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Lecture #7

Lecture #7. The stoichiometric matrix. Outline. The many facets of S Chemistry: S as a data matrix Network structure: S as a connectivity matrix Mathematics: S as a linear transformation Systems science: S as part of a network model. Matrix Attributes.

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Lecture #7

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  1. Lecture #7 The stoichiometric matrix

  2. Outline • The many facets of S • Chemistry: • S as a data matrix • Network structure: • S as a connectivity matrix • Mathematics: • S as a linear transformation • Systems science: • S as part of a network model

  3. Matrix Attributes

  4. Properties of the Stoichiometric Matrix

  5. S AS A DATA MATRIX

  6. Forming S math (bio) chemistry 1. 2. 2 3 4 3. S is a data matrix

  7. The GPRs: GenesProteinsReactions promiscuous isozymes gene protein GPRs reaction

  8. S as a data matrix S # compounds Smxn # reactions

  9. E. coli at the genome-scale: S for iAF1260

  10. S for Geobacter sulfurreducans Reguera, et al, Nature 435:1098-1101 (2005)

  11. QC/QA on S • Every column in S (si; reaction vector) represents a reaction • Chemically accurate, including charge and elemental balancing Elemental balance: E•S=0 Elemental matrix: E elements . . . . . . compounds the rows of E are in the left null of S

  12. QC/QA: quality control/quality assurance Elemental and charge balancing Time invariants in the left null space of S ES=0

  13. Elemental Composition of some Common Metabolites

  14. Example: S for Glycolysis E for Glycolysis

  15. QC/QA on S for glycolysis The product E•S is shown below. As can be seen E•S ≠ 0. Hydrogen needs to be added on the product side Hydrogen needs to be added on the reactant side Hydrogen needs to be added on the product side Hydrogen needs to be added on the product side and water on the reactant side Water has to be added to the product side to balance

  16. Balanced S matrix for Glycolysis after QC/QA E•S = 0, except for bi

  17. S AS AN INCIDENCE MATRIX

  18. S Represents a ‘Graph’ or ‘Map’ link, vj ( ) S: node, xi A particular flux vector 1/4 1/2 • Reaction Map • Flux Map: • a particular functional state • of the reaction map 1/4 1 1/2

  19. ST is also a ‘Map’ or ‘Graph’ 1 link, xi 1/6 1/2 ( ) v2 x3 x2 v1 x1 node, vj v3 - Compound map - Concentration map

  20. Duality, S, and ST

  21. Examples

  22. Reaction Map of iAF1260

  23. S AS A PART OF THE DYNAMIC MASS BALANCES

  24. Dynamic Mass Balance 100-106 molecules sec-min m3 v1 v3 xi: compound i v4 moles/vol/time vj: reaction j v5 v2 rate of formation rate of degradation

  25. Dynamic Mass Balance for a Network 1) steady state 0=S•v ; Systems Biology Vol I 2) dynamic states kinetic constants simulate Systems Biology Vol II

  26. Defining a ‘System’ Based on S outside I/O inside system boundary • physical; cell wall, nuclear membrane • virtual; ETS

  27. The different forms of S

  28. Example

  29. Reaction Map for Glycolysis as a System

  30. S for Glycolysis as a System

  31. S AS A MAPPING OPERATION

  32. S Represents a mapping Operation ;

  33. What is in the Four Subspaces? • (right) null space • 0=S•v; all steady state solutions • left null space • 0=l•S= time invariants

  34. The Column Space contains the directions of motion ( ) column space ;

  35. Row space contains the thermodynamic driving forces m x n concentration (compound) fluxes (reaction) m mxn n m ≤ n

  36. Summary • S is comprised of stoichiometric coefficients that are commonly integer numbers. The columns of the S represent chemical reactions while the rows represent compounds. • S has many important attributes: • 1) it is a data matrix, • 2) it gives the structure of a network, • 3) it is a mathematical mapping operations with four fundamental subspaces, and • 4) it forms a key part of in silico models describing the functional states of networks. • The column (reaction) vectorsrepresent chemical transformations and thus come with chemical information. • The reaction vectorsimply elemental and charge balance. The reaction vectors are thus orthogonal to the rows of E. These conservation quantities are in the left null space of S.

  37. Summary, con’t • Some quantities, such as free energy, are non-conserved during a chemical reaction. These quantities will be in the row space of S. • There are few basic forms of the elementary reactions. Most, if not all, biochemical reaction networks are either linear or bi-linear. • Structurally, or topologically, S represents a reaction map. The transpose of S represents a compound map. Both maps are topologically non-linear, as they contain joint edges between nodes. • The boundaries of a reaction network can be drawn in different ways and lead to three fundamental forms of S.

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