆ annot b e arbitrary in A 2 , and m ust in partiular on tain only arro ws of the form ( a, a ± e i ) . No w in the presen t, restrited, on text the equiv alen t linear programming problem desrib ed in [13 ℄ is v ery simple. F or ea h a ∈ A , the on trol problem ab o v e requires that the fo al p oin t φ ( a, u ( a )) b elongs to a ertain union of b o xes, i.e. its o ordinates m ust satisfy inequalities of the form θ j − ( a ) i < κ i ( a ) / ( γ 1 i ( a ) u ( a ) + γ 0 i ( a )) < θ j + ( a ) i , or equiv alen tly κ i ( a ) − γ 0 i ( a ) θ j + ( a ) i γ 1 i ( a ) θ j + ( a ) i < u ( a ) < κ i ( a ) − γ 0 i ( a ) θ j − ( a ) i γ 1 i ( a ) θ j − ( a ) i (9) if γ 1 i ( a ) > 0 , and in rev erse order otherwise. Hene, the solution set of the on trol problem is just the Cartesian pro dut of all in terv als of the form (9), when a v aries in A . It is th us iden tial to a retangle in R # A (where # denotes RR n ° 7130 10 F ar  ot & Gouzé ardinalit y), whi h is of full dimension if and only if the problem admits a so- lution. Thanks to the expliit desription (9), this set an b e omputed with a omplex- it y whi h is linear in # A . The latter gro ws exp onen tially with the dimension of the system, but in pratie, one will fae problems where E and E ⋆ only dier on a subset of initial v erties, sa y A ⋆ , and then the atual omplexit y will b e of order # A ⋆ . In addition to this t yp e of on trol, w e in tro due in this note some rst hin ts to w ard dynami feedba k on trol, where instead of a diret feedba k u , one uses some additional v ariable (here a single one), ev olving in time aording to a system of the form (1), and oupled to the initial system. This is suggested b y the retangular form of admissible inputs found in ( 9): instead of xing an arbitrary v alue in a retangle of an external input spae, one inreases the state spae dimension, whi h has the eet of adding new b o xes to the system. The dynamis of the supplemen tary v ariables is then dened b y analogy with the diret feedba k ase: when the initial v ariables are in a b o x a , this mak es additional v ariables tend to a b o x of the form (9). This raises a n um b er of questions, in large part due to the fat that instead of applying an input u ( a ) instan taneously when en tering b o x D a , the feedba k no w tends to w ard some v alue, whi h tak es some time. Instead of fully dev eloping a general theory , w e th us ha v e to  hosen to illustrate it on a simple example, in setion 5.1. 5 Examples W e no w illustrate with examples ho w it is p ossible to om bine results of the t w o previous setions, and ompute qualitativ e feedba k la ws ensuring (or prelud- ing) the existene and uniqueness of osillatory b eha viour of a system of the form (8 ). 5.1 Example 1: disapp earane of a limit yle Consider the follo wing t w o dimensional system: ( ˙ x 1 ( t ) = K 1 s − ( x 2 ) − ( γ 1 1 u + γ 0 1 ) x 1 ˙ x 2 ( t ) = K 2 [ s + ( x 1 , θ 1 1 ) s + ( x 2 ) + s + ( x 1 , θ 2 1 ) s − ( x 2 )] − γ 0 2 x 2 (10) where x 2 has a unique threshold, and s ± ( x 2 ) = s ± ( x 2 , θ 1 2 ) . W e assume moreo v er that the follo wing inequalities stand: γ 1 1 > 0 , K 1 > γ 0 1 θ 2 1 , K 2 > γ 0 2 θ 1 2 , (11) so that the rst dea y rate inreases with u . Also, the in terations are fun- tional : an ativ ation of a v ariable leads to the orresp onding fo al p oin t o ordi- nate b eing ab o v e a v ariable's threshold ( hosen as the highest one for x 1 , sine otherwise θ 2 1 annot b e rossed from b elo w). Remark that in this system, x 2 violates (H1) . Ho w ev er, it will app ear so on that this autoregulation is only eetiv e at a single w all, whi h is unstable, and th us an b e ignored safely . This system orresp onds to a negativ e feedba k lo op, where x 2 is moreo v er able INRIA Contr ol of limit yles in PW A gene networks 11 to mo dulate its ativ ation b y x 1 : when x 2 is ab o v e its threshold, the in teration is more eien t, sine it is ativ e at a lo w er threshold θ 1 1 < θ 2 1 . Biologially , this ma y happ en if the proteins o ded b y x 1 and x 2 form a dimer, whi h ativ ates x 2 more eien tly than x 1 protein alone. This is reminisen t of the mixe d fe e db ak lo op , a v ery widespread mo dule able to displa y v arious b eha viours [16 ℄. It migh t b e depited b y this graph 1 2 As seen in the equations, the salar input is assumed to aet the rst dea y rate, but not the seond (i.e. γ 1 2 = 0 ). No w, one readily omputes the fo al p oin ts of all b o xes: 00 01 10 1 1 20 21 φ 1 0 φ 1 0 φ 1 0 0 0 0 φ 2 φ 2 φ 2 (12) where φ 1 is an abbreviation for K 1 / ( γ 1 1 u + γ 0 1 ) , and φ 2 for K 2 /γ 0 2 . Under the onstrain ts (11 ), this readily leads to the transition graph in absene of input (i.e. u = 0 in all b o xes): TG (0) = 01 11 21 00 10 20 The dotted line represen ts an unstable w all, for whi h Filipp o v theory w ould b e required for full rigour. Ho w ev er, this w all is not rea hable, and w e ignore it afterw ards. No w, sine this graph has a yle, the t w o thresholds θ 1 , 2 1 are rossed, and (12 ) is easily seen to imply ondition (7), onlusion B ) of Theorem 1 applies : there is a unique stable limit yle. No w, in aordane with the setion's title, let us lo ok for a u leading to: TG ⋆ = 01 11 21 00 10 20 Clearly from TG ⋆ , the b o x D 10 attrats tra jetories from all other b o xes, and on tains its o wn fo al p oin t, whi h is th us a globally asymptotially stable equi- librium. The only states whose suessors dier in TG (0) , and TG ⋆ are 10 and 20 , hene w e assume u ( a ) = 0 for all other a ∈ A , or A ⋆ = { 10 , 20 } to reall the notations of previous setion. Then, Eq. (9) with θ j − ( a ) 1 = θ 1 1 and θ j + ( a ) 1 = θ 2 1 giv es: K 1 − γ 0 1 θ 2 1 γ 1 1 θ 2 1 < u ( a ) < K 1 − γ 0 1 θ 1 1 γ 1 1 θ 1 1 (13) for b oth a ∈ A ⋆ . This denes a nonempt y in terv al b y θ 1 1 < θ 2 1 , hene the Con trol Problem of previous setion an b e solv e under onstrain ts (11 ). An illustration on a n umerial example is sho wn Figure 2. No w, let us fo us on the question of realising an extended net w ork whi h solv es the same problem, b y adding a v ariable to system ( 10). In other w ords, RR n ° 7130 12 F ar  ot & Gouzé 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Figure 2: Dashed lines: with feedba k on trol. Plain lines: without. T w o initial onditions, (0 . 95 , 0 . 95 ) in b o x 21 (blue urv es) and (0 . 85 , 0 . 15 ) in 20 (red urv es). The on trolled and autonomous tra jetories only div erge in b o x 10, 20 where the feedba k is ativ e. See parameters in App endix A.1 one no w seeks to imp ose the dynamis desrib ed b y TG ⋆ using dynami feed- ba k. Biologially , this amoun ts to designing a geneti onstrut whose promoter dep ends transriptionaly on x 1 and x 2 , and inreases the degradation rate of x 1 . Let us denote b y y the expression lev el of this additional gene. The most ob vious v ersion of su h an extended system arises b y inreasing y pro dution rate exatly at b o xes in A ⋆ :        ˙ x 1 ( t ) = K 1 s − ( x 2 ) − ( γ 1 1 υ s + ( y ) + γ 0 1 ) x 1 ˙ x 2 ( t ) = K 2 [ s + ( x 1 , θ 1 1 ) s + ( x 2 ) + s + ( x 1 , θ 2 1 ) s − ( x 2 )] − γ 0 2 x 2 ˙ y ( t ) = s + ( x 1 , θ 1 1 ) s − ( x 2 ) − γ y y (14) υ a onstan t in the in terv al (13 ), so that foring s + ( y ) = 1 w ould lead us ba k to a stati feedba k solution. This use of a single onstan t υ is p ossible in this partiular example b eause onstrain ts (13 ) are iden tial for the t w o b o xes in A ⋆ , but it should b e noted that in general sev eral onstan ts migh t b e required. W e onsider without loss of generalit y that y ∈ [0 , 1 /γ y ] , sine higher v alues of y tend to 1 /γ y or 0 . Also, s + ( y ) is dened with resp et to a threshold θ y ∈ (0 , 1) . W e also assume θ y γ y < 1 , ensuring that y ma y ross its threshold when ativ ated. No w, (14) denes an autonomous systems of the form (1 ), whose transition graph has indeed a xed p oin t 101 : 011 111 211 001 101 201 010 110 210 000 100 200 (15) INRIA Contr ol of limit yles in PW A gene networks 13 This xed p oin t orresp onds the xed p oin t 10 of TG ⋆ : in fat, the upp er part of the graph ab o v e, where s + ( y ) = 1 is exatly TG ⋆ . Ho w ev er, it is not in v arian t, and some tra jetories an esap e to s + ( y ) = 0 , where w e see TG (0) , and th us the p ossibilit y of p erio di solutions. Besides, there are other yles in this graph. Unlik e stati feedba k on trol  and more realistially  the eet of y on γ 1 tak es some p ositiv e time, explaining wh y the situation is not a diret translation of previous ase. W e will no w sho w that under additional onstrain ts of the parameters go v erning y 's dynamis, it is p ossible to guaran tee that D 101 on tains a globally asymptotially stable equilibrium. T o a hiev e this, let us rephrase a lemma, pro v ed as Lemma 1 in [ 11 ℄: Lemma 1. F or any b ox, ther e is at most one p air of p ar al lel wal ls su  essively r osse d by solution tr aje tories of a system of the form (1). In other w ords, there is at most one diretion i su h that opp osite w alls, of the form x i = θ − i and x i = θ + i , are rossed. Moreo v er, su h an i is  haraterised, see [11 ℄, b y the ondition ∀ j 6 = i , τ i ( θ − i ) < τ j ( θ − j ) , (16) under the assumption I out = I + out (whi h simplies the desription without loss of generalit y), i.e. all exiting w alls o ur at higher threshold v alues, of the form θ + i , whi h is th us the threshold in v olv ed in the denition of τ i , Eq. (5). This allo ws us to pro v e the follo wing result: Prop osition 1. Supp ose that the p ar ameters of (14 ) satisfy, denoting φ 1 = K 1 γ 0 1 : (1 − γ y θ y ) 1 γ y >  φ 1 − θ 2 1 φ 1 − θ 1 1  1 γ 0 1 Then ther e the ste ady state in b ox D 101 attr ats the whole state sp a e of system (14 ). Pr o of. Sine ea h b o x is either on taining an asymptoti steady state, or has all its tra jetories esaping it to w ard a fo al p oin t, all limit set m ust b e on tained in a strongly onneted omp onen t of the transition graph, i.e. a olletion of yli paths sharing some v erties. A visual insp etion of the transition graph displa y ed in (15 ) sho ws that an y yli path in the transition graph TG m ust visit the state 100 . This state has only t w o suessors: 200 and 101 , the xed state. Hene it has t w o exit w alls, whi h w e denote b y: W + 1 = { θ 2 1 } × ( θ 0 2 , θ 1 2 ) × (0 , θ y ) and W + y = ( θ 1 1 , θ 2 1 ) × ( θ 0 2 , θ 1 2 ) × { θ y } . All other w alls are inoming. Denoting them b y ob vious analogy with the t w o exiting w alls, let us onsider ea h of them. First, b oth w alls W ± 2 are rep elling: this has already b een said for W + 2 when disussing auto-regulatory terms in (10). F or W − 2 , this follo ws from the fat that D 100 on tains only tra jetories esaping in nite time, and an b e extended to this w all b y on tin uit y . Moreo v er, it follo ws from TG that an y tra jetory esaping W ± 2 either rea hes D 101 (where the xed p oin t lies), or en ters D 100 again via the w all W − 1 . Th us, an y tra jetory whi h do es not en ter D 101 m ust ross the pair W ± 1 in suession. No w, from Lemma 1, among the t w o pairs of w alls W ± 1 , W ± y , only one an b e rossed in suession b y tra jetories. Moreo v er, the inequalit y in the statemen t is the exat translation of the ondition (16 ), in the ase where W ± y is the rossed pair of w alls, preluding an y attrator but the kno wn xed p oin t, φ (101) . RR n ° 7130 14 F ar  ot & Gouzé 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x 1 x 2 y Figure 3: Dashed line: inequalit y of prop osition 1 satised. Plain line: inequal- it y violated. T w o ommon initial onditions, ( x 1 , x 2 , y ) = (0 . 9 5; 0 . 95 , 0 . 1) (blue) and (0 . 95 , 0 . 95 , 0 . 95) (red). The v alue of y has b een divided b y 10 to k eep all v ariables in [0 , 1] . In b oth ases a limit yle is on trolled in to an equilibrium p oin t. See parameters in App endix A.2 Some elemen tary alulus sho ws that the left-hand side in the inequalit y of prop osition 1 is a inreasing funtion of γ y when γ y ∈ (0 , 1 /θ y ) , as assumed previously . Th us, this inequalit y is equiv alen t to requiring a lo w er b ound to γ y , ev en though this b ound do es not ha v e a simple expliit form. This fat an b e giv en an in tuitiv e explanation: γ y is in v ersely prop ortional to the  harateristi time of the v ariable y , in ea h b o x. Hene, prop osition 1 means that the dynamis of y m ust b e fast enough in order to retriev e the b eha viour of the stati feedba k on trol, whi h orresp onds to the limit of an instan taneous feedba k. See Figure 3 for a n umerial example. The results of this setion an b e summarised as Prop osition 2. A system of the form ( 10 ), with strutur al  onstr aints (11 ), has a unique, stable and glob al ly attr ative limit yle in absen e of input, i.e. u = 0 . Ther e exists a  ontr ol law ensuring a unique, stable and glob al ly attr ative e qui- librium p oint. This  ontr ol  an b e ahieve d in two ways: • ) Using a s alar pie  ewise  onstant fe e db ak u , suh that u ( a ) satises ( 13 ) for a ∈ { 10 , 2 0 } . • ) Using dynami fe e db ak with a single additional variable y , as in (14 ), whose de  ay r ate satises the  ondition in pr op osition 1, and with υ a solution of (13 ). INRIA Contr ol of limit yles in PW A gene networks 15 5.2 Example 2 : birth of a limit yle Let us no w onsider the follo wing system    ˙ x 1 ( t ) = K 1 s + ( x 2 ) − ( γ 1 1 u + γ 0 1 ) x 1 ˙ x 2 ( t ) = K 3 2 s − ( x 3 ) + K 1 2 s − ( x 1 , θ 2 1 ) − ( γ 1 2 u + γ 0 2 ) x 2 ˙ x 3 ( t ) = K 3 s + ( x 1 , θ 1 1 ) − ( γ 1 3 u + γ 0 3 ) x 3 (17) where s + ( x i ) abbreviates s + ( x i , θ 1 i ) for i = 2 , 3 . W e assume the follo wing on- strain ts to b e satised ( γ 1 i > 0 , i = 1 , 2 , 3 , K 1 > θ 2 1 γ 0 1 > θ 1 1 γ 0 1 K 3 > θ 3 γ 0 3 , K i 2 > θ 2 γ 0 2 , i = 1 , 3 . (18) This system is a partiular ase of t w o om bined negativ e feedba k lo ops, of the form: 3 2 1 Sine the b eha viour of a single lo op is w ell  haraterised b y theorem 2, it an b e onsidered as one of the simplest systems whose b eha viour migh t b e w orth in v estigating. Computing the fo al p oin ts of all b o xes, with the abbreviations φ i = K i / ( γ 1 i u + γ 0 i ) (with additional sup ersripts to φ i and K i for i = 2 ) and φ + 2 = φ 1 2 + φ 3 2 , leads to the follo wing table: 000 100 200 010 110 210 001 101 20 1 011 111 21 1 0 0 0 φ 1 φ 1 φ 1 0 0 0 φ 1 φ 1 φ 1 φ + 2 φ + 2 φ 3 2 φ + 2 φ + 2 φ 3 2 φ 1 2 φ 1 2 0 φ 1 2 φ 1 2 0 0 φ 3 φ 3 0 φ 3 φ 3 0 φ 3 φ 3 0 φ 3 φ 3 Under the indiated parameter onstrain ts, the follo wing transition graph is easily dedued: TG (0) = 011 111 21 1 001 10 1 201 010 11 0 210 000 10 0 200 (19) The region with b old arro ws  i.e. the whole graph in this ase  is in v arian t, and w e see that dep ending on the parameter v alues, the atual solutions of (17 ) ma y ha v e v arious b eha viours: to ea h p erio di path in TG (0) , a stable p erio di orbit ma y p ossibly orresp ond, and there is an innit y of su h paths. Some examples ha v e already b een pro vided of su h situations, where p erio di paths of arbitrary length an b e realised as stable limit yles, b y suitable  hoie of parameters [17 ℄. Although it do es not presen t a xed b o x, it ma y also ha v e a stable equilibrium, limit of damp ed osillations, as will b e illustrated so on. In order to guaran tee that the system osillates, w e x the follo wing ob jetiv e: TG ⋆ = 011 111 21 1 001 10 1 201 010 11 0 210 000 10 0 200 (20) RR n ° 7130 16 F ar  ot & Gouzé W e no w see that the in v arian t region in b old is a yle with no esaping edge. F urthermore, it lies in the region where s − ( x 1 , θ 2 1 ) = 1 , and it app ears from ( 17 ) that the system in this region is a negativ e feedba k lo op in v olving the three v ari- ables. Hene, from theorem 2 , w e an onlude that there exists a unique stable limit yle, whi h is globally attrativ e, as dedued from TG ⋆ . Y et, it remains to state the inequalities dening this graph. They follo w from the in v ersion of arro ws in on tat with some a ∈ A ⋆ = { 110 , 21 0 , 111 , 2 11 , 001 , 101 , 011 } , whi h leads to        θ 2 1 ( γ 0 1 + uγ 1 1 ) > K 1 > θ 1 1 ( γ 0 1 + uγ 1 1 ) K 1 2 + K 3 2 > K 3 2 > θ 1 2 ( γ 0 2 + uγ 1 2 ) K 1 2 < θ 1 2 ( γ 0 2 + uγ 1 2 ) K 3 > θ 1 3 ( γ 0 3 + uγ 1 3 ) This system, follo wing ( 9 ), an b e redued to: max  K 1 − γ 0 1 θ 2 1 γ 1 1 θ 2 1 , K 1 2 − γ 0 2 θ 1 2 γ 1 2 θ 1 2  < u ( a ) < min  K 3 2 − γ 0 2 θ 1 2 γ 1 2 θ 1 2 , K 1 − γ 0 1 θ 1 1 γ 1 3 θ 1 3 , K 3 − γ 0 3 θ 1 3 γ 1 3 θ 1 3  (21) for a ∈ A ⋆ . The problem is th us redued to the satisabilit y of the inequalit y b et w een the t w o extreme terms ab o v e. This fat holds for some parameter v alues satisfying onstrain ts (18 ), and the results of this setion an b e summarised as Prop osition 3. A system of the form ( 17 ), with strutur al  onstr aints (18 ), may pr esent a lar ge variety of asymptoti b ehaviours without input, i.e. when u = 0 . This inludes ste ady states, as shown Figur es 4 and 5 , as wel l as limit yles (not shown). If u is a s alar pie  ewise  onstant fe e db ak, suh that u ( a ) satises (21) for a ∈ A ⋆ , and u ( a ) = 0 elsewher e, then ther e exists a unique, stable and glob al ly attr ative limit yle. 6 Conlusion W e ha v e giv en, and illustrated b y t w o examples, a on trol metho dology to mak e unique stable limit yles app ear or disapp ear in h ybrid PW A systems. The ob- tained feedba k la ws are termed qualitativ e on trol b eause they dep end only on a qualitativ e abstration of the original system; its transition graph. F uture w ork suggested b y this study are mostly related to the question of dy- nami feedba k. A tually , the rst example sho ws the eetiv e p ossibilit y of using an additional v ariable to on trol a system, i.e. to design a on troller sys- tem to b e oupled to the original one. Moreo v er, the design of this dynami feedba k relied in a simple w a y on the stati feedba k problem. This te hnique should b e formalised in more general terms, and applied to other examples in the future. 7 A  kno wledgemen ts J.-L.G. w as partly funded b y the LSIA-INRIA Colage and the ANR MetaGenoR e g pro jets. E.F. w as partly funded b y the ANR-BBSR C Flower Mo del pro jet. INRIA Contr ol of limit yles in PW A gene networks 17 0.4 0.5 0.6 0.7 0.8 0.9 1 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x 1 x 2 x 3 Figure 4: Blue, dashed line: without feedba k on trol, spirals to w ards a xed p oin ts. Red, plain line: with on trol, tends to a limit yle. Common initial ondition ( x 1 , x 2 , x 3 ) = (0 . 95 , 0 . 95 , 0 . 1) . See parameters in App endix A.3 0 5 10 15 20 25 30 0.2 0.4 0.6 0.8 1 x 1 0 5 10 15 20 25 30 0.4 0.6 0.8 1 1.2 1.4 x 2 0 5 10 15 20 25 30 0 0.2 0.4 0.6 0.8 1 x 3 Figure 5: Same urv es as Figure 4 , vs time. The blue urv e stops b efore the red one for the follo wing reason. The n umerial solutions are omputed using the transition map, from w all to w all, and the unon trolled tra jetory rosses suessiv e thresholds within time in terv als tending to zero. RR n ° 7130 18 F ar  ot & Gouzé Referenes [1℄ E. Andrianan toandro, S. Basu, D. Karig, and R. W eiss. Syn theti biology: new engineering rules for an emerging disipline. Mol. Syst. Biol. , 2, 2006. [2℄ G. Batt, C. Belta, and R. W eiss. Mo del  he king geneti regulatory net- w orks with parameter unertain t y . In Hybrid systems:  omputation and  ontr ol , pages 6175, 2007. [3℄ C. Belta and L. Hab ets. Con trolling a lass of nonlinear systems on ret- angles. IEEE T r ansations On A utomati Contr ol , 51(11):1749, 2006. [4℄ R. Casey , H. de Jong, and J.-L. Gouzé. Pieewise-linear mo dels of ge- neti regulatory net w orks: Equilibria and their stabilit y . J. Math. Biol. , 52(1):2756, 2006. [5℄ H. de Jong, J. Geiselmann, G. Batt, C. Hernandez, and M. P age. Quali- tativ e sim ulation of the initiation of sp orulation in baillus subtilis. Bul l. Math. Biol. , 66(2):261300, 2004. [6℄ H. de Jong, J.-L. Gouzé, C. Hernandez, M. P age, T. Sari, and J. Geisel- mann. Qualitativ e sim ulation of geneti regulatory net w orks using pieewise-linear mo dels. Bul l. Math. Biol. , 66(2):301340, 2004. [7℄ Y. Dev eaux, A. P eauelle, G. R. Rob erts, E. Co en, R. Simon, Y. Mizuk ami, J. T raas, J. A. Murra y , J. H. Do onan, and P . Laufs. The ethanol swit h : a to ol for tissue sp ei gene indution during plan t dev elopmen t. Plant J. , 36:918930, 2003. [8℄ R. Edw ards. Analysis of on tin uous-time swit hing net w orks. Physi a D , 146:165199, 2000. [9℄ R. Edw ards, H. Siegelmann, K. Aziza, and L. Glass. Sym b oli dynamis and omputation in mo del gene net w orks. Chaos , 11(1):160169, 2001. [10℄ M. B. Elo witz and S. Leibler. A syn theti osillatory net w ork of transrip- tional regulators. Natur e , 403:335338, 2000. [11℄ E. F arot. Geometri prop erties of pieewise ane biologial net w ork mo d- els. J. Math. Biol. , 52(3):373418, 2006. [12℄ E. F arot and J.-L. Gouzé. P erio di solutions of pieewise ane gene net- w ork mo dels: the ase of a negativ e feedba k. Resear h Rep ort RR-6018, INRIA, 2006. http://hal.inria .f r/i nr ia -00 11 21 95/ en / . [13℄ E. F arot and J.-L. Gouzé. A mathematial framew ork for the on trol of pieewise-ane mo dels of gene net w orks. A utomati a , 44(9):23262332, 2008. [14℄ E. F arot and J.-L. Gouzé. Limit yles in pieewise-ane gene net w ork mo dels with m ultiple in teration lo ops. Int. J. Syst. Si. , 2009. to app ear. [15℄ E. F arot and J.-L. Gouzé. P erio di solutions of pieewise ane gene net- w ork mo dels: the ase of a negativ e feedba k lo op. A ta Biothe or eti a , 57(4):429455, 2009. INRIA Contr ol of limit yles in PW A gene networks 19 [16℄ P . F rançois and V. Hakim. Core geneti mo dule: The mixed feedba k lo op. Phys. R ev. E , 72:031908, 2005. [17℄ T. Gedeon. A ttrators in on tin uous time swit hing net w orks. Communi-  ations on Pur e and Applie d A nalysis (CP AA) , 2(2):187209, 2003. [18℄ L. Glass. Com binatorial and top ologial metho ds in nonlinear  hemial kinetis. J. Chem. Phys. , 63:13251335, 1975. [19℄ L. Glass and J. S. P asterna k. Stable osillations in mathematial mo dels of biologial on trol systems. J. Math. Biol. , 6:207223, 1978. [20℄ J.-L. Gouzé and T. Sari. A lass of pieewise linear dieren tial equations arising in biologial mo dels. Dynami al Systems , 17:299316, 2003. [21℄ C. Grilly , J. Stri k er, W. L. P ang, M. R. Bennett, and J. Hast y . A syn theti gene net w ork for tuning protein degradation in sa har omy es  er evisiae . Mol. Syst. Biol. , 3:127, 2007. [22℄ L. Hab ets and J. v an S h upp en. A on trol problem for ane dynamial systems on a full-dimensional p olytop e. A utomati a , 40:21 35., 2004. [23℄ P . A. Iglesias and B. P . Ingalls, editors. Contr ol The ory and Systems Biol- o gy . MIT Press, 2009. [24℄ H. K oba y ashi, M. Kaern, M. Araki, K. Ch ung, T. S. Gardner, C. R. Can tor, and J. J. Collins. Programmable ells: in terfaing natural and engineered gene net w orks. Pr o . Natl. A  ad. Si. U.S.A. , 101(22):84149, 2004. [25℄ L. Lu and R. Edw ards. Strutural priniples for p erio di orbits in glass net w orks. Journal of Mathemati al Biolo gy , 2009. DOI 10.1007/s00285- 009-0273-8, to app ear. [26℄ T. Mestl, E. Plah te, and S. W. Omholt. P erio di solutions of pieewise- linear dieren tial equations. Dyn. Stab. Syst. , 10(2):179193, 1995. [27℄ E. Plah te, T. Mestl, and S. W. Omholt. A metho dologial basis for de- sription and analysis of systems with omplex swit h-lik e in terations. J. Math. Biol. , 36:321348, 1998. [28℄ S. Shimizu-Sato, E. Huq, J. T epp erman, and P . H. Quail. A ligh t-swit hable gene promoter system. Nat. Biote hnol. , 20(10):10411044, 2002. [29℄ E. H. Snoussi. Qualitativ e dynamis of pieewise-linear dieren tial equa- tions: a disrete mapping approa h. Dyn. Stab. Syst. , 4(3-4):189207, 1989. [30℄ E. D. Son tag. Moleular systems biology and on trol. Eur. J. Contr ol , 11((4-5)):396435, 2005. [31℄ S. W yk e and M. Tisdale. Indution of protein degradation in sk eletal m us- le b y a phorb ol ester in v olv es upregulation of the ubiquitin-proteasome proteolyti path w a y . Life Sien es , 78(25):2898  2910, 2006. RR n ° 7130 20 F ar  ot & Gouzé A P arameter v alues A.1 P arameters for Figure 2 K 1 K 2 γ 0 1 γ 1 1 γ 0 2 θ 1 1 θ 2 1 θ 1 2 0 . 9 0 . 2 1 1 0 . 3 0 . 5 0 . 75 0 . 5 Moreo v er the v alue u ( a ) is omputed as the middle-p oin t of the in terv al dened b y (13 ). A.2 P arameters for Figure 3 K 1 K 2 γ 0 1 γ 1 1 γ 0 2 θ 1 1 θ 2 1 θ 1 2 θ y 0 . 9 0 . 2 1 1 0 . 3 0 . 5 0 . 7 5 0 . 5 0 . 5 T o  he k the inequalit y in prop osition 1 , w e need to ompute  φ 1 − θ 2 1 φ 1 − θ 1 1  1 γ 0 1 = 0 . 375 . Then, the t w o v alues of γ y w e ha v e tested are 0 . 1 and 1 . 7 , for whi h (1 − γ y θ y ) 1 γ y is resp etiv ely lose to 0 . 599 (inequalit y satised) and 0 . 328 (in- equalit y violated). A.3 P arameters for Figures 4 and 5 K 1 K 1 2 K 3 2 K 3 γ 0 1 γ 0 2 γ 0 3 γ 1 i θ 1 i θ 2 1 0 . 9 0 . 6 1 0 . 5 1 1 0 . 5 1 0 . 5 0 . 75 where i stands for all v alues in { 1 , 2 , 3 } F or these v alues, inequalit y ( 21) writes, term b y term: max { 0 . 2 , 0 . 2 } < u ( a ) < min { 1 , 0 . 8 , 0 . 5 } and w e ha v e  hosen u ( a ) = 0 . 3 in the sim ulations. INRIA Centre de recher che INRIA S oph ia Antipolis – Méditerranée 2004, route des Lucioles - BP 93 - 06902 Sophia Antipolis Cedex (France) Centre de recherc he INRIA Bordeaux – Sud Ouest : Domaine Unive rsitaire - 351, cours de la Libératio n - 33405 T alenc e Cedex Centre de recherc he INRIA Grenobl e – Rhône-Alpes : 655, ave nue de l’Europe - 38334 Montbonnot Saint-Ismier Centre de recherc he INRIA Lille – Nord Europe : Parc Scientifique de la Haute Borne - 40, ave nue Halley - 59650 V illeneuv e d’Ascq Centre de recherc he INRIA Nanc y – Grand Est : L ORIA, T echnopôle de Nancy-Bra bois - Campus scientifique 615, rue du Jardin Botani que - BP 101 - 54602 V ille rs-lès-Nanc y Cedex Centre de recherc he INRIA Pari s – Rocquencourt : Domaine de V oluceau - Rocquenco urt - BP 105 - 78153 Le Chesnay Cede x Centre de recherc he INRIA Renne s – Bretagne Atlantique : IRISA, Campus univ ersitai re de Beaulieu - 35042 Rennes Cedex Centre de recherc he INRIA Sacla y – Île-de-Franc e : Parc Orsay Uni versité - ZAC des V ignes : 4, rue Jacques Monod - 91893 Orsay Cede x Éditeur INRIA - Domaine de V olucea u - Rocquenc ourt, BP 105 - 78153 Le Chesnay Cede x (France) http://www.inria.fr ISSN 0249 -6399 0.2 0.3 0.4 0.5 0.6 0.7 0 0.1 0.2 0.3 0.4 0.5 0 .6 0 0.1 0.2 0.3 0.4 0.5 x x 2 0 0.1 0.2 0.3 0.4 0.5 0.6 0 0.1 0.2 0.3 0.4 0.5 x 1 x 2