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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

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Lecture #23

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  1. Lecture #23 Varying Parameters

  2. 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

  3. One parameter ROBUSTNESS ANALYSIS

  4. Robustness Analysis: the concept • 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

  5. 2. Robustness to Gene Deletions and Enzyme Defects 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. p Mathematics Biotechnol Prog., 16: 927-939, (2000).

  6. 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.

  7. Some history THE ORIGINAL CORE E. COLI MODEL

  8. Analysis of oxygen uptake rate Appl. Env. Micro 59: 2465 (1993)

  9. anaerobic aerobic 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.

  10. Shadow prices: Interpret changes in optimal solutions Formate & Acetate, Secreted ($0 shadow prices); Ethanol is not ($0.002) Formate, Acetate, Ethanol are Secreted ($0 shadow prices)

  11. partially anaerobic aerobic Flux distributions for different levels (or phases) of oxygenation Acetate is Secreted ($0 shadow prices)

  12. Results • Optimality principles and network reconstruction used to predict over all phenotypic states • Phenotypic functions interpreted using an econometric approach, ie the shadow prices

  13. today THE CURRENT CORE E. COLI MODEL

  14. Growth on glucose: similar results as historical 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. I II III IV V 5 distinct growth phases

  15. 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 By-products secreted during anaerobic ATP production Solution becomes infeasible at O2 uptake above 6 O2/Glc

  16. Growth on glucose:line of optimality (LO) Glucose uptake fixed at -10, O2 uptake variable O2 uptake fixed at -17, glucose uptake variable Partially anaerobic

  17. Growth on Different Substrates:illustrates partial anaerobic growth potential O2 uptake fixed at -17

  18. 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

  19. Publications EXAMPLES FROM THE LITERATURE

  20. Robustness in iJE660 • Vary the activity of essential genes • Look at the consequences of over-expression Biotech Prog 16:927 (2000)

  21. Effect of proton balancing on growth rate: prediction of H+ secretion Genome Biology 4:R54 (2003)

  22. Two parameters PHENOTYPIC PHASE PLANES

  23. Robustness Analysis: Projection of PhPP for Maximum Growth rate vs. Succinate uptake Robustness Analysis: Projection of PhPP for Maximum Growth rate vs. O2 uptake Phenotypic Phase Plane (PhPP) Biomass Production Biomass Production Biomass Production O2 uptake Succinate uptake Succinate uptake O2 uptake Line of Optimality (LO) PhPP vs. Robustness

  24. Phenotypic Phase Planes 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. 2.4 Line of Optimality 0.4 Oxygen Uptake Rate 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] Phase Plane Isoclines Carbon Uptake Rate Key References Edwards, 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). 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.

  25. Historical data Edwards et. al., Nat Biotech., 19, 2001 Ibarra et. al., Nature, 420, 2002

  26. Experimental data: acetate

  27. Experimental data: succinate

  28. CHARACTERISTICS OF THE PHPP

  29. Single Growth condition Metabolic Phenotype A Metabolic Infeasible Steady State Phenotype B Metabolic Flux B {Shadow Price A} {Shadow Price B} Infeasible Steady State Metabolic Flux A 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

  30. - Shadow Prices and Isoclines Shadow Price Relative shadow prices

  31. - Shadow prices and isoclines Dual Substrate Limitation Single Substrate Limitation Uptake B a “Futile” Region Uptake A

  32. Features of Phase Planes • Infeasible regions: fluxes don’t balance • Regions of single substrate limitations (a = 0 or infinity) • Regions of dual substrate limitations (a < 0) • Futile regions (a >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

  33. Line of Optimality: Max. Yx/s LO Metabolic Phenotype 1 Infeasible Steady State Oxygen Uptake B Metabolic Phenotype 2 Infeasible Steady State Carbon Source Uptake Rate

  34. CORE E. COLI CACULLATIONS

  35. Core E. coli model examples Growth on acetate with O2 Line of optimality acetate limited growth O2 limited growth Infeasible region (no growth)

  36. Core E. coli model examples Growth on glucose with O2 Line of optimality excess O2 acetate secreted acetate and formate secreted acetate, formate, and ethanol secreted

  37. Core E. coli model examples Growth on fumarate with O2 Line of optimality High uptake rate needed for fully anaerobic growth

  38. For whole organisms PHPP AT THE GENOME-SCALE

  39. The H. influenzae Metabolic Phase Plane J. Biol. Chem. 274(15):17410 (1999)

  40. E. Coli PhPP on Glucose

  41. BMC Genomics 2004, 5:63

  42. BMC Genomics 2004, 5:63

  43. 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

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