50 shades of rule composition

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

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

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

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


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


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


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


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

http://dx.doi.org/10.1016/j.molcel.2010.08.031

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

Rule composition in the DPO frameworkI rigorously grounded in category theoryI automatic coarse grainingI inference of

I all atom tracesI alternative relative timing

27

50 shades of rule composition

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