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Lecture #19

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.

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Lecture #19

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  1. Lecture #19 Alternative (equivalent) Optima

  2. 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

  3. 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: cv Such that S v = b =0 LB  v UB

  4. Equivalent Optimal Solutions Exist:How can we find & characterize them? Projected Solution Space FBA Growth Rate Some Flux

  5. 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

  6. Method 1: FLUX VARIABILITY ANALYSIS (FVA)

  7. Example: flux variability analysis Unique Optimal Solution maximize 3w1 + w2 such that w1 + w2 ≤ 2 w1, w2 ≥ 0 maximize 3w1 + 3w2 such that w1 + w2 ≤ 2 w1, w2 ≥ 0 w2 w2 max(w2) from FVA Alternate optimal solutions: 3w1 + 3w2 = 6 Unique optimal solution 1 1 min(w2) from FVA w1 w1 1 1 min(w1) from FVA max(w1) from FVA Alternate Optimal Solutions

  8. Flux Variability Analysis: • 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

  9. Flux Variability Analysis (FVA): the concept • Determine what is possible based on external measurements and network stoichiometry i = metabolites j = reactions Objective: maximize & minimize vj (for all j) subject to (be in the space of optimal solutions): S ∙ v = 0 <c ∙ v> = Zopt Solve series of 2 linear optimization for each reactions j. Metabolic Eng, 2003. 5(4): p. 264-76.

  10. 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.

  11. Core E. coli Metabolism

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

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

  14. Flux Variability for optimal 3PG yield on Glucose Flux Value

  15. flux = 0 always flux varies between 0 and 0.5 flux = 1 always flux varies between 1 and 1.5

  16. 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).

  17. 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

  18. 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

  19. 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).

  20. FVA Final thoughts Pseudo code FVA(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:n c = (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

  21. Method 2: EXTREME PATHWAY ANALYSIS

  22. metabolic flux (v3) metabolic flux (v1) Convex Analysis Cellular Biology Metabolic Phenotype Particular Solution Capabilities of a Metabolic Genotype Convex Hull Unique Generating Vectors Independent Extreme Pathways metabolic flux (v2) Positive Combination of Extreme Pathways Flux Vector Metabolic Genotype to PhenotypeDefined within the context of convex analysis

  23. Example From the core E. coli model – pyruvate yields from glucose, aerobic Optimal equivalent pathways Suboptimal equivalent pathways Non-producing pathways without regulation with regulation

  24. Equivalent sets exponentially increase the number of ExPas Set 1 Set 2 … Set n Pathway of n equivalent sets of size 2 forms 2n extreme pathways.

  25. 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.

  26. 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 Bioengineering71: 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).

  27. 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

  28. Method 3: Enumerating all Optimal Solutions

  29. 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.

  30. Metabolic Network Example Reaction List Metabolic Map v1: A → B b2 v2: A → C v1 B v3: B  C b1 b1: → A A v3 b3 v2 b2: B → C b3: C → Maximize Z = c·v = b3 Such that S·v = 0 0 v1,v2,b1,b2,b3 10 -10 v3 10

  31. Solution 1: B A 10 C 10 10 Solution 2: 10 B A 10 10 C 10 Intuitive Description of Algorithm • Find one valid flux distribution (Solution 1) • Repeatedly solve for optimal again: • 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.

  32. Number of Alternate Optima for various Metabolic Objectives (aerobic) in core E. coli ≥ # of ExPas *Did not converge

  33. 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 J98(10):2072-2081 (2010).

  34. 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.

  35. 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

  36. 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. 904 genes 931 rxns All optimal solutions

  37. 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

  38. 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

  39. 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.

  40. 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)

  41. 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

  42. Appendix Details of enumeration algorithm

  43. 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.

  44. Find non-zero fluxes (NZJ-1) at current solution, J-1. Pick at least one NZJ-1 flux to become zero at next solution (yi=1). Make sure that the set of non-zero fluxes haven’t been visited at previous k iterations. Constrain those selected fluxes to have zero flux. Find solution & repeat. Solution 1: B A 10 C 10 10 Solution 2: 10 B A 10 10 C 10  yi  1 i NZJ-1 yi+wi  1  wi  |NZk|-1 i NZk wi·vmin  vi wi·vmin

  45.  wi  |NZ1|-1 Find non-zero fluxes (NZJ-1) at current solution, J-1. Pick at least one NZJ-1 flux to become zero at next solution (yi=1). Make sure that the set of non-zero fluxes haven’t been visited at previous k iterations. Constrain those selected fluxes to have zero flux. Find solution & repeat. Solution 1: B A 10 C 10 10  yi  1 i NZJ-1 • My NZ1 fluxes are v2,b1,b3. • I will pick v2 to become zero at next solution: yv2=1 & yb1=yb3=0; • wv2 = 0; lets assume that wb1 and wb3 = 1. • v2 = 0 and the other fluxes can be between their normal vmin and vmax values. yi+wi  1  wi  |NZk|-1  wi  |NZ1|-1 (2  2) i NZk i NZ1 wi·vmin  vi wi·vmin

  46. Find non-zero fluxes (NZJ-1) at current solution, J-1. Pick at least one NZJ-1 flux to become zero at next solution (yi=1). Make sure that the set of non-zero fluxes haven’t been visited at previous k iterations. Constrain those selected fluxes to have zero flux. Find solution & repeat. Solution 1: B A 10 C 10 10 Solution 2: 10 B A 10 10 C 10  yi  1 i NZJ-1 yi+wi  1 yi+wi  1  wi  |NZk|-1  wi  |NZk|-1 i NZk i NZk wi·vmin  vi wi·vmin

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