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/lp/de-gruyter/a-cycle-of-enzymatic-reactions-with-some-properties-of-neuronal-ibebnQf0uF
- Publisher
- de Gruyter
- Copyright
- Copyright © 2015 by the
- ISSN
- 2300-7559
- eISSN
- 2300-7559
- DOI
- 10.1515/physica-2015-0005
- Publisher site
- See Article on Publisher Site
Abstract
A le u e la i n an u e e la i n ea i ns wi e essi n all s e i in ibi i n en es is nsi e e T e es n in na i al s s e is a a e ise b w a a e e s e su ea en n en a i ns C an e ai a e ns an s wa an ba kwa ea i ns k In a s e i al ase k=1 e s se as a uni ue e uilib iu A C>4 e e uilib iu is uns able an es se as s illa s lu i ns A k essen iall i e en 1, e s s e be es e i able be a es as a bis able i e Keywords: e a i a i , Rhodococcus erythropolis, s illa i ns INTR UCTI N Ba e ia Rhodococcus erythropolis an u ilise e a i a i as a a b n s u e 1, 2, 3 i s , e a i a i un e es w e e la i ns a e ui a i , w i an be e a b la e ka e l T e w la e subs ances can en e ea a i a i n pa hways. A he ce ain ch ice e a ic aci c ncen a i n an cell ensi y, scilla i ns e h yphen lic c p un s can be bse e in bac e ial cul u e 3, 4 . In e e plain bse e scilla i ns we ha e p p se a kine ic el a cycle u e hyla i n- e e hyla i n eac i ns 5, 6 . As a yna ical sys e , his el ha s e anal ies neu ne scilla p p se by DuninBa k sky 7 . The el was hi hly asy e ical because he w ll win assu p i ns. 1. e ea e e a ic aci as a ese i subs ance an i s c ncen a i ns as a pa a e e he sys e . 2. The a i he a e c ns an s e hyla i n an e e hyla i n eac i ns c ul be essen ially i e en uni y. I ha ne h ee e uilib iu p in s epen in n pa a e e s alues. hen he e was ne e uilib iu p in , he sys e ela e he s able e uilib iu scilla e a un he uns able e uilib iu . A h ee e uilib iu p in s he sys e beha e as a bis able i e as an e ci able sys e . M s hese ea u es a e c nse e als in a e sy e ical sys e . In his pape I c nsi e a el in which he nly s u ce asy e y ae essen ially i e en a e c ns an s wa an backwa eac i ns. C nsi e a i n his kin el appea e be necessa y an analysis synch nisa i n he p cesses akin place in i e en bac e ial cells in a cul u e. ORMULATION AND ENERAL RO ERTIES O T E MODEL The basic s uc u e he sys e i . 1 is he sa e as ha p esen e in p e i us pape 5 . Ve a ic aci i . 1b a e he cul u e bac e ia Rhodococcus erythropolis is e e hyla e w i es i ep ca echuic aci i 1c . C n e si ns a x an z y in i . 1a c esp n e e hyla i n- e hyla i n eac i ns a he p si i n 4. In a si ila way, c n e si ns x y an a z c esp n he sa e eac i ns a he p si i n 3. A C CLE O EN MATIC REACTIONS a a 1 z2 1 z2 x x kx kx 1 x 22 1 x ky ky 1 y2 2 xx 1 1 a 2a 2 COO COO COO COO 1 y 1 y2 1 kz 2 1 kzy a a 1 a2 OC OC 1 z2 1 z ky ky 1 x2 OC OC x2 b) a) c) a) b) c) ws esc ibe he a es es anilic aci an y FIG. 1. a) The cycle eac i ns. E p essi ns ne he a espec i e eac i ns, a ep esen s e a ic aci b), x an z is p ca echuic aci c). A he a ws in i . 1a he e a e i en e p essi ns ela in he a es c esp n in eac i ns he c ncen a i ns ea en s. These ela i ns a e base n he ll win assu p i ns: 1. Ve a ic aci a) ac s as a c ep ess he 3-O- e e hylase. 2. Vanilic aci x) ac s as a c ep ess he 4- e hylase. 3. ca echuic aci y) ac s as a c ep ess he 3- e hylase. 4. Is anilic aci ac s as a c ep ess he 4-O- e e hylase. ll win he au h s 8-13], we use he e p essi n 1/(1+rm) wi h m 2 esc ibe he in luence he c ep ess c ncen a i n n he c ncen a i n he c esp n in en y e. M e e aile iscussi n his ues i n ha e been p esen e ea lie 5]. ine ic ela i ns sh wn in i . 1a can al e na i ely esc ibe all s e ic inhibi i n en y es ins ea hei ep essi n. The sche e in i .1a can be als ela e e e sible subs i u i n eac i ns in w i e en p si i ns in s e he lecules, n necessa y in e a ic aci . Un e hese assu p i ns, he e lu i n he sys e can be esc ibe by he ll win se ina y i e en ial e ua i ns: da a kx kz a 2 2 2 dt 1 a 1 x 1 y 1 z 2 , 1) dx x kx ky a 2 2 2 dt 1 a 1 x 1 y 1 z 2 , dy x ky ky z 2 2 2 dt 1 a 1 x 1 y 1 z 2 , dz a ky kz z . 2 2 2 dt 1 a 1 x 1 y 1 z 2 E ua i ns 1-4) esc ibe he 5]. This i e, h we e , we ea a iable an n as a c ns an p si i ely in a ian an sa is ies b h p pe ies i en in he ea e ua i ns 1-4) ha 2) 3) 4) sa e se eac i ns as ha analyse ea lie e a ic aci c ncen a i n a) as a yna ical pa a e e . The yna ical sys e 1-4) is he ic e uilib iu p s ula e. s he lie pape 5] e ain ali . I ll ws d a x y z) 0 . dt S , he e is an in e al i n 5) const. 6) I eans ha he su c ncen a i ns all ea en s is c nse e . In ac , he sys e c ul be ea e as he sys e he e h ee. One yna ical a iable c ul be eli ina e usin he c nse a i n law 6). we e , hi hly sy e ical shape he e ua i ns 1-4) is e c n enien calcula i n han he c esp n in sys e he hi e . The sys e 1-4) has ne use ul p pe y. The subs i u i n: a y, x z, y in e ua i ns 1-4) an a, z x, k 1/k in he ela i n: 8) 7) escalin he i e acc t' t k esul s in he se e ua i ns which is i en ical wi h he s a in ne. S , a he sa e alue C, he sys e s wi h ela i e a e c ns an s k an 1/k ha e he sa e nu be e uilib iu p in s wi h he sa e s abili y p pe ies. E en e, he ll win ela i ns be ween s lu i ns b h sys e s ake place: A C CLE O EN MATIC REACTIONS 9) a1(t)=y2(t/k), x1(t)=z2(t/k), y1(t)=a2(t/k), z1(t)=x2(t/k), whe e subin e 1 an 2 e e s lu i ns wi h ela i e a e c ns an s k an 1/k espec i ely. O c u se, ela i ns 9) will be ali i ini ial c n i i ns sa is y ela i ns 7). E lu i n he sys e epen s n he alues pa a e e k an in e al i n C. In i . 2 he e a e sh wn w a eas A an B, a ke wi h uble h i n al lines, whe e he sys e has h ee e uilib iu p in s. In he a ea A he e a e w s able e uilib iu p in s sepa a e by he hi uns able e uilib iu sa le p in ). A k an C bel n in his a ea, he sys e beha es as a bis able i e . In he a ea B he sys e has he ll win e uilib iu p in s: s able n e, sa le-p in an uns able cus. A alues k an C he a ea B, he sys e beha es as an e ci able sys e . A A q q B B Relative rate constant (k) Relative rate constant (k) pp 40 80 Integral of motion (C) Integral of motion (C) FIG. 2. a a e e plane he sys e . The sys e has w s able an ne uns able e uilib iu p in s in he a ea A, ne s able an w uns able e uilib iu p in s h ee uns able e uilib iu p in s in he a ea B. Bey n he a eas A an B he sys e has ne e uilib iu . Cu e p sepa a es a eas wi h ne s able e uilib iu p in ha wi h n s able e uilib iu p in . Cu e sepa a es he a ea B wi h h ee e uilib iu p in s ha wi h sin le e uilib iu . The cu es a e base n nu e ically calcula e e uilib iu p in s usin Ma he a ica wi h achine p ecisi n 16 i i s) an p in in si si ni ican eci al i i s. e e ails see e . An e e plary phase portrait o this kin is shown in Fig. 6. There is a ery narrow part o the area B at its right-han e ge, where all three e uilibriu points are unstable. In this case, the autooscillations appear in the syste . Beyon the areas A an B the syste has one e uilibriu point. The single e uilibriu is stable or alues o C s aller than those alling in the cur e p in Fig. 2, an is unstable or the higher alues o C. All the entione areas, shown or k>1 in Fig. 2, appear also at k<1. Correspon ing cur es i i ing the para eter plane at k 1 are sy etrical on the logarith ic scale o k to those shown in Fig. 2 in respect to the straight line k 1. The bor ers o the areas shown in Fig. 2. ha e been eter ine nu erically with the precision o si signi icant eci al igits. EVOLUTION OF T E SYSTEM AUTOOSCILLATIONS Autooscillations appear in the syste at the alues o C higher than those correspon ing to the cur e p in Fig. 2. Insi e the area B the cur e p goes slightly abo e the cur e q. So, there is a narrow strip o the area B between p an q where the syste has three unstable e uilibriu points. For para eter alues belonging to this strip, the syste has oscillatory solutions with all three e uilibriu points place insi e the correspon ing li it cycle. Analytical linear analysis o stability is possible only in the ully sy etrical case with k=1. In this case the syste has an uni ue e uilibriu a x y z C . 4 10) point 10) are Eigen alues o the syste in the icinity o the e uilibriu gi en by the e pressions 11): 32 C 2 16 16 C 1,2 i 2C 2 . 4 , 1 a2 11) The ourth eigen alue is e ual to ero. Respecti e nor al ariables can be e ine by the trans or ation 12). Because o the conser ation law 6) the ariable 4 has a constant alue C/2, an the e olution o the syste can be represente in a 3- i ensional space 1 , 2 , 3 . 0 1 2 1 2 1 2 a x . y z 12) Nu erical solutions a(t) at k=1 an C 5 or 50 are shown in Fig. 3. Solutions x(t), y(t) an z(t) ha e the sa e shape as a(t) but are elaye in respect to a(t) by one, two an three uarters o the perio respecti ely. The perio o oscillations increases with growing alues o C see Fig. 4). At alues o k essentially i erent ro the unity, the shapes o oscillations o ariables a, x, y an z beca e i erent ro each other. As an e a ple, solutions a(t), x(t), y(t) an z(t) at k 40 an initial con itions a(0) 80, x(0)=y(0)=z(0)=0 are shown in Fig. 5. Accor ing to the substitution 9), solutions or k 1/40 0.025 an initial con itions y(0) 80, a(0)=x(0)=z(0) 0 can be obtaine ro Fig. 5. It is enough to change the ti e scale by substitution 2000 instea o 50 or the highest alue o ti e an rea Fig. 5a as y(t), 5b as z(t), 5c as a(t) an 5 as x(t). Depen ence o the oscillation perio on the su o reagent concentrations or a ew alues o k is shown in Fig. 4. The cur es presente in Fig. 4 were obtaine on the basis o nu erical solutions o e uations 1-4). Let us note that at high alues o k the perio o oscillations re ains al ost constant in a uite wi e range o C. For e a ple, at k 20 the perio changes its alue ro 9.60 at C 40 to 10.16 at C 100. So, at such alues o k the syste can work as a pace aker or so e rhyth s. EVOLUTION OF T E SYSTEM AT MULTIPLE EQUILIBRIUM POINTS At su iciently high alues o k the syste can ha e three e uilibriu points areas A an B in Fig. 2). In the area A, at k>8.2068, two o these e uilibriu points are stable an the thir one is unstable. Let us consi er, as an e a ple, the syste with k 40. At this alue o k, the area A inclu es the range o C ro 12.4932 to 21.0277. The alues o the ariables in e uilibriu an correspon ing eigen alues or k 40 an C 16.6 are gi en in Table 1. The yna ical ariables in the unstable e uilibriu point 2 in Table 1) ha e inter e iate alues in respect to those escribing the two stable e uilibriu points. Thus, in the state space o the syste , the unstable e uilibriu point is situate so ewhere between the stable e uilibriu points. Depen ing on initial con itions, the syste goes to the e uilibriu 1 with relati ely low alue o x or to the e uilibriu 3 with a high alue o x. The e uilibriu points 2 an 3 beco e closer an closer to each other when the alue o C ecreases ro 16.6 to 12.5. At the sa e ti e the o ain o attraction o the e uilibriu 1 is growing. E entually, at C 12.4932, the e uilibriu points 2 an 3 annihilate. In contrast, increasing alue o C ro 16.6 to 21 brings the e uilibriu points 1 an 2 closer an closer to each other. These e uilibriu points annihilate at C 21.0277. I i not e plore in etail the hypersur ace separating attraction o ains o the e uilibriu points 1 an 3. Ne ertheless, it is intuiti ely clear that the attraction o ain o the e uilibriu with a low alue o x point 1 in Table 1) is bigger at the le t-han e ge o the area A Fig. 2) than at the righthan e ge o this area. In the whole area A in all o the three e uilibriu points, the alues o y an z constitute only a inute part o the pool o all reagents C). Fully analogical iscussion can be applie to the area with k 1/8.2068 0.0840336 with respecti e changes in the roles o ariables an scale o ti e, accor ing to the substitutions 7) an 8). There is another area B in Fig. 2) with three e uilibriu points. This area appears at k>11.9 or at k 1/11.9 0.0840336. (a) (b) FIG. 3. Oscillations in a ully sy C=5 a) or C 50 b). etrical syste . Nu erical solutions a(t) at k 1 an Period 4 0 20 40 60 Integral of motion (C) 80 100 FIG. 4. Depen ence o the perio o oscillations on the su o reagent concentrations C). Values o the relati e rate constant k are shown at the respecti e cur es. Perio s were obtaine ro nu erical solutions o e uations 1-4), using Mathe atica . Let us return to the e a ple with k=40. For this alue o k an C belonging to the inter al 21.0277, 34.1035), the syste has a single stable e uilibriu characterise by three real an negati e eigen alues. As soon as C e cee s the alue o 34.1035, two a itional e uilibriu points appear. Both o the are unstable. The syste has three e uilibriu points or 34.1035 C 42.0551. Table 2 presents an e a ple o such three e uilibria or C=38. As can be seen ro Table 2, the sa le-point 2) is locate between the stable no e 1) an unstable ocus 3). In any case, the e olution o the syste lea s to e uilibriu 1. owe er, orbits attaining e uilibriu 1 can be essentially i erent or i erent initial con itions. In Fig. 6, there are shown pro ections o the two orbits on the plane a,x) or the alues o k=40 an C=38 as those use in Table 2. In the case o the orbit starting ro the point A initial con itions a(0), x(0), y(0), z(0)} = 1.134, 35.4, 1.408, 0.058}) changes in the alues o the ariables are ery s all. The syste rela es to e uilibriu 1 al ost onotonously. In contrast, on the orbit starting ro point B a(0), x(0), y(0), z(0)}= 1.66, 30.63, 5.33, 0.38}) the a plitu es o changes o the ariables are uch higher. Be ore attaining e uilibriu 1, the orbit goes aroun the unstable e uilibriu 3. So, we ha e to o with an e citable syste . E citation e ents are so ewhat i erent ro those escribe earlier 5]. eneration o positi e spikes o all ariables in the present syste is i possible because o the conser ation law 6). A ter start ro the point B, one can obser e i inishing alue o x, which reaches its ini u o 21.74 at ti e 3.6. In e pense o x, spikes o the re aining ariables are generate . The a i u alues appear in the ollowing se uence: y 2.97)=12.35, z 4.31)=1.72 an a 5.61)=4.92. E en ini u alue o x is essentially higher than a i u alues o y, z an a. (a) (b) (c) (d) FIG. 5. Oscillations in a highly asy C=80. etrical syste . Nu erical solutions at k=40 an points TABLE 1. Coor inates a,x,y,z) an respecti e eigen alues at k=40 an C=16.6 E E uilibriu no. uilibriu no. a x y z ) o the e uilibriu 1 1 16.0935 16.0935 0.504903 0.504903 4.85482 1010-5 4.85482 -5 1.54745 1010-3 1.54745 -3 -72.8782 -72.8782 -39.996 -39.996 -19.9246 -19.9246 2 2 14.1969 14.1969 2.40104 2.40104 2.96349 1010-4 2.96349 -4 1.75225 1010-3 1.75225 -3 -46.9177 -46.9177 -39.9947 -39.9947 3.16435 3.16435 3 3 2.9172 2.9172 13.6393 13.6393 3.59011 1010-2 3.59011 -2 7.67861 1010-3 7.67861 -3 -41.2675 -41.2675 -39.7528 -39.7528 -0.798068 -0.798068 a x y z TABLE 2. Coor inates a,x,y,z) an respecti e eigen alues at k=40 an C=38 E a x y z E uilibriu no. 1 1 uilibriu no. 2 2 1.16751 1.16751 1.09956 1.09956 a 34.4423 34.4423 36.3598 36.3598 x 2.31183 2.31183 0.524838 0.524838 y 0.0783655 0.0783655 0.0158716 0.0158716 z -7.75538 -7.75538 -32.844 -32.844 1 1 -0.731001 -0.731001 -17.7117 -17.7117 2 2 4.1651 4.1651 -1.02994 -1.02994 3 3 ) o the e uilibriu points 3 3 2.16193 2.16193 26.8251 26.8251 8.34074 8.34074 0.67221 0.67221 -1.48735 -1.48735 0.230103 0.829406i 0.230103 0.829406i 0.230103-0.829406i 0.230103-0.829406i The ariable x constitutes also a o inating raction o the whole reagent pool in all e uilibriu points see Table 2). So, in the e citable regi e area B in Fig. 2), the ariable x beha es like a reser oir substance. Coor inates o the sa le-point 2 eter ine the e citation threshol . The alue o C=38 use in Table 2 an Fig. 6 correspon s to the i le o the area B in Fig. 2 at k=40. At the le t-han e ge o this area, at C slightly higher than 34.1035, the sa lepoint 2 an unstable ocus 3 are ery close each to other an both are re ote ro stable no e 1. In such a situation the e citation threshol is relati ely high an a plitu es o generate spikes o the ariables are relati ely low. ith the growing alue o C e uilibriu points 1 an 2 co e closer an closer to each other. The e citation threshol beco es lower an lower. At C=42.0548, still insi e the area B in Fig. 2, estabilisation o the e uilibriu 1 takes place. So, in the inter al 42.0548 C 42.0551 the syste has three unstable e uilibriu points an oscillati e solutions. At C=42.0551 the e uilibriu points 1 an 2 annihilate. Again, analogical iscussion can be applie to k=1/40=0.025 with substitution 7). All processes will then be 40 ti es slower accor ing to ti e rescaling 8). 1 36 A 2 32 B 3 a FIG. 6. E citable syste . Pro ections o the two orbits on the plane a,x) at k=40 an C=38. Coor inates o stable no e 1), sa le-point 2) an unstable ocus 3) are gi en in Table 2. Orbit starting ro the point A 1.134, 35.4, 1.408, 0.058) correspon s to subthreshol rela ation. Orbit starting ro the point B 1.66, 30.63, 5.33, 0.38) correspon s to e citation. Orbits obtaine ro nu erical solutions o the e uations 1-4) using Mathe atica . CONCLUSIONS The yna ical syste 1-4) shares any o es o e olution with one analyse earlier 5]. Pro i ing a proper choice o para eter alues, both syste s can rela to an uni ue stable e uilibriu or beha e as bistable triggers. Both syste s can ha e autooscillatory solutions an can beha e as e citable syste s. But in contrast to the earlier presente syste with one reser oir substance, the syste 1-4) can not beha e as a trans ucer o sti ulus strength to re uency. Both syste s can escribe etabolic oscillations in a single bacterial cell. Such oscillations will appear in a bacterial culture i oscillations in i erent cells are synchronous. I suppose that both consi ere syste s will be use ul in searching a ourable con itions or synchronisation.
Journal
Annales UMCS, Physica – de Gruyter
Published: Mar 1, 2015