Structural Network Analysis (1)
Jörg Stelling
[email protected]
CSB Deterministic, SS 2015, 2
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.
2
Relations Between Spaces
Parameter
space
Flux
space
State
space
3
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.
4
Metabolic Networks: Definitions
R2
Aext
R1
B R4
A
Bext
R3

System definition: Internal versus external metabolites.

Reaction stoichiometry: Ratios of products / educts.

Reaction directionality: Reversibility / irreversibility.

Metabolic fluxes: Rates of metabolic reactions.
5
Metabolic Networks: Definitions
R2
Aext
R1
B R4
A
Bext
R3

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.
6
Metabolic Networks: Representation
R1 R2
R2
Aext
B R4
R1 A
R3
Bext
[
N=
R3
R4
1 −1 −1 0
0 1
1 −1
]

Network representation: Stoichiometric matrix N (n x q).

Rows → Internal metabolites i ; Columns → Reactions j.

Elements nij : Stoichiometric coefficients (>0 for products).
A
B
7
Metabolic Networks: Representation

Metabolic network:
6 (internal) metabolites,
10 reactions.

Reaction reversibilities
indicated by arrows.

Representation through
stoichiometric matrix N.
8
Metabolic Networks: Flux Distributions
R2
Aext
R1
B R4
A
Bext
R3

[] []
1
r= 1
0
1
1
r ' = −1
0
1
Flux distribution: Specification of all fluxes in the network
→ Vector r of q reaction rates.

Feasibility criterion: ri ≥ 0 for all irreversible reactions.
9
Metabolic Networks: Mass Balances
R2
Aext
R1 A
B R4
R3

Bext
dc A
= r 1−r 2 −r 3
dt
dc B
= r 2 r 3−r 4
dt
Mass balances for all internal metabolites; external
metabolites assumed to be sources or sinks.

Metabolite concentrations: dci/dt = fluxesi,in – fluxesi,out .
10
Metabolic Networks: Balancing Equation
d c t 
= N⋅r t
dt
Time-dependent
concentration
changes
Stoichiometric
matrix:
invariant
Flux distribution:
time-variant
r(t) = f(c(t),p,u(t))

Stoichiometric matrix N = well-known systems invariant.

Uncertainties in kinetic description of reaction rates.

Structural network analysis: Focus on the invariant N.
11
Conservation Relations: Principle
A
B

R1
C
D
dc A
dt
dc B
dt
dc C
dt
dc D
dt
= −r 1
= −r1
= r 1
= r1
Conservation relations: Weighted sums of metabolite
concentrations that are always constant in the network.

Example: [A]-[B]=const.; [A]+[C]=const.; [B]+[D]=const.
12
Conservation Relations: Analysis
A
B
R1
C
D
[] [ ]
−1
N = −1
1
1
1
y= −1
0
0
1
0
1
0
0
1
0
1
m-rank(N) = 4 -1 = 3

CRs y correspond to linearly dependent rows in N.

CRs lie in the (left) null-space (or: kernel) of matrix N:
T
N ⋅y = 0


T
y ⋅N = 0
T
Maximal number of linearly independent CRs given
by the dimension of the null space of N: n-rank(N) .
13
Conservation Relations: Applications
A
B
R1
C
D
[] [ ]
−1
N = −1
1
1
1
y= −1
0
0
1
0
1
0
0
1
0
1
m-rank(N) = 4 -1 = 3

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.
14
Balanced Networks: Quasi Steady State

Metabolic networks: Fast reactions (msec - seconds
timescale) and high turnover of reactands.

Quasi steady state → Metabolite balancing equation:
d c t 
= N⋅r t
dt

c t  , r t 
0 = N⋅r
const.
Homogeneous systems of linear equations:
Consumption of a metabolite equals production.
15
Balanced Networks: Null Space
0 = N⋅r

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).
16
Balanced Networks: Null Space
r1
Null space
0 = N⋅r
r2
17
Balanced Networks: Kernel Matrix
R2
R1 A
Aext
B R4
Bext
R3
N=

[
1 −1 −1 0
0 1
1 −1
]
rev = {R3}
irrev = {R1 , R2 , R4}
[ ]
1
0
K=
1
1
0
−1
1
0
q-rank(N) =
4 -2 = 2
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.
18
Kernel Matrix: Enzyme Subsets
R2
R1 A
Aext
B R4
Bext
R3
[
1 −1 −1 0
N=
0 1
1 −1

]
rev = {R3}
irrev = {R1 , R2 , R4}
[ ]
1
K= 0
1
1
0
−1
1
0
R1
R2
R3
R4
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.
19
Kernel Matrix: Limitations
R2
R1 A
Aext
B R4
Bext
R3
[
1 −1 −1 0
N=
0 1
1 −1
]
rev = {R3}
irrev = {R1 , R2 , R4}
[ ]
1
K= 0
1
1
0
−1
1
0
b= 1 −1
r T = 1 1 0 1

Caveat #1: Kernel matrix is not a unique representation.

Caveat #2: Reaction reversibilities are not considered.

Caveat #3: Network degrees of freedom ≤ dimension(K).
20
Metabolic Networks: (Manual) Reconstruction
21
J. Monk et al., Nat. Biotechnol. 32: 447 (2014).
Metabolic Networks: Stoichiometric Models

Nearly genome-scale level models of metabolic networks.

Reconstructions for several organisms including human.
22
Constraint-Based Network Analysis Methods
Today
N.E. Lewis et al., Nat. Rev. Microbiol. 10: 291 (2012).
23
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.
24
Flux Balance Analysis: Constraints

Quasi steady state assumption: 0 = N⋅r

Reaction reversibilities / capacities: i r i i

Optimal feasible steady state: wT⋅r → max !

Maximal growth rate.

Maximal energy (ATP) production.

Maximal yield of a desired product.

...
25
Flux Balance Analysis: Constraints
r1
Null space
0 = N⋅r
Feasible space
Reaction
reversibilities/
capacities
i r i i
r2
26
Flux Balance Analysis: Optimization

General optimization problem statement for FBA:
T
Z obj = w ⋅r → max !
s.t.
N⋅r = 0
i r i i

Linear objective function and system of linear
equality/inequality constraints → Linear program
→ Solution is computationally cheap.
27
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.
28
Flux Balance Analysis: Optimization
r1
Null space
0 = N⋅r
Max r1
Max (r1 + r2)
Feasible space
Reaction
reversibilities/
capacities
Max r2 i r i i
r2
29
M.A. Oberhardt et al., Molec. Syst. Biol. 5: 320 (2009).
Flux Balance Analysis: Applications
30
Oxygen uptake rate
J. Edwards et al., Nat. Biotech. 19: 125 (2001).
Application #1: Prediction of Phenotypes

Stoichiometric model
for bacterium E. coli
(n=436, q=720).
Acetate uptake rate

FBA: Maximization
of growth rate.

Prediction of growth
yields and uptake /
excretion rates.
31
growth mutant / wild type
Application #2: Prediction of Mutant Behavior
J. Edwards & B.O. Palsson, PNAS 97: 5528 (2000).

Stoichiometric model for bacterium E. coli (n=436, q=720).

FBA: Maximization of growth rate after gene deletions.

86% prediction accuracy (viability versus inviability).
32
M.W. Covert et al., Nature 429: 92 (2004).
Application #2: Prediction of Mutant Behavior

Integration of stoichiometric model with (discrete) representation of regulatory network: Improved predictions.

Identification of unknown knowledge gaps by evaluation
of the model against high-throughput experimental data.
33
Application #2: Prediction of Mutant Behavior
J. Edwards & B.O. Palsson, PNAS 97: 5528 (2000).

Prediction of flux
distributions: Wild
type (black fluxes)
versus mutant strain
(red).

Here: Gene knockout
(zwf) in the pentose
phosphate pathway
(red arrow).
34
Application #3: Mycoplasma pneumoniae

Model development:

Databases of
metabolic
reactions (KEGG).

Genome analysis.

Flux balance
analysis.

Growth
experiments
(variable
E. Yus et al., Science 326: 1263-68 (2009).
conditions).
35
Application #3: Mycoplasma pneumoniae
E. Yus et al., Science 326: 1263-68 (2009).
36
E. Yus et al., Science 326: 1263-68 (2009).
Application #3: Mycoplasma pneumoniae

M. pneumoniae
L. lactis
E. coli
B. subtilis
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.
37
J. Reed et al., PNAS 103: 17480 (2006).
Application #4: Systems Identification

Theory-experiment iteration: Identify missing network parts.
38
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.
39
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.
40
Caveat #2: Alternate Optima
r1
Null space
0 = N⋅r
Max r1
Feasible space
Reaction
reversibilities/
capacities
i r i i
r2
41
Flux Variability Analysis (FVA)

Identify minimal and maximal fluxes for (optimal)
objectiv function value using linear programming:
r j →max ! / min ! ∀ j ∈{1…q }
s.t.
N⋅r = 0
i r i i
T
w ⋅r = Z obj

Returns bounds on feasible fluxes; a solution is
unique when bounds for a given flux are identical.
42
Flux Variability Analysis (FVA)
r1
Null space
0 = N⋅r
r1min=r1max
Max r1
Feasible space
r2min
r2max
Reaction
reversibilities/
capacities
i r i i
r2
43
Flux Variability Analysis (FVA)
R. Mahadevan & C.H. Schilling, Metabolic Eng. 5: 264 (2003).

Example network with multiple inputs: Flux variability
depends on network structure and environment.
44
Caveat #2: Alternate Optima

E. coli metabolic
model: m=625, q=931.

Mixed integer linear
program (MILP):
LP:
J. Reed & B.O. Palsson, Genome Res. 14: 1797 (2004).
MI:
Solution matrix NZJ,
Decision variable yi
45
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).
46
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)
47