Post on 11-Feb-2017
50 Shades of Rule CompositionFrom Chemical Reactions to Higher Levels of Abstraction
Jakob L. Andersen, Christoph FlammDaniel Merkle, Peter Stadler
Department of Mathematics and Computer ScienceUniversity of Southern Denmark
FMMB 2014, NoumeaSeptember 22, 2014
1
Metabolic Flux Pattern and Atom Maps
Klunemann et al. (2014): Computational tools for modeling xenometabolism of the human gut microbiota, Trendsin Biotechnology, 32(3):157-165
2
Isotope Labeling Experiments
1. Introduce a isotope labeled substrate into thecell culture at metabolic steady state.
2. Allow the system to reach an isotopic steadystate.
3. Measure (e.g. NMR, MS) relative labeling inmetabolic intermediates and by products.
4. Estimate fluxes from these measurements.
I Quantitative interpretation requires a mathematical model,which relates metabolic flux to isotopomere abundance.
Figure adapted from [Wiechert, 2001]
3
Atom Transition NetworkEstimating fluxes from isotopomere patterns is an inverse problem.
The bijective atom-atom mappingbetween reaction educts and productsmust be known.
Getting this information is at least a graph isomorphism-hard problem.
C
C
C
C
C
C
CC
C
C
C
C
C
C
C
C
C
C
<=>
1 12
2
3
3
4
4
55
6
6
7 7
88
9
9
4
Outline
I Graph Rewriting and the Double Pushout Formalism(Elementary Reactions)
I Shades of Rule Composition
I Results and Atom Traces(Composed Reactions)
I An Enzymatic Reaction: β-LactamaseI A Pathway: GlycolysisI (An Autocatalytic Reaction: Formose)
5
Molecule EncodingA molecule is an undirected labelled graph.Vertex label ≡ atom type (e.g., “C” or “O-”)Edge label ≡ bond type (e..g, “-”, “=” or “#”)
OH -
C-
H-
C=
H
- O-
H-(a) Visualization of encoding
OH
C
H
C
HO
H(b) Prettified visualization
OHHO
(c) Open Babel visualization
Figure: 1,2-ethenediol
6
Reaction Patterns – Graph Transformation Rules
A reaction pattern is a graph transformation rule, in the DoublePushout Formalism: p = (L← K → R).
C C
O C
OH
LC C
O C
OH
KC C
O C
OH
R
Figure: Transformation rule for aldol addition
(As for the graphs: the rules are not restricted to chemistry.)
7
Reactions – Application of Transformation Rules1,2-ethenediol + formaldehyde aldol addition−−−−−−−−→ glyceraldehyde
CH
H
C
O
O
H
H
CH
OH
G
CH
H
C
O
O
H
H
CH
OH
D
CH
H
C
O
O
H
H
CH
OH
H
C C
O C
OH
LC C
O C
OH
KC C
O C
OH
R
8
Double Pushout Approach
Double pushout
L K R
G D H
l r
ρ λ
m k n
9
Pushout and Pullback : Rule Composition
p1 = (L1l1←− K1
r1−→ R1) p2 = (L2l2←− K2
r2−→ R2)
Rule composition
L1 K1 R1 L2 K2 R2
L C1 E C2 R
K
(1) (2)
(3)
u1 v1 e1 e2 v2 u2
w1 w2
l1
s1
r1
t1
l2
s2
r2
t2
A composition (L ql←− K qr−→ R) = p1 ∗E p2 can be defined1 and exists2, ...
1Ehrig et al. (1991): Parallelism and Concurrency in High-Level Replacement Systems. Math. Struct. Comp.C, 1:361–4042Golas (2010): Analysis and Correctness of Algebraic Graph and Model Transformations. Wiesbaden, Vieweg+Teubner
10
Pushout and Pullback : Rule Composition
p1 = (L1l1←− K1
r1−→ R1) p2 = (L2l2←− K2
r2−→ R2)
Rule composition
L1 K1 R1 L2 K2 R2
L C1 E C2 R
K
(1) (2)
(3)
u1 v1 e1 e2 v2 u2
w1 w2
l1
s1
r1
t1
l2
s2
r2
t2
A composition (L ql←− K qr−→ R) = p1 ∗E p2 can be defined1 and exists2, ...
1Ehrig et al. (1991): Parallelism and Concurrency in High-Level Replacement Systems. Math. Struct. Comp.C, 1:361–4042Golas (2010): Analysis and Correctness of Algebraic Graph and Model Transformations. Wiesbaden, Vieweg+Teubner
10
One Shade of Composing Rules (straight-forward)
R1 is a super-graph of L2
L1 R1 L2 R2
r1 r2=⇒ L ∼= L1 R R2
r
CH3
CCH2
CHCH2
CH
CHCH2
CH2
CH2
CH2
r1
CH3
CCH2
CHCH2
CH
CHCH2
CH2
CH2
CH2
CC
CC
C
C r2
CC
CC
C
C
CH3
CCH2
CHCH2
CH
CHCH2
CH2
CH2
CH2
r
CH3
CCH2
CHCH2
CH
CHCH2
CH2
CH2
CH2
11
One Shade of Composing Rules (straight-forward)
R1 is a super-graph of L2
L1 R1 L2 R2
r1 r2=⇒ L ∼= L1 R R2
r
CH3
CCH2
CHCH2
CH
CHCH2
CH2
CH2
CH2
r1
CH3
CCH2
CHCH2
CH
CHCH2
CH2
CH2
CH2
CC
CC
C
C r2
CC
CC
C
C
CH3
CCH2
CHCH2
CH
CHCH2
CH2
CH2
CH2
r
CH3
CCH2
CHCH2
CH
CHCH2
CH2
CH2
CH2
11
A Darker Shade of Composing Rules (masochistic)
R1 is a super-graph of a connected component of L2
L1 R1 L12
r2
L22
R2
r1
=⇒L1
r
L22
R R2
CH
CHCH2
CH2
CH2
CH2
r1
CH
CHCH2
CH2
CH2
CH2
CC
CC
C
C
r2
CC
CC
C
C
CH
CHCH2
CH2
CH2
CH2
CC
CC
r
CH
CHCH2
CH2
CH2
CH2
CC
CC
12
A Darker Shade of Composing Rules (masochistic)
R1 is a super-graph of a connected component of L2
L1 R1 L12
r2
L22
R2
r1
=⇒L1
r
L22
R R2
CH
CHCH2
CH2
CH2
CH2
r1
CH
CHCH2
CH2
CH2
CH2
CC
CC
C
C
r2
CC
CC
C
C
CH
CHCH2
CH2
CH2
CH2
CC
CC
r
CH
CHCH2
CH2
CH2
CH2
CC
CC
12
And an Even Darker Shade of Composing Rules (perverted)
The matching morphism is a common subgraph
L1 R1 L2 R2
r1 r2=⇒ L R
r
CC
CC
C
C r1
CC
CC
C
C
C
CC
C
CC
r2
C
CC
C
CC
CC
CC
C
CC
C
CC
r
CC
CC
C
CC
C
CC
13
And an Even Darker Shade of Composing Rules (perverted)
The matching morphism is a common subgraph
L1 R1 L2 R2
r1 r2=⇒ L R
r
CC
CC
C
C r1
CC
CC
C
C
C
CC
C
CC
r2
C
CC
C
CC
CC
CC
C
CC
C
CC
r
CC
CC
C
CC
C
CC
13
A Lighter Shade of Composing Rules
Parallel Composition
L1 R1
L2 R2
r1
r2=⇒ L ∼= L1 ∪ L2 R ∼= R1 ∪R2
r
14
β-Lactamase
+ →
Step 1 Lys73 deprotonates Ser70, which initiates a nucleophilic addition onto the carbonyl carbon of thebeta-lactam, forming a tetrahedral intermediate.
Step 2 The tetrahedral intermediate collapses, cleaving the C-N bond in the beta-lactam, the nitrogendeprotonates Ser130.
Step 3 Ser130 deprotonates Lys73.Step 4 Glu166 deprotonates water, which initiates a nucleophilic addition at the carbonyl carbon, forming a new
tetrahedral intermediate.Step 5 The tetrahedral intermediate collapses, cleaving the acyl-enzyme bond and liberating Ser70, which in turn
deprotonates the Glu166.
Holliday et al. (2005): MACiE: A Database of Enzyme Reaction Mechanisms. Bioinformatics 21 (2005) 4315–4316
15
β-Lactamase
+ →
Step 1 Lys73 deprotonates Ser70, which initiates a nucleophilic addition onto the carbonyl carbon of thebeta-lactam, forming a tetrahedral intermediate.
Step 2 The tetrahedral intermediate collapses, cleaving the C-N bond in the beta-lactam, the nitrogendeprotonates Ser130.
Step 3 Ser130 deprotonates Lys73.Step 4 Glu166 deprotonates water, which initiates a nucleophilic addition at the carbonyl carbon, forming a new
tetrahedral intermediate.Step 5 The tetrahedral intermediate collapses, cleaving the acyl-enzyme bond and liberating Ser70, which in turn
deprotonates the Glu166.
Holliday et al. (2005): MACiE: A Database of Enzyme Reaction Mechanisms. Bioinformatics 21 (2005) 4315–4316
15
β-Lactamase
r1 :
C
C
C
C
C
O
OH
NH2
L
C
C
C
C
C
O
O
H
NH2
K
C
C
C
C
C
O−
O
N+H3
R
16
β-Lactamase
r1 :
C
C
C
C
C
O
OH
NH2
L
C
C
C
C
C
O
O
H
NH2
K
C
C
C
C
C
O−
O
N+H3
R
r2 :C
C
CC
N
O−
OOH
L
C
C
CC
N
O
OO
H
K
C
C
CC
NH
O
OO−
R
r3 :
C
C O−
NH3+
LC
C O
NH2H
KC
C OH
NH2
R
r4 :
C
C
C
C
N
OO
H2O
O−
O
C
L
C
C
C
C
N
OO
OH
O
O
C H
K
C
C
C
C
N
O−O
OH
OH
O
C
R
r5 : C
C
CC
CN O−
O
OH
OH
O
C
L
C
C
CC
CN O
O
OH
O
O
CH
K
C
C
CC
CN O
OH
OH
O−
O
C
R
16
Rule Composition - β-Lactamase
ıG ◦ r1 ◦ r2 ◦ r3 ◦ r4 ◦ r5 ◦ ıH
17
Rule Composition - β-Lactamase
ıG ◦ r1 ◦ r2 ◦ r3 ◦ r4 ◦ r5 ◦ ıHAtom Traces:
O
S
H
H
O
O
H
H
CO 2 H
NH 2
CO 2 H
CO 2 H
N
C
O
O
NH
NH 2 Ph
O
NH
Ph
NH 2
NH 2
CO 2 H
NH 2
CO 2 H
CO 2-
N
C
OH
H
O
OHH
N
S
CO 2 H
OCO 2 H
CO 2 HN
N
H 2
H 2
CO 2 H
NH 2
NH 2
CO 2 H
CO 2-
8 10
H
C
C C
C
C
C
C
C
Catalytic amino acids: Serine, Lysine, Aspartate
17
Rule Composition - β-Lactamase
ıG ◦ r1 ◦ r2 ◦ r3 ◦ r4 ◦ r5 ◦ ıHAtom Traces:
O
S
H
H
O
O
H
H
CO 2 H
NH 2
CO 2 H
CO 2 H
N
C
O
O
NH
NH 2 Ph
O
NH
Ph
NH 2
NH 2
CO 2 H
NH 2
CO 2 H
CO 2-
N
C
OH
H
O
OHH
N
S
CO 2 H
OCO 2 H
CO 2 HN
N
H 2
H 2
CO 2 H
NH 2
NH 2
CO 2 H
CO 2-
8 10
H
C
C C
C
C
C
C
C
O
S
H
H
O
O
H
H
CO 2 H
NH 2
CO 2 H
CO 2 H
N
C
O
O
NH
NH 2 Ph
O
NH
Ph
NH 2
NH 2
CO 2 H
NH 2
CO 2 H
CO 2-
N
C
OH
H
O
OHH
N
S
CO 2 H
OCO 2 H
CO 2 HN
N
H 2
H 2
CO 2 H
NH 2
NH 2
CO 2 H
CO 2-
8 10
H
C
C C
C
C
C
C
C
Catalytic amino acids: Serine, Lysine, Aspartate
17
Rule Composition - β-Lactamase
For all permutations σ:
ıG ◦ rσ(1) ◦ . . . rσ(5) ◦ ıH
Well defined compositions, leading to the overall expected rule:
(r1, r2, r3, r4, r5)(r1, r2, r4, r3, r5)(r1, r2, r4, r5, r3)
⇒ r3 (H+-exchange reaction between amino acids) is the recyclingstep, which can be applied concurrently to steps r4 and r5.
18
Rule Composition - β-Lactamase
For all permutations σ:
ıG ◦ rσ(1) ◦ . . . rσ(5) ◦ ıH
Well defined compositions, leading to the overall expected rule:
(r1, r2, r3, r4, r5)(r1, r2, r4, r3, r5)(r1, r2, r4, r5, r3)
⇒ r3 (H+-exchange reaction between amino acids) is the recyclingstep, which can be applied concurrently to steps r4 and r5.
18
Rule Composition - β-Lactamase
Alternative for step 2:Protonation of the β-lactam nitrogen occurs before the C-N bondcleavage1
(r1, r1b, r3, r2b, r4, r5) (r1b, r1, r3, r2b, r4, r5)(r1, r1b, r2b, r3, r4, r5) (r1b, r1, r2b, r3, r4, r5)(r1, r1b, r2b, r4, r3, r5) (r1b, r1, r2b, r4, r3, r5)(r1, r1b, r2b, r4, r5, r3) (r1b, r1, r2b, r4, r5, r3)
1Atanasov et al. (2000): Protonation of the beta-lactam nitrogen is the trigger event in the catalytic action of class Abeta-lactamases. PNAS, 97(7) (2000) 3160–3165
19
Recap: Central Carbon Metabolism
Figure from Noor et al (2010) Central Carbon Metabolism as a Minimal Biochemical Walk between Precursors forBiomass and Energy, J Mol Cell 39:809-820 | DOI 10.1016/j.molcel.2010.08.031
20
Glycolysis
Transformation Rules:
r1 Pyranose-furanoser2 Furanose-linearr3 Ketose-aldoser4 ATP-phosphorylationr5 ATP-
dephosphorylationr6 NAD+-
phosphorylation
r7 Phosphomutaser8 Enolaser9 Keto-enol
r10 NAD+-oxoreductaser11 Lactonohydrolaser12 Hydrolyaser13 Reverse aldolase
21
Carbon Atom Trace of Glycolysis
Embden-Meyerhof-Parnas (EMP) pathway:
ıG(EMP) ◦Glucose → 2 GAP︷ ︸︸ ︷
r4 ◦ r1 ◦ r4 ◦ r2 ◦ r13 ◦ r3
◦ (r6 ◦∅ r6) ◦ (r5 ◦∅ r5) ◦ (r7 ◦∅ r7) ◦ (r8 ◦∅ r8) ◦ (r5 ◦∅ r5) ◦ (r9 ◦∅ r9)︸ ︷︷ ︸2 GAP → 2 Pyruvate
◦ ıH(EMP)
The Entner-Doudoroff (ED) pathway:
ıG(ED) ◦ r4 ◦ r10 ◦ r11 ◦ r12 ◦ r13︸ ︷︷ ︸Glucose → GAP + Pyruvate
◦ r6 ◦ r5 ◦ r7 ◦ r8 ◦ r5 ◦ r9︸ ︷︷ ︸GAP + Pyruvate → 2 Pyruvate
◦ ıH(ED)
22
Carbon Atom Trace of GlycolysisO OH
OH
OH
HO
HO
O OH
OH
OH
HO
PO
PO OH
OH
OHHO
O
PO
OH
OHHO
O
OP
O
PO
OH
OH
OP
O
+
66
66
6
5
5
5 5
5
44
44
4
33
33
22
22
2
3
11
11
1
O
PO
OH6
5
4
PO
O
PO
OH6
5
4
O
6
5
4
-O
OP
HO
O
6
5
4
OP
O
6
5
4
O
OP2
3
1HO
O
OP2
3
1HO
O
PO
2
3
1
O
PO
OH
2
3
1
O
PO
2
3
1
O
O
O
OH
OH
HO
PO
6
5
4
3
2
1
O
OH
OH
HO
PO
6
5
4
3
2
1
OOH
OH
OHHO
PO
6
5
4
3
2
1
OOH
OH
OP2
3
1HO
OPO
O
OH
O
1
2
3
HOHO
HOHO
HO
HO
Glucose ⇒ 2 Pyruvate
Reaction databases usu-ally list only products,educts, and (sometimes)the type of transforma-tion, but not the atommap itself.
Rule composition: all pos-sible atom traces, here forthe glycolysis EMP andED pathways
23
The Formose ChemistryFour Rules:
C
H
C
O
L
C
H
C
O
K
C
H
C
O
R
(a) Keto-to-enol (r1). (Enol-to-keto (r−11 ))
C C
O C
OH
LC C
O C
OH
KC C
O C
OH
R
(b) Aldol addition (r2). (Reverse aldol addition (r−12 ))
24
Formose Cycle(s)
O
HO HO
+
O
O
HO
OH
HO
OH
OH
OH
OH
O
HO
OH
OH
OH
OH
O
OH
O
O
+
+
OH
OH
OH
HO
OH
OH
HO
O
OH
HO
O
OH
+
OH
HOO
OH
OHOH
OH
OHOH
HO
O
OH
OH
HO
OH
OH
O
+
25
Carbon Atom Traces in Formose
26
Conclusions
27
Conclusions
27
Conclusions
Rule composition in the DPO frameworkI rigorously grounded in category theoryI automatic coarse grainingI inference of
I all atom tracesI alternative relative timing
27