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Solution Space? • In most cases lack of constraints provide a space of solutions • What can we do with this space? 1. Optimization methods (previous lesson) – May result in a single, unique solution – May still result in a (smaller) convex solution space 2. Explore alternative solutions in this space Lecture Outline 1. LP and MILP basic solution enumeration 2. Flux variability analysis (FVA) 3. Flux coupling Flux Variability Analysis • Determine for each reaction its range of possible flux (within feasible solutions) • Computed via 2 LP problems for each reaction (to find the lower and upper bounds) Flux Variability Analysis • For the E. coli metabolic network, 3% of the metabolic fluxes can vary and still allow for optimal biomass production on glucose • Assuming sub-optimal growth rate of above 95% of the maximal rate – up to 50% of the fluxes can vary! • This is a major issue with constraint-based modeling! • Various studies still ignore this and simply choose a single arbitrary FBA solution for their analysis Alternative MILP Solutions • Identify solutions with different integer values • The integer variables denoted yi and the number of reactions is M • Each “integer cut” excludes one previously found solution yj* • Which is equivalent to |yj* - yj|>0 Flux Coupling Analysis (FCA) • Used to check how pairs of fluxes affect one another • Done by calculating the minimum and maximum ratio between two fluxes • Transformation needed to make it a linear problem Types of coupling Identifying coupled reaction sets • A much higher percentage of reactions that are member of coupled sets in H. pylori (with the smaller network) compared to S. cerevisiae and E.coli • If the biomass production rate is fixed to its maximal rate, we get ~40% of the reactions coupled to the biomass production rate Alternative Optima: Hit and Run Sampling • • • • • Almaas, et. al, 2004 Based on a random walk inside the solution space polytope Choose an arbitrary solution Iteratively make a step in a random direction Bounce off the walls of the polytope in random directions 9 Alternative Optima: Uniform Random Sampling • Wiback, et. al, 2004 • The problem of uniform sampling a high-dimensional polytope is NP-Hard • Find a tight parallelepiped object that binds the polytope • Randomly sample solutions from the parallelepiped • Can be used to estimate the volume of the polytope 10 Biological Network Analysis: Regulation of Metabolism Tomer Shlomi Winter 2008 Lecture Outline 1. Transcriptional regulation 2. Steady-state Regulatory FBA (SR-FBA) 3. Regulatory FBA Transcriptional Regulation • RNA polymerase – protein machinery that transcribes genes • Transcription factors (TFs) bind to specific binding sites in the promoter region of a gene • After binding to DNA TFs either enhance (activator) or disrupt (repressor) RNA polymerase binding to DNA Transcriptional Regulatory Network • Nodes – transcription factors (TFs) and genes; • Edges – directed from transcription factor to the genes it regulates • Reflect the cell’s genetic regulatory circuitry • Derived through: ▲ Chromatin IP ▲ Microarrays S. cerevisiae 1062 TFs, X genes 1149 interactions 3. Steady-state Regulatory FBA (SR-FBA) Integrated Metabolic/Regulatory Models Genome-scale integrated model for E. coli (Covert 2004) • 1010 genes (104 TFs, 906 genes) • 817 proteins • 1083 reactions • The Extreme Pathways approach can’t work on such large-scale models • 16 The Steady-state Regulatory FBA Method • • • SR-FBA is an optimization method that finds a consistent pair of metabolic and regulatory steady-states Based on Mixed Integer Linear Programming Formulate the inter-dependency between the metabolic and regulatory state using linear equations g 0 1 1 v Regulatory state Metabolic state v2 v3 … … g1 = g2 AND NOT (g3) g3 = NOT g4 … v1 Stoichiometric matrix S·v = 0 vmin < v < vmax SR-FBA: Regulation → Metabolism • • • The activity of each reaction depends on the presence specific catalyzing enzymes For each reaction define a Boolean variable ri specifying whether the reaction can be catalyzed by enzymes available from the expressed genes Formulate the relation between the Boolean variable ri and the flux through reaction i g1 g2 g3 if (ri 0) then vi 0 Gene1 else i vi i Enzyme1 vi (1 ri ) i i i vi (1 ri ) i r1 Gene2 Gene3 Protein2 Protein3 OR Met1 Enzyme complex2 AND Met3 Met2 r1 = g1 OR (g2 AND g3) • • SR-FBA: Metabolism → Regulation The presence of certain metabolites activates/represses the activity of specific TFs For each such metabolite we define a Boolean variable mj specifying whether it is actively synthesized, which is used to formulate TF regulation equations TF2 = NOT(TF1) AND (MET3 OR TF3) if (vi 0) then m j 1 else m j 0 TF1 m j ( i ) vi TF2 TF3 Me1 Met3 Met2 Met4 m j ( i ) vi i 19 mj SR-FBA Formulation • Boolean variables – Regulatory state – g – Protein state – p – Reaction state – r – Reaction predicate - b Recursive formulation of regulatory logic as linear equations Formulation of Boolean G2R mapping Results: Validation of Expression and Flux Predictions • Prediction of expression state changes between aerobic and anaerobic conditions are in agreement with experimental data (p-value = 10-300) • Prediction of metabolic flux values in glucose medium are significantly correlated with measurements via NMR spectroscopy (spearman correlation 0.942) 21 The Functional Effects of Regulation on Metabolism • Metabolic constraints determine the activity of 45-51% of the genes depending of growth media (covering 57% of all genes) • The integrated model determines the activity of additional 13-20% of the genes (covering 36% of all genes) – 13-17% are directly regulated (via a TF) – 2-3% are indirectly regulated • The activity of the remaining 30% of the genes is undetermined 22 4. Regulatory FBA (rFBA) Regulatory Feedback • Many regulatory mechanisms cannot be described via steadystate description • Depends on – E synthesis rate – E degradation rate Dynamic FBA Profiles • Separation of time-scales – Transcriptional regulation: minutes – Metabolism: seconds • Divide experimental time to small steps • Regulatory changes are continuous across time intervals • Metabolic behavior is in steady-state within each timeinterval Δt1 Δt0 Regulatory state Metabolic state Metabolic state Δt2 Regulatory state Regulatory FBA • • Input: – Initial biomass, X0 – Initial extra-cellular concentrations So Method – Compute maximal metabolite uptake rates – Extra-cellular metabolite concentrations, Sc – Cell density (biomass), X – Growth rate, µ – Flux distribution, v – Gene expression state, g – Protein expression state, p – Apply FBA to compute a flux distribution, v, with maximal growth rate, µ (considering regulatory constraints, derived from protein exp. state p) – Compute new biomass: – Compute new extra-cellular concentrations: – Update gene expression state, g – Update protein expression state, p, based on protein synthesis and degradation constant Acetate re-utilization experiments