1. VL Netzwerke, WS 2007/08 Edda Klipp 1 Max Planck Institute Molecular Genetics Humboldt University Berlin Theoretical Biophysics Networks in Metabolism and Signaling Edda Klipp Humboldt University Berlin Lecture 5 / WS 2007/08 Stoichiometry in Metabolic Networks

2. VL Netzwerke, WS 2007/08 Edda Klipp 2 Max Planck Institute Molecular Genetics Humboldt University Berlin Theoretical Biophysics Stoichiometric Analysis of Cellular Reaction Systems 2AB + C 2D E F G v1v1 v3v3 v2v2 - Analysis of a system of biochemical reactions - Network properties - Enzyme kinetics not considered http://www.genome.ad.jp/kegg/pathway/map/map01100.html

3. VL Netzwerke, WS 2007/08 Edda Klipp 3 Max Planck Institute Molecular Genetics Humboldt University Berlin Theoretical Biophysics Stoichiometry and Graphs http://www.genome.ad.jp/kegg/pathway/map/map01100.html We consider a graph, e.g. a tuple (V,E) with V a set of n vertices and a set of m edges E : G=(V,E) Hypergraph

4. VL Netzwerke, WS 2007/08 Edda Klipp 4 Max Planck Institute Molecular Genetics Humboldt University Berlin Theoretical Biophysics Stoichiometric Coefficients Stoichiometric coefficients denote the proportions, with which the molecules of substrates and products enter the biochemical reactions. Example Catalase Stoichiometric coefficients for Hydrogenperoxid, water, oxygen -2 2 1 Stoichiometric coefficients can be chosen such that they agree with molecularity, but not necessarily. 11/2 2 -2 Their signs depend on the chosen reaction direction. Since reactions are usually reversible, one cannot distinguish between „substrate“ and „product“. v - v

5. VL Netzwerke, WS 2007/08 Edda Klipp 5 Max Planck Institute Molecular Genetics Humboldt University Berlin Theoretical Biophysics Time Course of Concentrations Usually described by ordinary differential equations (ODE) Example catalase for this choice of stoichiometric coefficienten: -22 1

6. VL Netzwerke, WS 2007/08 Edda Klipp 6 Max Planck Institute Molecular Genetics Humboldt University Berlin Theoretical Biophysics Time Course of Concentrations all rate equations must be considered at the same time. Several reactions at the same time S 1 S 2 S 3 1 2 3 4 Usually described by ordinary differential equations (ODE)

7. VL Netzwerke, WS 2007/08 Edda Klipp 7 Max Planck Institute Molecular Genetics Humboldt University Berlin Theoretical Biophysics Balance equations/Systems equations In general: We consider the substances S i and their stoichiometric coefficients n ij in the respective reaction j. If the biochemical reactions are the only reason for the change of concentration of metabolites, i.e. if there is no mass flow by convection, diffusion or similar Then one can express the temporal behavior of concentrations by the balance equations. r – number of reaction S i – metabolite concentration v j – reaction rate n ij – stoichiometric coefficient

8. VL Netzwerke, WS 2007/08 Edda Klipp 8 Max Planck Institute Molecular Genetics Humboldt University Berlin Theoretical Biophysics The Stoichiometric Matrix One can summarize the stoichiometric coefficients in matrix N. The rows refer to the substances, the columns refer to the reactions: Example S 1 S 2 S 3 1 2 3 4 4 321 S S S 1000 0110 1011 3 2 1              N Column: reaction Row: Substance External metabolite are not included in N.

9. VL Netzwerke, WS 2007/08 Edda Klipp 9 Max Planck Institute Molecular Genetics Humboldt University Berlin Theoretical Biophysics Summary Stoichiometric matrix Vector of metabolite concentrations Vector of reaction rates Parameter vector With N can one write systems equations clearly. Metabolite concentrations and reaction rates are dependent on kinetic parameters.

10. VL Netzwerke, WS 2007/08 Edda Klipp 10 Max Planck Institute Molecular Genetics Humboldt University Berlin Theoretical Biophysics The Steady State Reaction systems are frequently considered in steady state, Where metabolite concentrations change do not change with time. This describes an implicite dependency of concentrations and fluxes on the parameters. b.z.w. The flux in steady state is

11. VL Netzwerke, WS 2007/08 Edda Klipp 11 Max Planck Institute Molecular Genetics Humboldt University Berlin Theoretical Biophysics Concept of Steady States Restriction of modeling to essential aspects Analysis of the asymptotical time behavior of dynamic systemes (i.e. The behavior after sufficient long time span). Asymptotic behavior can be - oscillatory or - chaotic - in many relevant situations will the system reach a steady state. The conzept of steady state - important in kinetic modeling - mathematical idealization Time

12. VL Netzwerke, WS 2007/08 Edda Klipp 12 Max Planck Institute Molecular Genetics Humboldt University Berlin Theoretical Biophysics Concept of Steady States Separation of time constants fast and slow processes are coupled fast processes: initial transition period (often) quasi-steady state slow processes : change of some quantities in a certain period is often neglectable (Every steady state can be considered as quasi-steady state embedded in a larger non-stationary system). Biological organisms are characterized by flow of matter and energy time-independent regimes are usually non-equilibrium phenomena Fließgleichgewicht Mathematically: replace ODE system (for temporal behavior of variables (concentrations and fluxes)) by an algebraic equation system

13. VL Netzwerke, WS 2007/08 Edda Klipp 13 Max Planck Institute Molecular Genetics Humboldt University Berlin Theoretical Biophysics Example Unbranched pathway variabel Assumption: Linear kinetics System equations Matrix formalism dS 1 / dt = v 1 -v 2 dS 2 / dt = v 2 -v 3 d S 1 1 -1 0 dt S 2 0 1 -1 v1v2v3v1v2v3 = S Nv.. = Steady state Nv = 0 is usually a non-linear equation system, which cannot be solved analytically (necessitates knowledge of kinetic(). dS i /dt = 0

14. VL Netzwerke, WS 2007/08 Edda Klipp 14 Max Planck Institute Molecular Genetics Humboldt University Berlin Theoretical Biophysics The stoichiometric Matrix N - Characterizes the network of all reactions in the system - Contains information about possible pathways

15. VL Netzwerke, WS 2007/08 Edda Klipp 15 Max Planck Institute Molecular Genetics Humboldt University Berlin Theoretical Biophysics The Kernel Matrix K In steady state holds Non-trivial solutions exist only if the columns of N are linearly dependent. Mathematically, the linear dependencies can be expressed by a matrix K with the columns k which each solve K – null space (Kernel) of N The number of basis vectors of the kernel of N is

16. VL Netzwerke, WS 2007/08 Edda Klipp 16 Max Planck Institute Molecular Genetics Humboldt University Berlin Theoretical Biophysics Calculation of the Kernel matrix The Kernel matrix K can be calculated with the Gauss‘ Elimination Algorith for the solution of homogeneous linear equation systems. Example Alternative: calculate with computer programmes Such as „NullSpace[matrix]“ in Mathematica.

17. VL Netzwerke, WS 2007/08 Edda Klipp 17 Max Planck Institute Molecular Genetics Humboldt University Berlin Theoretical Biophysics Admissible Fluxes in Steady State: Examples S0S0 S1S1 S2S2 S3S3 v1v1 v2v2 v3v3 Unbranched pathway: one independent steady state flux

18. VL Netzwerke, WS 2007/08 Edda Klipp 18 Max Planck Institute Molecular Genetics Humboldt University Berlin Theoretical Biophysics Admissible Fluxes in Steady State: Examples S0S0 S1S1 S2S2 S3S3 v1v1 v2v2 v4v4 v3v3

19. VL Netzwerke, WS 2007/08 Edda Klipp 19 Max Planck Institute Molecular Genetics Humboldt University Berlin Theoretical Biophysics Admissible Fluxes in Steady State: Examples S v2v2 v1v1 v3v3

20. VL Netzwerke, WS 2007/08 Edda Klipp 20 Max Planck Institute Molecular Genetics Humboldt University Berlin Theoretical Biophysics Representation of Kernel matrix The Kernel matrix K is not uniquely determined. Every linear combination of columns is also a Possible solution. Matrix multiplication with a regular Matrix Q „from right“ gives another Kernel matrix. For some applications one needs a simple ("kanonical") representation of the Kernel matrix. A possible and appropriate choice is K contains many Zeros. I – Identity matrix

21. VL Netzwerke, WS 2007/08 Edda Klipp 21 Max Planck Institute Molecular Genetics Humboldt University Berlin Theoretical Biophysics Informations from Kernel matrix K -Admissible fluxes in steady state -Equilibrium reactions -Unbranched reaction sequences -Elementary modes

22. VL Netzwerke, WS 2007/08 Edda Klipp 22 Max Planck Institute Molecular Genetics Humboldt University Berlin Theoretical Biophysics Admissible Fluxes in steady state With the vectors k i (k 1, k 2,…) is also every linear combination A possible columns of K. for example: instead and also All admissible fluxes in steady state can be written as linear combinations of vectors k i : The coefficients  i have the respective units, eg. or. S v2v2 v1v1 v3v3 for In steady state holds

23. VL Netzwerke, WS 2007/08 Edda Klipp 23 Max Planck Institute Molecular Genetics Humboldt University Berlin Theoretical Biophysics Equilibrium reactions Case: all elements of a row in K are 0 Then: the respective reaction is in every steady state in equilibrium. Example

24. VL Netzwerke, WS 2007/08 Edda Klipp 24 Max Planck Institute Molecular Genetics Humboldt University Berlin Theoretical Biophysics Kernel matrix –Dead ends S0S0 S1S1 S2S2 S4S4 S3S3 v1v1 v2v2 v3v3 v4v4 S 1, S 2, S 3 intern, S 0, S 4 extern Necessary and sufficient condition for a „Dead end“: One metabolite has only one entry in the stoichiometric matrix (is only once Substrate or product). Flux in steady state through this reaction must vanish in steady state (J 4 = 0). Model reduction: one can neglect those reactants for steady state analyses.

25. VL Netzwerke, WS 2007/08 Edda Klipp 25 Max Planck Institute Molecular Genetics Humboldt University Berlin Theoretical Biophysics Unbranched Reaction Steps S0S0 S1S1 S2S2 S3S3 S4S4 v1v1 v2v2 v3v3 v4v4 The basis vectors of nullspace have the same entries for unbranched reaction sequences. Unbranched reaction sequences can be lumped for further analysis.

26. VL Netzwerke, WS 2007/08 Edda Klipp 26 Max Planck Institute Molecular Genetics Humboldt University Berlin Theoretical Biophysics Non-negative Flux Vectors In many biologically relevant situations have fluxes fixed signs. We can define their direction such that Sometimes is the value of individual rates fixed. Both conditions restrict the freedom for the choice of Basis vectors for K. Example: opposite uni-directional rates instead of net rates, - Description of tracer kinetics or dynamics of NMR labels - different isoenzymes for different directions of reactions - for (quasi) irreversible reactions

27. VL Netzwerke, WS 2007/08 Edda Klipp 27 Max Planck Institute Molecular Genetics Humboldt University Berlin Theoretical Biophysics Kernel Matrix – Irreversibility S0S0 S1S1 S2S2 S3S3 S4S4 v1v1 v2v2 v3v3 v4v4 Mathematically possible, biologically not feasible Other choice of basis vectors The basis vectors of a null space are not unique. The direction of fluxes (signs) do not necessarily agree with the direction of irreversible reactions. (Irreversibility limits the space of possible steady state fluxes.) S 1, S 2 internal, S 0, S 3, S 4 external

28. VL Netzwerke, WS 2007/08 Edda Klipp 28 Max Planck Institute Molecular Genetics Humboldt University Berlin Theoretical Biophysics Elementary Flux Modes Situation: some fluxes have fixed signes, others can operate in both directions. Which (simple) pathes connect external substrats? S P1P1 P2P2 P3P3 v1v1 v2v2 v3v3 S P1P1 P2P2 P3P3 v1v1 v3v3 v2v2

29. VL Netzwerke, WS 2007/08 Edda Klipp 29 Max Planck Institute Molecular Genetics Humboldt University Berlin Theoretical Biophysics Elementary Flux Modes -An elementary flux mode comprises all reaction steps, Leading from a substrate S to a product P. -Each of these steps in necessary to maintain a steady state. -The directions of fluxes in elementary modes fulfill the demands for irreversibility

30. VL Netzwerke, WS 2007/08 Edda Klipp 30 Max Planck Institute Molecular Genetics Humboldt University Berlin Theoretical Biophysics Elementary Flux Modes S2S2 S1S1 S3S3 P2P2 P1P1 P3P3

31. VL Netzwerke, WS 2007/08 Edda Klipp 31 Max Planck Institute Molecular Genetics Humboldt University Berlin Theoretical Biophysics Number of elementary flux modes S0S0 S1S1 S2S2 S3S3 S4S4 v1v1 v2v2 v3v3 v4v4 S0S0 S1S1 S2S2 S3S3 S4S4 v1v1 v2v2 v3v3 v4v4 The number of elementary modes is at least as high as the number of basis vectors of the null space.

32. VL Netzwerke, WS 2007/08 Edda Klipp 32 Max Planck Institute Molecular Genetics Humboldt University Berlin Theoretical Biophysics Flux Modes and Extreme Pathways vivi vjvj vkvk NK=0 Extreme pathways: All reactions are irreversible Flux cone

33. VL Netzwerke, WS 2007/08 Edda Klipp 33 Max Planck Institute Molecular Genetics Humboldt University Berlin Theoretical Biophysics Conservation relations: Matrix G If compounds or groups are not added to or deprived of a Reaction system, then must their total amount remain constant. Michaelis-Menten kinetics Isolated reaction: Pyruvatkinase, Na/K-ATPase Examples

34. VL Netzwerke, WS 2007/08 Edda Klipp 34 Max Planck Institute Molecular Genetics Humboldt University Berlin Theoretical Biophysics Conservation relations - calculation If there exist linear dependencies between the rows of the stoichiometric matrix, then one can find a matrix G such as N – stoichiometric matrix Due to holds The integration of this equation yields the conservation relations.

35. VL Netzwerke, WS 2007/08 Edda Klipp 35 Max Planck Institute Molecular Genetics Humboldt University Berlin Theoretical Biophysics Conservation relations – Properties of G The number of independent row vectors g (= number of Independent conservation relations) is given by (n = number of rows of the stoichiometric matrix = number of metabolites) G T is the Kernel matrix of N T, and can be found in the same way as K. (Gaussian elimination algorithm) The matrix G is not unique, with P regular quadratic matrix is again conservation matrix. Separated conservation conditions:

36. VL Netzwerke, WS 2007/08 Edda Klipp 36 Max Planck Institute Molecular Genetics Humboldt University Berlin Theoretical Biophysics Conservation relations – Examples

37. VL Netzwerke, WS 2007/08 Edda Klipp 37 Max Planck Institute Molecular Genetics Humboldt University Berlin Theoretical Biophysics Conservation relations – Examples Conservation of atoms or atom groups, e.g. Pyruvatdecarboxylase (EC 4.1.1.1) carbon oxygen hydrogen CH 3 CO-group Protons Carboxyl group Elektric charge

38. VL Netzwerke, WS 2007/08 Edda Klipp 38 Max Planck Institute Molecular Genetics Humboldt University Berlin Theoretical Biophysics Conservation relations – Examples v1v1 v2v2 v3v3 Glucose Gluc-6P Fruc-6PFruc-1,6P 2 ATP ADP

39. VL Netzwerke, WS 2007/08 Edda Klipp 39 Max Planck Institute Molecular Genetics Humboldt University Berlin Theoretical Biophysics Conservation Relations – Simplification of the ODE system If conservation relations hold for a reaction system, then the ODE system can be reduced, since some equations are linearly dependent. Rearrange N, L – Linkmatrix (independent upper rows, dependent lower rows) Rearrange S respectively (indep upper rows, dep lower rows) Reduced ODE system For dependent concentrations hold