1. Systems Biology Chapter 8: Fundamental Subspaces of S

2. Overview • Brief Review • Dimensions of the Fundamental Subspaces • Basics of Singular Value Decomposition • SVD of S for the Elementary Reactions • Interpretation of SVD: Systemic Reactions • Summary

3. y B (2,2) Space v2 A v1 (1,1) x Brief Review Chapter 7 - Topological properties of S Chapter 8 – Subspaces within S A basisfor a space is used to spanthe space This is an orthonormal basis Singular Value Decomposition (SVD) gives simultaneous orthonormal bases for all four fundamental subspaces (2,1)

4. Dimensions of the Fundamental Subspaces Famili & Palsson Biophys J. 83:16-26 2003 Rank r = linearly independent rows and columns of S dim(col S) = dim(row S) = r dim(null S) = n-r and dim(left null S) = m-r

5. Broad range of applications The Basics of SVD Unbiased combined characterization of metabolites and reactions Stoichiometric matrix is a “perfect” matrix http://departments.colgate.edu/chemistry/images/rowlett_diffpattern.jpg http://www.genetics.ucla.edu/labs/horvath/adag_files/image008.jpg

6. Reaction Singular value Metabolites = • • rxr S mxn U mxm mxn VTnxn Columns of U and rows of VT represent modes of S s values give weight to modes for reconstruction of S s values are rank-ordered in S

7. Column Space r Left null Space (m-r) Orthonormal Bases Row Space r Null Space (n-r) VT U

8. S v VT U S Mapping of Singular Vectors vVT = orthonormal combinations of fluxes SvVT=orthonormal combinations of concentrations USvVT=concentration time derivatives

9. Geometry of SVD Muller et al. SIAM Review, Vol 46(3) 2004, p 518-545.

10. 1 0.95 Fi si 1 1 2 2 3 3 4 4 5 5 r r Singular Value Spectrum

11. SVD of S for Elementary Rxns

12. v1 x1 x2 v2 OR Linear Algebra Textbook MATLAB! Reversible Conversion: Where x1 = CP and x2 = PC

13. Reversible Conversion

14. Numerical Example: S Given: VT U S

15. v1 x2 x3 x1 + v2 Bilinear Association Where x1 = C and x2 = P and x3 = CP

16. Interpretation of SVD: Systemic Reactions

17. Interpretation of SVD pools pathways

18. Systemic Reaction Reduces to For the k nonzero singular values Linear combination of compounds uniquely moved by a linear combination of fluxes • Eigen-reaction or system metabolic reaction uk are systemic stoichiometric coefficients vk are systemic participation numbers

19. Reaction Eigen-reaction Singular value = • • rxr S mxn U mxm mxn VTnxn Eigen-connectivity metabolite

20. Decomposition of Genome Scale Matrices

21. First Mode

22. Second Mode

23. Summary • SVD: unbiased information about all fundamental subspaces of S • Column and left null in U • Row and null in V • U and V contain orthonormal vectors • 1st r columns of U analogous to a single column of S • Corresponding column of V gives combination of reactions that drive systemic reaction column in U • Orthonormal basis vectors may not have biological or chemical meaning

24. Questions?