A cycle of enzymatic reactions with some properties of neuronal circuits

A N N A L E S

U N I V E R S I T A T I S M A R I A E C U R I E - S . à 2 ' 2 : S . A

L U B L I N – 3 2 L 2 N I A

V2L L;I; SECTI2 AAA 2014

A C<CLE 2) EN=<MATIC REACTI2NS :IT+ S2ME

3R23ERTIES 2) NEUR2NAL CIRCUITS

Jan Sielewiesiuk

'eSaUWPenW RI BiRSK\siFs, InsWiWuWe RI 3K\siFs, MaUia CuUie-SkáRGRwska UniYeUsiW\, Sl M CuUie-SkáRGRwskieM 1, 20-031 Lublin,

e-Pail sielewiesiuk#SRF]WauPFslublinSl

ABSTRACT

A F\Fle RI IRuU PeWK\laWiRn anG IRuU GePeWK\laWiRn UeaFWiRns wiWK UeSUessiRn RU allRsWeUiF inKibiWiRn RI en]\Pes is FRnsiGeUeG TKe FRUUesSRnGinJ G\naPiFal s\sWeP is FKaUaFWeUiseG b\ WwR SaUaPeWeUs WKe suP RI UeaJenW FRnFenWUaWiRns C anG WKe UaWiR RI UaWe FRnsWanWs RI IRUwaUG anG baFkwaUG UeaFWiRns k In a s\PPeWUiFal Fase k=1 WKe s\sWeP Kas a uniTue eTuilibUiuP AW C>4 WKe eTuilibUiuP is unsWable anG WKe s\sWeP Kas RsFillaWRU\ sRluWiRns AW k essenWiall\ GiIIeUenW IURP 1, WKe s\sWeP beFRPes e[FiWable RU beKaYes as a bisWable WUiJJeU

Keywords: YeUaWUiF aFiG, Rhodococcus erythropolis, RsFillaWiRns

INTR2'UCTI2N

BaFWeUia Rhodococcus erythropolis Fan uWilise YeUaWUiF aFiG as a FaUbRn sRuUFe >1, 2, 3@ )iUsW, YeUaWUiF aFiG unGeUJRes WwR GePeWK\laWiRns WR SURWRFaWeFKuiF aFiG, wKiFK Fan be GeFaUbR[\laWeG WR kaWeFKRl TKe WwR laWWeU

subsWances can enWeU GeaURPaWi]aWiRn paWhways. AW Whe ceUWain chRice RI YeUaWUic aciG cRncenWUaWiRn anG cell GensiWy, RscillaWiRns RI PeWhR[yphenRlic cRPpRunGs can be RbseUYeG in bacWeUial culWuUe >3, 4@. In RUGeU WR e[plain RbseUYeG RscillaWiRns we haYe pURpRseG a kineWic PRGel RI a cycle RI IRuU PeWhylaWiRn-GePeWhylaWiRn UeacWiRns >5, 6@. As a GynaPical sysWeP, Whis PRGel haG sRPe analRJies WR neuURne RscillaWRU pURpRseG by Dunin-BaUkRYsky >7@. The PRGel was hiJhly asyPPeWUical because RI Whe WwR IRllRwinJ assuPpWiRns. 1.:e WUeaWeG YeUaWUic aciG as a UeseUYRiU subsWance anG iWs cRncenWUaWiRns as a paUaPeWeU RI Whe sysWeP. 2. The UaWiR RI Whe UaWe cRnsWanWs IRU PeWhylaWiRn anG GePeWhylaWiRn UeacWiRns cRulG be essenWially GiIIeUenW IURP uniWy. IW haG Rne RU WhUee eTuilibUiuP pRinWs GepenGinJ Rn paUaPeWeUs Yalues. :hen WheUe was Rne eTuilibUiuP pRinW, Whe sysWeP Uela[eG WR Whe sWable eTuilibUiuP RU RscillaWeG aURunG Whe unsWable eTuilibUiuP. AW WhUee eTuilibUiuP pRinWs Whe sysWeP behaYeG as a bisWable WUiJJeU RU as an e[ciWable sysWeP.

MRsW RI Whese IeaWuUes aUe cRnseUYeG alsR in a PRUe syPPeWUical sysWeP. In Whis papeU I cRnsiGeU a PRGel in which Whe Rnly sRuUce RI asyPPeWUy aUe essenWially GiIIeUenW UaWe cRnsWanWs RI IRUwaUG anG backwaUG UeacWiRns. CRnsiGeUaWiRn RI Whis kinG RI PRGel appeaUeG WR be necessaUy IRU an analysis RI synchURnisaWiRn RI Whe pURcesses WakinJ place in GiIIeUenW bacWeUial cells in a culWuUe.

)ORMULATION AND *ENERAL 3RO3ERTIES O) T+E MODEL

The basic sWUucWuUe RI Whe sysWeP )iJ. 1 is Whe saPe as WhaW pUesenWeG in pUeYiRus papeU >5@. VeUaWUic aciG )iJ. 1b aGGeG WR Whe culWuUe RI bacWeUia Rhodococcus erythropolis is GePeWhylaWeG WwR WiPes WR JiYe pURWRcaWechuic aciG )iJ 1c. CRnYeUsiRns aļx anG zļy in )iJ. 1a cRUUespRnG WR GePeWhylaWiRn-PeWhylaWiRn UeacWiRns aW Whe pRsiWiRn 4. In a siPilaU way, cRnYeUsiRns xļy anG aļz cRUUespRnG WR Whe saPe UeacWiRns aW Whe pRsiWiRn 3.

A C<CLE O) EN=<MATIC REACTIONS :IT+ SOME 3RO3ERTIES« 81

x

2

1 z

a



_{1 a}2 x 

₂ 1 x kx 

_{1 y}

2

ky



a y

₂ 1 a a 

_{1 z}

2

z



2

1 y

kz



_{1 x}2 ky 

z

a) b) c)

FIG. 1. a) The cycle RI UeacWiRns. E[pUessiRns ne[W WR Whe aUURws GescUibe Whe UaWes RI

UespecWiYe UeacWiRns, a UepUesenWs YeUaWUic aciG b), x anG z isRPeUs RI Yanilic aciG anG y pURWRcaWechuic aciG c).

AW Whe aUURws in )iJ. 1a WheUe aUe JiYen e[pUessiRns UelaWinJ Whe UaWes RI cRUUespRnGinJ UeacWiRns WR Whe cRncenWUaWiRns RI UeaJenWs. These UelaWiRns aUe baseG Rn Whe IRllRwinJ assuPpWiRns:

1. VeUaWUic aciG a) acWs as a cRUepUessRU RI Whe 3-O-GePeWhylase. 2. Vanilic aciG x) acWs as a cRUepUessRU RI Whe 4-PeWhylase.

3. 3URWRcaWechuic aciG y) acWs as a cRUepUessRU RI Whe 3-PeWhylase. 4. IsRYanilic aciG acWs as a cRUepUessRU RI Whe 4-O-GePeWhylase.

)RllRwinJ RWheU auWhRUs >8-13], we use Whe e[pUessiRn 1/(1+rm_{) wiWh m 2 WR}

GescUibe Whe inIluence RI Whe cRUepUessRU cRncenWUaWiRn Rn Whe cRncenWUaWiRn RI Whe cRUUespRnGinJ en]yPe. MRUe GeWaileG GiscussiRn RI Whis TuesWiRn haYe been pUesenWeG eaUlieU >5]. .ineWic UelaWiRns shRwn in )iJ. 1a can alWeUnaWiYely GescUibe allRsWeUic inhibiWiRn RI en]yPes insWeaG RI WheiU UepUessiRn. The schePe in )iJ.1a can be alsR UelaWeG WR UeYeUsible subsWiWuWiRn UeacWiRns in WwR GiIIeUenW pRsiWiRns in sRPe RWheU PRlecules, nRW necessaUy in YeUaWUic aciG.

UnGeU Whese assuPpWiRns, Whe eYRluWiRn RI Whe sysWeP can be GescUibeG by Whe IRllRwinJ seW RI RUGinaUy GiIIeUenWial eTuaWiRns:

2 2 2 2

1

1

1

1

da

a

kx

kz

a

dt

a

x

y

z



_

_

_









, 1) COO+

OC+

3

OC+

3 COO+

O+

O+

x

2 1 z a  _{1 a}2 x  ₂ 1 x kx  _{1 y}2 ky 

a y

₂ 1 a a  _{1 z}2 z  2 1 y kz  _{1 x}2 ky 

z

a) b) c) COO+ OC+3 OC+3 COO+ O+ O+

2 2 2 2

1

1

1

1

dx

x

kx

ky

a

dt

a

x

y

z



_

_

_









, 2) 2 2 2 2

1

1

1

1

dy

x

ky

ky

z

dt



a





x





y





z

, 3) 2 2 2 2. 1 1 1 1 dz a ky kz z dt a  x  y  z 4)

ETuaWiRns 1-4) GescUibe Whe saPe seW RI UeacWiRns as WhaW analyseG eaUlieU >5]. This WiPe, hRweYeU, we WUeaW YeUaWUic aciG cRncenWUaWiRn a) as a GynaPical YaUiable anG nRW as a cRnsWanW paUaPeWeU. The GynaPical sysWeP 1-4) is pRsiWiYely inYaUianW anG saWisIies Whe PicUReTuilibUiuP pRsWulaWe. 3URRIs RI Whe bRWh pURpeUWies JiYen in Whe eaUlieU papeU >5] UePain YaliG. IW IRllRws IURP eTuaWiRns 1-4) WhaW

 ) 0

d

a x y z

dt    . 5)

SR, WheUe is an inWeJUal RI PRWiRn

.

a

  

x

y

z

C

const

6)

IW Peans WhaW Whe suP RI cRncenWUaWiRns RI all UeaJenWs is cRnseUYeG. In IacW, Whe sysWeP cRulG be WUeaWeG as Whe sysWeP RI Whe RUGeU WhUee. One GynaPical YaUiable cRulG be eliPinaWeG usinJ Whe cRnseUYaWiRn law 6). +RweYeU, hiJhly syPPeWUical shape RI Whe eTuaWiRns 1-4) is PRUe cRnYenienW IRU calculaWiRn Whan Whe cRUUespRnGinJ sysWeP RI Whe WhiUG RUGeU. The sysWeP 1-4) has Rne useIul pURpeUWy. The subsWiWuWiRn:

aĺy, xĺz, yĺa, zĺx, kĺ1/k 7)

inWR eTuaWiRns 1-4) anG UescalinJ Whe WiPe accRUGinJ WR Whe UelaWiRn: ' t

t

k 8)

UesulWs in Whe seW RI eTuaWiRns which is iGenWical wiWh Whe sWaUWinJ Rne. SR, aW Whe saPe Yalue RI C, Whe sysWePs wiWh UelaWiYe UaWe cRnsWanWs k anG 1/k haYe Whe saPe nuPbeU RI eTuilibUiuP pRinWs wiWh Whe saPe sWabiliWy pURpeUWies. EYen PRUe, Whe IRllRwinJ UelaWiRns beWween sRluWiRns RI bRWh sysWePs Wake place:

A C<CLE O) EN=<MATIC REACTIONS :IT+ SOME 3RO3ERTIES« 83

a1(t)=y2(t/k), x1(t)=z2(t/k), y1(t)=a2(t/k), z1(t)=x2(t/k), 9)

wheUe subinGe[ 1 anG 2 UeIeU WR sRluWiRns wiWh UelaWiYe UaWe cRnsWanWs k anG 1/k UespecWiYely. OI cRuUse, UelaWiRns 9) will be YaliG iI iniWial cRnGiWiRns saWisIy UelaWiRns 7). EYRluWiRn RI Whe sysWeP GepenGs Rn Whe Yalues RI paUaPeWeU k anG inWeJUal RI PRWiRn C. In )iJ. 2 WheUe aUe shRwn WwR aUeas A anG B, PaUkeG wiWh GRuble hRUi]RnWal lines, wheUe Whe sysWeP has WhUee eTuilibUiuP pRinWs. In Whe aUea A WheUe aUe WwR sWable eTuilibUiuP pRinWs sepaUaWeG by Whe WhiUG unsWable eTuilibUiuP saGGle pRinW). AW k anG C belRnJinJ WR Whis aUea, Whe sysWeP behaYes as a bisWable WUiJJeU. In Whe aUea B Whe sysWeP has Whe IRllRwinJ eTuilibUiuP pRinWs: sWable nRGe, saGGle-pRinW anG unsWable IRcus. AW Yalues k anG C IURP Whe aUea B, Whe sysWeP behaYes as an e[ciWable sysWeP.

0 40 80 120 Integral of motion (C) 1 10 100 R ela tiv e r ate c on sta nt (k ) 0 40 80 120 1 10 100 0 40 80 120 1 10 100 0 40 80 120 1 10 100 0 40 80 120 1 10 100 0 40 80 120 1 10 100 0 40 80 120 1 10 100 0 40 80 120 1 10 100 A B p q

FIG. 2. 3aUaPeWeU plane RI Whe sysWeP. The sysWeP has WwR sWable anG Rne unsWable

eTuilibUiuP pRinWs in Whe aUea A, Rne sWable anG WwR unsWable eTuilibUiuP pRinWs RU WhUee unsWable eTuilibUiuP pRinWs in Whe aUea B. BeyRnG Whe aUeas A anG B Whe sysWeP has Rne eTuilibUiuP. CuUYe p sepaUaWes aUeas wiWh Rne sWable eTuilibUiuP pRinW IURP WhaW wiWh nR sWable eTuilibUiuP pRinW. CuUYe T sepaUaWes Whe aUea B wiWh WhUee eTuilibUiuP pRinWs IURP WhaW wiWh sinJle eTuilibUiuP. The cuUYes aUe baseG Rn nuPeUically calculaWeG eTuilibUiuP pRinWs usinJ ³MaWhePaWica´ wiWh Pachine pUecisiRn 16 GiJiWs) anG pUinWinJ RI si[ siJniIicanW GeciPal GiJiWs. )RU PRUe GeWails see We[W.

0 40 80 120 Integral of motion (C) 1 10 100 Re la tiv e ra te co ns ta nt (k ) 0 40 80 120 1 10 100 0 40 80 120 1 10 100 0 40 80 120 1 10 100 0 40 80 120 1 10 100 0 40 80 120 1 10 100 0 40 80 120 1 10 100 0 40 80 120 1 10 100 A B p q

An e[ePplary phase portrait oI this kinG is shown in Fig. 6. There is a Yery narrow part oI the area B at its right-hanG eGge, where all three eTuilibriuP points are unstable. In this case, the autooscillations appear in the systeP. BeyonG the areas A anG B the systeP has one eTuilibriuP point. The single eTuilibriuP is stable Ior Yalues oI C sPaller than those Ialling in the curYe p in Fig. 2, anG is unstable Ior the higher Yalues oI C. All the PentioneG areas, shown Ior k>1 in Fig. 2, appear also at k<1. CorresponGing curYes GiYiGing the paraPeter plane at k1 are syPPetrical on the logarithPic scale oI k to those shown in Fig. 2 in respect to the straight line k 1. The borGers oI the areas shown in Fig. 2. haYe been GeterPineG nuPerically with the precision oI si[ signiIicant GeciPal Gigits.

EVOLUTION OF T+E SYSTEM

AUTOOSCILLATIONS

Autooscillations appear in the systeP at the Yalues oI C higher than those corresponGing to the curYe p in Fig. 2. InsiGe the area B the curYe p goes slightly aboYe the curYe q. So, there is a narrow strip oI the area B between p anG q where the systeP has three unstable eTuilibriuP points. For paraPeter Yalues belonging to this strip, the systeP has oscillatory solutions with all three eTuilibriuP points placeG insiGe the corresponGing liPit cycle.

Analytical linear analysis oI stability is possible only in the Iully syPPetrical case with k=1. In this case the systeP has an uniTue eTuilibriuP

.

4

C

a x y z

10)

EigenYalues oI the systeP in the Yicinity oI the eTuilibriuP point 10) are giYen by the e[pressions 11):





2 2 1,2 ₂ 2 16 32 2 . 16 C i C C O ª«  r º» «  » ¬ ¼ ₃ 2 4 , 1 a O   11)

The Iourth eigenYalue is eTual to ]ero. RespectiYe norPal Yariables can be GeIineG by the transIorPation 12). Because oI the conserYation law 6) the Yariable ȟ4 has a constant Yalue C/2, anG the eYolution oI the systeP can be

A CYCLE OF EN=YMATIC REACTIONS :IT+ SOME 3RO3ERTIES« 85 1 2 3 4 1 1 0 0 2 2 1 1 0 0 2 2 _. 1 1 1 1 2 2 2 2 1 1 1 1 2 2 2 2 a x y z [ [ [ [  ª º « » « » ª º _{«  »}ª º « » _{« »}« » « » _{« »}« » « » _{«   »}« » « » _{« » ¬ ¼}« » ¬ ¼ « » « » « » ¬ ¼ 12)

NuPerical solutions a(t) at k=1 anG C 5 or 50 are shown in Fig. 3. Solutions x(t), y(t) anG z(t) haYe the saPe shape as a(t) but are GelayeG in respect to a(t) by one, two anG three Tuarters oI the perioG respectiYely. The perioG oI oscillations increases with growing Yalues oI C see Fig. 4).

At Yalues oI k essentially GiIIerent IroP the unity, the shapes oI oscillations oI Yariables a, x, y anG z becaPe GiIIerent IroP each other. As an e[aPple, solutions a(t), x(t), y(t) anG z(t) at k 40 anG initial conGitions a(0) 80, x(0)=y(0)=z(0)=0 are shown in Fig. 5. AccorGing to the substitution 9), solutions Ior k 1/40 0.025 anG initial conGitions y(0) 80, a(0)=x(0)=z(0) 0 can be obtaineG IroP Fig. 5. It is enough to change the tiPe scale by substitution 2000 insteaG oI 50 Ior the highest Yalue oI tiPe anG reaG Fig. 5a as y(t), 5b as z(t), 5c as a(t) anG 5G as x(t). DepenGence oI the oscillation perioG on the suP oI reagent concentrations Ior a Iew Yalues oI k is shown in Fig. 4. The curYes presenteG in Fig. 4 were obtaineG on the basis oI nuPerical solutions oI eTuations 1-4). Let us note that at high Yalues oI k the perioG oI oscillations rePains alPost constant in a Tuite wiGe range oI C. For e[aPple, at k 20 the perioG changes its Yalue IroP 9.60 at C 40 to 10.16 at C 100. So, at such Yalues oI k the systeP can work as a pacePaker Ior soPe rhythPs.

EVOLUTION OF T+E SYSTEM AT MULTIPLE EQUILIBRIUM POINTS

At suIIiciently high Yalues oI k the systeP can haYe three eTuilibriuP points areas A anG B in Fig. 2). In the area A, at k>8.2068, two oI these eTuilibriuP points are stable anG the thirG one is unstable. Let us consiGer, as an e[aPple, the systeP with k 40. At this Yalue oI k, the area A incluGes the range oI C IroP 12.4932 to 21.0277. The Yalues oI the Yariables in eTuilibriuP anG corresponGing eigenYalues Ior k 40 anG C 16.6 are giYen in Table 1. The GynaPical Yariables in the unstable eTuilibriuP point 2 in Table 1) haYe interPeGiate Yalues in respect to those Gescribing the two stable eTuilibriuP points. Thus, in the state space oI the systeP, the unstable eTuilibriuP point is situateG soPewhere between the stable eTuilibriuP points. DepenGing on initial conGitions, the systeP goes to the eTuilibriuP 1 with relatiYely low Yalue oI x

or to the eTuilibriuP 3 with a high Yalue oI x. The eTuilibriuP points 2 anG 3 becoPe closer anG closer to each other when the Yalue oI C Gecreases IroP 16.6 to 12.5. At the saPe tiPe the GoPain oI attraction oI the eTuilibriuP 1 is growing. EYentually, at C 12.4932, the eTuilibriuP points 2 anG 3 annihilate. In contrast, increasing Yalue oI C IroP 16.6 to 21 brings the eTuilibriuP points 1 anG 2 closer anG closer to each other. These eTuilibriuP points annihilate at C 21.0277. I GiG not e[plore in Getail the hypersurIace separating attraction GoPains oI the eTuilibriuP points 1 anG 3. NeYertheless, it is intuitiYely clear that the attraction GoPain oI the eTuilibriuP with a low Yalue oI x point 1 in Table 1) is bigger at the leIt-hanG eGge oI the area A Fig. 2) than at the right-hanG eGge oI this area. In the whole area A in all oI the three eTuilibriuP points, the Yalues oI y anG z constitute only a Pinute part oI the pool oI all reagents C). Fully analogical Giscussion can be applieG to the area with k1/8.2068 0.0840336 with respectiYe changes in the roles oI Yariables anG scale oI tiPe, accorGing to the substitutions 7) anG 8). There is another area B in Fig. 2) with three eTuilibriuP points. This area appears at k>11.9 or at k1/11.9 0.0840336. 0 10 20 30 0 1 2 3 4 5

a

0 10 20 30

t

0 10 20 30 40 50

a

(a)

(b)

FIG. 3. Oscillations in a Iully syPPetrical systeP. NuPerical solutions a(t) at k 1 anG C=5 a) or C 50 b).

A CYCLE OF EN=YMATIC REACTIONS :IT+ SOME PROPERTIES« 87 0 20 40 60 80 100 Integral of motion (C) 4 8 12 16 Pe rio d 1 0 20 40 60 80 100 4 8 12 16 2 0 20 40 60 80 100 4 8 12 16 5 0 20 40 60 80 100 4 8 12 16 10 0 20 40 60 80 100 4 8 12 16 20 0 20 40 60 80 100 4 8 12 16 40

FIG. 4. DepenGence oI the perioG oI oscillations on the suP oI reagent concentrations

C). Values oI the relatiYe rate constant k are shown at the respectiYe curYes. PerioGs were obtaineG IroP nuPerical solutions oI eTuations 1-4), using ³MathePatica´. Let us return to the e[aPple with k=40. For this Yalue oI k anG C belonging to the interYal 21.0277, 34.1035), the systeP has a single stable eTuilibriuP characteriseG by three real anG negatiYe eigenYalues. As soon as C e[ceeGs the Yalue oI 34.1035, two aGGitional eTuilibriuP points appear. Both oI theP are unstable. The systeP has three eTuilibriuP points Ior 34.1035C42.0551. Table 2 presents an e[aPple oI such three eTuilibria Ior C=38. As can be seen IroP Table 2, the saGGle-point 2) is locateG between the stable noGe 1) anG unstable Iocus 3). In any case, the eYolution oI the systeP leaGs to eTuilibriuP 1. +oweYer, orbits attaining eTuilibriuP 1 can be essentially GiIIerent Ior GiIIerent initial conGitions. In Fig. 6, there are shown proMections oI the two orbits on the plane a,x) Ior the Yalues oI k=40 anG C=38 as those useG in Table 2. In the case oI the orbit starting IroP the point A initial conGitions ^a(0), x(0), y(0), z(0)} = ^1.134, 35.4, 1.408, 0.058}) changes in the Yalues oI the Yariables are Yery sPall. The systeP rela[es to eTuilibriuP 1 alPost Ponotonously. In contrast, on the orbit

starting IroP point B ^a(0), x(0), y(0), z(0)}=^1.66, 30.63, 5.33, 0.38}) the aPplituGes oI changes oI the Yariables are Puch higher. BeIore attaining eTuilibriuP 1, the orbit goes arounG the unstable eTuilibriuP 3. So, we haYe to Go with an e[citable systeP. E[citation eYents are soPewhat GiIIerent IroP those GescribeG earlier >5]. *eneration oI positiYe spikes oI all Yariables in the present systeP is iPpossible because oI the conserYation law 6). AIter start IroP the point B, one can obserYe GiPinishing Yalue oI x, which reaches its PiniPuP oI 21.74 at tiPe 3.6. In e[pense oI x, spikes oI the rePaining Yariables are generateG. The Pa[iPuP Yalues appear in the Iollowing seTuence: y2.97)=12.35, z4.31)=1.72 anG a5.61)=4.92. EYen PiniPuP Yalue oI x is essentially higher than Pa[iPuP Yalues oI y, z anG a.

0 20 40 60 80

a

0 20 40 60 80

x

0 20 40 60 80

y

0 20 40 60 80

z

0 10 20 30 40 50 t (a) (b) (c) (d)

FIG. 5. Oscillations in a highly asyPPetrical systeP. NuPerical solutions at k=40 anG C=80.

A CYCLE OF EN=YMATIC REACTIONS :IT+ SOME PROPERTIES« 89

TABLE 1. CoorGinates a,x,y,z) anG respectiYe eigenYalues Ȝ) oI the eTuilibriuP points

at k=40 anG C=16.6 ETuilibriuP no. 1 2 3 a x y z Ȝ1 Ȝ2 Ȝ3 16.0935 0.504903 4.85482[10-5 1.54745[10-3 -72.8782 -39.996 -19.9246 14.1969 2.40104 2.96349[10-4 1.75225[10-3 -46.9177 -39.9947 3.16435 2.9172 13.6393 3.59011[10-2 7.67861[10-3 -41.2675 -39.7528 -0.798068

TABLE 2. CoorGinates a,x,y,z) anG respectiYe eigenYalues Ȝ) oI the eTuilibriuP points

at k=40 anG C=38 ETuilibriuP no. 1 2 3 a x y z Ȝ1 Ȝ2 Ȝ3 1.09956 36.3598 0.524838 0.0158716 -32.844 -17.7117 -1.02994 1.16751 34.4423 2.31183 0.0783655 -7.75538 -0.731001 4.1651 2.16193 26.8251 8.34074 0.67221 -1.48735 0.2301030.829406i 0.230103-0.829406i The Yariable x constitutes also a GoPinating Iraction oI the whole reagent pool in all eTuilibriuP points see Table 2). So, in the e[citable regiPe area B in Fig. 2), the Yariable x behaYes like a reserYoir substance. CoorGinates oI the saGGle-point 2 GeterPine the e[citation thresholG. The Yalue oI C=38 useG in Table 2 anG Fig. 6 corresponGs to the PiGGle oI the area B in Fig. 2 at k=40. At the leIt-hanG eGge oI this area, at C slightly higher than 34.1035, the saGGle-point 2 anG unstable Iocus 3 are Yery close each to other anG both are rePote IroP stable noGe 1. In such a situation the e[citation thresholG is relatiYely high anG aPplituGes oI generateG spikes oI the Yariables are relatiYely low. :ith the growing Yalue oI C eTuilibriuP points 1 anG 2 coPe closer anG closer to each other. The e[citation thresholG becoPes lower anG lower. At C=42.0548, still insiGe the area B in Fig. 2, Gestabilisation oI the eTuilibriuP 1 takes place. So, in the interYal 42.0548C42.0551 the systeP has three unstable eTuilibriuP points anG oscillatiYe solutions. At C=42.0551 the eTuilibriuP points 1 anG 2 annihilate. Again, analogical Giscussion can be applieG to k=1/40=0.025 with substitution 7). All processes will then be 40 tiPes slower accorGing to tiPe rescaling 8). ETuilibriuP no. 1 2 3 a x y z Ȝ1 Ȝ2 Ȝ3 16.0935 0.504903 4.85482[10-5 1.54745[10-3 -72.8782 -39.996 -19.9246 14.1969 2.40104 2.96349[10-4 1.75225[10-3 -46.9177 -39.9947 3.16435 2.9172 13.6393 3.59011[10-2 7.67861[10-3 -41.2675 -39.7528 -0.798068 ETuilibriuP no. 1 2 3 a x y z Ȝ1 Ȝ2 Ȝ3 1.09956 36.3598 0.524838 0.0158716 -32.844 -17.7117 -1.02994 1.16751 34.4423 2.31183 0.0783655 -7.75538 -0.731001 4.1651 2.16193 26.8251 8.34074 0.67221 -1.48735 0.2301030.829406i 0.230103-0.829406i

1 2 3 4 5 a 24 28 32 36 x 1 2 3 4 5 24 28 32 36 1 2 3 4 5 24 28 32 36 1 2 3 A B

FIG. 6. E[citable systeP. ProMections oI the two orbits on the plane a,x) at k=40 anG C=38. CoorGinates oI stable noGe 1), saGGle-point 2) anG unstable Iocus 3) are giYen

in Table 2. Orbit starting IroP the point A 1.134, 35.4, 1.408, 0.058) corresponGs to subthresholG rela[ation. Orbit starting IroP the point B 1.66, 30.63, 5.33, 0.38) corresponGs to e[citation. Orbits obtaineG IroP nuPerical solutions oI the eTuations 1-4) using ³MathePatica´.

CONCLUSIONS

The GynaPical systeP 1-4) shares Pany PoGes oI eYolution with one analyseG earlier >5]. ProYiGing a proper choice oI paraPeter Yalues, both systePs can rela[ to an uniTue stable eTuilibriuP or behaYe as bistable triggers. Both

A CYCLE OF EN=YMATIC REACTIONS :IT+ SOME PROPERTIES« 91

systePs can haYe autooscillatory solutions anG can behaYe as e[citable systePs. But in contrast to the earlier presenteG systeP with one reserYoir substance, the systeP 1-4) can not behaYe as a transGucer oI stiPulus strength to IreTuency. Both systePs can Gescribe Petabolic oscillations in a single bacterial cell. Such oscillations will appear in a bacterial culture iI oscillations in GiIIerent cells are synchronous. I suppose that both consiGereG systePs will be useIul in searching IaYourable conGitions Ior synchronisation.

REFERENCES

1. Malarc]yk, E. 1989) Transformations of phenolic acids by Nocardia. Acta Microbiol. Polon. 38, 45-53.

2. Malarc]yk, E. anG .ochPaĔska-RGest, J. 1990) New aspects of co-regulation of

decarboxylation and demethylation activities in Nocardia. Acta BiochiP. Polon. 34,

145-148.

3. Malarc]yk, E. anG PaĨG]ioch-C]ochra, M. 2000) Multiple respiratory bursts as a

response to veratrate stress in Rhodococcus erythropolis. Cell Biol. Int. 24,

515-527.

4. PaĨG]ioch-C]ochra, M., Malarc]yk, E. anG Sielewiesiuk, J. 2003) Relationship of

demethylation processes and cell density in cultures of Rhodococcus erythropolis.

Cell Biol. Int. 27, 325-336.

5. Sielewiesiuk, J. anG Malarc]yk, E. 2002) A cycle of enzymatic reactions that

behaves as a neuronal circuit. J. Theor. Biol. 214, 255-262.

6. Sielewiesiuk, J., C]ubla, A., Malarc]yk, E. anG PaĨG]ioch, M. 1999) Kinetic model

for oscillations in a cycle of enzymatic reactions related to methoxyphenol transformation in Rhodococcus erythropolis culture. Cell. Mol. Biol. Lett. 4,

131-146.

7. Dunin-BarkoYsky, V.L. 1970) The oscillation of activity level in simple closed

neurone chains. Biophysics 15, 374-378 in Russian).

8. *ooGwin, B.C. 1966) An entrainment model for timed enzyme syntheses in

bacteria. Nature 209, 479-481.

9. +astings, S., Tyson, J. anG :ebster, D. 1977) Existence of periodic solutions for

negative feedback cellular control systems. J. DiIIerential ETuations 25, 39-64.

10. Tyson, J.J. 1975) On the e[istence oI oscillatory solutions in negatiYe IeeGback cellular control processes. J. Math. Biol. 1, 311-315.

11. Tyson, J.J. 1979) Periodic enzyme synthesis: reconsideration of the theory of

oscillatory repression. J. Theor. Biol. 80, 27-38.

12. *rigoroY, N.L., PolMakoYa, M.S. anG ChernaYsky, D.S. 1967) Model investigations

of the trigger schemes and the differentiation process. Molecular Biol. 1, 410-418,

in Russian).

13. RoPanoYsky, Yu.M., StepanoYa, N.V. anG ChernaYsky, D.S. 1975) Mathematical