1. The Optimal Metabolic Network Identification Paula Jouhten Seminar on Computational Systems Biology 21.02.2007

2. Introduction The capability to perform biochemical conversions is encoded in the genome Genome-scale metabolic network models Gene annotation information often incomplete Cell function is regulated on different levels What is the active set of reactions in an organism under specific conditions?

3. Constraint-based models Genome-scale metabolic network models for micro-organisms (Escherichia coli, Saccharomyces cerevisiae,...) Enzyme-metabolite connectivities Stoichiometric models Reaction stoichiometry specifies the reactants and their molar ratios a*metabolite1 + b*metabolite2 -> c*metabolite3 + d*metabolite4

4. Feasible flux distributions Metabolic flux = a rate at which material is processed through a reaction (mol/h), reaction rate Fluxome, flux distribution Stoichiometries define a feasible flux distribution solution space

5. Additional constraints Additional constraints are included as linear equations or inequalities Steady state: the metabolite pool sizes and the fluxes are constant Reaction capacity: upper bound for a reaction Reaction reversibility Measurements

6. Metabolic flux analysis Determination of the metabolic flux distribution Intracellular fluxes cannot be measured directly Stoichiometric model N: q x m Input data -> extracellular fluxes Steady-state assumption -> a homogenous system of linear mass balance equations Additional constraints: v i < v max A(ext)B(ext)P(ext)E(ext) ACP B DE v1 v2v3 v4 v5 v9 v6 v10 v7 v8 1 0 0 0 -1 -1 -1 0 0 0 0 1 0 0 1 0 0 -1 -1 0 0 0 0 0 0 1 0 1 0 -1 0 0 0 0 0 0 1 0 0 -1 = N 0 0 0 -1 0 0 0 0 0 1 0 0 -1 0 0 0 0 0 1 1

7. Example network A(ext)B(ext)P(ext)E(ext) ACP B DE v1 v2v3 v4 v5 v9 v6 v10 v7 v8 REV = {v2, v8} IRR = {v1, v3, v4, v5, v6, v7, v9, v10} 1 0 0 0 -1 -1 -1 0 0 0 0 1 0 0 1 0 0 -1 -1 0 0 0 0 0 0 1 0 1 0 -1 0 0 0 0 0 0 1 0 0 -1 = N 0 0 0 -1 0 0 0 0 0 1 0 0 -1 0 0 0 0 0 1 1 Steady state: Nv = 0 Flux constraints: Capacity Reversibility Measurements Steady state mass balance equations: A: v1 -v5 -v6 -v7 = 0 B: v2 f - v2 b +v5 -v8 f +v8 b -v9 = 0 C: v6 +v8 f -v8 b -v10 = 0...

8. Underdetermined systems Determined system? redundant system? Metabolism contains cycles etc -> the system is usually underdetermined Additional experimental constraints from isotopic-tracer experiments (carbon-13 labelling) Analysis of the feasible solution space Optimal solution

9. Flux balance analysis (FBA) Solely based on a constraint-based model and linear optimisation Objective function: maximising growth, ATP production,... Stoichiometry of growth: macromolecular composition of cell biomass Not all organisms optimise for growth subject to

10. Stoichiometry of growth Macromolecular composition of a cell can be determined experimentally Macromolecular composition is dependent on the growth conditions Macromolecule compositions? Constituent synthesis routes dependent on the conditions?

11. Optimal Metabolic Network Identification Model predictions and experimental data do not always agree (growth rate, fluxes) Errors in the model structure: gaps, conditionally inactive or down-regulated reactions, incorrect reaction mechanisms What is the active set of reactions (the best agreement between the model predictions and the experimental data) in an organism under specific conditions?

12. Inner problem solves the FBA for the particular networks structure Outer problem searches for an optimal network structure Bilevel-optimisation approach

13. Bilevel formulation Subject to minimisation of a weighted distance between the observed and predicted flux distributions optimal flux distribution given the constraints and y (the set of active reactions) K allowed reaction deletions y is a binary variable

14. Formulation as a MILP where Linear inner problem -> duality theory Inner problem is converted to a set of equalities and inequalities Alternative optimal flux vectors Searching for all the different active sets of reactions resulting in the same prediction

15. Application to evolved E. coli knock-out strains Knock-out strains with lower than optimal growth rates Transcriptional profiling 2-4 reaction deletions required for significant improvement of model predictions Regulation?