Modelling chemical kinetics

Transcription

1 Modelling chemical kinetics Nicolas Le Novère, Institute, EMBL-EBI

2 Systems Biology models ODE models Reconstruction of state variable evolution from process descriptions: Processes can be combined in ODEs (for deterministic simulations); transformed in propensities (for stochastic simulations) Systems can be reconfigured quickly by adding or removing a process A a p R B b q P Q substances are reaction R that substances A and B consumed produces P and Q by

3 ATP is consumed by processes 1 and 3, and produced by processes 7 and 10 (for 1 reactions 1 and 3, there are 2 reactions 7 and 10)

4 Chemical kinetics and fluxes E S2 S1 P

5 Statistical physics and chemical reaction

6 Statistical physics and chemical reaction Probability to find an object in a container within an interval of time

7 Statistical physics and chemical reaction Probability to find an object in a container within an interval of time

8 Law of Mass Action Waage and Guldberg (1864) activity rate-constant velocity stoichiometry

9 Law of Mass Action Waage and Guldberg (1864) activity rate-constant stoichiometry velocity gas solution

10 Evolution of a reactant Velocity multiplied by stoichiometry negative if consumption, positive if production Example of a unimolecular reaction

11 Evolution of a reactant Velocity multiplied by stoichiometry negative if consumption, positive if production Example of a unimolecular reaction

12 Evolution of a reactant Velocity multiplied by stoichiometry negative if consumption, positive if production Example of a unimolecular reaction [x]0 [x]0/2 [x]0/e ln2/k 1/k t

13 Reversible reaction is equivalent to

14 Reversible reaction is equivalent to

15 Example of an enzymatic reaction

16 Example of an enzymatic reaction

17 Example of an enzymatic reaction

18 Example of an enzymatic reaction [x] Not feasible in general Numerical integration t

19 Numerical integration (only for info. Not needed) Euler method: [x] [x]t+dt [x]t Dt t

20 Numerical integration (only for info. Not needed) Euler method: [x] [x]t+dt [x]t Dt t

21 Numerical integration (only for info. Not needed) Euler method: [x] [x]t+dt [x]t Dt t 4th order Runge-Kutta: with [x] Dt t

22 Choose the right formalism

23 Choose the right formalism irreversible catalysis

24 Choose the right formalism irreversible catalysis product escapes before rebinding

25 Choose the right formalism irreversible catalysis product escapes before rebinding quasi-steady-state

26 Enzyme kinetics Victor Henri (1903) Lois Générales de l'action des Diastases. Paris, Hermann. Leonor Michaelis, Maud Menten (1913). Die Kinetik der Invertinwirkung, Biochem. Z. 49: George Edward Briggs and John Burdon Sanderson Haldane (1925) A note on the kinetics of enzyme action, Biochem. J., 19:

27 Briggs-Haldane on Henri-Michaelis-Menten (only for info. Not needed) [E]=[E0]-[ES]

28 Briggs-Haldane on Henri-Michaelis-Menten (only for info. Not needed) [E]=[E0]-[ES] steady-state!!!

29 Generalisation: activators x y

30 Generalisation: activators d[y]/dt x y v 50%v Ka a x y [a]

31 Generalisation: activators d[y]/dt x y v 50%v [a] Ka d[y]/dt a v x y 50%v Ka log[a]

32 Generalisation: activators d[y]/dt x y v 50%v [a] Ka d[y]/dt a v x y 50%v Ka (NB: You can derive that as the fraction of target bound to the activator) log[a]

33 Phenomenological ultrasensitivity d[y]/dt v 50%v d[y]/dt Ka log[a] Ka log[a] Ka log[a] v 50%v d[y]/dt v 50%v

34 The Hill function Hill (1910) J Physiol 40: iv-vii.

35 The Hill function Hill (1910) J Physiol 40: iv-vii. 1 Y 0 1/K [x]

36 Generalisation: inhibitors x y d[y]/dt i v 50%v x y Ki (NB: You can derive that as the fraction of target not bound to the inhibitor) log[i]

37 Mathematics are beautiful

38 Generalisation: activators and inhibitors x y log [i] a x log [a] y i

39 absolute Vs relative activators a x y d[x]/dt v 50%v Ka log[a]

40 absolute Vs relative activators a x a y x y d[y]/dt d[y]/dt v(1+ v v 50%v Ka log[a] Ka log[a]

41 1 compartment

42 2 compartments

43 2 compartments A B Per unit of time

44 2 compartments with different volumes A B

45 2 compartments with different volumes A B Per unit of time

46 2 compartments with different volumes A B Per unit of time

47 2 compartments with different volumes A B Per unit of time

48 2 compartments with different volumes Stoichiometries (concentration change per Reaction events) are in fact scaling with volumes: A B Per unit of time

49 Homeostasis How can-we maintain a stable level with a dynamic system? Ø x Ø

50 Homeostasis How can-we maintain a stable level with a dynamic system? Ø x Ø

51 Homeostasis How can-we maintain a stable level with a dynamic system? Ø [x] time x Ø

52 Homeostasis How can-we maintain a stable level with a dynamic system? x Ø 0 [x] time Ø

53 Homeostasis How can-we maintain a stable level with a dynamic system? Ø x Ø 1 0 [x] time

54 Homeostasis How can-we maintain a stable level with a dynamic system? Ø x Ø 1 0 [x] time

55 Questions?

56 Conformational equilibrium

57 Binding equilibrium

58 How does a ligand activate its target?

59 How does a ligand activate its target?

60 How does a ligand activate its target? hint: K1>1

61 Add energies Multiply constants +1 quantum energy = constant divided by 10 Explore constants exponentially: Parameter space