Topological Properties of the Stoichiometric Matrix By: Linelle T Fontenelle
Dec 14, 2015
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
=
= =
=