Chapter 9: Metabolic Networks - Columbia · PDF fileComputer Science Department Columbia University Chapter 9: Metabolic Networks 9.1 Introduction 2 Overview Introduction ... (3*ATP)]

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Prof. Yechiam Yemini (YY)

Computer Science DepartmentColumbia University

Chapter 9: Metabolic Networks

9.1 Introduction

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Overview IntroductionMetabolic flux analysisReferences:

B. Palsson, “System Biology, Properties of Reconstructed Networks”Cambridge University Press, 2006.

Palsson’s on-line course notes:http://gcrg.ucsd.edu/classes/be203.htm

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Introduction

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Metabolism: Key Cellular FunctionAnabolism: synthesize molecules from simpler ones

e.g., amino-acids, nucleic acids…

Catabolism: breakdown molecules into simpler ones e.g., glycolysis…

NodeReactions form a network

Nodes=metabolitesEdges=reactionsEdge-labels:enzymes/genes

Network describes flow

Edge

Label

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Example: GlycolysisKey function: generates energy supply (ATP)ATP/ADP provide energy storage/release currency:

ATPADP+P releases energy ADP+PATP (Phosphorylation) stores energy

-1 -1

+2+2

ATP= -1-1+2+2=2

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Glycolysis: Transforming Sugar to EnergyKey elements of glycolysis

Break glucose molecules into two sugars [energy cost -2*ATP] Use these sugars to generate 4*ATP and 2*NADH: gain 2*ATP [The 2*NADH are oxidized by the citric acid cycle to generate 2*(3*ATP)]

Notes: Reaction “edge” can have multiple input and output nodes: network ~ hypergraph Semantics of metabolism is defined by reactions flux A pathway may be optimizing an objective function (e.g., ATP flux) Subject to constraints: conservation, enzymatic rates…

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Metabolic Databases Provide Details

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Example: Glycolysis Details

Gene labels

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9

Modular Organization

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Scale Free Structure

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Introduction toReaction Kinetics

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Simple Reaction KineticsUnimolecular reaction: AB

Bimolecular reaction:

!

v =d[B]

dt= "

d[A]

dt= k[A]

k

dt

Bd

dt

Advor

dt

Qd

dt

Pdv

][][][][!=!===

QPBA +!+

]][[][

BAkdt

Adv =!=

k

A(t)=a*exp(-kt)

t

[A] a

Exponential decay

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Oxidoreductase transfers electronsTransferase transfers moleculesHydrolase breaks only O-H bondsLyase breaks bondsIsomerase twists bondsLigase makes bonds

Enzymes and Functions

Reactions Are Modulated by Enzymes

How do enzymes work? Bind substrates to active sites Lower energy threshold for reaction

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How Do We Model Enzyme Kinetics?Unimolecular reaction flux depends linearly on [A].

Enzymes modify the flux-substrate relationshipAccelerates flux and bounds it

PA!

][][

Akdt

Adv =!=

v

[A]

PAEnzyme!! "!

v

[A]

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Michaelis-Menten Kinetics Enzyme reaction creates a substrate complex

Steady state assumption: [ES] remains constant

Under this assumption:

Yielding where

The total enzyme Et=[E]+[ES] (free+ bound)

Yielding

!

d([ES])

dt= 0

PEESSE ++k1

k-1

k2

!

d([ES])

dt= k

1[E][S]" k"1[ES]" k2[ES] = 0

!

[ES] =k1[E][S]

k"1 + k2

=[E][S]

Km

!

Km

=k"1 + k

2

k1

!

[ES] =([E

t]" [ES])[S]

Km

!

[ES] =[E

t][S]

Km

+ [S]

!

v = k2[ES] =

k2[E

t][S]

Km

+ [S]=V

max

[S]

Km

+ [S]

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Michaelis-Menten Formula Enzymes increase reaction flux and bound it

Can have orders of magnitude speed-up Km and Vmax are control parameters exerted by enzyme

[ ][ ]

mKS

Svv

+= maxS P

E

[S]

VVmax

Km

0.5Vmax 0≤V≤Vmax

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Metabolic Flux Analysis

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Building a Network Modelr1 A → Br2 2B → C + Yr3 2B + X → D + Yr4 D → E + Xr5 C + X → Dr6 C → Ee1 → Ae2 E →e3 Y →

e1 A

A Br1

2B C

Y

r2

Ee2

D

YX

r32B

Substrate

Reaction

Exchange: output to environment

Exchange: input from environmentCo-factor byproduct

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Building a Network Flux Model

v1 v2

v3

v4

v5

v62b1 b2

b3

b3

Reaction flux: outflow of metabolite

2A 2B C E

D

X

Y

Y

X

System BoundaryExchange flux

Internal flux

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The Stoichiometric Matrix

Changes in metabolite Xi = (inflow – outflow)

E.g., dXB/dt= v2+v6-v3-v5

Sij is the coefficient of Xi in reaction jSij is negative if Xi is an output of the reaction j (outflow),Sij is positive if Xi is an input of reaction j (inflow)

!=j

jiji vSt

X

d

][d A B C

D

v2 v3

v5

v1 v4

v6

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Stoichiometric Matrix Example

v1 v2

v3

v4

v5

v62b1 b2

b3

b3

2A 2B C E

D

X

Y

Y

X

YXEDCBA

b3b2b1v6v5v4v3v2v1

0-101010000000-11-100-100000110

00001-1100000-1-10010000000-2-2100100000-1

!=j

jiji vSt

X

d

][d

v1 A → Bv2 2B → C + Yv3 2B + X → D + Yv4 D → E + Xv5 C + X → Dv6 C → Eb1 → Ab2 E →b3 Y →

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Metabolic Flux Analysis (MFA)The differential equation is too difficult to solve

Parameters are not observable; e.g., Michaelis-Menten (Vmax,Km)

Consider the steady state flux distributionConcentration changes are much slower than reaction kinetics

Flux conservation law ~ Kirchoff’s current law

0=! vS= 0i

tXd

][d

0-101010000000-11-100-100000110

00001-1100000-1-10010000000-2-2100100000-1

111266

54212

0=v1=12 v2=2

v3=4

b1=122A 2B C E

D

X

Y

Y

X v5=1

v4=5

v6=1

b3=2

b3=4

b2=6

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Flux Vectors Belong to The Null Space of S6

12 6

0

0

0

612 6

0

2A 2B C E

D

X

Y

Y

X

12 2

4

5

1

112 6

4

2

2A 2B C E

D

X

Y

Y

X

6 0

3

3

0

06 3

3

0

2A 2B C E

D

X

Y

Y

X

0 0

00

0

00 0

0

0

2A 2B C E

D

X

Y

Y

X

12 -2

8

7-1

-112 6

8

-2

2A 2B C E

D

X

Y

Y

X

Not every solution of S v=0 is a valid flux

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Another Example

A B C

D

v1 v2

v3

b1 v6

v5

Eb4

v4

b3

b2

v7

-100011000000-100-101-110000-100-1-1001000000001-1-110001000000-1

A:B:C:D:E:

-v1+b1=0v1-v2-v3+v4=0v2-v5-v6-b2=0v3-v4+v5-v7-b3=0v6+v7-b4=0

Vb

=0

0

A B C

D

0 1

0

0 0

1E

0

1

0

0

A B C

D

2 90

1

2 1

90

E2

90

0

0

1

Reversible reaction may lead to circulation

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The Geometry of FluxDistributions

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Flux Vectors Form A Convex Cone Solutions of 0=Sv form a linear space (null space of S)

An admissible flux vector v must satisfy linear constraints: Thermodynamic constraints: vi>0 v>0 Michaelis-Menten bound: vi<Vi

max v<Vmax

The set F={v| Sv=0, 0<v} is a convex cone If v is a vector in F, so is λv for λ>0 (the ray in the direction of v) F is convex: If v,u are vectors in F, so is αu+(1-α)v for 0<α<1 The Michaelis-Menten bound v<Vmax slices the cone

Sv=0 v>0 v<Vmax

Linear space Convex cone Bounded cone

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How Does The Network Select A Flux? Flux transforms nutrients into products

Energy, biomass…

How does the network select a flux vector? Environment constraints: e.g., availability of nutrients.. Demand: products (e.g., energy) are needed by functions.. Network needs to regulate production

Does the network optimize production? E.g., does the glycolysis pathway select flux vectors to optimize ATP production?

Nut

rient

s

Pro

duct

s

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Constraint-Based Flux AnalysisHunt for flux states by

Reducing the flux cone through constraints Optimizing pathway objective function over the constrained cone

Optimizing: Max{Z=wTv: 0≤v≤Vmax} Max a linear function of the flux

There could be multiple optimal solutions The solutions are a face of the cone

growthgrowth

growth

One solution Optimal solutions Near-optimal solutions

W

Z=WTV

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Cone GeometryGiven a convex cone (body) F, we consider points u,v,w in F Convex combination:

A convex combination of v,u is a vector w=αu+(1-α)v (where 0<α<1) (α,1-α) may be considered as a unit mass distributed to v,u

with w= the center of mass

A point w is extreme in F, if w=αu+(1-α)v implies w=u or w=v W is not a convex combination of points in F

v1

v2

v3

Extreme ray

v

u αu+(1-α)vα=1

α=0

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Extreme Points Provide Useful InformationA convex body is spanned by its extreme points

Every point of F is a convex combination of the extreme points Extreme points play analogous role to basis of a linear space

Convex optimization yields extreme points Spanning the optimal face (set of optimal solutions)

v

u αu+(1-α)vα=1

α=0

W

Z=WTV

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Extreme Pathways Extreme Pathway (EP)= extreme point of F

Extreme flux cannot be obtained by combining fluxes A flux vector is a convex combination of extreme pathways

12 6

0

0

0

612 6

0

2A 2B C E

D

X

Y

Y

X

00633

3306

v1 v2

v3v4

v5

v62b1 b2

b3

b3

2A 2B C E

D

X

Y

Y

X

=3* 00211

1102

6 0

3

3

0

06 3

3

0

2A 2B C E

D

X

Y

Y

X

061266

00612

=6* 00211

0012

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Extreme Pathways

A B C

D

v1 v2

v3

b1 v6

v5

E b4

v4

b3

b2

v7

A B C

D

v1 v2

v3

b1 v6

v5

E b4

v4

b3

b2

v7

A B C

D

v1 v2

v3

b1 v6

v5

E b4

v4

b3

b2

v7

A B C

D

v1 v2

v3

b1 v6

v5

E b4

v4

b3

b2

v7

A B C

D

v1 v2

v3

b1 v6

v5

E b4

v4

b3

b2

v7

A B C

D

v1 v2

v3

b1 v6

v5

E b4

v4

b3

b2

v7

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Flux Based Analysis

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Consider Biomass GrowthNetwork selects flux to maximize biomass production

E.g., E. coli needs the following inputs to grow 1g of biomass

Define an objective function for flux selection Z = 41.2570 VATP - 3.547VNADH+18.225VNADPH + …. In general: Z=wTv= w1v1+w2v2+….. One can optimize different products by tuning w

Maximization problem:

Max Z=wTv subject to Vmin≤v≤Vmax

This may be solved using Linear Programming (LP) Predicts: growth rates, nutrient uptake rate… Vmax,Vmin may be estimated from observed data

Fell, et al (1986), Varma and Palsson (1993)

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(Edwards et al Nat Biotech vol 19 2001)Modeling E.coli Growth With FBA

Construct a metabolic network model 436 metabolites; 720 reactions

http://gcrg.ucsd.edu/organisms/ecoli/maps/central.jpg

(Edwards et al Nat Biotech 2001)

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Constructing an FBA Model Derive stoichiometric matrix S from network

Consider media with restricted inputs: acetate or succinate Flux equations: Sv=0

Establish constraints: αi<vi<βi For inorganic substrate (phosphate, CO2, sulfate…) assume no constraints For metabolites provided by the medium use 0≤vi≤vi

max

For metabolites not available in the medium constrain flux to 0 For exchange fluxes leaving the network assume no constraints for outbound flux

Compute optimal fluxes Use linear programming Project the flux cone on 3 dimensions:

oxygen & acetate uptake and growth flux

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Compare With Measurements Grow E.coli in a medium with acetate Measure metabolites

Oxygen uptakeAcetate/succinate uptakeDry mass growth

Compare with FBA

Acetate uptake rate

Oxy

gen

upta

ke r

ate

Measurements

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Succinate Metabolism 4 different operating regions

Input restrictions may lead todifferent regionsE.g., consider anaerobic operations

anaerobic

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Metabolism Can Adapt To InputsWith glycerol input

growth wassuboptimal

But after 40 days& 700 generationsE.coli evolved toachieve optimum

(Ibarra et al Nature 2001)

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NotesBacteria flux agrees with optimum growth predictions

How did the bacteria optimize its flux to adapt to input? Tune the regulatory network to adapt enzymes expression? E.g., adjust the Michaelis-Menten parameters Recall the diauxic shift (chapter 8.2) How will the bacteria adapt to genome changes? (chapter 9.2) How do the regulatory and metabolic networks interact?

Are there ways to narrow down the search for flux thatexplain metabolism? Lee et al (2000): optimal flux with minimal # of non-zero elements NP complete Mahadevan et al (2003): optimal extreme flux Random sampling (Almaas et al 2004, Wiback et al 2004)

Diauxic shift

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Historical Developments