Metabolic network analysis Marcin Imielinski University of Pennsylvania March 14, 2007

Genome sequencing Gene expression profiling Proteomics High-throughput phenotyping

The systems biology vision. Integrate quantitative and high-throughput experimental data to gain insight into basic biology and pathophysiology of cells, tissues, and organisms. Exploit knowledge of intracellular networks for drug design. Engineer organisms, e.g. for salvaging waste, synthesizing fuel.

Modeling cellular metabolism Raw Materials Products

Extracellular Compartment System Boundary production transport core metabolism steady state Other cellular processes Modeling cellular metabolism Legend: = reaction = biochemical species = reaction i/o Nutrients Protein Synthesis DNA replication Membrane Assembly

Glucose Pyruvate Pathway example: glycolysis carbon dioxide, water, energy

Genome scale metabolic models Most comprehensive summary of the current knowledge regarding the genetics and biochemistry of an organism Integrate functional genomic associations between genes, proteins, and reactions into single model Models have been built for 100+ bacterial organisms, yeast, human mitochondrion, liver cell, et al. Average model contains reactions and 300 – 1000 biochemical species. Sequenced Genome Functional Annotations Predicted Genes and Proteins Genome Scale Model

Genome Scale Metabolic Modeling - Approach Many parameters unknown on genome scale  kinetic constants  feedback regulation  cooperativity What is known  reaction stoichiometry  thermodynamic constraints  upper bounds on some reaction rates Constraints based approach:  start with minimal stoichiometric model  populate with constraints  restrict the range of feasible cellular behavior Adapted from Famili et al. (PNAS 2003)

stoichiometry matrix of all reactions in the system Entry S ij corresponds to the number of metabolite i produced in one unit of flux through reaction j v is a flux configuration of the network, which has implicit and nonlinear dependency on x S contains “true” reactions (e.g. enzyme catalyzed biochemical transformations) and “exchange fluxes” (representing exchange of material across system boundary and growth based dilution x= Sv Constraints Based Metabolic Modeling stoichiometry matrix (dimensionless) vector of rate of change of species concentrations mol/L/s vector of reaction rates (fluxes) mol/L/s

Example S= 1.0 A 1.0 E  1.0 B 1.0 F 1.0 C 1.0 F  1.0 D 1.0 E 1.0 B 1.0 D  1.0 G 123 A00 B10 C0 0 D01 E 10 F1 0 G001

Example S= 123 A00 B10 C0 0 D01 E 10 F1 0 G001 system boundary biochemical species reaction reaction I/O Legend

Example 123 A00 B10 C0 0 D01 E 10 F1 0 G001 system boundary = x v biochemical species reaction reaction I/O Legend

Example – flux configuration 123 A00 B10 C0 0 D01 E 10 F1 0 G001 system boundary = steady state species consumed species produced species reaction (inactive) reaction I/O Legend reaction (active)

Example – flux configuration 123 A00 B10 C0 0 D01 E 10 F1 0 G001 system boundary = steady state species consumed species produced species reaction (inactive) reaction I/O Legend reaction (active)

Example – flux configuration 123 A00 B10 C0 0 D01 E 10 F1 0 G001 system boundary = steady state species consumed species produced species reaction (inactive) reaction I/O Legend reaction (active)

Constraints Based Metabolic Modeling stoichiometry matrix of all reactions in the system quasi-steady state assumption  biochemical reactions are fast with respect to regulatory and environmental changes stoichiometry matrix (dimensionless) vector of rate of change of species concentrations mol/L/s vector of reaction rates (fluxes) mol/L/s

Example – steady state 123 A00 B10 C0 0 D01 E 10 F1 0 G001 system boundary = 0 v biochemical species reaction reaction I/O Legend

Example – steady state 123 A00 B10 C0 0 D01 E 10 F1 0 G001 system boundary = 0 0

Example – steady state A00100 B C D E F G00100 system boundary = 0 v biochemical species reaction reaction I/O Legend

Example – steady state A00100 B C D E F G00100 system boundary = steady state species consumed species produced species reaction (inactive) reaction I/O Legend reaction (active)

Example – expanded system boundary A00100 B C D E F G00100 A ext C ext G ext old system boundary = C ext A ext G ext expanded system boundary steady state species consumed species produced species reaction (inactive) reaction I/O Legend reaction (active)

Constraints Based Metabolic Modeling stoichiometry matrix (dimensionless) vector of rate of change of species concentrations mol/L/s vector of reaction rates (fluxes) mol/L/s stoichiometry matrix of all reactions in the system quasi-steady state assumption irreversibility constraints Irreversibility constraints

The Flux Cone chull (p 1, …, p q ) (extreme pathway decomposition) (polyhedral flux cone)

Minimal functional units of metabolism  i.e. “non-decomposable”  Network-based correlate of a biochemist’s notion of a “pathway” or “module” Uncover systems-level functional roles for individual genes / enzymes Capture flexibility of metabolism with respect to a particular objective Reactions participating in extreme pathways may be co-regulated Useful for drug design and metabolic engineering. Extreme pathways Wiback et al, Biophys J 2002

Ru5P FP F6P GAP R5P Extreme pathways: example system boundary biochemical species reaction reaction I/O Legend

system boundary Extreme pathway Extreme pathway Extreme pathway Extreme pathway Extreme pathway Extreme pathway Extreme pathways: example biochemical species reaction reaction I/O Legend

system boundary Extreme pathway Extreme pathway Extreme pathway Extreme pathway Extreme pathway Extreme pathway Extreme pathways: example biochemical species reaction reaction I/O Legend

system boundary Extreme pathway Extreme pathway Extreme pathway Extreme pathway Extreme pathway Extreme pathway Extreme pathways: example biochemical species reaction reaction I/O Legend

system boundary Extreme pathway Extreme pathway Extreme pathway Extreme pathway Extreme pathway Extreme pathway Extreme pathways: example biochemical species reaction reaction I/O Legend

system boundary Extreme pathway Extreme pathway Extreme pathway Extreme pathway Extreme pathway Extreme pathway Extreme pathways: example biochemical species reaction reaction I/O Legend

Extreme pathway Extreme pathway Extreme pathway Extreme pathway Extreme pathway Extreme pathway Extreme pathways: example system boundary biochemical species reaction reaction I/O Legend

system boundary Extreme pathway Extreme pathway Extreme pathway Extreme pathway Extreme pathway Extreme pathway Extreme pathways: example biochemical species reaction reaction I/O Legend

Extreme pathway Extreme pathway Extreme pathway Extreme pathway Extreme pathway Extreme pathway Objective: disable output of GAP (“exchange reaction” 5) system boundary biochemical species reaction reaction I/O Legend Knockout design

system boundary Extreme pathway Extreme pathway Extreme pathway Extreme pathway Extreme pathway Extreme pathway biochemical species reaction reaction I/O Legend Knockout design One solution: knockout reactions 1 and 4 (i.e. constrain v 1 =0 and v 4 =0)

system boundary Extreme pathway Extreme pathway Extreme pathway Extreme pathway Extreme pathway Extreme pathway biochemical species reaction reaction I/O Legend Knockout design One solution: knockout reactions 1 and 4 (i.e. constrain v 1 =0 and v 4 =0)

system boundary Extreme pathway Extreme pathway Extreme pathway Extreme pathway Extreme pathway Extreme pathway biochemical species reaction reaction I/O Legend Knockout design One solution: knockout reactions 1 and 4 (i.e. constrain v 1 =0 and v 4 =0)

system boundary Extreme pathway Extreme pathway Extreme pathway Extreme pathway Extreme pathway Extreme pathway One solution: knockout reactions 1 and 4 (i.e. constrain v 1 =0 and v 4 =0) biochemical species reaction reaction I/O Legend Knockout design

Extreme pathway Extreme pathway Extreme pathway Extreme pathway Extreme pathway Extreme pathway Objective: couple export of GAP to export of F6P system boundary biochemical species reaction reaction I/O Legend Knockout design

Extreme pathway Extreme pathway Extreme pathway Extreme pathway Extreme pathway Extreme pathway system boundary One solution: knockout reaction 2 (constrain v 2 = 0) biochemical species reaction reaction I/O Legend Knockout design

Tableau algorithm for EP computatioon Iteration 0: nonnegative orthant Extreme rays of K 0 = Euclidean basis vectors 1 … n Iteration i+1: Given and extreme rays of K i Compute extreme rays of K i+1

v1v1 v2v2 v3v3 Extreme ray of K i Tableau algorithm for EP computatioon

v1v1 v2v2 v3v3 S i+1 v = 0 Tableau algorithm for EP computatioon

v1v1 v2v2 v3v3 S i+1 v = 0 Sort extreme rays of K i with regards to which are on (+) side, (-) side, and inside hyperplane S i+1 v = 0 Tableau algorithm for EP computatioon

v1v1 v2v2 v3v3 S i+1 v = 0 Extreme rays of K i that are already in S i v=0 are automatically extreme rays of K i+1 Tableau algorithm for EP computatioon

v1v1 v2v2 v3v3 S i+1 v = 0 Combine pairs of extreme rays of K i that are on opposite sides of S i+1 v = 0 Tableau algorithm for EP computatioon

v1v1 v2v2 v3v3 S i+1 v = 0 From this new ray collection remove rays that are non-extreme. Tableau algorithm for EP computatioon

v1v1 v2v2 v3v3 S i+1 v = 0 Non-extreme rays are r for which there exists an r* in the collection such that NZ(r*) is a subset of NZ(r) Tableau algorithm for EP computatioon

 Applications Hemophilus influenzae (Schilling et al, J Theor Biol 2000) Human red blood cell (Wiback et al, Biophys J 2002) Helicobacter pylori (Schilling et al, J Bact 2002)  Limitations Combinatorial explosion of extreme rays Computational complexity of determining extremality Only directly applicable to “medium sized” networks (e.g. 200 species and 300 reactions)  Variants Elementary flux modes (Schuster Nat Biotech 2000) Minimal generating set (Wagner Biophys J 2005)  Approximate alternatives: Flux coupling analysis (Burgard et al Genome Res 2004) Sampling of flux cone (Wiback et al J Theor Biol 2004) Applications of network based pathway analysis

Flux Balance Analysis (Palsson et al.) Supplement metabolic network with a “biomass reaction” which consumes biomass substrates in ratios specified by chemical composition analysis of the cell. Model growth as flux through biomass reaction at steady state Use linear programming to predict optimum growth under a given set of mutations and nutrient conditions. Nutrients Biomass Edwards et al Nat Biotech 2001

0 ≤ v ≤ u vector of upper bounds corresponding to maximum rates of metabolic reactions 0= S | b Flux Balance Analysis: formulation stoichiometry matrix of metabolic network (dimensionless) vector of rate of change of species concentrations mol/L/s vector of metabolic fluxes (mol/L/s) biomass reaction (dimensionless) biomass flux (scalar) (mol/L/s) objective: maximize growth rate λ s.t. above constraints vλvλ 0 ≤ λ Nutrients Biomass

Modeling E. coli growth using FBA (Edwards et al Nat Biotech 2001) E coli model with 436 metabolites and 720 reactions Found that in vivo growth matched FBA- predicted optimal growth on minimal nutrient media employing acetate and succinate as carbon sources.

Modeling E. coli growth using FBA (Ibarra et al Nature 2001) In vivo growth was sub-optimal under glycerol However following over 40 days of culture and 700 generations of cell divisions, E. coli adaptively evolved to achieve optimum predicted growth rate

Modeling E. coli mutants using FBA Edwards et al 2004 E. coli model 436 species x 720 reactions  compared FBA predictions to published data on 36 E. coli gene deletion mutants in 4 nutrient media.  68 of 79 mutants agreed (qualitatively) between “simulation” and experiment. Covert et al 2004 E. coli model 761 species x 931 reactions  Compared FBA predictions to 13,750 mutant growth experiments in different nutrient media – gene deletion combination  Found 78.7% agreement Nutrients Biomass 0 ≤ v ≤ u 0= S | b 0 ≤ λ vλvλ maximize λ s.t. u i =0

FBA: concerns and limitations Assumes that a cell culture is optimized for growth.  Even bacteria like to do other things than just grow.  “Higher organisms” have even more complex “objectives” Even if we allow that a wild type organism is optimized for growth (because of years of evolution) a mutant may have difficulty “finding” the global optimum.  e.g. maybe a mutant bacteria will want to find the “closest” feasible state. What about alternative optima?  Optimal manifolds of these LPs are high-dimensional polyhedral sets. How will gene regulation influence the optimum?  Simple version: regulation will alter the upper bound constraints (u) on the fluxes.  Complicated version: modeling the interaction of metabolism and gene regulation will require including parameters and nonlinearities.

FBA: concerns and limitations Assumes that a cell culture is optimized for growth.  Even bacteria like to do other things than just grow.  “Higher organisms” have even more complex “objectives” Even if we allow that a wild type organism is optimized for growth (because of years of evolution) a mutant may have difficulty “finding” the global optimum.  e.g. maybe a mutant bacteria will want to find the “closest” feasible state. What about alternative optima?  Optimal manifolds of these LPs are high-dimensional polyhedral sets. How will gene regulation influence the optimum?  Simple version: regulation will alter the upper bound constraints (u) on the fluxes.  Complicated version: modeling the interaction of metabolism and gene regulation will require including parameters and nonlinearities.

Minimization of metabolic adjustment (MOMA) Segre et al PNAS 2002 Alternative method to FBA for computing mutant growth rates. Hypothesize that mutants will want to settle “close” to the wild type flux configuration. Find mutant flux distribution v that minimizes Euclidean distance to wild type growth state v wt Formulate as QP v wt can be obtained experimentally or computed using FBA. Segre et al PNAS 2002

Regulatory on-off minimization (ROOM) Shlomi et al PNAS 2005 Also hypothesize that mutants will want to be “close” to wild type flux configuration. However measure distance as the number of reactions whose flux bounds would have to be (significantly) changed from wild type Formulated as a MILP, with objective to minimize the number of reactions that need to be changed from wild type v wt (obtained using FBA). Performance on test data set of mutants: ROOM > FBA >> MOMA (ROOM has fewer false negatives) Shlomi et al PNAS 2005 MOMA wt (FBA) ROOM “biomass”

Review Stoichiometry matrix and constraints-based metabolic modeling Extreme pathway analysis Flux balance analysis Variants on FBA: MOMA and ROOM

Other topics not covered today Incorporating gene regulation to FBA  Covert et al Nature 2004 Analyzing alternative optima in FBA  Mahadevan et al Metab Eng 2003 Sampling the feasible flux region  Almaas et al Nature 2004  Wiback et al J Theor Biol 2004 Applying FBA to understand evolution  Papp et al Nature 2004  Pal et al Nat Genetics 2005 Modeling thermodynamic constraints  Beard et al J Theor Biol 2004  Qian et al Biophys Chem 2005 Conservation laws in metabolic networks  Famili et al Biophys J 2003  Imielinski et al Biophys J 2006