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) v x S=U  V T dx dt =Sv;; v VTVT  “stretches” x S U diagonal matrix VTvVTv linear combination of fluxes v VTVT  x U x =U( ) time derivatives are a linear combination ()

Singular Value Decomposition in Image Processing 5 values10 values303 values30 values52 valuesOriginal

Applications of Singular Value Decomposition Image processing Noise reduction Kinematics mRNA expression analysis

SVD: BASIC FEATURES

dx dt =Sv MATLAB: [U, S, V]= svd(A) Numerical check: ||A-USV T ||=0 ? dx dt =U  V T v

The Singular Values Diagonal entries in  The singular values  1,  2,…….  r largesmall singular value spectrum fractional singular values fi=fi= ii ii  r i=1 Fi=Fi=  i k=1 f k ; F r =1 cumulative fractional singular values

Orthonormal Basis Sets R(S) N(S) C(S) LN(S) S =U  V T x’x’ Dim =n Dim =m Dim (R) =r Dim(N) =n-r Dim(LN) =m-r Dim(C) = r v

SVD: KEY PROPERTIES

Property #1: Mode by Mode Reconstruction of S S=   i r i=1 =1=1 () ( ) + 22 () ( ) +…… =1=1 ( ) + 22 +…… m x n m x ll x n m x n ||||~1 ||u i ||=1 ||u j ||=1 =0 i≠j =1i=j definition of orthonormality  are scaling factors: i.e., S= ….

Property #2: S Maps the Right Singular Vector (v i ) to the Left Singular Vector (u i ) S = U  V T SV = U  (V T V) SV = U  S ( ) = ( )( ) |||| 0 0 k kk Sv k =  k u k ( )| = | (xV) =I U T U=I V T V=I |||| () ( ) =I U T =U -1 v k  k u k S m x m n x n Independent dimension in the Row space Independent dimension in the Columnspace Dimension in the Row space

Orthonormality and dynamic decoupling VTVT  x’x’ S U v 0 0 ( )(|) d(|) dt = x U T Decoupled motion

EXAMPLES

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

Bounded Spaces

Example #2

Orthonormal basis for Column and Left Null Left Null Col 3 3

An Alternative Set of Vectors for the Left Null Space 1 ( ) Col l 1 and l 2 are convex basis vectors (0,1,1)

GENOME-SCALE STOICHIOMETRIC MATRICES

SVD of S: global view

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

Mapping: chemical reaction interpretation

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

DECOMPOSITION OF THE CORE METABOLIC NETWORK IN E. COLI

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

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

DECOMPOSITION OF GENOME- SCALE MATRICES

The singular value spectrum

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

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

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

4 th mode Motion: stoichiometry Drivers: reactions

Angles as measures of similarity

ROTATING BASIS VECTORS

The effects of rotation: Rotation of the Basis Vectors for Col(S) Metabolites NADP, NADPH Q, QH 2 ATP,ADP

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

Applying to Metabolic Networks 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

Basis Rotation Methods The two major categories 1.Orthogonal Rotations: maintain all PCs perpendicular to each other Examples: varimax, orthomax, quartimax 2.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’,…)

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

The end

Extras

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.

Methods for Factor Rotation The two major categories 1.Orthogonal Rotations: maintain all PCs perpendicular to each other Examples: varimax, orthomax, quartimax 2.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’,…) CategoryMethod NameComments Orthnormal Quartimax Maximizes the sums of squares of the coefficients across the resultant vectors for each of the original variable Varimax Maximizes the sum of the variance of the loading vectors Equimax Spread the extracted variance evenly across the rotated factors Oblique Promax Uses an orthogonal solution as the basis for creating an oblique solution using a procrustes rotation Oblimax Maximizes the kurtosis of all the loadings across all variables and factors without consideration of the relative positon of the factors Direct Oblimin Similar to a quartimax approach, but minimizes and does away with reference vectors

APPLICATIONS OF FACTOR ROTATION TO METABOLIC NETWORKS 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

FACTOR ROTATION ON THE CORE E. COLI MODEL

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

Rotation of the Basis Vectors for Col(S) 1 st Mode High Energy Phosphate Bonds H,ATP,H 2 O, ADP, P i ATP, ADP Before Rotation After Rotation NAD, NADH, CoA, NADPH, CO 2, NADP, NADPH NAD, NADH 3 rd Mode NAD Redox metabolism

Rotation of the Basis Vectors for Col(S) Metabolites NADP, NADPH Q, QH 2 ATP,ADP

Rotation of the Basis Vectors for LN(S) Modes Metabolites AMP,ADP,ATPNADH,NAD Upon rotation, the time invariant pools are clearly resolved CoA,SuccCoA NADH,NAD,NADPH, NADP, QH 2

ROTATION OF BASIS VECTORS AT THE GENOME-SCALE

Rotating the bases vectors of LN(S) for iAF1260 The LN(S) basis vectors correspond to time invariant pools The pools found are: – Amino acyl tRNAs – tRNAs – Charge Carriers (NADH. NAD) – Co-factor Pools – Apolipoprotein-lipoprotein Factor Loading

TILTING OF BASIS VECTORS

Tilting the Left Null Space of iAF1260 The basis vectors correspond to the time invariant pools – Amino acyl tRNAs – tRNAs – Charge carriers (NADH + NAD) – Other co-factor pools – Apolipoprotein-lipoprotein