Integration of abduction and induction in biological networks using CF-induction Yoshitaka Yamamoto Graduate University for Advanced Studies Tokyo, Japan.

Integration of abduction and induction in biological networks

using CF-induction

Yoshitaka YamamotoGraduate University for Advanced Studies Tokyo, Japan.

Andrei DoncescuLAAS-CNRS Toulouse, France.

Katsumi InoueNational 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

3

OutlineLogical 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

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Abduction and Induction: Logical Framework

Input: – B : background theory. – E : (positive) examples / observations.

Output: H : hypothesis satisfying that

– B ∧ H ╞═ E– B ∧ H is consistent.

Inverse Entailment (IE)

B

E

ILP machin

e

H

Computing a hypothesis H can be done deductively by:

B ∧ ￢ E ╞═ ￢ H 　We use a consequence finding technique for (IE) computation.

Consequence findingGiven 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)

ThP(Σ) : the clauses entailed by Σ which belong to P.

• Characteristic clause C of Σ (wrt P ):

C belongs to ThP(Σ) ;

no other clause in ThP(Σ) subsumes C.

Carc(Σ, P) = μThP(Σ), where μ represents

“subsumption-minimal”.

IE for Abduction --- SOLARSOLAR(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)].

Production_field: [ ￢ arc(_,_)].

SOLAR outputs four consequences:

[ ￢ arc(a, d)] , [ ￢ arc(a, c)], [ ￢ arc(b, d)], [ ￢ arc(b, c)]

a c

b d

IE for Induction

• CF-induction CF-induction (Inoue 2004: Yamamoto, Ray & Inoue 2007)

• fc-HAILfc-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.

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 .

Principle of CF-induction

Algorithm1.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;

H is Skolem-free.

＊ Generalization

H ⊨ F － inverse Skolemization－ anti-instantiation－ dropping literals from clauses － addition of clauses － inverse resolution － Plotkin’s least generalization

9

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

Conclusion and future work

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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.

• Enzymes: accelerate the rate of a chemical reactionby up to three orders of magnitude

E : enzyme,S : substrate,P : product,ES : complex,k : the constant of the rate of a chemical reaction.

E + S ES E + Pk1

k-1

k2

k-2

E E

S

ES

P

11

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.

• Michaelis-Menten Reaction: - the relation between the activity of an enzyme and the concentration of a substrate- at steady state

Concentration of substrate

Time [T]

Activity

Concentration of Enzyme

E + S ES E + Pk1

k-1

k2

k-2

V = k2 [ES] - k-2 [E][P]

V = VmaxKm + [S]

[S]

V

[S]

Vmax

12

Effect on toxinsThere exists chemical compounds (inhibitors) which control activities of enzymes.

Higher the concentration of a inhibitor is, lower the activity of the enzyme controlled by the inhibitor becomes.

E E

E S

P

S

I

I

S

I

Activity

Concentration of substrate

—: without inhibitor

—: with inhibitor

E I

13

Logical modeling of inhibition[Tamaddoni-Nezda et al 2006]

Enz

S P

ToxinInhibited

Enz

S P

Toxin Not inhibited

Enz

S P

ToxinNot inhibited

concentration(P, down) ← reaction(S, Enz, P), ￢ inhibited(Enz, S, P), concentration(S, down).

concentration(P, up) ← reaction(S, Enz, P), ￢ inhibited(Enz, S, P), concentration(S, up).

concentration(P, down) ← reaction(S, Enz, P), inhibited(Enz, S, P).

concentration(S, up) ← reaction(S, Enz, P), inhibited(Enz, S, P).

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Prediction for inhibitory effect of a toxin

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”

The goal -Finding inhibitions of a metabolic pathway

Our approach-Using IE for abduction

Example 1/2

2-oxe-glutarate

l-lysine

l-2-aminoadipate

isocitrate

trans-aconitate

citrate

fumarate

taurine

succinate

nmnd

nmna

hippurate

acryloyl-coa

formate

formaldehyde

sarcosine

l-as

citrulline

ornithinearginin

eurea creatin

ecreatinin

emethylamin

etmao

lactate

beta-alanine

2.6.1.39; 4.2.1.36;...2.6.1.39; 4.2.1.36;...1.2.1.31; 1.5.1.7;...1.2.1.31; 1.5.1.7;...

4.3.2.14.3.2.1

3.6.3.33.6.3.3

Example 2/2

2-oxe-glutarate

l-lysine

l-2-aminoadipate

isocitrate

trans-aconitate

citrate

fumarate

taurine

succinate

nmnd

nmna

hippurate

acryloyl-coa

formate

formaldehyde

sarcosine

l-as

citrulline

ornithinearginin

eurea creatin

ecreatinin

emethylamin

etmao

lactate

beta-alanine

2.6.1.39; 4.2.1.36;...2.6.1.39; 4.2.1.36;...1.2.1.31; 1.5.1.7;...1.2.1.31; 1.5.1.7;...

4.3.2.14.3.2.1

3.6.3.33.6.3.3

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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

Conclusion and future work17

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Goals-Predicting the concentration of metabolites intracellular

-Discovering inductive rules augmenting incomplete background theory

Our Approaches- Using CF-induction

Prediction for intracellular fluxes

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

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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.

v1, v2, v3+, v3-, v4 : unknown fluxes at the steady state. rA, rC, rD, rE : metabolite extracellular accumulation rate.

A

E

DB

C

rA

rC

rE

rD

v3+v5

v4v3-

v2v1

( ) ( )[ ]⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧

−−+=−+=−−=

−=−=

−=

3v3vá

árCrE5v

árD1v5v

árC4v

á1v2v

rA1v

　　 E : concentration(d, up). concentration(e, down). concentration(c, down).

　　 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).

concentration(X, up) ← concentration(Y, up), reaction(Y, X), reaction(X, Z), ￢ inhibited(Y, X), inhibited(X, Z).

concentration(X, down) ← reaction(Y, X), ￢ inhibited(Y, X), concentration(Y, down).

Example 1:

Y X

Y X Z

H2 :￢ inhibited(a, b). inhibited(b, c).

￢ inhibited(b, d). inhibited(d, e).

concentration(X, down) ← concentration(Y, up), inhibited(Y, X).

H1 :￢ inhibited(a, b). inhibited(b, c).

￢ inhibited(e, c). inhibited(d, e).

￢ inhibited(b, d).

concentration(e, down) ← inhibited(d, e), ￢ inhibited(e, c). 　

Example 1: outputs of CF-induction

d e c

Y X

Example 2: the real metabolic pathway (Pyruvate) B:

reaction(pyruvate, acetylcoa).

reaction(pyruvate, acetaldehide).

reaction(glucose, glucosep).

reaction(glucosep, pyruvate).

reaction(acetaldehide, acetate).

reaction(acetate, acetylcoa).

reaction(acetaldehide, ethanol).

concentration(glucose, up).

terminal(ethanol).

blocked(X)←reaction(X,Z), inhibited(X,Z).

blocked(X)←terminal(X).

concentration(X,up) ←reaction(Y,X), ￢ inhibited(Y,X), blocked(X).

E : concentration(ethanol,up). concentration(pyruvate, up).

Acetate

AcetaldehidePyruvate

Glucose-P

Ethanol

Glucose

Acetylcoa

EC 4.1.1.1 EC 1.1.1.1

EC 1.2.1.10EC 1.2.4.1

X Z

X

H1:

￢ Inhibited(glucosep, pyruvate).

￢ inhibited(acetaldehide, ethanol).

inhibited(pyruvate, acetylcoa).

Example 2: outputs of CF-induction

Ethanol

Acetate

AcetaldehidePyruvate

Glucose-P

Glucose

Acetylcoa

Acetate

AcetaldehidePyruvate

Glucose-P

Glucose

Acetylcoa

Ethanol

　　 H2: ￢ inhibited(glucose, glucosep)

￢ Inhibited(glucosep, pyruvate).

￢ inhibited(acetaldehide, ethanol).

￢ inhibited(pyruvate, acetaldehide).

concentration(Y, up)←

￢ inhibited(X, Y), concentration(X, up).

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　　 E : concentration(d, up). concentration(e, down). concentration(c, down).

　　 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).

concentration(X, up) ← concentration(Y, up), reaction(Y, X), reaction(X, Z), ￢ inhibited(Y, X), inhibited(X, Z).

concentration(X, down) ← reaction(Y, X), ￢ inhibited(Y, X), concentration(Y, down).

Y X

Y X Z

H1 : ￢ 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

Conclusion

Introduction of inhibitions in metabolic pathways

Introduction of CF-inductionFull 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.