1. 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

2. Lecture Outline 1. LP and MILP basic solution enumeration 2. Flux variability analysis (FVA) 3. Flux coupling

3. 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)

4. 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

5. Alternative MILP Solutions Identify solutions with different integer values The integer variables denoted y i and the number of reactions is M Each “integer cut” excludes one previously found solution y j* Which is equivalent to |y j* - y j |>0

6. 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

7. Types of coupling

8. 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

9. 9 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

10. 10 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

11. Biological Network Analysis: Regulation of Metabolism Tomer Shlomi Winter 2008

12. Lecture Outline 1. Transcriptional regulation 2. Steady-state Regulatory FBA (SR-FBA) 3. Regulatory FBA

13. 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

14. 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: 1062 TFs, X genes 1149 interactions S. cerevisiae ▲ Chromatin IP ▲ Microarrays

15. 3. Steady-state Regulatory FBA (SR-FBA)

16. 16 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

17. 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 Regulatory state Metabolic state v v1v1 v2v2 v3v3 … g 0 1 1 … g 1 = g 2 AND NOT (g 3 ) g 3 = NOT g 4 … S·v = 0 v min < v < v max Stoichiometric matrix

18. SR-FBA: Regulation → Metabolism The activity of each reaction depends on the presence specific catalyzing enzymes For each reaction define a Boolean variable r i specifying whether the reaction can be catalyzed by enzymes available from the expressed genes Formulate the relation between the Boolean variable r i and the flux through reaction i Met1Met3 Met2 Gene2 Gene1Gene3 Protein2Protein3 Enzyme1 Enzyme complex2 AND OR ifthen else r1r1 r 1 = g 1 OR (g 2 AND g 3 ) g1g1 g2g2 g3g3

19. 19 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 m j specifying whether it is actively synthesized, which is used to formulate TF regulation equations Me1 Met2Met4 Met3 TF2TF3TF1 TF2 = NOT(TF1) AND (MET3 OR TF3) if then else mjmj

20. 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

21. 21 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)

22. 22 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

23. 4. Regulatory FBA (rFBA)

24. Regulatory Feedback Many regulatory mechanisms cannot be described via steady- state description Depends on –E synthesis rate –E degradation rate

25. 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 time- interval Δt0Δt0 Metabolic state Δt1Δt1 Regulatory state Metabolic state Δt2Δt2 Regulatory state

26. Regulatory FBA Input: –Initial biomass, X 0 –Initial extra-cellular concentrations S o Method –Compute maximal metabolite uptake rates –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 –Extra-cellular metabolite concentrations, S c –Cell density (biomass), X –Growth rate, µ –Flux distribution, v –Gene expression state, g –Protein expression state, p

27. Acetate re-utilization experiments