Systems Biology Study Group

1. Systems Biology Study Group Chapter 6: Basic Features of the Stoichiometric Matrix

2. Where have we been? • So far, we’ve talked about how to understand biology from a systems and network perspective • There are four fundamental steps in systems biology: • List the biological components that participate in the process of interest • Study the interactions between the components and construct linkages (network reconstruction) • Describe the reconstructed network mathematically and analyze its properties • Use model for analysis, prediction, and interpretation

3. Where are we going? Part II: Mathematical Representation of Reconstructed Networks • Key questions: • How do we go about taking information from chemical reactions in our network reconstruction and putting it into some kind of formal mathematical framework? • What are the characteristics (properties) of this mathematical framework? • What does this mathematical formulation tell us about the state of the biological/chemical network? Insights? • Does this modeling represent reality?

4. What am I doing? Chapter 6: Basic Features of the Stoichiometric matrix • What is the stoichiometric matrix? • Represented by S • Contains stoichiometric information from reactions in the chemical network • Associated with S is additional information about open reading frames, transcript levels, enzyme complex formation and protein localization • Represents a BIGG database; an interface between high throughput data and in silico analysis

5. Basic features of S • Formed from the stoichiometric coefficients of the reactions that comprise a reaction network • Entries are integers • Columns of S correspond to reactions • Rows of S correspond to a compound. By looking across the rows, one observes all the reactions a given compound participates in, and how the reactions are interconnected • S transforms the flux vector (reaction rates) into a new vector that contains the time derivatives of the concentrations; therefore S is a linear transformation of the flux vector

6. From R to R From R3 to R From Rn to R Quick review Mathematically, what is a transformation? • Start with basic functions: • f is a transformation from Rn to Rm • f maps Rn into Rm • When the equations are linear, then it’s a linear transformation

7. Quick review • For matrices, a linear transformation T:Rn Rm has the following form: • When n = m, the transformation is called a linear operator • A is called the standard matrix for the linear transformation

8. Flux vector (contains reaction rates) Concentration vector Basic features of S • S is a linear transformation of the flux vector into a vector of time derivatives of the concentration vector • S represents the standard matrix in this linear transformation

9. A quick dimensional analysis A few comments about how the units work out . . . • S contains the stoichiometric coefficients (among other items to be mentioned later) that are all dimensionless • The flux vector v contains reaction rates (mole s-1) • The concentration vector has units of moles • So, the transformation involves the following units:

10. Basic properties of S • Each individual equation represents a dynamic mass balance (a summation of the fluxes) • There are m metabolites and n reactions, so S has dimensions of m x n; generally n > m • S is not necessarily full rank, r < m • There are 4 fundamental subspaces of S: the row space, column space, nullspace, and left nullspace

11. What? Spaces and rank? • “If A is an m x n matrix, then the subspace of Rn spanned by the row vectors of A is called the row space of A and the subspace of Rm spanned by the column vectors is called the column space of A” (Anton 1994) • The solution space (a subspace of Rn) of Ax = 0 is called the nullspace of A • Now, consider AT. Its row space and column space already fall within A. That only leaves the nullspace of AT as a “new” space.

12. What? Spaces and rank? • Thus, we have the 4 fundamental spaces: • Rowspace of S • Columnspace of S • Nullspace of S (solution to Sv = 0) • Nullspace of ST (which Palsson calls the left nullspace of S) • From matrix theory: “The dimension of a finite-dimensional vector space V, denoted dim(V), is defined to be the number of vectors in a basis for V.” (Anton 1994)

13. What? Spaces and rank? Also from matrix theory: • For any matrix A, the row space and column space have the same dimension. This common dimension is called the rank of A. • For a matrix A with n columns: rank(A) + nullity(A) = n • Rank = 2 and nullity = 4

14. Returning to properties of S • These 4 subspaces are important in analyzing biochemical networks (detailed in later chapters) • Column and left null spaces • Reaction vectors (column vectors of S) are structural (fixed) features • Fluxes (vi) are scalar quantities (variables) as they represent the flux through a reaction • Vectors in the left null space (li) are orthogonal to the column space • lirepresents a mass conservation

15. Properties of S • Row and null spaces • The flux vector consists of two components v = vdyn + vss Where vss is the solution to: Svss = 0 (nullspace of S) • vdyn is orthogonal to the nullspace • All of these subspaces will be discussed in more detail in later chapters

16. S is a connectivity matrix • All of the reaction information in S can be drawn into the standard reaction maps that we’re familiar with • A node corresponds to a row in S • A link corresponds to a column in S

17. S is a connectivity matrix • -ST represents a compound map • Nodes represent reactions • Connections represent compounds

18. S is a connectivity matrix • Let’s look at a simple reaction: This column vector represents a connection between all the compounds participating in this reaction If compounds participate in more than 1 reaction, there are interactions among columns of S • The higher the number of independent reactions, the smaller the orthogonal left nullspace becomes

19. Reversible conversion • Transformation between 2 compounds made up of the same 2 things • Bimolecular association • Combination of things to form a new compound • Cofactor-coupled reaction • One compound donates something to another compound The fundamental reactions C = primary metabolite; P = phosphate group; A = cofactor

20. A couple definitions • The topological structure of the reaction maps is important in determining the properties of a network • Linear maps—links that have only 1 input and one output • Seldom seen in biochemistry • Nonlinear maps—links with more than one input or more than one output • Most common number of metabolites: 4 • Metabolic cofactors create nonlinearity in S • Metabolites that participate in more than 2 reactions create nonlinearity in ST

21. C O H P Gluc G6P The elemental matrix • The in silico network must follow the rules of chemistry • The elemental matrix E contains the composition of all the compounds in a network • Columns corresponds to a compound • Rows corresponds to the elements • For glucose, C6H12O6 to G6P:

22. ES = 0 The elemental matrix • Matter conservation must be observed, so: • Nonconserved quantities (osmotic coefficients, etc.) can be appended as a row to E • For compounds and reactions consisting of 2 or 3 elements, they can be visualized as a point in a 2D or 3D elemental space. Reaction vectors, then, connect the points.

23. Open and closed networks • A systems boundary defines the boundary between the system and the external environment • A boundary can be open or closed • Where the boundary is drawn has direct consequences on the form S takes • For a biochemical network: • bi = exchange flux • vi = internal flux • ci = external concentration • xi= internal concentration

24. vi bi xi ci Forms of S • Stot includes everything: internal and external components and is a general form, and by definition everything would be internal. However, there may be a smaller, localized system of interest and thus we may divide the components among internal and external members.

25. vi vi bi xi xi Forms of S • Sexch contains only the external fluxes and exchange fluxes, but not the external compounds. Generally used in pathway analysis (covered in a later chapter). • Sint eliminates external fluxes and thus contains only internal ones. Useful to define conserved pools of compounds. Sint can be partitioned even further into primary & secondary metabolites . . .

26. ? Thank you! Source: Anton, H. (1994). Elementary Linear Algebra. New York, John Wiley and Sons.