Metabolic network analysis Marcin Imielinski University of Pennsylvania March 14, 2007.

Metabolic network analysis

Marcin Imielinski

University of Pennsylvania

March 14, 2007

Genome sequencing

Gene expression profiling Proteomics

High-throughput phenotyping

The systems biology vision.

Integrate quantitative and high-throughput experimental data to gain insight into basic biology and pathophysiology of cells, tissues, and organisms.

Exploit knowledge of intracellular networks for drug design.

Engineer organisms, e.g. for salvaging waste, synthesizing fuel.

Modeling cellular metabolism

Raw Materials Products

ExtracellularCompartment

System Boundary

System Boundary

production

transport

core metabolism

steady state

Other cellularprocesses

Modeling cellular metabolism

Legend: = reaction = biochemical species = reaction i/o

Nutrients

Protein Synthesis

DNA replication

MembraneAssembly

Glucose

Pyruvate

Pathway example: glycolysis

carbon dioxide, water, energy

Genome scale metabolic models Most comprehensive summary of the current knowledge

regarding the genetics and biochemistry of an organism Integrate functional genomic associations between genes,

proteins, and reactions into single model Models have been built for 100+ bacterial organisms, yeast,

human mitochondrion, liver cell, et al. Average model contains 500-1500 reactions and 300 – 1000

biochemical species.

SequencedGenome

FunctionalAnnotations

Predicted Genes and

Proteins

GenomeScaleModel

Genome Scale Metabolic Modeling - Approach Many parameters

unknown on genome scale kinetic constants feedback regulation cooperativity

What is known reaction stoichiometry thermodynamic

constraints upper bounds on some

reaction rates

Constraints based approach: start with minimal

stoichiometric model populate with constraints restrict the range of feasible

cellular behavior

Adapted from Famili et al. (PNAS 2003)

stoichiometry matrix of all reactions in the system

Entry Sij corresponds to the number of metabolite i produced in one unit of flux through reaction j

v is a flux configuration of the network, which has implicit and nonlinear dependency on x

S contains “true” reactions (e.g. enzyme catalyzed biochemical transformations) and “exchange fluxes” (representing exchange of material across system boundary and growth based dilution

x= Sv

Constraints Based Metabolic Modeling

stoichiometry matrix (dimensionless)

vector of rate of change of species concentrations mol/L/s

vector of reactionrates (fluxes)mol/L/s

Example

S=

1.0 A 1.0 E 1.0 B 1.0 F 1.0 C 1.0 F 1.0 D 1.0 E 1.0 B 1.0 D 1.0 G

1 2 3

A -1 0 0

B 1 0 -1

C 0 -1 0

D 0 1 -1

E -1 1 0

F 1 -1 0

G 0 0 1

Example

S=1 2 3

A -1 0 0

B 1 0 -1

C 0 -1 0

D 0 1 -1

E -1 1 0

F 1 -1 0

G 0 0 1

systemboundary

1

2

3biochemical species

reaction

reaction I/O

Legend

Example

1 2 3

A -1 0 0

B 1 0 -1

C 0 -1 0

D 0 1 -1

E -1 1 0

F 1 -1 0

G 0 0 1

systemboundary

1

2

3

= xv

biochemical species

reaction

reaction I/O

Legend

Example – flux configuration

1 2 3

A -1 0 0

B 1 0 -1

C 0 -1 0

D 0 1 -1

E -1 1 0

F 1 -1 0

G 0 0 1

systemboundary

1

2

3

= 1

0

0

-1

1

0

0

-1

1

0

steady state species

consumed species

produced species

reaction (inactive)

reaction I/O

Legend

reaction (active)


1 2 3

A -1 0 0

B 1 0 -1

C 0 -1 0

D 0 1 -1

E -1 1 0

F 1 -1 0

G 0 0 1

systemboundary

1

2

3

= 1

1

0

-1

1

-1

1

0

0

0


consumed species

produced species

reaction (inactive)

reaction I/O

Legend

reaction (active)


1 2 3

A -1 0 0

B 1 0 -1

C 0 -1 0

D 0 1 -1

E -1 1 0

F 1 -1 0

G 0 0 1

systemboundary

1

2

3

= 1

1

1

-1

0

-1

0

0

0

1


consumed species

produced species

reaction (inactive)

reaction I/O

Legend

reaction (active)

Constraints Based Metabolic Modeling stoichiometry matrix of all reactions in the

system quasi-steady state assumption

biochemical reactions are fast with respect to regulatory and environmental changes




Example – steady state

1 2 3

A -1 0 0

B 1 0 -1

C 0 -1 0

D 0 1 -1

E -1 1 0

F 1 -1 0

G 0 0 1

systemboundary

1

2

3

= 0v

biochemical species

reaction

reaction I/O

Legend


1 2 3

A -1 0 0

B 1 0 -1

C 0 -1 0

D 0 1 -1

E -1 1 0

F 1 -1 0

G 0 0 1

systemboundary

1

2

3

= 00


1 2 3 4 5 6

A -1 0 0 1 0 0

B 1 0 -1 0 0 0

C 0 -1 0 0 1 0

D 0 1 -1 0 0 0

E -1 1 0 0 0 0

F 1 -1 0 0 0 0

G 0 0 1 0 0 -1

systemboundary

1

2

3

5

4

6

= 0v

biochemical species

reaction

reaction I/O

Legend


1 2 3 4 5 6

A -1 0 0 1 0 0

B 1 0 -1 0 0 0

C 0 -1 0 0 1 0

D 0 1 -1 0 0 0

E -1 1 0 0 0 0

F 1 -1 0 0 0 0

G 0 0 1 0 0 -1

systemboundary

1

2

3

= 1

1

1

1

1

1

0

0

0

0

0

0

0

5

4

6


consumed species

produced species

reaction (inactive)

reaction I/O

Legend

reaction (active)

Example – expanded system boundary

1 2 3 4 5 6

A -1 0 0 1 0 0

B 1 0 -1 0 0 0

C 0 -1 0 0 1 0

D 0 1 -1 0 0 0

E -1 1 0 0 0 0

F 1 -1 0 0 0 0

G 0 0 1 0 0 -1

Aext 0 0 0 -1 0 0

Cext 0 0 0 0 -1 0

Gext 0 0 0 0 0 1

oldsystemboundary

1

2

3

= 1

1

1

1

1

1

0

0

0

0

0

0

0

-1

-1

1

4

5

6

Cext

Aext

Gext

expandedsystemboundary


consumed species

produced species

reaction (inactive)

reaction I/O

Legend

reaction (active)

Constraints Based Metabolic Modeling




stoichiometry matrix of all reactions in the system

quasi-steady state assumption irreversibility constraints

Irreversibilityconstraints

The Flux Cone

chull (p1, …, pq)

(extreme pathway decomposition)

(polyhedral flux cone)

Minimal functional units of metabolism i.e. “non-decomposable” Network-based correlate of a

biochemist’s notion of a “pathway” or “module”

Uncover systems-level functional roles for individual genes / enzymes

Capture flexibility of metabolism with respect to a particular objective

Reactions participating in extreme pathways may be co-regulated

Useful for drug design and metabolic engineering.

Extreme pathways

Wiback et al, Biophys J 2002

1 2 3 4 5 6 7 8 9 10 11 12 13

Ru5P 0 0 -1 -2 0 1 0 0 0 0 0 1 2

FP2 0 -1 0 0 0 0 1 -1 0 0 1 0 0

F6P 1 0 0 2 0 0 -1 1 0 -1 0 0 -2

GAP 0 2 0 1 -1 0 0 0 0 0 -2 0 -1

R5P 0 0 1 -1 0 0 0 0 -1 0 0 -1 1

Extreme pathways: example

system boundary1 10

8

7

5 9

2

11

123

4

13

6

biochemical species

reaction

reaction I/O

Legend

system boundary1 10

8

7

5 9

2

11

123

4

13

6

1 2 3 4 5 6 7 8 9 10 11 12 13

Extreme pathway 1 1 1 0 0 2 0 1 0 0 0 0 0 0

Extreme pathway 2 0 0 1 0 0 1 0 0 1 0 0 0 0

Extreme pathway 3 0 0 1 1 1 3 0 0 0 2 0 0 0

Extreme pathway 4 0 0 2 2 0 6 0 1 0 5 1 0 0

Extreme pathway 5 0 2 1 1 5 3 2 0 0 0 0 0 0

Extreme pathway 6 5 1 4 0 0 0 1 0 6 0 0 0 2


biochemical species

reaction

reaction I/O

Legend

1 2 3 4 5 6 7 8 9 10 11 12 13

Extreme pathway 1 1 1 0 0 2 0 1 0 0 0 0 0 0

Extreme pathway 2 0 0 1 0 0 1 0 0 1 0 0 0 0

Extreme pathway 3 0 0 1 1 1 3 0 0 0 2 0 0 0

Extreme pathway 4 0 0 2 2 0 6 0 1 0 5 1 0 0

Extreme pathway 5 0 2 1 1 5 3 2 0 0 0 0 0 0

Extreme pathway 6 5 1 4 0 0 0 1 0 6 0 0 0 2


system boundary1 10

8

7

5 9

2

11

123

4

13

6

biochemical species

reaction

reaction I/O

Legend

system boundary1 10

8

7

5 9

2

11

123

4

13

6

1 2 3 4 5 6 7 8 9 10 11 12 13

Extreme pathway 1 1 1 0 0 2 0 1 0 0 0 0 0 0

Extreme pathway 2 0 0 1 0 0 1 0 0 1 0 0 0 0

Extreme pathway 3 0 0 1 1 1 3 0 0 0 2 0 0 0

Extreme pathway 4 0 0 2 2 0 6 0 1 0 5 1 0 0

Extreme pathway 5 0 2 1 1 5 3 2 0 0 0 0 0 0

Extreme pathway 6 5 1 4 0 0 0 1 0 6 0 0 0 2


biochemical species

reaction

reaction I/O

Legend

1 2 3 4 5 6 7 8 9 10 11 12 13

Extreme pathway 1 1 1 0 0 2 0 1 0 0 0 0 0 0

Extreme pathway 2 0 0 1 0 0 1 0 0 1 0 0 0 0

Extreme pathway 3 0 0 1 1 1 3 0 0 0 2 0 0 0

Extreme pathway 4 0 0 2 2 0 6 0 1 0 5 1 0 0

Extreme pathway 5 0 2 1 1 5 3 2 0 0 0 0 0 0

Extreme pathway 6 5 1 4 0 0 0 1 0 6 0 0 0 2

Objective: disable output of GAP (“exchange reaction” 5)

system boundary1 10

8

7

5 9

2

11

123

4

13

6

biochemical species

reaction

reaction I/O

Legend

Knockout design

system boundary1 10

8

7

5 9

2

11

123

4

13

6

1 2 3 4 5 6 7 8 9 10 11 12 13

Extreme pathway 1 1 1 0 0 2 0 1 0 0 0 0 0 0

Extreme pathway 2 0 0 1 0 0 1 0 0 1 0 0 0 0

Extreme pathway 3 0 0 1 1 1 3 0 0 0 2 0 0 0

Extreme pathway 4 0 0 2 2 0 6 0 1 0 5 1 0 0

Extreme pathway 5 0 2 1 1 5 3 2 0 0 0 0 0 0

Extreme pathway 6 5 1 4 0 0 0 1 0 6 0 0 0 2

biochemical species

reaction

reaction I/O

Legend

Knockout design

One solution: knockout reactions 1 and 4 (i.e. constrain v1=0 and v4=0)

system boundary1 10

8

7

5 9

2

11

123

4

13

6

1 2 3 4 5 6 7 8 9 10 11 12 13

Extreme pathway 1 1 1 0 0 2 0 1 0 0 0 0 0 0

Extreme pathway 2 0 0 1 0 0 1 0 0 1 0 0 0 0

Extreme pathway 3 0 0 1 1 1 3 0 0 0 2 0 0 0

Extreme pathway 4 0 0 2 2 0 6 0 1 0 5 1 0 0

Extreme pathway 5 0 2 1 1 5 3 2 0 0 0 0 0 0

Extreme pathway 6 5 1 4 0 0 0 1 0 6 0 0 0 2

One solution: knockout reactions 1 and 4 (i.e. constrain v1=0 and v4=0)

biochemical species

reaction

reaction I/O

Legend

Knockout design

1 2 3 4 5 6 7 8 9 10 11 12 13

Extreme pathway 1 1 1 0 0 2 0 1 0 0 0 0 0 0

Extreme pathway 2 0 0 1 0 0 1 0 0 1 0 0 0 0

Extreme pathway 3 0 0 1 1 1 3 0 0 0 2 0 0 0

Extreme pathway 4 0 0 2 2 0 6 0 1 0 5 1 0 0

Extreme pathway 5 0 2 1 1 5 3 2 0 0 0 0 0 0

Extreme pathway 6 5 1 4 0 0 0 1 0 6 0 0 0 2

Objective: couple export of GAP to export of F6P

system boundary1 10

8

7

5 9

2

11

123

4

13

6

biochemical species

reaction

reaction I/O

Legend

Knockout design

1 2 3 4 5 6 7 8 9 10 11 12 13

Extreme pathway 1 1 1 0 0 2 0 1 0 0 0 0 0 0

Extreme pathway 2 0 0 1 0 0 1 0 0 1 0 0 0 0

Extreme pathway 3 0 0 1 1 1 3 0 0 0 2 0 0 0

Extreme pathway 4 0 0 2 2 0 6 0 1 0 5 1 0 0

Extreme pathway 5 0 2 1 1 5 3 2 0 0 0 0 0 0

Extreme pathway 6 5 1 4 0 0 0 1 0 6 0 0 0 2

system boundary1 10

8

7

5 9

2

11

123

4

13

6

One solution: knockout reaction 2 (constrain v2 = 0)

biochemical species

reaction

reaction I/O

Legend

Knockout design

Tableau algorithm for EP computatioon

Iteration 0: nonnegative orthant

Extreme rays of K0 = Euclidean basis vectors 1 … n

Iteration i+1:

Given

and extreme rays of Ki

Compute extreme rays of Ki+1

v1

v2

v3

Extreme ray of Ki


v1

v2

v3

Si+1v = 0


v1

v2

v3

Si+1v = 0

Sort extreme rays of Ki with regards to which are on (+) side, (-) side, and inside hyperplane Si+1v = 0


v1

v2

v3

Si+1v = 0

Extreme rays of Ki that are already in Siv=0 are automatically extreme rays of Ki+1


v1

v2

v3

Si+1v = 0

Combine pairs of extreme rays of Ki that are on opposite sides of Si+1v = 0


v1

v2

v3

Si+1v = 0

From this new ray collection remove rays that are non-extreme.


v1

v2

v3

Si+1v = 0

Non-extreme rays are r for which there exists an r* in the collection such that NZ(r*) is a subset of NZ(r)


Applications Hemophilus influenzae (Schilling et al, J Theor Biol 2000) Human red blood cell (Wiback et al, Biophys J 2002) Helicobacter pylori (Schilling et al, J Bact 2002)

Limitations Combinatorial explosion of extreme rays Computational complexity of determining extremality Only directly applicable to “medium sized” networks (e.g. 200 species

and 300 reactions) Variants

Elementary flux modes (Schuster Nat Biotech 2000) Minimal generating set (Wagner Biophys J 2005)

Approximate alternatives: Flux coupling analysis (Burgard et al Genome Res 2004) Sampling of flux cone (Wiback et al J Theor Biol 2004)

Applications of network based pathway analysis

Flux Balance Analysis (Palsson et al.) Supplement metabolic

network with a “biomass reaction” which consumes biomass substrates in ratios specified by chemical composition analysis of the cell.

Model growth as flux through biomass reaction at steady state

Use linear programming to predict optimum growth under a given set of mutations and nutrient conditions.

Nu

trie

nts

Bio

ma

ss

Edwards et al Nat Biotech 2001

0 ≤ v ≤ u vector of upper boundscorresponding to maximum ratesof metabolic reactions

0= S | b

Flux Balance Analysis: formulationstoichiometry matrix

of metabolic network (dimensionless)


vector ofmetabolic fluxes (mol/L/s)

biomassreaction (dimensionless)

biomassflux (scalar)(mol/L/s)

objective: maximize growth rate λ s.t. above constraints

vλ

0 ≤ λ

Nu

trie

nts

Bio

ma

ss

Modeling E. coli growth using FBA (Edwards et al Nat Biotech 2001)

E coli model with 436 metabolites and 720 reactions

Found that in vivo growth matched FBA-predicted optimal growth on minimal nutrient media employing acetate and succinate as carbon sources.

Modeling E. coli growth using FBA (Ibarra et al Nature 2001)

In vivo growth was sub-optimal under glycerol

However following over 40 days of culture and 700 generations of cell divisions, E. coli adaptively evolved to achieve optimum predicted growth rate

Modeling E. coli mutants using FBA

Edwards et al 2004 E. coli model 436 species x 720 reactions compared FBA predictions to published data on 36 E. coli gene deletion

mutants in 4 nutrient media. 68 of 79 mutants agreed (qualitatively) between “simulation” and

experiment. Covert et al 2004 E. coli model 761 species x 931 reactions

Compared FBA predictions to 13,750 mutant growth experiments in different nutrient media – gene deletion combination

Found 78.7% agreement

Nu

trie

nts

Bio

ma

ss

0 ≤ v ≤ u

0= S | b

0 ≤ λ

vλ

maximize λ s.t.

ui=0

FBA: concerns and limitations Assumes that a cell culture is optimized for growth.

Even bacteria like to do other things than just grow. “Higher organisms” have even more complex “objectives”

Even if we allow that a wild type organism is optimized for growth (because of years of evolution) a mutant may have difficulty “finding” the global optimum. e.g. maybe a mutant bacteria will want to find the “closest” feasible

state. What about alternative optima?

Optimal manifolds of these LPs are high-dimensional polyhedral sets.

How will gene regulation influence the optimum? Simple version: regulation will alter the upper bound constraints (u)

on the fluxes. Complicated version: modeling the interaction of metabolism and

gene regulation will require including parameters and nonlinearities.

FBA: concerns and limitations Assumes that a cell culture is optimized for growth.

Even bacteria like to do other things than just grow. “Higher organisms” have even more complex “objectives”

Even if we allow that a wild type organism is optimized for growth (because of years of evolution) a mutant may have difficulty “finding” the global optimum. e.g. maybe a mutant bacteria will want to find the “closest”

feasible state. What about alternative optima?

Optimal manifolds of these LPs are high-dimensional polyhedral sets.

How will gene regulation influence the optimum? Simple version: regulation will alter the upper bound constraints (u)

on the fluxes. Complicated version: modeling the interaction of metabolism and

gene regulation will require including parameters and nonlinearities.

Minimization of metabolic adjustment (MOMA)Segre et al PNAS 2002 Alternative method to FBA

for computing mutant growth rates.

Hypothesize that mutants will want to settle “close” to the wild type flux configuration.

Find mutant flux distribution v that minimizes Euclidean distance to wild type growth state vwt

Formulate as QP vwt can be obtained

experimentally or computed using FBA.

Segre et al PNAS 2002

Regulatory on-off minimization (ROOM)Shlomi et al PNAS 2005 Also hypothesize that mutants

will want to be “close” to wild type flux configuration.

However measure distance as the number of reactions whose flux bounds would have to be (significantly) changed from wild type

Formulated as a MILP, with objective to minimize the number of reactions that need to be changed from wild type vwt (obtained using FBA).

Performance on test data set of mutants: ROOM > FBA >> MOMA (ROOM has fewer false negatives)

Shlomi et al PNAS 2005

MO

MA

wt

(F

BA

)R

OO

M

“biomass”

“biomass”

“biomass”

Review

Stoichiometry matrix and constraints-based metabolic modeling

Extreme pathway analysis Flux balance analysis Variants on FBA: MOMA and ROOM

Other topics not covered today Incorporating gene regulation to FBA

Covert et al Nature 2004 Analyzing alternative optima in FBA

Mahadevan et al Metab Eng 2003 Sampling the feasible flux region

Almaas et al Nature 2004 Wiback et al J Theor Biol 2004

Applying FBA to understand evolution Papp et al Nature 2004 Pal et al Nat Genetics 2005

Modeling thermodynamic constraints Beard et al J Theor Biol 2004 Qian et al Biophys Chem 2005

Conservation laws in metabolic networks Famili et al Biophys J 2003 Imielinski et al Biophys J 2006

Metabolic network analysis Marcin Imielinski University of Pennsylvania March 14, 2007.

Documents

Transcript of Metabolic network analysis Marcin Imielinski University of Pennsylvania March 14, 2007.