7. robustness & phase planes february 6th, 2008 no … · 7. robustness & phase planes...

UW-Madison, Chemical & Biological Engineering

Constraint-Based Workshops

7. Robustness & Phase PlanesFebruary 6th, 2008

No Meeting Next Week!!


Constraint-Based MethodsOptimal Solutions

1. FBA2. Flux Variability

Flux Dependencies1. Robustness2. Phase Planes3. Flux Coupling

All Allowable Solutions1. Extreme Pathways2. Elementary Modes3. Sampling

Altering Phenotypes1. Genetic Mutations2. Strain Design

Application of AdditionalConstraints1. Regulation2. Energy Balance

Price, Reed, and Palsson Nat. Reviews Microbiol. 2004


Robustness Analysis

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 w/slope equal to Shadow Price


Example

In this example we vary the maximum allowable uptake rate of oxygen. The whole range of oxygenation is shown, 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


Review of Shadow Prices & Reduced Costs

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

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

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

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


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)


Robustness Analysis

Define Number of Steps

Define Flux to Vary


Robustness Analysis Calculations

• Calculate the sensitivity of the objective function to changes in, use glucose uptake rate of 5 and aerobic conditions:– PGL (pentose phosphate flux)– GAPD (glycolytic flux)– ICDHyr (TCA cycle flux)

Graph results in excel!


Robustness Analysis

0

0.1

0.2

0.3

0.4

0.5

0.6

0 5 10 15 20 25 30 35

Flux Value (mmol/gDW/hr)

Gro

wth

Rat

e (1

/hr)

PGLICDHyrGAPD


What Does this Mean?

• Which reaction(s) are essential (note that FBA, MOMA, and ROOM would all predict the same lethal phenotype)?


Phase Plane Analysis:

Varying multiple fluxes simultaneously


Parameter Variation

Phenotypic 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 uptake Succinate uptake

Bio

mas

s P

rodu

ctio

n

Bio

mas

s P

rodu

ctio

n

Bio

mas

s P

rodu

ctio

n

O2 uptake

Succinate uptake


Phenotype Phase Plane

• 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 Pr

ice A

}M

etabo

lic

Phen

otype

A

{Shadow Price B}

Metabolic

Phenotype B

SingleGrowthcondition


Shadow Prices and Isoclines

Shadow Price

Relative shadow prices

boundaryii b

Z⎥⎦

⎤∂∂

−=γ

A

B

B

A

B

A

dbdb

dbdZ

dbdZ

=−=−=γγ

α -


Isoclines:Lines w/ Constant Objective Values

Upt

ake

B

Uptake A

Dual S

ubst

rate

Lim

itatio

n

Sin

gle

Subs

trat

eLi

mit

atio

n

“Futile”

Region

α

A

B

B

A

B

A

dbdb

dbdZ

dbdZ

=−=−=γγ

α -


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

infinity)• Regions of dual substrate limitations (α < 0) • Futile regions (α >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


Line of Optimality: Max. Yx/sO

xyge

n U

ptak

e B

Carbon Source Uptake Rate

Infe

asib

le S

tead

y St

ate

Infeasible Steady State

Met

abol

ic

Phen

otyp

e 1

Metabolic

Phenotype 2

LO


Acetate Phase Plane for E. coli

Line ofOptimality


Acetate PhPP: Two Futile Regions

0

5

10

15

20

0 5 10 15 20

2.0

0.3

1

2

Acetate Uptake Rate

Oxy

gen

Upt

ake

Rat

e

↑ growth rate↑ growth rate

Hypothesis:Metabolic regulation will drive the operation of the metabolic network toward the line of optimality

Lines of constant growth rate‘isoclines’


Acetate PhPP & Experimental Data


Growth on Acetate 3D Phase Plane:

The Phase Plane


Succinate Phenotype Phase Plane

Succinate Uptake Rate

Oxy

gen

Upt

ake

Rat

e

0

5

10

15

20

25

0 5 10 15 20

SUR

OU

R

0.0

1.0

2.0

3.0

4.0

5.0

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Point

APR

Experimental FBA

1

4

3

2-also works for:malate, glucose,fumarate

-does not work for glycerol


Growth on Succinate

LO

Dual substrate limited region


Application:Predicting

complex biology;adaptive evolution

and picking optimal growth

states


Methods – adaptive evolution

• Cultures grown in 250ml minimal medium supplemented with 2g/L carbon source

• Serial passage during exponential growth

• Stable growth rate achieved at end of evolution

• Cells frozen throughout evolution for phenotype testing

Wild type Day 1 Day 2

Phenotype testing

Day …

Phenotype testing

0

0.1

0.2

0.3

0.4

0.5

0.6

0 10 20 30 40 50 60 70 80 90 100

H o u r


Cellular Evolution: Growth rates on Glycerol

EVOLVING STABLE

Pre- EvolutionPost-Evolution

During Evolution

Ibarra et al, 2002, Nature, 420: 186-189

7. robustness & phase planes february 6th, 2008 no … · 7. robustness & phase planes...

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