Basics of biochemical systems theory

1. Basics of biochemicalsystems theory I. Graphical representation of biochemical networks

2. Mass transport Glucoseout Glucosein Glucose6-P Fructose6-P Chemical conversion Glucose Glucose6-P ATP ADP Material flow

3. Glucose Glucose6-P ATP ADP Information flow FDP - + HK

4. Networks Glc G6P F6P FDP - - + PGI + + ATP ADP ATP ADP PFK HK

5. Fundamental types of variables • State variables: any quantities that may directly influence a process and whose values might change (eg. concentrations, temperature) • Flux variables: reaction rates v2 Glc G6P F6P FDP - - v4 + v3 v1 PGI + + ATP ADP ATP ADP PFK HK v5

6. Other types of variables Independent variables fixed or “isolated” variables Glc G6P F6P FDP - - + PGI + + ATP ADP ATP ADP PFK HK

7. Other types of variables Dependent variables Dynamic variables Glc G6P F6P FDP - - + PGI + + ATP ADP ATP ADP PFK HK

8. Other types of variables Aggregated variables Linear combinations variables Glc G6P F6P FDP - - + PGI + + ATP ADP ATP ADP PFK HK ATP + ADP = ANPtot v2f - v2r = v2

9. Other types of variables Constrained variables Variables whose values depends on the values of other variables not through the interactions of the various processes but through imposed algebraic relationships (moiety conservation, equilibria) Glc G6P F6P FDP - - + PGI + + ATP ADP ATP ADP PFK HK ATP = ANPtot – ADP – AMP F6P = Keq G6P

10. Other types of variables Implicit variables Variables that, in a given context, can be assumed constant or not influencing “subsystem” behavior, and are thus neglected X Glc G6P F6P FDP - - + PGI + + ATP ADP ATP ADP PFK HK Z Y

11. Symbology State variables: Xi - Dependent variables - Independent variables - Aggregated variables Flux variables: vij v12 X6 X1 X2 X7 X3 v21 v37 - - + v23 v61 X9 + + X5 X4 X5 X4 X10 X8 v45 X5 + X4 = X11

12. Strategy for developinga graphical representation • Sketch graphic in familiar terms • Potential variables • Important processes • Principal interactions • Make conversion table • Redraw graph in symbolic terms • Analyze and refine model

13. X2 X1 X3 Common errors and ambiguities Failure to explicitly account forconsumption/dilution of components Non-obvious point: “expansion fluxes”: dilution of components owing to volume expansion (e.g. cell growth).

14. X2 X2 X1 X1 X3 X3 X2 X2 X1 X1 X4 X4 X3 X3 Common errors and ambiguities Confusing convergent processeswith a multi-substrate reaction

15. X1 X2 X1 X2 Common errors and ambiguities Failure to account for the actual molecularity of each species in a given reaction

16. X3 X1 X2 Common errors and ambiguities Confusion between material and information flow -

17. System and environment X7 X4 X5 - - X3 X1 X2 X6

19. Realism and accuracy vs. feasibility “We should make things as simple as possible, but not simpler.” Albert Einstein “Any intelligent fool can make things bigger, more complex, and more violent. It takes a touch of genius — and a lot of courage — to move in the opposite direction.” Albert Einstein

20. Sources of simplification • Spatial or topological features (compartmentalization, channeling) • Time-scale separation (evolutionary, developmental, biochemical, biomolecular) and quasi-steady-state approximation • Functional simplifications (operating ranges and linearizations, non-linearity)

21. Basics of biochemicalsystems theory II. Mathematical representation of processes

22. Rate laws Mathematical functions representing the instantaneous rate of a process as an explicit function of all the state variables that have a direct influence on the process Mass action rate laws: in a well-mixed homogeneous environment, the rate of an elementary reaction a1X1 + a2X2 + … + anXn … is given by: Kinetic orders Rate constant Mass action rate laws may also approximate well the kinetics ofsome more-complex reaction mechanisms

23. Michaelis constant Rate laws Mathematical functions representing the instantaneous rate of a process as an explicit function of all the state variables that have a direct influence on the process Michaelis-Menten rate laws: initial rates of enzyme-catalyzed reactions in absence of cooperative effects. Maximal rate

24. Rate laws Mathematical functions representing the instantaneous rate of a process as an explicit function of all the state variables that have a direct influence on the process Michaelis-Menten rate laws: initial rates of enzyme-catalyzed reactions in absence of cooperative effects.

25. Rate laws Mathematical functions representing the instantaneous rate of a process as an explicit function of all the state variables that have a direct influence on the process Hill rate laws: approximation to initial rates of enzyme-catalyzed reactions with cooperativity. Maximal rate Hill number Km: Half-saturation constant

26. Rate laws Mathematical functions representing the instantaneous rate of a process as an explicit function of all the state variables that have a direct influence on the process Many other rate laws possible Could we find a convenient (even if only approximated) generic representation?

27. Taylor series Linear approximation

28. First-order (linear) Taylor series approximation for multiple variables

29. Advantages and limitationsof linearized representation • Good enough approximation in many cases (small operating ranges, linear by design) • Strong theory and methods available However: • Biochemical processes are strongly nonlinear

30. Power-law representation Taylor series in logarithmic space First-order truncation:

31. Kinetic orders Rate constant Power-law representation Taylor series in logarithmic space First-order truncation for n variables

32. Power-law representation Meaning of the kinetic orders gi With all other conc. constant gi>(<)0  v increases (decreases) with Xi gi = x  a 1% change in Xi causes a x% change in v

33. X X v = k X2 g = 2 v = k X g = 1 Ranges of values of kinetic orders Kinetic orders for mass action kinetics v Log-log plot X v = k Xcg = c

34. g0 g=0.5 g1 Ranges of values of kinetic orders Kinetic orders for substrates under Michaelis-Menten kinetics v/Vmax X/KM 0 ≤ g ≤ 1

35. g0 g-1 Ranges of values of kinetic orders Kinetic orders for inhibitors under Michaelis-Menten kinetics v/v(X=0) g=-0.5 X/KI -1 ≤ g ≤ 0

36. g0 g1 Ranges of values of kinetic orders Kinetic orders for substrates under cooperative kinetics v/Vmax g>1 X/KM 0 ≤ g ≤ (4)

37. Parameter estimation • From general considerations • From experimentally characterized rate expression • Top-down modeling

38. Basics of biochemicalsystems theory III. Mathematical representation of networks

39. X2 v12 v41 X1 X4 v13 X3 Mass balance Chemical processes do not destroy or create matter Instantaneous rate of X1 accumulation= Instantaneous rate of X4 conversion into X1– (Instantaneous rate of X1 conversion into X2+ Instantaneous rate of X1 conversion into X3)

40. X2 v12 v41 X1 X4 v13 X3 Steady state No net accumulation or dilution O= Instantaneous rate of X4 conversion into X1– (Instantaneous rate of X1 conversion into X2+ Instantaneous rate of X1 conversion into X3)

41. Mass balance equationsfor a network v51 - X5 X1 v12 v14 v40 v13 X4 v23 X2 X3 v20

42. Mass balance equationsfor a network General form vijk: unidirectional rate of utilization of Xi for the production of Xj via the kth parallel process jik: number of molecules of Xj created in each reaction jik: number of molecules of Xi consumed in each reaction

43. X2 X2 X2 X2 X1 X1 X1 X1 X4 X4 X4 X4 X3 X3 X3 X3 Flux aggregation Major types Emphasis: Fluxes Pools Strategies Parallel Antiparallel

44. X2 X2 X2 X2 X1 X1 X1 X1 X4 X4 X4 X4 X3 X3 X3 X3 Flux aggregation Power-law representation under various types of aggregation Emphasis: Fluxes Pools Strategies Parallel Antiparallel

45. Generalized Mass Action systems Power-law representation on the basis of flux aggregation emphasizing processes More simply: Maintains separate identity of fluxes at branch points

46. X2 X2 X2 X2 X1 X1 X1 X1 X4 X4 X4 X4 X3 X3 X3 X3 Flux aggregation Power-law representation under various types of aggregation Emphasis: Fluxes Pools Strategies Parallel Antiparallel

47. Synergistic systems Power-law representation on the basis of flux aggregation emphasizing pools X1, ..., Xn: Dependent variables Xn+1, ..., Xn+m: Independent variables Kinetic orders Rate constants

48. Some advantages of S systems • More compact representation • Fewer parameters • More mathematically tractable • Frequently more accurate

49. Ranges of values of kinetic orders Relationship between kinetic orders for the S systems representation and those for the GMA representation V1+ v211+ 2v212+v213 v211 X2 v212 X1 v213 X3 For “unit” enzyme-catalyzed steps, gij is normaly -4gij 4

50. v1 - X5 X1 v2 v5 v6 v4 X4 v3 X2 X3 v7 Setting up an S system 1. For each dynamic metabolite, aggregate the production fluxes and the removal fluxes separately.