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=UVT dxdt =Sv; ;

•

••

v

VT

“stretches”

x•S

U

diagonal matrix

VTvlinear combination of fluxes

vVT

• •

x•U

x• =U(•)time derivatives are a linear combination (•)

Singular Value Decomposition in Image Processing

5 values 10 values

303 values30 values 52 values

Original

http://peter.wreck.org/reports/Math4305/

Applications of Singular Value Decomposition

Image processing

http://antwrp.gsfc.nasa.gov/apod/image/0011/earthlights_dmsp_big.jpg

Noise reduction

Kinematics

mRNA expression analysis

SVD: BASIC FEATURES

dxdt =Sv

MATLAB:[U, S, V]= svd(A)

Numerical check:||A-USVT||=0?

dxdt =UVTv

The Singular ValuesDiagonal entries in

The singular values1, 2,…….r

large smallsingular value spectrum

fractional singular values

fi=i

i

r

i=1

Fi=i

k=1fk ; Fr=1

cumulative fractionalsingular values

Orthonormal Basis Sets

R(S)

N(S)

C(S)

LN(S)

S=UVT x’

Dim =n Dim =m

Dim (R) =r

Dim(N) =n-rDim(LN) =m-r

Dim(C) = rv

SVD: KEY PROPERTIES

Property #1: Mode by Mode Reconstruction of S

S=i<uiviT>

r

i=1

=1() ( )+ 2() ( )

+……

=1( ) + 2( ) +……

m x n

m x l l x n

m x n

||•||~1

||ui||=1||uj||=1

<ui•uj>=0 i≠j=1 i=j

definition oforthonormality

are scaling factors: i.e., S= 100+ 10 + 1 + ….

Property #2: S Maps the Right Singular Vector (vi) to the Left Singular Vector (ui)

S = UVT

SV = U(VTV)SV = U

S ( ) = ( )( )|||| |||| 00

k k k

Svk = kuk

( )| = •|

(xV)

=I

UTU=I

VTV=I

||||( ) ||||( )=I

UT=U-1

vk kuk

S

m x m

n x n

Independent dimension in the Row space

Independentdimension in the Columnspace

Dimension in the Row space

Orthonormality and dynamic decoupling

•

••

VT

x’S

U

v

00( )(|)d(|)

dt =

x UT

Decoupled motion

EXAMPLES

Example #1

m = n=2; r=1

Bounded Spaces

Example #2

Orthonormal basis for Column and Left Null

-1-11

0 21 -11 1

Left Null

Col

3

3

An Alternative Set of Vectors for the Left Null Space

-1-11( )Col

l1 and l2 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 andAcCoA charging

Systemic reactions w/ biomass

Translocation of a protonAnd ATP synthesis

Growth

Transhydrogenation andAcCoA charging

DECOMPOSITION OF GENOME-SCALE MATRICES

The singular value

spectrum

1st mode:high energy phosphate bonds

Motion: stoichiometry

Drivers: reactions

2nd mode: NADPH redox metabolism

Motion: stoichiometry

Drivers: reactions

3rd Mode: translocated protonMotion: stoichiometry

Drivers: reactions

4th modeMotion: stoichiometry

Drivers: reactions

ROTATING BASIS VECTORS

The effects of rotation:Rotation of the Basis Vectors for Col(S)

Met

abol

ites

NADP, NADPH Q, QH2ATP,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

Basis Rotation Methods

• The two major categories1. 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=UVT 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 VT 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,

quartimax2. 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’,…)

Category Method Name Comments

Orthnormal

Quartimax

Maximizes the sums of squares of the coefficients across the resultant vectors for each of the original variable

VarimaxMaximizes 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

Fi

Fi: Cumulative fractional singular value

Rotation of the Basis Vectors for Col(S) 1st Mode High Energy Phosphate Bonds

H,ATP,H2O, ADP, Pi

ATP, ADP

Before Rotation After Rotation

NAD, NADH, CoA, NADPH, CO2, NADP, NADPH

NAD, NADH

3rd Mode NAD Redox metabolism

Rotation of the Basis Vectors for Col(S) M

etab

olite

s

NADP, NADPH Q, QH2ATP,ADP

Rotation of the Basis Vectors for LN(S)

Modes

Met

abol

ites

AMP,ADP,ATP NADH,NAD

Upon rotation, the time invariant pools are clearly resolved

CoA,SuccCoANADH,NAD,NADPH,NADP, QH2

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

Fac

tor

Lo

adin

g

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