"Dogmatic" P Systems

“Dogma ic” P Sys ems? Jos´e M. Sempe e Depa amen o de Sis emas In o m´a icos y Compu aci´on Uni e sidad Poli ´ecnica de Valencia [email p o ec ed] Summa y. In his wo k we p opose a a ian o P sys ems based on he Cen al Dogma o Molecula Biology which es ablishes he ans o ma ion o DNA s ands in o p o ein p oduc s by applying di e en s ing ans o ma ion such as ansduc ions and ansc ip- ions. We in oduce a new kind o wo m objec ules o ca y ou ansducion ope a ions. Finally, we es ablish he uni e sali y o he p oposed model by simula ing I e a ed ini e s a e sequen ial ansduce s (IFTs). 1 In oduc ion P sys ems [15] we e in oduced as a compu a ional model inspi ed by he in- o ma ion and biochemical p oduc p ocessing o li ing cells h ough he use o memb ane communica ion. In mos o he wo ks abou P sys ems, in o ma ion is ep esen ed as mul ise s o symbol/objec s which can in e ac and e ol e acco ding o p ede ined ules. Ne e heless, he use o s ings o ep esen he in o ma ion and he use o ules o ans o m s ings ins ead o mul ise s o objec s ha e always been p esen in he li e a u e o his scien i ic a ea. So, in his mos ly e e ed book [15], Gh. P˘aun o e iews he use o s ing ules in P sys ems. Di e en a ian s o s ing-based P sys ems ha e been p oposed along he ime. We can men ion ew i ing P sys ems [11], e e ed as memb ane sys ems wi h wo m objec s [2] in he case o genomic ope a ions, inse ion-dele ion P sys ems [6] and splicing P sys ems [14], among o he s. Obse e ha mos o hese models ha e been used o language gene a ion [12]. In [5, 7], he p oposal o hyb id P sys ems in oduces he use o con ex ual ules and Chomsky ules o achie e uni e sali y by gene a ing all he ecu si ely enume able languages. Recen ly, in [13] a a ian o P sys ems wi h wo m objec s and e olu iona y based ope a ions has been in oduced o simula e Ne wo ks o E olu iona y P ocesso s, hence o achie e uni e sali y. In his wo k, we p opose a a ian o P sys ems wi h wo m objec s and a new kind o wo m ules based on he cen al dogma o molecula biology which se s ?Wo k suppo ed by he Spanish Minis e io de Educaci´on y Ciencia unde p ojec TIN2007-60769 292 J.M. Sempe e he amewo k o ob ain p o ein p oduc s om DNA s ands by applying, among o he s, ansduc ion and ansc ip ion ope a ions. The s uc u e o his wo k is as ollows: In sec ion 2 we in oduce basic concep s and no a ion on o mal language heo y, i e a ed ansduc ions, P sys ems and molecula biology ela ed o he Cen al Dogma. Then, we will de ine he dogma ic ules in egions which ansduce ( agmen s o ) wo m objec s in o ( agmen s o ) wo m objec s. We will p opose a simula ion o i e a ed ansduc ions wi h he new p oposed model in o de o achie e uni e sali y. Finally, we will ou line u u e esea ch ela ed o his wo k. 2 Basic Concep s We s a by summa izing he no ions used h oughou his wo k. An alphabe is a ini e and nonemp y se o symbols. Any ini e sequence o symbols om an alphabe Vis called wo d o s ing o e V. The se o all wo ds o e Vis deno ed by V∗. A language o e he alphabe Vis any subse o V∗. A g amma is a cons uc G= (N, Σ, P, S) whe e Nand Σa e he alphabe s o auxilia y and e minal symbols wi h N∩Σ=∅,S∈Nis he axiom o he g amma and Pis a ini e se o p oduc ions in he o m α→β, whe e α∈ (N∪Σ)∗N(N∪Σ)∗and β∈(N∪Σ)∗. The language o he g amma is deno ed by L(G) and i is he se o e minal s ings ha can be ob ained om Sby applying symbol subs i u ions acco ding o P. Fo mally, w1⇒ Gw2i w1=uα ,w2=uβ and α→β∈P. We will deno e by ∗ ⇒ G he e lexi e and ansi i e closu e o ⇒ G. So, he language gene a ed by Gis de ined by he se L(G) = {w∈Σ∗:S∗ ⇒ Gw}. Fou la ge amilies o languages gene a ed by g amma s can be de ined: REG ( egula ), CF (con ex - ee), CS (con ex -sensi i e) and RE ( ecu si ely enume - able). The de ini ion o hese amilies comes om he es ic ion o e he p oduc- ion o ms in he g amma . The well known Chomsky’s hie a chy es ablishes he inclusions REG ⊂CF ⊂CS ⊂RE. I e a ed T ansduc ions In he ollowing, we will in oduce I e a ed ini e s a e sequen ial ansduce s (IFT) as i was de ined in p e ious wo ks ([1, 8, 10]). An IFT is de ined by he uple T= (Q, Σ, q0, a0, F, P ), whe e Qis a ini e se o s a es,Σis an alphabe , q0∈Qis an ini ial s a e,a0∈Σis a s a ing symbol, F⊆Qis he se o inal s a es and Pis a ini e se o ansduc ion ules in he o m (q, a, p, x) wi h q, p ∈Q,a∈Σand x∈Σ∗which we will w i e as qa →xp. The ansduc ion ule qa →xp means ha i he ini e con ol is in s a e qand i eads he symbol a hen i changes o s a e pand w i es x. We de ine a di ec ansi ion s ep as ollows uqa `uwp i qa →wp ∈P “Dogma ic” P Sys ems 293 The e lexi e and ansi i e closu e o `will be deno ed by `∗. We say ha w de i es x, and i will be deno ed by w=⇒x, i q0w`∗xp, o p∈Q(obse e ha pis any s a e in Qno necessa ily inal). We will deno e he e lexi e and ansi i e closu e o =⇒by =⇒∗. I in he p e ious de i a ion he p ocess s ops in a inal s a e we will w i e =⇒ins ead o =⇒. Tha is, w =⇒x, i q0w`∗xp, o p∈F. The language gene a ed by Tis de ined as ollows L(T) = {x∈Σ∗:a0=⇒∗w =⇒x, w ∈Σ∗} We deno e by IF Tn he amily o languages gene a ed by IFT wi h a mos n s a es. The hie a chy o amilies in IFTnhas been comple ely explo ed, and i has been p o ed ha i collapses a le el ou . We ha e he ollowing esul s Lemma 1.[10] RE =IF T4; [1] CS ⊂IF T3; [10] CF ⊂IFT2. In addi ion, IFTs ha e been ela ed o he compu ing by ca ing pa adigm [9] as a way o gene a e e en non- ecu si ely enume able languages. The Cen al Dogma o Molecula Biology The Cen al Dogma o Molecula Biology is ou sou ce o inspi a ion o he a ian o P sys em which we will p opose la e . We ollow he ideas exposed in [4]. Mainly, he cen al dogma o molecula biology es ablishes a me apho o how DNA s ands in he li ing cell a e ans o med in o p o ein p oduc s by means o in o ma ion s o age and ans o ma ion. Mainly, a sec ion o DNA ( he gene) is ansc ibed o a molecule o messenge RNA and he mRNA is ansla ed by he ibosome in o a p o ein. In he euka y- o ic o ganisms he mRNA molecule is p ocessed, be o e ansla ion, by splicing ou ce ain subsequences called in ons. The DNA is eplica ed be o e he ansc ip- ion. The ansc ip ion is made by complemen ing he single DNA s and, and by subs i u ing he hymine nucleo ide by he u acil one in he RNA molecule. The ansla ion om (spliced) mRNA o p o eins is based on a mapping o nucleo ide iple s called codons o amino acids wi h he help o ans e RNA ( RNA). Unde a compu e science poin o iew, he cen al dogma can be iewed as a sequence o well known ope a ions o e s ings such as mo phisms, ansduc ions and splicing. The main ing edien s ha we will conside in he subsequen P sys em ha we will p opose a e he ollowings: •The e a e di e en p ocesses in di e en egions. DNA duplica ion and DNA ansc ip ion o mRNA occu s in he nucleus o he cell, while mRNA ans- la ion o amino acids occu s in some cases in he endoplasmic e iculum wi h he memb ane ibosome. •The e a e di e en alphabe sizes and symbols in ol ed in he ope a ions. The DNA s ands is a sequence o ou di e en nucleo ides: adenine (A), hymine (T), cy osine (C) and guanine (G), in he RNA he hymine (T) is subs i u ed by he u acil (U), while he p o eins a e sequences o e a wen y-le e alphabe ( he amino acids) 294 J.M. Sempe e Fig. 1. The Cen al Dogma o Molecula Biology. (This pic u e has been aken om accessexcellence.o g) •T ansc ip ion and ansla ion can be pe o med by alphabe ic homomo phisms and ini e ansduc ions. •The e a e di e en p oduc s a e e y s age which in e ac s in o di e en e- gions. The DNA duplica ion, ansc ip ion and splicing needs he p esence o di e en p o eins and o he molecula compounds. The p o eins a e he inal p oduc o he cycle DNA-RNA-p o ein. 3 Dogma ic P sys ems In his sec ion, we will p opose a a ian o P sys ems ha wo k wi h wo m objec s in a ansduc ion-like app oach. Fi s , we will in oduce a new kind o egion ules o wo k wi h. Adogma ic ule is de ined as ollows u: pos →wad1,ad2,··· ,adk,whe e u, a e s ings (wo m objec s), pos ∈ {l, , ∗} and o all i: 1 ≤i≤k adi∈ {he e, ou , inj}. The meaning is he ollowing: P o ided ha he e exis a wo m objec uin he egion (we can omi he p esence o u), all he wo m objec s wi h subs ing a posi ion pos (which means, igh mos one ( ), le mos one (l) o “Dogma ic” P Sys ems 295 a bi a y posi ion (∗)) change subs ing by wand send a copy o he new wo m objec a he egions de ined by adia e elimina ing he o iginal wo m objec om he egion. Example 1. Le he egion Rha e he ule 1de ined as eee :al→bbhe e and he wo m objec s eee and abbcbaa. Then a e applying 1in he egion, he wo m objec s a e eee and bbbbcbaa. I he ule 1is de ined as eee :a →bbhe e, we ob ain abbcbabb as a new wo m objec . Finally, i he ule is de ined as eee :a∗→bbhe e hen we ob ain he se o new s ings {bbbbcbaa, abbcbbba, abbcbabb}. Obse e ha , in his case, we ha e p e iously ob ained h ee copies o he ini ial s ing be o e applying he ule. The ule al→bbhe e can be applied o e baa and i ob ains he new s ing bbba. He e, we ha e omi ed he p esence o an addi ional s ing and he ule changes he le mos appea ance o a symbol a.¤ The add essing label inj, can be di ec ly applied o con iguous egions a he same le el. Tha is, i he e exis egions jand iinside he same egion, hen a ule a egion ican send wo m objec s o egion jdi ec ly. We can obse e ha he dogma ic ules cap u e he ollowing aspec s om he Cen al Dogma o Molecula Biology: •The ules ans o m pa s o a s ing in o a new subs ing as in ansc ip ion and ansduc ion. •The ules make copies o he a ge s ing be o e ans o ma ion as in DNA eplica ion. •The ules need he p esence o o he objec s o be applied. •The ules can add ess con iguous egions (i.e. RNA mo ing om nucleus o ibosomes). Now, we will de ine a Dogma ic Psys em2as he ollowing cons uc Π= (V, µ, A1,· · · , Am,(R1, ρ1),··· ,(Rm, ρm), i0), whe e: •V is an alphabe •µis a memb ane s uc u e consis ing o mmemb anes •Ai, 1 ≤i≤mis a ini e se o s ings associa ed wi h he egion i( he axioms) •Ri, 1 ≤i≤mis a ini e se o dogma ic ules o e Vassocia ed wi h he i h egion and ρiis a pa ial o de ela ion o e Rispeci ying a p io i y •i0is a numbe be ween 1 and mand i speci ies he ou pu memb ane o Π(in he case ha i equals o ∞ he ou pu is ead ou side he sys em). 2Di e en ac onyms we e candida es o naming Dogma ic P sys ems. Among o he s, dP sys ems we e conside ed bu i was p e iously used by o he au ho s in a di e - en con ex . Ano he ac onym was dogP bu he au ho hinks ha , in such a case, ca alyze s will ne e be used in his con ex gi en ha ”dogs” and ”ca s” could no coope a e and li ing in he same egions. We lea e open he sea ch o a good ac onym o he p oposed Dogma ic Psys ems. 296 J.M. Sempe e Ini ially, he sys em holds he se o axioms a e e y egion. Then, in a ully pa allel manne all he ules a e applied o e he s ings de ined a e e y egion. The sys em hal s whene e no ule can be applied a any egion. The language gene a ed by Πis he se o wo m objec s collec ed a egion i0. In he case ha i0=∞, he language is collec ed in ex e nal mode as he se o s ings in he en i onmen . The language gene a ed by Πis deno ed by L(Π). Obse e ha i he language is in ini e hen he sys em will ne e hal so i will add new wo m objec s o he ou pu egion o he en i onmen . Obse e ha his p oposal is di e en om [3] whe e he au ho s p opose a memb ane sys em amewo k wi h sympo /an ypo ules o pe o m di e en ypes o ansduc ions. In ha wo k he p oposed sys em ope a es wi h s ings by aking e e y symbol o he inpu s ing o he en i onmen (ou side he memb ane sys em) and pu ing e e y symbol o he ansduced s ing in he en i onmen . He e, we will a oid sympo /an ypo ules and we will wo k wi h s ings in a wo m objec app oach. 4 A Simula ion o I e a ed T ansduc ions by Dogma ic P Sys ems In his sec ion, we will show a simula ion o IFTs wi h ns a es by dogma ic P Sys ems. Ou app oach will use n egions inside he skin one in o de o simula e he ns a es o he IFT. The ansi ions o he IFT will be simula ed by using he di ec add ess inj. We will need o ma k some symbols in o de o ca y ou he ansduc ion om le o igh . In addi ion, we will use di e en alphabe s o a oid a w ong applica ion o he ansduc ion ules a di e en symbols, and o p e en ha he simula ion goes on e en i he IFT canno ca y ou a comple e ansduc ion. Le T= (Q, Σ, q0, a0, F, P ) be an IFT wi h Q={q0,· · · , qn}. Then, we p opose he ollowing dogma ic P sys em Π= (V, µ, A, A0,· · · , An,(R, ρ),(R0, ρ0),· · · ,(Rn, ρn),∞),whe e •V=Σ∪ˆ Σ∪˘ Σ∪ {#}, whe e ˆ Σ={ˆa:a∈Σ}and ˘ Σ={˘a:a∈Σ} •µ= [[0]0,· · · ,[n]n] (we ha e omi ed a label o he skin egion). •A0={#a0},A=∅, and o all i: 1 ≤i≤n Ai=∅. •Type (a) ules: Fo e e y ule q0a→ qj∈P, we add he ule #al→#ˆ inj i qj6=q0o he ule #al→#ˆ he e i qj=q0 o R0 •Type (b) ules: Fo e e y ule qia→ qj∈P, and o e e y symbol ˆ b∈ˆ Σ we add he ule ˆ bal→ˆ bˆ inji qi6=qjo he ule ˆ bal→ˆ bˆ he e i qi=qj o Ri •Type (c) ules: Fo e e y egion Riand o e e y pai o symbols ˆa∈ˆ Σand b∈Σadd he ollowing ule ˆabl→ˆabhe e •Type (d) ules: Fo e e y egion Risuch ha qi∈F, and o e e y symbol ˆa∈ˆ Σadd he ollowing ule ˆa →˘aou “Dogma ic” P Sys ems 297 •Type (e) ules: Fo e e y egion Risuch ha qi6∈ F, and o e e y symbol ˆa∈ˆ Σadd he ollowing ule ˆa →ˆaou •Type ( ) ules: Add o R he ules {ˆal→ahe e :a∈Σ} •Type (g) ules: Add o R he ules {˘al→ain0,ou } •Type (h) ule: #l→#in0 We will explain he ules in he sys em as ollows: Type (a) ules s a he ansduc ion o he s ing om he ini ial s a e. Hence, we use he # symbol as a le delimi e o he s ing o be ansduced. The alphabe ˆ Σis used o ma k he symbols ha ha e been ansduced du ing a de i a ion p ocess. Type (b) ules simula e he ansi ions in he ansduce . Obse e ha we use he add ess inj o change he s a e in he ini e con ol and he add ess he e o simula e he ansduce loops. Type (c) ules a e used o block he s ings ha canno be comple ely ansduced (obse e ha he IFT can be non comple e and i would no inish he de i a ion p ocess). Type (d) ules a e used o ou pu he ansduced s ings ha a i e o a inal s a e. He e, we use he alphabe ˘ Σ o ma k he s ings ha belong o he language gene a ed by he ansduce . Type (e) ules a e used o ou pu he ansduced s ings ha a i e o a non inal s a e. The p io i ies o he ules in egions Rikeep he ollowing o de : Type (a) ules >Type (b) ules >Type (c) ules >Type (d) and Type (e) ules. The ules o he skin egion a e explained as ollows: Type ( ) ules a e used o es o e he s ing symbols o he ansduced s ing in o de o eed-back he ansduce wi h a new inpu s ing (hence, i pe o ms he i e a ion in he ans- duc ion). Type (g) ules a e used o es o e he symbols om hose ansduced s ings ha come om a inal s a e (hence, hey belong o he language gene a ed by i e a ing he ansduce ). In such a case, one copy o he s ing is sen ou he en i onmen while ano he copy is sen in he egion ze o in o de o eed-back he ansduce . Finally, he ule o ype (g) is used o send he ansduced s ing in o he ini ial egion o i e a e a new ansduc ion. I a s ing w∈L(T), hen #w∈L(Π). We can obse e ha he ansi ions om Ta e simula ed by he P sys em by means o he ules o ype (a) and (b). The i e a ion is ca ied ou a he skin egion by applying ules o ype (g) o (h) (a e es o ing he symbols wi h ules o ype ( ). I he ansduced s ing a i es o a inal s a e, hen ules o ype (g) a e applied and he s ing wi h he le ma k # ou pu s he sys em. Example 2. Le us conside he ini e ansduce de ined h ough he ollowing ansi ion diag am, wi h aas he s a ing symbol The p oposed dogma ic P sys em is de ined wi h a memb ane s uc u e [[0]0,[1]1,[2]2], and he ollowing dogma ic ules Skin egion ules 1: ˆal→ahe e 4: ˘al→ain0,ou 45 : #l→#in0 2:ˆ bl→bhe e 5:˘ bl→bin0,ou 3: ˆcl→che e 6: ˘cl→cin0,ou 298 J.M. Sempe e wi h ρde ined as { 1, 2, 3}>{ 4, 5, 6}> 45, and A=∅. Region 0 ules 7: #al→#ˆ bˆ bin1 9: ˆaal→ˆaˆ bˆ bin1 12 : ˆabl→ˆaˆcˆcin2 8: #bl→#ˆcˆcin2 10 :ˆ bal→ˆ bˆ bˆ bin1 13 :ˆ bbl→ˆ bˆcˆcin2 11 : ˆcal→ˆcˆ bˆ bin1 14 : ˆcbl→ˆcˆcˆcin2 15 : ˆaal→ˆaahe e 18 :ˆ bal→ˆ bahe e 21 : ˆcal→ˆaahe e 16 : ˆabl→ˆabhe e 19 :ˆ bbl→ˆ bbhe e 22 : ˆcbl→ˆcbhe e 17 : ˆacl→ˆache e 20 :ˆ bcl→ˆ bche e 23 : ˆccl→ˆcche e 24 : ˆa →ˆaou 25 :ˆ b →ˆ bou 26 : ˆc →ˆcou wi h ρ0de ined as { 7, 8}>{ 9, 10, 11, 12, 13, 14}>{ 15, 16, 17, 18, 19, 20, 21, 22, 23}>{ 24, 25, 26}, and A0={#a} Region 1 ules 27 : ˆaal→ˆaˆ bˆ bhe e 30 : ˆabl→ˆaˆcˆcin2 28 :ˆ bal→ˆ bˆ bˆ bhe e 31 :ˆ bbl→ˆ bˆcˆcin2 29 : ˆcal→ˆcˆ bˆ bhe e 32 : ˆcbl→ˆcˆcˆcin2 33 : ˆaal→ˆaahe e 36 :ˆ bal→ˆ bahe e 39 : ˆcal→ˆcahe e 34 : ˆabl→ˆabhe e 37 :ˆ bbl→ˆ bbhe e 40 : ˆcbl→ˆcbhe e 35 : ˆacl→ˆache e 38 :ˆ bcl→ˆ bche e 41 : ˆccl→ˆcche e 42 : ˆa →ˆaou 43 :ˆ b →ˆ bou 44 : ˆc →ˆcou wi h ρ1de ined as { 27, 28, 29, 30, 31, 32}>{ 33, 34, 35, 36, 37, 38, 39, 40, 41}>{ 42, 43, 44}, and A1=∅ Region 2 ules “Dogma ic” P Sys ems 299 45 : ˆabl→ˆaˆcˆche e 46 :ˆ bbl→ˆ bˆcˆche e 47 : ˆcbl→ˆcˆcˆche e 48 : ˆaal→ˆaahe e 51 :ˆ bal→ˆ bahe e 54 : ˆcal→ˆcahe e 49 : ˆabl→ˆabhe e 52 :ˆ bbl→ˆ bbhe e 55 : ˆcbl→ˆcbhe e 50 : ˆacl→ˆache e 53 :ˆ bcl→ˆ bche e 56 : ˆccl→ˆcche e 57 : ˆa →˘aou 58 :ˆ b →˘ bou 59 : ˆc →˘cou wi h ρ2de ined as { 45, 46, 47}>{ 48, 49, 50, 51, 52, 53, 54, 55, 56}> { 57, 58, 59}, and A2=∅ ¤ F om he p e ious p oposed P sys em and o he wo ks p e iously e e ed we ge he ollowing esul . Theo em 1. E e y ecu si ely enume able language can be gene a ed by a dog- ma ic P sys em. P oo . The esul comes om he simula ion o IFTs by dogma ic P sys ems ha we ha e p oposed be o e. Gi en ha any ecu si ely enume able can be gene a ed by an IFT wi h ou s a es [10] hen we ha e he esul . ¤ 5 Conclusions and u u e wo k In his pape we ha e p oposed new kinds o ules o P sys em in which we ha e been inspi ed by he Cen al Dogma o Molecula Biology. The P sys ems ha we ha e p oposed a e a sui able amewo k o gene a e languages. We hink ha hese kind o ules will help in he cons uc ion o sys ems o biological simula ions due o i s inspi a ion om na u e. Ou u u e esea ch will ocus on he powe o hese sys ems o ansduce o mal languages wi h no i e a ion. Hence, we will s udy he simula ion o a ional and ecognizable ansduc ions and he simula ion o ( es ic ed) gsms. In addi ion, he amewo k o accep languages o s ings o hei Pa ikh mappings (which is he na u al amewo k o P sys ems) should be explo ed oo. Finally, due o he ela ion be ween IFTs and Compu ing by ca ing we should explo e he possibili y o applying memb ane sys ems o ha pa adigm, as a con inua ion o a p e ious wo k [16].