Transcript Chapter 10: The Left Null Space of S

Chapter 10: The Left Null
Space of S
- or Now we’ve got S. Let’s do some
Math and see what happens.
A review of S
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Every column is a
reaction
Every row is a
compound
S transforms a flux
vector v into a
concentration time
derivative vector,
dx/dt = Sv
Networks from S
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S: a network showing
how reactions link
metabolites
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-ST: a network
showing how
compounds link
reactions
Introducing L
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LS = 0
Dimension of L is m-r
Rows are:
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linearly independent
span L
Are orthogonal to the
reaction vectors of S
(columns)
Finding L
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“The convex basis for the left null space
can be computed in the same way as the
right null space by transposing S”
- Palsson p. 155
What we really do to find L:
a little bit of math
Remember we’re trying to find L from LS = 0.
We might try to say that since SR = 0 and LS = 0,
S = R. But matrix multiplication is generally not
commutative. That is, LS  SL, so that’s wrong.
BUT, we can use the identity that (LS)T=STLT to
make some progress:
LS = 0
(LS)T = 0T = 0
STLT = 0
Matlab: why we’re not afraid of
a big S
STLT = 0 means that LT is the basis for the null
space of ST.
Let b = ST. Then the Matlab command a =
null(b) will return a basis for the null space of
LT.
Once we have a, the Matlab command L = a’
will return L.
Note that this L is not a unique basis - there are
infinitely many.
So? What does L mean?
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We’ve found a matrix, L, that when multiplied by
S, gives the 0 matrix:
LS = 0
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Recall the definition of S as a transformation:
dx/dt = Sv
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Let’s do more math!
Doing Math to find the
meaning of L
dx/dt = Sv
L dx/dt = LSv
since LS = 0,
L dx/dt = 0
Palsson writes this as d/dt Lx = 0 (eq 10.5)
We can integrate to find Lx = a
Pools are like Pathways.
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Chapter 9: Using R (the
right null space), found
with the rows of S, to find
extreme pathways on
flux maps.
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Chapter 10: Using L (the
left null space), found
with the columns of S, to
find pools.
Pathways and pools
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3 types of extreme
pathways
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through fluxes
futile cycles + cofactors
internal cycles
3 types of pools
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primary compounds
primary and secondary
compounds internal to
system
only secondary
compounds
Back to the Math:
the reference state of x
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In L x = a, there’s a few ways we can get x and
a. For example, we can pick either initial or
steady-state conditions to set the pool sizes, ai
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L x = a is true for many different values of x,
such as Lxref = a. So whatever x we pick, we
can also pick a xref such that L (x - xref) = 0.
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This transformation changes the basis of the
concentration space. Whereas x is not
orthogonal to L, x - xref is.
The reference state of x
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The new basis of the
concentration space
from (x - xref) allows
us to transform our
choice of x to a
closed, or bounded,
concentration space
that has end points
representing the
extreme
concentration states.
Intermission…
Until next week?