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: v1  x
1
 t

2 v
1

2 v
2

3 v
3 v2
X
1
X
2 x
3
 x
2
 t

4 v
1

5 v
2

6 v
3 v3  x
3
 t

7 v
1

8 v
2

9 v
3


Stoichiometric Matrix
In matrix notation: d x

Sv dt
S is a linear transformation of the flux vector, v , to a vector of time derivatives of the concentration vector
Where: d x
 dt dx dt dx dt dx
3 dt
1
2
















S






2 2 3


4 5 6
7 8 9


 v





 v
1 v v
2
3






S Matrix as a Linear Map
Flux Solution Space
-Row space
-Null space d x
 dt
Sv
Concentration Solution Space
-Column Space
-Left null space http://www.biophysj.org

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 Sv ss
= 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
Reactions
2 x2







 s
.
.
.
11 s m 1
.
.
.
.
.
.
s ij
.
.
.
.
.
.
s
1
.
.
.
s mn n








6 x1 x3
3 x4
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
 Based on nonzero elements of S
The Binary form of S








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



S








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 



Elementary Topological Properties
Participation Number i m 


1 s  ij
 j








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 


Connectivity Number
 i
 j n 

 s
1 ij

Expanded Elementary Topological
Properties
T Compound Adjacency Matrix: A x
=
Diagonal elements summation gives the no. of rxns in which compound x i participates. The off diagonal elements gives the number of rxns in which both compounds x i and x j participate and shows how extensively the 2 compounds are topologically connected in the network
T
A x






1 0 0

1 0 1

1 1 1


1 0 1 
 x





1 1 1 1

0 0 1 0
0 1 1 1 



= 





1 1 1 1

1 2 2 2 

1
1
2
2
3
2
2
2


 n x m = 3 x 4 m x m = 4 x 4 m x n = 4x 3
( ) ii

 s  2 k ik
( ) ij

 s s kj

Expanded Elementary Topological
Properties
Reaction Adjacency Matrix: A v =
T
T A v





1 1 1 1

0 0 1 0
0 1 1 1 


 x




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
( a v
) ii
 m x n = 4x 3

2 k s ki
( ) ij
 n x n = 3 x 3
 s s ki
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:
C + AP
CP + A








1
1
1 1
1 1

1





1


S




1 1

1 1

1 1


1 1



1 1 1 1
 1 1 1 1
T




Example of Adjacency Matrix
 A x
=
=




1 1

1 1

1 1


1 1

T



1 1 1 1

1 1 1 1 

= 





2 2 2 2
2 2 2 2
2 2 2 2
2 2 2 2






=
T A v
=


1 1 1 1
 1 1 1 1







1 1

1 1

1 1


1 1

=


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)