Network Properties
Copyright © 2008: Sauro
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Network Properties
Rates of Change
Network Structure
(Stoichiometry Matrix)
Reaction Rate Vector
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Network Properties
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Network Properties
Relationships among the columns -Flux Analysis
Relationships among the
rows -Moiety Analysis
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Conserved Moieties
Over a small enough time scale, the amount
of round subgroup will remain constant.
Subgroups that remain constant during the
evolution of a pathway are called moieties.
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Conserved Moieties
a. Conserved Cycles
b. Moiety Conserved Cycles
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Conserved Moieties
AMP
ADP
ATP
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Conserved Moieties
AMP
ADP
ATP
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Conserved Moieties
Fast
AMP
Fast
ADP
ATP
Slow
Synthetic and Degradation
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Conserved Moieties
Protein Phosphorylation
ATP
ADP
Phosphate
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Conserved Moieties
Protein Phosphorylation
ATP
ADP
Fast
Slow
Synthetic and Degradation
+
= T
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Examples
1. ATP, ADP, AMP
2. NADH, NAD
3. CoA, Acetyl-CoA
4. Protein, Phosphorylated Protein
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Examples
1. ATP, ADP, AMP
2. NADH, NAD
3. CoA, Acetyl-CoA
4. Protein, Phosphorylated Protein
So what?
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Examples
1. ATP, ADP, AMP
2. NADH, NAD
3. CoA, Acetyl-CoA
4. Protein, Phosphorylated Protein
1. Cycles add redundancies to the model equations.
2. They introduce a new parameter into the model:
Total mass in the cycle
3. Can cause numerical instability in certain calculations
(Jacobian becomes non-invertible)
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Simple Example
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Simple Example
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Simple Example
The sum of S1 and S2
remains constant in time
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Simple Example
The sum of S1 and S2
remains constant in time
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Simple Example
The sum of S1 and S2
remains constant in time
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How many
For large metabolic maps, 5% to
10% of the differential equations
are redundant due to conservation
cycles.
http://www.iubmb-nicholson.org/
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How are conserved cycle identified?
Conserved cycles in a pathway appear as linearly
dependent rows in the stoichiometry matrix.
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Linearly Dependent Rows
A linearly dependent set of vectors is one where at
least one of the vectors can be written as a linear
combination of the other vectors.
This means there is redundant information in the vector set.
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The Rank
The Rank of a set of vectors arranged in a matrix
is the number of linearly independent vectors in
the matrix.
The Rank for this matrix is two
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How many conserved cycles are there?
Conserved cycles in a pathway appear as linearly
dependent rows in the stoichiometry matrix.
What’s the Rank
of this matrix?
Type rank(N) in
Matlab or Jarnac.
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How many conserved cycles are there?
Conserved cycles in a pathway appear as linearly
dependent rows in the stoichiometry matrix.
Rank(N) = 2
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How many conserved cycles are there?
Conserved cycles in a pathway appear as linearly
dependent rows in the stoichiometry matrix.
The Rank for this matrix is two
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How many conserved cycles are there?
Conserved cycles in a pathway appear as linearly
dependent rows in the stoichiometry matrix.
The Rank for this matrix is two
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How many conserved cycles are there?
Conserved cycles in a pathway appear as linearly
dependent rows in the stoichiometry matrix.
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How many conserved cycles are there?
Conserved cycles in a pathway appear as linearly
dependent rows in the stoichiometry matrix.
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How many conserved cycles are there?
Conserved cycles in a pathway appear as linearly
dependent rows in the stoichiometry matrix.
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How many conserved cycles are there?
Conserved cycles in a pathway appear as linearly
dependent rows in the stoichiometry matrix.
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How many conserved cycles are there?
Conserved cycles in a pathway appear as linearly
dependent rows in the stoichiometry matrix.
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What are the conserved relationships?
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Echelon Forms
There is a particular kind of matrix that one frequently encounteres in
studying the stoichiometry matrix: The Reduced Echelon Form
Examples of Matrices in Reduced Echelon Form
1.
2.
3.
4.
All rows that consist entirely of zeros are at the bottom of the matrix.
In each non-zero row, the first non-zero entry is a one, called the leading one.
The leading 1 in each row is to the immediate right of all leading 1’s above it.
Each column that contains a leading one has zeros above and below it.
If not, then the matrix is in echelon form, otherwise it is in reduced echelon34form
Echelon Forms
There is a particular kind of matrix that one frequently encountered in
studying the stoichiometry matrix: The Reduced Echelon Form
Examples of Matrices in Reduced Echelon Form
A reduced echelon form has the following general block structure:
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Echelon Forms
Any matrix can be transformed to its reduced echelon form by a series of:
1. Interchanging rows (exchange)
2. Multiplying a row throughout by a nonzero number (scaling)
3. Adding one row, one or more times to another row (replacement)
Reduction to reduced echelon form can be used to find conservation laws.
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Echelon Forms
Let us start with this system:
Recast in Matrix Form:
Identity Matrix
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Echelon Forms
Let us start with this system:
Reduce the stoichiometry matrix to reduced echelon form. To
preserve the right-hand side, we apply the same operations to
the left-side.
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Echelon Forms
Let us start with this system:
Partition the M matrix in to X and Y, along the same row line
as the reduced echelon form:
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Echelon Forms
Multiplying out the lower partition one obtains:
This is the answer we seek. It describes relationships among the rates of
change that do not change in time (the definition of a conserved cycle).
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Let Try an Example
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Let Try an Example
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Let Try an Example
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Let Try an Example
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Let Try an Example
Y Partition
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Let Try an Example
Independent
Species
Dependent
Species
Y Partition
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Let Try an Example
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Advanced Analysis
Reorder the rows in the stoichiometry matrix so that the top m_o rows include
the independent species and the bottom m – m_o rows include the dependent
species.
Since the bottom No dependent rows can be derived by linear combinations of
the top NR rows, we can define a matrix, the link matrix (Lo), that can carry out
this operation:
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Advanced Analysis
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Advanced Analysis
Si are the independent species, and Sd the dependent species.
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Advanced Analysis
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Advanced Analysis
“Y” Matrix =
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Advanced Analysis
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Advanced Analysis
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Advanced Analysis
The conservation laws can be obtained by computing
the null space (kernel) of the transpose of the
stoichiometry matrix.
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What Happens in Simulators
1. From the stoichiometry matrix obtain: m and the set of
dependent and independent species
2. From
obtain
In your ODE solver: Use
compute:
and
to compute dSi/dt
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Advanced Analysis
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