Lecture #19

Alternative (equivalent) Optima

Summary

• LP statement and the occurrence of many optimal solutions

• Methods to study degenerate solutions– Flux variability analysis (FVA)– Extreme pathways and multiple

optima– Enumerating all the alternative

optima

FBA Optimization Problem Statement

• Objective Function: A function that is maximized or minimized to identify optimal solutions

• Constraints: Place limits on the allowable values the solutions can take on.

Maximize: cv

Such that S v = b =0LB v UB

v1

v3

v2

Equivalent Optimal Solutions Exist:How can we find & characterize them?

Gro

wth

Rat

e

Projected Solution Space

FBA

Some Flux

Example of two equivalent optima

•Two equivalent (same input/output state) flux distributions for optimally generating malate from succinate in the core E. coli model.•All non-negative combinations of them give the same optima

FLUX VARIABILITY ANALYSIS (FVA)

Method 1:

maximize 3w1 + 3w2

such that w1 + w2 ≤ 2 w1, w2 ≥ 0w2

w1

1

1

Alternate optimal solutions:3w1 + 3w2 = 6

min(w1) from FVA max(w1) from FVA

min(w2) from FVA

max(w2) from FVA

Alternate Optimal Solutions

maximize 3w1 + w2

such that w1 + w2 ≤ 2 w1, w2 ≥ 0w2

w1

1

1

Unique optimal solution

Unique Optimal Solution

Example: flux variability analysis

Flux Variability Analysis:

Flux Variabilityfor V1

Alternate orEquivalent Optima

Incr

ease

in

Sta

ted

Obj

ectiv

e F

unct

ion

V1

V2

• First, identify the maximum value of the objective function and constrain objective function to this value.

• Second, minimize and maximize each flux independently to identify flexibility in the fluxes across alternate optima.

If we have n fluxes, we solve ≈ 2n FBA problems

Flux Variability Analysis (FVA): the concept

• Determine what is possible based on external measurements and network stoichiometry

Solve series of 2 linear optimization for each reactions j.

Objective: maximize & minimize vj (for all

j) subject to (be in the space of optimal solutions):

i = metabolitesj = reactionsjv

+ ∞

max

“FeasibleRegion”

- ∞

jvmin

Metabolic Eng, 2003. 5(4): p. 264-76.

maxminjjj v

S ∙ v = 0

<c ∙ v> = Zopt

Reduced Cost

• Definition: – dZ/dvi =0; (and more specifications)

• In order for a flux to be variable, the reduced cost must be equal to zero. The converse is not necessarily true.

Core E. coli Metabolism

Metabolite YieldCarbon

Conversion

# of Variable

Fluxes3PG 2 100% 24PEP 2 100% 24Pyr 2 100% 24OA 2 133.33% 24G6P 0.8916 89.16% 16F6P 0.8916 89.16% 16R5P 1.0571 88.10% 44E4P 1.2982 86.55% 25G3P 1.6818 84.09% 13AcCoA 2 66.67% 36αKG 1 83.33% 36SuccCoA 1.64 109.33% 2

Yield on Glucose (Aerobic):E. coli core model (95 fluxes)

Yield on Glucose (Anaerobic):E. coli core model

Metabolite YieldCarbon

Conversion

# of Variable

Fluxes3PG 1 50% 38PEP 1 50% 38Pyr 1 50% 38OA 1 66.67% 38G6P 0.625 62.50% 2F6P 0.625 62.50% 2R5P 0.72 60.00% 26E4P 0.8491 56.60% 21G3P 1.0345 51.72% 21AcCoA 1 33.33% 38αKG 0.4 33.33% 38SuccCoA 1.434 95.60% 2

Flux Variability for optimal 3PG yield on Glucose

flux = 0 always

flux = 1 always

flux varies between 0 and 0.5

flux varies between 1 and 1.5

Studies using FVA• Mahadevan R, Schilling CH.

The effects of alternate optimal solutions in constraint-based genome-scale metabolic models. Metab Eng. 2003 Oct;5(4):264-76.

• Duarte, N.C., Palsson, B.Ø., and Fu, P., "Integrated Analysis of Metabolic Phenotypes in ''Saccharomyces cerevisiae'', BMC Genomics, 5:63 (2004).

• Reed, J.L. and Palsson, B.Ø., "Genome-scale in silico models of ''E. coli'' have multiple equivalent phenotypic states: assessment of correlated reaction subsets that comprise network states”, Genome Research, 14:1797-1805(2004).

• Vo, T.D., Greenberg, H.J., and Palsson, B.Ø., "Reconstruction and functional characterization of the human mitochondrial metabolic network based on proteomic and biochemical data", Journal of Biological Chemistry, 279(38):39532-40 (2004).

• Teusink B, Wiersma A, Molenaar D, Francke C, de Vos WM, Siezen RJ, Smid EJ. “Analysis of growth of Lactobacillus plantarum WCFS1 on a complex medium using a genome-scale metabolic model.” J Biol Chem. 281(52):40041-8 (2006).

Literature Example: Vo et. al. 2004Reconstruction and functional characterization of the human mitochondrial metabolic network based on

proteomic and biochemical data

• Study on human mitochondria under various optimality conditions:– Condition 1: max ATP synthesis– Condition 2: max Heme production– Condition 3: max Phospholipid production

FVA of the human mitochondria in the cardiomyocyte

Ordered set of reactions showing decrease in variability as additional optimization criteria are added.

Highly variable fluxes

FVA as a method for evaluating effect of constraints

• L. plantarum model.• Blue – range of fluxes

in unconstrained model.• Green – range of fluxes

in ATP-constrained model:– ATP production equals

ATP consumption

J Biol Chem. 281(52):40041-8 (2006).

FVA Final thoughts

Pseudo codeFVA(S, vmin, vmax, k)

% optimization 1: maximize reaction k

c_k = (0 … 0,1,0, 0) % a 1 at position k

f_opt = max(c_k∙v | S∙v = 0, vmin < v < vmax)

% the actual FVA. max/min every other rxn

for i = 1:nc = (0 … 0,1,0, 0) % a 1 at position i

FVAmini = min(c∙v | S∙v = 0, vmin < v < vmax, vk = f_opt)

FVAmaxi = max(c∙v | S∙v = 0, vmin < v < vmax, vk = f_opt)

end

return (FVAmin, FVAmax)

• Pros:– Easy to compute– Easy to interpret

• Cons:– FVA does not give any

information about correlated reactions

– Each reaction is treated independently

EXTREME PATHWAY ANALYSIS

Method 2:

Metabolic Genotype to PhenotypeDefined within the context of convex analysis

Convex AnalysisConvex Analysis Cellular BiologyCellular Biology

Unique GeneratingVectors

IndependentExtreme Pathways

Flux VectorPositive Combination of Extreme Pathways

me

tab

olic

flu

x (v

1)

metabolic flux (v

2 )

metabolic flux (v 3)

Convex HullCapabilities of a Metabolic Genotype

Particular SolutionMetabolic Phenotype

ExampleFrom the core E. coli model – pyruvate yields from glucose, aerobic

without regulation with regulation

1.48 1.5 1.52 1.54 1.56 1.58 1.6 1.62 1.64 1.66 1.68

x 104

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Extreme Pathway Number

Pyr

uvat

e Y

ield

1450 1500 1550 1600 1650 1700 1750 1800 1850 1900

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Extreme Pathway Number

Pyr

uvat

e Y

ield

Optimal equivalent pathways

Suboptimal equivalent pathways

Non-producing pathways

Equivalent sets exponentially increase the number of ExPas

X1

v1X1a

X1b

X2

X2a

X2b

v1a

v1b

v2a

v2b

X3 … XnVy

Xna

Xnb

vna

vnb

Xy

v1c

v1d

v2c

v2d

v3a

v3b

V(n-1)c

V(n-1)d

vnc

vnd

Pathway of n equivalent sets of size 2 forms 2n extreme pathways.

Set 1 Set 2 … Set n

Equivalent Sets

Two equivalent sets in E. coli. A – Two alternative and equivalent ways to import a proton using two succinate transporters or two transhydrogenases. B - two equivalent ways to produce pyruvate, CO2 and NADH from Malate and NAD.

Some Extreme Pathway Literature• 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).

• Price, N.D., Reed, J.L., Papin, J.A., Famili, I. and Palsson, B.Ø.; "Analysis of Metabolic Capabilities using Singular Value Decomposition of Extreme Pathway Matrices ", Biophysical Journal 84:794-804 (2003).

• Papin, J.A., Price, N.D., and Palsson, B.Ø.; "Extreme Pathway Lengths and Reaction Participation in Genome-Scale Metabolic Networks ," Genome Research, 12: pp. 1889-1900 (2002).

• Price, N.D., Famili, I., Beard, D.A., and Palsson, B.Ø., "Extreme Pathways and Kirchhoff's Second Law ", Biophysical Journal, 83: pp. 2879-2882. (2002).

• Schilling, C. H., Covert, M.W., Famili, I., Church, G.M., Edwards, J.S., and Palsson, B.Ø., "Genome-scale metabolic model of Helicobacter pylori 26695 ", Journal of Bacteriology, 184(16): pp. 4582-4593 (2002).

• Wiback, S.J., and Palsson, B.Ø., "Extreme Pathway Analysis of Human Red Blood Cell Metabolism ", Biophysical Journal, 83(2): pp. 808-818 (2002).

• Papin, J.A., Price, N.D., Edwards, J.S., and Palsson, B.Ø., "The Genome-Scale Metabolic Extreme Pathway Structure in Haemophilus influenzae Shows Significant Network Redundancy ", Journal of Theoretical Biology, 215(1): pp. 67-82 (2002).

• Price, N.D., Papin, J.A., Palsson, B.Ø., "Determination of Redundancy and Systems Properties of the Metabolic Network of Helicobacter pylori Using Genome-Scale Extreme Pathway Analysis",Genome Research, 12: 760-769 (2002).

Extreme Pathways Final thoughts

Implementation:Actual implementation is quite difficult. A

rough sketch:Begin with matrix T(0) ≈ ST

Through a series of transformations T(0) → T(1) → … → T(n) zero out certain elements in T.

Read off Extreme Pathways from T(n)

As this happens, size of T increases.

Several implementations available at:

http://systemsbiology.ucsd.edu/Downloads/Extreme_Pathway_Analysis

• Pros:– Provides biologically

meaningful pathways– Form a mathematically

relevant convex basis

• Cons:– Computation scales

very poorly with network size.

– Numbers grow with networks size

Enumerating all Optimal Solutions

Method 3:

Algorithm For Identifying Different “Corner” Points

• GOAL: given your past solutions, find a new one that uses a different set of non-zero fluxes in the solution.

• The result is that you will identify all the different corner point solutions that have the same objective function value.

• Any optimal solution, can be written as the weighted sum of the corner point optimal solutions (aka we have a convex basis) of the optimal solution space.

Metabolic Network Example

v1: A → Bv2: A → C

b1: → Ab2: B →b3: C →

Reaction List

AB

Cb3

b2

b1v1

v2

Metabolic Map

Maximize Z = c·v = b3

Such that S·v = 0 0 v1,v2,b1,b2,b3 10

-10 v3 10

v3: B Cv3

AB

C 10

10

10

Solution 1:

AB

C 10

10

10

10

Solution 2:

1. Find one valid flux distribution (Solution 1)

2. Repeatedly solve for optimal again:

1. At each Iteration, add additional constraints that new solution must be “different” from any previous solution.

In this Example there are only 2 optimal solutions so the algorithm must only be run for two iterations.

Intuitive Description of Algorithm

Metabolite YieldCarbon

Conversion

# of Variable

FluxesAlternate

optima3PG 2 100% 24 11PEP 2 100% 24 11Pyr 2 100% 24 81OA 2 133.33% 24 11G6P 0.8916 89.16% 16 2F6P 0.8916 89.16% 16 2R5P 1.0571 88.10% 44 2E4P 1.2982 86.55% 25 2G3P 1.6818 84.09% 13 2AcCoA 2 66.67% 36 162αKG 1 83.33% 36 >500*SuccCoA 1.64 109.33% 2 1

Number of Alternate Optima for various Metabolic Objectives (aerobic) in core E. coli

*Did not converge

≥ # of ExPas

Studies using enumeration of AO• Lee, S. , Palakornkule, C., Domach, M. M., and Grossmann, I.E.,

“Recursive MILP model for finding all the alternate optima in LP models for metabolic networks”, Computers & Chemical Engineering, 24(2-7): 711-716 (2000)

• Mahadevan R, Schilling CH. “The effects of alternate optimal solutions in constraint-based genome-scale metabolic models.” Metab Eng. 2003 Oct;5(4):264-76.

• Reed, J.L. and Palsson, B.Ø., "Genome-scale in silico models of ''E. coli'' have multiple equivalent phenotypic states: assessment of correlated reaction subsets that comprise network states”, Genome Research, 14:1797-1805(2004).

• Thiele, I., Fleming, R.M.T., Bordbar, A., Schellenberger, J., and Palsson, B.Ø., "Functional Characterization of Alternate Optimal Solutions of Escherichia coli's Transcriptional and Translational Machinery", Biophys J 98(10):2072-2081 (2010).

Literature Analysis:Genome-Scale In Silico Models of E. coli Have Multiple Equivalent Phenotypic States:

Assessment of Correlated Reaction Subsets That Comprise Network States

• Simulated E. coli iJR904 model under 136 growth conditions (88 aerobic, 48 anaerobic)

• Generated up to 500 alternate optima for each growth condition using MILP algorithm.

Result 1: Only few alternate optima needed to describe range of FVA solutions

• Comparisons of properties for sampled optima with all optima.

• The number of variable fluxes and the allowable ranges for these fluxes across all optima were calculated using FVA

• Each line is for one of the 88 carbon sources capable of supporting aerobic growth.

– (A) shows that as the number of calculated optima increases, the number of variable fluxes found in these sampled optimal solutions approaches the total number of variable fluxes.

– (B) shows how the magnitude of the flux variations is represented by the sampled optima relative to the actual flux variability across all optima.

Finding variable fluxes

Finding the range of variable fluxes

Result 2: Reactions usage in optimal flux distributions

• For each reaction in the metabolic network, what fraction of the optimal flux distributions utilize this reaction (fopt).

• The reactions are then rank-ordered by frequency of use in optimal flux distributions.

• Each reaction in the model was previously classified into one of 30 subsystems.

• Different subsystems are used with different frequency.

All optimal solutions

904 genes931 rxns

Result 3: Correlated sets and regulon structure

• All optimal distributions were combined and the reaction correlations determined.

• 66 correlated reaction sets emerged, most of size 2.

• Reactions in sets tend to be controlled by the same set of genes as defined by the EcoCyc regulon structures

Results 4: Comparing to expression data

• Used 20 expression data sets to see if it correlated with: – A) Genes in correlated

reaction sets.– B) Genes in the same

transcription units.

Significant clusters

Main Results

• Only a small subset of reactions in the network have variable fluxes across optima;

• Sets of reactions that are always used together in optimal solutions, correlated reaction sets, showed moderate agreement with the currently known transcriptional regulatory structure in E. coli and available expression data, and

• Reactions that are used under certain environmental conditions can provide clues about network regulatory needs.

Enumeration: Final thoughts

Implementation:Iterative MILP method where each

iteration produces one additional flux distribution.

Implemented in GAMS and Matlab.

For exact specifications see:Lee, S. , Palakornkule, C., Domach, M. M., and

Grossmann, I.E., “Recursive MILP model for finding all the alternate optima in LP models for metabolic networks”, Computers & Chemical Engineering, 24(2-7): 711-716 (2000)

Reed, J.L. and Palsson, B.Ø., "Genome-scale in silico models of ''E. coli'' have multiple equivalent phenotypic states: assessment of correlated reaction subsets that comprise network states”, Genome Research, 14:1797-1805(2004).

• Pros:– Easier to compute (MILP)

than ExPas– Can terminate early without

computing all alternatives (not possible for ExPas)

– Forms mathematically “nice” convex set of optimal flux distributions (tighter than FVA)

• Cons:– Number of equivalent

solutions may be quite large (> 500)

Summary• Biologically relevant alternate optima exist• Linear programming often does not define

unique optimum.• Several techniques exist for studying alternate

optima and sub-optima– Flux variability analysis– Extreme pathways– Enumeration of all equivalent optimal solutions

• These methods have their pros and cons and you need to choose one that suits your needs

Appendix

Details of enumeration algorithm

Algorithm Details

• GOAL: given your past solutions, find a new one that uses a different set of non-zero fluxes in the solution.

• The result is that you will identify all the different corner point solutions that have the same objective function value.

• Any optimal solution, can be written as the weighted sum of the corner point optimal solutions.

AB

C 10

10

10

Solution 1:

AB

C 10

10

10

10

Solution 2:

1. Find non-zero fluxes (NZJ-

1) at current solution, J-1.

2. Pick at least one NZJ-1 flux to become zero at next solution (yi=1).

3. Make sure that the set of non-zero fluxes haven’t been visited at previous k iterations.

4. Constrain those selected fluxes to have zero flux.

5. Find solution & repeat.

yi 1

i NZJ-1

wi |NZk|-1i NZk

yi+wi 1

wi·vmin vi wi·vmin

AB

C 10

10

10

Solution 1: 1. Find non-zero fluxes (NZJ-

1) at current solution, J-1.

2. Pick at least one NZJ-1 flux to become zero at next solution (yi=1).

3. Make sure that the set of non-zero fluxes haven’t been visited at previous k iterations.

4. Constrain those selected fluxes to have zero flux.

5. Find solution & repeat.

yi 1

i NZJ-1

wi |NZk|-1i NZk

yi+wi 1

wi·vmin vi wi·vmin

1. My NZ1 fluxes are v2,b1,b3.

2. I will pick v2 to become zero at next solution: yv2=1 & yb1=yb3=0;

3. wv2 = 0; lets assume that wb1 and wb3 = 1.

4. v2 = 0 and the other fluxes can be between their normal vmin and vmax values.

wi |NZ1|-1 (2 2)i NZ1

AB

C 10

10

10

Solution 1:

AB

C 10

10

10

10

Solution 2:

1. Find non-zero fluxes (NZJ-

1) at current solution, J-1.

2. Pick at least one NZJ-1 flux to become zero at next solution (yi=1).

3. Make sure that the set of non-zero fluxes haven’t been visited at previous k iterations.

4. Constrain those selected fluxes to have zero flux.

5. Find solution & repeat.

yi 1

i NZJ-1

wi |NZk|-1i NZk

yi+wi 1

wi·vmin vi wi·vmin

wi |NZk|-1i NZk

yi+wi 1