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Transcript of Integration of abduction and induction in biological networks using CF-induction Yoshitaka Yamamoto...
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).
14
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
17
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
19
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).
24
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