Lecture #23

Varying Parameters

Outline• Varying a single parameter

– Robustness analysis– Old core E. coli model– New core E. coli model– Literature examples

• Varying two parameters– The phenotypic phase plane (PhPP)– Characteristics of the PhPP– Core E. coli computations– Genome-scale computations– PhPPs and experimental design

ROBUSTNESS ANALYSISOne parameter

Robustness Analysis: the concept

O2 uptake rateG

row

th R

ate

•Used to calculate how the objective function changes to incremental changes in a particular flux.

•Curves are piecewise linear with slope equal to Shadow Price

Mathematics

Biological Significance:The impairment of an enzyme can have a system wide effect and affect the optimal growth rate achievable by an organism.

Example: Fluxes in E. coli have been analyzed to study how a continuous impairment of the enzyme will affect the predicted optimal growth rate.

2. Robustness to Gene Deletions and Enzyme Defects

p

Biotechnol Prog., 16: 927-939, (2000).

Shadow Prices (pi) & Reduced Costs (ri)

• Shadow Prices (pi): – One for each constraint or metabolite– pi=dZ/dbi

– pi<0 means adding metabolite (ie. change b=0 to b<0) would increase Z.

– pi>0 means removing metabolite (ie. change b=0 to b>0) would increase Z.

• Reduced Costs (ri):– One for each variable or flux.– dZ/dvj (for zero fluxes)– ri < 0 means increasing flux (vj) would reduce Z.

THE ORIGINAL CORE E. COLI MODELSome history

Analysis of oxygen uptake rateAppl. Env. Micro 59: 2465 (1993)

Historic Example

In this example we vary the maximum allowable uptake rate of oxygen.

The optimal growth solution is computed for the whole range of oxygenation, from fully aerobic conditions to fully anaerobic conditions.

The growth rate is graphed in the upper panel and the by-product secretion rates in the lower.

anaerobic aerobic

Shadow prices: Interpret changes in optimal solutions

Formate, Acetate,Ethanol are Secreted ($0 shadow prices)

Formate & Acetate,Secreted ($0 shadow prices); Ethanol is not ($0.002)

Flux distributions for different levels (or phases) of oxygenation

partially anaerobic aerobic

Acetate is Secreted ($0 shadow prices)

Results

• Optimality principles and network reconstruction used to predict over all phenotypic states

• Phenotypic functions interpreted using an econometric approach, ie the shadow prices

THE CURRENT CORE E. COLI MODELtoday

Growth on glucose: similar results as historical

0 5 10 15 20 250

2

4

6

8

10

12

14

16

18

O2 uptake rate (mmol/g DW-hr)

by-p

rodu

ct s

ecre

tion

(mm

ol/g

DW

-hr)

AcetateEthanolFormate

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

10

20

30

40

50

60

70

80

growth rate (1/hr)

nutri

ent u

ptak

e an

d by

-pro

duct

sec

tretio

n ra

tes

(mm

ol/g

DW

-hr))

GlucoseO2

AcetateEthanolFormate

Glucose and O2 uptake and byproduct secretion rates at different growth rates. Max O2 uptake rate = -17.

Byproduct secretion rates at different O2 uptake rates. Glucose uptake rate = -10.

5 distinct growth phases I II III IV V

ATP production in core E. coli:Contrast with growth function

• Robustness analysis of ATP production from glucose• Vary O2 uptake from 0 to fully aerobic and compute

maximum ATP production

0 1 2 3 4 5 6 70

2

4

6

8

10

12

14

16

18

O2 uptake rate (mmol/g DW-hr)

atp

yiel

d

Solution becomes infeasible at O2 uptake above 6 O2/Glc

0 1 2 3 4 5 60

0.5

1

1.5

2

2.5

3

3.5

4

O2 uptake rate (mmol/g DW-hr)

by-p

rodu

ct s

ecre

tion

(mm

ol/g

DW

-hr)

AcetateEthanolFormate

H+

By-products secreted during anaerobic ATP production

Growth on glucose:line of optimality (LO)

0 5 10 15 20 250.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

O2 uptake rate (mmol/g DW-hr)

grow

th ra

te (1

/hr)

0 5 10 15 200

0.2

0.4

0.6

0.8

1

1.2

1.4

glucose uptake rate (mmol/g DW-hr)

grow

th ra

te (1

/hr)

Glucose uptake fixed at -10, O2 uptake variable

O2 uptake fixed at -17, glucose uptake variable

Partially anaerobic

Growth on Different Substrates:illustrates partial anaerobic growth potential

0 5 10 15 20 25 30 35 40 45 500

0.05

0.1

0.15

0.2

0.25

0.3

0.35

acetate uptake rate (mmol/g DW-hr)

grow

th ra

te (1

/hr)

0 10 20 30 40 50 60 70 800

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

-ketoglutarate uptake rate (mmol/g DW-hr)

grow

th ra

te (1

/hr)

0 2 4 6 8 10 12 14 16 18 200

0.2

0.4

0.6

0.8

1

1.2

1.4

glucose uptake rate (mmol/g DW-hr)

grow

th ra

te (1

/hr)

0 5 10 15 20 25 30 35 400

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

succinate uptake rate (mmol/g DW-hr)

grow

th ra

te (1

/hr)

O2 uptake fixed at -17

Points of interest

• Updated core model has similar oxygen response characteristics

• Can predict the trafficking of protons• Illustrates the LO as the best biomass

yield/growth achieved when oxygen is fully utilized

• Can contrast growth properties of different substrates

• You now try your own ideas

EXAMPLES FROM THE LITERATUREPublications

Robustness in iJE660• Vary the activity of essential genes • Look at the consequences of over-expression

Biotech Prog 16:927 (2000)

Effect of proton balancing on growth rate: prediction of H+ secretion

Genome Biology 4:R54 (2003)

PHENOTYPIC PHASE PLANES

Two parameters

PhPP vs. RobustnessPhenotypic Phase Plane (PhPP)

Robustness Analysis: Projection of PhPP for Maximum Growth rate vs. O2 uptake

Robustness Analysis: Projection of PhPP for Maximum Growth rate vs. Succinate uptake

Line of Optimality (LO)

O2 uptakeSuccinate uptake

Biom

ass P

rodu

ction

Biom

ass P

rodu

ction

Biom

ass P

rodu

ction

O2 uptake

Succinate uptake

Mathematics: Shadow prices from the dual solution are calculated for different uptake rates. Shadow prices are constant within a region, changes in shadow prices delineate the different regions.

Biological Significance: Can determine what the optimal nutrient uptake rates to allow for maximal biomass production (Line of Optimality) and what uptake rates are not feasible.

Example: Comparison of experimentally measured uptake rates shows that E. coli uses its metabolic network to maximize biomass for some carbon sources (operates along the line of optimality) [Edwards NBT, Ibarra Nature]

Key ReferencesEdwards, J.S., Ibarra, R.U., and Palsson, B.Ø., "In silico predictions of Escherichi coli metabolic capabilities are consistent with experimental data", Nature Biotechnology 19: 125-130(2001).

Edwards, J.S., Ramakrishna R., Palsson, B.Ø., “Characterizing the metabolic phenotype: A phenotype phase plane analysis",Biotechnology and Bioengineering, 77(1): pp. 27-36 (2002).

Schilling,C.H., Edwards, J.S., Letscher, D.L., and Palsson, B.Ø., "Combining pathway analysis with flux balance analysis for the comprehensive study of metabolic systems", Biotechnology and Bioengineering 71: 286-306 (2001).

Ibarra, R.U., Edwards, J.S., and Palsson, B.Ø.; "Escherichia coli K-12 undergoes adaptive evolution to achieve in silico predicted optimal growth," Nature, 420: pp. 186-189 (2002).

Carbon Uptake Rate

Oxy

gen

Upt

ake

Rate

Line of Optimality

0.4

2.4

IsoclinesPhase Plane

Phenotypic Phase Planes

Edwards et. al., Nat Biotech., 19, 2001 Ibarra et. al., Nature, 420, 2002

Historical data

Experimental data: acetate

Experimental data: succinate

CHARACTERISTICS OF THE PHPP

Phenotypic Phase Planes• 2-dimensional region

– Spanned by 2 metabolic fluxes

• Typically uptake rates– lines to demarcate phase of

constant shadow price– By definition, metabolic

pathway utilization is different in each region of the phase plane

Met

abol

ic F

lux

B

Metabolic Flux A

Infe

asib

le S

tead

y St

ate

Infeasible Steady State

{Sha

dow P

rice A

}M

etabo

lic

Phen

otyp

e A

{Shadow Price B}

Metabolic

Phenotype B

SingleGrowthcondition

Shadow Prices and Isoclines

Shadow Price

Relative shadow prices

boundaryii b

Z

A

B

B

A

B

A

dbdb

dbdZdb

dZ

-

Shadow prices and isoclinesU

ptak

e B

Uptake A

Dual S

ubstr

ate

Limita

tion

Sing

le S

ubst

rate

Lim

itatio

n

“Futile”

Region

A

B

B

A

B

A

dbdb

dbdZdb

dZ

-

Features of Phase Planes

• Infeasible regions: fluxes don’t balance• Regions of single substrate limitations ( = 0 or

infinity)• Regions of dual substrate limitations ( < 0) • Futile regions ( >0 )• Isoclines (like constant height in topography maps)• Line of optimality: corresponds to maximal biomass

yield (g cells/mmol carbon source)– You find this by fixing carbon uptake rate and the optimize

for biomass using FBA, this will give you one point on the LO unless oxygen is limiting

Line of Optimality: Max. Yx/s

Oxy

gen

Upt

ake

B

Carbon Source Uptake Rate

Infe

asib

le S

tead

y St

ate

Infeasible Steady State

Met

aboli

c

Phen

otyp

e 1

Metabolic

Phenotype 2

LO

CORE E. COLI CACULLATIONS

Core E. coli model examples

Growth on acetate with O2

Line of optimality

Infeasible region (no growth)

aceta

te lim

ited gr

owth

O 2 limited growth

Core E. coli model examples

Growth on glucose with O2

Line of optimality

acetate, formate, and ethanol secretedacetate and formate secreted

aceta

te se

creted

exce

ss O

2

Core E. coli model examples

Growth on fumarate with O2

Line of optimality

High uptake rate needed for fully anaerobic growth

PHPP AT THE GENOME-SCALEFor whole organisms

The H. influenzae Metabolic Phase Plane

J. Biol. Chem. 274(15):17410 (1999)

E. Coli PhPP on Glucose

BMC Genomics 2004, 5:63

BMC Genomics 2004, 5:63

Summary

• A parameter in an in silico model can be varied and repeated optimization computations performed

• Changing one parameter is called ‘robustness analysis’

• Changing two parameters is called ‘phenotypic phase plane analysis’

• Optimal growth properties have been productively analyzed with these methods