Structural analysis of metabolic network models Leonid Chindelevtich Simon Fraser University Wednesday, April 19, 2017

Metabolic network models Example: a network for Escherichia coli

Constraint-based models Stoichiometric matrix Allowed fluxes: {v: Sv = 0 and vI ≥ 0} I: irreversible reactions Toy example taken from Klamt, S and Gilles, E. Bioinformatics (2004).

Blocked reactions: GSMN-TB Beste et al. GSMN-TB: a web-based genome-scale network model of Mycobacterium tuberculosis metabolism. Genome Biology 2007, 8:R89.

Deleting blocked reactions: consistency issues

Trying out different : reproducibility issues growth no growth Thiele et al. Expanded metabolic reconstruction of Helicobacter pylori (iIT341 GSM/GPR): an in silico genome-scale characterization of single and double-deletion mutants.

Our approach: exact arithmetic maximize x1 - x2 subject to x1 - x2 ≤ -1 - x1 + x2 ≤ -1 x1≥ 0, x2 ≥ 0 maximize x1 - x2 subject to x1 - x2 ≤ -1 -1.000001 x1 + x2 ≤ -1 x1≥ 0, x2 ≥ 0 The feasible region is empty, so no solution. A small perturbation makes this feasible! Chindelevitch et al. Metabolic Network Analysis Demystified. RECOMB 2012.

Concept: Elementary flux mode

Concept: Elementary flux mode

Dual concept: minimal cut set Size = 1: essential reaction Size = 2: synthetic lethal pair

Constraints define cut sets Theorem: Let S be the stoichiometric matrix of a fully irreversible network. Then X is a (minimal) cut set for reaction j iff there exists a v in Row(S) such that R-(v) = X and v has a positive entry in position j (and X is minimal) Summing the last 3 rows gives -[0,0,0,0,-1,-1,0,0,1] Hence - v5 - v6 + v9 = 0 so if v5 = v6 = 0, then v9 = 0. Chindelevitch et al. An exact arithmetic toolbox for a consistent and reproducible structural analysis of metabolic network models. Nature Communications, in press

Classifying blocked reactions Definition: reaction j is topology-blocked if it contains a unique metabolite, possibly after other topology-blocked reactions are deleted. Detection: iterative graph traversal. Definition: reaction j is stoichiometry-blocked if condition Sv = 0 implies vj = 0. Detection: Gauss-Jordan reduction. Definition: reaction j is irreversibility-blocked if conditions Sv = 0 and vI ≥ 0 imply vj = 0. Detection: linear program. Definition: reaction j is semi-blocked if conditions Sv = 0 and vI ≥ 0 imply vj ≥ 0 or vj ≤ 0. Detection: linear program.

An unexpected simplification Theorem. To eliminate all the blocked reactions in a general metabolic network S, it suffices to identify the irreversibility-blocked irreversible reactions of S, then identify the stoichiometry-blocked reactions remaining after those are deleted and delete them.

An unexpected simplification Proof. After all the irreversibility-blocked irreversible reactions are removed from S, all remaining irreversibility-blocked reactions are stoichiometry-blocked. Indeed, if reaction i is blocked, then neither of its directions is feasible, so by Farkas’ lemma we get: Adding them together yields: However, none of the remaining irreversible reactions are blocked, so all the coefficients vanish and reaction i is indeed stoichiometry-blocked, completing the proof.

With a little bit more work… Theorem. If S is the stoichiometric matrix of a network with no blocked reactions, then reactions j and k are constrained to have proportional fluxes (form an enzyme subset) iff the corresponding rows in the null space matrix of S are proportional. Proof sketch. Apply the approach from the previous proof to the composite flux constraint to deduce, by a similar reasoning, that

Blocked growth in silico Before correcting “typos”, 44/89 models are blocked; after correcting, 33/89 are blocked! Mongoose can correct the rest automatically. Source: UCSD Systems Biology website, http://gcrg.ucsd.edu/InSilicoOrganisms

Mongoose: key contributions A complete structural analysis of a genome-scale metabolic network in exact arithmetic. First robust and reproducible approach for performing metabolic network model analysis. Parsed over 100 existing metabolic models and resolved two important open problems. Next steps: help correct the issues Mongoose identified and support network reconstruction.

Future work Help correct the issues Mongoose identified. Support metabolic network reconstruction. Integrate metabolism with gene regulation. mongoose.csail.mit.edu