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 – Genome-scale computations – PhPPs and experimental design

ROBUSTNESS ANALYSIS One parameter

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

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

Shadow Prices (  i ) & Reduced Costs (  i ) Shadow Prices (  i ): – One for each constraint or metabolite –  i =dZ/db i –  i <0 means adding metabolite (ie. change b=0 to b<0) would increase Z. –  i >0 means removing metabolite (ie. change b=0 to b>0) would increase Z. Reduced Costs (  i ): – One for each variable or flux. – dZ/dv j (for zero fluxes) –  i < 0 means increasing flux (v j ) would reduce Z.

THE ORIGINAL CORE E. COLI MODEL Some history

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

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. anaerobicaerobic

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

Flux distributions for different levels (or phases) of oxygenation partially anaerobicaerobic Acetate is Secreted ($0 shadow prices)

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

THE CURRENT CORE E. COLI MODEL today

Growth on glucose: similar results as historical Glucose and O 2 uptake and byproduct secretion rates at different growth rates. Max O 2 uptake rate = -17. Byproduct secretion rates at different O 2 uptake rates. Glucose uptake rate = distinct growth phases I II III IV V

ATP production in core E. coli: Contrast with growth function Robustness analysis of ATP production from glucose Vary O 2 uptake from 0 to fully aerobic and compute maximum ATP production Solution becomes infeasible at O 2 uptake above 6 O 2 /Glc By-products secreted during anaerobic ATP production

Growth on glucose: line of optimality (LO) Glucose uptake fixed at -10, O 2 uptake variable O 2 uptake fixed at -17, glucose uptake variable Partially anaerobic

Growth on Different Substrates: illustrates partial anaerobic growth potential O 2 uptake fixed at -17

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

EXAMPLES FROM THE LITERATURE Publications

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

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

PHENOTYPIC PHASE PLANES Two parameters

PhPP vs. Robustness Phenotypic Phase Plane (PhPP) Robustness Analysis: Projection of PhPP for Maximum Growth rate vs. O 2 uptake Robustness Analysis: Projection of PhPP for Maximum Growth rate vs. Succinate uptake Line of Optimality (LO) O 2 uptake Succinate uptake Biomass Production O 2 uptake Succinate uptake

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. 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. 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] 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: (2001)."In silico predictions of Escherichi coli metabolic capabilities are consistent with experimental data" Edwards, J.S., Ramakrishna R., Palsson, B.Ø., “Characterizing the metabolic phenotype: A phenotype phase plane analysis",Biotechnology and Bioengineering, 77(1): pp (2002).Characterizing the metabolic phenotype: A phenotype phase plane analysis" 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: (2001)."Combining pathway analysis with flux balance analysis for the comprehensive study of metabolic systems" 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 (2002).Escherichia coli K-12 undergoes adaptive evolution to achieve in silico predicted optimal growth Carbon Uptake Rate Oxygen Uptake Rate Line of Optimality Isoclines Phase Plane Phenotypic Phase Planes

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

Experimental data: acetate

Experimental data: succinate

CHARACTERISTICS OF THE PHPP

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

Shadow Prices and Isoclines Shadow Price Relative shadow prices -

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

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

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

CORE E. COLI CACULLATIONS

Core E. coli model examples Growth on acetate with O 2 Line of optimality Infeasible region (no growth) acetate limited growth O 2 limited growth

Core E. coli model examples Growth on glucose with O 2 Line of optimality acetate, formate, and ethanol secreted acetate and formate secreted acetate secreted excess O 2

Core E. coli model examples Growth on fumarate with O 2 Line of optimality High uptake rate needed for fully anaerobic growth

PHPP AT THE GENOME-SCALE For whole organisms

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

E. Coli PhPP on Glucose

BMC Genomics 2004, 5:63

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