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Systems Biology & Uncertainty
Factors that limit the power of mechanistic (ODE-based)
model development for biological systems:
Factor 1: Unknown parameter values and model
structures for (nearly all) biological systems of interest.
....
Answer by structural network analysis methods:
Focus on network structure to derive conditions for
possible qualitative behavior (parameter-free).
Involves radical simplifications of models and scope.
3
Relations Between Spaces
Parameterspace
Fluxspace
Statespace
4
Structural Analysis: Approach
Idea: With well-characterized reaction stoichiometries
and reversibilities constrain the possible network
behavior (→ constraint-based approaches).
Based on first principles: Conservation of mass (and
energy and possibly other constraints).
Application primarily to metabolic networks.
Mathematics: Linear algebra, convex analysis.
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Metabolic Networks: Definitions
System definition: Internal versus external metabolites.
Reaction stoichiometry: Ratios of products / educts.
Reaction directionality: Reversibility / irreversibility.
Metabolic fluxes: Rates of metabolic reactions.
A B BextAext R4
R2
R3
R1
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Metabolic Networks: Definitions
Number of (internal) metabolites {A,B}: n = 2.
Number of metabolic reactions {R1-R4}: q = 4.
Sets of reversible and irreversible reactions:
rev = {R3}, irrev = {R1,R2,R4}, rev ∩ irrev = 0.
A B BextAext R4
R2
R3
R1
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Metabolic Networks: Representation
Network representation: Stoichiometric matrix N (n x q).
Rows → Internal metabolites i ; Columns → Reactions j.
Elements nij : Stoichiometric coefficients (>0 for products).
A B BextAext R4
R2
R3
R1
N =[1 −1 −1 00 1 1 −1 ]
R2 R3 R4
B
R1
A
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Metabolic Networks: Representation
Metabolic network:
6 (internal) metabolites,
10 reactions.
Reaction reversibilities
indicated by arrows.
Representation through
stoichiometric matrix N.
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Metabolic Networks: Flux Distributions
Flux distribution: Specification of all fluxes in the network
→ Vector r of q reaction rates.
Feasibility criterion: ri ≥ 0 for all irreversible reactions.
A B BextAext R4
R2
R3
R1 r '=[1
−101
]r=[1101
]
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Metabolic Networks: Mass Balances
Mass balances for all internal metabolites; external
metabolites assumed to be sources or sinks.
Metabolite concentrations: dci/dt = fluxesi,in – fluxesi,out .
dcA
dt= r1−r2−r3
dcB
dt= r2r3−r4
A B BextAext R4
R2
R3
R1
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Metabolic Networks: Balancing Equation
Stoichiometric matrix N = well-known systems invariant.
Uncertainties in kinetic description of reaction rates.
Structural network analysis: Focus on the invariant N.
d c t dt
= N⋅r t
Flux distribution:time-variant
r(t) = f(c(t),p,u(t))
Time-dependentconcentration
changes
Stoichiometricmatrix:
invariant
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Conservation relations: Weighted sums of metabolite
concentrations that are always constant in the network.
Example: [A]-[B]=const.; [A]+[C]=const.; [B]+[D]=const.
Conservation Relations: Principle
D
R1C
B
A
dcA
dt= −r1
dcB
dt= −r1
dcC
dt= r1
dcD
dt= r1
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Conservation Relations: Analysis
CRs y correspond to linearly dependent rows in N.
CRs lie in the (left) null-space (or: kernel) of matrix N:
Maximal number of linearly independent CRs given
by the dimension of the null space of N: n-rank(N) .
N T⋅y = 0 yT
⋅N = 0T
y=[1 1 0
−1 0 10 1 00 0 1
]D
R1C
B
AN =[
−1−111
]m-rank(N) = 4 -1 = 3
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Conservation Relations: Applications
Interpretation: [A]-[B]=const.; [A]+[C]=const.; [B]+[D]=const.
Conserved moieties: Positive sum of metabolite concentrations.
CRs shrink the possible dynamic behavior: Model reduction.
y=[1 1 0
−1 0 10 1 00 0 1
]D
R1C
B
AN =[
−1−111
]m-rank(N) = 4 -1 = 3
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Balanced Networks: Quasi Steady State
Metabolic networks: Fast reactions (msec - seconds
timescale) and high turnover of reactands.
Quasi steady state → Metabolite balancing equation:
Homogeneous systems of linear equations:
Consumption of a metabolite equals production.
d c t dt
= N⋅r t c t , r t
const.0 = N⋅r
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Balanced Networks: Null Space
Trivial solution r = 0 → Thermodynamic equilibrium.
Metabolic networks: q >> n → Degrees of freedom of
the network → Infinite number of compliant vectors r.
Linear algebra: All possible solutions lie in the (vector)
null space (or: kernel) of N with dimension q-rank(N).
0 = N⋅r
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Balanced Networks: Null Space
0 = N⋅r
r1
r2
Null space
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Balanced Networks: Kernel Matrix
Task: Find q-rank(N) linearly independent solutions
→ Arrange in a kernel matrix K.
Reconstruction of all r by linear combination b of
the columns of the kernel matrix: r = Kb.
K=[1 00 −11 11 0
]q-rank(N) =
4 -2 = 2
A B BextAext R4
R2
R3
R1
rev = {R3}
irrev = {R1 , R2 , R4}N =[1 −1 −1 0
0 1 1 −1]
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Kernel Matrix: Enzyme Subsets
Enzyme subset: Set of reactions that (in steady
state) always operate together in a fixed ratio.
Detection: Rows in K differ only by scalar factor.
Strong coupling → Indication of common regulation.
A B BextAext R4
R2
R3
R1
rev = {R3}
irrev = {R1 , R2 , R4}N =[1 −1 −1 0
0 1 1 −1]
K=[1 00 −11 11 0
] R2
R1
R4
R3
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Kernel Matrix: Limitations
Caveat #1: Kernel matrix is not a unique representation.
Caveat #2: Reaction reversibilities are not considered.
Caveat #3: Network degrees of freedom ≤ dimension(K).
rT= 1 1 0 1
A B BextAext R4
R2
R3
R1
rev = {R3}
irrev = {R1 , R2 , R4}N =[1 −1 −1 0
0 1 1 −1]
K=[1 00 −11 11 0
]b= 1 −1
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Metabolic Networks: (Manual) Reconstruction
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Metabolic Networks: Stoichiometric Models
Nearly genome-scale level models of metabolic networks.
Reconstructions for several organisms including human.
J. M
on
k e
t a
l., N
at.
Bio
tech
no
l. 3
2:
44
7 (
20
14
).
23
Constraint-Based Network Analysis Methods
N.E. Lewis et al., Nat. Rev. Microbiol. 10: 291 (2012).
Today
24
Flux Balance Analysis: Principles
Idea: Incorporate further constraints to
limit network behaviour with respect to
feasible steady state flux distributions →
More predictive models.
Examples for constraints:
Quasi steady state assumption.
Reaction reversibilities / capacities.
Optimal feasible steady state.
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Flux Balance Analysis: Constraints
Quasi steady state assumption:
Reaction reversibilities / capacities:
Optimal feasible steady state:
Maximal growth rate.
Maximal energy (ATP) production.
Maximal yield of a desired product.
...
wT⋅r→max!
irii
0 = N⋅r
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Flux Balance Analysis: Constraints
0 = N⋅r
r1
r2
Feasible space
Null space
Reaction reversibilities/
capacitiesirii
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Flux Balance Analysis: Optimization
General optimization problem statement for FBA:
Linear objective function and system of linear
equality/inequality constraints → Linear program
→ Solution is computationally cheap.
Z obj = wT⋅r→max !
s.t.
N⋅r = 0
irii
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Flux Balance Analysis: Optimization
Simplex algorithm: Solutions lie on vertices → Start
on vertex → Evaluate gradients + move along edges
→ Continue search or stop at optimal solution.
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Flux Balance Analysis: Optimization
0 = N⋅r
r1
r2
Feasible space
Null space
Reaction reversibilities/
capacitiesirii
Max (r1 + r2)
Max r2
Max r1
30
Flux Balance Analysis: Applications
M.A
. O
be
rha
rdt
et
al.,
Mo
lec.
Sys
t. B
iol.
5:
32
0 (
20
09
).
31
Application #1: Prediction of Phenotypes
Stoichiometric model
for bacterium E. coli
(n=436, q=720).
FBA: Maximization
of growth rate.
Prediction of growth
yields and uptake /
excretion rates.
J. E
dw
ard
s e
t a
l., N
at.
Bio
tech
. 1
9:
12
5 (
20
01
).
Oxy
gen
upt
ake
rat
e
Acetate uptake rate
32
Application #2: Prediction of Mutant Behavior
Stoichiometric model for bacterium E. coli (n=436, q=720).
FBA: Maximization of growth rate after gene deletions.
86% prediction accuracy (viability versus inviability).
J. Edwards & B.O. Palsson, PNAS 97: 5528 (2000).gro
wth
mu
tan
t / w
ild ty
pe
33
Application #2: Prediction of Mutant Behavior
Integration of stoichiometric model with (discrete) repre-
sentation of regulatory network: Improved predictions.
Identification of unknown knowledge gaps by evaluation
of the model against high-throughput experimental data.
M.W
. C
ove
rt e
t a
l., N
atu
re 4
29
: 9
2 (
20
04
).
34
Application #2: Prediction of Mutant Behavior
Prediction of flux
distributions: Wild
type (black fluxes)
versus mutant strain
(red).
Here: Gene knockout
(zwf) in the pentose
phosphate pathway
(red arrow).
J. E
dw
ard
s &
B.O
. P
als
son
, P
NA
S 9
7:
55
28
(2
00
0).
35
Application #3: Mycoplasma pneumoniae
Model development:
Databases of
metabolic
reactions (KEGG).
Genome analysis.
Flux balance
analysis.
Growth
experiments
(variable
conditions).E. Yus et al., Science 326: 1263-68 (2009).
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Application #3: Mycoplasma pneumoniae
E. Yus et al., Science 326: 1263-68 (2009).
37
Application #3: Mycoplasma pneumoniae
Model analysis in combination with experimental data:
Detailed results: Slow growth (8h doubling time) not
caused by energy but protein synthesis limitations.
Global results: Genome reduction facilitated by
multifunctional metabolic enzymes.
E.
Yu
s e
t a
l., S
cie
nce
32
6:
12
63
-68
(2
00
9). M. pneumoniae L. lactis
E. coli B. subtilis
38
Application #4: Systems Identification
J. R
ee
d e
t a
l., P
NA
S 1
03
: 1
74
80
(2
00
6).
Theory-experiment iteration: Identify missing network parts.
39
Caveat #1: Objective Function
Strong dependence of results on objective function.
Use of 'natural' objective functions such as growth:
Not applicable to all organisms (e.g. cells in
multicellular organisms → cancer ...).
Not applicable under all conditions (e.g. after
perturbation of an organism).
Alternative / conflicting approaches to optimization.
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Caveat #2: Alternate Optima
Linear programming problem implies that finding a
solution can be guaranteed, but:
Unique value of the objective function ('growth').
Existence of infinitely many solutions with
optimal value of objective function possible.
Without incorporating further constraints: Poor
performance in predicting flux distributions.
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Caveat #2: Alternate Optima
0 = N⋅r
r1
r2
Feasible space
Null space
Reaction reversibilities/
capacitiesirii
Max r1
42
Flux Variability Analysis (FVA)
Identify minimal and maximal fluxes for (optimal)
objectiv function value using linear programming:
Returns bounds on feasible fluxes; a solution is
unique when bounds for a given flux are identical.
r j→max ! /min! ∀ j∈{1…q}
s.t.
N⋅r = 0
irii
wT⋅r = Z obj
43
Flux Variability Analysis (FVA)
0 = N⋅r
r1
r2
Feasible space
Null space
Reaction reversibilities/
capacitiesirii
Max r1
r2min r2
max
r1min=r1
max
r2min
44
Flux Variability Analysis (FVA)
Example network with multiple inputs: Flux variability
depends on network structure and environment.
R. Mahadevan & C.H. Schilling, Metabolic Eng. 5: 264 (2003).
45
E. coli metabolic
model: m=625, q=931.
Mixed integer linear
program (MILP):
J. Reed & B.O. Palsson, Genome Res. 14: 1797 (2004).
MI:
Solution matrix NZJ,
Decision variable yi
LP:
Caveat #2: Alternate Optima
46
Structural Network Analysis
Network structure: Stoichiometric (& other) constraints on behavior → Predictive models rooted in first principles and comprehensive (and available) biological knowledge.
Basic definitions: Stoichiometric matrix, mass balances, quasi steady-state assumption.
Analysis relying on linear algebra: Null space, kernel matrix, ... → Feasible flux distributions, enzyme subsets, ...
Flux balance analysis (FBA) → Optimality assumption → Linear program → Predictions of phenotype, mutants, ...
Possible alternative optima → flux variability analysis (FVA).
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Further Reading
J.D. Orth, I. Thiele & B.O. Palsson. What is flux balance analysis?
Nature Biotechnology 28: 245-248 (2010).
N.E. Lewis, H. Nagarajan & B.O. Palsson. Constraining the metabolic
genotype-phenotype relationship using a phylogeny of in silico methods.
Nature Reviews Microbiology 10: 291-305 (2012).
M. Terzer, N.D. Maynard, M.W. Covert & J. Stelling. Genome-scale
metabolic networks. Wiley Interdisciplinary Reviews Systems Biology
(2009).
(http://www3.interscience.wiley.com/journal/122456827/abstract)