Positive and negative feedback loops coupled by common transcription activator and repressor

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

32SITIVE AN' NE*ATIVE )EE'BAC. L223S

C2U3LE' B< C2MM2N TRANSCRI3TI2N ACTIVAT2R

AN' RE3RESS2R

-DQ SLeleZLeVLXN AJDWD àRSDFLXN

'eSDUWPeQW RI BLRSK\VLFV IQVWLWXWe RI 3K\VLFV MDULD CXULe-SNáRGRZVND UQLYeUVLW\ LXElLQ 3RlDQG, e-PDLl VLeleZLeVLXN#SRF]WDXPFVlXElLQSl

ABSTRACT

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1. INTRODUCTION

CiUcaGian UhythPs in euNaUyRtic RUganisPs aUe geneUateG Ey special PRtiIs in tUanscUiptiRn UegulatiRn netZRUNs. The PRst iPpRUtant RI these PRtiIs aUe clRseG negatiYe IeeGEacN lRRps. TUanscUiptiRn RI a gene in such a lRRp giYes PRNA, ZhRse tUanslatiRn pURGuces a pURtein. The pURtein unGeUgRes a nuPEeU RI tUansIRUPatiRns anG EecRPes the UepUessRU RI its RZn gene. The schePe RI this NinG can alsR ZRUN in a chain RI en]yPatic UeactiRns, iI the enG pURGuct RI the chain inhiEits an en]yPe cataly]ing Rne RI the IiUst UeactiRns. As a PatteU RI Iact, this systeP, NnRZn as *RRGZin¶s RscillatRU, Zas RUiginally applieG tR en]yPatic UeactiRns *RRGZin 1966. In the siPplest YaUiant, the systeP cRnsists RI Rne cRRpeUatiYe pURcess tUanscUiptiRn, allRsteUic en]yPe, PePEUane UeceptRU anG seYeUal UeactiRns Zith the lineaU PRnRPRleculaU Ninetics. The systePs Zhich geneUate RscillatiRns in vivo haYe PRUe cRPple[ stUuctuUe -RlPa et al. 2010, )eng anG La]aU 2012, Dunlap 1999, AlRn 200. CiUcaGian clRcNs, as a Uule, cRntain inteUlRcNeG negatiYe anG pRsitiYe IeeGEacN lRRps SaithRng et al. 2010. The e[istence RI RscillatiRns anG theiU peUiRG aUe GeteUPineG Ey the negatiYe IeeGEacN lRRp. On the RtheU hanG, the pRsitiYe IeeGEacN lRRp can tR sRPe GegUee PRGiIy the peUiRG, aPplituGe anG UREustness RI the RscillatiRns. It can eYen Iully GaPp the RscillatiRns in sRPe special cases.

:e cRnsiGeU a hypRthetical systeP RI tZR genes ZhRse tUanscUiptiRn is gRYeUneG Ey the saPe tUanscUiptiRn IactRUs. One RI these genes encRGes a pURtein Zhich can EecRPe a UepUessRU. The RtheU gene encRGes a pUecuUsRU pURtein RI the actiYatRU. TUanscUiptiRn RI ERth genes is pRssiEle pURYiGeG that the cRncentUatiRn RI the UepUessRU is suIIiciently lRZ anG the cRncentUatiRn RI the actiYatRU is suIIiciently high. )URP the YieZpRint RI PathePatical PRGeling, the essential IeatuUe RI the systeP is lRgic RI the cRupling EetZeen the UepUessRU anG actiYatRU IeeGEacN lRRps. The lRgical cRnMunctiRn aUepUessiRn ŀ actiYatiRn Rught tR Ee satisIieG tR staUt tUanscUiptiRn.

SiPultaneRus tUanscUiptiRn RI Pany genes, inGuceG Ey cRPPRn IactRUs, RccuUs in pURNaUyRtic anG euNaUyRtic cells. In lactRse RpeURn RI Escherichia

coli the saPe pURPRteU anG RpeUatRU cRntURl tUanscUiptiRn RI pRlycistURnic genes

encRGing thUee pURteins +Rgg 2005. 3RlycistURnic genes anG PRNA aUe Tuite usual phenRPena in EacteUial RpeURns. In euNaUyRtes, as a Uule, the syne[pUessiRn RI pURteins IuctiRning in the saPe pURcess is cRRUGinateG Ey tUans-acting IactRUs NiehUs anG 3Rllet 1999. The alteUnatiYe splicing RI pUe-PRNA PaNes it pRssiEle tR REtain tZR GiIIeUent pURteins Rn the Easis RI the saPe tUanscUipt. In such a case, e[pUessiRn RI these pURteins is inGuceG in the saPe pURPRteU MaMeUcaN, :en-)eng Chen anG EGeUy 2004. BRth PechanisPs RccuU in ciUcaGin clRcNs RippeUgeU anG BURZn 2010, SteigeU anG .|steU 2011.

The systeP is UepUesenteG in RuU PRGel Ey a set RI RUGinaUy GiIIeUential eTuatiRns. LineaU staEility analysis NayIeh anG BalachanGUaP 1995 anG nuPeUical sRlutiRns ZeUe useG as tRRls in RuU seaUch. :e tUieG tR GeteUPine hRZ the phase pRUtUait RI the systeP GepenGs Rn paUaPeteUs. :e haYe IRunG geneUal UelatiRns EetZeen paUaPeteU Yalues Zhen the nuPEeU RI eTuiliEUiuP pRints Zas changeG in a saGGle-nRGe RU tUanscUitical EiIuUcatiRn. The cRnGitiRns RI the +RpI EiIuUcatiRn haYe Eeen IRunG in a less geneUal IRUP in a IeZ special, syPPetUical, cases.

2. STRUCTURE O) T+E S<STEM

AND ITS MAT+EMATICAL MODEL

:e Giscuss a speciIic hypRthetic systeP Zhich cRulG Uegulate e[pUessiRn RI genes. The systeP cRntains tZR genes Zhich aUe tUanscUiEeG siPultaneRusly. TheiU tUanscUiptiRn is UegulateG Ey tZR tUanscUiptiRn IactRUs, UepUessRU anG actiYatRU. The tUanscUiptiRn is gRing Rn Zhen cRncentUatiRn RI the UepUessRU is lRZ anG cRncentUatiRn RI the actiYatRU is suIIiciently high. Such UelatiRns EetZeen tUancUiptiRns RI the tZR genes can Ee Ueali]eG Ey a cRPPRn pURPRteU in an e[tUePaly siPpliIieG PRGel, as in )ig. 2.1. 4ualitatiYely, the saPe situatiRn is IRUPeG in ciUcaGian systePs Ey E-ER[es, Zhich aUe actiYateG Ey an actiYatRU. In such a case, a UepUessRU inteUacting Zith the actiYatRU pUeYents its inteUactiRn Zith E-ER[. TZR pURteins encRGeG in the UegulateG genes unGeUgR Pany tUansIRUPatiRns Zith the tUanscUiptiRn IactRUs as enG pURGucts. The systeP is UepUesenteG Ey a schePe in )ig. 2.1 anG a set RI q RUGinaUy GiIIeUential eTuatiRns 1. OuU PRGel is cRnsistent Zith highly siPpliIieG schePes RI the PaPPalian ciUcaGian clRcN pUesenteG Ey -RlPa et al. 2010, )ig.1 anG Ey OsteU 2010, )ig. 5..

FIG. 2.1. SchePe RI the UegulatRUy systeP. 3 – pURPRteU, *R – gene RI the UepUessRU, *A – gene RI the actiYatRU.

X1 -Xp

Xp+1 – Xq

GR













1 1 1 1 1 1 1 1 1 1 , 1 1 , 2,..., , , 1 1 , 2,..., . n q m n p q i i i i i n p q p p m n p q j j j j j ax dx k x dt x x dx h x k x i p dt dx bx k x dt x x dx h x k x j p q dt                

1

In eTuatiRns 1, xp anG xq aUe cRncentUatiRns RI the UepUessRU anG actiYatRU

UespectiYely. The YaUiaEles x1 anG xp+1 aUe cRncentUatiRns RI PRNA encRGing

UepUessRU¶s anG actiYatRU¶s pUecuUsRUs. The Uest RI YaUiaEles UeIeU tR cRncentUatiRns RI tUansient IRUPs RI the pURteins. ETuatiRns Zith inGices 1 anG

p+1 cRUUespRnG tR PRNA synthesis anG Gecay. :e use GiPensiRnless YaUiaEles.

The unit Yalue IRU the IiUst p YaUiaEles 1 is the UepUessRU cRncentUatiRn Zhich UeGuces the Uate RI tUanscUiptiRn tR the halI RI its Yalue in the UepUessRU aEsence. The unit Yalue RI the YaUiaEles xp+1 …xq cRUUespRnGs tR such actiYatRU

cRncentUatiRn at Zhich the Uate RI tUanscUiptiRn EecRPes eTual tR the halI RI its Pa[iPuP Yalue. +ill¶s cReIIicients RI cRRpeUatiYity IRU the UepUessRU anG actiYatRU aUe UespectiYely m anG n. The ki anG hi aUe Uate cRnstants. The cRnstants

a anG b aUe Pa[iPuP Uates RI tUanscUiptiRn at the PRst IaYRUaEle cRnGitiRns

xpĺ[qĺ’.

The systeP 1 is pRsitiYely inYaUiant at pRsitiYe Yalues RI the Uate cRnstants. II all the YaUiaEles haYe negatiYe initial Yalues then they Zill Uest nRn-negatiYe GuUing the eYRlutiRn RI the systeP. In seaUching eTuiliEUiuP pRints, all the YaUiaEles e[cept IRU xp anG xq can Ee eliPinateG. CRnGitiRns IRU eTuiliEUiuP

can Ee giYen the IRllRZing shape 2:



1



1



,



1



1



, n n m n m n ay by x y x y

D

x y

E

   

2

1 1 1 1 1 1

,

.

q p i i i p i p q i i i i p

k

k

h

h

D

E

   

–

–

–

–

ETuiliEUiuP Yalues RI the UePaining YaUiaEles aUe

1 1 1 1 , 1,...., 1, , 1,...., 1. p i i j j p i i j q i i j j q i i j k x x j p h k x y j p q h       

–

–

–

–



ETuatiRns 2 haYe Rne REYiRus sRlutiRn

x

y

0

. It Peans, in YieZ RI , that the RUigin RI phase cRRUGinates is an eTuiliEUiuP pRint RI the systeP 1. It IRllRZs IURP 2 that the nuPEeU RI eTuiliEUiuP pRints anG theiU cRRUGinates GepenGs Rn tZR paUaPeteUs A a/D anG B b/E, Zhich aUe siPple IunctiRns RI the2q Uate cRnstants. 3RssiEle eTuiliEUiuP pRints EeyRnG the RUigin \can Ee IRunG IURP eTuatiRns 2 in the IRllRZing IRUP 4:







1

1 ,

.

1

1

n m n

By

B

y

x

A

x

y







4

ETuatiRns 4 can Ee UeGuceG tR the eTuiYalent pRlynRPial eTuatiRn

1

_0.

n m n n m n n n n n

B x



_

A x

_

B x

_

AB x



_

A

4’ In the case RI n=1, the eTuatiRn 4’ takes the shape:





1

₁

_0.

m m

Bx



_

Ax

_

Bx A

_

_

B

AccRUGing tR DescaUtes Uule, it has then Rne B>1  RU nRne B<1 Ueal anG pRsitiYe URRt. SR, at n=1, the systeP 1 has Rne RU tZR eTuiliEUiuP pRints. The nuPEeU RI eTuiliEUiuP pRints Zith nRn-negatiYe phase cRRUGinates is changeG

thURugh tUanscUitical EiIuUcatiRn at B=1. In the case RI Q•, theUe aUe twR sign changes in the pRlynRPial 4’ anG it can haYe twR RU nRne pRsitiYe URRts. The systeP can haYe Rne, twR RU thUee eTuiliEUiuP pRints. The nuPEeU RI eTuiliEUiuP pRints changes IURP Rne tR thUee in a saGGle-nRGe EiIuUcatiRn. BiIuUcatiRnal Yalues RI the paUaPeteUs we will IinG using chaUacteUistic eTuatiRn RI the systeP.

. C+ARACTERISTIC E4UATIONS O) T+E S<STEM

AND T+E NUMBER O) E4UILIBRIUM 3OINTS

TwR eTuatiRns with inGices 1 anG p+1 RI the systeP 1 haYe nRnlineaU IunctiRns in theiU Uight hanG paUts. Let us call these IunctiRns f1 anG fp+1:













1 1 1 1 1 1 1 1

 , , 

,

1

1

 ,

, 

1

1

n q p q m n p q n q p p p q m n p p p q

ax

f x x x

k x

x

x

bx

f

x x

x

k x

x

x

   













5

anG wUite theiU paUtial GeUiYatiYes 6.



 











 









1 1 1, 2 1 1 1, 2 1 1 1, 2 1 1 1, 2

,

1

1

,

1

1

,

1

1

.

1

1

m n p q p _{m n} p _{p q} n q q _{m n} q _{p q} m n p p q p p _{m n} p _{p q} n p q p q _{m n} q _{p q}

amx

x

f

g

x

_x

_x

anx

f

g

x

_x

_x

f

bmx

x

g

x

_x

_x

f

bnx

g

x

_x

_x

       



w

w

_

_

w

w

_

_

w



w

_

_

w

w

_

_

6

InGices in the leIt paUt RI 6 UeIeU tR pRsitiRns RI paUticulaU GeUiYatiYes in the -acREian RI the systeP 6’.

1 1, 1, 1 2 1 1 1, 1 1, 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 p q i i p p p p p p q j j q q k g g h k h k h k J g k g h k h k         ª º « _ » « » «  » « _ » « » « _ » « »  « » « _ » ¬ ¼

6’

,...,

1

2,...,

1.

i

p

j

p

q







In a geneUal case, the chaUacteUistic eTuatiRn RI the systeP 1 has the shape :

 

 

 









1 1 1, 1 1 1 1 1 1, 1 1 1 1, 1, 1, 1, 1,

1

1

1

0,

,

q _{q p} q p i p q i i i i p i p q p p i i i i p q q p q p q p p i i i p i i

u

g

h

u

g

h

u

g

g

g g

h

u

k

O

           z

 



 



 







–

– –

– –

–



wheUe Ȝ is an eigenYalue RI the PatUi[ J 6’. As it IRllRws IURP 6, the teUP





1 1, 1, 1, 1, 1, q p q p q p p i i i p g _ g g g _  h z



_–

in the eTuatiRn  Yanishes.

:ith any n>1 all the GeUiYatiYes 6 aUe eTual tR ]eUR in the RUigin, anG chaUacteUistic eTuatiRn appeaUs tR Ee Tuite siPple anG sRlYaEle in this eTuiliEUiuP pRint 8.

 





1

1

q q i

0,

i i

,

1,..., .

i

k

O

O

k

i

q



_–





8

All eigenYalues IRU this pRint RI eTuiliEUiuP aUe Ueal anG negatiYe. The eTuiliEUiuP at RUigin is staEle at any PeaningIul Yalues RI paUaPeteUs.

 

 

 









1 1 1, 1 1 1 1 1 1, 1 1 1 1, 1, 1, 1, 1,

1

1

1

0

q _{q p} q p i p q i i i i p i p q p p i i i i p q q p q p q p p i i i p i i

u

g

h

u

g

h

u

g

g

g g

h

u

k

O

           z

 



 



 







–

– –

– –

–

In the case RI n=1, g_p1,q b, g1,q=a. :e can use GeIinitiRns RI paUaPeteUs ȕ

anG B anG giYe tR the chaUacteUistic eTuatiRn IRU an eTuiliEUiuP at RUigin the shape RI 9.









1 1 1

0.

p q q i i i i i p i p

k

O

k

O

B

k

 

ª

º



_«





_»

¬

¼

–

–

–

9

TheUe aUe p negatiYe eigenYalues Oi ki, i 1,...,p. All RI the UePaining

eigenYalues haYe theiU Ueal paUts negatiYe at B<1, Rne RI theP EecRPes eTual tR ]eUR at B=1 anG at least Rne RI theP is pRsitiYe at B>1. NRw n 1, the eTuiliEUiuP at RUigin is staEle at B<1 anG unstaEle at B>1. TheUe is a tUanscUitical EiIuUcatiRn at B=1. An aGGitiRnal pRint RI eTuiliEUiuP with pRsitiYe cRRUGinates appeaUs anG eTuiliEUiuP at RUigin EecRPes unstaEle.

In RUGeU tR e[aPine staEility RI the eTuiliEUiuP pRints EeyRnG the RUigin, Rne shRulG sRlYe eTuatiRns 2 RU 4, suEstitute REtaineG x anG y IRU xp anG xq in 6

anG IinG eigenYalues RI the Uesulting -acREian. But, it is UatheU iPpRssiEle tR REtain a cleaU syPERlic sRlutiRn RI the PentiRneG eTuatiRns. InsteaG, we use eTuatiRns 2 tR e[pUess paUaPeteUs a anG b as IunctiRns RI x anG y anG inseUt REtaineG e[pUessiRns intR 6. Ne[t, we use Uesulting e[pUessiRns as UespectiYe entUies RI the -acREian. Such a pURceGuUe Uesults in the IRllRwing chaUacteUistic eTuatiRn 10:













1 1 1 1 1

0.

1

1

m q q p p q i n i i m i i i i p i i i p

n

mx

k

k

k

k

k

y

x

O

O

O

 















–

– –

– –

10

The eTuatiRn is YaliG in the pRints RI eTuiliEUiuP which aUe situateG EeyRnG the RUigin RI cRRUGinates,



xz0

 

š yz0



, incluGing n=1. :e haYe nRt useG any aGGitiRnal assuPptiRn in GeUiYatiRn RI the eTuatiRn 10.

It is easy tR IinG cRnGitiRns IRU saGGle-nRGe EiIuUcatiRn using eTuatiRn 10. In the pRint RI such EiIuUcatiRn, at least Rne RI eigenYalues is eTual tR ]eUR. :ith

Ȝ , eTuatiRn 10 is UeGuceG tR the UelatiRn 11:

1

0.

1

1

m n m

n

mx

y

x









11

This UelatiRn cRnstitutes the cRnGitiRn which is tR Ee satisIieG Ey cRRUGinates RI an eTuiliEUiuP pRint at a saGGle-nRGe EiIuUcatiRn. It can Ee wUitten as a IunctiRn \[ 12:









1 1 . 1 1 m n m n m n x y m x       12

)RUPula 12 allRws us tR calculate y IRU an aUEitUaUy chRsen Yalue RI x. Ne[t, ERth x anG y can Ee useG tR calculate the paUaPeteUs A anG B accRUGing tR eTuatiRns 4. In such a way, we can IinG at what Yalues RI A anG B the saGGle-nRGe EiIuUcatiRn takes place.





















2 2 1 1 1 1 , 1 1 1 . 1 1 1 1 m m m m n m n nx x A n n m x n x B n n m x  m x             ª º ª º ¬ ¼ ¬ ¼ 1

In RUGeU tR get physically PeaningIul, pRsitiYe, Yalues RI y, A anG B IURP 12 anG 1 , we shRulG cRnIine RuU chRice RI m, n anG x Yalues tR:

1

0

,

1 1.

1

m

n

x

m n

m

n







!  t

 

14

:e can nRw GUaw a paUaPetUic plRt %[>$[@ accRUGing tR 1 with the paUaPeteU x anG any paUticulaU paiU RI +ill cReIIicients. The plRt wRulG GiYiGe the plane $%intR aUeas with Rne RU thUee eTuiliEUiuP pRints. It IRllRws IURP 14 that theUe is nR EiIuUcatiRn with Ȝ=0 at n=1 anG nRn]eUR Yalues RI YaUiaEles.

It will Ee useIul tR intURGuce PRUe cRPple[ paUaPeteUs

,

.

1

1

m n m

n

mx

y

x

J

P





15

In teUPs RI the paUaPeteUs ȖanG ȝ, chaUacteUistic eTuatiRn 10 anG cRnGitiRn IRU saGGle-nRGe EiIuUcatiRn 11 REtain sRPewhat siPpleU anG PRUe geneUal IRUPs RI 16 anG 1:













1 1 1 1 1

0

q q p p q i i i i i i i p i i i p

k

O J

k

k

O

P

k

k

O

 











–

– –

– –

16 1. J P  1

+RweYeU, the new paUaPeteUs haYe a sPall GisaGYantage. A paiU RI Yalues Ȗ anG ȝ chaUacteUi]es Rne speciIic pRint RI eTuiliEUiuP, nRt the whRle systeP. Each paUticulaU systeP is UepUesenteG in the plane ȝ Ȗ Ey a nuPEeU RI pRints eTual tR the nuPEeU RI eTuiliEUiuP pRints. In any case theUe is a pRint 0, n cRUUespRnGing tR the pRint RI eTuiliEUiuP in the RUigin RI the phase cRRUGinates. An unstaEle saGGle pRint can Ee IRunG at J P! 1. The thiUG pRint RI eTuiliEUiuP can Ee IRunG at J P 1. AutRRscillatiRns aURunG this pRint can Ee geneUateG aIteU a pRssiEle +RpI EiIuUcatiRn.

4. +OP) BI)URCATION AND OSCILLATIONS IN T+E S<STEM

4.1. T+E CASE O) T+E +I*+EST S<MMETR< :IT+ ALL KI=1 AND Q=2P

:e cRulG IinG cRnGitiRns RI +RpI biIuUcatiRn in the IRUP RI analytical UelatiRns Rnly in sRPe special cases. Let us cRnsiGeU the systeP satisIying twR essential liPitatiRns. Rate cRnstants RI Gecay ki IRU all the YaUiables in the

systeP aUe eTual tR each RtheU. The UepUessRU’s lRRp cRnsists RI the saPe nuPbeU RI substances as the actiYatRU’s lRRp T S. The assuPptiRn that all Gecay cRnstants aUe eTual tR 1 will Pake RuU calculatiRns PRUe siPple but it will nRt Pake theP less geneUal. UnGeU such assuPptiRns the chaUacteUistic eTuatiRn 16 aGRpts the siPple anG sRlYable IRUP 18.



1

O

 

pª_¬ 1

O



p  

J P

º_¼ 0. 18 TheUe aUe p URRts Ȝ -1. The UePaining p eigenYalues satisIy the eTuatiRn 19

 

1



O

p

R

,

R



J P

,

  

m R n

.

19

The ineTuality in 19 IRllRws IURP GeIinitiRns RI 5ȖanG ȝ In the case RI

1

R  all URRts RI the eTuatiRn 19

1

1

p

R

O

 ₂₀

haYe negatiYe Ueal paUts anG UespectiYe pRint RI eTuilibUiuP is stable. At R=1, Rne Ueal eigenYalue Yanishes anG a saGGle- nRGe biIuUcatiRn takes place. :ith R=1, the secRnG paUt RI 19, leaGs tR the UelatiRn, which we haYe alUeaGy RbtaineG in the Iully geneUal case 1.

In the case RI negatiYe R, eTuatiRn 19 can be giYen the IRUP RI 21



1

O



p R



cRs

S

isin

S



, R  

P J

m. 21 The eTuatiRn has URRts:









1 2 1 1 2 1 cRs 1 sin , 0,1,..., 1. p p j j j R i R j p p p S S O      22

EigenYalues with j=0 RU j=p-1 haYe the highest Ueal paUts. The eTuilibUiuP will be Gestabili]eG when the Ueal paUt RI these twR eigenYalues becRPes pRsitiYe. It takes place at

1 cRsp . R p

S

P J

 § ·  t ¨ ¸ © ¹ 2

The case RI eTuality in 2 giYes a UelatiRn between the cRRUGinates RI eTuilibUiuP pRints at +RpI biIuUcatiRn.





cRs 1 cRs 1 . cRs 1 1 p p m n p m n m n x p p y m x p

S

S

S

ª º  _«   _» ¬ ¼ § _ · _ ¨ ¸ © ¹ 24

In RUGeU tR Rbtain PeaningIul Yalues RI y, paUaPeteUs RI the systeP anG the cRRUGinate x shRulG satisIy the IRllRwing liPitatiRns:

1 cRs 1 1 cRs  , . cRs 1  cRs 1 p p m p p n p m n x p _{m m n} p p S S S S    !      25

Let us nRte that the +RpI biIuUcatiRn is pRssible when GiIIeUence between +ill cReIIicients RI UepUessiRn anG actiYatiRn e[ceeGs a ceUtain Yalue shRwn in the IiUst paUt RI 25. The saPe Yalue bRunGeG +ill cReIIicient at +RpI biIuUcatiRn in a lRRp with Rne UepUesseG gen InYeUni]]i anG TUeu 1991, as well as a pURGuct RI +ill cReIIicients RI all cRRpeUatiYe pURcess in a single lRRp cRntaining Pany UepUesseG RU actiYateG genes Sielewiesiuk anG àRpaciuk 2012.

The UelatiRn 24 is satisIieG in eTuilibUiuP pRints unGeUgRing a +RpI biIuUcatiRn. Using this UelatiRn anG cRnGitiRns RI eTuilibUiuP 2 we can e[pUess paUaPeteUs $ DĮ anG % E ȕ as IunctiRns RI x - UepUessRU’s cRncentUatiRn in eTuilibUiuP 26.

















2 2 1 1 1 1 cRs , cRs 1 cRs 1 1 cRs . cRs 1 cRs 1 cRs 1 1 m p p p m m p n n p p m p m nx x p A n m n x p p n x p B n m n x m x p p p S S S S S S  S  ª º  _«   _» ¬ ¼  ­ _{ }ª _{ } º ½ ª§ _ · _ º ® «_¬ »_¼ ¾ «¨_© ¸_¹ » ¯ ¿ ¬ ¼ 26

ETuatiRns 26 GescUibe a paUaPetUic cuUYe %[>$[@ which sepaUates in the plane $% the aUea with pRssible RscillatRUy sRlutiRns IURP the aUea wheUe RscillatiRn aUe iPpRssible.

FIG. 1. PaUaPeteU plane RI the systeP with ^p, q, m, n`=^10, 20, 4, 2`. LRweU cuUYe cRUUespRnGs tR the saGGle-nRGe biIuUcatiRn, the uppeU Rne – tR +RpI biIuUcatiRn.

2 4 6 8 A 2.5 5 7.5 10 12.5 15 B

AutRRscillatiRns

Bistable tUiggeU

FIG. 2. PaUaPeteU plane RI the systeP with ^p, q, m, n`=^10, 20, 4, 1`. LRweU cuUYe % cRUUespRnGs tR the tUanscUitical biIuUcatiRn, the uppeU Rne – tR +RpI biIuUcatiRn.

SystePs with a cRRpeUatiYe n>1, )ig. 1 anG nRn-cRRpeUatiYe n=1, )ig. 2 actiYatiRn haYe sRPe TualitatiYe siPilaUities anG GiIIeUences. SystePs RI bRth the kinGs haYe a pRint RI eTuilibUiuP at the RUigin RI cRRUGinates at any Yalues RI A anG B. TwR RtheU, nRn]eUR, pRints RI eTuilibUiuP appeaU thURugh saGGle-nRGe biIuUcatiRn in systePs with n>1, anG Rne aGGitiRnal eTuilibUiuP appeaUs thURugh tUanscUitical biIuUcatiRn when n=1. The eTuilibUiuP at the RUigin is stable at all Yalues RI paUaPeteUs with n>1, but it becRPes unstable IRU B>1 with n=1. +RpI biIuUcatiRn takes place in bRth the kinGs RI actiYatiRn at suIIiciently high Yalues RI the paUaPeteUs A anG B.

4.2. REPRESSION AND ACTIVATION LOOPS :IT+ DI))ERENT NUMBERS O) ELEMENTS

Let us cRnsiGeU sRPe systePs with slightly lRweU syPPetUy, wheUe the twR lRRps cRnsist RI GiIIeUent nuPbeUs RI pURtein tUansIRUPatiRns. As in sectiRn 4.1, we cRntinue tR use all the Gecay Uate cRnstants eTual tR Rne anG the chaUacteUistic eTuatiRn 16 in the IRUP:



1

O



q 

J



1

O



p 

P



1

O



q p 0. 2 In RUGeU tR IinG Yalues RI the paUaPeteUs Ȗ anG ȝ at +RpI biIuUcatiRn, we suppRseG a puUely iPaginaUy eigenYalue

O

i

Z

anG intURGuceG a new cRPple[ YaUiable





1 1 cRs sin z  

O

i

Z

z

M

i

M

28 2 4 6 8 10 A 2 4 6 8 10 B

One stable eTuilibUiuP 0,0 One unstable 0,0

anG Rne stable eTuilibUiuP

TwR unstable eTuilibUiuP pRints. AutRRscillatiRns.

with

tan

,

1

2

1

.

cRs

z

M Z

Z

M



The substitutiRn RI the YaUiable z IRU Ȝ in 2 Uesults in eTuatiRns 29:









cRs

cRs

cRs

cRs

cRs

cRs

sin

cRs

sin

sin

q p p q p p

p

q p

q

p

q p

q

J

M

M P

M

M

M

J

M

M P

M

M

M

 









29

anG biIuUcatiRnal Yalues RI the paUaPeteUs ȖanG ȝ:













sin

sin

,

.

cRs

q p

sin 2

cRs

p

sin 2

q

p

p

p q

p q

M

M

J

P

M



M

M

M







0

These UelatiRns allRweG us tR cRnstUuct paUaPetUic plRts Ȗȝ using ij as a paUaPeteU. Sets RI such plRts aUe shRwn in )ig.  TS anG in )ig. 4 T!S. In bRth the IiguUes, the stUaight line J P 1 cRUUespRnGs tR pRints RI the saGGle nRGe biIuUcatiRn. At least Rne eigenYalue RI these pRints RI eTuilibUiuP is eTual tR ]eUR in all systePs unGeU cRnsiGeUatiRn. TheUe is at least Rne Ueal anG pRsitiYe eigenYalue at J P! 1. The UePaining cuUYes cRUUespRnG tR eTuilibUiuP pRints haYing a paiU RI cRPple[ eigenYalues with ]eUR Ueal paUt.

FIG. 3. CuUYes ȖȝIRU +RpI biIuUcatiRn in systePs haYing UepUessiRn lRRp cRnsisting RI p=10 Ueagents. ActiYatiRn lRRps cRntain q-Ueagents with q=14, 15, 16, 1, 18, 19, 20,

in the RUGeU IURP the leIt tR the Uight. DasheG line Ȗ ȝ cRUUespRnGs tR saGGle-nRGe biIuUcatiRn. 1 2 3 4 5 6 1 2 3 4 5 6

ȝ

14 20 OscillatiRns

NuPeUical calculatiRns weUe GRne IRU the nuPbeU RI Ueagents in the UepUessiRn lRRp p=10 anG that in the actiYatiRn lRRp is q-p. In the cases RI q-p= 4, 5, 6, , 8 anG 9, the cuUYes enG Rn the stUaight line Ȗ ȝ1 with ij 0. As it was shRwn eaUlieU 2, the stUaight line J P 1.6512 cRUUespRnGs tR +RpI biIuUcatiRn in the PRst syPPetUical systeP with bRth lRRps RI eTual si]es p=10 anG q-p=10. OscillatiRns aUe pRssible aURunG the eTuilibUiuP pRints with Ȗȝ+1 belRnging tR the aUea belRw saGGle-nRGe line anG tR the Uight IURP the cuUYes RI +RpI biIuUcatiRn. Let us UePinG that any paUticulaU systeP has in the plane Ȗȝ an accessible aUea with Pdm anG Jdn. It is pRssible, at high n, that the cuUYe RI +RpI biIuUcatiRn GRes nRt GiYiGe the accessible Uectangle m ɯ n intR twR sepaUate paUts. An e[aPple RI such situatiRn can be seen in )ig.  IRU T=14 RU 15 anG n=2, wheUe theUe aUe nR stable RscillatiRns in spite RI e[isting cRPple[ eigenYalues with pRsitiYe Ueal paUt. The systeP gRes tR the stable eTuilibUiuP with ]eUR Yalues RI all YaUiables. It appeaUs that RscillatiRns aUe iPpRssible, when inGuctiYe lRRp is tRR shRUt.

FIG. 4. CuUYes ȖȝIRU +RpI biIuUcatiRn in systePs haYing UepUessiRn lRRp cRnsisting RI p=10 Ueagents. ActiYatiRn lRRps cRntain q-p Ueagents with q=25, 24, 2, 22, 21, 20.

Again, the stUaight line Ȗ ȝcRUUespRnGs tR saGGle-nRGe biIuUcatiRn

.

On the cRntUaUy, lRng lRRps RI actiYatiRn GR nRt suppUess RscillatiRns. :e pUesent in )ig. 5 Yalues RI the peUiRG RI RscillatiRns RbtaineG IURP nuPeUical sRlutiRns. The RscillatiRns haYe the highest IUeTuency when the lRRp RI actiYatiRn is slightly lRngeU than the lRRp RI UepUessiRn S  T-S . The peUiRG appURaches a cRnstant Yalue with incUeasing nuPbeU RI elePents in the lRRp RI actiYatiRn. At q<p, the shRUteU lRRp RI actiYatiRn slRws GRwn the RscillatiRns. In the pRint with _cRs p

p S P



anG Ȗ , puUe iPaginaUy eigenYalue is eTual tR WDQʌS anG iPplies the peUiRG RI 19.4 IRU S cRnIURnt IRUPulas

1 2 3 4 5 6 2 4 6 8

ȝ

Ȗ

25 20

22 anG 2. As can be seen in )ig. 5, the peUiRG RI RscillatiRns has Rnly slightly higheU Yalue with lRng actiYatiRn lRRps.

16 20 24 28 32 q 16 20 24 28 32

Pe

rio

d T

FIG. 5. DepenGence RI RscillatiRns peUiRG Rn the nuPbeU RI elePents in the lRRp RI actiYatiRn q-. The unit RI tiPe is the UecipURcity RI Gecay Uate cRnstants, RU Pean liIe tiPe RI PRNA x1P Q 

Let us cRnsiGeU eTuatiRns 29 anG UelatiRns 0 in a special case RI

/ p

I S . SRlutiRns 0 giYe then Ȗ  anG cRs

p

p

S P



in a geneUal case RI any

q>p. These cRRUGinates ȝȖ UeIeU tR the pRint which is cRPPRn IRU all RI the

biIuUcatiRnal cuUYes in )igs  anG 4. In cases RI lRng actiYatiRn lRRps, when q is an integeU Pultiple RI p:

,

2,,...

q

jp j

_1

the secRnG Rne RI the twR eTuatiRns 29 is satisIieG with ij ʌSIRU any Yalue RI ȝ anG Ȗ SubstitutiRn RI these Yalues RI q anG ijintR the IiUst eTuatiRn RI 29 giYes UelatiRn Ȗȝwhich shRulG be satisIieG at +RpI biIuUcatiRn

 2   1 cRs cRs . j p p j p p S S J P  _§  _·  ¨_¨  ¸_¸ © ¹ 2

RelatiRn 2 is illustUateG in )ig. 6. The stUaight line with M  UepURGuces e[actly the cRnGitiRn RI +RpI biIuUcatiRn in the PRst syPPetUical systePs 2. The biggest paUt RI

FIG. 6. +RpI biIuUcatiRn cuUYes in systePs wheUe q is an integeU Pultiple RI ST MS The Yalues RI j aUe shRwn at UespectiYe stUaight tlines. DRtteG line cRUUespRnGs tR the saGGle-nRGe biIuUcatiRn Ȗ ȝ

the plane ȝ Ȗ, cRUUespRnGing tR autRRscillatiRns, haYe the systePs with q=3 when the actiYatiRn lRRp is twice as lRng as the lRRp RI UepUessiRn. In the liPit RI YeUy high Yalues RI j at YeUy lRng lRRps RI actiYatiRn the IunctiRn 2 gRes tR the stUaight line with an inIinite slRpe

cRs p p

S

P

 

anG the e[istence RI RscillatRUy sRlutiRns GRes nRt PRUe GepenG Rn the paUaPeteU Ȗ RelatiRns RbtaineG in the sectiRn 4.2. enable us tR IinG eTuilibUiuP pRints with a paiU RI puUe iPaginaUy eigenYalues. In +RpI biIuUcatiRn, the Ueal paUt RI this paiU RI cRPple[ eigenYalues shRulG change its sign. :e useG nuPeUical calculatiRns anG checkeG in seYeUal cases that the change RI sign GRes Ueally take place. UnIRUtunately, we haYe nRt IRunG any cleaU pURRI RI this change in

1 2 3 4 5 5 10 15 20

ȝ

Ȗ

2



4

5

6



1 2 3 4 5 5 10 15 20

ȝ

Ȗ

3

5

7

6

4

2

general case. In the case oI T S, relations 22 anG 23 iPply an analytical e[pression Ior the biIurcating pair oI eigenYalues:





1





1

cos

1

sin .

p

i

p

p

p

S

S

O

P J



 r

P J



₃₄

In this case, the change oI the sign oI 5HȜat crossing biIurcational Yalue oI

ȝ-Ȗ curYe 2 in Fig. 6 is eYiGent. The characteristic eTuation 27 has a relatiYely

siPple analytical solution also in the case oI q=3p. The biIurcating pair oI eigenYalues is then giYen by 35:

1 1 2

₄

2

₄

cos

1

sin .

2

2

p p

i

p

p

P

P

J

S

P

P

J

S

O

§

_¨

¨





·

_¸

¸

 r

¨

§

_¨





¸

·

_¸

©

¹

©

¹

35

The real part oI 35 is an icreasing Iunction oI ȝanG ȖanG Yanishes at ȝ anG ȖsatisIying relation 32 with j=3 curYe 3 in Fig. 6. So, it Pust change its sign by crossing this curYe.

4.3. REPRESSION AND ACTIVATION LOOPS :IT+ DIFFERENT RATE CONSTANTS OF DECA<

Let us now consiGer the systeP whose both loops consist oI the saPe nuPber oI elePents T S, but they GiIIer in the rate oI Gecay oI their reagents. :e assuPe that the constants ki in the repression loop i=1,..,p haYe the unit

Yalue anG those in the actiYation loop are eTual to k. In such a case the characteristic eTuation 16 can be written as:



1

_

O

 

p

_k

_

O



p

_

J

_k

p



1

_

O



p

_

P



_k

_

O



p

0.

36 IntroGucing S  anG Ȝ LȦ into eTuation 36 allows us to e[press paraPeters Ȗ anG ȝ as Iunctions oI Ȧ_{. ParaPetric plots ȖȦ}2_>ȝȦ2_{@ show the}

Yalues oI Ȗ anG ȝ, which corresponG to +opI biIurcation. The plots are presenteG in Figs 7 anG 8. Oscillatory solutions are possible in areas below the straight line Ȗ ȝ anG Yalues oI ȝ higher than those in biIurcation curYes.

All oI the consiGereG asyPPetrical systePs haYe the loop oI repression consisteG oI 10 reagents Gecaying with the rate constant eTual to 1. Their actiYation loops GiIIer one IroP another by nuPbers oI elePents in the loops oI actiYation or by the rate constants oI Gecay. There is obYious TualitatiYe siPilarity between the systePs with elongateG actiYation loops anG the systePs with slower Gecay oI reagents in these loops Figs 4 anG 7. Finite iPaginary

eigenYalues appear on both siGes oI the straight line Ȗ ȝ In spite oI this Iact, oscillations arounG eTuilibriuP points with Ȗ!ȝ are iPpossible. All eTuilibriuP points IroP this part oI the ȝȖ plane haYe one real positiYe eigenYalue. EYolution oI the systeP takes it away IroP the Yicinity oI such an eTuilibriuP. Let us note that in both cases the turnoYer oI actiYator is slower than that oI the repressor.

FIG. 7. CurYes Ȗȝ Ior +opI biIurcation in the systePs with eTual nuPber oI elePents in both loops. Rates oI Gecay constants are eTual to unity in the loop oI repression. The Yalues oI Gecay rate constants in the loop oI actiYation N are shown at curYes.

FIG. 8. CurYes Ȗȝ Ior +opI biIurcation in the systePs with eTual nuPber oI elePents in both loops. Rate oI Gecay constants are eTual to unity in the loop oI repression. The curYes corresponGs to Gecay rate constants in the loop oI actiYation k=2, 1.8, 1.7, 1.6, 1.5, 1.4, 1.3, 1.2, anG 1.1. 1 2 3 4 5 6 7 2 4 6 8 10 1.5 2 2.5 3 3.5 4 1 2 3 4 5 6

ȝ

0.6

0.7

0.8

0.9

1

ȝ

Ȗ

2

1.1

1.5 2 2.5 3 3.5 4 1 2 3 4 5 6

ȝ

2

1.1

Ȗ

On the other hanG, biIurcational curYes oI the systePs with shorteneG loops oI actiYation are Puch siPilar to those oI the systePs with higher rate constants oI Gecay in these loops Figs 3 anG 8. The latter systePs haYe pure iPaginary eigenYalues only at Ȗȝ At Ȗ ȝ+1, iPaginary eigenYalue attains ]ero. Possible oscillations arounG the eTuilibriuP points with Ȗ slightly lower than

ȝ shoulG haYe e[trePely low IreTuency. In this pair oI asyPPetrical cases,

actiYator’s loop has shorter tiPe oI turnoYer than the repressor’s loop.

All biIurcation curYes in Figs 3, 4, 7 anG 8 represent eTuilibriuP points with a pure iPaginary eigenYalue They haYe one coPPon point with cos

p p S P  § · ¨ ¸ © ¹ ,

ȝ§1.65172 at p=10, anG Ȗ=0. It can be inIerreG IroP relations 22 anG 23 that

in this biIurcation point iPaginary eigenYalue tan

p

S

Z anG we can e[pect oscillations with a perioG



2



2

tan /

T p

p

S

S | . NuPerical solutions suggest that this eYaluation oI the perioG is better than those baseG on the actual iPaginary part oI coPple[ eigenYalue. The turnoYer tiPe oI the repression loop is the Pain Iactor GeterPining the perioG oI oscillations. The actiYation loop with short turnoYer tiPe e[cluGes oscillations or Pakes theP slower. The slower actiYation loop has no signiIicant inIluence on the perioG oI oscillations. These rules are illustrateG in Figs 5 anG 9.

0 0.4 0.8 1.2 1.6

Decay constants k in the loop of activator

18 20 22 24 26 28 Pe riod of os ci llat ions T

FIG. 9. PerioG oI oscillations obtaineG IroP nuPerical solutions in systePs with 10 elePents in both loops, ki=1 in the loop oI repression, ki=k in the loop oI actiYation,

5. CONCLUSIONS

:e consiGereG a PoGel systeP oI gene e[pression regulateG by two transcription Iactors, repressor anG actiYator. :e assuPeG that transcription oI the gene takes place proYiGeG that the concentration oI the repressor is low anG, at the saPe tiPe, the concentration oI the actiYator is high. Repression anG actiYation are both cooperatiYe processes with respectiYe +ill coeIIicients m anG n.

+igh Yalues oI m proPotes oscillations anG high Yalues oI n Pake theP less probable. Oscillatory solutions are possible when GiIIerence m-n is suIIiciently high, see relation 25. :e introGuceG paraPeters Ȗ anG ȝ, Iunctions oI coorGinates oI eTuilibriuP points, which enableG us to GeriYe the characteristic eTuation anG the saGGle-noGe biIurcation in a Tuite general way 16,17. The +opI biIurcation was analy]eG in a Iew special cases. Oscillations are generateG in our systeP by a negatiYe IeeGback loop oI the *ooGwin’s type. A coupleG loop with positiYe IeeGback Goes not Gisturb oscillations when its turnoYer tiPe is longer than the turnoYer tiPe oI the negatiYe IeeGback loop. Oscillations are sloweG Gown or eYen Iully GaPpeG, when the loop oI actiYation has the turnoYer tiPe shorter than that oI the repression loop.

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