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Positive And Negative Feedback Loops Coupled By Common Transcription Activator And Repressor

Positive And Negative Feedback Loops Coupled By Common Transcription Activator And Repressor l ee C e e ee leel e e ee el e ee e e ee T e e e e l l e e l e e e l l e e e e l l C e e e e e e e e e e e l e ll l l e e ll ll lll e e e e e e e l ll le T e e e e e e e e l e e e l e A ll e el T l ee e e ee l le e e e e e e ee e l T e l l e e e e e e Keywords el le l ll . SIELE IESIU A. O ACIU 1. INTRODUCTION Ci ca i hyth s in eu a y tic gis s a e gene ate y special ti s in t sc ipti n egulati n net s. The st i p tt these ti s a e cl se negati e ee ac l ps. T sc ipti n a gene in such a l p gi es RNA, h se t slati n p uces a p tein. The p tein un e g es a nu e t s ati ns ec es the ep ess its n gene. The sche e this in c als in a chain en y atic eacti ns, i the en p uct the chain inhi its en y e cataly ing ne the i st eacti ns. As a atte act, this syste , n n as in s scillat , as iginally applie t en y atic eacti ns in 1966 . In the si plest a it, the syste c nsists ne c pe ati e p cess t sc ipti n, all ste ic en y e, e e ecept se e al eacti ns ith the linea n lecula inetics. The syste s hich gene ate scillati ns in vivo ha e e c ple st uctu e l a et al. 2010, eng La a 2012, Dunlap 1999, Al n 200 . Ci ca i cl , as a ule, c ntain inte l c e negati e p siti e ee ac l ps Saith ng et al. 2010 . The e istence scillati ns thei pe i a e ete ine y the negati e ee ac l p. On the the h , the p siti e ee ac l p c t s e eg ee i y the pe i , a plitu e ustness the scillati ns. It c e en ully a p the scillati ns in s e special cases. e c nsi e a hyp thetical syste t genes h se t sc ipti n is g e ne y the sa e t sc ipti n act s. One these genes enc es a p tein hich c ec e a ep ess . The the gene enc es a p ecu s p tein the acti at . T sc ipti n th genes is p ssi le p i e that the c ncent ati n the ep ess is su iciently l the c ncent ati n the acti at is su iciently high. the ie p int athe atical eling, the essential eatu e the syste is l gic the c upling et een the ep ess acti at ee ac l ps. The l gical c n uncti n ep essi n acti ati n ught t e satis ie t sta t t sc ipti n. Si ulte us t sc ipti n y genes, in uce y c n act s, ccu s in p a y tic eu a y tic cells. In lact se pe n Escherichia coli the sa e p te pe at c nt l t sc ipti n p lycist nic genes enc ing th ee p teins gg 2005 . lycist nic genes RNA a e uite usual phen ena in acte ial pe ns. In eu a y tes, as a ule, the syne p essi n p teins ucti ning in the sa e p cess is c inate y t sacting act s Nieh s llet 1999 . The alte nati e splicing p e- RNA a es it p ssi le t tain t i e ent p teins n the asis the sa e t sc ipt. In such a case, e p essi n these p teins is in uce in the sa e p te Ma e ca , en- eng Chen E e y 2004 . B th echis s ccu in ci ca in cl Rippe ge B n 2010, Steige ste 2011 . IT COMMON... The syste is ep esente in u el y a set ina y i e ential e uati ns. Linea sta ility alysis Nay eh Balach a 1995 nu e ical s luti ns e e use as t ls in u sea ch. e t ie t ete ine h the phase p t ait the syste epen s n pa a ete s. e ha e un gene al elati ns et een pa a ete alues hen the nu e e uili iu p ints as chge in a sa le-n e t sc itical i u cati n. The c n iti ns the p i u cati n ha e een un in a less gene al in a e special, sy et ical, cases. 2. STRUCTURE O T E S STEM D ITS MAT EMATICAL MODEL e iscuss a speci ic hyp thetiyste hich c ul egulate e p essi n genes. The syste c ntains t genes hich a e t sc i e si ulte usly. Thei t sc ipti n is egulate y t t sc ipti n act s, ep ess acti at . The t sc ipti n is g ing n hen c ncent ati n the ep ess is l c ncent ati n the acti at is su iciently high. Such elati ns et een t c ipti ns the t genes c e eali e y a c n p te in e t e aly si pli ie el, as in ig. 2.1. ualitati ely, the sa e situati n is e in ci ca i syste s y E- es, hich a e acti ate y acti at . In such a case, a ep ess inte acting ith the acti at p e ents its inte acti n ith E- . T p teins enc e in the egulate genes un e g y t s ati ns ith the t sc ipti n act s as en p ucts. The syste is ep esente y a sche e in ig. 2.1 a set q ina y i e ential e uati ns 1 . Ou el is c nsistent ith highly si pli ie sche es the a ali ci ca i cl c p esente y l a et al. 2010 , ig.1 y Oste 2010 , ig. 5. . X1 -Xp GR GA Xp+1 ­ Xq FIG. 2.1. Sche e the egulat y syste . A ­ gene the acti at . ­p te , R ­ gene the ep ess , dx1 dt dxi dt dx p dt dx j dt axqn 1 x m 1 xqn p hi 1 xi k1 x1 , 2,..., p, k p 1 xp 1 , p 2,..., q. ki xi , i bxqn 1 x m 1 xqn p hj 1 x j k j xj , j In e uati ns 1 , xp xq a e c ncent ati ns the ep ess acti at especti ely. The a ia les x1 xp+1 a e c ncent ati ns RNA enc ing ep ess s acti at s p ecu s s. The est a ia les e e t c ncent ati ns t sient s the p teins. E uati ns ith in ices 1 p+1 c esp n t RNA synthesis ecay. e use i ensi nless a ia les. The unit alue the i st p a ia les s the ep ess c ncent ati n hich e uces the ate t sc ipti n t the hal its alue in the ep ess a sence. The unit alue the a ia les xp+1 ...xq c esp n s t such acti at c ncent ati n at hich the ate t sc ipti n ec es e ual t the hal its a i u alue. ill s c e icients c pe ati ity the ep ess acti at a e especti ely m n. The ki hi a e ate c nstts. The c nstts a b a e a i u ates t sc ipti n at the st a a le c n iti ns xp . q The syste s p siti ely in a it at p siti e alues the ate c nstts. I all the a ia les ha e n n-negati e initial alues then they ill est n nnegati e u ing the e luti n the syste . In sea ching e uili iu p ints, all the a ia les e cept xp xq c e eli inate . C n iti ns e uili iu c e gi en the ll ing shape 2 : ay n 1 y n he e x y a e e uili iu alues x, by n 1 y n xp xq especti ely. y, p 1 q IT COMMON... ki hi p 1 q 1 ki hi p 1 E uili iu alues the e aining a ia les a e xj j q j ki x, 1,...., p 1, hi ki xj j 1 q j y, 1,...., q hi E uati ns 2 ha e ne i us s luti n x y 0 . It es, in ie , that the igin phase c inates is e uili iu p int the syste 1 . It ll s 2 that the nu e e uili iu p ints thei c inates epen s n t pa a ete s A a / B b / , hich a e si ple uncti ns the 2q ate c nstts. ssi le e uili iu p ints ey n the igin c e un e uati ns 2 in the ll ing 4: By n 1 1 y n 1, B x. A ial e uati n E uati ns 4 c e e uce t the e ui alent p lyn Bn xm In the case x m Bn xn AB n x n 4' n=1, the e uati n 4' takes the shape: Bx m Acc p siti e nu e Ax m Bx A1 B ing t Desca tes ule, it has then ne B>1 n ne B<1 eal t. S , at n=1, the syste 1 has ne t e uili iu p ints. The e uili iu p ints ith n n-negati e phase c inates is chge th ugh t sc itical i u cati n at B=1. In the case , the e a e tw sign chges in the p lyn ial 4' it c ha e tw n ne p siti e ts. The syste c ha e ne, tw th ee e uili iu p ints. The nu e e uili iu p ints chges ne t th ee in a sa le-n e i u cati n. Bi u cati nal alues the pa a ete s we will in using cha acte istic e uati n the syste . . C ARACTERISTIC E UATIONS O T E S STEM D T E NUMBER O E UILIBRIUM OINTS Tw e uati ns with in ices 1 p+1 the syste 1 ha e n nlinea uncti ns in thei ight h pa ts. Let us call these uncti ns f1 fp+1: f1 x1 , x p , xq fp x p , x p 1 , xq axqn 1 x m 1 xqn p bxqn 1 x m 1 xqn p k1 x1 , k p 1 xp w ite thei pa tial e i ati es 6 . g1, p g1, q gp gp f1 xp f1 xq fp n amx m 1 x p n 1 xq 1 2 , , , . n xq n 1 x m 1 x n bmx m 1 x 1, p xp fp p n 1 xq 1 2 n bnxq 1, q xq n 1 x m 1 x In ices in the le t pa t ac i the syste 6' . 6 e e t p siti ns pa ticula e i ati es in the IT COMMON... k1 h1 0 0 0 0 0 0 k2 hi 1 0 0 0 0 0 0 ki hp 1 0 0 0 g1, p 0 0 kp g p 1, p 0 0 0 0 0 0 kp 1 hj 1 0 0 0 0 0 0 kj hq 1 g1, q 0 0 0 g p 1, q 0 kq 6' i j ,..., p 2,..., q 1. the syste pp In a gene al case, the cha acte istic e uati n q 1 p 1 1 1 has the shape i 1i 1 uiui 11 p p 1 1 p p 11 g pp 1, qq g 1, h hii i i 11 u ui i gg1, p 1, p p p 1 i 11 hi i h ii p 1 ui qq1 1 1 1 gg p1,1, qg1,1,pp p q g ui u i whe e gp 1, q g1, q g pp 1, pp 1, 1, i i 1,1, i p p i hhi 0,0 i kiki the at i J 6' . As it ishes. ll ws 6 , the te is eigen alue g1, p g1, q g p q 1 1, , i p hi in the e uati n ith y n>1 all the e i ati es 6 a e e ual t e in the igin, cha acte istic e uati n appea s t e uite si ple s l a le in this e uili iu p int 8 . ki 0, ki , i 1,..., q. All eigen alues this p int e uili iu at igin is sta le at y e uili iu a e eal negati e. The eing ul alues pa a ete s. In the case n=1, g p 1, q b , g1,q=a. e c use e initi ns B gi e t the cha acte istic e uati n e uili iu shape 9. pa a ete s at igin the ki ki ki The e a e p negati e eigen alues i the e aining ki , i 1,..., p . All eigen alues ha e thei eal pa ts negati e at B<1, ne the ec es e ual t e at B=1 at least ne the is p siti e at B>1. N w n 1 , the e uili iu at igin is sta le at B<1 unsta le at B>1. The e is a t sc itical i u cati n at B=1. a iti nal p int e uili iu with p siti e c inates appea s e uili iu at igin ec es unsta le. In e t e a ine sta ility the e uili iu p ints ey n the igin, ne sh ul s l e e uati ns 2 4 , su stitute taine x y xp xq in 6 in eigen alues the esulting ac i. But, it is athe i p ssi le t tain a clea sy li luti n the enti ne e uati ns. Instea , we use e uati ns 2 t e p ess pa a ete s a b as uncti ns x y inse t taine e p essi ns int 6 . Ne t, we use esulting e p essi ns as especti e ent ies the ac i. Such a p ce u e esults in the ll wing cha acte istic e uati n 10 : ki n 1 yn i 1 ki ki mx m ki ki The e uati n is ali in the p ints e uili iu which a e situate ey n the igin c inates, x 0 y 0 , inclu ing n=1. e ha e n t use y a iti nal assu pti n in e i ati n the e uati n 10 . It is easy t in c n iti ns sa le-n e i u cati n using e uati n 10 . In the p int such i u cati n, at least ne eigen alues is e ual t e . ith , e uati n 10 is e uce t the elati n 11 : n 1 yn mx m This elati n c nstitutes the c n iti n which is t e satis ie y c inates e uili iu p int at a sa le-n e i u cati n. It c e w itten as a uncti n 12 : IT COMMON... n 1 1 m 1 n xm m ula 12 all ws us t calculate y a it a y ch sen alue x. Ne t, th x y c e use t calculate the pa a ete s A B acc ing t e uati ns 4 . In such a way, we c in at what alues A B the sa len e i u cati n takes place. A B nx 1 n 1 n xm 1 m xm n 1 , xm 1 1 n 2 1 n In 12 m xm e t get physically eing ul, p siti e, alues y, A B 1 , we sh ul c n ine u ch ice m, n x alues t : 0 xm n 1 , m 1 n m n 1 1. e c n w aw a pa a et ic pl t acc ing t 1 with the pa a ete x y pa ticula pai ill c e icients. The pl t w ul i i e the ple int a eas with ne th ee e uili iu p ints. It ll ws 14 that the e is n i u cati n with =0 at n=1 n n e alues a ia les. It will e use ul t int uce e c ple pa a ete s n , 1 yn In te s the pa a ete s sa le-n e i u cati n 11 s 16 1 : mx m . , cha acte istic e uati n 10 c n iti n tain s ewhat si ple e gene al ki ki ki ki ki we e , the new pa a ete s ha e a s all isa tage. A pai alues cha acte i es ne speci ic p int e uili iu , n t the wh le syste . Each pa ticula syste is ep esente in the ple y a nu e p ints e ual t the nu e e uili iu p ints. In y case the e is a p int 0, n c esp n ing t the p int e uili iu in the igin the phase c inates. unsta le sa le p int c e un at p int 1 . The thi e uili iu c e un at 1 . Aut scillati ns a un this p int c e gene ate a te a p ssi le p i u cati n. 4. OP BI URCATION D OSCILLATIONS IN T E S STEM 4.1. T E CASE O T E I EST S MMETR IT ALL KI=1 D Q=2P e c ul in c n iti ns p bi u cati n in the alytical elati ns nly in s e special cases. Let us c nsi e the syste satis ying tw essential li itati ns. Rate c nstts ecay ki all the a iables in the syste a e e ual t each the . The ep ess 's l p c nsists the sa e nu be substces as the acti at 's l p . The assu pti n that all ecay c nstts a e e ual t 1 will ake u calculati ns e si ple but it will n t ake the less gene al. Un e such assu pti ns the cha acte istic e uati n 16 a pts the si ple s l able 18 . The e a e p 19 ts -1. The e aining p eigen alues satis y the e uati n R, m R n. 19 In the case The ine uality in 19 ll ws ts the e uati n 19 R 1 all e initi ns Rp ha e negati e eal pa ts especti e p int e uilib iu is stable. At R=1, ne eal eigen alue ishes a sa le- n e bi u cati n takes place. ith R=1, the sec n pa t 19 , lea s t the elati n, which we ha e al ea y btaine in the ully gene al case 1 . IT COMMON... In the case negati e R, e uati n 19 c be gi en the The e uati n has R ts: i sin m. Rp 2j R p sin 2j j 0,1,..., p 1. Eigen alues with j=0 j=p-1 ha e the highest eal pa ts. The e uilib iu will be estabili e when the eal pa t these tw eigen alues bec es p siti e. It takes place at The case e uality in 2 gi es a elati n between the c e uilib iu p ints at p bi u cati n. inates np yn 1 m m n p 1 x p 1 . In e t btain eing ul alues y, pa a ete s inate x sh ul satis y the ll wing li itati ns: np x p 1 the syste the m n 1 mp 1 . p 1 m n p Let us n te that the p bi u cati n is p ssible when i e ence between ill c e icients ep essi n acti ati n e cee s a ce tain alue sh wn in the i st pa t 25 . The sa e alue b un e ill c e icient at p bi u cati n in a l p with ne ep esse gen In e ni i T eu 1991 , as well as a p uct ill c e icients all c pe ati e p cess in a single l p c ntaining y ep esse acti ate genes Sielewiesiuk paciuk 2012 . The elati n 24 is satis ie in e uilib iu p ints un e g ing a p bi u cati n. Using this elati n c n iti ns e uilib iu 2 we c e p ess pa a ete s as uncti ns x - ep ess 's c ncent ati n in e uilib iu 26 . nx 1 x m A np p 1 m p p p n p p n B np p 1 m n p p 1 n 1 n mp E uati ns 26 esc ibe a pa a et ic cu e which sepa ates in the ple the a ea with p ssible scillat y s luti ns the a ea whe e scillati n a e i p ssible. B 15 12.5 10 7.5 5 2.5 Aut scillati ns Bistable t igge One stable e uilib iu 0,0 FIG. 1. Pa a ete ple the syste with p, q, m, n = 10, 20, 4, 2 . L we cu e c esp n s t the sa le-n e bi u cati n, the uppe ne ­ t p bi u cati n. IT COMMON... B 10 8 6 4 2 One unstable 0,0 ne stable e uilib iu One stable e uilib iu 0,0 Tw unstable e uilib iu Aut scillati ns. p ints. FIG. 2. Pa a ete ple the syste with p, q, m, n = 10, 20, 4, 1 . L we cu e c esp n s t the t sc itical bi u cati n, the uppe ne ­ t p bi u cati n. Syste s with a c pe ati e n>1, ig. 1 n n-c pe ati e n=1, ig. 2 acti ati n ha e s e ualitati e si ila ities i e ences. Syste s b th the kin s ha e a p int e uilib iu at the igin c inates at y alues A B. Tw the , n n e , p ints e uilib iu appea th ugh sa le-n e bi u cati n in syste s with n>1, ne a iti nal e uilib iu appea s th ugh t sc itical bi u cati n when n=1. The e uilib iu at the igin is stable at all alues pa a ete s with n>1, but it bec es unstable B>1 with n=1. p bi u cati n takes place in b th the kin s acti ati n at su iciently high alues the pa a ete s A B. 4.2. REPRESSION D ACTIVATION LOOPS IT DI ERENT NUMBERS O ELEMENTS Let us c nsi e s e syste s with slightly l we sy et y, whe e the tw l ps c nsist i e ent nu be s p tein t s ati ns. As in secti n 4.1, we c ntinue t use all the ecay ate c nstts e ual t ne the cha acte istic e uati n 16 in the : In e t in alues the pa a ete s at p bi u cati n, we supp se a pu ely i agina y eigen alue i int uce a new c ple a iable z 1 z i sin with t 1 in 2 The substituti n the a iable z esults in e uati ns 29 : p sin p q sin q p p q sin q bi u cati nal alues the pa a ete s sin p sin 2 p sin q p q sin 2 p These elati ns all we us t c nst uct pa a et ic pl ts using as a pa a ete . Sets such pl ts a e sh wn in ig. in ig. 4 . In b th the igu es, the st aight line 1 c esp n s t p ints the sa le n e bi u cati n. At least ne eigen alue these p ints e uilib iu is e ual t e in all syste s un e c nsi e ati n. The e is at least ne eal p siti e eigen alue at 1 . The e aining cu es c esp n t e uilib iu p ints ha ing a pai c ple eigen alues with e eal pa t. Oscillati ns 14 FIG. 3. Cu es p bi u cati n in syste s ha ing ep essi n l p c nsisting p=10 eagents. Acti ati n l ps c ntain qeagents with q=14, 15, 16, 1 , 18, 19, 20, in the e the le t t the ight. Dashe line c esp n s t sa le-n e bi u cati n. IT COMMON... Nu e ical calculati ns we e ne the nu be eagents in the ep essi n l p p=10 that in the acti ati n l p is q-p. In the cases q-p= 4, 5, 6, , 8 9, the cu es en n the st aight line 1 with 0. As it was sh wn ea lie 2 , the st aight line p 1.651 2 c esp n s t bi u cati n in the st sy et ical syste with b th l ps e ual si es p=10 q-p=10 . Oscillati ns a e p ssible a un the e uilib iu p ints with +1 bel nging t the a ea bel w sa le-n e line t the ight the cu es p bi u cati n. Let us e in that y pa ticula syste has in the ple accessible a ea with m n . It is p ssible, at high n, that the cu e p bi u cati n es n t i i e the accessible ectgle m n int tw sepa ate pa ts. e a ple such situati n c be seen in ig. =14 15 n=2, whe e the e a e n stable scillati ns in spite e isting c ple eigen alues with p siti e eal pa t. The syste g es t the stable e uilib iu with e alues all a iables. It appea s that scillati ns a e i p ssible, when in ucti e l p is t sh t. FIG. 4. Cu es p bi u cati n in syste s ha ing ep essi n l p c nsisting p=10 eagents. Acti ati n l ps c ntain q-p eagents with q=25, 24, 2 , 22, 21, 20. Again, the st aight line c esp n s t sa le-n e bi u cati n. On the c nt a y, l ng l ps acti ati n n t supp ess scillati ns. e p esent in ig. 5 alues the pe i scillati ns btaine nu e ical s luti ns. The scillati ns ha e the highest e uency when the l p acti ati n is slightly l nge th the l p ep essi n . The pe i app aches a c nstt alue with inc easing nu be ele ents in the l p acti ati n. At q p. These c inates bi u cati nal cu es in igs intege ultiple p: e e t the p int which is c n all the 4. In cases l ng acti ati n l ps, when q is jp, j 2, ,... the sec n ne the tw e uati ns 29 is satis ie with y alue Substituti n these alues q int the i st e uati n 29 gi es elati n which sh ul be satis ie at p bi u cati n 2 j p IT COMMON... Relati n 2 is illust ate in ig. 6. The st aight line with ep uces e actly the c n iti n p bi u cati n in the st sy et ical syste s 2 . The biggest pa t FIG. 6. p bi u cati n cu es in syste s whe e q is intege ultiple The alues j a e sh wn at especti e st aight tlines. D tte line c esp n s t the sa le-n e bi u cati n the ple , c esp n ing t aut scillati ns, ha e the syste s with q=3 when the acti ati n l p is twice as l ng as the l p ep essi n. In the li it e y high alues j at e y l ng l ps acti ati n the uncti n 2 g es t the st aight line with in inite sl pe the e istence scillat y s luti ns es n t e epen n the pa a ete Relati ns btaine in the secti n 4.2. enable us t in e uilib iu p ints with a pai pu e i agina y eigen alues. In p bi u cati n, the eal pa t this pai c ple eigen alues sh ul chge its sign. e use nu e ical calculati ns checke in se e al cases that the chge sign es eally take place. Un tunately, we ha e n t un y clea p this chge in general case. In the case o , relations 22 e pression or the bi urcating pair o eigen alues: 23 i ply alytical sin . p In this case, the chge o the sign o at crossing bi urcational alue o - cur e 2 in Fig. 6 is e i ent. The characteristic e uation 27 has a relati ely si ple alytical solution also in the case o q=3p. The bi urcating pair o eigen alues is then gi en by 35 : sin . p The real part o 35 is icreasing unction o ishes at satis ying relation 32 with j=3 cur e 3 in Fig. 6 . So, it ust chge its sign by crossing this cur e. 4.3. REPRESSION D ACTIVATION LOOPS IT DIFFERENT RATE CONSTTS OF DECA Let us now consi er the syste whose both loops consist o the sa e nu ber o ele ents , but they i er in the rate o ecay o their reagents. e assu e that the constts ki in the repression loop i=1,..,p ha e the unit alue those in the acti ation loop are e ual to k. In such a case the characteristic e uation 16 c be written as: kp 1 Intro ucing into e uation 36 allows us to e press 2 2 para eters as unctions o . Para etric plots show the alues o , which correspon to op bi urcation. The plots are presente in Figs 7 8. Oscillatory solutions are possible in areas below the straight line alues o higher th those in bi urcation cur es. All o the consi ere asy etrical syste s ha e the loop o repression consiste o 10 reagents ecaying with the rate constt e ual to 1. Their acti ation loops i er one ro other by nu bers o ele ents in the loops o acti ation or by the rate constts o ecay. There is ob ious ualitati e si ilarity between the syste s with elongate acti ation loops the syste s with slower ecay o reagents in these loops Figs 4 7 . Finite i aginary POSTITIVE D NE ATIVE FEEDBACK LOOPS IT COMMON... eigen alues appear on both si es o the straight line In spite o this act, oscillations aroun e uilibriu points with are i possible. All e uilibriu points ro this part o the ple ha e one real positi e eigen alue. E olution o the syste takes it away ro the icinity o such e uilibriu . Let us note that in both cases the turno er o acti ator is slower th that o the repressor. FIG. 7. Cur es or op bi urcation in the syste s with e ual nu ber o ele ents in both loops. Rates o ecay constts are e ual to unity in the loop o repression. The alues o ecay rate constts in the loop o acti ation are shown at cur es. FIG. 8. Cur es or op bi urcation in the syste s with e ual nu ber o ele ents in both loops. Rate o ecay constts are e ual to unity in the loop o repression. The cur es correspon s to ecay rate constts in the loop o acti ation k=2, 1.8, 1.7, 1.6, 1.5, 1.4, 1.3, 1.2, 1.1. On the other h , bi urcational cur es o the syste s with shortene loops o acti ation are uch si ilar to those o the syste s with higher rate constts o ecay in these loops Figs 3 8 . The latter syste s ha e pure i aginary eigen alues only at At +1, i aginary eigen alue attains ero. Possible oscillations aroun the e uilibriu points with slightly lower th shoul ha e e tre ely low re uency. In this pair o asy etrical cases, acti ator's loop has shorter ti e o turno er th the repressor's loop. All bi urcation cur es in Figs 3, 4, 7 8 represent e uilibriu points with a pure i aginary eigen alue They ha e one co 1.65172 at p=10, =0. It c be in erre ro on point with relations 22 t p 23 that in this bi urcation point i aginary eigen alue oscillations with a perio T we c e pect 2 2 p . Nu erical solutions suggest that t / p this e aluation o the perio is better th those base on the actual i aginary part o co ple eigen alue. The turno er ti e o the repression loop is the ain actor eter ining the perio o oscillations. The acti ation loop with short turno er ti e e clu es oscillations or akes the slower. The slower acti ation loop has no signi ict in luence on the perio o oscillations. These rules are illustrate in Figs 5 9. Period of oscillations T Decay constts k in the loop of activator FIG. 9. Perio o oscillations obtaine ro nu erical solutions in syste s with 10 ele ents in both loops, ki=n the loop o repression, ki=k in the loop o acti ation, POSTITIVE D NE ATIVE FEEDBACK LOOPS IT COMMON... 5. CONCLUSIONS e consi ere a o el syste o gene e pression regulate by two trscription actors, repressor acti ator. e assu e that trscription o the gene takes place pro i e that the concentration o the repressor is low , at the sa e ti e, the concentration o the acti ator is high. Repression acti ation are both cooperati e processes with respecti e ill coe icients m n. igh alues o m pro otes oscillations high alues o n ake the less probable. Oscillatory solutions are possible when i erence m-n is su iciently high, see relation 25 . e intro uce para eters , unctions o coor inates o e uilibriu points, which enable us to eri e the characteristic e uation the sa le-no e bi urcation in a uite general way 16,17 . The op bi urcation was aly e in a ew special cases. Oscillations are generate in our syste by a negati e ee back loop o the oo win's type. A couple loop with positi e ee back oes not isturb oscillations when its turno er ti e is longer th the turno er ti e o the negati e ee back loop. Oscillations are slowe own or e en ully a pe , when the loop o acti ation has the turno er ti e shorter th that o the repression loop. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Annales UMCS, Physica de Gruyter

Positive And Negative Feedback Loops Coupled By Common Transcription Activator And Repressor

Sielewiesiuk, Jan; Łopaciuk, Agata
Annales UMCS, Physica , Volume 69 (1) – Mar 1, 2015
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Abstract

l ee C e e ee leel e e ee el e ee e e ee T e e e e l l e e l e e e l l e e e e l l C e e e e e e e e e e e l e ll l l e e ll ll lll e e e e e e e l ll le T e e e e e e e e l e e e l e A ll e el T l ee e e ee l le e e e e e e ee e l T e l l e e e e e e Keywords el le l ll . SIELE IESIU A. O ACIU 1. INTRODUCTION Ci ca i hyth s in eu a y tic gis s a e gene ate y special ti s in t sc ipti n egulati n net s. The st i p tt these ti s a e cl se negati e ee ac l ps. T sc ipti n a gene in such a l p gi es RNA, h se t slati n p uces a p tein. The p tein un e g es a nu e t s ati ns ec es the ep ess its n gene. The sche e this in c als in a chain en y atic eacti ns, i the en p uct the chain inhi its en y e cataly ing ne the i st eacti ns. As a atte act, this syste , n n as in s scillat , as iginally applie t en y atic eacti ns in 1966 . In the si plest a it, the syste c nsists ne c pe ati e p cess t sc ipti n, all ste ic en y e, e e ecept se e al eacti ns ith the linea n lecula inetics. The syste s hich gene ate scillati ns in vivo ha e e c ple st uctu e l a et al. 2010, eng La a 2012, Dunlap 1999, Al n 200 . Ci ca i cl , as a ule, c ntain inte l c e negati e p siti e ee ac l ps Saith ng et al. 2010 . The e istence scillati ns thei pe i a e ete ine y the negati e ee ac l p. On the the h , the p siti e ee ac l p c t s e eg ee i y the pe i , a plitu e ustness the scillati ns. It c e en ully a p the scillati ns in s e special cases. e c nsi e a hyp thetical syste t genes h se t sc ipti n is g e ne y the sa e t sc ipti n act s. One these genes enc es a p tein hich c ec e a ep ess . The the gene enc es a p ecu s p tein the acti at . T sc ipti n th genes is p ssi le p i e that the c ncent ati n the ep ess is su iciently l the c ncent ati n the acti at is su iciently high. the ie p int athe atical eling, the essential eatu e the syste is l gic the c upling et een the ep ess acti at ee ac l ps. The l gical c n uncti n ep essi n acti ati n ught t e satis ie t sta t t sc ipti n. Si ulte us t sc ipti n y genes, in uce y c n act s, ccu s in p a y tic eu a y tic cells. In lact se pe n Escherichia coli the sa e p te pe at c nt l t sc ipti n p lycist nic genes enc ing th ee p teins gg 2005 . lycist nic genes RNA a e uite usual phen ena in acte ial pe ns. In eu a y tes, as a ule, the syne p essi n p teins ucti ning in the sa e p cess is c inate y t sacting act s Nieh s llet 1999 . The alte nati e splicing p e- RNA a es it p ssi le t tain t i e ent p teins n the asis the sa e t sc ipt. In such a case, e p essi n these p teins is in uce in the sa e p te Ma e ca , en- eng Chen E e y 2004 . B th echis s ccu in ci ca in cl Rippe ge B n 2010, Steige ste 2011 . IT COMMON... The syste is ep esente in u el y a set ina y i e ential e uati ns. Linea sta ility alysis Nay eh Balach a 1995 nu e ical s luti ns e e use as t ls in u sea ch. e t ie t ete ine h the phase p t ait the syste epen s n pa a ete s. e ha e un gene al elati ns et een pa a ete alues hen the nu e e uili iu p ints as chge in a sa le-n e t sc itical i u cati n. The c n iti ns the p i u cati n ha e een un in a less gene al in a e special, sy et ical, cases. 2. STRUCTURE O T E S STEM D ITS MAT EMATICAL MODEL e iscuss a speci ic hyp thetiyste hich c ul egulate e p essi n genes. The syste c ntains t genes hich a e t sc i e si ulte usly. Thei t sc ipti n is egulate y t t sc ipti n act s, ep ess acti at . The t sc ipti n is g ing n hen c ncent ati n the ep ess is l c ncent ati n the acti at is su iciently high. Such elati ns et een t c ipti ns the t genes c e eali e y a c n p te in e t e aly si pli ie el, as in ig. 2.1. ualitati ely, the sa e situati n is e in ci ca i syste s y E- es, hich a e acti ate y acti at . In such a case, a ep ess inte acting ith the acti at p e ents its inte acti n ith E- . T p teins enc e in the egulate genes un e g y t s ati ns ith the t sc ipti n act s as en p ucts. The syste is ep esente y a sche e in ig. 2.1 a set q ina y i e ential e uati ns 1 . Ou el is c nsistent ith highly si pli ie sche es the a ali ci ca i cl c p esente y l a et al. 2010 , ig.1 y Oste 2010 , ig. 5. . X1 -Xp GR GA Xp+1 ­ Xq FIG. 2.1. Sche e the egulat y syste . A ­ gene the acti at . ­p te , R ­ gene the ep ess , dx1 dt dxi dt dx p dt dx j dt axqn 1 x m 1 xqn p hi 1 xi k1 x1 , 2,..., p, k p 1 xp 1 , p 2,..., q. ki xi , i bxqn 1 x m 1 xqn p hj 1 x j k j xj , j In e uati ns 1 , xp xq a e c ncent ati ns the ep ess acti at especti ely. The a ia les x1 xp+1 a e c ncent ati ns RNA enc ing ep ess s acti at s p ecu s s. The est a ia les e e t c ncent ati ns t sient s the p teins. E uati ns ith in ices 1 p+1 c esp n t RNA synthesis ecay. e use i ensi nless a ia les. The unit alue the i st p a ia les s the ep ess c ncent ati n hich e uces the ate t sc ipti n t the hal its alue in the ep ess a sence. The unit alue the a ia les xp+1 ...xq c esp n s t such acti at c ncent ati n at hich the ate t sc ipti n ec es e ual t the hal its a i u alue. ill s c e icients c pe ati ity the ep ess acti at a e especti ely m n. The ki hi a e ate c nstts. The c nstts a b a e a i u ates t sc ipti n at the st a a le c n iti ns xp . q The syste s p siti ely in a it at p siti e alues the ate c nstts. I all the a ia les ha e n n-negati e initial alues then they ill est n nnegati e u ing the e luti n the syste . In sea ching e uili iu p ints, all the a ia les e cept xp xq c e eli inate . C n iti ns e uili iu c e gi en the ll ing shape 2 : ay n 1 y n he e x y a e e uili iu alues x, by n 1 y n xp xq especti ely. y, p 1 q IT COMMON... ki hi p 1 q 1 ki hi p 1 E uili iu alues the e aining a ia les a e xj j q j ki x, 1,...., p 1, hi ki xj j 1 q j y, 1,...., q hi E uati ns 2 ha e ne i us s luti n x y 0 . It es, in ie , that the igin phase c inates is e uili iu p int the syste 1 . It ll s 2 that the nu e e uili iu p ints thei c inates epen s n t pa a ete s A a / B b / , hich a e si ple uncti ns the 2q ate c nstts. ssi le e uili iu p ints ey n the igin c e un e uati ns 2 in the ll ing 4: By n 1 1 y n 1, B x. A ial e uati n E uati ns 4 c e e uce t the e ui alent p lyn Bn xm In the case x m Bn xn AB n x n 4' n=1, the e uati n 4' takes the shape: Bx m Acc p siti e nu e Ax m Bx A1 B ing t Desca tes ule, it has then ne B>1 n ne B<1 eal t. S , at n=1, the syste 1 has ne t e uili iu p ints. The e uili iu p ints ith n n-negati e phase c inates is chge th ugh t sc itical i u cati n at B=1. In the case , the e a e tw sign chges in the p lyn ial 4' it c ha e tw n ne p siti e ts. The syste c ha e ne, tw th ee e uili iu p ints. The nu e e uili iu p ints chges ne t th ee in a sa le-n e i u cati n. Bi u cati nal alues the pa a ete s we will in using cha acte istic e uati n the syste . . C ARACTERISTIC E UATIONS O T E S STEM D T E NUMBER O E UILIBRIUM OINTS Tw e uati ns with in ices 1 p+1 the syste 1 ha e n nlinea uncti ns in thei ight h pa ts. Let us call these uncti ns f1 fp+1: f1 x1 , x p , xq fp x p , x p 1 , xq axqn 1 x m 1 xqn p bxqn 1 x m 1 xqn p k1 x1 , k p 1 xp w ite thei pa tial e i ati es 6 . g1, p g1, q gp gp f1 xp f1 xq fp n amx m 1 x p n 1 xq 1 2 , , , . n xq n 1 x m 1 x n bmx m 1 x 1, p xp fp p n 1 xq 1 2 n bnxq 1, q xq n 1 x m 1 x In ices in the le t pa t ac i the syste 6' . 6 e e t p siti ns pa ticula e i ati es in the IT COMMON... k1 h1 0 0 0 0 0 0 k2 hi 1 0 0 0 0 0 0 ki hp 1 0 0 0 g1, p 0 0 kp g p 1, p 0 0 0 0 0 0 kp 1 hj 1 0 0 0 0 0 0 kj hq 1 g1, q 0 0 0 g p 1, q 0 kq 6' i j ,..., p 2,..., q 1. the syste pp In a gene al case, the cha acte istic e uati n q 1 p 1 1 1 has the shape i 1i 1 uiui 11 p p 1 1 p p 11 g pp 1, qq g 1, h hii i i 11 u ui i gg1, p 1, p p p 1 i 11 hi i h ii p 1 ui qq1 1 1 1 gg p1,1, qg1,1,pp p q g ui u i whe e gp 1, q g1, q g pp 1, pp 1, 1, i i 1,1, i p p i hhi 0,0 i kiki the at i J 6' . As it ishes. ll ws 6 , the te is eigen alue g1, p g1, q g p q 1 1, , i p hi in the e uati n ith y n>1 all the e i ati es 6 a e e ual t e in the igin, cha acte istic e uati n appea s t e uite si ple s l a le in this e uili iu p int 8 . ki 0, ki , i 1,..., q. All eigen alues this p int e uili iu at igin is sta le at y e uili iu a e eal negati e. The eing ul alues pa a ete s. In the case n=1, g p 1, q b , g1,q=a. e c use e initi ns B gi e t the cha acte istic e uati n e uili iu shape 9. pa a ete s at igin the ki ki ki The e a e p negati e eigen alues i the e aining ki , i 1,..., p . All eigen alues ha e thei eal pa ts negati e at B<1, ne the ec es e ual t e at B=1 at least ne the is p siti e at B>1. N w n 1 , the e uili iu at igin is sta le at B<1 unsta le at B>1. The e is a t sc itical i u cati n at B=1. a iti nal p int e uili iu with p siti e c inates appea s e uili iu at igin ec es unsta le. In e t e a ine sta ility the e uili iu p ints ey n the igin, ne sh ul s l e e uati ns 2 4 , su stitute taine x y xp xq in 6 in eigen alues the esulting ac i. But, it is athe i p ssi le t tain a clea sy li luti n the enti ne e uati ns. Instea , we use e uati ns 2 t e p ess pa a ete s a b as uncti ns x y inse t taine e p essi ns int 6 . Ne t, we use esulting e p essi ns as especti e ent ies the ac i. Such a p ce u e esults in the ll wing cha acte istic e uati n 10 : ki n 1 yn i 1 ki ki mx m ki ki The e uati n is ali in the p ints e uili iu which a e situate ey n the igin c inates, x 0 y 0 , inclu ing n=1. e ha e n t use y a iti nal assu pti n in e i ati n the e uati n 10 . It is easy t in c n iti ns sa le-n e i u cati n using e uati n 10 . In the p int such i u cati n, at least ne eigen alues is e ual t e . ith , e uati n 10 is e uce t the elati n 11 : n 1 yn mx m This elati n c nstitutes the c n iti n which is t e satis ie y c inates e uili iu p int at a sa le-n e i u cati n. It c e w itten as a uncti n 12 : IT COMMON... n 1 1 m 1 n xm m ula 12 all ws us t calculate y a it a y ch sen alue x. Ne t, th x y c e use t calculate the pa a ete s A B acc ing t e uati ns 4 . In such a way, we c in at what alues A B the sa len e i u cati n takes place. A B nx 1 n 1 n xm 1 m xm n 1 , xm 1 1 n 2 1 n In 12 m xm e t get physically eing ul, p siti e, alues y, A B 1 , we sh ul c n ine u ch ice m, n x alues t : 0 xm n 1 , m 1 n m n 1 1. e c n w aw a pa a et ic pl t acc ing t 1 with the pa a ete x y pa ticula pai ill c e icients. The pl t w ul i i e the ple int a eas with ne th ee e uili iu p ints. It ll ws 14 that the e is n i u cati n with =0 at n=1 n n e alues a ia les. It will e use ul t int uce e c ple pa a ete s n , 1 yn In te s the pa a ete s sa le-n e i u cati n 11 s 16 1 : mx m . , cha acte istic e uati n 10 c n iti n tain s ewhat si ple e gene al ki ki ki ki ki we e , the new pa a ete s ha e a s all isa tage. A pai alues cha acte i es ne speci ic p int e uili iu , n t the wh le syste . Each pa ticula syste is ep esente in the ple y a nu e p ints e ual t the nu e e uili iu p ints. In y case the e is a p int 0, n c esp n ing t the p int e uili iu in the igin the phase c inates. unsta le sa le p int c e un at p int 1 . The thi e uili iu c e un at 1 . Aut scillati ns a un this p int c e gene ate a te a p ssi le p i u cati n. 4. OP BI URCATION D OSCILLATIONS IN T E S STEM 4.1. T E CASE O T E I EST S MMETR IT ALL KI=1 D Q=2P e c ul in c n iti ns p bi u cati n in the alytical elati ns nly in s e special cases. Let us c nsi e the syste satis ying tw essential li itati ns. Rate c nstts ecay ki all the a iables in the syste a e e ual t each the . The ep ess 's l p c nsists the sa e nu be substces as the acti at 's l p . The assu pti n that all ecay c nstts a e e ual t 1 will ake u calculati ns e si ple but it will n t ake the less gene al. Un e such assu pti ns the cha acte istic e uati n 16 a pts the si ple s l able 18 . The e a e p 19 ts -1. The e aining p eigen alues satis y the e uati n R, m R n. 19 In the case The ine uality in 19 ll ws ts the e uati n 19 R 1 all e initi ns Rp ha e negati e eal pa ts especti e p int e uilib iu is stable. At R=1, ne eal eigen alue ishes a sa le- n e bi u cati n takes place. ith R=1, the sec n pa t 19 , lea s t the elati n, which we ha e al ea y btaine in the ully gene al case 1 . IT COMMON... In the case negati e R, e uati n 19 c be gi en the The e uati n has R ts: i sin m. Rp 2j R p sin 2j j 0,1,..., p 1. Eigen alues with j=0 j=p-1 ha e the highest eal pa ts. The e uilib iu will be estabili e when the eal pa t these tw eigen alues bec es p siti e. It takes place at The case e uality in 2 gi es a elati n between the c e uilib iu p ints at p bi u cati n. inates np yn 1 m m n p 1 x p 1 . In e t btain eing ul alues y, pa a ete s inate x sh ul satis y the ll wing li itati ns: np x p 1 the syste the m n 1 mp 1 . p 1 m n p Let us n te that the p bi u cati n is p ssible when i e ence between ill c e icients ep essi n acti ati n e cee s a ce tain alue sh wn in the i st pa t 25 . The sa e alue b un e ill c e icient at p bi u cati n in a l p with ne ep esse gen In e ni i T eu 1991 , as well as a p uct ill c e icients all c pe ati e p cess in a single l p c ntaining y ep esse acti ate genes Sielewiesiuk paciuk 2012 . The elati n 24 is satis ie in e uilib iu p ints un e g ing a p bi u cati n. Using this elati n c n iti ns e uilib iu 2 we c e p ess pa a ete s as uncti ns x - ep ess 's c ncent ati n in e uilib iu 26 . nx 1 x m A np p 1 m p p p n p p n B np p 1 m n p p 1 n 1 n mp E uati ns 26 esc ibe a pa a et ic cu e which sepa ates in the ple the a ea with p ssible scillat y s luti ns the a ea whe e scillati n a e i p ssible. B 15 12.5 10 7.5 5 2.5 Aut scillati ns Bistable t igge One stable e uilib iu 0,0 FIG. 1. Pa a ete ple the syste with p, q, m, n = 10, 20, 4, 2 . L we cu e c esp n s t the sa le-n e bi u cati n, the uppe ne ­ t p bi u cati n. IT COMMON... B 10 8 6 4 2 One unstable 0,0 ne stable e uilib iu One stable e uilib iu 0,0 Tw unstable e uilib iu Aut scillati ns. p ints. FIG. 2. Pa a ete ple the syste with p, q, m, n = 10, 20, 4, 1 . L we cu e c esp n s t the t sc itical bi u cati n, the uppe ne ­ t p bi u cati n. Syste s with a c pe ati e n>1, ig. 1 n n-c pe ati e n=1, ig. 2 acti ati n ha e s e ualitati e si ila ities i e ences. Syste s b th the kin s ha e a p int e uilib iu at the igin c inates at y alues A B. Tw the , n n e , p ints e uilib iu appea th ugh sa le-n e bi u cati n in syste s with n>1, ne a iti nal e uilib iu appea s th ugh t sc itical bi u cati n when n=1. The e uilib iu at the igin is stable at all alues pa a ete s with n>1, but it bec es unstable B>1 with n=1. p bi u cati n takes place in b th the kin s acti ati n at su iciently high alues the pa a ete s A B. 4.2. REPRESSION D ACTIVATION LOOPS IT DI ERENT NUMBERS O ELEMENTS Let us c nsi e s e syste s with slightly l we sy et y, whe e the tw l ps c nsist i e ent nu be s p tein t s ati ns. As in secti n 4.1, we c ntinue t use all the ecay ate c nstts e ual t ne the cha acte istic e uati n 16 in the : In e t in alues the pa a ete s at p bi u cati n, we supp se a pu ely i agina y eigen alue i int uce a new c ple a iable z 1 z i sin with t 1 in 2 The substituti n the a iable z esults in e uati ns 29 : p sin p q sin q p p q sin q bi u cati nal alues the pa a ete s sin p sin 2 p sin q p q sin 2 p These elati ns all we us t c nst uct pa a et ic pl ts using as a pa a ete . Sets such pl ts a e sh wn in ig. in ig. 4 . In b th the igu es, the st aight line 1 c esp n s t p ints the sa le n e bi u cati n. At least ne eigen alue these p ints e uilib iu is e ual t e in all syste s un e c nsi e ati n. The e is at least ne eal p siti e eigen alue at 1 . The e aining cu es c esp n t e uilib iu p ints ha ing a pai c ple eigen alues with e eal pa t. Oscillati ns 14 FIG. 3. Cu es p bi u cati n in syste s ha ing ep essi n l p c nsisting p=10 eagents. Acti ati n l ps c ntain qeagents with q=14, 15, 16, 1 , 18, 19, 20, in the e the le t t the ight. Dashe line c esp n s t sa le-n e bi u cati n. IT COMMON... Nu e ical calculati ns we e ne the nu be eagents in the ep essi n l p p=10 that in the acti ati n l p is q-p. In the cases q-p= 4, 5, 6, , 8 9, the cu es en n the st aight line 1 with 0. As it was sh wn ea lie 2 , the st aight line p 1.651 2 c esp n s t bi u cati n in the st sy et ical syste with b th l ps e ual si es p=10 q-p=10 . Oscillati ns a e p ssible a un the e uilib iu p ints with +1 bel nging t the a ea bel w sa le-n e line t the ight the cu es p bi u cati n. Let us e in that y pa ticula syste has in the ple accessible a ea with m n . It is p ssible, at high n, that the cu e p bi u cati n es n t i i e the accessible ectgle m n int tw sepa ate pa ts. e a ple such situati n c be seen in ig. =14 15 n=2, whe e the e a e n stable scillati ns in spite e isting c ple eigen alues with p siti e eal pa t. The syste g es t the stable e uilib iu with e alues all a iables. It appea s that scillati ns a e i p ssible, when in ucti e l p is t sh t. FIG. 4. Cu es p bi u cati n in syste s ha ing ep essi n l p c nsisting p=10 eagents. Acti ati n l ps c ntain q-p eagents with q=25, 24, 2 , 22, 21, 20. Again, the st aight line c esp n s t sa le-n e bi u cati n. On the c nt a y, l ng l ps acti ati n n t supp ess scillati ns. e p esent in ig. 5 alues the pe i scillati ns btaine nu e ical s luti ns. The scillati ns ha e the highest e uency when the l p acti ati n is slightly l nge th the l p ep essi n . The pe i app aches a c nstt alue with inc easing nu be ele ents in the l p acti ati n. At q p. These c inates bi u cati nal cu es in igs intege ultiple p: e e t the p int which is c n all the 4. In cases l ng acti ati n l ps, when q is jp, j 2, ,... the sec n ne the tw e uati ns 29 is satis ie with y alue Substituti n these alues q int the i st e uati n 29 gi es elati n which sh ul be satis ie at p bi u cati n 2 j p IT COMMON... Relati n 2 is illust ate in ig. 6. The st aight line with ep uces e actly the c n iti n p bi u cati n in the st sy et ical syste s 2 . The biggest pa t FIG. 6. p bi u cati n cu es in syste s whe e q is intege ultiple The alues j a e sh wn at especti e st aight tlines. D tte line c esp n s t the sa le-n e bi u cati n the ple , c esp n ing t aut scillati ns, ha e the syste s with q=3 when the acti ati n l p is twice as l ng as the l p ep essi n. In the li it e y high alues j at e y l ng l ps acti ati n the uncti n 2 g es t the st aight line with in inite sl pe the e istence scillat y s luti ns es n t e epen n the pa a ete Relati ns btaine in the secti n 4.2. enable us t in e uilib iu p ints with a pai pu e i agina y eigen alues. In p bi u cati n, the eal pa t this pai c ple eigen alues sh ul chge its sign. e use nu e ical calculati ns checke in se e al cases that the chge sign es eally take place. Un tunately, we ha e n t un y clea p this chge in general case. In the case o , relations 22 e pression or the bi urcating pair o eigen alues: 23 i ply alytical sin . p In this case, the chge o the sign o at crossing bi urcational alue o - cur e 2 in Fig. 6 is e i ent. The characteristic e uation 27 has a relati ely si ple alytical solution also in the case o q=3p. The bi urcating pair o eigen alues is then gi en by 35 : sin . p The real part o 35 is icreasing unction o ishes at satis ying relation 32 with j=3 cur e 3 in Fig. 6 . So, it ust chge its sign by crossing this cur e. 4.3. REPRESSION D ACTIVATION LOOPS IT DIFFERENT RATE CONSTTS OF DECA Let us now consi er the syste whose both loops consist o the sa e nu ber o ele ents , but they i er in the rate o ecay o their reagents. e assu e that the constts ki in the repression loop i=1,..,p ha e the unit alue those in the acti ation loop are e ual to k. In such a case the characteristic e uation 16 c be written as: kp 1 Intro ucing into e uation 36 allows us to e press 2 2 para eters as unctions o . Para etric plots show the alues o , which correspon to op bi urcation. The plots are presente in Figs 7 8. Oscillatory solutions are possible in areas below the straight line alues o higher th those in bi urcation cur es. All o the consi ere asy etrical syste s ha e the loop o repression consiste o 10 reagents ecaying with the rate constt e ual to 1. Their acti ation loops i er one ro other by nu bers o ele ents in the loops o acti ation or by the rate constts o ecay. There is ob ious ualitati e si ilarity between the syste s with elongate acti ation loops the syste s with slower ecay o reagents in these loops Figs 4 7 . Finite i aginary POSTITIVE D NE ATIVE FEEDBACK LOOPS IT COMMON... eigen alues appear on both si es o the straight line In spite o this act, oscillations aroun e uilibriu points with are i possible. All e uilibriu points ro this part o the ple ha e one real positi e eigen alue. E olution o the syste takes it away ro the icinity o such e uilibriu . Let us note that in both cases the turno er o acti ator is slower th that o the repressor. FIG. 7. Cur es or op bi urcation in the syste s with e ual nu ber o ele ents in both loops. Rates o ecay constts are e ual to unity in the loop o repression. The alues o ecay rate constts in the loop o acti ation are shown at cur es. FIG. 8. Cur es or op bi urcation in the syste s with e ual nu ber o ele ents in both loops. Rate o ecay constts are e ual to unity in the loop o repression. The cur es correspon s to ecay rate constts in the loop o acti ation k=2, 1.8, 1.7, 1.6, 1.5, 1.4, 1.3, 1.2, 1.1. On the other h , bi urcational cur es o the syste s with shortene loops o acti ation are uch si ilar to those o the syste s with higher rate constts o ecay in these loops Figs 3 8 . The latter syste s ha e pure i aginary eigen alues only at At +1, i aginary eigen alue attains ero. Possible oscillations aroun the e uilibriu points with slightly lower th shoul ha e e tre ely low re uency. In this pair o asy etrical cases, acti ator's loop has shorter ti e o turno er th the repressor's loop. All bi urcation cur es in Figs 3, 4, 7 8 represent e uilibriu points with a pure i aginary eigen alue They ha e one co 1.65172 at p=10, =0. It c be in erre ro on point with relations 22 t p 23 that in this bi urcation point i aginary eigen alue oscillations with a perio T we c e pect 2 2 p . Nu erical solutions suggest that t / p this e aluation o the perio is better th those base on the actual i aginary part o co ple eigen alue. The turno er ti e o the repression loop is the ain actor eter ining the perio o oscillations. The acti ation loop with short turno er ti e e clu es oscillations or akes the slower. The slower acti ation loop has no signi ict in luence on the perio o oscillations. These rules are illustrate in Figs 5 9. Period of oscillations T Decay constts k in the loop of activator FIG. 9. Perio o oscillations obtaine ro nu erical solutions in syste s with 10 ele ents in both loops, ki=n the loop o repression, ki=k in the loop o acti ation, POSTITIVE D NE ATIVE FEEDBACK LOOPS IT COMMON... 5. CONCLUSIONS e consi ere a o el syste o gene e pression regulate by two trscription actors, repressor acti ator. e assu e that trscription o the gene takes place pro i e that the concentration o the repressor is low , at the sa e ti e, the concentration o the acti ator is high. Repression acti ation are both cooperati e processes with respecti e ill coe icients m n. igh alues o m pro otes oscillations high alues o n ake the less probable. Oscillatory solutions are possible when i erence m-n is su iciently high, see relation 25 . e intro uce para eters , unctions o coor inates o e uilibriu points, which enable us to eri e the characteristic e uation the sa le-no e bi urcation in a uite general way 16,17 . The op bi urcation was aly e in a ew special cases. Oscillations are generate in our syste by a negati e ee back loop o the oo win's type. A couple loop with positi e ee back oes not isturb oscillations when its turno er ti e is longer th the turno er ti e o the negati e ee back loop. Oscillations are slowe own or e en ully a pe , when the loop o acti ation has the turno er ti e shorter th that o the repression loop.

Journal

Annales UMCS, Physica – de Gruyter

Published: Mar 1, 2015

References

  • Circadian oscillators in eukaryotes and
    Jolma
  • Splicing of the period gene terminal intron is regulated by light circadian clock factors and phospholipase C
    Majercak
  • The contribution of the interlocking loops and extensive nonlinearity to the properties of circadian clock models ONE
    Saithong
  • Molecular bases for circadian clocks
    Dunlap
  • References Network motifs : theory and experimental approaches
    Alon
  • Clocks metabolism and the epigenome
    Feng
  • Synexpression groups in eukaryotes
    Niehrs
  • reu Quantitative analysis of the Hopf bifurcation in the Goodwin n - dimensional metabolic control system
    Invernizzi
  • Regulation of gene expression by Goodwin s loop with many genes Annales Universitatis Mariae Curie - Skłodowska sectio AAA
    Sielewiesiuk
  • lamm chuster A generalized model of the repressilator
    Müller
  • Oscillatory behavior in enzymatic control processes in Enzyme Regulation
    Goodwin
  • öster Spotlight on post - transcriptional control in the circadian system
    Steiger