Humboldt- Universität zu Berlin Edda Klipp Systembiologie 3 - Stoichiometry Sommersemester 2010 Humboldt-Universität zu Berlin Institut für Biologie Theoretische Biophysik

Humboldt- Universität zu Berlin 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

Humboldt- Universität zu Berlin 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 Stoichiometric coefficients can be chosen such that they agree with molecularity, but not necessarily. 11/ Their signs depend on the chosen reaction direction. Since reactions are usually reversible, one cannot distinguish between „substrate“ and „product“. v - v

Humboldt- Universität zu Berlin Time Course of Concentrations Usually described by ordinary differential equations (ODE) Example catalase for this choice of stoichiometric coefficienten: -22 1

Humboldt- Universität zu Berlin Time Course of Concentrations Several reactions at the same time  all rate equations must be considered at the same time. S 1 S 2 S Usually described by ordinary differential equations (ODE) One can summarize the stoichiometric coefficients in matrix N. The rows refer to the substances, the columns refer to the reactions S S S              N Column: reaction Row: Substance External metabolites are not included in N.

Humboldt- Universität zu Berlin 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

Humboldt- Universität zu Berlin 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.

Humboldt- Universität zu Berlin 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

Humboldt- Universität zu Berlin 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 dt S v1v2v3v1v2v3 = SNv.. = Steady state Nv = 0 is usually a non-linear equation system, which cannot be solved analytically (necessitates knowledge of kinetic(). dS i /dt = 0

Humboldt- Universität zu Berlin The Stoichiometric Matrix N - Characterizes the network of all reactions in the system - Contains information about possible pathways

Humboldt- Universität zu Berlin 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

Humboldt- Universität zu Berlin Calculation of the Kernel Matrix The Kernel matrix K can be calculated with the Gaussian Elimination Algorith for the solution of homogeneous linear equation systems. Example Alternative: calculate with computer programmes Such as „NullSpace[matrix]“ in Mathematica.

Humboldt- Universität zu Berlin 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

Humboldt- Universität zu Berlin Informations from Kernel Matrix K -Admissible fluxes in steady state -Equilibrium reactions -Unbranched reaction sequences -Elementary modes

Humboldt- Universität zu Berlin Admissible Fluxes in Steady State S0S0 S1S1 S2S2 S3S3 v1v1 v2v2 v3v3 S0S0 S1S1 S2S2 S3S3 v1v1 v2v2 v4v4 v3v3 S v2v2 v1v1 v3v3 Examples

Humboldt- Universität zu Berlin 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 für In steady state holds

Humboldt- Universität zu Berlin Equilibrium Reactions Case: all elements of a row in K are 0 Then: the respective reaction is in every steady state in equilibrium. Example

Humboldt- Universität zu Berlin 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.

Humboldt- Universität zu Berlin 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.

Humboldt- Universität zu Berlin 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

Humboldt- Universität zu Berlin 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

Humboldt- Universität zu Berlin 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.

Humboldt- Universität zu Berlin 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:

Humboldt- Universität zu Berlin Conservation Relations – Examples Conservation of atoms or atom groups, e.g. Pyruvatdecarboxylase (EC ) carbon oxygen hydrogen CH 3 CO-group Protons Carboxyl group Elektric charge

Humboldt- Universität zu Berlin 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

Humboldt- Universität zu Berlin Basic Elements of Biochemical Networks Systems equations r – number of reactions S i – metabolite concentrations v j – reaction rates n ij – stoichiometric coefficients Network properties Individual reaction properties Matrix representation S1S1 S2S2 S4S4 S3S3 v1v1 v2v2 v3v3 v4v4 v5v5

Humboldt- Universität zu Berlin Nachtrag vom 10. Mai 2010

Humboldt- Universität zu Berlin 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

Humboldt- Universität zu Berlin 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

Humboldt- Universität zu Berlin Elementary Flux Modus -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

Humboldt- Universität zu Berlin 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.