© 2015 H. Scott Hinton Lesson: Robustness and Phenotype Phase Plane AnalysisBIE 5500/6500Utah State University Constraint-based Metabolic Reconstructions

© 2015 H. Scott Hinton

Lesson: Robustness and Phenotype Phase Plane AnalysisBIE 5500/6500Utah State University

Constraint-based Metabolic Reconstructions & Analysis

Robustness Analysis & Phenotype Phase Plane Analysis




Learning Objectives

• Explain the capabilities of robustness analysis

• Explain how shadow prices can be used in metabolic modeling

• Explain how reduced costs can be used in metabolic modeling

• Explain the capabilities of phenotype phase plane analysis




Lesson Outline

• Robustness Analysis

• Shadow Prices

• Reduced Costs

• Phenotype Phase Plane

Analysis




ROBUSTNESS ANALYSIS• The flux through one reaction is varied and the optimal

objective value is calculated as a function of this flux. This reveals how sensitive the objective is to a particular reaction.

• Example: Determine the effect of varying glucose uptake on growth

Clear;

% Input model

model=readCbModel('ecoli_textbook');

% Set oxygen uptake rate

model = changeRxnBounds(model,'EX_o2(e)',-17,'l');

% Set the upper bound for glucose uptake

model = changeRxnBounds(model,'EX_glc(e)',-18.5,'l');

% Set optimization objective

model = changeObjective(model,'Biomass_Ecoli_core_N(w/GAM)_Nmet2');

% Use robustnessAnalysis for glucose uptake rate

robustnessAnalysis(model,'EX_glc(e)',100);

AerobicGlucoseBioMassRA.m




Robustness Analysis ExampleImpact of Increasing Glucose

Growth remains at 0 hr-1 until a glucose uptake rate of about 0.48 mmol gDW-1 hr-1, because with such a small amount of glucose, the system cannot make 8.39 mmol gDW-1 hr-1 of ATP needed to meet the default lower bound of the ATP maintenance reaction (ATPM)

Oxygen uptake limits growth! Excess glucose cannot be fully oxidized, so the acetate fermentation pathways is used.

X

X





Robustness Analysis Example Maps(AerobicGlucoseBioMassRA.m)

EX_glc(e) =-7 mmol gDW-1 hr-1 EX_glc(e) =-10 mmol gDW-1 hr-1




Robustness Analysis Example Impact of Increasing Oxygen

At an oxygen uptake of about 21.80 mmol gDW-1 hr-1, growth actually begins to decrease as oxygen uptake increases. This is because glucose becomes limiting at this point, and glucose that would have been used to produce biomass must instead be used to reduce excess oxygen.

EX_glc(e) is set at -10 mmol gDW-1 hr-1


XX




Robustness Analysis Example Maps Impact of Increasing Oxygen

EX_o2(e) = -20 EX_o2(e) = -25

AerobicGlucoseBioMassRA_Map.m




Maximum Anaerobic Ethanol ProductionAnaerobicEthanolRA.m

% AnaerobicEthanolRA.mclear;

% Input the E.coli core modelmodel=readCbModel('ecoli_textbook');

% Set uptake ratesmodel = changeRxnBounds(model,'EX_glc(e)',-10,‘b'); model = changeRxnBounds(model,'EX_o2(e)',-0,'b');

% Set optimization objective to Biomass_Ecoli_core_N(w/GAM)_Nmet2model = changeObjective(model,'Biomass_Ecoli_core_N(w/GAM)_Nmet2');

% Using robustnessAnalysis, plot the objective function as a function % of the ethanol secretion rate

[controlFlux, objFlux] = robustnessAnalysis(model,'EX_etoh(e)',100);

Maximum Growth Rate

(8.283, 0.2117)

Maximum Ethanol

Production

Fix Glucose and Oxygen Uptake

Rates




Ethanol Production PhenotypesAnaerobicEthanolRA.m & AnaerobicEthanolRA_Maps.m

% Draw a map of the different production phenotypesclear;model=readCbModel('ecoli_textbook');

model = changeRxnBounds(model,'EX_glc(e)',-10,'b'); model = changeRxnBounds(model,'EX_o2(e)',-0,'b');

model = changeRxnBounds(model,'EX_etoh(e)',4.242,'b'); % Low %model = changeRxnBounds(model,'EX_etoh(e)',12.53,'b'); % Medium %model = changeRxnBounds(model,'EX_etoh(e)',18.38,'b'); % High


FBAsolution = optimizeCbModel(model,'max',0,0)

map=readCbMap('ecoli_Textbook_ExportMap');options.zeroFluxWidth = 0.1;options.rxnDirMultiplier = 10;

drawFlux(map, model, FBAsolution.x, options);

printFluxVector(model, FBAsolution.x, true)

Low Production(4.242, 0.1757)

High Production(18.38, 0.1063)

Medium Production

(12.53, 0.2002)




Ethanol Production Phenotype MapsAnaerobicEthanolRA_Maps.m

Low Production Medium Production High Production




Lesson Outline


• Shadow Prices

• Reduced Costs


Analysis




Shadow Prices: Metabolites

• Shadow prices, πi, are the derivative of the

objective function, Z, with respect to the flux,

bi, of a metabolite.

• The shadow prices define the incremental

change in the objective function if a

constraining flux is incrementally changed.

• The sensitivity of an FBA solution is indicated

by shadow prices. They indicate how much

the addition of a given metabolite will

increase or decrease the objective.

• In the COBRA Toolbox, shadow prices can be

calculated by optimizeCbModel. The vector

of y shadow prices is solution.y (glpk

solver)

-0.0360

-0.0325

-0.0230

+0.0229

ii

Z

b

slope

ShadowPricesExampleO2.m

EX_glc(e) = -10




Shadow PricesSince

If the objective is set to maximize cell growth rate (ZGR), and the

shadow price (πi), of oxygen (bEX_o2(e)) is -0.0230, it means that an

additional flux unit of oxygen, within the EX_o2(e) uptake region

of

-15 and -21, will increase the growth rate by 0.0230.

Each steady state solution (phenotype) will have different

shadow prices. This example is based on EX_glc(e) = -10

(AerobicGlucoseBioMassShadowPrices.m)

i i i NADHi NADH

i

Z ZZ b

b b

Z

ATPATP

ATP

1 unit of flux of bNADH

-0.0360

-0.0325

-0.0230

+0.0229

ii

Z

b

slope

EX_glc(e) = -10

ShadowPricesExampleO2.m




Shadow Prices for Growth on Glucose

(ShadowPricesAerobicGrowthRateData.m)

clear;


changeCobraSolver('glpk'); % Use Matlab solver

% Set the lower bounds for oxygen and glucose uptake

model = changeRxnBounds(model,'EX_o2(e)',-20,‘b ');

model = changeRxnBounds(model,'EX_glc(e)',-10,'l');

% Set optimization objective to Biomass_Ecoli_core_N(w/GAM)_Nmet2

model =

changeObjective(model,'Biomass_Ecoli_core_N(w/GAM)_Nmet2');

% Optimize objective function

FBAsolution = optimizeCbModel(model,'max'); % Must allow loops

% Print flux values


% Print shadow prices

'Shadow prices'

printShadowPriceVector(model, FBAsolution.y, true)

Metabolite SP13dpg[c] -0.0372pg[c] -0.0273pg[c] -0.0276pgc[c] -0.0576pgl[c] -0.055ac[c] -0.003acald[c] -0.011acald[e] -0.011acon-C[c] -0.040actp[c] -0.013akg[c] -0.033akg[e] -0.031amp[c] 0.010atp[c] -0.010cit[c] -0.040coa[c] 0.010dhap[c] -0.036e4p[c] -0.043etoh[c] -0.013etoh[e] -0.010f6p[c] -0.059fdp[c] -0.071for[c] -0.004for[e] -0.004fru[e] -0.046fum[c] -0.028fum[e] -0.023g3p[c] -0.036g6p[c] -0.059

Metabolite SPglc-D[e] -0.046gln-L[c] -0.047gln-L[e] -0.037glu-L[c] -0.037glu-L[e] -0.034glx[c] -0.015h[c] 0.003icit[c] -0.040lac-D[c] -0.015lac-D[e] -0.013mal-L[c] -0.028mal-L[e] -0.023nadh[c] -0.004nadph[c] -0.006o2[c] -0.023o2[e] -0.023oaa[c] -0.027pep[c] -0.027pi[c] -0.003pyr[c] -0.014pyr[e] -0.011q8h2[c] 0.006r5p[c] -0.051ru5p-D[c] -0.051s7p[c] -0.066succ[c] -0.022succ[e] -0.024succoa[c] -0.019xu5p-D[c] -0.051




Glucose Shadow Prices: Aerobic Growth Rate Example(ShadowPricesAerobicGrowthRate_glc.m)

If the objective is set to maximize cell growth rate (ZGR), and the

shadow price (πglc-D[e]), of glucose is -0.046 when

EX_o2(e) = -20, it means that an additional molecule of glucose will

increase the growth rate by 0.046.

clear;



model = changeRxnBounds(model,'EX_glc(e)',-10,‘b ');


FBAsolution = optimizeCbModel(model,'max')printFluxVector(model, FBAsolution.x, true)

Biomass (EX_glc(e) = -10)

0.832 0.878

Biomass (EX_glc(e) = -11) ΔBiomass

0.046

-0.046

-0.137

-0.033

Change between -10 & -11




Lesson Outline


• Shadow Prices

• Reduced Costs


Analysis




Reduced Costs: Reactions• Reduced costs are the derivatives of the objective function (Z) with respect to an internal reaction (vi) with 0 flux. Reduced costs indicate how much each particular reaction affects the objective.

• The reduced costs are associated with each flux (vi) and signify the amount by which the objective function is decreased if vi is increased. For instance, if the input flux of glucose shows a reduced cost of -x, it means that increasing that flux by one unit will increase of the objective function by x units.

• In the COBRA Toolbox, reduced costs can be calculated by optimizeCbModel. The vector of reduced costs is FBAsolution.w (glpk solver)

i i i ii

Z ZZ v Z

v v ATP

1 unit of flux

EX_glc(e) = -10+0.0229

-0.0230

-0.0325-0.0360

ii

Z

v

slope

ReducedCostsExampleO2.m




Glucose Reduced Costs: Aerobic Growth Rate Example(ReducedCostsAerobicGrowthRate_glc.m)

If the objective is set to maximize cell growth rate

(ZGR), and the reduced costs (ρEX_glc(e)), of glucose

(EX_glc(e)) is -0.046, it means that an additional unit

of glucose will increase the growth rate by 0.046.

clear;



model = changeRxnBounds(model,'EX_glc(e)',-10,‘b ');


FBAsolution = optimizeCbModel(model,'max')


Biomass (EX_glc(e) = -10)

0. 832 0. 878

Biomass (EX_glc(e) = -11) ΔBiomass

0.046

-0.046

-0.137

-0.033

Change between -10 & -11




Lesson Outline


• Shadow Prices

• Reduced Costs


Analysis




Phenotype Phase Plane

Price, N. D., J. L. Reed, et al. (2004). "Genome-scale models of microbial cells: evaluating the consequences of constraints." Nature reviews. Microbiology 2(11): 886-897.




Phenotype Phase Plane Analysis

• It is also possible to vary two parameters simultaneously and plot the results as a phenotypic phase plane. These plots can reveal the interactions between two reactions.

• Matlab Cobra function: phenotypePhasePlane(model, ‘rxn1’,’rxn2’)

Phenotype Phase Plane Shadow Prices of rxn1 Shadow Prices of rxn2




Phenotype Phase Plane Analysis ExampleAerobicGlucoseBioMassPPP.m

% Input model


% Set oxygen and glucose uptake rates

model = changeRxnBounds(model,'EX_o2(e)',-20,'l');

model = changeRxnBounds(model,'EX_glc(e)',-20,'l');

% Set optimization objective


% Phenotype phase plane analysis

phenotypePhasePlane(model,'EX_glc(e)', 'EX_o2(e)‘);




Phenotype Phase Plane Analysis Example (II)

Variables: EX_o2(e) & EX_glc(e)

Phase 1

Phase 3

Phase 2

Phase 4

Phase 5

AerobicGlucoseBioMassPPP.m




Phenotype Phase Plane Analysis Example (Phase 1)

>> AerobicGlucoseBioMassPhase1

FBAsolution =

f: 0x: []stat: -1origStat: 110solver: 'glpk'time: 0.0210

No Growth

No growth; not enough glucose

AerobicGlucoseBioMassPhase1.m

EX_glc(e)= 1EX_o2(e)= 10

Ferm

TCA

PPP

Glyc

Ana

OxP





ACONTa 3.41586ACONTb 3.41586AKGDH 2.69395ATPM 8.39ATPS4r 21.278Biomass 0.183018CO2t -10.2116CS 3.41586CYTBD 20ENO 5.23969EX_co2(e) 10.2116EX_glc(e) -3EX_h2o(e) 11.5448EX_h(e) 3.67134EX_nh4(e) -0.997959EX_o2(e) -10EX_pi(e) -0.673268


EX_glc(e)

Growth is limited by excess oxygen; not enough glucose to reduce all the oxygen and produce biomass optimally

FBA 2.81795FUM 3.2184GAPD 5.51348GLCpts 3GLNS 9.47587GLUDy -0.951162GLUN 9.42907H2Ot -11.5448ICDHyr 2.89141ICL 0.524456MALS 0.524456MDH 3.74286NADH16 16.7816NH4t 0.997959O2t 10PDH 4.62623PFK 2.81795

EX_glc(e)= -3EX_o2(e)= -10

PGI 2.96248PGK -5.51348PGM -5.23969PIt2r 0.673268PYK 2.14468RPE -0.131553RPI -0.131553SUCDi 3.2184SUCOAS -2.69395TALA -0.032741THD2 0.444093TKT1 -0.032741TKT2 -0.098811TPI 2.81795

Ferm

TCA

PPP

Glyc

Ana

OxP





ACKr -1.83668ACONTa 2.106ACONTb 2.106ACt2r -1.83668AKGDH 1.70302ATPM 8.39ATPS4r 21.5061Biomass 0.373508CO2t -10.4318CS 2.106CYTBD 20ENO 7.66481EX_ac(e) 1.83668EX_co2(e) 10.4318EX_glc(e) -5EX_h2o(e) 13.1526EX_h(e) 9.32925


EX_nh4(e) -2.03666EX_o2(e) -10EX_pi(e) -1.37402FBA 3.84494FUM 1.70302G6PDH2r 2.35059GAPD 8.22357GLCpts 5GLNS 0.095506GLUDy -1.94116GND 2.35059H2Ot -13.1526ICDHyr 2.106MDH 1.70302NADH16 18.297NH4t 2.03666O2t 10

PDH 5.34252PFK 3.84494PGI 2.57284PGK -8.22357PGL 2.35059PGM -7.66481PIt2r 1.37402PPC 1.07032PTAr 1.83668PYK 1.40059RPE 1.29858RPI -1.05201SUCDi 1.70302SUCOAS -1.70302TALA 0.71671TKT1 0.71671TKT2 0.581873TPI 3.84494

EX_glc(e)

EX_glc(e)= -3EX_o2(e)= -10

EX_glc(e)

Ferm

TCA

PPP

Glyc

Ana

OxP

Not enough oxygen to fully oxidize glucose; acetate produced through fermentation

EX_glc(e)= -5EX_o2(e)= -10





ACKr -9.90568ACONTa 0.60316ACONTb 0.60316ACt2r -9.90568ATPM 8.39ATPS4r 18.8064Biomass 0.559051CO2t -4.89467CS 0.60316CYTBD 20ENO 16.0799EX_ac(e) 9.90568EX_co2(e) 4.89467EX_for(e) 11.5033EX_glc(e) -10EX_h2o(e) 8.96702EX_h(e) 32.6235EX_nh4(e) -3.04839


EX_o2(e) -10EX_pi(e) -2.05658FBA 7.84632FORti 11.5033G6PDH2r 4.79277GAPD 16.9163GLCpts 10GLNS 0.142949GLUDy -2.90544GND 4.79277H2Ot -8.96702ICDHyr 0.60316NADH16 20NH4t 3.04839O2t 10PDH 1.10076PFK 7.84632PFL 11.5033

PGI 5.09263PGK -16.9163PGL 4.79277PGM -16.0799PIt2r 2.05658PPC 1.60202PTAr 9.90568PYK 4.18773RPE 2.79333RPI -1.99944TALA 1.49758TKT1 1.49758TKT2 1.29576TPI 7.84632

EX_glc(e)

Ferm

TCA

PPP

Glyc

Ana

OxPEX_glc(e)= -10EX_o2(e)= -10

Not enough oxygen to fully oxidize glucose; acetate and formate are produced and secreted





ACALD -2.12502ACKr -23.0206ACONTa 0.912497ACONTb 0.912497ACt2r -23.0206ALCD2x -2.12502ATPM 8.39ATPS4r 5.95415Biomass 0.845766CO2t 1.51113CS 0.912497CYTBD 20ENO 34.4864ETOHt2r -2.12502EX_ac(e) 23.0206EX_co2(e) -1.51113EX_etoh(e)2.12502EX_for(e) 29.2279


EX_glc(e) -19EX_h2o(e) 2.52474EX_h(e) 69.2146EX_nh4(e) -4.61179EX_o2(e) -10EX_pi(e) -3.11132FBA 18.1587FORti 29.2279GAPD 35.7517GLCpts 19GLNS 0.216262GLUDy -4.39553H2Ot -2.52474ICDHyr 0.912497NADH16 20NH4t 4.61179O2t 10PFK 18.1587

PFL 29.2279PGI 18.8266PGK -35.7517PGM -34.4864PIt2r 3.11132PPC 2.42363PTAr 23.0206PYK 12.6238RPE -0.607936RPI -0.607936TALA -0.151307THD2 14.5016TKT1 -0.151307TKT2 -0.456629TPI 18.1587

EX_glc(e)

Ferm

TCA

PPP

Glyc

Ana

OxPEX_glc(e)= -19EX_o2(e)= -10

Not enough oxygen to fully oxidize glucose; acetate, formate and ethanol are produced and secreted.




Phenotype Phase Plane Analysis Example (VI)


Phase 1

Phase 3

Phase 2

Phase 4

Phase 5










Phenotype Phase Plane Analysis Example (VII)

There is a different phenotype for each phase region

Ph

ase

1Ph

ase

2Ph

ase

3

Phase 4

Phase 5

+0.02290.00 -0.0230 -0.0325

-0.0581 -0.0360

Oxygen Shadow Prices

Ph

ase

1Ph

ase

2Ph

ase

3

Phase 4

Phase 5

-0.13730.00 -0.0459 -0.0325

-0.0305

Glucose Shadow Prices




Phenotype Phase Plane Analysis Example (VIII)

(Robustness Analysis setting O2 = -5 mmol/g DW-hr)


Given: O2 = -5 mmol/g DW-hr

PhaseBoundaries

Robustness Analysis

Ph

ase

1Ph

ase

2Ph

ase

3

Phase 4

Phase 5

-0.13730.0000 -0.0459 -0.0325

-0.0305





Phenotype Phase Plane Analysis Example (IX)

(Robustness Analysis setting O2 = -15 mmol/g DW-hr)


Given: O2 = -15 mmol/g DW-hr

PhaseBoundaries

Robustness Analysis

Ph

ase

1Ph

ase

2Ph

ase

3

Phase 4

Phase 5

-0.13730.00 -0.0459 -0.0325

-0.0305





Phenotype Phase Plane Analysis Example (X)

(Robustness Analysis setting O2 = 10 mmol/g DW-hr)


Given: O2 = 10 mmol/g DW-hr

PhaseBoundaries

Robustness Analysis

Ph

ase

1Ph

ase

2Ph

ase

3

Phase 4

Phase 5

-0.13730.00 -0.0459 -0.0325

-0.0305





Line of Optimality• The line of optimality (LO) is defined as a line

representing the optimal relation between the two metabolic fluxes used to create a phenotype phase plane.

• The line of optimality is determined by specifying an uptake rate of the substrate along the x-axis and then allowing any value for the flux along the y-axis. Linear Programming can then be used to calculate the optimal value of the objective as a function of the y-axis flux. Once the objective is determined, the corresponding flux value for the y-axis is used to plot the line of optimality (LO).

• The LO defines the optimal utilization of the metabolic pathways without limitations on the availability of the substrates.

LO




Phenotype Phase Plane Analysis Example (XI)

(Robustness Analysis setting Glucose = -2.5 mmol/gDW-hr)

Ph

ase

1Ph

ase

2Ph

ase

3

Phase 4

Phase 5

-0.13730.0000 -0.0459 -0.0325

-0.0305 There is a different phenotype for each phase region

Given: Glucose = -2.5 mmol/gDW-hr

PhaseBoundaries

Robustness AnalysisLO

LO





Phenotype Phase Plane Analysis Example (XII)

(Robustness Analysis setting Glucose = -5 mmol/gDW-hr)


Given: Glucose = -5 mmol/gDW-hr

PhaseBoundaries

Robustness Analysis

LO

Ph

ase

1Ph

ase

2Ph

ase

3

Phase 4

Phase 5

-0.13730.0000 -0.0459 -0.0325

-0.0305

LO





Phenotype Phase Plane Analysis Example (XIII)


Phase 1

Phase 3

Phase 2

Phase 4

Phase 5






Line of Optimality (LO)





Adaptive Laboratory Evolution

A phenotypic phase plane is a representation of how two fluxes in a metabolic network relate to each other and affect in silico-predicted optimal growth. Distinct planes are represented by several colors. Here, the line of optimality (LO, yellow) defines the ratio of glycerol uptake rate to oxygen uptake rate (OUR) that leads to optimal biomass production. On glycerol, wild-type E. coli initially has a phenotype that maps to a suboptimal region of the portrait. After a growing for several hundred generations on glycerol, the E. coli phenotype migrates to the line of optimality.

(Conrad, T. M., N. E. Lewis, et al. (2011). "Microbial laboratory evolution in the era of genome-scale science." Molecular Systems Biology 7: 509.)




Lesson Outline


• Shadow Prices

• Reduced Costs


Analysis




Extra Slides




Phenotype Phase Plane Analysis Example (III)

EX_glc(e) = -1 mmol/g DW-hr

Phase 1EX_glc(e)

Ferm

TCA

PPP

Glyc

Ana

OxP



Phase 2EX_glc(e)

Ferm

TCA

PPP

Glyc

Ana

OxP





Phenotype Phase Plane Analysis Example (IV)

Phase 3


EX_glc(e)

Ferm

TCA

PPP

Glyc

Ana

OxP


Phase 4


EX_glc(e)

Ferm

TCA

PPP

Glyc

Ana

OxP





Phenotype Phase Plane Analysis Example (V)

Phase 5


EX_glc(e)

Ferm

TCA

PPP

Glyc

Ana

OxP


Documents

© 2015 H. Scott Hinton Lesson: Robustness and Phenotype Phase Plane AnalysisBIE 5500/6500Utah State University Constraint-based Metabolic Reconstructions