Recent Computability Models Inspired from Biology: DNA and Membrane Computing

Recen Compu abili y Models Inspi ed om Biology: DNA and Mem- b ane Compu ing Gheo ghe PÃUN, Ma io J. PÉREZ-JIMÉNEZ ABSTRACT: We b ie ly p esen wo a eas o na u al compu ing, i idly in es iga ed in he ecen yea s: DNA com- pu ing and memb ane compu ing. Bo h o hem ha e he oo s in cellula biology and a e a he de eloped a he heo e ical le el (new concep s, models, pa adigms o compu e science, wi h ma hema ical and epis- emological signi icance ha e been conside ed in his amewo k), bu bo h a eas a e s ill looking o im- plemen a ions o a p ac ical in e es . Keywo ds: Compu e Science, Ma hema ics, Tu ing compu abili y, Biochemis y, DNA compu ing, Memb ane Com- pu ing. 1 In oduc ion In a g ea ex en , he his o y o ( heo e ical) compu e science is he his o y o a - emp ing o model ( o malize) he compu a ions pe o med in na u e, s a ing wi h he way he humans compu e ( his was, o ins ance, he explici goal o Leibniz, a he u n o se en een h and eigh een h cen u ies, and o Tu ing, in 1935–36), going h ough he supposed o ganiza ion and unc ioning o he b ain, o he ne wo ks o neu ons ( his is he o igin o ini e au oma a, see McCulloch & Pi s 1943, Kleene 1956, and o neu al ne wo ks a ea, see, e.g., Ande sson 1996), and ge ing close and close o he molecula biology, o he cell and i s cons i uen s. The gene al assump- ion/obse a ion is ha li e has used o millions o yea s many p ocesses, aking places in speci ic ma e ial en i onmen s and using speci ic ma e ial s uc u es, which may be conside ed as compu ing p ocesses and de ices. This asse ion is deba able: Wha is a compu a ion? Does na u e compu es? A which le els? Which p ocesses a e and which a e no compu a ions? And so on and so o h. We do no en e he e his deba e (we adhe e o he opinion ha na u e jus e ol es, he goal o li e is li e i sel , and a p ocess can be conside ed a compu a ion only by a human being, ‘compu ing’ is no a na u al ac i i y, bu an a i ac ) and we adop a ma hema ical pe spec i e: compu ing means Tu ing compu ing (an inpu - ou pu ela ion, es ablished by “mechanically” ollowing an algo i hm: a p ecise se- quence o ins uc ions which always hal s). Mo eo e , when we look o a piece o eal- i y we sea ch sugges ions o wo basic ing edien s o a compu ing model: da a s uc- u es (suppo s o compu ing) and ope a ions abou hese da a s uc u es. O cou se, hese elemen s o any compu ing model do no exis as such, we abs ac hem om he biochemical objec s/s uc u es and p ocesses. (We an icipa e, illus a ing his dis- cussion wi h he s uc u e o he DNA molecules and he many ope a ions which a e possible wi h hese molecules.) A e abs ac ing a da a s uc u e and some ope a ions abou i , we can p oceed in a a he s anda d way in o de o ob ain a compu ing de- ice whe e he da a s uc u e can be p ocessed by means o he conside ed ope a ions: conside an ini ial con igu a ion o ou de ice, consis ing o one o se e al se s o op- e a ion-based ‘ins uc ions’ and a gi en collec ion o da a; by using he ins uc ions in a speci ied manne we pass o a new con igu a ion; by i e a ing he ope a ions we ob- Gheo ghe PÃUN, Ma io J. PÉREZ-JIMÉNEZ 72 ain a compu a ion; de ine in some way he no ion o a success ul compu a ion; s a - ing om he ini ial con igu a ion o he ‘de ice’, whe e he inpu o he compu a ion is placed, and p oceeding along a success ul compu a ion, we can ge he ou pu o he compu a ion, i s esul . This scena io is ollowed bo h in DNA and memb ane compu ing, he wo a eas o na u al compu ing we a e going o discuss in he nex sec ions. Be o e going in o some de ails, some gene ali ies a e s ill wo h men ioning. Na u al compu ing is an impo an end o compu e science looking o compu ing models – pe haps also compu e s – inspi ed om he way na u e “compu es”, and i con ains a eas which we e al eady p o ed as success ul bo h heo e ically and p ac ically. The examples o neu al ne wo ks and o gene ic algo i hms (mo e gene ally, o e olu iona y compu - ing) a e illus a i e in his espec . Especially in e es ing om ou poin o iew a e gene ic algo i hms (see, e.g., Beye 2001): op imiza ion p oblems a e o mula ed in e ms o imp o ing a i ness unc ion o e a popula ion o “ch omosomes”, ep e- sen ed by s ings o digi s, which e ol e by means o ope a ions known om he ge- ne ics, such as ecombina ion and poin mu a ion. A he i s sigh , his is jus a an- dom walk h ough he space o possible solu ions, a b u e o ce app oach, which highly con as s wi h he su p ising success o such algo i hms in a la ge numbe o applica ions. The e is no ma hema ical “explana ion” o his success, bu his obse a- ion b ings op imism o na u al compu ing esea ch: he ac ha li e uses ce ain ools (again, a an abs ac le el: da a s uc u es and ope a ions), which we e imp o ed du - ing e olu ion, du ing a long in e al o ime, can be an indica ion ha hese ools ha e some ea u es which make hem use ul also o compu ing. I ecombina ion, poin mu a ion, selec ion, e c, we e use ul and so e icien in gene ic algo i hms, why no as- suming ha o he simila ly use ul ools and compu ing ideas can be ound a he ge- ne ic le el o a uppe le els, such as he li ing cell. Insis ing a he DNA le el we ge DNA compu ing, insis ing a he cellula le el we ge he memb ane compu ing. Some di e ences be ween he abo e men ioned a eas a e isible and in e es ing o wha ollows. Neu al ne wo ks and gene ic algo i hms a e inspi ed om biology and implemen ed on usual elec onic compu e s. They aim a inding new ypes o al- go i hms, o a be e use o he classic compu e , maybe o also imp o ing he a chi- ec u e o he classic compu e . In u n, DNA compu ing has a di e en goal, a new ambi ion: o use DNA molecules as a suppo o compu ing, o eplace/supplemen he elec onic chip wi h a “we chip”. The need o his a ises om he obse a ion ha he elec onic chips canno con inue oo much o become smalle , cheape , as e , because, i hey will do i as in he las decades (in six ies was o mula ed he so-called Moo e law, saying ha e e y yea he compu e s will become wice smalle , cheape , and as e ; la e he in e al was inc eased o 18 mon hs, hen o wo yea s, now he law seems o be abandoned), hey will soon each he quan um ba ie . Mo eo e , he sequen ial compu e s ha e in insic e iciency limi s ( hey canno sol e p oblems o exponen ial complexi y in a easible ime); a possible way o o e pass his in insic limi is he pa allelism, bu his aises se ious di icul ies in he elec onic case, o in- s ance, in wha conce ns he con ol o p ocesso s, he ene gy hey dissipa e, e c. A p omise comes om biology, om DNA and cellula le els. The pa allelism made Recen Compu abili y Models Inspi ed om Biology: DNA and Memb ane Compu ing 73 possible by DNA molecules is eally massi e: billions o molecules – hence “p oces- so s” – can ind oom in a iny biochemical es ube; mo eo e , biochemis y is highly nonde e minis ic, which means ha he con ol o compu a ions could be o a new ype, “cheap” om se e al poin s o iew. DNA compu ing deals wi h compu ing in i o, wi h implemen ing algo i hms in a labo a o y ( he e also a e esea ches abou DNA compu ing in i o, mainly dealing wi h he compu ing-like p ocesses which ake place in ce ain cells a he gene ic le el, o ins ance, in cilia es). Wha abou he cell, he smalles li ing uni we know? The cell is a e y complex biochemical “ ac o y”, whe e many p ocesses ake place, some o hem o an in o ma ional ype. Can he li ing cell be conside ed/in e p e ed/used as a compu ing de ice? The ques ion is no simple: Wha is a cell ( om a ma hema ical poin o iew)? Which ea u es o he cell s uc u e and which p ocesses aking place in he cell compa men s a e use ul/essen ial o a compu ing model? A e abs ac - ing a cell-like compu ing model, as we will immedia ely see, whe e should we y o implemen i , on a usual elec onic compu e (like in he case o gene ic algo i hms) o on a bio-suppo (as i is he goal o DNA compu ing)? We do no answe he e hese ques ions; he aim o his pape is only o le he eade ha ing a i s con ac wi h DNA and memb ane compu ing a eas, a an in oduc o y and in o mal le el. De ails (including u he ques ions) can be ound in he i les men ioned in he bibliog aphy, and he web pages de o ed o he wo discussed domains. 2 A Glimpse o DNA Compu ing The ac ha he DNA molecules can be used as a suppo o compu ing has been specula ed since se e al decades (Ch. Benne , M. Con ad, e c), bu he i s (success- ul) expe imen has been epo ed in 1994, by L.M. Adleman. The p oblem deal wi h was he so-called Hamil onian pa h p oblem o di ec ed g aphs: whe he o no a pa h exis s which isi s each node o a g aph exac ly once. The p oblem is known o be compu a ionally di icul (i is NP-comple e), bu he solu ion was e y e icien (ob ained in linea ime). Seen a e eigh yea s, he expe imen looks simple om a biochemical poin o iew: one encodes he nodes by single s anded DNA molecules o leng h 20 (consis ing o 20 nucleo ides), hen one encodes he edges o he g aph by single s anded DNA molecules which a e complemen a y, in he Wa son-C ick sense, o he node-codes and consis o he second hal o he code o he eme ging node and he i s hal o he a ge node o he edge, one places billions o such molecules in a es ubes and one le s hem o anneal and o m double s anded mole- cules; in his way, he edge-codes ac as splin s o node-codes, hence he possible pa hs in he g aph a e encoded by chains o node-codes and edge-codes; by well- known il e ing p ocedu es one selec s he pa hs which isi all nodes exac ly once ( he Hamil onian ones); i any molecule exis s which encodes such a pa h, hen he p oblem has he answe “yes”. Adleman’s expe imen was a g ea e en in compu e science, in spi e o he ac ha he g aph conside ed was a small one, wi h only se en nodes and hi een edges. Howe e , he expe imen was he i s one o his ype, i p o ed ha DNA compu - ing is possible; in he e ms o Ha manis (1994), his was a demo. Many expe imen s Gheo ghe PÃUN, Ma io J. PÉREZ-JIMÉNEZ 74 ha e ollowed, in USA, Japan, Eu ope, con e ences we e ini ia ed in his a ea, a la ge numbe o pape s we e published. Howe e , up o now no compu a ion o a p ac ical in e es was epo ed. The passing om a oy-p oblem o a p oblem o a signi ican size is no a all easy, because o he quan i y o equi ed DNA (and o o he bio- chemical ools), and, mainly, because o he di icul y o coping wi h e o s. The bio- chemical eac ions canno (ye ) be pe ec ly con olled, he algo i hms based on hem a e e o -p one, and his eques s bo h p og esses in bioenginee ing and in imp o ing he heo e ical models (and in inding he adequa e classes o p oblems o be a acked in his a ea, e.g., wi h e o esis an solu ions). To abs ac a li le bi , Adleman has used as da a s uc u e he DNA molecule (sin- gle o double s anded) and he annealing ope a ions as he basic ope a ion. In a mas- si ely pa allel manne , his ope a ion ensu es he gene a ion o all candida e solu ions o he p oblem (p o iding ha “enough” DNA is p esen ). The il e ing phase, when i was checked whe he o no a solu ion exi s, can be conside ed as he phase o ead- ing he esul ; i was done manually, bu he numbe o s eps was o he same o de o magni ude as he numbe o nodes o he g aph – hence he linea ime o sol ing he p oblem. The many expe imen s which we e epo ed in he mean ime use simila da a s uc u es (some imes, ci cula molecules, o molecules wi h o he shapes, such as hai pins), bu se e al o he ope a ions. One o he mos in e es ing case is ha o he splicing ope a ion, conside ed o he i s ime by T. Head, in 1987, in a heo e ical amewo k no di ec ly dealing wi h compu ing. This ope a ion was explici ly used in a compu ing model by Pãun, Rozenbe g, and Salomaa (1996), whe e he no ion o an H sys em was in oduced. Because hese sys ems a e among he mos in es iga ed DNA compu ing models and because hey a e ypical o his a ea, we will p esen hem wi h some de ails in he nex sec ion. Fo u he models o DNA compu ing we e e o he web page www.wi.liacs.nl/home/pie /aaa, o he monog aph Pãun, Rozenbe g, and Salomaa, 1998, o he p oceedings olumes o he se ies o con e ences DNA Based Compu e s, ini ia ed in 1995 in P ince on, as well as o he new Kluwe jou nal Na u al Compu ing and he new Sp inge se ies o books wi h he same i le. 3 Compu ing by Splicing: H Sys ems The abs ac splicing ope a ion was in oduced in 1987 by T. Head, as a ma hema ical model o he ecombina ion o DNA molecules unde he in luence o es ic ion en- zymes (and ligases) – he e o e ( heo e ical in es iga ions o ) compu ing by splicing has been ini ia ed se en yea s be o e Adleman’s expe imen . The splicing o wo DNA molecules co esponds o wo ope a ions: cu ing he molecules by es ic ion enzymes and pas ing oge he he agmen s ob ained in his way, p o iding ha hey ha e ma ching s icky ends. Fo example, conside he ollow- ing wo (double s anded) DNA molecules: 5' - CCCCCTCGACCCCC - 3' 3' - GGGGGAGCTGGGGG - 5' and 5' - AAAAAGCGCAAAAA - 3' Recen Compu abili y Models Inspi ed om Biology: DNA and Memb ane Compu ing 75 3' - TTTTTCGCGTTTTT - 5' and he es ic ion enzymes TaqI and SciNI, o which he ecogni ion si es a e: T C G A G C G C A G C T and C G C G espec i ely (we ha e also indica ed he cu s ha hese enzymes make wi hin hei ec- ogni ion si es). These enzymes will cu he abo e wo molecules p oducing he ollow- ing ou molecules: 5' - CCCCCT CGACCCCC - 3' 3' - GGGGGAGC, TGGGGG - 5' 5' - AAAAAG CGCAAAAA - 3' 3' - TTTTTCGC, GTTTTT - 5' Because he agmen s ob ained in his way ha e complemen a y s icky ends, he annealing o s icky ends ollowed by liga ion will ei he ep oduce he wo o iginal molecules, o he ollowing wo new molecules will be o med: 5' - CCCCCTCGCAAAAA - 3' 3' - GGGGGAGCGTTTTT - 5' 5' - AAAAAGCGACCCCC - 3' 3' - TTTTTCGCTGGGGG - 5' As a model o he abo e biochemical ope a ion, T. Head conside ed a s ing op- e a ion (passing om double s anded sequences o s ings is allowed due o he p e- cise Wa son- C ick complemen a i y o nucleo ides) which was u he abs ac ed in (Pãun 1996a). In sho , one conside s splicing ules o he o m =u1#u2$u3#u4, whe e u1, u2, u3, u4 a e s ings o e a gi en alphabe . Gi en such a ule , and wo s ings w1u1u2w2, z1u3u4z2, by he splicing o hese s ings we ge he s ings w1u1u4z2, z1u3u2w2. The ela ion wi h he biochemical ope a ion o ecombina ion is clea : each s ing u1u2, u3u4 co esponds o he si e o a es ic ion enzyme, wo si es s ay oge he in he same ule i hey p oduce ma ching s icky ends, while he c ossing is supposed o be included in he con ex s ings (hence i can be emp y). F om s ing ope a ions we pass o language ope a ions in he na u al manne : con- side a se R o splicing ules (o any ype) and a se L o s ings; by splicing any wo possible s ings om L we ge a new se o s ings, R(L); he p ocess can be i e a ed, s a ing ei he om R(L) o om L∪R(L) ( he la e case co esponds o he obse a- ion ha when a DNA molecule is p esen , we may assume ha a bi a ily many cop- ies o i a e p esen , ob ained by ampli ica ion; in he i s case we may assume ha he eac ion is comple e, all old s ings ha ing been p ocessed). In his way, a compu ing (language gene a ing) de ice is ob ained, o he o m γ=(V, A, R), whe e V is an alphabe , A is a se o s ings o e V , and R is a se o splicing ules o e V . Such a machine y gene a es a language in he ollowing way: s a om he s ings in A, splice hem in all possible ways wi h espec o he ules in R, add all esul ing s ings o A and i e a e he p ocess. The language gene a ed by , deno ed by L(γ), consis s o all s ings which can be ob ained in his way. A s anda d ex ension is o also conside a e minal alphabe , T⊆V , and o accep in L(γ) only he s ings consis ing o symbols om T. Gheo ghe PÃUN, Ma io J. PÉREZ-JIMÉNEZ 76 Splicing sys ems γ=(V, A, R) wi h ini e se s A and R (hence wi hou a e minal alphabe ) canno gene a e all egula languages, bu , con e sely, he language L(γ) gen- e a ed by such a sys em is egula . Fo Head ules his was p o ed by K. Culik II and T. Ha ju al eady in 1991, while o Pãun ules i was p o ed by D. Pix on (1996). Splicing sys ems o he o m γ=(V, T, A, R), wi h ini e A and R, cha ac e ize he amily o egula languages, hence he powe o ini e au oma a. F om a compu a- ional poin o iew, he compe ence o ini e au oma a is oo limi ed. A cha ac e iza- ion o ecu si ely enume able languages (hence o he powe o Tu ing machines) is ob ained when using a se o splicing ules which is a egula languages ( he ules a e w i en as s ings, hence i makes sense o speak abou he ype o hei language). F om a compu a ional poin o iew, he abo e men ioned esul s a e qui e “ us- a ing”: ini e H sys ems compu e only a he le el o ini e au oma a, while he com- pu a ional uni e sali y is ob ained by using an in ini e se o splicing ules. Fo una ely, he p oo o he uni e sali y (Pãun 1996b) indica es a numbe o ways o o e coming his d awback. This p oo goes as ollows. S a ing om a ype-0 Chomsky g amma G, one cons uc s an equi alen ex ended H sys em γ whose sen en ial o ms a e ci - cula ly pe mu ed e sions o he sen en ial o ms o G, and he simula ion o he ules o G akes place wi hin su ixes o he sen en ial o ms o γ ( he ci cula pe mu a ion ensu es ha each de i a ion s ep in G can be simula ed in his way). Ve y c ucial o his “ o a eand- simula e” p ocedu e a e he i s and he las symbols o each sen en- ial o m, which in ac a e ma ke s, holding some in o ma ion abou he cu en s age o he simula ion. Tha is, we can igno e he s ings we splice as long as we know hei i s and las symbols, and he splicing si es. In o he wo ds, i is su icien o ha e a ini e numbe o splicing ules, and o associa e wi h each ule ce ain “p omo e s”, which a e symbols whose p esence allows he splicing o a gi en s ing. This obse a ion leads o ex ended H sys ems wi h pe mi ing con ex s, whose ules ha e associa ed ini e se s o symbols such ha a ule is applicable only o s ings which con ain he associa ed symbols. Ac ually, many o he ypes o con olled H sys ems we e conside ed. Abou a dozen such sys ems can be ound in he li e a u e, in gene al imi a ing he ypes o con ols known om he “classic” egula ed ew i ing a ea in o mal language heo y. In pa icula , he ollowing con ols we e in es iga ed: o bid- ding con ex s (symbols a e associa ed wi h ules and a s ing canno be spliced i i con- ains such a symbol), a ge languages ( he splicing o wo s ings is allowed only i he esul ing s ings belong o a gi en egula language which is associa ed wi h he ule o associa ed wi h he whole se o ules; in he o me case we say ha we ha e local a - ge s, and in he la e case we ha e a global a ge ), p og ammed con ol (a nex mapping is gi en on he se o ules, which indica es he sequencing o ules), e ol ing se s o ules (a each s ep, a di e en se o ules is p oduced, by poin mu a ion ules which ac on he splicing ules hemsel es), double splicing ( he s ings esul ing om he splicing o wo s ings a e immedia ely spliced again by any a ailable ule), conside ing mul ise s o s ings ( he s ings a e coun ed, by splicing hey a e consumed, he mul iplici y o s ings esul ing om a splicing ope a ion is inc eased by one). Re e ences can be ound in he sou ces men ioned a he end o he p e ious sec ion. Recen Compu abili y Models Inspi ed om Biology: DNA and Memb ane Compu ing 77 In all hese cases one ge s cha ac e iza ions o ecu si ely enume able languages. Simila esul s a e ob ained o a ious ypes o dis ibu ed H sys ems, whe e he s ings and he se o splicing ules a e sepa a ed in a ious ways, so ha a sys em o se e al “simple” H sys- ems is ob ained, wo king in a pa allel manne and coope a ing in ob aining a com- mon esul . All he p oo s a e cons uc i e, hence, s a ing om uni e sal ype-0 g amma s, one can ob ain uni e sal H sys ems, hence p og ammable H sys ems able o compu ing a he le el o Tu ing machines. These esul s ha e some gene al consequences. The ac ha i e a ed splicing wi h espec o a ini e se o ules compu es only egula languages indica es ha wha we can compu e in he ee mode (hence in he mode encoun e ed in na u e) is no oo much. The e o e, we need a u he ing edien in o de o achie e he desi ed uni e - sali y. A se ies o possibili ies a e sugges ed by he con olled and he dis ibu ed H sys- ems men ioned abo e. Howe e , he con ols and he dis ibu ed a chi ec u es con- side ed up o now seem no easy o be implemen ed wi hin he p esen day bio- echnology. So, an impo an dilemma a ises: should we con ine a he le el o ini e au oma a and hope o implemen a weak compu ing de ice based on splicing soon, o , p o iding ha we need a highe compu ing powe , should we ha e o look o im- p o ed models and o imp o ed bio- echnologies in o de o, hope ully, implemen a uni e sal DNA compu e ? A ques ion o a g ea in e es , o be add essed in an in e - disciplina y eam, in he nea u u e. 4 Memb ane Compu ing DNA compu ing deals wi h p ocesses aking place a he gene ic le el, wi h he hope o epea hem in i o. Howe e , many p ocesses which a e obse ed in i o canno be epea ed a all, o hey de elop in a di e en way in i o. A possible solu ion is o use he cell i sel as he en i onmen o a compu a ion, and his is he s a ing poin o memb ane compu ing. The a ea is a he young – i was ini ia ed by (Pãun 2000) ( he pape was ci cula ed on web a he end o 1998) –bu i is a he de eloped om a ma hema ical poin o iew. We will ecall he e only he mos basic ideas and esul s, he main classes o mem- b ane sys ems and hei p ope ies; o u he de ails, we e e he eade o he web page h p://psys ems.disco.unimib.i and o he monog aph (Pãun 2002). Memb ane compu ing s a s om he assump ion ha he p ocesses aking place in he compa men al s uc u e o a li ing cell can be in e p e ed as compu a ions. Ab- s ac ing om he biochemical de ails, one ge s memb ane sys ems (called also P sys- ems), which, oughly speaking, consis s o a cell-like memb ane s uc u e, in he com- pa men s o which one places mul ise s o objec s which e ol e acco ding o gi en ules in a synch onous, pa allel, and non-de e minis ic manne . The objec s can be de- sc ibed by symbols o by s ings o symbols om a gi en alphabe . The objec s can also pass h ough memb anes, he memb anes can be dissol ed, di ided, c ea ed. An e olu ion o a memb ane sys em is a compu a ion; we conside as success ul only he Gheo ghe PÃUN, Ma io J. PÉREZ-JIMÉNEZ 78 hal ing compu a ions, wi h which a esul is associa ed. Many classes o P sys ems we e al- eady conside ed in he li e a u e. Mos o hem a e compu a ionally comple e, i.e., equal in powe o Tu ing machines. I an exponen ial wo kspace can be c ea ed (in polynomial ime), by di iding memb anes, o by eplica ing s ing-objec s, o by c ea - ing memb anes om objec s which can be eplica ed, hen polynomial ime solu ions o NP-comple e p oblems can be ob ained. The memb anes appea ing in a P sys em y o mimic he ole and he unc ioning o memb anes om li ing cells. The basic unc ion o biological memb anes is o de- ine compa men s and o ela e compa men s o hei en i onmen , including neighbou ing compa men s. The cu en ly accep ed model o he memb ane s uc u e is he so- called luid-mosaic model, p oposed in 1972 by S. Singe and G. Nicolson. Acco ding o his model, a memb ane is a phospholipid bilaye in which p o ein molecules (as well as o he molecules) a e o ally o pa ially embedded. Figu e 1: A memb ane s uc u e The (plasma) memb ane is only pa ially pe meable (in gene al, o small non- cha ged molecules), bu a ious molecules can pass h ough memb anes by means o p o ein channels. The ansmemb ane ans e can ake place in a passi e manne , e.g., by di usion owa ds he egion o lowe concen a ion, and in an ac i e (media ed) man- ne . Ac ually, he e a e wo main ypes o p o ein channels: hose which jus selec he mo ing objec s by hei size, and hose which in e ac wi h speci ic molecules when helping hem o c oss he memb ane; he la e ype is called ca ie p o ein. The p o ein channels a e also impo an o he in e -cellula communica ion: neighbou ing cells can link hei p o ein channels, and in his way, a complex commu- nica ion ne wo k can be es ablished among cells. I is impo an o no e ha he p o- ein channels can be open o closed, depending on he con en s o he adjacen com- pa men s. Fo ins ance, i one o he cells is in aded by “undesi ed” molecules, hen he cell isola es i sel om he neighbou ing cells by closing he passage channels – hey may be e-opened again, once he eme gency si ua ion has been esol ed. Memb ane Sys ems (wi h Symbol-Objec s) – An In o mal In oduc ion. The memb ane s uc u e o a P sys em is a hie a chical a angemen o memb anes (un- de s ood as h ee dimensional esicles), embedded in a skin memb ane, he one which sepa a es he sys em om i s en i onmen . A memb ane wi hou any memb ane inside is called elemen a y. Each memb ane de ines a egion. Fo an elemen a y memb ane his is Recen Compu abili y Models Inspi ed om Biology: DNA and Memb ane Compu ing 79 he space enclosed by i , while he egion o a non-elemen a y memb ane is he space inbe ween he memb ane and he memb anes di ec ly included in i . Figu e 1 illus- a es hese no ions. We label memb anes (by posi i e in ege s in Figu e 1) in o de o be able o add ess hem in p og amming compu a ions by memb ane sys ems. Since each egion is delimi ed (“ om he ou side”) by a unique memb ane, we will use he labels o memb anes o also iden i y (label) he egions hey delimi . Each egion con ains a mul ise o objec s, and a se o (e olu ion) ules. The objec s a e ep esen ed by symbols om a gi en alphabe . Typically, an e olu ion ule om egion is o he o m ca→cbinjdou ehe e, and i “says” ha a copy o he objec a, in he p esence o a copy o he ca alys c ( his is an objec which is ne e modi ied, i only as- sis s he e olu ion o o he objec s), is eplaced by a copy o he objec b and wo cop- ies o he objec d. Mo eo e , he copy o b has o en e “immedia ely” he inne memb ane o egion labeled by j (hence o en e egion j), one copy o objec d is sen ou h ough he memb ane o egion , and one copy o e emains in egion . No e ha he conside ed e olu ion ule can be applied in he egion only i his e- gion includes he memb ane j. Memb ane sys ems a e synch onous, in he sense ha a global clock is assumed, i.e., he same clock holds o all egions o he sys em. In each ime uni a ans o ma ion o a con igu a ion o he sys em akes place by applying he ules in each egion, in a non- de e minis ic and maximally pa allel manne . This means ha he objec s o e ol e and he ules go e ning his e olu ion a e chosen in a nonde e minis ic way; his choice is “exhaus i e” in he sense ha , a e he choice was made, no ule can be applied any- mo e in he same e olu ion s ep ( he e a e no enough objec s a ailable anymo e o any ule o be applied now – his is he maximali y o applica ion). In his way, one ge s ansi ions be ween he con igu a ions o he sys em. A se- quence o ansi ions is called a compu a ion. A con igu a ion is hal ing, i no ule is ap- plicable in any egion. A compu a ion is hal ing i i eaches a hal ing con igu a ion. The esul o a (hal ing) compu a ion is he numbe o objec s sen ( h ough he skin memb ane) o he en i onmen du ing he compu a ion. Many modi ica ions/ex ensions o his e y basic model ske ched abo e a e dis- cussed in he li e a u e. We will b ie ly men ion he e only a ew o hem. The i s ex ension is o conside a p io i y ela ion among ules. This means ha in each egion a s ic pa ial o de ela ion on he se o ules om his egion is gi en – hen, a ule can be chosen ( o p ocess a mul ise o objec s) in a gi en s ep only i no ule o a highe p io i y is applicable. Ano he use ul “con ol de ice” is he possibili y o modi y he memb ane pe me- abili y. Thus, a memb ane can be made hinne (ac ion δ) o hicke (ac ion τ). A memb ane o no mal hickness is dissol ed by ac ion δ ( he objec s o a dissol ed memb ane emain in he egion su ounding i , while he ules a e emo ed; he skin memb ane canno be dissol ed), o made impe meable (no objec can pass h ough such a memb ane) by ac ion τ. An impe meable memb ane is e u ned o no mal hickness (hence i is again pe meable) by ac ion δ. Many possibili ies a e o e ed by he communica ion commands. Fo ins ance, he e a e a numbe o ways o weakening he p og amming powe p o ided by inj : o