56:198:582 Biological Networks
Lecture 4
Left Null Space of S
Definition
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The left null space of S is the set of all vectors l such that
l · sj = 0
for all j = 1, . . . , n, where sj are the column vectors of S
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Let li , i = 1, 2, . . . , m − r be a set of basis vectors that span
the left null space, and let L be the (m − r ) × m matrix whose
rows are li
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Then
LS = 0
The conservation relationships
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Since LS = 0, we have
dx
=0
dt
d
⇒ (Lx) = 0
dt
LSv = 0 ⇒ L
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Thus, the vectors in the left null space represent conservation
relationships, i.e. summation of compount concentrations,
called pools, that are time invariant. The compound
combinations corresponding to vectors in the left null space
are always conserved at constant values, regardless of what
the flux v is.
Pool sizes
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Because d(Lx)/dt = 0, we have the mass conservation
equation
Lx = a,
where a is a vector that gives the sizes (the total
concentration) of the pools
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This equation defines an affine hyperplane, i.e. a hyperplane
not passing through the origin. This hyperplane is the
concentration space in which the concentration vector x
resides
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Since the concentrations xi are non-negative, the left null
space can be described using a convex basis
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A convex basis for the left null space can be computed by
computing a convex basis for the (right) null space of S0
Reference states
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Since Lx = a for all allowable concentration vectors x, we have
Lx = Lxref = a
or
L(x − xref ) = 0
for some reference concentration vector xref
Reference states
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The choice of xref is more or less arbitrary. We choose xref
according to the following two criteria
1. xref is orthogonal to the column vectors of S (this allows the
affine concentration space to be spanned using vectors from
columns of S)
2. x − xref is orthogonal to the rows of L (this is a required
condition because L(x − xref ) = 0)
An example: Simple reversible reaction
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Consider the reaction system
v1 : x1 → x2
v2 : x2 → x1
What is S?
An example: Simple reversible reaction
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Consider the reaction system
v1 : x1 → x2
v2 : x2 → x1
What is S?
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What is dim(Left null(S))?
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An example: Simple reversible reaction
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Consider the reaction system
v1 : x1 → x2
v2 : x2 → x1
What is S?
What is dim(Left null(S))?
L = 1 1 . This says that the total concentration of x1 + x2
is conserved
An example: Simple reversible reaction
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Consider the reaction system
v1 : x1 → x2
v2 : x2 → x1
What is S?
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What is dim(Left null(S))?
L = 1 1 . This says that the total concentration of x1 + x2
is conserved
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Suppose a = (1). What is the affine concentration space?
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An example: Simple reversible reaction
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I
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Consider the reaction system
v1 : x1 → x2
v2 : x2 → x1
What is S?
What is dim(Left null(S))?
L = 1 1 . This says that the total concentration of x1 + x2
is conserved
Suppose a = (1). What is the affine concentration space?
1
−1
x=
+ξ
, ξ ∈ [0, 1]
0
1
An example: Simple reversible reaction
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What is xref ?
An example: Simple reversible reaction
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What is xref ?
xref
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1/2
=
1/2
Finally, we have
x − xref =
1/2
−1
+ξ
,
−1/2
1
ξ ∈ [0, 1]
An example: Bilinear assocation
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Consider the reaction system
v1 :
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What is S?
x1 + x2 → x3
An example: Bilinear assocation
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Consider the reaction system
v1 :
x1 + x2 → x3
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What is S?
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What is dim(Left null(S))?
An example: Bilinear assocation
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Consider the reaction system
v1 :
x1 + x2 → x3
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What is S?
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What is dim(Left null(S))?
1 0 1
L=
. What are the conserved pools?
0 1 1
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An example: Bilinear assocation
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Consider the reaction system
v1 :
x1 + x2 → x3
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What is S?
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What is dim(Left null(S))?
1 0 1
L=
. What are the conserved pools?
0 1 1
0
Suppose a = 1 2 . What is the affine concentration space?
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An example: Bilinear assocation
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Consider the reaction system
v1 :
x1 + x2 → x3
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What is S?
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What is dim(Left null(S))?
1 0 1
L=
. What are the conserved pools?
0 1 1
0
Suppose a = 1 2 . What is the affine concentration space?
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 
 
1
−1
x = 2 + ξ −1 ,
0
1
ξ ∈ [0, 1]
An example: Bilinear assocation
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We can determine xref from s1 · xref = 0 and L(x − xref ) = 0
to obtain
 
0
xref = 1
1
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Thus

x − xref

 
1
−1
=  1  + ξ −1 ,
−1
1
ξ ∈ [0, 1]
Row and Column Spaces of S
The Column Space
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The column space is the space of all possible concentration
derivatives in the network. It is spanned by the column
vectors si of S:
dx
= s1 v1 + s2 v2 + · · · + sn vn ,
dt
where 0 ≤ vi ≤ vi,max
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The column space can be used to understand how
concentrations change in the network
An example
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Consider the reaction
2H2 O2 → O2 + 2H2 O,
which has elemental matrix
O
H
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H2 O2 O2 H2 O
2
2
1
2
0
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Consider the compounds −2H2 O2 + O2 + 2H2 O,
2H2 O2 + 2O2 + H2 O, and 2H2 O2 + 2H2 O. The first is
derived from the reaction itself, and the second and third from
the elemental matrix. We expect compound 1 changes in
time, while the other two are conserved
An example
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We can see this because

  
−2H2 O2 + O2 + 2H2 O
9
d 


2H2 O2 + 2O2 + H2 O
= 0 v1
dt
2H2 O2 + 2H2 O
0
0
The column space is spanned simply by 9 0 0 , describing
the time variation of the compounds of interest
The row space
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The row space is orthogonal to vectors in the null space.
Hence vectors in the row space are flux vectors v, and
Sv 6= 0
for vectors v in the row space
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Vectors in the row space are constrained by 0 ≤ vi ≤ vi,max