Download - Lecture #23
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