1. Lecture 5: Chemical Reactions Outline: basic concepts Nonlinearities: saturation: Michaelis-Menten kinetics switching: Goldbeter-Koshland

2. Basics Simple reaction

3. Basics Simple reaction Number of AB pairs in volume v :

4. Basics Simple reaction Number of AB pairs in volume v : Reaction equation:

5. Basics Simple reaction Number of AB pairs in volume v : Reaction equation:

6. Basics Simple reaction Number of AB pairs in volume v : Reaction equation: Note: r has units 1/t, k has units volume/t

7. Reversible reactions, stoichiometry Reaction can go both ways:

8. Reversible reactions, stoichiometry Reaction can go both ways:

9. Reversible reactions, stoichiometry Reaction can go both ways: Equilibrium:

10. Reversible reactions, stoichiometry Reaction can go both ways: Equilibrium: Stoichiometry:

11. Reversible reactions, stoichiometry Reaction can go both ways: Equilibrium: Stoichiometry:

12. Reversible reactions, stoichiometry Reaction can go both ways: Equilibrium: Stoichiometry:

13. Michaelis-Menten Enzyme + substrate complex -> enzyme + product reversible  rates a,d

14. Michaelis-Menten Enzyme + substrate complex -> enzyme + product

15. Michaelis-Menten Enzyme + substrate complex -> enzyme + product

16. Michaelis-Menten Enzyme + substrate complex -> enzyme + product reversible  rates a,d

17. Michaelis-Menten Enzyme + substrate complex -> enzyme + product reversible   irreversible rates a,d rate k

18. Michaelis-Menten Enzyme + substrate complex -> enzyme + product reversible   irreversible rates a,d rate k Rate equations:

19. Michaelis-Menten Enzyme + substrate complex -> enzyme + product reversible   irreversible rates a,d rate k Rate equations:

20. Michaelis-Menten Enzyme + substrate complex -> enzyme + product reversible   irreversible rates a,d rate k Rate equations:

21. Michaelis-Menten Enzyme + substrate complex -> enzyme + product reversible   irreversible rates a,d rate k Rate equations:

22. Reduction: Eliminate E :

23. Reduction: Eliminate E : To solve:

24. Reduction: Eliminate E : To solve: Initial conditions:

25. Initial regime: Lots of S, [S] hardly changes from S 0

26. Initial regime: Lots of S, [S] hardly changes from S 0

27. Initial regime: Lots of S, [S] hardly changes from S 0

28. Initial regime: Lots of S, [S] hardly changes from S 0

29. Initial regime: Lots of S, [S] hardly changes from S 0

30. Initial regime: Lots of S, [S] hardly changes from S 0

31. Initial regime: Lots of S, [S] hardly changes from S 0 Michaelis constant

32. Slow dynamics If [S] changes slowly compared with τ fast,

33. Slow dynamics If [S] changes slowly compared with τ fast,

34. Slow dynamics If [S] changes slowly compared with τ fast,

35. Slow dynamics If [S] changes slowly compared with τ fast,

36. Slow dynamics If [S] changes slowly compared with τ fast,

37. Slow dynamics If [S] changes slowly compared with τ fast,

38. Slow dynamics If [S] changes slowly compared with τ fast, consistent

39. Result: Reaction rate

40. Result: Reaction rate saturation

41. Cooperative binding

42. Equations ->

43. Cooperative binding After fast transient: Equations ->

44. Cooperative binding Hill coefficient n After fast transient: Equations ->

45. Goldbeter-Koshland switching 2 MM reactions, 1 in each direction:

46. Goldbeter-Koshland switching 2 MM reactions, 1 in each direction:

47. Goldbeter-Koshland switching 2 MM reactions, 1 in each direction:

48. Goldbeter-Koshland switching 2 MM reactions, 1 in each direction:

49. Goldbeter-Koshland switching 2 MM reactions, 1 in each direction:

50. Goldbeter-Koshland switching 2 MM reactions, 1 in each direction:

51. Goldbeter-Koshland switching 2 MM reactions, 1 in each direction:

52. Goldbeter-Koshland switching 2 MM reactions, 1 in each direction: Steady state (add 1 st 2 or 2 nd 2 eqns):

53. Steady state:

58. Quadratic equation for S 1

59. Solution: where

60. Solution: where S 1, 1-S 1

61. Solution: where S 1, 1-S 1 Sharp switching for small κ