Automatic Network Reconstruction using ASP
Building biological models by inferring functional dependencies from experimental data is an im- portant issue in Molecular Biology. To relieve the biologist from this traditionally manual process, various approaches have been proposed to increase th…
Authors: Max Ostrowski, Torsten Schaub, Markus Durzinsky
Under conside ratio n for public ation in Theory and Practice of Logic Pro grammi ng 1 A utomatic Network Reconstruct ion using ASP Markus Durzinsky and W olfgang Marwan Magde bur g Centr e for Systems Biolo gy , Univer sit ¨ at Magdebu rg Max Ostrowski and T orsten Schau b ∗ Univer sit ¨ at P otsdam Annegret W agler Univer sit ´ e Blaise P ascal, Clermont-F err and submitte d [TB A]; re vised [TBA]; ac cepted [TBA] Abstract Building biological models by inferring functional dependencies from expe rimental data is an im- portant issue in Molecular Biology . T o relieve the biologist from this traditionally manu al process, v arious app roaches ha ve been propos ed to in crease the degree of automation. Ho we ver , av ailable ap- proaches often yield a single model only , rely on specific assumptions, and/or use dedicated, heuris- tic algorithms that are intolerant to changing circumstances or r equirements in the view of the rapid progress made in Biotechnology . Our aim is to pro vide a declarative solution to the problem by ap- peal to Answer Set Programming (ASP) overcoming these difficulties. W e build upon an existing approach to Automatic Netw ork Reconstruction propo sed by part of the authors. This approach has firm mathematical foundations and is well suited for ASP due to its combinatorial flavo r providing a characterization of all models explaining a set of experiments. The usage of ASP has several ben- efits over the e xisting heuristic algorithms. First, it is declarativ e and thus transpa rent for biological experts. Second, it is elaboration tolerant and thus allows for an easy e xploration and incorporation of biological constraints. Third, it allows for e xploring the entire space of possible models. Finally , our approach of fers an excellent performance, matching existing, special-purpo se systems. 1 Introduction The creation of b iological models by in ferring functional dependencie s from experimental data is a key issue in molecu lar biolo gy . A c ommon ap proach is to construct descr iptiv e models from series of e xperimen ts. This (manual) process u sually starts fro m a mo del defined using existing biologica l knowledge which is then gr adually refine d by ap peal to data gathe red in sub sequent exp eriments. A mod el obtained th is way is howe ver me rely consistent with the gather ed e xperimen tal data, and, besides simulation, no true indication can b e g iv en how well the r esulting mo del ca ptures the biological sy stem. For instance, it is u nclear whether the obtain ed mod el is on e amo ng many or few a lternative m odels. Moreover , it is o f great interest to know the difference amo ng alternative models in o rder to design ne w experiments for further discriminating the best fitti ng model. This problem is addressed in the area of A utomatic Network Reconstruc- tion (ANR) (Gifford and Jaakkola 2001; Y eung et al. 2002; Liang et al. 199 8; ∗ Affili ated with Simon Fraser Univ ersity , Canada, and Griffit h Unive rsity , Australia. 2 Max Ostr owski et al. Repsilber et al. 2002). Howe ver , the av ailable appro aches often yield a sing le mode l only , rely on specific assumption s, a nd/or use d edicated, heu ristic alg orithms for con- structing a m odel from experimental d ata. Moreover , all these approac hes ar e intolerant to chang ing c ircumstances o r requ irements in the view of the rapid p rogress made in Biotechnolo gy . Unlike this, we p rovide a declarative solution to the problem by appea l to Answer Set Program ming (ASP; (Baral 2003)). T o this en d, we build upon the appro ach to ANR pr oposed in (Du rzinsky et al. 2010; Marwan et al. 2008). This appr oach has firm math ematical fo undation s and is well suited for ASP due to its comb inatorial flav or providing a characterizatio n of all models explaining a set of experiments. The usage o f ASP has sev eral benefits over the existing heur istic alg orithms. First, it is de clarative and thus transpa rent for bio logical exper ts. Secon d, it is elaboratio n to lerant an d thus allo ws for an easy exploratio n and inc orpor ation of biolog ical constrain ts. Third , it allows for exploring the entire space of p ossible models. Fin ally , our appr oach offers an excellent perfor mance, matching e x isting, special-pur pose systems. The next section gives a formal intro duction to ANR, as pr ovided in (Durzinsky et al. 2010; Marwan et al. 2008), followed by a brief introd uction to ASP in Section 3. Section 4 is d edicated to o ur solutio n to ANR in ASP . W e e mpirically evaluate our appro ach in Section 5 a nd conc lude with a discussion an d a su mmary in Section 6 and 7. 2 A utomat ic Netw ork Reconstruction Automatic Network Reco nstruction a ims at co nstructing all m odels explaining a set of (pertur bation) e xperimen ts reflecting a c ertain biolog ical process. Our approach starts f rom experimental time-series data and gene rates all interaction n etworks that acco unt for th e observed mass or signal flo w . W e br iefly d escribe the steps of this appr oach pro posed in (Durzinsky et al. 2010; Durzinsky et a l. 2008; Marwan et al. 2008). W e represen t a collection S o f n o bservable species as a vector ( s 1 , . . . , s n ) b eing con- sidered to be c rucial for descr ibing the studied biolog ical p henome non, along w ith a cor- respond ing vector ( D 1 , . . . , D n ) of associated capacities over N 0 . Accordingly , species s i is assigned a value f rom capacitiy D i for 1 ≤ i ≤ n . A state x of species ( s 1 , . . . , s n ) is a vector ( x 1 , . . . , x n ) su ch that x i ∈ D i for 1 ≤ i ≤ n . Thus, x i provides the value of species s i in state x for 1 ≤ i ≤ n . No te that our conc ept of a state is only partial beca use it is confined to the observable species in S . In what follows, we leave the set S of species implicit whenever clear fro m the context. A (perturbatio n) experiment E ( x 0 ) = ( x 0 ; x 1 , . . . , x k ) over S is a sequence of states reflecting the time-d ependen t respon se ( x 1 , . . . , x k ) of a biologic al system to a (specific) perturb ation of th e system in state x 0 . W e associate with eac h respo nse state x i ∈ E ( x 0 ) its terminal state x k ∈ E ( x 0 ) a nd define t ( x i ) = x k for all 1 ≤ i < k . T ypically , several exper iments E ( x 0 ) starting from d ifferent initial states x 0 are nece s- sary to describe a b iological phe nomeno n. W e encode a set E of d ifferent exp eriments in terms of an experiment g raph G ( E ) = ( X , E P ∪ E R ) over S , which is a d irected g raph such that X is the multi- set of states in E , and E P and E R are disjoint sets o f p erturba - tion and resp onse edges, resp ectiv ely . That is, for each E ( x 0 ) = ( x 0 ; x 1 , . . . , x k ) ∈ E , we have ( x 0 , x 1 ) ∈ E P and ( x i , x i +1 ) ∈ E R for 1 ≤ i < k . For illustration , con sider Fig- Network Reconstruction in ASP 3 x 0 0 0 0 x 1 1 0 0 x 2 0 0 0 x 3 0 0 0 x 4 0 0 1 x 5 0 1 0 x 6 0 1 0 x 7 0 0 0 x 8 0 0 1 Fig. 1. An Experimen t G raph G ( E ) . ure 1 showing an experiment graph over species { fr , r , sp o } , en coding thr ee exper iments E ( x 0 ) = ( x 0 ; x 1 , . . . , x 4 ) , E ( x 2 ) = ( x 2 ; x 5 , x 0 ) , and E ( x 3 ) = ( x 3 ; x 6 , . . . , x 8 ) . Th e en- tries in each state vector g i ve the r espective v alues of each spec ies; co ntinuou s arrows represent response edges, dashed ones giv e perturbation edges. An experiment graph G ( E ) = ( X, E P ∪ E R ) is valid if I . e very state x ∈ X has at most one outgoin g arc in E R , II . x = x ′ implies t ( x ) = t ( x ′ ) for all x, x ′ ∈ X an d III . ( x ′ − x ) 6∈ N n holds for all ( x, x ′ ) ∈ E R , Condition I stipulates tha t an experimen t graph is d eterministic, while II req uires that no equal 1 states lead to different term inal states. III demands that ther e must be at least o ne species that decreases be tween two con secutive respon se states. In fact, the experimen t graph in Figure 1 violates two validity conditions: II is vio lated thro ugh states x 5 and x 6 , as these states are eq ual but lead to differing terminal states x 0 and x 8 , resp ectiv ely . III is violated by response edge ( x 2 , x 3 ) . For the reconstruction, we use the paradig m that system states can be ch anged by ap - plying reactions. A r eaction over n species is described by a vector r ∈ Z n , where r i < 0 for some 1 ≤ i ≤ n . So, a reaction m ust have at least on e negative en try to co nsume at least one species. A r eaction r is enabled in a state x over n species with c apacities ( D 1 , . . . , D n ) , if we have x i + r i ∈ D i for all 1 ≤ i ≤ n , i. e. if neither n onnegativity nor capacity co nstraints are violated. For instance, reaction r = (0 , − 1 , 0) is enabled in x 6 because x 6 + r = (0 , 0 , 0 ) belongs to the species’ capacity . Giv en an experiment grap h ( X , E P ∪ E R ) and a respo nse edge ( x, x ′ ) ∈ E R , we say that this response is r ealized by a sequen ce σ (( x, x ′ )) = ( r 1 , . . . , r l ) of reactio ns, if IV . y i + r i = y i +1 for all 1 ≤ i ≤ l , and V . ( y 1 , y 2 , . . . , y l +1 ) is a seque nce of states such that x = y 1 and x ′ = y l +1 , VI . r i k · r j k ≥ 0 for all 1 ≤ i, j ≤ l an d all 0 ≤ k ≤ n . All reactions subseq uently ap plied to state x fulfill the respo nse edge and ultimately lead to the con secutiv ely observed state x ′ in E R . For example, the r eaction r = (0 , − 1 , 0) constitutes a singleton sequen ce σ (( x 6 , x 7 )) as x 6 + r = x 7 realizes ( x 6 , x 7 ) . Note 1 Recal l that X is a multi-set; two states are equal if their vect or of s pecie s is equal 4 Max Ostr owski et al. that VI stipu lates that all reactions in such a sequen ce must be mo notone ; 2 at micro- scopic lev el, a species cannot be produced and consumed (or vice versa) by tw o reactions, see (Durzinsky et al. 2008) for details. T o also account for the experimentally observed mass o r sig nal flow , (Marwan et al. 2008) propose to use a partial orde r on th e set of r eactions to reflec t their relati ve rates. A sequence ( r 1 , . . . , r l ) o f reactions is said to respect such a partial order ≺ , if r i is the u nique ≺ -minimal reactio n enabled in an (intermediate) state y i for each 1 ≤ i < l . No te that the reaction ord er ≺ must be sufficiently strong to g uarantee a unique fastest reaction at each step. Th is implies for each state to have a uniq ue successor state, ensuring the system’ s determinism. Follo wing (Marwan et al. 2008), a re gulato ry structu r e ( R , ≺ ) over species S co nsists of a set of reactions R and a partial order ≺ among them. 3 A regulatory structure ( R , ≺ ) is conformal with a valid e x perimen t g raph ( X, E P ∪ E R ) , if VII . for all r ∈ R , r is not enabled in any terminal s tate of X , VIII . for all e ∈ E R , there is a ≺ -respectin g realizing sequence 4 σ ( e ) ⊆ R , and IX . there exists no r ∈ R where r is n ot an element of some σ ( e ) . As defin ed in (Durzinsky et al. 2010), the Network Recon struction Pr ob lem for a valid experiment graph consists in finding all regulatory s tructures conf ormal with the graph. An inv alid experim ent g raph ca n be r ecovered b y ad ding new , artificial species to S . 5 Giv en an (in valid) experiment g raph ( X, E P ∪ E R ) , an extension ( X ′ , E P ∪ E R ) with a species is obtaine d by replacing each state ( x 1 , . . . , x n ) in X with ( x 1 , . . . , x n , x n +1 , . . . , x n + a ) su ch that x n + i ∈ { 0 , 1 } for 1 ≤ i ≤ a ; all other capa ci- ties and edges are left intact. Note that an experiment graph has 2 a extensions. An extension ( X ′ , E P ∪ E R ) of an experimen t graph with a species is valid , if X . ( X ′ , E P ∪ E R ) is a valid e xperimen t g raph and XI . x n + i = x ′ n + i for each ( x, x ′ ) ∈ E P and 1 ≤ i ≤ a . The latter condition stipulates that additional species are not direct targets of experime ntal perturb ations, but they certainly respond in succ essi ve states. Similarly , we w an t to reduce the ch anges o f ad ditional species in r esponse edges: A r esponse edge ( x, x ′ ) ∈ E R is subject to an additiona l chan ge , if x n + j 6 = x ′ n + j for some 1 ≤ j ≤ a . At last, g i ven an in valid experiment graph , the N etwork Recon struction Pr oblem co nsists in solving the NRP for all valid extensions of that gr aph, first, a dding a minimum num ber of add itional spec ies and , secon d, com prising a minimum number o f a dditional chan ges. For bre v ity , such e xtensions are called minimal valid e xten sions . Figure 2 and 3 sh ow th e two 6 valid extensions of th e inv alid experiment graph in Fig- ure 1. The node s in th e figures are th e vectors from Fig ure 1 extended b y the two ad ditional species. Both e xtensions differ in th e values attributed to the two a dditional specie s in state 2 This is a sig nificant constrai nt on the qualit y of t ime series data. The response of the system must be measured with suffic ient time resolution, such that oscillat ion betwe en measurements can be e xcluded. 3 R is also refe rred to as a net work because such reactio n sets are easily con verted to P etri net s , as done in (Marwa n et al. 2008). T his is ho wev er beyond the scope of this paper . 4 W e sligh tly abuse notatio n, and tak e σ ( e ) ⊆ R to mean that eac h element of σ ( e ) is also in R . 5 This all o ws for dif ferenti ating seeming ly equal yet dif ferent s tates, ena bling new rea ctions by decre asing addi- tional species, or av oiding reac tions in te rminal states. 6 Actuall y , there are four ex tensions with symmetric behavi or on the additional species. Network Reconstruction in ASP 5 x 0 0 0 x 1 0 0 x 2 1 0 x 3 0 1 x 4 0 0 x 5 1 0 x 6 0 1 x 7 0 1 x 8 0 0 Fig. 2. First Extension G ( E 1 ) of the Exp eriment Graph G ( E ) in Figu re 1. x 0 0 0 x 1 0 0 x 2 1 1 x 3 0 1 x 4 0 0 x 5 1 1 x 6 0 1 x 7 0 1 x 8 0 0 Fig. 3. Second Extension G ( E 2 ) of the Exp eriment Graph G ( E ) in Figu re 1. x 2 and x 5 . Th e extension s are co nform al with the regulatory structures with th e network s depicted in Figu re 4 and 5, respectiv ely . The additional spec ies a re referred to as x and y . Reactions are given as boxes, species as cir cles. All r eaction en tries h av e th e cap acity {− 1 , 0 , 1 } . An arrow fr om a spe cies to a reactio n stands f or a − 1 in the reac tion vector , one from a reaction to a species stands for +1 . According ly , the regulato ry stru cture in Figur e 4 over species ( fr , r , sp o , x , y ) com- prises the following reactions: r 1 = ( − 1 , 0 , 0 , 1 , 0) , r 2 = ( 0 , − 1 , 0 , 0 , 0) , r 3 = (0 , − 1 , 0 , − 1 , 0) , r 4 = (0 , 0 , 0 , − 1 , 1 ) , and r 5 = (0 , 0 , 1 , 0 , − 1 ) and the order ing ≺ = { ( r 4 , r 3 ) , ( r 3 , r 2 ) , ( r 5 , r 2 ) } . Th e regulatory stru cture in Figure 5 comp rises the r eactions: r 1 = ( − 1 , 0 , 0 , 1 , 1) , r 2 = (0 , − 1 , 0 , − 1 , − 1) , r 3 = (0 , − 1 , 0 , 0 , 0) , r 4 = (0 , 0 , 1 , − 1 , 0 ) , and r 5 = (0 , 0 , 0 , 0 , − 1) and the ordering ≺ = { ( r 4 , r 5 ) , ( r 5 , r 2 ) , ( r 4 , r 3 ) , ( r 3 , r 2 ) } . 3 Answer Set Programming W e rely on th e input language of the ASP g roun der gringo (Gebser et al. ) (extendin g the languag e of lp arse (Syrj ¨ anen )) and introd uce o nly infor mally the basics of ASP . A com - prehen si ve, forma l introduction to ASP can be found in (Baral 2003; Gelfond 2008). W e consider extended logic pr ograms as in troduc ed in (Simons et al. 2002). A rule r is fr x y sp o r r 1 r 4 r 5 r 2 r 3 Fig. 4. Regulatory Structure conformal with the e xtended Experiment Graph in Figure 2. 6 Max Ostr owski et al. fr x y sp o r r 1 r 5 r 4 r 2 r 3 Fig. 5. Regulatory Structure conformal with the e xtended Experiment Graph in Figure 3. of the following f orm: H ← B 1 , . . . , B m , ∼ B m +1 , . . . , ∼ B n . By he ad ( r ) = H and b o dy ( r ) = { B 1 , . . . , B m , ∼ B m +1 , . . . , ∼ B n } , we deno te the head and the bod y of r , respe ctiv ely , whe re “ ∼ ” stands fo r default negation. Th e head H is an atom a belong ing to some alphabe t A , the falsum ⊥ , or a # su m constraint L # sum [ ℓ 1 = w 1 , . . . , ℓ k = w k ] U . In the latter, ℓ i = a i or ℓ i = ∼ a i is a literal and w i a non -negative integer weight f or a i ∈ A and 1 ≤ i ≤ k ; L and U are integers pr oviding a lower and an upper bound. Either or both o f L and U can be om itted, in which case they are identified with the ( trivial) bou nds 0 an d ∞ , re spectiv ely . Whenever all weights eq ual one, the # sum constraint L # sum { ℓ 1 = 1 , . . . , ℓ k = 1 } U be comes a “ # count ” co nstraint and is simply written as L { ℓ 1 , . . . , ℓ k } U . A r ule r such th at he ad ( r ) = ⊥ is an integrity constraint . Each body componen t B i is either an atom or a # sum constraint for 1 ≤ i ≤ n . If b o dy ( r ) = ∅ , r is called a fact , and we skip “ ← ” wh en wr iting facts belo w . W e a dhere to the definition of answer sets pr ovided in (Simons et al. 2002), wh ich applies to logic prog rams containing extended constructs ( # sum constraints) under “choice semantics”. In addition to rules, a logic program can contain # minimiz e statements of the form # minimiz e { ℓ 1 = w 1 , . . . , ℓ k = w k } . Besides literals ℓ j , a # minimiz e statement includ es inte ger weights w j for 1 ≤ j ≤ k . A # minimiz e statemen t d istinguishes optim al answer sets of a pro gram as the o nes yielding the smallest weighted sum for the tru e literals among ℓ 1 , . . . , ℓ k . F o r a formal introduction, we refer the interested reader to (Simons et al. 2002). Like wise, first-or der rep resentations, commonly used to encod e prob lems in ASP , are only informally introdu ced. In fact, gringo requires programs to be safe , that is, each vari- able must occur in a po siti ve bod y literal. Formally , we only rely on th e function gr ound to denote the set of all ground instan ces, gr ound (Π) , of a program Π con taining first-or der variables. Further langu age constructs of interest, inclu de c ondition al literals, like “ a : b ”, the range and pooling oper ator “.. ” and “;” as well a s standard arithmetic opera tions. The “:” con nective expan ds to the list of all instances o f its lef t-hand side suc h that co rre- sponding instances of literals on th e right-hand side hold (Syr j ¨ anen ; Geb ser et al. ). While “.. ” a llows fo r specify ing integer intervals, “;” allo ws for pooling alternative terms to be used a s a rguments within an atom. For instance, p (1 .. 3) as well as p (1; 2; 3) stand f or the three facts p (1) , p (2) , a nd p (3) . Given this, q ( X ) : p ( X ) results in q (1) , q (2) , q (3) . See (Gebser et al. ) for a detailed description of the input language of the ground er gringo . Network Reconstruction in ASP 7 4 Declarative A utomatic Network Reconstruction Our app roach expresses the ANR Pro blem in form o f a logic pro gram und er answer set semantics. I n the next section s, we explain the encod ing in detail, startin g with the r epre- sentation of the experiment graph. 4.1 Representing experiment instances As comm on in ASP , a p roblem instance is g i ven as facts. W e define the instance o f an experiment graph ( X , E P ∪ E R ) over S with dom ain D , as a set of facts I ( X, E P ∪ E R ) = { sp e cies ( s ) | s ∈ S } ∪ { c ap acity ( s i , d i ) | s i ∈ S a nd d i ∈ D } ∪ { state ( x ) | x ∈ X } ∪ { e dge ( p, x, x ′ ) | ( x, x ′ ) ∈ E P } ∪ { e dge ( r, x, x ′ ) | ( x, x ′ ) ∈ E R } ∪ { terminalState ( x ′ ) | x ′ = t ( x ) , x ∈ X } ∪ { value ( x, s i , x i ) | x = ( x 1 , . . . , x n ) ∈ X, 1 ≤ i ≤ n } . Predicates sp e cies ( s i ) and c ap acity ( s i , d i ) denote the species s i ∈ S over their associated capacities d i ∈ D . Similarly , we use state ( x ) to denote each state x ∈ X and mark termi- nal states with predic ate terminalState ( x ) . For the edge s of the gr aph, we use predicate e dge ( T , x, x ′ ) , wh ere T ca n be either p o r r to indicate that ( x, x ′ ) is a pertu rbation o r a response edge, respectiv ely . As an example, T able 1 gives the specification of the experiment graph in Figure 1. sp e cies ( f r ) . sp e cies ( r ) . sp e cies ( s po ) . c ap acity ( f r, 0 .. 1 ) . c ap acity ( r , 0 .. 1) . c ap acity ( spo, 0 .. 1) . state ( x 0 ) . state ( x 1 ) . stat e ( x 2 ) . state ( x 3 ) . state ( x 4 ) . stat e ( x 5 ) . state ( x 6 ) . state ( x 7 ) . state ( x 8 ) . e dge ( p, x 0 , x 1 ) . e dge ( p, x 2 , x 5 ) . e dge ( p, x 3 , x 6 ) . e dge ( r, x 1 , x 2 ) . e dge ( r, x 2 , x 3 ) . e dge ( r, x 3 , x 4 ) . e dge ( r, x 5 , x 0 ) . e dge ( r, x 6 , x 7 ) . e dge ( r, x 7 , x 8 ) . terminalState ( x 0 ) . t erminalState ( x 4 ) . t erminalState ( x 8 ) . value ( x 0 , f r, 0) . value ( x 1 , f r, 1) . value ( x 2 , f r, 0) . value ( x 3 , f r, 0) . value ( x 0 , r, 0) . value ( x 1 , r, 0) . value ( x 2 , r, 0) . value ( x 3 , r, 0) . value ( x 0 , spo, 0) . value ( x 1 , spo, 0) . value ( x 2 , spo, 0) . value ( x 3 , spo, 0) . value ( x 4 , f r, 0) . value ( x 5 , f r, 0) . value ( x 6 , f r, 0) . value ( x 7 , f r, 0) . value ( x 8 , f r, 0) . value ( x 4 , r, 0) . value ( x 5 , r, 1) . value ( x 6 , r, 1) . value ( x 7 , r, 0) . value ( x 8 , r, 0) . value ( x 4 , spo, 1) . value ( x 5 , spo, 0) . value ( x 6 , spo, 0) . value ( x 7 , spo, 0) . value ( x 8 , spo, 1) . T able 1. Specification of Experiment Graph in Figure 1. Giv en the instan ce in T able 1, th e solution s of our final logic progr am r epresent all regulatory structures that are confor mal with the extensions of the exper iment g raph. W e start with checking the v alidity of the experiment graph. 8 Max Ostr owski et al. 4.2 Checking V alidity In Section 2, three co nditions were specified fo r v alid ity . Checkin g the se conditions can be done with the following logic program. Condition I , ensuring that each state has only one outgoin g arc, i s given in (1). ← e dge ( r, X 1 , X 2 ) , e dge ( r , X 1 , X 3 ) , X 2 6 = X 3 . (1) Rules (2) to (5) account for Cond ition II . W e first collect all pairs of states that are not equal and th en co mpute the associa ted term inal state o f each state which can be d etermined deterministically . Rule (5) ensures that no two equal states lead to unequal terminal states. ne q ( X 1 , X 2 ) ← value ( X 1 , S, V ) , ∼ value ( X 2 , S, V ) , state ( X 2 ) . (2) t ( X, T ) ← e dge ( r , X , T ) , terminalState ( T ) . (3) t ( X 1 , T ) ← e dge ( r, X 1 , X 2 ) , t ( X 2 , T ) . (4) ← ∼ ne q ( X 1 , X 2 ) , ne q ( T 1 , T 2 ) , t ( X 1 , T 1 ) , t ( X 2 , T 2 ) . (5) The rules in (6) and (7) ensure a decrease in each response as required in Condition III . de cr e ase ( X 1 ) ← e dge ( r , X 1 , X 2 ) , value ( X 1 , S, V 1 ) , value ( X 2 , S, V 2 ) , V 2 − V 1 < 0 . (6) ← ∼ de cr e ase ( X 1 ) , e dge ( r , X 1 , X 2 ) . (7) The next proposition ensures correctness and completeness of this logic program. Pr opo sition 1 Let ( X, E P ∪ E R ) be th e experiment graph and Π 1 be the log ic progr am g r ound ( I ( X , E P ∪ E R ) ∪ { (1) , . . . , (5) } ) . Then, the experiment graph ( X, E P ∪ E R ) is valid if f the re exists an answer set of Π 1 . The pr oof o f this an d all fo llowing results f ollow f rom th e c onstruction o f the r espective logic progr ams. R ecall that the experimen t g raph in Figure 1 is inv alid, so the corresp ond- ing program has no answer set. 4.3 Building Regulatory Structures W e n ow pro ceed by defining a finite logic prog ram that allows us to find all regulatory structures conform al with a valid experiment graph. For guar anteeing the finiteness of the groun d prog ram, we need to k now the maximu m number of r eactions that shall be u sed. As each reaction has to consum e at least one species, the number of reactions is bound by the total number of decreases during a response edge. Although ther e exist better ap proxim ations, for simplicity , we define the log ic prog ram B ( X , E P ∪ E R ) that r esults in the single answer set A , such th at maxA dds ( n ) ∈ A , maxR e actions ( m ) ∈ A an d length ( x, x ′ , m ) ∈ A for all ( x, x ′ ) ∈ E R , where n, m ∈ N 0 are suf ficie nt bounds. Given th ese b ound s, we now cho ose a cer tain number o f reactions in (9) to be part of the regulatory s tructure. r (1) . (8) { r ( N +1) } ← r ( N ) , maxRe actio ns ( M ) , N
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