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How to control a biological switch: a mathematical framework for the control of piecewise affine models of gene networks

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(1)How to control a biological switch: a mathematical framework for the control of piecewise affine models of gene networks Etienne Farcot, Jean-Luc Gouzé. To cite this version: Etienne Farcot, Jean-Luc Gouzé. How to control a biological switch: a mathematical framework for the control of piecewise affine models of gene networks. [Research Report] RR-5979, INRIA. 2006, pp.22. �inria-00094853v2�. HAL Id: inria-00094853 https://hal.inria.fr/inria-00094853v2 Submitted on 21 Sep 2006. HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés..

(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. How to control a biological switch: a mathematical framework for the control of piecewise affine models of gene networks Etienne Farcot — Jean-Luc Gouzé. N° 5979 Septembre 2006. ISSN 0249-6399. apport de recherche. ISRN INRIA/RR--5979--FR+ENG. Thème BIO.

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(135) lnˆ^rur_4me``<mecqœ_a: `MDl‚¡6→}eqsme`Vmeme\{\^`a`

(136) ‡p{rkq†k~¢`M`

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(170) Ÿ-†kln ?`<`Mvw‡<}qs„w`§†^l%„1qsmeˆRqœrw¯wˆrwtœ¯w}o„`Mme†W`Ml(qsˆ rŸJme\{le`”rw_alejk`l¢moln ?`<_ `§‡qœ¡6`MlMqsmoª)\{c€rwˆXzkrwm }qœ†kˆ{`< {}6zkmom§r:ª”m[-„\{£w``MjX„tœ_at·moqœvw\{\u`Mmle` „w {tœletyr'„zO\^ln„,qœp{¯wtœ`a` „w`ˆ °±`M`‡¹°?m`§l‡¹qsm ˆFmerwr Ÿ _`_ba cdDeVfe.

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(172) rrmeŸ³\{Ÿme`amo\^\{z{``% {tœŸ´qs ?rwvu`<}o\u}%_ m p?qœˆurwmoz{`<ˆOˆ^†{leqsl‚m€jarwŸ¶ru}J`M„ume‡o`M\H_aqœ ?ˆ{`< {}zk„1m mez^‡}erR`uruª}oc€†{ˆµqsˆ^„w„ˆFmeja`‡MpF„wj le`wp¤ p +. Uj. U=. p Y. [0, Uj ].. ^zqœc­lm2ln`§‡<le†Xrw`<_a`M\{_a {`<l®}otœ``M}omo„`M`<Ÿy„utœmojµ`<lnru}§meª‚ˆ^z{„}oc­p{Ÿ)ˆ{tœ`§†k` †qœlome‡rw\O}oˆ·„1`m¤Omo`rwp}¯6r„°'}oqœnK„w¤¤Op{qs„µtsm`§lŸ ¡6„w„‡qœts}om%t`le¡ p?zk`:`©”mn„wmo‡lo`<`2ln}2z{meln_arµz{`4‡<qsmerw`§qsˆ^ˆW†›leqymeme†k\{r'`<`:} †kln`§E^`§leˆ^xF‡<q²z{}emoqœ`<`

(173) p?tº`

(174) ª

(175) lnz{le[-p^rw\{le_a`qœml2`l(‡qœrˆ{ruŸˆu {memez{qœ\^m2ˆR` z{xurwqœzOˆuz^„mol%ˆu`<}omole¯1q²`mo„m2qsty`§l rl Ÿ IËqsŸ´ˆ^ru {}®zk`mo¦{¤?l%„„w_a¡6ˆ^qœ†' {tœt‹tœ`}o„w`Mtsqsln¡-Ÿ me}o„µ„§qyjk‡¹vwml‚`M„wp?ˆ{ts`` t me\{`4me}o`M„me_a`<ˆFmol‚mo\^¤„1m2¡6qs¡ me\`a {rw}e}ru†k ?`M}eru`§le† ` mor;! melM\{¤`§rwlnˆ{`:`;lnz{ln\OpO„ln`<tœt molMleª `m A{ru}®`<¦k„w_a {¤Jts`ulnr›¤³qsmoŸ¶\^mo„1\{m `aqsˆ{ {z{m u mo„w£wªH`Ml®[-¯1\{„w[0,`<tsz^ˆ·`M¤‹Ul‚rwqœˆ{]ˆ `µle_:rw_a„§j!` lnE^ruˆ{ts¯wqsme`:`:mole\{``m U = {u , . . . u } ‡„arwln}oru}ots`Mzklemo Oqsruruˆ^ˆ;†kruqsˆ^}-v%ˆ^‡rrumMˆuª moqsˆRz{rwz^lJ {}erup{tœ`<_u¤u„wˆ^†a‡\{`M‡£

(176) „ŸËme`M}e¡)U„w}o†^l=¡6u\{`mo\{`<} „1m¶ts`§„wUlnmJrw⊂ˆ{`([0,rŸ?Umo\{`] †kqyle‡<}e`<me`6¯1„wtsz^`Ml‹qsˆ U qœl¶„tyler mec€lnˆ®\{¡6`”qsmomo\{}e‡`J`§\{l¢ {qœmoˆ{}o}evu`Mqy‡¹lMlemo`<¤{qœˆF‡<rwm·„ˆ ˆ'¡ u|rupO}e`2£±†{¤Mqy¡ `l me`‹`§‡‡¹Ÿ´rurkmoˆ^`M‡ln†'z^mol·„„uˆFrw‡<m%ˆ2‡z^Ÿ´ {ru}oqs„} `§me‡`M`<`M„wtœ¡6ju‡¤Rqœ\ le„`JDˆO‡†µru∈ˆ^¡6lnqsmoDme„\'ˆFª;m³ˆ^Ÿ´ra[-`<`§le\{†kqsqyvul2p^ˆ{„w}oLq ‡E?`<£®tœ‡<qœ„`M‡l2ˆFrum)ˆurwmemoˆ }oqsrw_amet\{`2ty„§`µ†{¡(„w`<lMtylo„§ª·lejuz{c€ªˆ2_aM rw kmeme\{‡qœrwru`Mˆ ˆ^}·le¡ me`M\^rwxF}„z{†{m2`<l ˆ^mou‡<\{`‚}e=`§rwlnŸu(x) \^morw\{ty† qyl6¤,‡„‡o}o\^ˆOrurw†®loqyle‡_aqs`®ˆ{rwvFqy}ol)l<`<¤ mer1\O¯w<\„1`<=m} < meŸ´rw\{}6`_aqœˆ{rw {}ozk`%m®†{_a`mo„,„wj;qst·p?rw`ˆ;z{moˆ^\{„w` _†kp{V` EOqœvwˆ{z{q²moruqsz^rulnˆtœj'rŸln`M`<ªˆX„wl„aŸ´z{ˆ^‡meqœrwˆ V → U ¤?le`<`le`M‡meqœrwˆ^l‚Ž{ªF2¤³Žkª Ž

(177) Ÿ´ru}vw`<ˆ^`<}„t·ˆ{rwmo„1moqsruˆ^l„ˆ^† WOªF2 u [}e-`§\{„`4tœqs»§ {„}op{`Mtœ‡<`%qœleqs`ˆ›Ÿ´}orw`M}o„_ tlerwjRŸ lnmeκ(x,  „ ^ ˆ † ¤le`<`MˆW„ulŸ´z{ˆO‡¹meqœrwˆOl®rwŸJmo\{`aqsˆ{ ^zkm ¤·_a„,j›ˆ{rwm%p?`a„w}ep{qsme}„}ojw¤±„wˆ^†Wln\^rwz{ty†HpO` ·) Γ(x, ·) M ` : _ < l S›rw}o`% {}o`M‡<qœle`%Ÿ´rw}o_al6rwŸme\^`Mle`%Ÿ´z{ˆ^‡meqœrwˆ^l¡6qstœtp?u`„wlolnz{_a`M†'ty„1mo`<}(rwˆ·¤^p{z{m¡ ` E^}olnm( {}or1¯Fqy†k` ª me[-\{\{`2`p^‡„u\{lnrwqy‡‚qy‡Ÿ `'„w‡¹rmŸ®l)„Hmo\^Ÿ´„1`<m`M†{†kp^r:„wˆ{‡£+rmtœrR†krw`< ? `M†{ˆ^`<†' ?†{`<ˆ^qs}o†{`Mqs‡ˆ{mev!tœj”rwrwˆŸmeme\{\{qy`l(ln}o ?`<vw`§z^‡FqtœEO„me‡‚ruŸ´}eruj!}e_†kruª _a„wqsˆ rŸme\^`‡<z{}e}o`<ˆFmalnmo„me`u¤¶}o„me\{`<}4me\^„wˆ rwˆ q²ml {`¦{}e`§„w‡‡qymelntœ`µjXxFlez^qs_a„ˆFqœtyme„qs}®mo„momer'qœ¯w¡6`”\O¯1„1„m2tœz{qy`wl2¤†k\O`M„wlol ‡}oˆ{qœpOqy‡`§`;†H‡qœrwˆ!ˆOlnln`§`M‡¹xFmoz{qsru`Mˆ^ˆ!‡Žk`§l2¤`rw¦{ˆ!‡`Mmo k\{m%`;me†k\^jR„ˆ^m%„Ÿ´_ark‡Mqy‡<„lMt‹ª ?M6rwqœ‡ˆumemz^l2„w†ktstœ`<jw ?¤`<meˆO\{†W`µrwp?ˆW`M\^me„§\^¯R`aqœrwqœz{ˆ{} {zkrwm Ÿ-me\^ª:`µ[-le\{jk`al¢moty`<„1_meme`<qœ}l p?‡<„w`<leqœˆ{`w¤Rv4mo‡\{ru`%ˆ^Ÿ´lnrkmo‡<„„wˆFt³m) ?qsˆ;rwqœ`§ˆu„wm(‡r\”Ÿ}ole`<z^vu‡oz{\›ty„1„ameru†k}erwja_:†k„rwqœ_:ˆ'„p?qœ`<ˆ qœˆ{Dvaˆ{¤Fr1mo¡\{9` r:^Ÿr1mo¡\{`%rwŸ´?Ÿ rwI}o@$_ K ‡<G ts`§„}otsj4mo„w£w`Ml¶mo\{`%le„w_a`(Ÿ´ru}e_„wl qœˆµme\{`%u „wzkmeruˆ{rw_arwzOl j=1. j. j. j. 1 j. k j. i j. j. k j. j. j. j. D. r. a. φ(a, u) =. . κ1 (a, u) κn (a, u) ··· γ1 (a, u) γn (a, u). . DŒ. I $K. ,. ¡6[-\{\{`<` }o`®me\{`2  ¯w \`§ 9‡¹meru FN} Vu,\O „w l6„  E{qy¦kl®`Mmo†'\{`a¯1„¡6tœz{\{`uru¤Rts`:me\^ln„`<mm2‡<qœˆ „ˆ'¡6p?\{` qœ‡‡o\W\{rFŸ´rkln`M‡Mˆ'„t„u ?‡<ru‡<rwqsˆF}mo†kl2qsˆ{‡Mv

(178) „ˆWmer”p?ln`:ru_4‡\{`2rFlnle`M ?ˆ·`M¤‡Fq E^<\`§,†;< mo {\{z{`:}o ?leru`m2le`wrwª Ÿ)„tœtŸ´rk‡<„wt ?ruqsˆFmol rw¡6pk\{m`<„ˆ:qœˆ{z^`Mle† qsˆ^pRXv jWA¯1qœts„qœ}o {jR ?qsr1ˆ^¯4v;lnmoru\{ts`”z{meqœqœˆ{rw {ˆ^zkl m

(179) 892,qs= ˆ+:º¤uqs¡6mol qsmo¡6\”\{„2rutsxF`”z{qs†kmeru`_:†k„qs°±qœˆ `<}o`<G ˆuφ(a, ªqsˆ{vOM(¤wtsme¡ \{`ruz{le\^vw„\!tœtOmeln\{ru`:_4mo`<`<me}oqœ_ _a`M VlJ,z^ln '`(qsm  „wl¶qyl „rˆ”ŸËme„`Mp{ˆ!p{}o`M`<_a¯R {qy„1tsmor1qsjwru`Mˆ·† ¤ U). m a _ M `  „ ^ ˆ rwhR_aqsˆOq²‡me` meqœˆ{v ¡6qœ tst± _a& rF \l¢ Fm)/Vrw FŸË meª `Mˆµ[-p?\{qy`%l-„

(180) ¡6‡qœtœrut·_aˆ{ ^`<¯w„w`M‡}-m-p?lnz{` pO„ln_ ^„up^‡qs`‚vuz{rrwŸ z^l-qsˆ'¤{„meˆ^\^† `2 {}e`§ln`Mˆum(„

(181) ln‡<merwz^ˆu†{mojwqsˆRª z{rwzOl _:„ ·¤Fmo\{`‚Ÿ´rk‡M„t±le`m-¡6qstœt±vw`Mˆ{`<}„tœtsj pme?\{`

(182) `2„ }o`<pvwU¨­z^†{tœqs„w_a}-`<†kˆOrulnqœ_:rwˆ^„„wqœˆ^tl-‡¡6rw_aq²mo ^\›„uln‡¹z{m®p^leloz{‡p^}oqœle k ^m6„wqœ‡<ˆ;`mor\{Ÿ `%RŸ´rwtœª%tsr1[-¡6\{qœˆ{`

(183) Rva ?leru`lom=leG qsp{tœ`

(184) φ(a, lez^‡<‡·)`§lelerw}l6rwŸ‹mo\{`4†kru_a„wqsˆ D ¡6qstœtmo\{`<ˆHp?`

(185) vwqœ¯w`Mˆ›pRj I2§$ K S(a) = {a + sign(b − a) | b ∈ V, D ∩ φ(a, U) 6= ∅} . ¡6[-\{\{`<`:}o`z^lnsign vu`<qsˆO¯w`§†kl `M† mo\{me` r›ln†{qœvwV` ˆ'E^ˆ{r`”Ÿq²lemz^l6‡M„w‡}e`Mvuloz{lnru_a}ol®`<ˆFrwmMˆ{ª tœjW„_arwˆ{v}e`Mvwz{ty„}2†krw_:„wqsˆ^l%me\^„m„}o`a„w†1~n„w‡`Mˆum2mer ¤ : R → {−1, 0, 1} : `  r Ÿ y q 2 l s q F ˆ e m `<[-¯w\{`<qœˆ'l;qœŸ´ˆ'r{‡<me„\{t2` lnsign(b ‡M`<„wm'le`®qyl¡6−‡\{ru`Mˆ^a)ˆ l¢moD}o„wqœˆ{qyl-`M†ˆ{rwpRmMj>ª me\^`!ln ?`§‡Fq EO‡ ‡o\{ruqœ‡<`WrŸ

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