Towards a Logical Analysis of Biochemical Reactions
Patrick Doherty and Steve Kertes and Martin Magnusson and Andrzej Szalas
4
Introduction
Example Continued
hypothesis generation using weakest sufficient (wsc) and strongest
necessary conditions (snc) may be used to provide additional information in the context of an incomplete model of metabolic pathways.
It is assumed that any reaction is specified by:
1
Introduction
α(n)
n2 : C00493 + C00002
YDR127W
−→
C03175 + C00008
TEMS
SYS
ER
UT
Linköpings universitet, Artificial Intelligence and Integrated Computer Systems (AIICS)
ItSweden
is assumed that reaction
We provide
a logicaland
model
of biochemical
reactions
and23
show
how
Department
of Computer
Information
Science,
SE-587
Linköping,
ICIAL INTELLI
F
I
GE
T
N
AR
1
INTEGRATED C
D
N
OM
A
P
CE
depicted by the dashed box is, in fact, missing.
The above set of reactions is expressed by formulas using the re
lation
symbol
prec(R,
R
), meaning that reaction node R directly
4 Example Continued
precedes reaction node R , and av(C, R), meaning that compound
It
reaction represented by reaction node R.
C isis assumed
availablethat
for reaction
YDR127W
For
example,
the
first
reaction
is expressed
n2 : C00493 + C00002 −→
C03175by:
+ C00008
(1)
n : c1 + . . . + ck −→ c1 + . . . + cl ,
We provide a logical model of biochemical reactions and show how
hypothesis
usingofweakest
sufficient
a label (name)
the reaction,
c1 , .(wsc)
. . , ck and
are strongest
reactants
where n is generation
necessary
may
be usedoftonprovide
additional
inforproducts
and α(n)
is a formula
(inputs forconditions
, cl are
n), c1 , . . .(snc)
react(n
→ box is, in fact, missing.
1 , R)
depicted
by
the
dashed
mation
in the context
of an
incomplete
model offor
metabolic
pathways.
that
specifies
additional
conditions
necessary
the
reaction,
such
av(ydr127w,
R)∧
4 Example
Continued
1 Introduction
The
above
set
of
reactions
is expressed by formulas using the reIt
is
assumed
that
any
reaction
is
specified
by:
as temperature, pressure, presence of catalyzers, etc.
av(c02652,
R)R∧ ),av(c00005,
R)∧
lation
symbol
prec(R,
meaning
that
reaction
node R directly
4
Example
Continued
1 Introduction
It
is
assumed
that
reaction



We
a
logical
model
of
biochemical
reactions
and
show
how
4
Example
1 provide
Introduction
Continued
∀R
.
[prec(R,
R
), and
→ av(c00006,
R )]∧ that compound
precedes
reaction
node
R
av(C,
R),
meaning
α(n) 

YDR127W


,
(1)
+
.
.
.
+
c
−→
c
n
:
c
+
.
.
.
+
c
hypothesis
generation
using
weakest
sufficient
(wsc)
and
strongest
1
k
It
is
assumed
that
reaction
l
1
∀R
.
[prec(R,
R
)
→
av(c00493,
R
)].
n
:
C00493
+
C00002
−→
C03175
+ C00008
2
We provide a logical model of biochemical reactions and show how
C
is
available
for
reaction
represented
by
reaction
node
R.
It
is
assumed
that
reaction
We provide
a logical (snc)
modelmay
of biochemical
reactions
and showinforhow
2hypothesis
Hypotheses
Generation
necessary
conditions
be
used
to
provide
additional
YDR127W
generation using weakest sufficient (wsc) and strongest
For
example,
the
first
reaction
is expressed
by:
YDR127W
n
:
C00493
+
C00002
−→
C03175
+
C00008
The
reaction
is
also
present,
among
many
other reactions, in
2missing
depicted
by
the
dashed
box
is,
in
fact,
missing.
hypothesis
generation
using
weakest
sufficient
(wsc)
and
strongest
is
a label
(name)
of
the
reaction,
c
,
.
.
.
,
c
are
reactants
where n mation
1
k
n
:
C00493
+
C00002
−→
C03175
+ C00008
in the(snc)
context
ofbe
anused
incomplete
model
of metabolic
pathways.
2
necessary
conditions
may
to
provide
additional
infor

the database,
and
is→expressed
by:
Suppose
one
is given
a
(incomplete)
specification
of
a
set
of
interactThe
above
set
of
reactions
is expressed by formulas using the renecessary
conditions
(snc)
may
be
used
to
provide
additional
inforare
products
of
n
and
α(n)
is
a
formula
(inputs for
,
.
.
.
,
c
n),
c
react(n
,
R)
1
depicted
by
the
dashed
box
is,
in
fact,
missing.
1
l that any reaction is specified by:
It
is
assumed
 is, in fact, missing.
mation
in the context
of anuse
incomplete
model
of metabolic
pathways.
depicted
by
the
dashed
box
ing
reactions.
We
would
this
set
of
formulas
as
the
background
lation
symbol
prec(R,
R
), meaning
that reaction
node
R directly
mation
in the context
of an necessary
incompletefor
model
of metabolic
that specifies
additional
conditions
the reaction,
suchpathways.
av(ydr127w,
R)∧ is expressed
react(n
R)of→reactions
The
above
by
formulas
using
the
re2 ,set

It is assumed
that additionally,
any reaction
is specified
by: of observations are
The
above
set
of
reactions
is
expressed
by
formulas
using
the
re
theory
T
.
Suppose
that
a
number
precedes
reaction
node
R
,
and
av(C,
R),
meaning
that
compound
α(n)


It :iscpressure,
assumed
that any reaction
is specified
as temperature,
presence
of catalyzers,
etc. by:
av(c02652,
R)R)∧
R)∧
av(ydr127w,
lation
prec(R,
R∧),av(c00005,
meaning
that
reaction node R directly

(1) symbol
n
1 + . . . + ck −→ c1 + . . . + cl ,
lation
symbol
prec(R,
R
),
meaning
thatreaction
reactionnode
nodeR.R directly
 available for reaction represented
 by
made referring to reactions
known
to
have
occurred,
or
compounds
C
is
. [prec(R,
→ av(c00006,
R )]∧ that compound
av(c00493,
R)R∧),av(c00002,
R)∧meaning
precedes ∀R
reaction
node
and
av(C,
R),
α(n) 


precedes
reaction
R
, and av(C,
that compound
(1)
. . . + cl ,in a reaction,
cparticipation
n : cto
. .available
. + ck −→

 node
 R), meaning

1 +
1 +α(n)
known
be
for
etc.
Let
α
deFor
example,
the
first
reaction
is
expressed
by:
∀R
.
[prec(R,
R
)
→
av(c00493,
R
)].
av(c03175,
)]∧ node R.
. . .reaction,
+ cl ,
−→ c1of+the
n : cn1 is
+ a. .label
. + ck(name)
C(1)
is available for reaction represented
by reaction
c1 , . . . , ck are reactants
where
C is
2 Hypotheses
Generation
 available for
 reaction represented
 by reaction node R.
note
the formula
representing
these
observations.
Generally,
it
will


∀R reaction
.react(n
[prec(R,
R
)→
→ av(c00008,
R
)]. other reactions, in
For
example,
the first
reaction
is expressed
by:
are products
of
n
and
α(n)
is
a
formula
(inputs
for(name)
. . .the
, cl reaction,
n), c1 ,of
,
R)
1
The
missing
is
also
present,
among
many
n
is
a
label
c
,
.
.
.
,
c
are
reactants
where
1
kan incomplete
For example, the first reaction is expressed by:
not be thethat
case
that
T
|=
α
because
T
only
provides
n
is
a
label
(name)
of
the
reaction,
c
,
.
.
.
,
c
are
reactants
where
1for the kreaction, such


specifies
additional
conditions
necessary
av(ydr127w,
R)∧
the
database,
and
is
expressed
by:
We
assume
that
the
underlying
database
contains partial information
are
products
of
n
and
α(n)
is
a
formula
(inputs
for
,
.
.
.
,
c
n),
c
Suppose
one
is
given
a
(incomplete)
specification
of
a
set
of
interactreact(n
,
R)
→


1
1 reactions.
l
specification
of
the
are products
of n andetc.
α(n) is a formula
(inputs
for n), cpressure,
react(n
1 , R) → R) ∧ av(c00005, R)∧
1 , . . . , clpresence
as
temperature,
of
catalyzers,
av(c02652,
about
theav(ydr127w,
observed
chain
of reactions:
that
specifies
additional
conditions
for as
thethe
reaction,
such
ingWe
reactions.
Wetowould
use athis
setnecessary
of(candidate
formulas
background
R)∧
react(n
,
R)
→
would
like
generate
formula
hypotheses)
φ
in
a



2
that specifies additional conditions necessary for the reaction, such
av(ydr127w,
R)∧
∀R
.
[prec(R,
R
)
→
av(c00006,
R
)]∧
as
temperature,
pressure,
presence
of
catalyzers,
etc.
theory
T
.
Suppose
additionally,
that
a
number
of
observations
are
av(c02652,
R)
∧
av(c00005,
R)∧
av(ydr127w,
such that φetc.
together
restricted as
subset
of the language
ofpresence
reactionsofPcatalyzers,
 R)∧3 , rR)
react(n
react(n
∧
) R )].
temperature,
pressure,
av(c02652,
∧ )react(n
av(c00005,
1 , r1 ) ∧∀R
3 )R
4 , r4R)∧

.
[prec(R,
→
av(c00493,
madethe
referring
to reactions
known
toentail
have the
occurred,
or compounds
∀R
. [prec(R,
)av(c00002,
→ av(c00006,
R )]∧
 R


av(c00493,
R)
∧
R)∧
T
does
observations
α.
It
is
with
background
theory
2 Hypotheses Generation
∀R
.
[prec(R,
R
)
→
av(c00006,
R
)]∧other re a description of reactions n1 , n3 ,n4 and many
together
with
known to be
for participation
a reaction,
Letjust
α de∀R
[prec(R,
R
)].
The ..missing
reaction
also
other reactions, in
 R
 present,

∀R
[prec(R,
R )) →
→isav(c00493,
av(c03175,
Ramong
)]∧ many
important
thatavailable
we do not
over commit in
otherwise
weetc.
could
as
∀R
.
[prec(R,
R
)
→
av(c00493,
R
)].
2 Hypotheses
Generation
n2 . Let
the
consideredby:
knowledge
base be denoted
actions, including
database,
 is expressed

note
the
formula
representing
these
observations.
Generally,
it
will
the
and
Suppose
one
is
given
a
(incomplete)
specification
of
a
set
of
interact2
Hypotheses
Generation
∀R reaction
. [prec(R,
R ) present,
→ av(c00008,
)]. other reactions, in
easily choose α itself as the hypothesis which wouldn’t do much
The missing
is also
among R
many
by K DB. The missing reaction is also present, among many other reactions, in
not
be
the
case
that
T
|=
α
because
T
only
provides
an
incomplete
ing
reactions.
We
would
use
this
set
of
formulas
as
the
background
(α; T ; P ) does
just the right
we
good.
In one
fact,isthe
W SC
the database,
andthe
is underlying
expressed
by:
react(n
,
R)
→
Suppose
given
a (incomplete)
specification
of athing
set ofsince
interact2
We
assume
that
database
contains
information
SC (α; Kby:
DB ; av), partial
where
We
can
now
consider,
e.g.,
W
the
database,
and
is
expressed
Suppose
one
is
given
a
(incomplete)
specification
of
a
set
of
interactspecification
of
the
reactions.
. (α;
Suppose
a number
of formula
observations are
know
thattheory
T ∧W
SCwould
T ;use
P ) additionally,
|= αset
andofitformulas
isthat
the weakest
av(ydr127w,
R)∧
ing
reactions.
WeT
this
as thesuch
background
about
the
observed
chain
of
reactions:
react(n
,
R)
→
2
ing like
reactions.
We
would
useknown
this
settoof
formulas
as the
background
def
We
would
to generate
a formula
(candidate
hypotheses)
φorincompounds
a
made
referring
to
reactions
have
occurred,
react(n
→prec(r
by
definition.
2 ,rR)
av(c00493,
R) ∧ av(c00002,
R)∧2 , r3 )],
theory T . Suppose additionally, that a number of observations are
α ≡ ∃N.[react(N,
)
∧
2
1 , r2 ) ∧ prec(r
av(ydr127w,
R)∧
theory
T
. be
Suppose
additionally,
thatP asuch
number
observations
are
that
φoftogether
restricted
subset
ofsnc’s
the
language
of reactions
react(n1 , r1 ) ∧∀R
react(n
∧

 react(n4 , r4 )

3 , r3 )R)∧
known
to
available
for
participation
in
a
reaction,
etc.
Let
α
deav(ydr127w,
The
wsc’s
and
can
be
calculated
efficiently
for
a
large
class
.
[prec(R,
R
)
→
av(c03175,
R
)]∧
made referring to reactions known to have occurred, or compounds
av(c00493,
R)
∧
av(c00002,
R)∧
made
referring
to
reactions
known
to
have
occurred,
or
compounds
providing
one
with
the
weakest
requirement
expressed
in
T does
entail
theobservations.
observations
α. It an
is it will
with
the note
background
theory

 av(c00002, R)∧  terms of av
the
formula
representing
these
Generally,
av(c00493,
R)
∧
of
formulas
by
expressing
them
as
second-order
formulas,
using
 a description
 of reactions
∀R . [prec(R,
R)→
)]. other reknown to be available for participation in a reaction, etc. Let α den
,
n
many
together with
1av(c00008,
3 ,n
4 andR
∀R
.
[prec(R,
R
)
→
av(c03175,
R
)]∧



known
to do
be
available
forαparticipation
inwe
a provides
reaction,
etc.as
Let αonly,
de- making α true,
provided
that
the
background
theory
given by
important
that
we
not
over
commit
otherwise
could just
not
be
the
case
that
T
|=
because
T
only
an
incomplete
∀R
.
[prec(R,
R
)
→
av(c03175,
R
)]∧
equivalence
proved
in
[3],
and
obtaining
logically
equivalent
first


note the formula representing these observations. Generally, it will
n2 .that
Let
the
considered
knowledge
base
bepartial
denoted
actions, including
∀R
.
[prec(R,
R
)
→
av(c00008,
R
)].
We
assume
the
underlying
database
contains
information



note
the
formula
representing
observations.
Generally,
it
will
K
DB holds.
∗ these
easily
choose
α
itself
as
the
hypothesis
which
wouldn’t
do
much
specification
of
the
reactions.
∀R
.
[prec(R,
R
)
→
av(c00008,
R
)].
order
by T
applying
the D LS
algorithm
e.g.,
not
be formulas
the case that
|= α because
T only
providesdescribed,
an incomplete
by K DB. about the observed chain of reactions:
notWe
be
theWcase
that
α because
onlything
provides
incomplete
In
our case,
the
generated hypotheses
will contain
the
disjunct
SC
(α;
T generate
;TP )|=does
just
theTright
sinceanwe
good.
In fact,
the
We
assume
that
the
underlying
database
contains
partial
information
would
like
to
a
formula
(candidate
hypotheses)
φ
in
a
in
[2].
specification of the reactions.
; av), where
We canWe
now
consider,
e.g.,
W SC(α; K DB
assume
that
the
underlying
database
contains
partial
information
specification
of
the
reactions.
av(c00002,
r
),
reflecting
sufficient
conditions
for
reaction
r
ocknow thatrestricted
T ∧W SCsubset
(α; T ; P
)the
|= language
α and it isofthereactions
weakestPsuch
formula
2
2
about
the
observed
chain
of
reactions:
such
that
φ
together
of
react(n
,
r
)
∧
react(n
,
r
)
∧
react(n
,
r
)
1
1
3
3
4
4
We would like to generate a formula (candidate hypotheses) φ in a
def about the observed chain of reactions:
We
would
like
to
generate
a
formula
(candidate
hypotheses)
φ
in
a
(α;
K DB;{out})
will concuring
under
the prec constraints.
The1 ,SrNC
by definition.
T
does
entail
the
observations
α.
It
is
with
the
background
theory
α
≡
∃N.[react(N,
r
)
∧
prec(r
)
∧
prec(r
,
r
)],
2
2
2
3
restricted subset of the language of reactions P such that φ together
react(n
react(n
react(n
1 , r1 ) ∧
3 , r3 ) ∧ of
4 , rn
4 )1 , n3 , n4 and many other retogether
with
a
description
reactions
P
such
that
φ
together
restricted
subset
of
the
language
of
reactions
tain
the
disjunct
The
wsc’s
and
snc’s
can
be
calculated
efficiently
for
a
large
class
react(n
,
r
)
∧
react(n
,
r
)
∧
react(n
C00006
3
A
Metabolic
Pathway
Example
1 1
3 3
4 , r4 )
important
that
we
do
not
over
commit
otherwise
we
could
just
as
α. It is
with theC02652
background theory T does entail the observations
providing
oneawith
the weakest
requirement
expressed
in terms
of reav
n2reactions
. Let
the considered
knowledge
base
be denoted
actions,
including
C00008
T
does
entail
the
observations
α.
It
is
with
the
background
theory
n
,
n
,
n
and
many
other
together
with
description
of
of formulas
by
expressing
them
as
second-order
formulas,
using
an
1
3
4
easily
choose
α itself
as the hypothesis
which
wouldn’t
do much
out(N,
c03175)
∧ provided
out(N,
c00008),
n1 , n
many
other retogether
with
a
description
of reactions
important
that
we
do
not
over
commit
otherwise
we
could
just
as
3 , n4 and
only,
making
α
true,
that
the
background
theory
given
by
DB
.
by
K
Consider
a
fragment
of
the
aromatic
amino
acid
pathway
of
yeast
important
that
weW
doSC
not
over
commit
otherwise
wething
could
just
as including n2 . Let the considered knowledge base be denoted
C00493
actions,
equivalence
proved
in [3],
and
obtaining
logically
equivalent
first(α;
T
;
P
)
does
just
the
right
since
we
good.
In
fact,
the
n2 . for
Lete.g.,
thereaction
considered
knowledge
base be denoted
actions,
including
easily
choose
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REFERENCES
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Journal
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REFERENCES
Consider
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essary
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REFERENCES
edge from a compound node YDR127W
to a reaction node denotes a substrate.
revisited’,
Journal of Automated Reasoning, 18(3), 297–336, (1997).
edge
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REFERENCES
n3reaction.
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essary andR.D.
weakest
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formulas’, Interof the
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allow conditions
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King,
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logic programming,
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of
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additionally
allow
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YGL148W
[1] C.H.
Bryant,
S.H.
Muggleton,
D.B.
Kell,
P.(2000).
Reiser, Electronic
and
national
Joint
Conference
on AI S.G.
(IJCAI’2001),
145
– 151,Linkoping
that
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[1]
C.H.
Bryant,
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Reiser,
and
R.D.
King,
‘Combining
inductive
logic
programming,
active
learning
and
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case
rectangles
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The
figure
depicts
the
following
reactions:
ticles
in
Computer
and Information
Science,
6(12), (2001).
R.D.
King,
‘Combining
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learning
that a respective enzyme is to be available for reaction.
and
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to
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genes’,
Linkoping
Electronic
Arthat a respective enzyme is to be available for reaction.
[2] P.
Doherty,
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and A. Szałas,
‘Computing
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and
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ticles
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Science,
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revisited’,
Journal of
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(1997).
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Science,
6(12),
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n1 figure
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[2] P. Doherty, W. Łukaszewicz, and A. Szałas, ‘Computing circumscription
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[3]
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Doherty, W. Łukaszewicz, and
and A.
A.Szałas,
Szałas,‘Computing
‘Computingcircumscription
strongest nec[2]
P.
revisited’, essary
Journaland
of Automated
Reasoning,
18(3), 297–336,
(1997).
weakest
sufficient
conditions
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first-order
formulas’,
Interrevisited’,
Journal
of
Automated
Reasoning,
18(3),
297–336,
(1997).
[3] P. Doherty,national
W. Łukaszewicz,
and A. on
Szałas,
‘Computing 145
strongest
necJoint
Conference
AI
(IJCAI’2001),
–
151,
(2000). nec[3]
P.
Doherty,
W.
Łukaszewicz,
and
A.
Szałas,
‘Computing
strongest
essary and weakest sufficient conditions of first-order formulas’, Interessary
and weakest
conditions
first-order
national Joint
Conference
on AIsufficient
(IJCAI’2001),
145 –of151,
(2000). formulas’, International Joint Conference on AI (IJCAI’2001), 145 – 151, (2000).