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Lecture #9

Lecture #9. The four fundamental subspaces. Outline. SVD and its uses SVD: basic features SVD: key properties Examples: simple reactions & networks Genome-scale stoichiometric matrices Examples Tilting of basis vectors. SVD AND ITS USES. The Singular Value Decomposition (SVD). dx. •.

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Lecture #9

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  1. Lecture #9 The four fundamental subspaces

  2. Outline • SVD and its uses • SVD: basic features • SVD: key properties • Examples: simple reactions & networks • Genome-scale stoichiometric matrices • Examples • Tilting of basis vectors

  3. SVD AND ITS USES

  4. The Singular Value Decomposition(SVD) dx • • • • v x x x x ; ; =Sv S=USVT dt VT S U v • • v S • VT =U(•) U VTv “stretches” S time derivatives are a linear combination (•) linear combination of fluxes • • diagonal matrix

  5. Original 5 values 10 values 30 values 52 values 303 values Singular Value Decomposition in Image Processing http://peter.wreck.org/reports/Math4305/

  6. Image processing http://antwrp.gsfc.nasa.gov/apod/image/0011/earthlights_dmsp_big.jpg Noise reduction Kinematics mRNA expression analysis Applications of Singular Value Decomposition

  7. SVD: BASIC FEATURES

  8. dx =Sv dt MATLAB: [U, S, V]= svd(A) Numerical check: ||A-USVT||=0 ? dx =USVTv dt

  9. The Singular ValuesDiagonal entries in S The singular values s1, s2,……. sr large small singular value spectrum fractional singular values fi= si si cumulative fractional singular values Fi= fk ; Fr=1 r i S S i=1 k=1

  10. Orthonormal Basis Sets Dim =m Dim =n S C(S) R(S) x’ =USVT v Dim(C) = r Dim (R) =r LN(S) N(S) Dim(N) =n-r Dim(LN) =m-r

  11. SVD: KEY PROPERTIES

  12. Property #1: Mode by Mode Reconstruction of S r m x n ||ui||=1 S=S si<uiviT> ||uj||=1 i=1 m x l l x n ( ) ( ) <ui•uj> =s1 + s2 +…… =0 i≠j =1 i=j ( ( ) ) ( ) m x n ( ) definition of orthonormality =s1 + s2 +…… • are scaling factors: • i.e., S= 100+ 10 + 1 + …. ||•||~1

  13. Property #2: S Maps the Right Singular Vector (vi) to the Left Singular Vector (ui) S = USVT SV = US(VTV) SV = US (xV) UTU=I m x m VTV=I =I n x n k k k ( ) ( )=I S ( ) = ( )( ) |||| |||| 0 |||| |||| 0 UT=U-1 Svk = skuk S ( )| = •| Independent dimension in the Row space Dimension in the Row space Independent dimension in the Columnspace vkskuk

  14. Orthonormality and dynamic decoupling Decoupled motion x UT d(|) dt = v S x’ • VT U S • • ( )(|) 0 0

  15. EXAMPLES

  16. Example #1 m = n=2; r=1

  17. Bounded Spaces

  18. Example #2

  19. Orthonormal basis for Column and Left Null 3 3 -1 -1 1 02 1 -1 1 1 Col Left Null

  20. An Alternative Set of Vectors for the Left Null Space ( ) -1 -1 1 (0,1,1) Has positive values for the concentrations Col l1 and l2 are convex basis vectors

  21. GENOME-SCALE STOICHIOMETRIC MATRICES

  22. SVD of S: global view

  23. Mapping: from fluxes to concentration time derivatives • Dynamic equation • Flux drivers • Motion of concentrations

  24. Mapping: chemical reaction interpretation

  25. Systemic chemical reactions • Systems rate equation • Systems pseudo-elementary reactions: v_ki are ‘systems’ partition numbers • Systems (eigen) reaction: u_ki are ‘systems’ stoichiometric coefficients

  26. DECOMPOSITION OF THE CORE METABOLIC NETWORK IN E. COLI

  27. Systemic reactions w/o biomass Translocation of a proton ATP synthesis Transhydrogenation and AcCoA charging

  28. Systemic reactions w/ biomass Growth Translocation of a proton And ATP synthesis Transhydrogenation and AcCoA charging

  29. DECOMPOSITION OF GENOME-SCALE MATRICES

  30. The singular value spectrum

  31. 1st mode:high energy phosphate bonds Motion: stoichiometry Drivers: reactions

  32. 2nd mode: NADPH redox metabolism Motion: stoichiometry Drivers: reactions

  33. 3rd Mode: translocated proton Motion: stoichiometry Drivers: reactions

  34. 4th mode Motion: stoichiometry Drivers: reactions

  35. ROTATING BASIS VECTORS

  36. The effects of rotation:Rotation of the Basis Vectors for Col(S) NADP, NADPH ATP,ADP Q, QH2 Metabolites

  37. Interpreting the basis vectors • Interpreting the variable loadings on the basis vectors can be hard due to the maximal variance characteristic of SVD. • In order to gain biological insights from the principal components, the basis vectors can be rotated. • Rotation is just a change of basis. • There is no gain or loss of information From Barrett et al

  38. Applying to Metabolic Networks

  39. Basis Rotation Methods • The two major categories • Orthogonal Rotations: • maintain all PCs perpendicular to each other • Examples: varimax, orthomax, quartimax • Oblique Rotations: • Relax the orthogonality constraint • gain simplicity in the interpretation. • Allow PCs to be correlated • Examples: promax, oblimin • In MATLAB • A=rotatefactors(B,’Method’,…)

  40. Summary • S=USVT is the most fundamental decomposition of a matrix • S has the singular values and gives the “effective” dimensionality of the mapping that S represents • U and VT have orthonormal basis vectors for the four subspaces • We may want oblique basis vectors to represent chemistry/biology

  41. The end

  42. Extras

  43. Summary (detailed) • SVD provides unbiased and decoupled information about all the fundamental subspaces of S simultaneously. • The first r columns of the left singular matrix U contain a basis for the column space of S, and the remaining m-r columns contain a basis for the left null space. • The first r columns of the right singular matrix r contain a basis for the row space of S and the remaining n-r columns contain a basis for the null space. • The sets of basis vectors in U and V are orthonormal. • The first r columns of U give systemic reactions, analogous to a single column of S, representing a single reaction. • The corresponding column of V gives the combination of the reactions that drive a systemic reaction. • Orthonormal basis vectors are mathematically convenient but not necessarily biologically or chemically meaningful.

  44. Methods for Factor Rotation The two major categories • Orthogonal Rotations: • maintain all PCs perpendicular to each other • Examples: varimax, orthomax, quartimax • Oblique Rotations: • Relax the orthogonality constraint • Gain simplicity in the interpretation. • Allow PCs to be correlated • Examples: promax, oblimin • In MATLAB • A=rotatefactors(B,’Method’,…)

  45. Compute bases vectors for the subspaces of S Rotate the PC’s and interpret biochemical basis Identify Reaction and compound sets that define the basis APPLICATIONS OF FACTOR ROTATION TO METABOLIC NETWORKS

  46. FACTOR ROTATION ON THE CORE E. COLI MODEL

  47. Singular Value Spectrum of the core E. coli Fi: Cumulative fractional singular value Fi • 14 Modes account for >50 % of the network. • 43 out of 72 modes account for > 90% of the network. Modes

  48. Rotation of the Basis Vectors for Col(S) 1st Mode High Energy Phosphate Bonds Before Rotation After Rotation ATP, ADP H,ATP,H2O, ADP, Pi 3rd Mode NAD Redox metabolism NAD, NADH NAD, NADH, CoA, NADPH, CO2, NADP, NADPH

  49. Rotation of the Basis Vectors for Col(S) NADP, NADPH ATP,ADP Q, QH2 Metabolites

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