1. Max-Planck-Institut für molekulare Genetik Metabolic control analysis 1 Network Analysis Graph theorety : Node + Edges  Routes, (Substances + Reactions) Measure for connectivity Stoichiometry: + Molecule numbers  Conservation relations, Flux distributions, Elementary modes Kinetics: + Kinetics + Concentrations  Control analysis

2. Max-Planck-Institut für molekulare Genetik Metabolic control analysis 2 Metabolic Control Theory Change of activity of an enzyme, e.g. PFK ? Change of concentration of metabolites, e.g. pyruvat ? ? Change of steady-state fluxes eg. within TCA cycle ?

3. Max-Planck-Institut für molekulare Genetik Metabolic control analysis 3 Example: Flux Control

4. Max-Planck-Institut für molekulare Genetik Metabolic control analysis 4 Metabolic Control Theory Quantification of metabolic control Quantification of the impact of small parameter changes on the variables of a metabolic system. Problem: Relations between steady-state variables and parameters are usually non-linear and can not be expressed analytically. There exists no theorie, which permits quantitative prediction of the effect of large changes of enzyme activity on fluxes. Restriction to small (infinitesimal) changes. ( Linearisation of the system in vicinity of steady state ). Controlling parameters: kinetic constants, enzyme concentrations,... Controlled variables: fluxes, substrate concentrations Wanted: mathematical function quantifying control.

5. Max-Planck-Institut für molekulare Genetik Metabolic control analysis 5 Metabolic Control Theory Relevant questions: - Many mechanisms and regulatory properties of isolated enzyme reactions are known  what is their quantitative meaning for metabolism in vivo? -Which step of a metabolic systems controls a given flux? (Is there a rate-limiting step?) -Which effectors or modifiers have the most influence on the reaction rate? Example: biotechnological production of a substance, Increase of turnover rate Question: which enzyme to activate in order to yield the most effect? Example: disease of metabolism, overproduction of a substance Question: which reaction to modify to reduce overproduction in a predictable way?

6. Max-Planck-Institut für molekulare Genetik Metabolic control analysis 6 Coefficients used in Control Theory Metabolic systems are networks; their behavior depends on the structure of the network and the properties of the individual components. There are two types of coefficients : local and global ones Elasticity coefficientsControl coefficients Response coefficients quantify the sensitivity of a rate for the change of a concentration Or a parameter value directly, immediate (no steady state) Quantitative measure for change of steady-state variables Assume reaching new steady states Depends on network structure

7. Max-Planck-Institut für molekulare Genetik Metabolic control analysis 7 Locally: Elasticity Coefficients S1S1 S2S2 v1v1 v2v2 v3v3 ? ? ? Question: How sensitive is a rate of an enzyme reaction with respect to small changes of a metabolite concentration? Consider enzyme as isolated, Wanted: immediate effect Elasiticity coefficient of reaction rate k with respect to metabolite concentration S i

8. Max-Planck-Institut für molekulare Genetik Metabolic control analysis 8 Parameter Elasticity  -elasticities comprise derivatives with respect to a metabolite concentration (a variable!).  -elasticities comprise derivatives with respect to parameter values (kinetic constants, enzyme concentrations,...) S1S1 S2S2 v1v1 v 2 ( K m, V max )v3v3 ?

9. Max-Planck-Institut für molekulare Genetik Metabolic control analysis 9 Globally: Control Coefficients 1. The system of metabolic reactions is in steady state. J = v(S(p),p) S = S(p) 2. A small perturbation of a reaction is performed (Addition of enzyme, addition of metabolite,....) 3. The system approaches a new (nearby) steady state. J J+  J S S+  S What is the change of steady state-variables (fluxes, concentrations) due to the perturbation of a single reaction?

10. Max-Planck-Institut für molekulare Genetik Metabolic control analysis 10 Definition of Control Coefficients Flux control coefficient - Change of rate of the k-th reaction under isolated fixed conditions - Resulting change of steady state flux through the j-th reaction - Normalization factor S1S1 S2S2 v1v1 v2v2 v3v3 ? ? ?

11. Max-Planck-Institut für molekulare Genetik Metabolic control analysis 11 Definition of Control Coefficients Concentration Control coefficient - Change of rate of the k-th reaction under isolated fixed conditions - Normalization factor - Resulting change of steady state concentration of S i S1S1 S2S2 v1v1 v2v2 v3v3 ? ?

12. Max-Planck-Institut für molekulare Genetik Metabolic control analysis 12 Choice of Perturbation Parameter The change of v k is based on a change of some parameters p k, which influences only this k-th reaction. (Enzymkonzentration, Inhibitoren, Aktivatoren,....) Extended expression of flux control coefficient: Important: Perturbation of p k influences directly only v k and no further reaction The flux control coefficients are then independent of the choice of the perturbed parameters p k. The can be interpreted as measure for the degree to which reaction k controls a given flux in steady state.

13. Max-Planck-Institut für molekulare Genetik Metabolic control analysis 13 Response Coefficients, global Consider: complete system in steady state. This state is determined by the values of the parameters; Parameter changes influence the steady state. The sensitivity of steady-state variables with respect to parameter Perturbations is expressed by response coefficients. S1S1 S2S2 v1v1 v 2 ( K m, V max, I )v3v3 ? ? ?

14. Max-Planck-Institut für molekulare Genetik Metabolic control analysis 14 Response Coefficients, Additivity Additivity: If several reactions are sensitive for this parameter if parameter m influences Only one reaction j : S1S1 S2S2 v1v1 v2(p)v2(p)v3(p)v3(p)

15. Max-Planck-Institut für molekulare Genetik Metabolic control analysis 15 Example 1 Experimentelly measurable quantities: Flux control coefficient Add inhibitor to a biochemical reaction

16. Max-Planck-Institut für molekulare Genetik Metabolic control analysis 16 Non-normalized Coefficients Non-normalized flux and Concentration control coefficients Non-normalized elasticities Examples Control of second reaction? Non-normalized flux control coeff. Be Glycolysis: Glucose 2 Lactat Normalized coefficients

17. Max-Planck-Institut für molekulare Genetik Metabolic control analysis 17 Matrix Notation

18. Max-Planck-Institut für molekulare Genetik Metabolic control analysis 18 Theorems of Control Theory The problem: The fluxes J usually cannot be expressed as mathematical functions Of the reaction rates. How can one calculate the global control coefficients from the local (measurable) changes ?? The solution: Use of theoremes. Here, the theoremes are only given and described with examples. The mathematical derivation is given for your information on the following pages.

19. Max-Planck-Institut für molekulare Genetik Metabolic control analysis 19 The Summation Theorems Thought experiment: What happens, if we induce by experimental manipulation the same fractional change  in the local rate of all steps of the system? Result: Flux J must also increase by factor . Since all rates increase in the same ratio, remain the concentration of the variable metabolites S 1 and S 2 unchanged. The combined effect of all changes in the local rates on the systems variables J, S 1 and S 2 can be described As the sum of all individual effects caused by the change of each local rate. For flux J holds: Thus holds Analog for S 1 and S 2 It follows:

20. Max-Planck-Institut für molekulare Genetik Metabolic control analysis 20 The Summation Theorems The Flux control coefficients of a metabolic pathway Add up to 1. The enzymes share the control over flux. The concentration control coefficients for a substance Add up to zero. Some enzymes increase a metabolite concentration Others decrease it. Matrix notation:

21. Max-Planck-Institut für molekulare Genetik Metabolic control analysis 21 Connectivity Theorems – General Relations Connectivity between flux control coefficients and elasticities Connectivity between concentration control coefficients and elasticities

22. Max-Planck-Institut für molekulare Genetik Metabolic control analysis 22 Example: Calculate flux control coefficients Summation theorem: Connectivity theorem: Result: Since in general: andfollows (i.a.!): Both reaction exert positive control over the flux.

23. Max-Planck-Institut für molekulare Genetik Metabolic control analysis 23 Example: Calculate concentration control coefficients It holds: and Producing reactions have positive control, consuming reactions have negative control. Summation theorem: Connectivity theorem: Result:

24. Max-Planck-Institut für molekulare Genetik Metabolic control analysis 24 Example: Linear pathway P 1 P 2 S 1 S i-1 S r-1 v 1 v i+1 v r S i... v i... Concentration Control Coefficients Producing reactions have positive control, consuming reactions have negative control. Reaction S5S5 S9S9 S1S1

25. Max-Planck-Institut für molekulare Genetik Metabolic control analysis 25 Linear Metabolic Pathway Each rate is a function of the concentrations of substrates and productes Assuming mass action kinetics With the equilibrium constants One can derive an equation for the Steady state flux P 1 P 2 S 1 S i-1 S r-1 v 1 v i+1 v r S i... v i...

26. Max-Planck-Institut für molekulare Genetik Metabolic control analysis 26 Linear Metabolic Pathway – Flux Control r flux control coefficients 1 summation theorem r-1 connectivity theorems General Expression for flux control coefficients (if ) P 1 P 2 S 1 S i-1 S r-1 v 1 v i+1 v r S i... v i...

27. Max-Planck-Institut für molekulare Genetik Metabolic control analysis 27 Linear Pathway - Properties P 1 P 2 S 1 S i-1 S r-1 v 1 v i+1 v r S i... v i... Ratio of two successive flux control coeff.: Flux control coefficients: Summation theorem Since sum of all flux control coeff is 1, and Ratio of two successive flux control coeff. Is positiv, are all flux control coeffizients in an unbranched pathway positiv.

28. Max-Planck-Institut für molekulare Genetik Metabolic control analysis 28 Linear Pathway - Properties P 1 P 2 S 1 S i-1 S r-1 v 1 v i+1 v r S i... v i... Ratio of two successive flux control coeff.: Flux control coefficients tend to be larger at the beginning than at the end. Case 1: Be the kinetic constants of all involved enzymes equal and the equilibrium constants larger than 1

29. Max-Planck-Institut für molekulare Genetik Metabolic control analysis 29 Linear Pathway - Properties P 1 P 2 S 1 S i-1 S r-1 v 1 v i+1 v r S i... v i... Case 2: with Using Relaxation time  as measure for the velocity of an enzyme: with or holds and therefore All enzymes are involved in control. Slow enzymes exert more control. There is no „rate-limiting step“ holds:

30. Max-Planck-Institut für molekulare Genetik Metabolic control analysis 30 Flux increase – how? P 1 P 2 S 1 S 2 v 1 v 2 v 4 S 3 v 3 Simple case: 1234 0.1 0.2 0.3 0.4 0.5 Flux control coefficients Reaction E 1  E 1 + 1% J  J + C 1 * 1%  1.0053

31. Max-Planck-Institut für molekulare Genetik Metabolic control analysis 31 Flux increase P 1 P 2 S 1 S 2 v 1 v 2 v 4 S 3 v 3 P 1 P 2 S 1 S 2 v 1 v 2 v 4 S 3 v 3 P 1 P 2 S 1 S 2 v 1 v 2 v 4 S 3 v 3 P 1 P 2 S 1 S 2 v 1 v 2 v 4 S 3 v 3

32. Max-Planck-Institut für molekulare Genetik Metabolic control analysis 32 Irreversibility and Feedback P 1 P 2 S 1 S 2 v 1 v 2 v 4 S 3 v 3 P 1 P 2 S 1 S 2 v 1 v 2 v 4 S 3 v 3 P 1 P 2 S 1 S 2 v 1 v 2 v 4 S 3 v 3

33. Max-Planck-Institut für molekulare Genetik Metabolic control analysis 33 Branching System P0P0 S1S1 S2S2 P3P3 P4P4 P5P5 v1v1 v2v2 v3v3 v4v4 v5v5

34. Max-Planck-Institut für molekulare Genetik Metabolic control analysis 34 Branching System with ATP/ADP-Exchange G0 Rang(N) = 2 < r Conservation relation ATP + ADP = const. Reduced stoichiometric matrix Basis vector for admissible steady state fluxes P 1 S P 3 P 2 v 1 v 2 v 3 ATP ADP ATP

35. Max-Planck-Institut für molekulare Genetik Metabolic control analysis 35 Mathematical Derivation of the Theorems Start with equation Implicite Differentiation w.r.t. parameter vector p regular Jacobi matrix M Rearrange to

36. Max-Planck-Institut für molekulare Genetik Metabolic control analysis 36 Mathematical Derivation of Theorems, 2 Start with equation Implicite differentiation w.r.t. parameter vector p Rearrange to Non-normalized Flux CC Both non-normalized CC are independent of the choice of perturbed parameter. They depend only on stoichiometry (N) and kinetics (dv/dS) !!

37. Max-Planck-Institut für molekulare Genetik Metabolic control analysis 37 Theorems, Normalized Control Coefficients

38. Max-Planck-Institut für molekulare Genetik Metabolic control analysis 38 Reaction System with Conservation Relations Problem: Jacobi-Matrix is not regular Rearrange rows of N and S, Such that dependent rows are at bottom. Implicite Differentiation of independent steady-state equations w.r.t parameter vector p The non-singular Jacobi matrix Of the reduced systems : Non-normalized concentrations cc Non-normalized flux cc

39. Max-Planck-Institut für molekulare Genetik Metabolic control analysis 39

40. Max-Planck-Institut für molekulare Genetik Metabolic control analysis 40 Glycolysis – Concentration Control Coefficients

41. Max-Planck-Institut für molekulare Genetik Metabolic control analysis 41 Glycolysis – Flux Control Coefficients

42. Max-Planck-Institut für molekulare Genetik Metabolic control analysis 42 Hierarchical Control

43. Max-Planck-Institut für molekulare Genetik Metabolic control analysis 43

44. Max-Planck-Institut für molekulare Genetik Metabolic control analysis 44

45. Max-Planck-Institut für molekulare Genetik Metabolic control analysis 45 due to steady state:

46. Max-Planck-Institut für molekulare Genetik Metabolic control analysis 46

47. Max-Planck-Institut für molekulare Genetik Metabolic control analysis 47 Systems Equations: an Example ODEs d[S 1 ]/dt = v 1  v 2 d[S 2 ]/dt = v 3  v 4 d[S 3 ]/dt = v 5 d[S 4 ]/dt =  v 3 + v 4 S1S2S3S4S1S2S3S4 S = v1v2v3v4v5v1v2v3v4v5 v = N = S1S2S3S4S1S2S3S4 1  1 0 0 0 0 0 1  1 0 0 0 0 0 1 0 0  1 1 0 Stoichiometric Matrix v 1 v 2 v 3 v 4 v 5 1  1 0 0 0 0 0 1  1 0 0 0 0 0 1 0 0  1 1 0 X v1v2v3v4v5v1v2v3v4v5 = v 1  v 2 +0 +0 +0 0 +0 +v 3  v 4 +0 0 +0 +0 +0 +v 5 0 +0  v 3 +v 4 +0 N v d[S]/dt X = S1S1 S2S2 S4S4 S3S3 v1v1 v2v2 v3v3 v4v4 v5v5

48. Max-Planck-Institut für molekulare Genetik Metabolic control analysis 48 Stoichiometric matrix N - Information or Steady state: Linear equation system, Non-trivial solutions only for Feasible steady state fluxes Elementary modes Balanced fluxes example S1S1 S2S2 S4S4 S3S3 v1v1 v2v2 v3v3 v4v4 v5v5

49. Max-Planck-Institut für molekulare Genetik Metabolic control analysis 49 Stoichiometric matrix N - Information Conservation relations: example S1S1 S2S2 S4S4 S3S3 v1v1 v2v2 v3v3 v4v4 v5v5

50. Max-Planck-Institut für molekulare Genetik Metabolic control analysis 50 Systemic Properties: Response and Control S1S1 S2S2 S4S4 S3S3 v1v1 v2v2 v3v3 v4v4 v5v5 p1p1 p2p2 p3p3 p5p5 p4p4 S1[0] = 0 S2[0] = 0 S3[0] = 0 S4[0] = 1 p1 = 1 p2 = 1 p3 = 1 p4 = 0.5 p5 = 0.5 p6 = 0.5 v6v6 p6p6

51. Max-Planck-Institut für molekulare Genetik Metabolic control analysis 51 Non-Steady State Trajectories What is the effect of parameter perturbations on time courses ? S[t] S1S1 S2S2 S3S3 S4S4 Time p2p2 p4p4 p1,3 p4p4 p5p5 S 1[0] S 3[0] S 2[0] S 4[0] p2p2 S 2[0] S 4[0] p1,3 S 1[0] p5p5 S 3[0] R S3 R S2 B.P. Ingalls, H.M. Sauro, JTB, 222 (2003) 23–36 S1[0] = 0 S2[0] = 0 S3[0] = 0 S4[0] = 1 p1 = 1 p2 = 1 p3 = 1 p4 = 0.5 p5 = 0.5 S1S1 S2S2 S4S4 S3S3 v1v1 v2v2 v3v3 v4v4 v5v5 p1p1 p2p2 p3p3 p5p5 p4p4

52. Max-Planck-Institut für molekulare Genetik Metabolic control analysis 52 Experimental Methods to Determine Control Coefficients - Titration with purified enzyme - Addition of specific inhibitors - Overexpression of an enzyme using genetic techniques - Downregulation of individual genes / Reduction of enzyme amount

53. Max-Planck-Institut für molekulare Genetik Metabolic control analysis 53 Metabolic Control Analysis - History/People 1973 Kacser /Burns 1974 Heinrich /Rapoport - Definition of coefficients about 1980 Discovery by Experimentalists (Westerhoff) 1988 Reder – Matrix Formulation BTK - Models and Experiments (Fell, Cornish-Bowden, Hofmeyr, Bakker, Schuster,….)

54. Max-Planck-Institut für molekulare Genetik Metabolic control analysis 54 Skalare, Vektoren, Matrizen S 1 = 1 Rechengesetze: Addition: a 11 a 12 a 21 a 22 b 11 b 12 b 21 b 22 a 11 +b 11 a 12 +b 12 a 21 +b 21 a 22 +b 22 ( ( ( ) ) ) += 1 Spalte, n Zeilen m Spalten, n Zeilen (m x n) Multiplikation: a 11 a 12 a 21 a 22 b 11 b 12 b 21 b 22 a 11 b 11 + a 12 b 21 a 11 b 12 + a 12 b 22 a 21 b 11 + a 22 b 21 a 21 b 12 + a 22 b 22 ( ( ) ) ). = (m x n)(n x p)(m x p) ( a 11 a 12 a 21 a 22 ( ) k. = ka 11 ka 12 ka 21 ka 22 ( ) Inverse Matrix: A B = CA B B -1 = C B -1 A = C B -1 B B -1 = I = 1 0 0 0 1 0 0 0 1 ( ) B quadratisch, nicht singulär