Structural Properties of Networks
Systems and Synthetic Biology
498A
Copyright © 2008: Sauro
1
Network Equation
Rates of Change
Network Structure
(Stoichiometry Matrix)
Reaction Rate Vector
2
Stoichiometry Matrix
3
Stoichiometry Matrix
Relationships among the columns -Flux Analysis
Relationships among the
rows -Moiety Analysis
4
Conserved Moieties
In the above cycle the molar amount of the round
subgroup remains constant.
Subgroups whose molar amounts remain constant during
the evolution of a pathway are called moieties.
5
Conserved Moieties
Also called:
Conserved Cycles
Or
Moiety Conserved Cycles
6
Conserved Moieties
AMP
ADP
ATP
7
Conserved Moieties
AMP
ADP
ATP
8
Conserved Moieties
Fast
AMP
Fast
ADP
Slow
ATP
Over long timescales the
moieties are not conserved.
Synthetic and Degradation
9
Conserved Moieties
Protein Phosphorylation
ATP
ADP
Phosphate
Protein
10
Conserved Moieties
Protein Phosphorylation
ATP
ADP
Fast
+
= T
Slow
Synthetic and Degradation
11
Examples
1. ATP, ADP, AMP
2. NADH, NAD
3. CoA, Acetyl-CoA
4. Protein, Phosphorylated Protein
5. etc.
12
Examples
1. ATP, ADP, AMP
2. NADH, NAD
3. CoA, Acetyl-CoA
4. Protein, Phosphorylated Protein
So what?
13
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 moles in the cycle
3. Can cause numerical instability in certain calculations
(Jacobian becomes non-invertible)
4. Can have profound effects on the behavior of models
14
Redundancies
15
Redundancies
16
Redundancies
The sum of S1 and S2
remains constant in time
17
Redundancies
The sum of S1 and S2
remains constant in time
18
Redundancies
The number of equations
has been reduced by one
Blackboard
19
New Parameters in the Model
The number of equations
has been reduced by one
20
New Parameters in the Model
Et = E + ES
Vmax
In a simple cycle, the total molarity in the cycle acts as a linear scaling factor.
New Parameters in the Model
K
A
B
P
Three Conservation Cycles:
A/B/C
K
P
C
New Parameters in the Model
K
Increase
A
B
P
Three Conservation Cycles:
A/B/C
K
P
C
New Parameters in the Model
K
Increase
A
B
P
Three Conservation Cycles:
A/B/C
K
P
C
New Parameters in the Model
Sequestration of K
K
Increase
A
B
C
P
Three Conservation Cycles:
A/B/C
K
P
The net effect is that increases in A inhibit the
B -> C reaction. Similar arguments apply to C
New Parameters in the Model
A
B
C
The end result is a toggle switch that can display two steady states
How many are there?
For large metabolic maps, 5% to 10%
of the species are involved in
moiety conserved cycles.
That is 5% to 10% of the differential
equations are redundant.
http://www.iubmb-nicholson.org/
27
How are conserved cycles
identified?
Conserved cycles in a pathway appear as linearly
dependent rows in the stoichiometry matrix.
28
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.
29
How are conserved cycles
identified?
Conserved cycles in a pathway appear as linearly
dependent rows in the stoichiometry matrix.
v1
v2
S1
S2
30
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
31
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.
32
How many conserved cycles are there?
Conserved cycles in a pathway appear as linearly
dependent rows in the stoichiometry matrix.
Rank(N) = 2
33
How many conserved cycles are there?
Conserved cycles in a pathway appear as linearly
dependent rows in the stoichiometry matrix.
Rank(N) = 2
E * -1 = ES
S2 = -1 * (ES + S1)
34
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
35
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
36
How many conserved cycles are there?
Conserved cycles in a pathway appear as linearly
dependent rows in the stoichiometry matrix.
37
How many conserved cycles are there?
Conserved cycles in a pathway appear as linearly
dependent rows in the stoichiometry matrix.
38
How many conserved cycles are there?
Conserved cycles in a pathway appear as linearly
dependent rows in the stoichiometry matrix.
39
How many conserved cycles are there?
Conserved cycles in a pathway appear as linearly
dependent rows in the stoichiometry matrix.
40
How many conserved cycles are there?
Conserved cycles in a pathway appear as linearly
dependent rows in the stoichiometry matrix.
41
Echelon Forms
There is a particular kind of matrix that one frequently encounteres in the
study of 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 echelon42form
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:
43
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.
44
Echelon Forms
Let us start with this system:
Recast in Matrix Form:
Identity Matrix
45
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.
46
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:
47
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).
48
Let’s Try an Example
49
Let’s Try an Example
50
Let’s Try an Example
51
Let’s Try an Example
52
Let’s Try an Example
Y Partition
53
Let’s Try an Example
Independent
Species
Dependent
Species
Y Partition
54
Let’s Try an Example
Independent
Species
Dependent
Species
Y Partition
I
55
Let’s Try an Example
Independent Species
Dependent Species
Expressing the dependent species in terms of the independent species.
56
Advanced Analysis
Independent Species
Dependent Species
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:
m = number of rows in the stoichiometry matrix.
57
Advanced Analysis
58
Advanced Analysis
Si are the independent species, and Sd the dependent species.
59
Advanced Analysis
60
Advanced Analysis
“Y” Matrix =
cf. Slide 554
61
Advanced Analysis
“Y” Matrix =
cf. Slide 55
62
Advanced Analysis
63
Advanced Analysis
64
Advanced Analysis
The conservation laws can be obtained by computing
the null space (kernel) of the transpose of the
stoichiometry matrix.
65
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
66
Advanced Analysis
67
Practical Consequences
68
Sleeping Sickness
Parasitic Protozoa
Trypanosoma, causative
agent for an often fatal
disease called sleeping
sickness. Half a million
people are thought to have
the disease at any one time.
Sleeping Sickness – Chagas’
disease
Reduviid bug, otherwise known
as the assassin bugs” or
“kissing bugs”
Treatment
The availability of drugs for the treatment of sleeping
sickness is poor:
1.
Eflornithine is effective for the late stage of the disease
and is very expensive.
2.
Pentamidine and Suramin have serious side effects.
3.
Melarsoprol is very toxic, if not fatal in the wrong dose.
Finding Suitable Targets
In general, there are two ways to kill or impair a parasite
metabolically:
1.
Reduce an important flux to a very low value
2. Cause some intermediate metabolite/protein to accumulate to
such an extent that is becomes toxic
Physiology when in Bloodstream
Tryps depends entirely on
glycolysis for all energy needs,
it has neither a functional
Krebs cycle or oxidative
phosphorylation. It doesn’t
even have a capacity to store
carbohydrate.
While in the blood, it consumes
glucose equivalent to it’s own
body mass every hour.
Glycolytic rates are probably of
the order of 10 times host
rates.
Conservation Laws
There are four conservation laws:
NAD
AMP (Gly)
AMP (Cyt)
Phosphate
http://bip.cnrs-mrs.fr/bip10/eise2.htm
http://bip.cnrs-mrs.fr/bip10/eise3.htm#2
Nature (1996) Supplement on intelligent drug design, 384 (suppl.), 1-26.
Singular Jacobian Matrix
Blackboard, simple cycle example
75
Newton Raphson Method
or
How p.ss.eval works
Refer to other notes
76