Lecture #7

The stoichiometric matrix

Outline

• The many facets of S• Chemistry:

– S as a data matrix• Network structure:

– S as a connectivity matrix• Mathematics:

– S as a linear transformation• Systems science:

– S as part of a network model

Matrix Attributes

Properties of the Stoichiometric Matrix

S AS A DATA MATRIX

Forming S

1.

2.

3.

S is a data matrix

(bio) chemistry

2 3 4

math

The GPRs: GenesProteinsReactions

gene

protein

reaction

GPRs

isozymespromiscuous

S as a data matrix

S# compounds

# reactionsSmxn

E. coli at the genome-scale: S for iAF1260

S for Geobacter sulfurreducans

Reguera, et al, Nature 435:1098-1101 (2005)

QC/QA on S• Every column in S (si; reaction vector)

represents a reaction– Chemically accurate, including charge and

elemental balancing

elem

ents

compounds

. . . . . .

the rows of E are in the left null of S

Elemental balance:E•S=0

Elemental matrix: E

QC/QA: quality control/quality assurance

ES=0

Time invariants in the left null space of S

Elemental and charge

balancing

Elemental Composition of some Common Metabolites

Example: S for Glycolysis vhk vpgi vpfk vtpi vald vgapdh vpgk vpglm veno vpk vldh vamp vapk vpyr vlac vatp vnadh vgluin vampinGLC -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0G6P 1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0F6P 0 1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0FBP 0 0 1 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0

DHAP 0 0 0 -1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0

GA3P 0 0 0 1 1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0

13DPG 0 0 0 0 0 1 -1 0 0 0 0 0 0 0 0 0 0 0 03PG 0 0 0 0 0 0 1 -1 0 0 0 0 0 0 0 0 0 0 02PG 0 0 0 0 0 0 0 1 -1 0 0 0 0 0 0 0 0 0 0PEP 0 0 0 0 0 0 0 0 1 -1 0 0 0 0 0 0 0 0 0PYR 0 0 0 0 0 0 0 0 0 1 -1 0 0 -1 0 0 0 0 0LAC 0 0 0 0 0 0 0 0 0 0 1 0 0 0 -1 0 0 0 0NAD 0 0 0 0 0 -1 0 0 0 0 1 0 0 0 0 0 1 0 0

NADH 0 0 0 0 0 1 0 0 0 0 -1 0 0 0 0 0 -1 0 0

AMP 0 0 0 0 0 0 0 0 0 0 0 -1 1 0 0 0 0 0 1ADP 1 0 1 0 0 0 -1 0 0 -1 0 0 -2 0 0 1 0 0 0ATP -1 0 -1 0 0 0 1 0 0 1 0 0 1 0 0 -1 0 0 0Pi 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 1 0 0 0

GLC G6P F6P FBP DHAP GA3P 13DPG 3PG 2PG PEP PYR LAC NAD NADH AMP ADP ATP PiC 6 6 6 6 3 3 3 3 3 3 3 3 21 21 10 10 10 0H 12 11 11 10 5 5 4 4 4 2 3 5 27 28 13 13 13 1O 6 9 9 12 6 6 10 7 7 6 3 3 14 14 7 10 13 4N 0 0 0 0 0 0 0 0 0 0 0 0 7 7 5 5 5 0P 0 1 1 2 1 1 2 1 1 1 0 0 2 2 1 2 3 1

E for Glycolysis

QC/QA on S for glycolysis

vhk vpgi vpfk vtpi vald vgapdh vpgk vpglm veno vpk vldh vamp vapk vpyr vlac vatp vnadh vgluin vampinC 0 0 0 0 0 0 0 0 0 0 0 -10 0 -3 -3 0 0 6 10H -1 0 -1 0 0 -1 0 0 -2 1 1 -13 0 -3 -5 1 -1 12 13O 0 0 0 0 0 0 0 0 -1 0 0 -7 0 -3 -3 1 0 6 7N 0 0 0 0 0 0 0 0 0 0 0 -5 0 0 0 0 0 0 5P 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 1

The product E•S is shown below. As can be seen E•S ≠ 0.

Hydrogen needs to be added on the product side

Water has to be added to the product side to balance

Hydrogen needs to be added on the reactant side

Hydrogen needs to be added on the product side and water on the reactant side

Hydrogen needs to be added on the product side

Balanced S matrix for Glycolysis after QC/QA

vhk vpgi vpfk vtpi vald vgapdh vpgk vpglm veno vpk vldh vamp vapk vpyr vlac vatp vnadh vgluin vampinGLC -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0G6P 1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0F6P 0 1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0FBP 0 0 1 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0DHAP 0 0 0 -1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0GA3P 0 0 0 1 1 -1 0 0 0 0 0 0 0 0 0 0 0 0 013DPG 0 0 0 0 0 1 -1 0 0 0 0 0 0 0 0 0 0 0 03PG 0 0 0 0 0 0 1 -1 0 0 0 0 0 0 0 0 0 0 02PG 0 0 0 0 0 0 0 1 -1 0 0 0 0 0 0 0 0 0 0PEP 0 0 0 0 0 0 0 0 1 -1 0 0 0 0 0 0 0 0 0PYR 0 0 0 0 0 0 0 0 0 1 -1 0 0 -1 0 0 0 0 0LAC 0 0 0 0 0 0 0 0 0 0 1 0 0 0 -1 0 0 0 0NAD 0 0 0 0 0 -1 0 0 0 0 1 0 0 0 0 0 1 0 0NADH 0 0 0 0 0 1 0 0 0 0 -1 0 0 0 0 0 -1 0 0AMP 0 0 0 0 0 0 0 0 0 0 0 -1 1 0 0 0 0 0 1ADP 1 0 1 0 0 0 -1 0 0 -1 0 0 -2 0 0 1 0 0 0ATP -1 0 -1 0 0 0 1 0 0 1 0 0 1 0 0 -1 0 0 0Pi 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 1 0 0 0H 1 0 1 0 0 1 0 0 0 -1 -1 0 0 0 0 1 1 0 0H2O 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 -1 0 0 0

E•S = 0, except for bi

S AS AN INCIDENCE MATRIX

S Represents a ‘Graph’ or ‘Map’

( )S:link, vj

node, xi

•Reaction Map

•Flux Map:a particular functional stateof the reaction map

1

1/2

1/21/4

1/4A particular flux vector

ST is also a ‘Map’ or ‘Graph’

( )link, xi

node, vj

1/21

x1v1

x3x2

v2

v3

- Compound map- Concentration map

1/6

Duality, S, and ST

Examples

Reaction Map of iAF1260

S AS A PART OF THE DYNAMIC MASS BALANCES

Dynamic Mass Balance

v5

v4

v3

v2

v1

rate of formation rate of degradation

xi: compound i

vj: reaction jmoles/vol/time

100-

106 m

olecu

les

3 sec-

min

Dynamic Mass Balance for a Network

1) steady state 0=S•v ; Systems Biology Vol I

2) dynamic states

simulate

Systems Biology Vol II

kineticconstants

Defining a ‘System’ Based on S

systemboundary

inside

outside I/O

•physical; cell wall, nuclear membrane

•virtual; ETS

The different forms of S

Example

Reaction Map for Glycolysis as a System

S for Glycolysis as a System

S AS A MAPPING OPERATION

S Represents a mapping Operation

;

What is in the Four Subspaces?

• (right) null space– 0=S•v; all steady state solutions

• left null space– 0=l•S= time invariants

The Column Space contains the directions of motion

column space ;

( )

Row space contains the thermodynamic driving forces

m mxn n

m x n

concentration(compound)

fluxes(reaction)

m ≤ n

Summary• S is comprised of stoichiometric coefficients that are

commonly integer numbers. The columns of the S represent chemical reactions while the rows represent compounds.

• S has many important attributes: – 1) it is a data matrix, – 2) it gives the structure of a network, – 3) it is a mathematical mapping operations with four fundamental subspaces,

and – 4) it forms a key part of in silico models describing the functional states of

networks.

• The column (reaction) vectors represent chemical transformations and thus come with chemical information.

• The reaction vectors imply elemental and charge balance. The reaction vectors are thus orthogonal to the rows of E. These conservation quantities are in the left null space of S.

Summary, con’t• Some quantities, such as free energy, are non-conserved

during a chemical reaction. These quantities will be in the row space of S.

• There are few basic forms of the elementary reactions. Most, if not all, biochemical reaction networks are either linear or bi-linear.

• Structurally, or topologically, S represents a reaction map. The transpose of S represents a compound map. Both maps are topologically non-linear, as they contain joint edges between nodes.

• The boundaries of a reaction network can be drawn in different ways and lead to three fundamental forms of S.