slides - FMMB 2014
50 Shades of Rule Composition From Chemical Reactions to Higher Levels of Abstraction
Jakob L. Andersen, Christoph Flamm Daniel Merkle, Peter Stadler Department of Mathematics and Computer Science University of Southern Denmark
FMMB 2014, Noumea September 22, 2014
1
Metabolic Flux Pattern and Atom Maps
Kl¨ unemann et al. (2014): Computational tools for modeling xenometabolism of the human gut microbiota, Trends in Biotechnology, 32(3):157-165 2
Isotope Labeling Experiments 1. Introduce a isotope labeled substrate into the cell culture at metabolic steady state. 2. Allow the system to reach an isotopic steady state. 3. Measure (e.g. NMR, MS) relative labeling in metabolic 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 Network Estimating fluxes from isotopomere patterns is an inverse problem.
The bijective atom-atom mapping between reaction educts and products must be known.
Getting this information is at least a graph isomorphism-hard problem. 6
9
C
C
2 C C
C
4
C
8
1 C
5
8C
C C
7
C1
C
9
7C
5 C
C
2
3 C
C
4
3 C
6
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 I I
An Enzymatic Reaction: β-Lactamase A Pathway: Glycolysis (An Autocatalytic Reaction: Formose)
Molecule Encoding A molecule is an undirected labelled graph. Vertex label ≡ atom type (e.g., “C” or “O-”) Edge label ≡ bond type (e..g, “-”, “=” or “#”)
H -
-
O -
H
H
O
C = C O H
H C C
- H
H
(a) Visualization of encoding
HO
O
H
(b) Prettified visualization
OH
(c) Open Babel visualization Figure: 1,2-ethenediol 6
Reaction Patterns – Graph Transformation Rules A reaction pattern is a graph transformation rule, in the Double Pushout Formalism: p = (L ← K → R). L C
K C
O
C C
H
O
R C
O
C C
H
O
C
O
C H
O
Figure: Transformation rule for aldol addition
(As for the graphs: the rules are not restricted to chemistry.)
7
Reactions – Application of Transformation Rules aldol addition
1,2-ethenediol + formaldehyde −−−−−−−−→ glyceraldehyde L
K C
C O
O
C O
H
R C
C
O
H O
H
O
H
G
H
H
O
D
H
C
C O
C
H
O
H H
C
C O
C
H
O
H H
C
O
H
O C
C H
H H
O
C H
C
C
C
H H
O
H
H
8
Double Pushout Approach
Double pushout L
l
m
G
K
r
k ρ
D
R n
λ
H
9
Pushout and Pullback : Rule Composition l
r
l
1 1 p1 = (L1 ← − K1 − → R1 )
r
2 2 p2 = (L2 ← − K2 − → R2 )
Rule composition L1
l1
u1 L
s1
K1
r1
v1
(1)
C1
R1
L2 e1
e2
t1
E
s2
w1
(3)
w2
l2
K2
(2)
v2 C2
r2
R2 u2
t2
R
K
10
Pushout and Pullback : Rule Composition l
r
l
1 1 p1 = (L1 ← − K1 − → R1 )
r
2 2 p2 = (L2 ← − K2 − → R2 )
Rule composition L1
l1
u1 L
s1
K1
r1
v1
(1)
C1
R1
L2 e1
e2
t1
E
s2
w1
(3)
w2
l2
K2
(2)
v2 C2
r2
R2 u2
t2
R
K
q
qr
l A composition (L ← − K −→ R) = p1 ∗E p2 can be defined1 and exists2 , ...
1 2
Ehrig et al. (1991): Parallelism and Concurrency in High-Level Replacement Systems. Math. Struct. Comp.C, 1:361–404 Golas (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
R1
L2
r2
R2
=⇒
L∼ = L1
r
R
R2
One Shade of Composing Rules (straight-forward) R1 is a super-graph of L2 L1
H2 C H2 C H2 C C H2
r1
H2 C CH CH H2 C
R1
L2
r2
H2 C CH
r1
C
H2 C H2 C
CH3 H2 C
H2 C H2 C C H2
H2 C CH CH H2 C
C H2
H2 C CH CH H2 C
r
C
C
C
R2
C
H2 C
C C
CH
CH
C C H2
r2
C
C
C
C C
H2 C CH
C H2
C C
CH3
H2 C CH3
R
C CH
H2 C CH
r
L∼ = L1
=⇒
R2
CH3
A Darker Shade of Composing Rules (masochistic) R1 is a super-graph of a connected component of L2 L1
r1
R1
L12
L1 r2
L22
R2
r
=⇒
R
R2
L22
12
A Darker Shade of Composing Rules (masochistic) R1 is a super-graph of a connected component of L2 L1
r1
R1
L12
L1 r2
r
=⇒
R2
L22
H2 C H2 C
H2 C CH
H2 C
H2 C
r1
H2 C
CH C H2
C CH
C
CH
C
H2 C
C
CH C H2
C C
r2
C
H2 C
C CH
H2 C
R2
C C
r
H2 C
C
C C
CH C H2
C
C
CH
H2 C
C C
C
C H2
H2 C
R
L22
C C
12
And an Even Darker Shade of Composing Rules (perverted)
The matching morphism is a common subgraph
L1
r1
R1
L2
r2
R2
=⇒
L
r
R
13
And an Even Darker Shade of Composing Rules (perverted)
The matching morphism is a common subgraph
L1
r1
R1
r2
L2
C C
=⇒
C C
C
R2
C
r1
C
C
C
C C
C
C
C
C C
C C
C
C
C C
r2
C
r
C
C
C
C
C
C
C
C C
C
C
C
C
R
C C
C
C
C
r
L
C C C
13
A Lighter Shade of Composing Rules
Parallel Composition L1 L2
r1 r2
R1 =⇒ R2
L∼ = L1 ∪ L2
r
R∼ = R1 ∪ R2
14
β-Lactamase
+
→
15
β-Lactamase
+
→
Step 1 Lys73 deprotonates Ser70, which initiates a nucleophilic addition onto the carbonyl carbon of the beta-lactam, forming a tetrahedral intermediate. Step 2 The tetrahedral intermediate collapses, cleaving the C-N bond in the beta-lactam, the nitrogen deprotonates 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
L r1 :
K
R
C
C
C
C
C
C
OH
C
O
C
C O
H2 N
C
C
H
O
H2 N
C O C
C
O−
H3 N+
C
β-Lactamase L r1 :
K
C
C
C
C
C
C
OH
C
O
C
O
C O
H2 N
C
C
H
O
H2 N
L
C
C
N
C
C
O
C
H
−
NH3 + C
O
C O−
C N
C
C
r5 :
C O
C
N
C
OH
C
H
C
O
HO
O
HO
C
C
OH
O
C
N
R N OH
O
C C
C
C
C N
O
C C
OH
O−
O
O
K
O−
C
OH
C
L N
NH2
R
O
C
C C
C
NH2
K O
O
C
C
C H2 O
C O
O
L O
C
R
H
O−
C NH O−
K C
O
O O
L r3 :
C
C N
O O
C
H3 N+
R
C C
OH
r4 :
O−
C
K
C C
r2 :
R
C
C O
H
O
C
C C O
C OH
C
OH O−
C C O
Rule Composition - β-Lactamase ıG ◦ r1 ◦ r2 ◦ r3 ◦ r4 ◦ r5 ◦ ıH
Rule Composition - β-Lactamase ıG ◦ r1 ◦ r2 ◦ r3 ◦ r4 ◦ r5 ◦ ıH Atom Traces: N H2
N H2
CO 2 H H
C O
C
8
O
N H2
O C
H
O
CO 2 H H
N
H NH 2
H N
O
O
C
Ph
H
CO 2 H
10
O
NH 2
CO 2 H
O
O
N H
S
H N
C
N C
H
Ph
H O
H
S
CO 2 H
C
CO 2 H
CO 2 H CO 2 H
CO 2 H
CO 2 H
NH 2
C
NH 2 C NH 2
CO 2
-
Catalytic amino acids: Serine, Lysine, Aspartate
NH 2
C CO 2
-
Rule Composition - β-Lactamase ıG ◦ r1 ◦ r2 ◦ r3 ◦ r4 ◦ r5 ◦ ıH Atom Traces: N H2
N H2
CO 2 H N H C2
CO H 2 HO
O
H
H NH 2
O
8
O
NH 2 C
O
N
CO 2 H
H
NH 2H
C
10 O O
C
C
CO 2 2
NH 2
-
NH 2
CO 2 -
Catalytic amino acids: Serine, Lysine, Aspartate
Ph
H N
O
N
O
H N
S
H CO 2 H S
CO 2 H
CO 2 H
CO 2 H
NH 2
C
NH 2
H N
O C
O
10
HN
C
O
CO 2 H
CO 2 H
NH 2
CNH
O
O H
H
H
CO 2 HC
CO 2 H
CO 2 H
C
C
Ph
H
H
O
NH 2
CO 2 H
CO 2 H
N H2
CO 2 H
C S
CO 2 H
N H2
CO 2 H
C
N
S
2
C
O O
O CO H H
CO 2 H
Ph H N
N C
H
H
H
O
8
C
H N
O C
N H2
Ph
H
CO 2 H
NH 2
C C
CO 2
CO 2 -
-
NH 2
Rule Composition - β-Lactamase
For all permutations σ: ıG ◦ rσ(1) ◦ . . . rσ(5) ◦ ıH
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 recycling step, 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 bond cleavage1 (r1 , r1b , r3 , r2b , r4 , r5 ) (r1 , r1b , r2b , r3 , r4 , r5 ) (r1 , r1b , r2b , r4 , r3 , r5 ) (r1 , r1b , r2b , r4 , r5 , r3 )
(r1b , r1 , r3 , r2b , r4 , r5 ) (r1b , r1 , r2b , r3 , r4 , r5 ) (r1b , r1 , r2b , r4 , r3 , r5 ) (r1b , r1 , r2b , r4 , r5 , r3 )
1
Atanasov et al. (2000): Protonation of the beta-lactam nitrogen is the trigger event in the catalytic action of class A beta-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 for Biomass and Energy, J Mol Cell 39:809-820 | DOI 10.1016/j.molcel.2010.08.031
20
Glycolysis
Transformation Rules: r1 Pyranose-furanose
r7 Phosphomutase
r2 Furanose-linear
r8 Enolase
r3 Ketose-aldose
r9 Keto-enol
r4 ATP-phosphorylation
r10 NAD+-oxoreductase
r5 ATPdephosphorylation
r11 Lactonohydrolase
r6 NAD+phosphorylation
r12 Hydrolyase r13 Reverse aldolase
21
Carbon Atom Trace of Glycolysis Embden-Meyerhof-Parnas (EMP) pathway: Glucose → 2 GAP
}|
z
{
ıG(EMP) ◦ r4 ◦ r1 ◦ r4 ◦ r2 ◦ r13 ◦ r3
◦ (r6 ◦∅ r6 ) ◦ (r5 ◦∅ r5 ) ◦ (r7 ◦∅ r7 ) ◦ (r8 ◦∅ r8 ) ◦ (r5 ◦∅ r5 ) ◦ (r9 ◦∅ r9 ) ◦ ıH(EMP) {z
|
}
2 GAP → 2 Pyruvate
The Entner-Doudoroff (ED) pathway: ıG(ED) ◦ r4 ◦ r10 ◦ r11 ◦ r12 ◦ r13 ◦ r6 ◦ r5 ◦ r7 ◦ r8 ◦ r5 ◦ r9 ◦ ıH(ED) |
{z
}
Glucose → GAP + Pyruvate
|
{z
}
GAP + Pyruvate → 2 Pyruvate
22
Carbon Atom Trace of Glycolysis O
O
6
PO
5
1
4
2
HO
O
6
PO
OH
3
1 2
HO
5
O
PO
1
5
OH
O 1 2
5
OH
1
OH
HO
1
OP
2
O
4
3
HO
O
+
OH
OH
6
PO
3
OP
6
HO
5 4
O
PO
1
PO
OH
OP
6
5
PO
O
1
HO
O
O
6
1 2
5
OH 2 3
4
O
3
O HO
HO
1
OP
2 3
O
HO
1
O
HO
3
O
4
O
PO
2
5 4
-
O
HO
6
PO
OP
3
HO
O
PO
Reaction databases usually list only products, educts, and (sometimes) the type of transformation, but not the atom map itself.
1 2
3
O
4
HO OH
2
5
O
OP
2
3
OH
O
OH
2 4
OH
OH
6
PO
OH 3
1
5
3
HO
5
PO
OH
OH
4
O
6
OP
2
OH 4
HO
OH
Glucose ⇒ 2 Pyruvate 6
2 3
6
OH
3
O OH
PO
2
1
4
HO
1
HO
OH
OH
5 4
OH
OH
6
O
6
HO
3
OH
PO
OH
5 4
O
Rule composition: all possible atom traces, here for the glycolysis EMP and ED pathways
The Formose Chemistry Four Rules: L H C
O C
C
K
R
H
H
O C
C
O C
(a) Keto-to-enol (r1 ). (Enol-to-keto (r1−1 )) L C
K C
O
C C
H
O
R C
O
C C
H
O
C
O
C H
O
(b) Aldol addition (r2 ). (Reverse aldol addition (r2−1 ))
24
Formose Cycle(s) OH
O
O
HO
HO
OH
OH
HO
O
HO
+
+
O O
OH
OH
OH
OH
O
OH
OH HO
HO
+ OH
OH
OH
OH
O
OH
OH
OH
O
OH HO
HO
O
HO
OH
OH
HO
O
HO
OH
+
+ OH
O OH
OH
OH
OH OH
OH
OH O
25
Carbon Atom Traces in Formose
26
Conclusions
27
Conclusions
27
Conclusions
Rule composition in the DPO framework I
rigorously grounded in category theory
I
automatic coarse graining inference of
I
I I
all atom traces alternative relative timing
27
Jakob L. Andersen, Christoph Flamm Daniel Merkle, Peter Stadler Department of Mathematics and Computer Science University of Southern Denmark
FMMB 2014, Noumea September 22, 2014
1
Metabolic Flux Pattern and Atom Maps
Kl¨ unemann et al. (2014): Computational tools for modeling xenometabolism of the human gut microbiota, Trends in Biotechnology, 32(3):157-165 2
Isotope Labeling Experiments 1. Introduce a isotope labeled substrate into the cell culture at metabolic steady state. 2. Allow the system to reach an isotopic steady state. 3. Measure (e.g. NMR, MS) relative labeling in metabolic 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 Network Estimating fluxes from isotopomere patterns is an inverse problem.
The bijective atom-atom mapping between reaction educts and products must be known.
Getting this information is at least a graph isomorphism-hard problem. 6
9
C
C
2 C C
C
4
C
8
1 C
5
8C
C C
7
C1
C
9
7C
5 C
C
2
3 C
C
4
3 C
6
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 I I
An Enzymatic Reaction: β-Lactamase A Pathway: Glycolysis (An Autocatalytic Reaction: Formose)
Molecule Encoding A molecule is an undirected labelled graph. Vertex label ≡ atom type (e.g., “C” or “O-”) Edge label ≡ bond type (e..g, “-”, “=” or “#”)
H -
-
O -
H
H
O
C = C O H
H C C
- H
H
(a) Visualization of encoding
HO
O
H
(b) Prettified visualization
OH
(c) Open Babel visualization Figure: 1,2-ethenediol 6
Reaction Patterns – Graph Transformation Rules A reaction pattern is a graph transformation rule, in the Double Pushout Formalism: p = (L ← K → R). L C
K C
O
C C
H
O
R C
O
C C
H
O
C
O
C H
O
Figure: Transformation rule for aldol addition
(As for the graphs: the rules are not restricted to chemistry.)
7
Reactions – Application of Transformation Rules aldol addition
1,2-ethenediol + formaldehyde −−−−−−−−→ glyceraldehyde L
K C
C O
O
C O
H
R C
C
O
H O
H
O
H
G
H
H
O
D
H
C
C O
C
H
O
H H
C
C O
C
H
O
H H
C
O
H
O C
C H
H H
O
C H
C
C
C
H H
O
H
H
8
Double Pushout Approach
Double pushout L
l
m
G
K
r
k ρ
D
R n
λ
H
9
Pushout and Pullback : Rule Composition l
r
l
1 1 p1 = (L1 ← − K1 − → R1 )
r
2 2 p2 = (L2 ← − K2 − → R2 )
Rule composition L1
l1
u1 L
s1
K1
r1
v1
(1)
C1
R1
L2 e1
e2
t1
E
s2
w1
(3)
w2
l2
K2
(2)
v2 C2
r2
R2 u2
t2
R
K
10
Pushout and Pullback : Rule Composition l
r
l
1 1 p1 = (L1 ← − K1 − → R1 )
r
2 2 p2 = (L2 ← − K2 − → R2 )
Rule composition L1
l1
u1 L
s1
K1
r1
v1
(1)
C1
R1
L2 e1
e2
t1
E
s2
w1
(3)
w2
l2
K2
(2)
v2 C2
r2
R2 u2
t2
R
K
q
qr
l A composition (L ← − K −→ R) = p1 ∗E p2 can be defined1 and exists2 , ...
1 2
Ehrig et al. (1991): Parallelism and Concurrency in High-Level Replacement Systems. Math. Struct. Comp.C, 1:361–404 Golas (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
R1
L2
r2
R2
=⇒
L∼ = L1
r
R
R2
One Shade of Composing Rules (straight-forward) R1 is a super-graph of L2 L1
H2 C H2 C H2 C C H2
r1
H2 C CH CH H2 C
R1
L2
r2
H2 C CH
r1
C
H2 C H2 C
CH3 H2 C
H2 C H2 C C H2
H2 C CH CH H2 C
C H2
H2 C CH CH H2 C
r
C
C
C
R2
C
H2 C
C C
CH
CH
C C H2
r2
C
C
C
C C
H2 C CH
C H2
C C
CH3
H2 C CH3
R
C CH
H2 C CH
r
L∼ = L1
=⇒
R2
CH3
A Darker Shade of Composing Rules (masochistic) R1 is a super-graph of a connected component of L2 L1
r1
R1
L12
L1 r2
L22
R2
r
=⇒
R
R2
L22
12
A Darker Shade of Composing Rules (masochistic) R1 is a super-graph of a connected component of L2 L1
r1
R1
L12
L1 r2
r
=⇒
R2
L22
H2 C H2 C
H2 C CH
H2 C
H2 C
r1
H2 C
CH C H2
C CH
C
CH
C
H2 C
C
CH C H2
C C
r2
C
H2 C
C CH
H2 C
R2
C C
r
H2 C
C
C C
CH C H2
C
C
CH
H2 C
C C
C
C H2
H2 C
R
L22
C C
12
And an Even Darker Shade of Composing Rules (perverted)
The matching morphism is a common subgraph
L1
r1
R1
L2
r2
R2
=⇒
L
r
R
13
And an Even Darker Shade of Composing Rules (perverted)
The matching morphism is a common subgraph
L1
r1
R1
r2
L2
C C
=⇒
C C
C
R2
C
r1
C
C
C
C C
C
C
C
C C
C C
C
C
C C
r2
C
r
C
C
C
C
C
C
C
C C
C
C
C
C
R
C C
C
C
C
r
L
C C C
13
A Lighter Shade of Composing Rules
Parallel Composition L1 L2
r1 r2
R1 =⇒ R2
L∼ = L1 ∪ L2
r
R∼ = R1 ∪ R2
14
β-Lactamase
+
→
15
β-Lactamase
+
→
Step 1 Lys73 deprotonates Ser70, which initiates a nucleophilic addition onto the carbonyl carbon of the beta-lactam, forming a tetrahedral intermediate. Step 2 The tetrahedral intermediate collapses, cleaving the C-N bond in the beta-lactam, the nitrogen deprotonates 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
L r1 :
K
R
C
C
C
C
C
C
OH
C
O
C
C O
H2 N
C
C
H
O
H2 N
C O C
C
O−
H3 N+
C
β-Lactamase L r1 :
K
C
C
C
C
C
C
OH
C
O
C
O
C O
H2 N
C
C
H
O
H2 N
L
C
C
N
C
C
O
C
H
−
NH3 + C
O
C O−
C N
C
C
r5 :
C O
C
N
C
OH
C
H
C
O
HO
O
HO
C
C
OH
O
C
N
R N OH
O
C C
C
C
C N
O
C C
OH
O−
O
O
K
O−
C
OH
C
L N
NH2
R
O
C
C C
C
NH2
K O
O
C
C
C H2 O
C O
O
L O
C
R
H
O−
C NH O−
K C
O
O O
L r3 :
C
C N
O O
C
H3 N+
R
C C
OH
r4 :
O−
C
K
C C
r2 :
R
C
C O
H
O
C
C C O
C OH
C
OH O−
C C O
Rule Composition - β-Lactamase ıG ◦ r1 ◦ r2 ◦ r3 ◦ r4 ◦ r5 ◦ ıH
Rule Composition - β-Lactamase ıG ◦ r1 ◦ r2 ◦ r3 ◦ r4 ◦ r5 ◦ ıH Atom Traces: N H2
N H2
CO 2 H H
C O
C
8
O
N H2
O C
H
O
CO 2 H H
N
H NH 2
H N
O
O
C
Ph
H
CO 2 H
10
O
NH 2
CO 2 H
O
O
N H
S
H N
C
N C
H
Ph
H O
H
S
CO 2 H
C
CO 2 H
CO 2 H CO 2 H
CO 2 H
CO 2 H
NH 2
C
NH 2 C NH 2
CO 2
-
Catalytic amino acids: Serine, Lysine, Aspartate
NH 2
C CO 2
-
Rule Composition - β-Lactamase ıG ◦ r1 ◦ r2 ◦ r3 ◦ r4 ◦ r5 ◦ ıH Atom Traces: N H2
N H2
CO 2 H N H C2
CO H 2 HO
O
H
H NH 2
O
8
O
NH 2 C
O
N
CO 2 H
H
NH 2H
C
10 O O
C
C
CO 2 2
NH 2
-
NH 2
CO 2 -
Catalytic amino acids: Serine, Lysine, Aspartate
Ph
H N
O
N
O
H N
S
H CO 2 H S
CO 2 H
CO 2 H
CO 2 H
NH 2
C
NH 2
H N
O C
O
10
HN
C
O
CO 2 H
CO 2 H
NH 2
CNH
O
O H
H
H
CO 2 HC
CO 2 H
CO 2 H
C
C
Ph
H
H
O
NH 2
CO 2 H
CO 2 H
N H2
CO 2 H
C S
CO 2 H
N H2
CO 2 H
C
N
S
2
C
O O
O CO H H
CO 2 H
Ph H N
N C
H
H
H
O
8
C
H N
O C
N H2
Ph
H
CO 2 H
NH 2
C C
CO 2
CO 2 -
-
NH 2
Rule Composition - β-Lactamase
For all permutations σ: ıG ◦ rσ(1) ◦ . . . rσ(5) ◦ ıH
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 recycling step, 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 bond cleavage1 (r1 , r1b , r3 , r2b , r4 , r5 ) (r1 , r1b , r2b , r3 , r4 , r5 ) (r1 , r1b , r2b , r4 , r3 , r5 ) (r1 , r1b , r2b , r4 , r5 , r3 )
(r1b , r1 , r3 , r2b , r4 , r5 ) (r1b , r1 , r2b , r3 , r4 , r5 ) (r1b , r1 , r2b , r4 , r3 , r5 ) (r1b , r1 , r2b , r4 , r5 , r3 )
1
Atanasov et al. (2000): Protonation of the beta-lactam nitrogen is the trigger event in the catalytic action of class A beta-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 for Biomass and Energy, J Mol Cell 39:809-820 | DOI 10.1016/j.molcel.2010.08.031
20
Glycolysis
Transformation Rules: r1 Pyranose-furanose
r7 Phosphomutase
r2 Furanose-linear
r8 Enolase
r3 Ketose-aldose
r9 Keto-enol
r4 ATP-phosphorylation
r10 NAD+-oxoreductase
r5 ATPdephosphorylation
r11 Lactonohydrolase
r6 NAD+phosphorylation
r12 Hydrolyase r13 Reverse aldolase
21
Carbon Atom Trace of Glycolysis Embden-Meyerhof-Parnas (EMP) pathway: Glucose → 2 GAP
}|
z
{
ıG(EMP) ◦ r4 ◦ r1 ◦ r4 ◦ r2 ◦ r13 ◦ r3
◦ (r6 ◦∅ r6 ) ◦ (r5 ◦∅ r5 ) ◦ (r7 ◦∅ r7 ) ◦ (r8 ◦∅ r8 ) ◦ (r5 ◦∅ r5 ) ◦ (r9 ◦∅ r9 ) ◦ ıH(EMP) {z
|
}
2 GAP → 2 Pyruvate
The Entner-Doudoroff (ED) pathway: ıG(ED) ◦ r4 ◦ r10 ◦ r11 ◦ r12 ◦ r13 ◦ r6 ◦ r5 ◦ r7 ◦ r8 ◦ r5 ◦ r9 ◦ ıH(ED) |
{z
}
Glucose → GAP + Pyruvate
|
{z
}
GAP + Pyruvate → 2 Pyruvate
22
Carbon Atom Trace of Glycolysis O
O
6
PO
5
1
4
2
HO
O
6
PO
OH
3
1 2
HO
5
O
PO
1
5
OH
O 1 2
5
OH
1
OH
HO
1
OP
2
O
4
3
HO
O
+
OH
OH
6
PO
3
OP
6
HO
5 4
O
PO
1
PO
OH
OP
6
5
PO
O
1
HO
O
O
6
1 2
5
OH 2 3
4
O
3
O HO
HO
1
OP
2 3
O
HO
1
O
HO
3
O
4
O
PO
2
5 4
-
O
HO
6
PO
OP
3
HO
O
PO
Reaction databases usually list only products, educts, and (sometimes) the type of transformation, but not the atom map itself.
1 2
3
O
4
HO OH
2
5
O
OP
2
3
OH
O
OH
2 4
OH
OH
6
PO
OH 3
1
5
3
HO
5
PO
OH
OH
4
O
6
OP
2
OH 4
HO
OH
Glucose ⇒ 2 Pyruvate 6
2 3
6
OH
3
O OH
PO
2
1
4
HO
1
HO
OH
OH
5 4
OH
OH
6
O
6
HO
3
OH
PO
OH
5 4
O
Rule composition: all possible atom traces, here for the glycolysis EMP and ED pathways
The Formose Chemistry Four Rules: L H C
O C
C
K
R
H
H
O C
C
O C
(a) Keto-to-enol (r1 ). (Enol-to-keto (r1−1 )) L C
K C
O
C C
H
O
R C
O
C C
H
O
C
O
C H
O
(b) Aldol addition (r2 ). (Reverse aldol addition (r2−1 ))
24
Formose Cycle(s) OH
O
O
HO
HO
OH
OH
HO
O
HO
+
+
O O
OH
OH
OH
OH
O
OH
OH HO
HO
+ OH
OH
OH
OH
O
OH
OH
OH
O
OH HO
HO
O
HO
OH
OH
HO
O
HO
OH
+
+ OH
O OH
OH
OH
OH OH
OH
OH O
25
Carbon Atom Traces in Formose
26
Conclusions
27
Conclusions
27
Conclusions
Rule composition in the DPO framework I
rigorously grounded in category theory
I
automatic coarse graining inference of
I
I I
all atom traces alternative relative timing
27
