Positive circuits and maximal number of fixed points in discrete dynamical systems

P ositiv e circuits and maximal n um b er of fixed p oin ts in discrete dynamical systems Adrien Ric ha rd L ab o r atoir e I3S, UMR 6070 CNRS & Universit´ e de Nic e-Sophia Antip olis, 2000 r oute des Luci oles, 06903 Sophia Ant i p olis, F r anc e. Email: richard@i3s.uni ce.fr Abstract W e consider the Cartesian p ro duct X o f n fi n ite in terv als of integ ers and a map F from X to i tself. As main result, w e establish an upp er b ound on the num b er of fixed p oint s for F whic h only dep ends on X a n d on the top ology of the p ositiv e circu its of the in teraction graph asso ciated with F . Th e pro of uses and strongly ge n eralizes a theorem of Richard and Comet whic h corr esp onds to a discrete version of the Thomas’ conjecture: if the in teraction graph asso ciated with F has no p ositiv e circuit, t hen F has at most one fix ed p oin t. The obtained upp er b ound on the n u m b er of fixed p oin ts also strongly generalizes the one established b y Aracena et al for a particular class of Bo olean net works. Key wor ds: Discrete dynamical system, Discrete Jacobian matrix, Interact ion graph, P ositiv e circuit, Fixed p oint. 1 In tro duction W e are in terested b y the n umber of fixed p o ints for maps that op erate on the Cartesian pro duct of n finite in terv als of in tegers (when this pro duct is { 0 , 1 } n , suc h maps are usually called Bo ole an network ). Our motiv ation comes f rom biology , where these maps a re extensiv ely used to describe the b ehav io r of gene net w orks. The contex t is then the following. When studying gene netw or ks, biolog ists often illustrate their results by inter- action gr aphs . These are directed graphs where v ertices corresp o nd to g enes and where edges are lab elled with a sign: a po sitiv e (resp. negative) edge from j to i means that the protein enco ded b y gene j activ a tes (resp. represses ) the syn thesis of the protein enco ded by gene i . These graphs are then used as ba sis Preprint su bmitted to Elsevier 3 June 2018 to g enerate dynamic al mo dels describing the temp oral ev olution of the concen- tration of the encoded proteins (see [1] fo r a literature rev iew). Unfortunately , these mo dels require, in most cases, una v ailable information on the strength of the inte r actions. One is thus faced with the follo wing difficult question: Which dynamic al pr op erties of a gene network c an b e inferr e d fr om its inter action gr aph (in the ab senc e of information on the str ength of the inter actions)? In this pap er, we fo cus on this question in a general discrete mo deling frame- w ork. The set of states of a net w or k of n genes is represen ted by the pro duct X = Q n i =1 X i of n finite in terv als of in tegers. Eac h in terv al X i then corresp onds to the set of p ossible concen tration lev els for the prot ein enco ded by gene i . On one hand, the dynamics of the net work is describ ed b y the succes sive iter- ations of a map F from X to itse lf whose fixed p oin ts corr espo nd to the stable states of the netw o r k. A t this stage, it is w orth noting that the n umber of stable states is a k ey feature of gene net w orks dynamics: according to an idea of D elbr ¨ uc k [2], the presence of multiple stable states is one p ossible mec ha- nism fo r biological differen tiation. One the other hand, the in teraction g raph of the netw ork is deduced from F in tw o steps . First, to eac h state x ∈ X and to eac h directional v ector v ∈ {− 1 , 1 } n suc h tha t x + v ∈ X is asso ciated a lo c al in ter action gr aph G F ( x, v ) whic h con tains a p ositiv e (resp. negativ e) edge from j to i if f i ( x 1 , . . . , x j + v j , . . . , x n ) − f i ( x ) v j is p ositiv e (resp. negativ e) ( f i denotes the i th comp onen t of F ). Then, the glob al inter action gr aph G ( F ) of the net w o rk is defined to b e the union of all the lo cal in teraction graphs. Note that eac h lo cal interaction graph is a subgraph of the global one, and t ha t the global in teraction graph can ha ve b oth a p ositive and a negative edge fro m one ve rt ex to another. In this setting, Richard and Comet [3] partialy answ er t he previous question b y pro ving a w ell kno wn conjecture of Ren ´ e Thomas relating the stable states of the net w ork to the p ositiv e circuits of its lo cal in teraction graphs (a circuit is p ositiv e if it ha s an ev en n umber of negative edges). A w eak form of their result follo ws (the original statemen t needs a dditio nal definitions and is giv en latter in the pap er): Theorem 1 [3] L et X b e a pr o duct of n finite interval of inte ge rs , and let F b e a ma p fr om X t o itself. If al l the lo c al inter action gr aphs G F ( x, v ) ar e without p ositive cir cuit, then F has at most one fixe d p oint. Aracena, D emongeot and Goles [4,5] pro ve d another theorem relating stable states to p ositiv e circuits. They establish, in a b o olean con text, an upp er b ound on the num b er of stable stat es whic h o nly dep ends on the p o sitiv e circuits of the global in teraction graph of the net work: 2 Theorem 2 [4,5] L et F b e a map fr om { 0 , 1 } n to itself such that G ( F ) has no b oth a p ositive and a ne gative e dge fr om one vertex to another. If I is a subset of { 1 , . . . , n } such that e ach p ositive c ir c uit of G ( F ) has a t le ast one vertex in I , then the numb er of fixe d p oints fo r F is less than or e qual to 2 | I | . The main result of this pa p er is a significativ e generalization of b ot h Theorem 1 and Theorem 2. A w eak form of t his result is: Theorem 3 L et X = Q n i =1 X i b e a pr o duct of n finite interval of inte gers, a nd let F b e a map fr om X to itself. I f I is a subset of { 1 , . . . , n } such that e ach p ositive cir cuit of e ach lo c a l inter action gr aph G F ( x, v ) has at le ast on e vertex in I , then the numb er of fixe d p oints for F is les s than or e qual to Q i ∈ I | X i | . Theorem 3 implies Theorem 1 , since if all the lo cal in teraction graphs G F ( x, v ) are without p ositiv e circuit then I = ∅ satisfies the conditions of Theorem 3. So the corresp onding b ound is 1 and Theorem 1 is reco v ered. Theorem 3 also implies Theorem 2. Indeed, let F b e a map from X to it self and supp ose I to b e suc h tha t eac h p ositiv e circuit of G ( F ) has at least one v ertex in I . Then I satisfies the conditions of Theorem 3 b ecause each lo cal interaction graphs G F ( x, v ) is a subgraph of G ( F ). So the corresp onding b ound is Q i ∈ I | X i | and it equals 2 | I | in the particular case where X is the n -cub e { 0 , 1 } n . W e th us reco v er the conclusion o f The o rem 2 ( even if G ( F ) has b oth a p o sitiv e and a negative edge f r o m one v ertex to a nother). The pro of of Theorem 3, which is done by induction on I with Theorem 1 as ba se case, is indep enden t of the pro of of Theorem 2 give n in [4,5]. Not e also that Theorem 2 do es not imply Theorem 1 ev en if this la tter is stated for maps F fr o m { 0 , 1 } n to itself suc h that G ( F ) has no b oth a p ositiv e and a negativ e edge from o ne v ertex to a nother. The pap er is organized as fo llo w. In Section 2, in order to o btain a bo und stronger tha n the one men tioned ab o v e and more relev ant from a bio lo gical p oin t of view, w e fo cus o n the async hrono us iterations of F that Thomas use to describ e the dynamics of gene netw or ks [6,7,8,9]. First, w e represen t these iterations under the form of a directed gra ph Γ( F ) on X usually called a s yn- chr onous state tr ansition gr aph . Then, w e define the attr actors of Γ( F ) to b e the smallest subsets of states without output edges in Γ ( F ). The fixed p oin ts of F then corresp ond t o part icular attra ctors. In Section 3, w e characterize a subgraph G F ( x, v ) of G F ( x, v ) whic h only dep ends on Γ( F ) and whic h is, for this reason, w ell suited to the study of Γ( F ). In Section 4 w e establish our main result: an upp er b ound on the numb er of attr actors in Γ( F ) which only dep ends on the map G F and whic h has Theorem 3 as immediate consequence. Final commen ts are given in Section 5. These ar e ab out the influence of con- nections b et ween p ositiv e circuits and the interes t of the established bo und in the con text of the so called Tho m as’ lo gic al me tho d [6,7,8,9] whic h is, in practice, one of the most usual discrete mo deling metho d of gene netw orks. 3 2 Async hronous state transition graph and attractors Let X = Q n i =1 X i b e the pro duct of n finite interv als of integers of cardinality strictly greater than 1, and consider a map F f rom X to itself, x = ( x 1 , . . . , x n ) ∈ X 7→ F ( x ) = ( f 1 ( x ) , . . . , f n ( x )) ∈ X. In the following definition, w e atta ch to F a directed graph on X called asynchr onous state tr ansition gr aph . According to Thomas [6,7,8,9], this state graph can b e seen as a mo del for the dynamics of a net work of n g enes: the set of v ertices X is the set of p o ssible states for the netw or k (each in terv al X i cor- resp onds to the p o ssible concen tration of the pr o tein enco ded b y gene i ), a nd eac h path corresp onds to a p ossible ev olution of the system. [Async hronous state tr a nsition graphs can also b e seen as discretizations of piecewis e- linear differen tial systems, see [10,11] for instance.] Definition 1 The async hronous stat e transition graph of F is the dir e cte d gr aph Γ( F ) whose set of vertic es is X a nd w h ich c ontains an e dge fr om x to y if ther e e x ists i ∈ { 1 , . . . , n } such that f i ( x ) 6 = x i and y = x + sign( f i ( x ) − x i ) · e i , wher e e i denotes the n -tuple whose i th c omp onent is 1 and whose other c om- p onents ar e 0 , and wher e sign( a ) = a/ | a | for al l inte ger a 6 = 0 . [F ollo wing this description of the dynamics, f i ( x ) can b e seen a s the v alue to ward whic h the concen tratio n x i of the pr o tein enco ded by gene i evolv es: at state x , there exists a state transition allow ing the i th comp onen t of the system to increase (resp. de crease) if and only if x i < f i ( x ) (resp . x i > f i ( x )).] The fixed p oints o f F ha ve no success o r in Γ( F ) and naturally corresp ond to the stable states of the s ystem. In the next definition, we intro duce the notion of attr actor whic h extends, in a natural w ay , the notion of stable stat e. Definition 2 A trap domain of Γ( F ) is a non-empty subset A of X such that, for al l e dges ( x, y ) of Γ( F ) , if x ∈ A then y ∈ A . A n a ttractor of Γ( F ) is a smal lest tr ap doma i n with r esp e ct to the inclusion r elation. In o t her words, the attr actors of Γ( F ) are the smallest set of states that we cannot leav e. They extend the notion o f stable state in the sense that x is a fixed p o in ts of F if and only if { x } is an attr a ctor of Γ( F ). Note a lso that there alwa ys exists at least one attr a ctor (since X is a trap domain). Other easy observ ations follow: (1) F rom eac h state, there is a path whic h leads to an attractor (this is wh y one can say that a t t r actors p erfo rm, in w eak sense, a n 4 attraction); (2) A t t ractors are strongly connected comp onents; (3) A ttractors are m utually disjoin ted (this p oint used in the pro of of our main result). 3 Discrete Jacobian matrix and interaction graph In this section, w e in t r o duce a notion o f lo cal in teraction graph w ell suited to the study of Γ( F ). W e pro ceed as in [3] by first in tro ducing a discrete Jacobian matrix for F based on a not io n of discrete directional deriv a tiv e. Let X ′ b e the set of couples ( x, v ) suc h that x ∈ X , v ∈ {− 1 , 1 } n and x + v ∈ X . Definition 3 F or al l ( x, v ) ∈ X ′ , we c al l Ja cobia n matrix of F e v aluate d at x along the dir e ctional ve ctor v the n × n matrix F ′ ( x, v ) = ( f ij ( x, v )) define d by f ij ( x, v ) = f i ( x + v j e j ) − f i ( x ) v j ( i, j = 1 , . . . , n ) . [If v j is po sitiv e (resp. negativ e), then f ij ( x, v ) ma y b e seen as the rig h t (resp. left) partial deriv ativ e o f f i with respect to the j th v a r iable ev aluated at x . In b oth cases, f ij ( x, v ) is a natural discrete analogue of ( ∂ f i /∂ x j )( x ).] An inter action gr aph is here a directed graph whose set of v ertices is { 1 , . . . , n } and where eac h edge is pro vided with a sign. More formally , eac h edge is c haracterized by a triple ( j, s, i ) where j (resp. i ) is the initial (resp. final) v ertex and where s ∈ {− 1 , 1 } is the sign of the edge. The set of edges of an in teraction gr a ph G is denoted E ( G ). An in tera ctio n graph G is a sub gr aph of an in teraction graph G ′ if E ( G ) ⊆ E ( G ′ ). Definition 4 We c al l in t era ctio n graph of F ev aluated at ( x, v ) ∈ X ′ , and we denote d by G F ( x, v ) , the inter action gr aph which c ontains a p ositive ( r e s p . ne gative) e dge fr om j to i if f ij ( x, v ) is p ositive (r esp. ne gative). [T o illustrate this definition, a ssume that f ij ( x, v ) is p ositiv e and t ha t v j = 1. Then, f i ( x ) < f i ( x + e j ) so w e can sa y that, at stat e x , an increase of x j induces an increase of f i , that is, an increase of the v alue tow a r d whic h the i th comp onen t of t he system ev olv es. In other w ords, j a cts as an a ctiv ator of i , and w e hav e a p ositive edge from j to i in G F ( x, v ).] In our con t ext, the o b vious fa ct that G F ( x, v ) do es not only dep end on Γ( F ) is not satisfactory sinc e it is commonly accepte d that the in teraction gra ph of a net w ork only depends on its dynamics, whic h is here c haracterized b y Γ( F ). This lead us, as in [3 ], to sligh tly mo dify the definition of G F ( x, v ) in order to obtain an in teraction graph G F ( x, v ) whic h o nly dep ends Γ( F ). 5 Definition 5 We c al l in t era ctio n graph of F ev aluated at ( x, v ) ∈ X ′ with thresholds , an d w e denote by G F ( x, v ) , the inter action gr aph which c ontains a p ositive (r esp. ne gative) e dge fr om j to i if f ij ( x, v ) is p ositive (r esp. ne gative) and if f i ( x ) and f i ( x + v j e j ) ar e on b oth sides of (the thr eshold) x i + v i / 2 . [ a and b are on b oth sides of c if a < c < b or b < c < a .] Remark 1 G F ( x, v ) is a subgraph of G F ( x, v ) (often strict sinc e the addi- tional c ondition “on b oth sides of the thr eshold” is r a ther str on g ). Remark 2 The intr o duction of G F ( x, v ) has b e en motivate d by ar gumen ts c oming fr om the mo deling c ontext. Another r elevant ar gument is the fol lowing: b ecause G F ( x, v ) is a subgraph of G F ( x, v ), all the incoming results remains v alid but b ecomes less strong when stated with G F ( x, v ) instead of G F ( x, v ) . Remark 3 In the B o ole an c ase, i.e. when X = { 0 , 1 } n , G F ( x, v ) = G F ( x, v ) . Definition 6 We c al l glo bal in teraction graph of F , and we denote by G ( F ) , the inter action gr aph whose set of e dges i s S ( x,v ) ∈ X ′ E ( G F ( x, v )) . Ob viously , G ( F ) only depends on Γ( F ) and can th us b e seen as the globa l in teraction g raph of the net w ork of dynamics Γ( F ). Note that G ( F ) can ha v e b oth a p ositive and a negative edge fro m one ve rt ex to another. No w, w e recall the notion of p ositiv e circuit a nd the notion of p o sitiv e feedbac k v ertex set. This ha s b een in tro duced b y Aracena et al [4,5] to study the fixed p oin t s o f Bo olean net w o rks. Definition 7 A p ositiv e circuit i n an inter action gr aph G is a non-empty se quenc e of e dges, say ( j 1 , s 1 , i 1 ) , ( j 2 , s 2 , i 2 ) , . . . , ( j r , s r , i r ) , such that: i k = j k +1 for 1 ≤ k < r (the se quenc e is a p ath); i r = j 1 (the p ath is a cir cuit); the vertic es j k ar e mutual ly distinc t (the cir cuit is elementary); the pr o duct of the signs s k is p ositive (ev e n numb er of ne gative e dges). Definition 8 [4] A p ositiv e feedbac k v ertex set of an inter action gr aph G is a subset I ⊆ { 1 , . . . , n } such that e ach p ositive cir cuit of G has a vertex in I . One can remark that: (1) The set of v ertices of G is alwa ys a p ositiv e feedbac k v ertex set of G ; (2) The empty set is a p ositiv e feedbac k v ertex set of G if and only if G has no p ositiv e circuit; (3) If G ′ is a subgraph of G then all the p ositiv e feedbac k ve rt ex sets of G are p ositiv e feedbac k vertex sets of G ′ . 6 4 P ositive circuits and att ractors As pr eviously , let X = Q n i =1 X i b e the pro duct o f n finite in terv als of integers of cardinalit y strictly greater than 1, and let F b e a map f r om X to it self. W e ar e interes t ed by the relations b et we en the map G F (defined on X ′ ) and the num b er of attractor s in Γ( F ). The following theorem, presen ted in [3] as solution of a disc r ete v ersion o f the Thomas’ conjecture, give s suc h a relation. Theorem 4 [3] If G F ( x, v ) has no p ositive cir cuit for al l ( x, v ) ∈ X ′ , then Γ( F ) has a unique attr actor. The follo wing theorem extends the previous one by providing, without a n y condition on the map G F , an upp er b ound on the n umber of attracto r s in Γ( F ) whic h o nly dep ends o n G F . Theorem 5 (main result) F or e ach i ∈ { 1 , . . . , n } , let T i ( G F ) b e the set of r e al numb ers t for which ther e exists ( x, v ) ∈ X ′ such that t = x i + v i / 2 and such that i b e l o n gs to a p ositive cir cuit of G F ( x, v ) . Supp ose I to b e, for al l ( x, v ) ∈ X ′ , a p ositive fe e db ack vertex set of G F ( x, v ) . Then , the numb er of attr actors in Γ( F ) is less than Y i ∈ I h | T i ( G F ) | + 1 i . Pr o of − W e reason by induc t ion on I . Supp ose I to b e, for an y ( x, v ) ∈ X ′ , a p ositiv e feedbac k ve rt ex set of G F ( x, v ). Base c ase. If I = ∅ it means that there is no ( x, v ) ∈ X ′ suc h that G F ( x, v ) has a p o sitiv e circuit. So, follo wing Theorem 4, Γ( F ) has at most one attractors and the theorem holds. Induction step. Supp ose that I 6 = ∅ . The induc tio n hy p othesis is the fo llo wing: Induction hyp othesis: Let ˜ F b e a map from X to itself. If ˜ I is, for all ( x, v ) ∈ X ′ , a p ositiv e feedbac k v ertex set of G ˜ F ( x, v ), a nd if ˜ I is strictly include d in I , then Γ( ˜ F ) has at most Q i ∈ ˜ I | T i ( G ˜ F ) | + 1 a t t ractors. Without lo ss of generality , supp ose that 1 ∈ I . Let P b e the part it ion of X 1 whose elemen ts Y are the maximal interv als of X 1 (with resp ect to the inclusion relation) ve rif ying ∀ t ∈ T i ( G F ) , t < min( Y ) or max( Y ) < t. (1) 7 Remark that, b y definition, | P | = | T 1 ( G F ) | + 1 . (2) Let Y be an y interv al of P , and consider the map ˜ F = ( ˜ f 1 , . . . , ˜ f n ) : X → X defined b y ˜ f i = f i for i > 1 and by ∀ x ∈ X, ˜ f 1 ( x ) =              min( Y ) if f 1 ( x ) < min( Y ) f 1 ( x ) if f 1 ( x ) ∈ Y max( Y ) if f 1 ( x ) > max( Y ) . Then, for all x, y ∈ X , ˜ f i ( x ) < ˜ f i ( y ) ⇒ f i ( x ) ≤ ˜ f i ( x ) < ˜ f i ( y ) ≤ f i ( y ) ( i = 1 , . . . , n ) . (3) Indeed, this is ob vious for i > 1, and for i = 1 it is sufficien t to remark that ˜ f 1 ( x ) < ˜ f 1 ( y ) ⇒ ˜ f 1 ( x ) < max( Y ) ⇒ f 1 ( x ) ≤ ˜ f 1 ( x ) , and that ˜ f 1 ( x ) < ˜ f 1 ( y ) ⇒ min( Y ) < ˜ f 1 ( y ) ⇒ ˜ f 1 ( y ) ≤ f 1 ( y ) . No w, w e prov e that, for all ( x, v ) ∈ X ′ , G ˜ F ( x, v ) is a subgraph of G F ( x, v ) . (4) Let ( x, v ) ∈ X ′ and supp ose ( j, s, i ) to b e an edge of G ˜ F ( x, v ). According to (3), ˜ f ij ( x, v ) and f ij ( x, v ) ha ve the same sign (here s ), and f i ( x ) and f i ( x + v j e j ) are on b oth sides of x i + v i / 2 since ˜ f i ( x ) and ˜ f i ( x + v j e j ) are. In other w ords, ( j, s, i ) is an edge of G F ( x, v ). So (4) is prov ed and, a s an immediate consequence, T i ( G ˜ F ) ⊆ T i ( G F ) ( i = 1 , . . . , n ) . (5) Then, for all ( x, v ) ∈ X ′ , w e hav e the following: V ertex 1 b elongs to none p ositiv e circuit of G ˜ F ( x, v ). (6) Indeed, supp ose, b y con tradiction, that vertex 1 b elongs to a p o sitive circuit of G ˜ F ( x, v ). Let j b e the predec essor o f 1 in this circuit, and let t = x 1 + v 1 / 2. By definition, t ∈ T 1 ( G ˜ F ) and from (5) it comes that t ∈ T 1 ( G F ). W e then deduce, f rom (1) and the fact that the imag es of ˜ f 1 are in Y , that ˜ f 1 ( x ) and ˜ f 1 ( x + v j e j ) are not on b oth sides of t . In o t her w ords, there is no edge from j to 1 in G ˜ F ( x, v ), a con tradiction. 8 Let ˜ A b e the set of attr a ctors of Γ( ˜ F ) and let ˜ I = I \ { 1 } . (7) Let ( x, v ) b e an y elemen t of X ′ . Since I is a p ositiv e feedbac k vertex set of G F ( x, v ) and since G ˜ F ( x, v ) is a subgraph of G F ( x, v ), I is also a p ositiv e feedbac k ve rtex set of G ˜ F ( x, v ). W e then deduce from (6) that ˜ I is a p osi- tiv e f eedbac k v ertex set of G ˜ F ( x, v ). Since this holds f o r a ll ( x, v ) ∈ X ′ , b y induction hy p othesis, | ˜ A | ≤ Y i ∈ ˜ I | T i ( G ˜ F ) | + 1 , and from (5) w e o btain: | ˜ A | ≤ Y i ∈ ˜ I | T i ( G F ) | + 1 . (8) No w, let A b e the set of attractors of Γ( F ), a nd let A Y b e the set of A ∈ A con taining a p oint x such that x 1 ∈ Y . W e claim that: ∀ A ∈ A Y , there exists ˜ A ∈ ˜ A suc h t ha t ˜ A ⊆ A. (9) So let A ∈ A Y , and consider the set ¯ A of x ∈ A suc h that x 1 ∈ Y . W e pro ve that ¯ A is a tra p domain of Γ( ˜ F ). Supp ose ( x, y ) to b e an edge of Γ( ˜ F ) suc h that x ∈ ¯ A . By definition, there exists index i suc h tha t ˜ f i ( x ) 6 = x i and y = x + sign ( ˜ f i ( x ) − x i ) e i . W e consider tw o cases: (1) Case i > 1. Then, y 1 = x 1 ∈ Y . In a dditio n, ˜ f i ( x ) = f i ( x ) so ( x, y ) is a n edge of Γ ( F ). Hence y ∈ A (since x ∈ A ) a nd we deduce that y ∈ ¯ A . (2) Case i = 1. Supp o se that x 1 < ˜ f 1 ( x ) (the pro of is similar if x 1 > ˜ f 1 ( x )). Then, x 1 < y 1 ≤ ˜ f 1 ( x ) and since x 1 and ˜ f 1 ( x ) are in Y we hav e y 1 ∈ Y . In addition, min( Y ) ≤ x 1 < ˜ f 1 ( x ) so x 1 < ˜ f 1 ( x ) ≤ f 1 ( x ). Th us ( x, y ) is an egde of Γ( F ). Hence y ∈ A (since x ∈ A ) a nd w e deduce tha t y ∈ ¯ A . Since y ∈ ¯ A in b oth cases, ¯ A is trap domain of Γ( ˜ F ). Th us there exists at least one attractor ˜ A ∈ ˜ A suc h that ˜ A ⊆ ¯ A , and ( 9 ) holds since ¯ A ⊆ A . F ollo wing (9), there exists a map H : A Y → ˜ A suc h that H ( A ) ⊆ A for a ll A ∈ A Y . Since the attractors of Γ( F ) a r e mu t ua lly disjoin t ed, the elemen ts of A Y are m utually disjointed , and w e deduces that the images of H are also m utually disjointe d. Conseq uently , H is an injection. So | A Y | ≤ | ˜ A | and w e deduce from (8) that | A Y | ≤ Y i ∈ ˜ I | T i ( G F ) | + 1 . 9 Since this inequalit y holds for all Y ∈ P , and since A = ∪ Y ∈ P A Y , w e hav e: | A | ≤ X Y ∈ P | A Y | ≤ X Y ∈ P h Y i ∈ ˜ I | T i ( G F ) | + 1 i = | P | Y i ∈ ˜ I | T i ( G F ) | + 1 . Using (2) and (7) w e conclude: | A | ≤ h | T 1 ( G F ) | + 1 i Y i ∈ ˜ I | T i ( G F ) | + 1 = Y i ∈ I | T i ( G F ) | + 1 .  Corollary 1 I f I is a p ositive fe e db ack vertex set of G ( F ) , then the numb er of attr actors in Γ( F ) and, in p articular, the numb er of fix e d p oints for F ar e less than Q i ∈ I | X i | . Pr o of − It is sufficien t to apply Theorem 5 b y noting that: (1) eac h G F ( x, v ) is a subgraph of G ( F ); (2 ) | T i ( G F ) | + 1 ≤ | X i | ; (3) The n umber of fixed p oints for F is less than the num b er of attractor s in Γ( F ) .  Remark 4 The b ound on the numb er of fixe d p oints fo r F g i v e n Cor ol lary 1 has b e e n pr ove d by Ar ac ena et al [4,5] in the B o ole an c ase and under the str on g hyp othesis that G ( F ) do es no t c ontain b oth a p ositive and a ne gative e dge fr om one vertex to another (that is, the entries of the Jac obian matrix of F ar e everywher e ≥ 0 or everywher e ≤ 0 ); se e the The or em 2 state d in the intr o duction. Remark 5 The or ems 1 and 3 state d in the intr o duction ar e obtaine d f r om The or e ms 4 and 5 by no ting that G F ( x, v ) is a sub gr aph of G F ( x, v ) and by using the p oints (2) and (3 ) in the pr o of of Cor ol lary 1. 5 Commen ts 5.1 Influenc e of c onne ctions b etwe en p ositive cir cuits Corollary 1 is sufficien t to highligh t the fact that: “A high level o f c onne ction b etwe en p ositive cir cuits le a ds to a smal l numb er of fixe d p oints” . Supp ose, for sak e o f simplicit y , that all the in t erv als X i are of cardinalit y q , and let r b e the smallest n umber of v ertices that a p ositiv e feedbac k v ertex set of G ( F ) can contain. Then, t he smallest upp er b ound for the n umber o f fixed p oin ts for F give n b y Corollary 1 is q r , and the more the p ositive circuits of G ( F ) are connected, the more r is small. Indeed, let us say that a vertex r epr es e nts a circuit when it b elongs to this circuit. Then, r corresp onds to the smallest n um b er of ve r t ices allowin g the represen tation of eac h p ositiv e circuit. So, 10 the mo r e the p ositiv e circuit are connected, the more it is p o ssible t o c ho ose v ertices represen ting a n um b er of p ositiv e circuits, and the more r is small. F or instance, r is alw ays ≤ to the n um b er p of p ositiv e circuits that G ( F ) con tains, but r < p whe neve r G ( F ) has connected p ositiv e circuits, and in the ex tremal case where a ll the p ositiv e circuits of G ( F ) share a same ve rtex, r = 1. 5.2 Thomas’ lo gic al metho d In practice, the dynamics of a g ene net w ork is often mo deled from its in terac- tion graph G , typically b y using the w ell kno wn Thomas’ lo gi c al metho d [7,8,9]. In few w o rds, Thomas a sso ciates to G a finite state s pa ce X and describes the b eha vior of the interactions of G by lo gic al p ar ameters . Then, he dedu ces from the v alue of these parameters a map F from X to itself whose async hronous state transition g r a ph describes a p ossible dynamics for the net w or k; see [1 2] for a f o rmal presen tation. This mo deling metho d is coherent with our notion of in teraction graph in the sense that, for all parameters v alues, the resulting map F has the prop ert y to b e suc h that G ( F ) is a subgraph of G [1 3]. So , thanks to Corollary 1, o ne can sa y , in the tota l absence of inf o rmation on the v alue of the parameters, that follo wing Thomas’ logical metho d, the num b er of at t r actors in the dynamics of the net w ork is less than µ ( G , X ) = min I ∈I ( G ) Y i ∈ I | X i | , where I ( G ) is the set o f smallest p ositiv e feedbac k v ertex sets of G (with resp ect to the inclusion relat io n). This result is of practical in terest since the v alue of the parameters is most often unknow n a nd difficult to estimate, a nd since the n umber of attractors is an imp ortant featur e of the dynamics of the net w o rk. F or instance, if the netw or k is kno wn t o con tr o l a differen t iation pro cess into k cell types, one often considers t ha t the dynamics of the net work has to con t a in a t least k attra ctors. The b ound µ ( G , X ) can then b een used in order to c hec k if the data of G and X is consisten t with the presence of k attractors (there is inconsistence whenev er µ ( G , X ) < k ). 5.3 F e e db ack cir cuit functiona l i ty Finally , Theorem 5 is related to one of the main concept raised b y the Thomas’ logical metho d: the concept of fe e db ack cir cuit functionali ty [11,8,9 ,1 4]. Roughly sp eaking, it has b een observ ed that some inequality constrain ts o n the lo g ical parameters describing the b eha vior of the in teractions of a p ositiv e ( r esp. neg- ativ e) circuit of G often lead to a dynamics c on taining sev eral attracto r s ( resp. 11 describing oscillations). F or that reason, when these constrain ts are satisfied, the corresp onding circuit is said functional. Ev en if this notion is not w ell un- derstand and often informally stated, it is o ften used in practice to establish the v alue o f the lo g ical parameters, see [15,16,17,18,19,20] for instance. A natura l fo r malization o f the no tion o f f unctional circuit, also pro p osed in [13,21], is the following: giv en a map F from X to itself whose interaction graph G ( F ) is a subgraph of G , a circuit C of G is functiona l at ( x, v ) ∈ X ′ if C is a circuit of G F ( x, v ). It is t hen easy to see that the upp er b ound for the num b er o f attractors giv en by Theorem 5 only dep ends on the localizatio n (inside X ′ ) and on the connections of the functional p ositiv e circuits o f t he system. In our knowled g e, this is one of the first mathematical result relating the functional circuits o f the system to its glob al dynamical prop erties (for relations b et w een functional circuits and lo c al dynamical prop erties, see the recen t parer [21]). Ac knowledgem ent I wish to thank Christophe Soul´ e for his precious suggestions. References [1] H. d e Jong, Mo d eling and simulatio n of genetic regulatory systems: a literature review, Journal of Computation al Biolo gy , 9 (2002) 67-10 3. [2] O. Delbr ¨ uc k, Discussion, i n U ni t´ es Biolo giqu e s Dou ´ ees de Continuit ´ es G ´ en´ etiques , V olume 33, Edition CNRS, Ly on , 1949. [3] A. Richard, J.-P . 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