A dynamical modeling to study the adaptive immune system and the influence of antibodies in the immune memory*

Transcription

1 dnmcl modelng to stud te dptve mmune sstem nd te nfluence of ntbodes n te mmune memor* lexndre de Cstro 1,2**, Crlos rederco ronz 2 nd Domngos lves 2,3 *Publsed n CMES, vol 45(1) pp (2009) + revsons

2 bstrct. Immunologcl sstems ve been n bundnt nsprton to contemporr computer scentsts. Problem solvng strteges, stemmng from known mmune sstem penomen, ve been successfull ppled to cllengng problems of modern computng (MONROY, SB, GODÍNEZ, 2004). Smulton sstems nd mtemtcl modelng re lso begnnng use to nswer more complex mmunologcl questons s mmune memor process nd durton of vccnes, were te regulton mecnsms re not stll known suffcentl (LUNDEGRD, LUND, KESMIR, BRUNK, NIELSEN, 2007).In ts rtcle we studed n mcn pproc to smulte te process of ntgenc mutton nd ts mplctons for te process of memor. Our results ve suggested tt te durblt of te mmune memor s ffected b te process of ntgenc mutton nd b popultons of soluble ntbodes n te blood. Te results lso strongl suggest tt te decrese of te producton of ntbodes fvors te globl mntennce of mmune memor. Kewords: mmune memor, ntbodes, B- cells, 1. INTRODUCTION Nturl computng brngs togeter nture nd computng to develop new computtonl tools for problem solvng (CSTRO,2006). pproces s evolutonr computng, neurocomputng nd mmunocomputng ve found numerous pplctons n vret of growng felds ncludng engneerng, computer scence, nd bologcl modelng (LIN, LIOU, 2005; YOSHIMUR, 2006; YMERICH, SERR, 2006; LCERD, SILV, 2006; OISHI, YOSHIMUR, 2007; SINGH, MNI. GNGULI, 2007; YNG, TNG, HTSUKMI, ZHENG, WOODRD, 2007; OISHI, YOSHIMUR, 2008; KERH, LI, GUNRTNM, SUNDERS, 2008). 1 Centro Nconl de Pesqus Tecnológc em Informátc pr grcultur, Empres Brsler de Pesqus gropecuár (EMBRP), Cmpns , Brzl. 2 Deprtmento de Informátc em Súde, Unversdde ederl de São Pulo (UNIESP), São Pulo , Brzl 3 culdde de Medcn de Rberão Preto, Unversdde de São Pulo (USP), Rberão Preto , Brzl. **Correspondng utor: lexndre.cstro@embrp.br In ts scenro, mmunocomputng m be n mportnt tool to better to better understnd te mmmls defense sstem nd t cn provde results dffcult to be observed n vvo. s bref ntroducton to te mmune sstem, we consder te mmune responses medted mnl b te lmpoctes B nd T responsble b te specfc recognton of te ntgens (strnge molecules to te orgnsm cpble of beng recognzed b te mmune sstem) nd b soluble molecules tt ts B lmpoctes secrete, te ntbodes (ROITT, BROSTO, MLE, 1998). In generl, te mmune sstem must present vrgn, mmune nd tolernt sttes nd m present lmts of memorzton. In te vrgn stte te popultons re ll t ver low level, wc mens, wt vlues of te order of te qunttes rndoml produced b te bone mrrow. In te mmune stte te populton of B cells tt specfcll recognzes knd of ntgen remns t determned level, even fter te suppresson of ts ntgen. In te tolernt stte te popultons of ntbodes nd of B cells dd not respond to te presence of ntgens or self-ntgens (protens from te ndvdul s orgnsm tself). However, tere cn be postve selecton of B cells nturll self-rectve, ndctng te exstence of subgroup of B lmpoctes subject to self rectvt (ROITT, BROSTO, MLE, 1998). mong mllons of knds of B lmpoctes of te orgnsm, ec one wt ts specfc ntbod eld on te membrne, onl tose wc recognze specc ntgen re stmulted. Wen ts occurs, te B lmpocte multples, orgntng lnege of cells (clones) ble to produce specfc ntbodes gnst te ntgen tt nduced ts multplcton. Te ntbodes produced b mture B lmpocte known s plsmcte re relesed n gret mount n te blood. Te multplcton contnues s long s tere re ntgens ble to ctvte tem. s determned knd of ntgen s beng elmnted from te bod, te number of lmpoctes speclzed n

3 bttle t lso dmnses. However, smll populton of tese lmpoctes remns n te orgnsm for te rest of te ndvdul s lfe, consttutng wt s denomnted mmune memor. Durng te evoluton of te mmune sstem, n orgnsm fnds gven ntgen repetedl. Te effcenc of te dpttve response to secondr encounters could be consderbl ncresed b te storng of popultons of cells tt produce ntbodes wt g ffnt to tt ntgen, denomnted memor cells. Insted of strtng ll over ec tme gven ntgenc stmulus s presented ts strteg gurntees tt te speed nd effcenc of te mmune response s ennced fter ec nfecton (ROITT, BROSTO, MLE, 1998). To better understnd ts process, two fundmentl teores for te mmune memor were developed. Te frst one consders tt, fter te expnson of te B cells, te formton of plsm cells nd memor cells occur. ccordng to. M. Burnet (BURNET,1959), tese memor cells would be remnng cells from n mmunologc response tt, supposedl, survve untl te end of te ndvdul s lfe terefore wt lfe longer tn te oter cell of te orgnsm. Te second teor, due to N. K. Jerne (JERNE, 1974), consders tt te mmune sstem presents memor nd cpct of response for second nvson of te sme ntgen, wt self-orgnzton of te sstem, llowng te formton of cellulr popultons tt lst for long tme. In oter words, ts utor teorzes tt te popultons survve, nd not onl specfc knd of cell wt lfespn longer tn te one from oter cells of te orgnsm. 2. MTERILS ND METHODS In ts pper, computtonl model s presented to smulte te bevor of te mmune sstem, consderng structurl mecnsms of regulton tt were not ncluded n te smplfed model proposed b Lgrec (LGREC, LMEID, SNTOS, 2001). In our pproc we consdered not onl te ntbodes lnked to te surfce of te B cells (surfce receptors), but lso te popultons of ntbodes soluble n te blood (ntbodes secreted b mture B cells), mkng ts model closer to te rel bevor of te mmune sstem (CSTRO, 2005; CSTRO, 2006; CSTRO, 2007; CSTRO, RONZ, LVES, 2009; CSTRO, RONZ, GICHETTO, LVES, 2009). Besdes te dfferentton of te B cells, te model exposed ere llows representng te generton, mntennce nd regulton of te mmune memor n more complete w, troug memor network, tt combnes te crcterstcs of Burnet s clonl selecton teor nd Jerne s network potess, consderng onl dotpc ntdotpc nterctons (BURNET, 1959; JERNE, 1974). In te model dscussed ere, te moleculr receptors of te B cells re represented troug bt-strngs wt dverst of 2 B, were B s te number of bts n te strng (CSTRO, 2005; CSTRO, 2006; CSTRO, 2007; CSTRO, RONZ, LVES, 2009; CSTRO, RONZ, GICHETTO, LVES, 2009; PERELSON, WEISBUCH, 1997). Te ndvdul components of te mmune sstem represented n te model re te B cells, te ntbodes nd te ntgens. Te B cells (clones) re crcterzed b ter surfce receptor nd modeled b bnr strng. Te eptopes portons of n ntgen tt cn be lnked b te B cell receptors (BCR) re lso represented b bt strngs. Te ntbodes ve receptors (prtopes) tt re represented b te sme bt-strng tt models te BCR of te B cell wc produced tem. Ec strng (spe) s ssocted to n nteger σ ( 0 σ M= 2 B 1 ) wc represents ec on of te clones, ntgens or ntbodes. Te negbors to gven σ re expressed b te Boolen functon σ = ( 2 1xorσ ). Te complementr form of σ s obtned b σ =M σ nd te tme evoluton of te concentrtons of te severl popultons s obtned s functon of te nteger vrnts σ nd t, troug drect terton. Te equtons tt descrbe te bevor of te clonl popultons re clculted troug n tertve process, for dfferent prmeters nd ntl condtons:

4 (1) m + (1 d) + b (t) ( σ, t) wt te complementr spes ncluded n te term (σ, t), (σ,t) = ( 1 )[(σ,t) + (σ,t) + (σ,t)] + B 1 = [(σ, t) + (σ, t) + (σ, t)]. In tese equtons, nd re, respectvel, te popultons of ntbodes nd ntgens; b s te prolferton rte of te B cells; σ nd σ re te complementr spes of σ nd of te nerest B negbors n te percube (wt te t bt flpped). Te frst term (m), nsde te curled brckets n equton (1), represents te producton of te cells b te bone mrrow nd t s stocstc vrble. Ts term s smll, but non-zero. Te second term nsde te curled brckets descrbes te popultons tt ve survved to nturl det (d), nd te trd term represents te clonl prolferton due to terton wt complementr spes (oter clones, ntgens or ntbodes). Te prmeter s te reltve connectvt mong determned btstrng nd te negborood of ts mrror mge or complementr spe. Wen = 0.0, onl perfect complementr spes re llowed. Wen = 0.5, strng cn equll recognze ts mrror mge nd ts frst negbors. Te fctor (t) s expressed b: ( ) ( ) ( ) ( ) t = [ σ, t + σ, t + σ, t ] σ (2) Te tme evoluton of te ntgens s determned b:, (σσt +1) = k (t) ( 1 )[(σ, t) + (σ, t)] + B [(σ = 1 (3), t) + were k s te speed n wc te popultons of ntgens or ntbodes decrese to zero, wc mens, te ntgen removl rte due to tertons wt te popultons of B cells nd ntbodes. Te popultons of ntbodes s descrbed b group of vrble 2 B defned n B-dmensonl percube, nterctng wt te ntgenc popultons: (4) ( ) ( ) σ, t +1 = σ, t + b ( ) σ, t ( t) B ( 1 ) ( σ, t) + ( σ, t) =1 k ( ) σ, t ( t) were te contrbuton of te complementr spes (σ, t) s gn ncluded n te lst term, b s te ntbod prolferton, nd k s te ntbod removl rte, wc mesures ts tertons wt te oter popultons. Te popultons of ntbodes (tt represent te l number of ntbodes) depend on te noculted dosge of ntgens. Te fctors nd re te (t) (t) responsble b te control nd decrese of te popultons of ntgens nd ntbodes, wle (σ(t) te fctor s te correspondng fctor (t) for te ccumulton of te clone popultons n te formton of te mmune memor. Te clonl populton (σ( t) (normlzed l number of clones) cn vr snce te vlue produced b te bone mrrow (m) untl ts (σ, t)], (σ, t)

5 mxmum vlue (n our model, e unt), snce te Verust fctor lmts ts growt. Te Verust fctor produces locl control of te popultons of clones (B cells), consderng te severl regulton mecnsms (CSTRO, 2009). However, te popultons of B cells re strongl ffected b te popultons of ntbodes soluble n te blood. Ts s te reson tt leds us to nclude te term (σ,t) s n extr contrbuton n te set of mps prevousl coupled proposed b Lgrec (LGREC, LMEID, SNTOS, 2001). In order to properl stud te tme evoluton of te components of te mmune sstem, we defne clone s beng onl set of B cells. So, te populton of ntbod s treted seprtel n te present model. Te equtons (1) to (4) form set of coupled mps tt descrbes te mn nterctons of te mmune sstem mong enttes tt nterct troug connectons ke-lock tpe, wc mens, enttes tt recognze ec oter specfcll. Ts set of equtons s solved tertvel, consderng dfferent ntl condtons. 3. RESULTS Te smultons performed n ts pper sow te generton, mntennce nd te regulton mecnsms of te mmune memor, nd te cellulr dfferentton, troug dotpc ntdotpc nterctons, tt combne te crcterstcs of te clonl selecton teor nd te mmune network teor. To sow te extenson of te vldt of te model, te results of some smultons re presented. Immunzton experments, n wc te severl ntgens, wt fxed concentrtons re njected n te bod n n ntervl of 1000 tme steps, n order to stmulte mmune response. Wen new ntgen s ntroduced, ts ntercton wt ll te oter components n te sstem s obtned troug rndom number genertor. Te lengt of te B bt strng ws fxed n 12, correspondng to potentl repertor of 4096 cells nd dstnctve receptors. Injectons of dfferent ntgens, n tme ntervls, correspondng to one perod of lfe or to te entre lfe of te ndvdul were gven. In te smultons, te vlue d = 0.99 ws consdered for te rte of nturl det of te cells (poptoss), nd te prolferton rte of clones nd ntbodes, s beng 2 nd 100, respectvel. or te connectvt prmeter te vlue 0,01 ws cosen; nd te ntbodes nd ntgens removl rte (k) ws fxed n 0,1, so tt n ec ntervl of 1000 tme steps, te popultons of ntgens nd ntbodes dspper before te next ntgen be noculted. In ec noculton, te sme seed for te rndom number genertor ws used, so te dfferent ntgens re noculted t te sme order n ll te smultons. Mn smultons were performed, wt ntgen doses vrng between nd 1.5. Next, te results of some smultons wll be sown, wt specl empss to two ntermedte vlues for te doses 0.08 nd 0.10 n te regon of coverge of te smultons. ltoug ver close, tese vlues present dstnct results for te mmune memor nd, consequentl, durton of vccnes. Lttle ltertons n te ntl condtons of te sstem ffect sgnfcntl te evoluton, sowng tt te modelng of te mmune sstem, troug non-lner coupled mps, represents good reproducton of complex bologcl sstem, suc s te mmune memor. Te results for doses t extreme lmts, wt peculr bevors, wll be treted n te contnuton of ts pper. In gure 1 te tme evoluton of te frst clonl populton tt recognzes te frst noculted ntgen, s sown: () wt te ddton of ntbodes populton, nd (b) wtout consderng ntbodes n te set of coupled mps. Te ddton of ntbodes to te sstem do not orgnte consderble locl dsturb, owever, n gures 1 to 3, t s sown tt te ddton of ntbodes lters te globl cpct of te mmune memor, for dfferent ntgenc concentrtons. Te results obtned wtout te presence of te term referrng to te ntbodes. gure 1(b), correspond to te smplfed model (LGREC, LMEID, SNTOS, 2001), n wc te ntbodes soluble n te blood re not consdered.

6 In te gures 1 nd 2 te memor cpct s represented, consderng te sstem wt or wtout te presence of ntbodes. or bot concentrtons of ntgens 0.08 nd 0.10 wen te popultons of ntbodes re consdered (gures 1() nd 2()), te cpct of te mmune memor network s smller tn n te bsence of popultons of ntbodes (gures 1(b) nd 2(b)). In te gure 4, wt g ntgenc dosge, t s possble to notce clerl tt te bgger te ntbod prolferton rte, te smller te network cpct. In te bsence of specfc model for te ntbodes, te popultons rec ger levels. Tkng nto ccount tt te popultons of ntbodes soluble n te blood elp n te regulton of te B cells dfferentton, we cn nfer from te results not onl te mportnt role of te ntbodes n te mecnsm of regulton of te prolferton of te B cells, but lso n te mntennce of te mmune memor. gure 2 Memor cpct for te concentrton of ntgens equls 0.08: () wt te ddton of ntbod populton, nd (b) wtout ntbodes. gure 1 Tme evoluton of te frst clonl populton tt recognzes te frst noculted ntgen wt () ddton of ntbod populton nd (b) wtout ntbod. gure 3 Memor cpct for ntgenc concentrton equls 0.10: () wt te ddton of ntbod popultons, nd (b) wtout ntbodes.

7 gure 4 Memor cpct for te concentrton of ntgens equls 1.0: () wtout te ddton of ntbod popultons; (b) wt ntbod prolferton rte equls 100 nd (c) wt ntbod prolferton rte equls Te decrese of ctve popultons cn be explned troug te ntercton between ntbodes nd B cells, wc s ccordng to te mmune network teor. Te results suggest tt, despte promotng te fgtng to nfectons, te g producton of ntbodes cn destro te memor clonl popultons produced b prevous nfectons. Toug te dnmcl model proposed, we lso ve reproduced n mqun, experments to stud te bevor of te sstem fcng ntgenc mutton. Usng 10 smples tt represent orgnsms wt te sme ntl condtons, te severl popultons of ntgens re noculted wt concentrtons fxed n 0,1nd njected n ntervls of 1000 tme steps. Wen new ntgen s ntroduced, ts nterctons (connectons) wt ll te oter components n te sstem re obtned ccordng to rndom number genertor. Cngng te seed of te rndom number genertor, te bts n te bt-strngs re ltered (flpped) nd, s te bt-strngs represent te ntgenc vrblt, te ltertons of bts, consequentl, represent te respectve muttons. To stud te bevor of te sstem fcng mutton, we fxed n 350, 250 nd 110 te number of njectons of dfferent mutted ntgens. Te sme vlues of te prmeters prevousl used were consdered, wc mens, te poptoss or cell s nturl det rte d = 0.99, clonl prolferton rte equls 2.0 nd ntbod prolferton rte equls 100. Te connectvt prmeter ws consdered equls 0.01 nd te bone mrrow term m ws fxed n gure 5 sows te verge lfespn of te clonl popultons tt specfcll recognzed te mutted ntgens, consderng 350, 250 nd 110 njectons. Te verges, clculted over te 10 smples, ndcte tt, pprentl, te frst popultons tend to survve more tn te oters, ndependentl of te number of te nocultons. gure 5 verge of te lfespn of te popultons tt recognzed te ntgens, for () 350, (b) 250 nd (c) 110 nocultons.

8 mmune response, but lso elp n te regulton of te B cells dfferentton. Te presence of soluble ntbodes cnge te globl propertes of te network, nd ts bevor cn onl be observed wen te popultons re treted seprtel. Te pproc proposed lso sows tt t s mpossble to foresee te populton tt wll survve for long perod. Tese results re resonble becuse t s mpossble to predct te durton of vccne. On te oter nd, our results ve strongl suggested tt te bsence or decrese of ntbod producton promotes te globl mntennce of mmunztons 5. REERENCES gure 6 Lfespn of te clonl popultons n ec smple. However, n gure 6()-(j) t s possble to vsulze te bevor of ec one of te 10 smples seprtel, wen we dmnster n slco 110 njectons. It s clerl notced, ccordng to gure 6, tt we cn not ffrm tt te frst clonl populton lsts longer tn te popultons subsequentl recognzed oter ntgens, snce te smultons ndcte tt onl n 2 smples te frst clonl populton ve survved for long tme (gures 6(b) nd ()). It s mportnt to glgt tt ts dscrepnc between te results of gures 5 nd 6 s due to te fct tt n two smples te lfespn of te frst clonl populton excted ws long, so, te rtmetc men ws g even tt n oter smples te frst clonl popultons ve not survved for long perod. 4. DISCUSSION ND CONCLUSIONS Te results presented n ts rtcle suggest tt te process of ntgenc mutton ve relton wt te durblt of te mmune memor nd te popultons of ntbodes soluble n te blood not onl prtcpte of te YMERICH ; SERR M. (2006). n nt Colon Optmzton lgortm for Stckng Sequence Desgn of Composte Lmntes; CMES, Vol. 13, No. 1, pp BURNET, M. (1959) Te Clonl Selecton Teor of cqured Immunt. Cmbrdge Unverst Press. CSTRO, (2005). network model for clonl dfferentton nd mmune memor. Psc Stt. Mec. pplc., v. 355, pp CSTRO, (2006). ntbodes producton nd te mntennce of te mmunologcl memor. Eur. Ps. J. ppl. Ps. 33, p CSTRO,. (2007) Rndom bevors n te process of Immunologcl memor. Smulton Modellng Prctce nd Teor 15, p CSTRO, ; RONZ, C.; LVES, D. (2009) Vrl mutton nd ts nfluence n te tme evoluton of te mmunztons. Eur. Ps. J. ppl. Ps. v. 47, pp CSTRO, ; RONZ, C.; GICHETTO, P. LVES, D. (2009). Influence of te ntgenc muttons n tme evoluton of te mmune memor: dnmc modelng. Lecture Notes n Computer Scence, v. 5676, pp CSTRO, L.N. undmentls of Nturl Computng: Bsc Concepts, lgortms, nd pplctons. Cpmn nd Hll/CRC (2006) JERNE, NK. (1974) Towrds Network Teor of te Immune Sstem. nn. Immunol., v. 125C, p KERH T.; LI, JS. ;GUNRTNM, D; SUNDERS, R. (2008) Evluton of Sesmc Desgn Vlues n te Twn Buldng Code b Usng rtfcl Neurl Network. CMES, Vol. 26, No. 1, pp LCERD L.; SILV JM (2006). Dul BEM Genetc lgortm Sceme for te Identfcton of Polrzton Curves of Bured Slender Structures. CMES, Vol. 14, No. 3, pp , 2006

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