carbon tracing in chemical reaction networksmy thesishow to model a metabolic networkemu...

My thesis How to model a metabolic network EMU Model Considerations Complexity Conclusion

Carbon Tracing in Chemical Reaction Networks

Jonas Nyrup

University of Southern Denmark

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Outline

My thesis

How to model a metabolic network

EMU Model

Considerations

Complexity

Conclusion

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My thesis

I Where can a specific carbon in substrates end up?I Where could a specific carbon in products come from?I Complexity questions

Extensions:I For k specific carbons. . .I Expand to probabilistic likelihood modelI Substrates with labeling distribution

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Reaction network

Weitzel, Wiechert, and Nöh, 2007

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Atom transition network

“What is the probability that a carbon is labeled?”


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Isotopomer network

“What is the probability that a subset of carbons are labeled?”


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Why not just use isotopomer model?

I In the atom model the amount of information per molecule isn, where n is the carbon number.

I In the isotopomer model this number explodes into 2n.

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Limitations of the atom labeling modelFrom isotopomer distribution to positional label enrichment:

A = ( , , , ) = (60%, 25%, 15%, 0%)⇓

A1 = + = 25%A2 = + = 15%

A1 = 25%A2 = 15%

Now what is the probability of ?

A1 · A2 = 3.75%A1 and A2 are not independent.Information is lost!

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Limitations of the atom labeling model

A1 = 25%A2 = 15%



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A1 = 25%A2 = 15%


A1 · A2 = 3.75%


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A1 = 25%A2 = 15%


A1 · A2 = 3.75%

A1 and A2 are not independent.Information is lost!

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A1 = 25%A2 = 15%



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EMU Model

Here, we present a novel framework for the modeling ofisotopic labeling systems that significantly reduces thenumber of system variables without any loss ofinformation. . . We define an EMU of a compound to be amoiety comprising any distinct subset of the compound’satoms.

I The isotopomer model usually has ∼10x more informationthan needed.

I Find the minimal amount of information needed to simulateisotopic labeling.

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Antoniewicz, Kelleher, and Stephanopoulos, 2007

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Stoichiometry matrices for steady state

−v4 v4 0 0 00 −v1−v3 v3 0 00 v2 −v2−v5 v5 00 0 0 −v1−v3 v3v5 0 0 v2 −v2−v5

C1

B2D2B3D3

=

0 0−v1 0

0 00 −v10 0

[ A2A3

][

−v5−v2 v2v3 −v1−v3

][D23B23

]=[

−v5 00 −v1

][B3×C1

A23

][−v6 v6 0

0 −v5−v2 v20 v3 −v1−v3

][ F123D123B123

]=[ 0 0

−v5 00 −v1

][B23×C1

A123

]

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EXAMPLE

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Limitations of the isotopomer modelO OH

OH

OH

HO

HO O OH

OH

OH

HO

PO

PO OH

OH

OH HO

O PO

OH

OH HO

O

OP

O

PO

OH

OH

OP

O

+

6 6

6 6

6

5

5

5 5

5

4 4

4 4

4

3 3

3 3

2 2

2 2

2

3

1 1

1 1

1

O

PO

OH 6

5

4

PO

O

PO

OH 6

5

4

O

6

5

4

- O

OP

HO

O

6

5

4

OP

O

6

5

4

O

OP 2

3

1 HO

O

OP 2

3

1 HO

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

O OH

OH

OH HO

PO 6

5

4

3

2

1

O OH

OH

OP 2

3

1 HO

O O

O

OH

O

1

2 3

HO HO

HO HO

HO HO

O OH

OH

OH

HO

HO O OH

OH

OH

HO

PO

PO OH

OH

OH HO

O PO

OH

OH HO

O

OP

O

PO

OH

OH

OP

O

+

6 6

6 6

3

5

2

5 5

5

4 4

4 4

1

3 3

3 3

2 2

2 2

2

3

1 1

1 1

1

O

PO

OH 3

2

1

PO

O

PO

OH 3

2

1

O

3

2

1

- O

OP

HO

O

3

2

1

OP

O

3

2

1

O

O

OH

OH

HO

PO 6

5

4

3

2

1

O

OH

OH

HO

PO 6

5

4

3

2

1

O OH

OH

OH HO

PO 6

5

4

3

2

1

O OH

OH

HO

HO

HO

−−−A = ED−−−A = EMP

Andersen et al., 201413 / 24


Limitations of the isotopomer modelIdeal Model EMU Model

PO

OH

OHHO

O

OP

O

PO

OH

OH

OP

O

+

6

6

5

5

4

4

3

2

2

3

1

1

O

PO

OH6

5

4

PO

OP2

3

1HO

O

OP2

3

1HO

O

PO

PO

OH

OHHO

O

OP

O

PO

OH

OH

OP

O

+

6

3

2

5

4

1

3

2

2

3

1

1

O

PO

OH3

2

1

PO

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Limitations of the isotopomer model

The fundamental problem

A→ B,Bab → a, b

⇒ 2B1

When backtracking from a B1, does it origin from A1 or A2?

Some sort of global labeling is missing, information is lost

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Limitations of the isotopomer model

The fundamental problem

A→ B,Bab → a, b

⇒ 2B1

When backtracking from a B1, does it origin from A1 or A2?Some sort of global labeling is missing, information is lost

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Could be modelled with 2nP(t, n) = 2nt!(t−n)! states per molecule,

where t is the sum of carbon numbers of all input substrates.

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ConsiderationsFalse positives

Be aware of false positives (if they can exist?)A C

DA can go to both C or D.

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ConsiderationsFalse positives

Be aware of false positives (if they can exist?)A C

DBA can go to both C or D.B can only go to D, so A must go to C .

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ConsiderationsIdentity reactions

Identity reactions can be removed:

abcd → abcd → abdc ⇒ abcd → abdc

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ConsiderationsAutomorphisms

Indistinguishable carbons, due to automorphisms:

C1 C2 C3 = C3 C2 C1

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(Bio)chemical equivalence

Antoniewicz, Kelleher, and Stephanopoulos, 2007

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Complexity

A digraph with infinite many non-simple paths.

A B C Er1 r2

r3

r4

r1 = (3, 4)r2 = (1, 5)r3 = (1, 2)r4 = ()

If we model reactions with permutations there are a finite number ofnon-simple paths before repeating itself.

r312 = r3r1r2 = (1, 2, 5)(3, 4)|r312| = lcm(3, 2) = 6

There are 6 ways to take the loop with distinct outcome.

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ComplexityGetting more complicated

A

F

B C Er2

r3a

r3b

r4r1

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Flow of computation

I Generative chemistryI Input: substrates, targets, reactionsI Output: reaction networkI E.g. derive all glycolysis pathways.

I Compute integer flow solution.I Chemistry happens sequentially, use Petri nets to capture time

aspectI Do reachability analysis with Petri nets

I If possible: A concurrency graph is returnedI Trace carbon in each concurrency graph

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Conclusion

I Current models are lacking information about global labeling.I By adding this, we can trace atoms through a reaction network.I Improve simulation of real chemistry

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References I

Michael Weitzel, Wolfgang Wiechert, andKatharina Nöh. “The topology of metabolic isotopelabeling networks”. In: BMC Bioinformatics 8.1(2007), p. 315. doi: 10.1186/1471-2105-8-315.url:http://dx.doi.org/10.1186/1471-2105-8-315.Maciek R. Antoniewicz, Joanne K. Kelleher, andGregory Stephanopoulos. “Elementary metabolite units(EMU): A novel framework for modeling isotopicdistributions”. In: Metabolic Engineering 9.1 (Jan.2007), pp. 68–86. doi:10.1016/j.ymben.2006.09.001. url: http://dx.doi.org/10.1016/j.ymben.2006.09.001.

http://dx.doi.org/10.1186/1471-2105-8-315

http://dx.doi.org/10.1186/1471-2105-8-315

http://dx.doi.org/10.1016/j.ymben.2006.09.001



References II

Jakob Lykke Andersen et al. “50 Shades of RuleComposition”. In: Formal Methods in Macro-Biology.Springer Science + Business Media, 2014, pp. 117–135.doi: 10.1007/978-3-319-10398-3_9. url: http://dx.doi.org/10.1007/978-3-319-10398-3_9.

http://dx.doi.org/10.1007/978-3-319-10398-3_9

http://dx.doi.org/10.1007/978-3-319-10398-3_9

http://dx.doi.org/10.1007/978-3-319-10398-3_9

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