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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. - PowerPoint PPT Presentation
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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