Integration of abduction and induction in biological networks

Integration of abduction and induction in biological networks using CF-induction Yoshitaka Yamamoto Graduate University for Advanced Studies Tokyo, Japan. Andrei Doncescu LAAS-CNRS Toulouse, France. Katsumi Inoue National Institute of Informatics Tokyo, Japan. FJ’ 07

Our goal • Modeling of biological systems: – Explain and predict the metabolic pathway into the cell – Generic Model: • Saccharomyces Cerevisiae • E-coli – Inductive/Abductive Logic Programming: can explain the biological knowledge

Outline Logical setting of abduction and induction CF-induction (CFI) Consequence finding Procedure of CF-induction Features of CF-induction Inhibition in metabolic networks Simplification of metabolic networks How enzymes work Effect on toxins Prediction for inhibition in metabolic networks Integration of abduction and induction on the inhibitory effect using CFI System demonstration Conclusion and future work 3

Abduction and Induction: Logical Framework Input: – B : background theory. – E : (positive) examples / observations. Output: l H : hypothesis satisfying that B – B ∧ H ╞═ E – B ∧ H is consistent. E Inverse Entailment (IE) ILP machine H Computing a hypothesis H can be done deductively by: B ∧ ￢E ╞═ ￢H We use a consequence finding technique for (IE) computation.

Consequence finding Given an axiom set, the task of consequence finding is to find out some theorems of interest. • Theorems to find out are not given in an explicit way, but are characterized by some properties. Restricted consequence finding – How to find only interesting conclusions? [Inoue 91] Production field and characteristic clauses Production field P = <L, Cond > L : the set of literals to be collected – Cond : the condition to be satisfied (e. g. length) Th. P(Σ) : the clauses entailed by Σ which belong to P. Characteristic clause C of Σ (wrt P ): C belongs to Th. P(Σ) ; – no other clause in Th. P(Σ) subsumes C. Carc(Σ, P) = μTh. P(Σ), where μ represents “subsumption-minimal”.

IE for Abduction --- SOLAR (Nabeshima, Iwanuma & Inoue 2003) • • • B: full clausal theory E: conjunction of literals (￢E is a clause) H: conjunctions of literals (￢H is a clause) Example: graph completion problem – pathway finding Find an arc which enables a path from a to d. Axioms: [￢node(X), ￢node(Y), ￢arc(X, Y), path(X, Y)]) [￢node(X), ￢node(Y), ￢node(Z), ￢arc(X, Y), ￢path(Y, Z), path(X, Z)]. [node(a)]. [node(b)]. [node(c)]. [node(d)]. [arc(a, b)]. [arc(c, d)]. Negated Observation: [￢path(a, d)]. a c b d Production_field: [￢arc(_, _)]. SOLAR outputs four consequences: [￢arc(a, d)] , [￢arc(a, c)], [￢arc(b, d)], [￢arc(b, c)]

IE for Induction • CF-induction (Inoue 2004: Yamamoto, Ray & Inoue 2007) • fc-HAIL (Inoue & Ray 2007) B, E, H: full clausal theory • Note: CF-induction is the only existing ILP system that is complete for full clausal theories.

Principle of CF-induction B ∧ ￢E ⊨ ￢H (IE) ⇔ B ∧ ￢E ⊨ Carc(B ∧ ￢E, P) ⊨ ￢H. CC ⊨ ￢H where CC ⊆ Instances(Carc(B ∧ ￢E, P)). H ⊨ F where F is ￢CC in CNF. n. Algorithm 1. Compute Carc(B ∧ ￢E , P). 2. Construct a bridge formula CC. 3. Convert ￢CC into CNF F. 4. Generalize F to H such that B ∧ H is consistent; 1. H is Skolem-free. ＊Generalization H ⊨F － inverse Skolemization － anti-instantiation － dropping literals from clauses － addition of clauses － inverse resolution － Plotkin’s least generalization

Outline Logical Setting of Abduction and Induction CF-induction (CFI) Consequence finding Procedure of CF-induction Features of CF-induction Inhibition in metabolic networks Simplification of metabolic networks How enzymes work Effect on toxins Prediction for inhibition in metabolic networks Integration abduction and induction on the inhibitory effect using CFI System demonstration 9 Conclusion and future work

Simplification of metabolic networks Metabolic pathway: sequences of enzyme-catalyzed reaction steps, converting substrates to a variety of products to meet the needs of the cell. Mono-molecular enzymes catalyzed reactions: mediated by enzymes—proteins that encourage a chemical change. • E : enzyme, S : substrate, k 1 k 2 E+S ES E + P P : product, k-1 k-2 ES : complex, k : the constant the rate of a Enzymes: accelerate the rate of a chemical reaction of chemical reaction. by up to three orders of magnitude S E ES E P

How enzymes work Activity of an enzyme: - the rate of the chemical reaction catalyzed by the enzyme. 1 unit (U) ≡ the amount of the enzyme for changing the substrate whose amount is 1 μmol to the product over one minute. proportionate to the amount of the enzyme. Activity Concentration of substrate E+S k 1 k-1 ES k 2 E+P k-2 V = k 2 [ES] - k-2 [E][P] Concentration o Enzyme Time [T] Michaelis-Menten Reaction: - V Vmax the relation between the activity of an enzyme and the concentration of a substrate - at steady state [S] V = Vmax Km + [S] 11 [S]

Effect on toxins There exists chemical compounds (inhibitors) which control activities of enzymes. E S S I E P E I I S Higher the concentration of a inhibitor is, lower the activity of the enzyme controlled by the inhibitor becomes. Activity —: without inhibitor —: with inhibitor Concentration of 12 substrate

Logical modeling of inhibition [Tamaddoni-Nezda et al 2006] Toxin Inhibited concentration(P, down) ← reaction(S, Enz, P), inhibited(Enz, S, P). Enz S P Not inhibited Toxin S concentration(P, down) ← reaction(S, Enz, P), ￢inhibited(Enz, S, P), concentration(S, down). Enz S Not inhibited concentration(S, up) ← reaction(S, Enz, P), inhibited(Enz, S, P). P Enz Toxin concentration(P, up) ← reaction(S, Enz, P), ￢inhibited(Enz, S, P), concentration(S, up). P 13

Prediction for inhibitory effect of a toxin The goal -Finding inhibitions of a metabolic pathway Our approach -Using IE for abduction Examples E : changes (up or down) of concentrations of metabolites in treated cases (injected with a toxin) • Background Theory B : - chemical reactions in a metabolic networks - four clauses concerning the inhibitory effect of a toxin • Hypothesis H : a conjunction of literals whose predicate is “inhibition” 14

Example 1/2 2. 6. 1. 39; 4. 2. 1. 36; . . . 1. 2. 1. 31; 1. 5. 1. 7; . . . l-2 -aminoadipate 2 -oxe-glutarate isocitrate succinate fumarate hippurate nmnd trans-aconitate taurine 4. 3. 2. 1 nmna l-as citrulline arginine ornithine formate urea formaldehyde sarcosine methylamine tmao acryloyl-coa beta-alanine lactate 3. 6. 3. 3 creatinine creatine l-lysine

Example 2/2 2. 6. 1. 39; 4. 2. 1. 36; . . . 1. 2. 1. 31; 1. 5. 1. 7; . . . l-2 -aminoadipate 2 -oxe-glutarate isocitrate succinate fumarate hippurate nmnd trans-aconitate taurine 4. 3. 2. 1 nmna l-as citrulline arginine ornithine formate urea formaldehyde sarcosine methylamine tmao acryloyl-coa beta-alanine lactate 3. 6. 3. 3 creatinine creatine l-lysine

Outline Logical Setting of Abduction and Induction CF-induction (CFI) Consequence finding Procedure of CF-induction Features of CF-induction Inhibition in metabolic networks Simplification of metabolic networks How enzymes work Effect on toxins Prediction for inhibition in metabolic networks Integration abduction and induction on the inhibitory effect using CFI System demonstration 17 Conclusion and future work

Prediction for intracellular fluxes Goals -Predicting the concentration of metabolites intracellular -Discovering inductive rules augmenting incomplete background theory Our Approaches - Using CF-induction Examples E : changes (up or down) of concentrations of metabolites extracelluar • Background theory B : - chemical reactions in a metabolic networks - two clauses concerning the inhibitory effect • Hypothesis H : - a clausal theory which consists of both lierals whose predicate is “inhibition” and clauses corresponding to inductive rules

Metabolite Balancing • Intracellular fluxes are determined as a function of the measurable extracellular fluxes using a stoichiometric model for major intracellular reactions and applying a mass balance around each intracellular metabolite. v 1, v 2, v 3+, v 3 -, v 4 : unknown fluxes at the steady state. r. A, r. C, r. D, r. E : metabolite extracellular accumulation rate. 19

Example 1: B: concentration(a, up). reaction(a, b). reaction(b, d). reaction(d, e). reaction(e, c). reaction(c, b). reaction(b, c). ￢concentration(X, up) ← concentration(X, down) ← reaction(Y, X), ￢inhibited(Y, X), concentration(Y, down). Y X concentration(X, up) ← concentration(Y, up), reaction(Y, X), reaction(X, Z), ￢inhibited(Y, X), inhibited(X, Z). E: Y X Z concentration(d, up). concentration(e, down). concentration(c, down).

Example 1: outputs of CF-induction H 1 : ￢inhibited(a, b). inhibited(b, c). ￢inhibited(e, c). inhibited(d, e). ￢inhibited(b, d). concentration(e, down) ← inhibited(d, e), ￢inhibited(e, c). d e c H 2 : ￢inhibited(a, b). inhibited(b, c). ￢inhibited(b, d). inhibited(d, e). concentration(X, down) ← concentration(Y, up), inhibited(Y, X). Y X

Example 2: the real metabolic pathway (Pyruvate) B: reaction(pyruvate, acetylcoa). Glucose reaction(pyruvate, acetaldehide). reaction(glucose, glucosep). Glucose-P reaction(glucosep, pyruvate). reaction(acetaldehide, acetate). reaction(acetate, acetylcoa). reaction(acetaldehide, ethanol). concentration(glucose, up). Pyruvate Acetaldehide EC 1. 2. 4. 1 EC 1. 2. 1. 10 Acetylcoa terminal(ethanol). blocked(X)←reaction(X, Z), inhibited(X, Z). blocked(X)←terminal(X). EC 1. 1 EC 4. 1. 1. 1 X Acetate Z X concentration(X, up) ←reaction(Y, X), ￢inhibited(Y, X), blocked(X). E : concentration(ethanol, up). concentration(pyruvate, up). Ethanol

Example 2: outputs of CF-induction H 2: H 1: ￢Inhibited(glucosep, pyruvate). ￢inhibited(acetaldehide, ethanol). inhibited(pyruvate, acetylcoa). Glucose ￢inhibited(glucose, glucosep) ￢Inhibited(glucosep, pyruvate). ￢inhibited(acetaldehide, ethanol). ￢inhibited(pyruvate, acetaldehide). concentration(Y, up)← ￢inhibited(X, Y), concentration(X, up). Glucose-P Glucose Pyruvate Acetaldehide Ethanol Glucose-P Acetylcoa Acetate Pyruvate Acetylcoa Acetaldehide Acetate Ethanol

B: concentration(a, up). reaction(a, b). reaction(b, d). reaction(d, e). reaction(e, c). reaction(c, b). reaction(b, c). ￢concentration(X, up) ← concentration(X, down) ← reaction(Y, X), ￢inhibited(Y, X), concentration(Y, down). Y X concentration(X, up) ← concentration(Y, up), reaction(Y, X), reaction(X, Z), ￢inhibited(Y, X), inhibited(X, Z). E: Y X Z concentration(d, up). concentration(e, down). concentration(c, down). H 1 : ￢inhibited(a, b). inhibited(b, c). ￢inhibited(e, c). inhibited(d, e). ￢inhibited(b, d). concentration(e, down) ← inhibited(d, e), ￢inhibited(e, c). d e c 24

Conclusion Introduction of inhibitions in metabolic pathways Introduction of CF-induction Full clausal theories (non-Horn clauses) for B, E and H Completeness of hypotheses finding Integration of abduction and induction on inhibitory effects using CF-induction.