Oscillations in Michaelis-Menten Systems

Transcription

1 Oscillations in Michaelis-Menten Systems Hwai-Ray Tung Brown University July 18, 2017 Hwai-Ray Tung (Brown University) Oscillations in Michaelis-Menten Systems July 18, / 22

2 Background A simple example of genetic oscillation Gene makes compound (transcription factor) Transcription factor binds to promoter Circadian Clocks Hwai-Ray Tung (Brown University) Oscillations in Michaelis-Menten Systems July 18, / 22

3 Background A simple example of genetic oscillation Gene makes compound (transcription factor) Transcription factor binds to promoter Circadian Clocks Mass Action Kinetics Write chemical reaction network as a system of differential equations Michaelis-Menten (MM) Approximation for Biochemical Systems Assume low concentration of intermediates Allows for elimination of variables, reducing differential equations from mass action Michaelis-Menten approximation has oscillations implies original has oscillations Hwai-Ray Tung (Brown University) Oscillations in Michaelis-Menten Systems July 18, / 22

4 Michaelis-Menten System (Dual Futile Cycle) Phosphorylation Dephosphorylation S 0 + E k 1 S 0 E k 3 S 1 + E k 4 S 1 E k 6 S 2 + E k2 k5 S 2 + F l 1 S 2 F l 3 S 1 + F l 4 S 1 F l 6 S 0 + F l2 l5 Oscillatory behavior is unknown. Wang and Sontag (2008) showed Michaelis-Menten approximation has no oscillations Bozeman and Morales (REU 2016) showed Michaelis-Menten approximation has no oscillations with more elementary techniques Hwai-Ray Tung (Brown University) Oscillations in Michaelis-Menten Systems July 18, / 22

5 Processive vs Distributive Distributive: S 0 + E k 1 S 0 E k 3 S 1 + E k 4 S 1 E k 6 S 2 + E k2 k5 S 2 + F l 1 S 2 F l 3 S 1 + F l 4 S 1 F l 6 S 0 + F l2 l5 Processive: S 0 + E k 1 S 0 E k 7 S 1 E k 6 S 2 + E k2 S 2 + F l 1 S 2 F l 7 S 1 F l 6 S 0 + F l2 Hwai-Ray Tung (Brown University) Oscillations in Michaelis-Menten Systems July 18, / 22

6 This Year s Project S 1 + F S 2 + F S 2 F S 1 F S 0 + F S 0 + E S 0 E S 1 E S 2 + E S 1 + E Does it have oscillations? Does its Michaelis-Menten approximation have oscillations? Hwai-Ray Tung (Brown University) Oscillations in Michaelis-Menten Systems July 18, / 22

7 Mass Action Kinetics Write chemical reaction network as a system of differential equations Example for rate of [S 0 E] S 0 + E k 1 S 0 E gives Hwai-Ray Tung (Brown University) Oscillations in Michaelis-Menten Systems July 18, / 22

8 Mass Action Kinetics Write chemical reaction network as a system of differential equations Example for rate of [S 0 E] S 0 + E k 1 S 0 E gives d[s 0 E] = k 1 [S 0 ][E] Hwai-Ray Tung (Brown University) Oscillations in Michaelis-Menten Systems July 18, / 22

9 Mass Action Kinetics Write chemical reaction network as a system of differential equations Example for rate of [S 0 E] S 0 + E k 1 S 0 E gives S 0 + E k 2 S 0 E gives d[s 0 E] = k 1 [S 0 ][E] Hwai-Ray Tung (Brown University) Oscillations in Michaelis-Menten Systems July 18, / 22

10 Mass Action Kinetics Write chemical reaction network as a system of differential equations Example for rate of [S 0 E] S 0 + E k 1 S 0 E gives S 0 + E k 2 S 0 E gives d[s 0 E] d[s 0 E] = k 1 [S 0 ][E] = k 2 [S 0 E] Hwai-Ray Tung (Brown University) Oscillations in Michaelis-Menten Systems July 18, / 22

11 Mass Action Kinetics Write chemical reaction network as a system of differential equations Example for rate of [S 0 E] S 0 + E k 1 S 0 E gives S 0 + E k 2 S 0 E gives d[s 0 E] = k 1 [S 0 ][E] S 0 + E k 1 k2 S 0 E gives d[s 0 E] = k 2 [S 0 E] Hwai-Ray Tung (Brown University) Oscillations in Michaelis-Menten Systems July 18, / 22

12 Mass Action Kinetics Write chemical reaction network as a system of differential equations Example for rate of [S 0 E] S 0 + E k 1 S 0 E gives S 0 + E k 2 S 0 E gives d[s 0 E] = k 1 [S 0 ][E] S 0 + E k 1 k2 S 0 E gives d[s 0 E] = k 2 [S 0 E] d[s 0 E] = k 1 [S 0 ][E] k 2 [S 0 E] Hwai-Ray Tung (Brown University) Oscillations in Michaelis-Menten Systems July 18, / 22

13 Corresponding ODEs d[s 0] = l 6[S 1F ] k 1[S 0][E] + k 2[S 0E] d[s 2] = k 6[S 1E] l 1[S 2][F ] + l 2[S 2F ] d[s 1] = k 3[S 0E] k 4[S 1][E] + k 5[S 1E] + l 3[S 2F ] + l 5[S 1F ] l 4[S 1][F ] d[e] = (k 2 + k 3)[S 0E] + (k 5 + k 6)[S 1E] k 1[S 0][E] k 4[S 1][E] d[f ] = (l 2 + l 3)[S 2F ] + (l 5 + l 6)[S 1F ] l 1[S 2][F ] l 4[S 1][F ] d[s 0E] = k 1[S 0][E] (k 2 + k 3)[S 0E] k 7[S 0E] d[s 1E] = k 4[S 1][E] (k 5 + k 6)[S 1E]+k 7[S 0E] d[s 2F ] = l 1[S 2][F ] (l 2 + l 3)[S 2F ] l 7[S 2F ] d[s 1F ] = l 4[S 1][F ] (l 5 + l 6)[S 1F ]+l 7[S 2F ] Hwai-Ray Tung (Brown University) Oscillations in Michaelis-Menten Systems July 18, / 22

14 Michaelis-Menten Approximation of ODEs: Conservation Equations Conservation equations refer to conservation of mass. S T = [S 0 ] + [S 1 ] + [S 2 ] + [S 0 E] + [S 1 E] + [S 2 F ] + [S 1 F ] E T = [E] + [S 0 E] + [S 1 E] F T = [F ] + [S 2 F ] + [S 1 F ] S 1 + F S 2 + F S 2 F S 1 F S 0 + F S 0 + E S 0 E S 1 E S 2 + E S 1 + E Hwai-Ray Tung (Brown University) Oscillations in Michaelis-Menten Systems July 18, / 22

15 Michaelis-Menten Approximation of ODEs: Conservation Equations Conservation equations refer to conservation of mass. S T = [S 0 ] + [S 1 ] + [S 2 ] + [S 0 E] + [S 1 E] + [S 2 F ] + [S 1 F ] E T = [E] + [S 0 E] + [S 1 E] F T = [F ] + [S 2 F ] + [S 1 F ] d[e] d[s 0 E] d[s 1 E] = (k 2 + k 3 )[S 0 E] + (k 5 + k 6 )[S 1 E] k 1 [S 0 ][E] k 4 [S 1 ][E] = k 1 [S 0 ][E] (k 2 + k 3 )[S 0 E] k 7 [S 0 E] = k 4 [S 1 ][E] (k 5 + k 6 )[S 1 E]+k 7 [S 0 E] Hwai-Ray Tung (Brown University) Oscillations in Michaelis-Menten Systems July 18, / 22

16 Michaelis-Menten Approximation of ODEs: Assumption We assume enzyme concentrations are small: E T = εẽt, F T = ε F T, [E] = ε[ẽ], [F ] = ε[ F ], [S 0 E] = ε[ S 0 E], [S 1 E] = ε[ S 1 E], [S 2 F ] = ε[ S 2 F ], [S 1 F ] = ε[ S 1 F ], τ = εt Hwai-Ray Tung (Brown University) Oscillations in Michaelis-Menten Systems July 18, / 22

17 Michaelis-Menten Approximation of ODEs: Plug in d[s 0] = l 6[ S 1F ] k 1[S 0][Ẽ] dτ + k2[ S 0E] d[s 2] = k 6[ S 1E] l 1[S 2][ dτ F ] + l 2[ S 2F ] X d[s1] = k 3[S 0E] k 4[S 1][E] + k 5[S 1E] + l 3[S 2F ] + l 5[S 1F ] l 4[S 1][F ] X d[e] = (k 2 + k 3)[S 0E] + (k 5 + k 6)[S 1E] k 1[S 0][E] k 4[S 1][E] X d[f ] = (l 2 + l 3)[S 2F ] + (l 5 + l 6)[S 1F ] l 1[S 2][F ] l 4[S 1][F ] ε d[ S 0E] dτ ε d[ S 1E] dτ ε d[ S 2F ] dτ ε d[ S 1F ] dτ = k 1[S 0][Ẽ] (k2 + k3)[ S 0E] k 7[ S 0E] = k 4[S 1][Ẽ] (k5 + k6)[ S 1E]+k 7[ S 0E] = l 1[S 2][ F ] (l 2 + l 3)[ S 2F ] l 7[ S 2F ] = l 4[S 1][ F ] (l 5 + l 6)[ S 1F ]+l 7[ S 2F ] Hwai-Ray Tung (Brown University) Oscillations in Michaelis-Menten Systems July 18, / 22

18 Michaelis-Menten Approximation of ODEs d[s 0 ] dτ d[s 2 ] dτ = (a 1[S 1 ] + a 2 [S 2 ])[ F T ] 1 + c 2 [S 1 ] + d 2 [S 2 ] = (a 4[S 1 ] + a 5 [S 0 ])[ẼT ] 1 + b 1 [S 0 ] + c 1 [S 1 ] [S 1 ] = S T [S 0 ] [S 2 ] a 3 [S 0 ][ẼT ] 1 + b 1 [S 0 ] + c 1 [S 1 ] a 6 [S 2 ][ F T ] 1 + c 2 [S 1 ] + d 2 [S 2 ] b 1 = k 1(k 5 + k 6 + k 7) (k 2 + k 3 + k 7)(k 5 + k 6) k4 c 1 = k 5 + k 6 l4l6 a 1 = l 5 + l 6 l 1l 6l 7 a 2= (l 2 + l 3 + l 7)(l 5 + l 6) k1(k3 + k7) a 3 = (k 2 + k 3 + k 7) d 1 = l 1(l 5 + l 6 + l 7) (l 2 + l 3 + l 7)(l 5 + l 6) l4 c 2 = l 5 + l 6 k4k6 a 4 = k 5 + k 6 k 1k 6k 7 a 5= (k 2 + k 3 + k 7)(k 5 + k 6) l1(l3 + l7) a 6 = (l 2 + l 3 + l 7) Hwai-Ray Tung (Brown University) Oscillations in Michaelis-Menten Systems July 18, / 22

19 Dulac s Criterion f ([S 0 ], [S 2 ]) = d[s 0] dτ g([s 0 ], [S 2 ]) = d[s 2] dτ = (a 1[S 1 ] + a 2 [S 2 ])[ F T ] 1 + c 2 [S 1 ] + d 2 [S 2 ] = (a 4[S 1 ] + a 5 [S 0 ])[ẼT ] 1 + b 1 [S 0 ] + c 1 [S 1 ] a 3 [S 0 ][ẼT ] 1 + b 1 [S 0 ] + c 1 [S 1 ] a 6 [S 2 ][ F T ] 1 + c 2 [S 1 ] + d 2 [S 2 ] Theorem (Dulac) If the sign of df d[s 0 ] + dg d[s 2 ] does not change across a simply connected domain in R 2, the system does not exhibit oscillations in the domain. Hwai-Ray Tung (Brown University) Oscillations in Michaelis-Menten Systems July 18, / 22

20 Results Theorem (T.) In order for the reduced system to exhibit oscillations, one of the following must be true: (a 5 c 1 a 4 b 1 a 3 c 1 )S T a 3 + a 4 (a 2 c 2 a 1 d 1 a 6 c 2 )S T a 1 + a 6 Hwai-Ray Tung (Brown University) Oscillations in Michaelis-Menten Systems July 18, / 22

21 Results Theorem (T.) In order for the reduced system to exhibit oscillations, one of the following must be true: (a 5 c 1 a 4 b 1 a 3 c 1 )S T a 3 + a 4 Corollary (a 2 c 2 a 1 d 1 a 6 c 2 )S T a 1 + a 6 The MM approximation of the distributive model does not have oscillations, as shown earlier by Bozeman & Morales (2016) and Wang & Sontag (2008). Corollary The MM approximation of the processive model does not have oscillations, as immediately implied from work by Conradi et al. (2005) and Conradi & Shiu (2015). Hwai-Ray Tung (Brown University) Oscillations in Michaelis-Menten Systems July 18, / 22

22 Results Theorem (T.) In order for the reduced system to exhibit oscillations, one of the following must be true: (a 5 c 1 a 4 b 1 a 3 c 1 )S T a 3 + a 4 Corollary (a 2 c 2 a 1 d 1 a 6 c 2 )S T a 1 + a 6 The MM approximation of the distributive model does not have oscillations, as shown earlier by Bozeman & Morales (2016) and Wang & Sontag (2008). Corollary The MM approximation of the processive model does not have oscillations, as immediately implied from work by Conradi et al. (2005) and Conradi & Shiu (2015). Hwai-Ray Tung (Brown University) Oscillations in Michaelis-Menten Systems July 18, / 22

23 Results Theorem (T.) In order for the reduced system to exhibit oscillations, one of the following must be true: (a 5 c 1 a 4 b 1 a 3 c 1 )S T a 3 + a 4 Corollary (a 2 c 2 a 1 d 1 a 6 c 2 )S T a 1 + a 6 The MM approximation of the distributive model does not have oscillations, as shown earlier by Bozeman & Morales (2016) and Wang & Sontag (2008). Corollary The MM approximation of the processive model does not have oscillations, as immediately implied from work by Conradi et al. (2005) and Conradi & Shiu (2015). Hwai-Ray Tung (Brown University) Oscillations in Michaelis-Menten Systems July 18, / 22

24 Results Theorem (T.) In order for the reduced system to exhibit oscillations, one of the following must be true: (a 5 c 1 a 4 b 1 a 3 c 1 )S T a 3 + a 4 (a 2 c 2 a 1 d 1 a 6 c 2 )S T a 1 + a 6 Corollary The MM approximation of the mixed-mechanism model does not have oscillations. S 0 + E k 1 S 0 E k 3 S 1 + E k 4 S 1 E k 6 S 2 + E k2 k5 S 2 + F l 1 S 2 F l 7 S 1 F l 6 S 0 + F l2 Hwai-Ray Tung (Brown University) Oscillations in Michaelis-Menten Systems July 18, / 22

25 Future Directions: Does MM approx ever oscillate? My guess: Rarely, if ever. Hwai-Ray Tung (Brown University) Oscillations in Michaelis-Menten Systems July 18, / 22

26 Future Directions: Does MM approx ever oscillate? My guess: Rarely, if ever. Substitute each parameter with integer between 0 and 10. Hwai-Ray Tung (Brown University) Oscillations in Michaelis-Menten Systems July 18, / 22

27 Future Directions: Does MM approx ever oscillate? My guess: Rarely, if ever. Substitute each parameter with integer between 0 and 10. About 95% cannot oscillate by Dulac Hwai-Ray Tung (Brown University) Oscillations in Michaelis-Menten Systems July 18, / 22

28 Future Directions: Does MM approx ever oscillate? My guess: Rarely, if ever. Substitute each parameter with integer between 0 and 10. About 95% cannot oscillate by Dulac Of remaining 5%, about 99% reflect pattern below Blue Red Yellow Green Hwai-Ray Tung (Brown University) Oscillations in Michaelis-Menten Systems July 18, / 22

29 Future Directions: Does MM approx ever oscillate? My guess: Rarely, if ever. Substitute each parameter with integer between 0 and 10. About 95% cannot oscillate by Dulac Of remaining 5%, about 99% reflect pattern below Blue Red Yellow Green Possible proof from Wang & Sontag s approach (monotone systems theory) Hwai-Ray Tung (Brown University) Oscillations in Michaelis-Menten Systems July 18, / 22

30 Future Directions: Does MM approx ever oscillate? Hopf Bifurcations: Equilibrium changes stability type when parameters are changed Implies existence of oscillations generally Routh-Hurwitz Criterion gives necessary conditions for Hopf bifurcation Hwai-Ray Tung (Brown University) Oscillations in Michaelis-Menten Systems July 18, / 22

31 Future Directions: Does MM approx ever oscillate? Ex: Selkov model dx = x + ay + x 2 y dy = b ay x 2 y Hwai-Ray Tung (Brown University) Oscillations in Michaelis-Menten Systems July 18, / 22

32 Future Directions: Issues with MM Approximation Networks Can Oscillate MM Approx Can Oscillate Distributive Unknown No Processive No No Mixed-Mechanism Yes No Hwai-Ray Tung (Brown University) Oscillations in Michaelis-Menten Systems July 18, / 22

33 Future Directions: Issues with MM Approximation Conservation law S T S 0 + S 2 is violated. Hwai-Ray Tung (Brown University) Oscillations in Michaelis-Menten Systems July 18, / 22

34 Summary Hwai-Ray Tung (Brown University) Oscillations in Michaelis-Menten Systems July 18, / 22

35 Summary Wanted to examine oscillations in systems with processive and distributive elements Hwai-Ray Tung (Brown University) Oscillations in Michaelis-Menten Systems July 18, / 22

36 Summary Wanted to examine oscillations in systems with processive and distributive elements Applied Michaelis-Menten approximation to reduce number variables Hwai-Ray Tung (Brown University) Oscillations in Michaelis-Menten Systems July 18, / 22

37 Summary Wanted to examine oscillations in systems with processive and distributive elements Applied Michaelis-Menten approximation to reduce number variables Obtained a necessary condition for oscillations Hwai-Ray Tung (Brown University) Oscillations in Michaelis-Menten Systems July 18, / 22

38 Summary Wanted to examine oscillations in systems with processive and distributive elements Applied Michaelis-Menten approximation to reduce number variables Obtained a necessary condition for oscillations Discovered mixed-mechanism network has no oscillations, differing from original Hwai-Ray Tung (Brown University) Oscillations in Michaelis-Menten Systems July 18, / 22

39 Summary Wanted to examine oscillations in systems with processive and distributive elements Applied Michaelis-Menten approximation to reduce number variables Obtained a necessary condition for oscillations Discovered mixed-mechanism network has no oscillations, differing from original Future Directions Monotone Systems Theory if you think no oscillations Hopf Bifurcations if you think there are oscillations When does Michaelis-Menten Approximation preserve oscillations? Hwai-Ray Tung (Brown University) Oscillations in Michaelis-Menten Systems July 18, / 22

40 Thanks for Listening! Special thanks to Advisor: Dr. Anne Shiu Mentors: Nida Obatake, Ola Sobieska, Jonathan Tyler Host: Texas A&M University Funding: National Science Foundation Hwai-Ray Tung (Brown University) Oscillations in Michaelis-Menten Systems July 18, / 22