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Problem 6 (comes with worked problems that are a hint). Please
only solve if you will try to answer question in complete
fashion.
Problem 6 (20pts) Consider the 2D model of the activator-repressor clock given by n KA KA À =f(A,B) = n m - YAA, δΑ 1+ B A KA + KB 2.4,0 +QA (A)" () () (A) (A)" n KA B =g(A,B) AB,0 +aB KB dB 1+ - YBB. n KA Assuming that the activator and the repressor act as a dimer and as a monomer (n = 2, m= 1), respectively, identify parameter regions for the clock to produce sustained oscillations. =
Hint Sketch how the nullclines look like, linearize the dynamics, and use the implicit function theorem to conclude stability properties of the equilibria by analyzing the sign of the trace and determinant of the system matrix obtained via the linearization. Use the Poincare-Bendixson theorem to derive sufficient conditions between 7A and YB to ensure the emergence of oscillations. See Example 4–5 under BFS_Practice_Problems/Oscillations and Example 2 under BFS_Practice_Problems/Modules.
CHAPTER #3 Use the Porin care - Beudixsou theorence to denve conditious about when it is reasonable to expect that the Brusselator witle the dynamics below will oscillate. 2 at + xy - (b+1)x f(ty) bx-x²4 0 - at 0 bx- 02 a - X Summing them > a 을 Ý = g(x,y) ① Find the equilibrium points X Y - (5+1)x x²4 yields substituting it back yields y = 2/4 = 1 librium: (a, /) ② Linearize the depamics around the unique equilisnim (2xy - (5+1) x 6 A A 2 16-2xy T= trace (A)- (6-1)-a? 1,2 = { (+ 2 ST240) o = det (A) = -all-1 tabra ) ) There is a unique equi of 2x Ha 21 -6 ra +
- we the Stability depends real part of eigenvalues expect where the of oscillatious emergence unique equilibriucu switches froce stable node to unstable spical at TO a²=b-1 This is confirmed by simulatious below: - 10 a=1 b=1.9 2 a ² <b-1 stable mode oscillations no 2 3 10 al b=2.1 2 a > b-l a unstable spiral stable limit cycle 6 7 8 9 10
CHAPTER #3 Use the Poincare - Bendixsou the ocene to derive conditions about when the dioxide - iodine - malodive reaction wifle the dynamics below will oscillate. systeem s sa-x- 4 xy = f(x,y ) - , a>o 17 bx (1 - 1 12 ) agerry) ( b>o Itx' ① Localize the equilisria exy a-x- Cound It x bx (1-4 itxe) x 20 can't be at au equilibrium assume that xo second constraint becomes 0- 1- 1+x
exy 3 a - 2 Ito 0 : 1 - hx - exy Itx? It? - Subtract the second from the first exy 0 a - X -4x + 4xy exy = a-5x Itx² It x² At аш equilibrium x = och substitute back into second constraint ya 1+x² = 2 a I + 25 Unique equilibrium is (x,y)- (§, 10) 2 25 Position / existence of equilibrium is independent of b
☆ stab ③ stability analysis of equilisrium Linearize the dynamics around equilibrium of ex of ay 3 a ²-125 - 202 alcon apot a²+25 2a2b -sab Trace : 3a²-sab- 125 a²+25 Determinants -Sab (30²125) + 4023b (2²+25)? 625ab -15 as thoa²b (a +257² 62506 + 25 ab labla+25) (a+25² Ca²+25)? a 2sab 20 a2+25
Therefore have to understand how the we sign of the mace depends bu a and b 5 5 30²-125 sa unique >0 trace >o det unstable eq oscillatiour a a=10 b=5 125 ñ 6.46 50 stable eq oscillatious 45 40 35 30 25 20 15 10 5 0 0 10 20 30 40 50 60 70 80 90 3 4 5 6 7 uustable oscillations -
3 To easure need the emergence of oscillatious, we to construct trapping region that invoke the Poincare - Bendixsou then a So we can ។ y +x LO y = bx (1 - 4 tahun) 1 I want these to hold → seek for sy & * You x=0 ²= a- x- 4xy Hx? <o ý=axo to х y = 0 -> ý = bx>0 I want 4x4 & 。 l-t co a-X- 1+x Itx d ' y> lpx" 4x (1+x²) a- * - a-x-hx co 1+x² a <5x < => *x Х yle y > Hai د > It 2 а 25
CHAPTER #5 AR-clock to Ideutify parameter regious for the produce sustained oscillatious. (A (B 1. Dynamics in 4D mRNA dynamics are fast compared to protein dynamics 2. Dynamics in 2D QSS approximation: as if mRNA were constantly at their equilibrium linearization AR clock implicit function thm 3. Activator as monomer det>O, tr<0 stable equilibrium linearization implicit function thm 4. Activator as dimer det>O tr>0 if ... unstable equilibrium
4D 0 Dynamics in ш, . f(A8) - Дид Ғ ( А= 24 А up - – і - 2,4 - 13 F, (А) - Удшg B 2 5 F (A,B) (*)(, ) КАО + од (e)" LA 4 1 + t о-во F, (А) = + в (4) 1+ се, ч () - in 2D (ass approximation ) ② Dynamics ů RO A АША F. (A,B) 4 F2(A) 53 шg мо са Å = ZA F, (A18) - 8AA F) - E с f(AB) fa (A,B) B ї. Il 2 др F() - За Ж 1) 8BB g (A,B) fz (A)
3 ® Activator acts as a monomer (mol) luplicit function theorem. f(A, B) = 0 8(A, B) = 0 dB dA af 24 24 B f (A,B) = 0 OB ag 2A dB dA A - O g(AIB) - 4 23 Linearization about the Cunknown) equilibrium ӘҒ DA of ƏB JE J af + of ag ZA 28 Te a T = trace (J) - Dedet())- 2 - JA ag Frou f(A,B) and g(A,B) - ag f ) >0 colco مونتا zo DA af சில < o
Froue implicit function theorem: dB 2 O 4 dB dA dA Ғ () - о 8*18). о со (әҒ 24 СОҒ ЭТ Ә 8 Оe 4 oz 24 ЭТ 20 24 25 әА ə К- 24 2 ЯА Ә О. ӘР Әя әА Ә0 4 Әf ЯА → equilibrium is stable т. 25 ЭН + + Әз 93 + о as Q when the activentor acts шолошег Си = 1), can't guarantee stable periodic abits Cascillations) we
3 © Activator acts as dimer (n=2) luplicit function theorem. f(A, B) = 0 g(A, B) = 0 dB dA f (ALB) - 4 toldala B ag 24 dB dA g(AIB) - 4 o ag 2B A Linearization about the Cunknown) equilibrium . af DA of ƏB JE told apa 器 Dodet(). ? 29 - Froue f(A,B) and g(4,8) ► f( T = trace (J) - af JA af aa + (= JA 28 -> ag DA >0 colco موا ag LO ӘҒ DB <o
From ieplicit fuuchion theorem : de dA dB dA V о 3 0.1) - о ҒА ) = 0 Ә8 Оe af Оe Of ӘК о д Zo 20 25 Əа ар 20 24- Ә0 90 9. Әя Da - में ƏS 2 O ә) Ә0 Т- + Gl818 ора) T>0 we need to ensure о Т? afi га -8A - 8B > о *, Әe ӘҒ. -8A РА 8B > Хе > |
- We af, ZA -8A > 82 want for oscillatious Repressor timescale needs to be su sufficiently slow compared to that of the actuator decrease 80 increase stability of repressor 183 50 50 = 7.5 40 40 А 30 30 A, B A, B 20 20 IB B 10 10 0 0 20 80 100 0 0 20 80 100 40 60 time 40 60 time