Topological Properties of the Stoichiometric Matrix

By:

Linelle T Fontenelle

Topology

Some geometric problems do not depend on the exact shape of the object, but rather on the way that points are connected in space

Two spaces are topologically equivalent if one can be deformed into another without tearing space apart or sticking distinct parts together

Stoichiometric Matrix

Consider the following system:

x

tv v v

x

tv v v

x

tv v v

1

1 2 3

2

1 2 3

3

1 2 3

2 2 3

4 5 6

7 8 9

X1

X2

x3

v1

v2

v3

Stoichiometric Matrix In matrix notation:

S is a linear transformation of the flux vector, v, to a vector of time derivatives of the concentration vector

Where:

d

d t

xSv

d

d t

dx

d tdx

d tdx

d t

x

1

2

3

S

2 2 3

4 5 6

7 8 9

v

v

v

v

1

2

3

S Matrix as a Linear Map

http://www.biophysj.org

d

d t

xSv

Flux Solution Space

-Row space

-Null space

Concentration Solution Space

-Column Space

-Left null space

Row Space of S

Spanned by all the independent rows of S

Space in which changes in the concentration values contribute to the flux rates

http://gcrg.ucsd.edu/classes/4_slides_Smatrix.pdf

Null Space of S

Consists of all vectors that satisfy Svss = 0

i.e. dx/dt = 0

Spans the steady state pathway space of a biochemical network http://gcrg.ucsd.edu/classes/4_slides_Smatrix.pdf

Left Null Space of S

Consists of all vectors that define the dependencies of the rows in S

Constrains the conserved relationships

http://gcrg.ucsd.edu/classes/4_slides_Smatrix.pdf

Column Space of S

Spanned by all the independent columns of S

Defines the dynamic concentration space in which metabolites are formed and consumed http://gcrg.ucsd.edu/classes/4_slides_Smatrix.pdf

Connectivity Properties of the S Matrix

s s

s s

n

m m n

11 1

1

. . . . . .

. .

. .

. .

. . . . . .

Reactions

Met

abol

ites

sij

x1

x2

x3

x4

6

2

3

2

Every column of S is a chemical rxn with a defined set of values. The fluxes however are the values that represent the activity of the rxns and indicates how much is going thru them. S connects all the metabolites in a defined metabolic system. Metabolic networks must make all the biomass components of the cell in order for it to grow.

Metabolite Connectivity

Metabolite Connectivity of E.coli, H. influenzae, H. pylori and S. cerevisiae

http://gcrg.ucsd.edu/classes/4_slides_Smatrix.pdf

Elementary Topological Properties

2 0 1 2 3 0 1 1 0

3 0 3 2 1 1 0 0 2

1 0 2 1 0 3 1 6 1

0 1 2 5 4 1 3 1 0

0 1 1 0 2 0 4 3 0

1 0 1 1 1 0 1 1 0

1 0 1 1 1 1 0 0 1

1 0 1 1 0 1 1 1 1

0 1 1 1 1 1 1 1 0

0 1 1 0 1 1 1 1 0

SS

The Binary form of S

Based on nonzero elements of S

Elementary Topological Properties

s ij j

i

m

1

i ijsj

n

1

1 0 1 1 1 0 1 1 0

1 0 1 1 1 1 0 0 1

1 0 1 1 0 1 1 1 1

0 1 1 1 1 1 1 1 0

0 1 1 0 1 1 1 1 0

Participation Number Connectivity Number

S

Expanded Elementary Topological Properties

1 0 0

1 0 1

1 1 1

1 0 1

1 1 1 1

0 0 1 0

0 1 1 1

m x n = 4x 3n x m = 3 x 4

=

1 1 1 1

1 2 2 2

1 2 3 2

1 2 2 2

m x m = 4 x 4

Compound Adjacency Matrix: Ax =

S S T

S S T

Ax

( ) a x ii iks 2

k( ) a x ij ik k js s

x

Diagonal elements summation gives the no. of rxns in which compound xi participates. The off diagonal elements gives the number of rxns in which both compounds xi and xj participate and shows how extensively the 2 compounds are topologically connected in the network

Expanded Elementary Topological Properties

1 1 1 1

0 0 1 0

0 1 1 1

1 0 0

1 0 1

1 1 1

1 0 1

3 1 3

1 1 1

3 1 3

n x m = 3 x 4 m x n = 4x 3 n x n = 3 x 3

=x

Reaction Adjacency Matrix: Av = S T S

( )a vk

ii k is 2 ( ) a v ij jk k is s

S T S Av

Diagonal elements give participation no. Off diagonal elements count how many compounds 2 rxns have in common

Example of Adjacency Matrix

A cofactor-coupled reaction:

1 1

1 1

1 1

1 1

C + AP CP + A

S

1 1

1 1

1 1

1 1

S

1 1 1 1

1 1 1 1

S T

Example of Adjacency Matrix

Ax = S S T

Av = S T S1 1

1 1

1 1

1 1

1 1

1 1

1 1

1 1

1 1 1 1

1 1 1 1

1 1 1 1

1 1 1 1

2 2 2 2

2 2 2 2

2 2 2 2

2 2 2 2

4 4

4 4

=

= =

=

Conclusion/Future Direction

Elementary topological properties give network connectivity and component participation information

Leads to combined and simultaneous characterization of metabolites and reactions: SVD (decouples and decorrelates systemic properties)