Seminar in bioinformatics

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Seminar in bioinformatics

• Computation of elementary modes a unifying
  framework and the new binary approach

Julien Gagneur and Steffen Klamt
BMC Bioinformatics 2004, 5175
Elad Gerson, Spring 2006, Technion.
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Agenda

• Quick overview of last weeks lecture.
• Extension of the EP concept.
• Enter EM.
• General framework for EM computation.
• Reversible reactions split.
• Network compression.
• Post processing.
• Some implementation tweaks.

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Last week on bioinformatics seminar!

• Given a metabolic network we wish to find all the
  possible flux distributions which results in a
  steady state.
• Meaning, the overall flux in a pathway is 0.

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Last week on bioinformatics seminar!

• This is done by describing the pathway as a
  stoichiometric matrix S, solving the equation

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Last week on bioinformatics seminar!

• Notice that we are interested only in solutions
  where
• (sign suggests reactions direction).
• Solution space is spanned by linearly independent
  vectors.
• We look for a spanning set s.t. every solution
  can be written as a linear combination of the
  spanning vector where all coefficients are
  non-negative (Genetically independent).
• Those solutions are called
• Extreme pathways (EP).
• Can be found using the Null
• Space Approach (NSA)
• Algorithm.

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Problem

• Biology suggests some reaction are reversible.
• Consider the following network for instance
• R5 can work in both directions (Not
  simultaneously!)

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Solution ?

• Remove the restriction, signs suggests
  direction ..
• Bad idea ..
• Not all reactions are reversible.
• Solutions no longer take the form of a polyhedral
  cone.

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Solution !

• Split the reversible reactions ..
• Find Extreme Pathways using the NSA algorithm.
• Post process found EPs, merge split reactions
  (opposite direction should be set with a
  negative sign).
• Post processed EPs are now called - Elementary
  Modes (EM).

R5a
R5b
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Compressing the network

• Removing redundancies

Can be united..
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Compressing the network

• Removing redundancies

R1 is null in any feasible steady state
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Compressing the network

• Removing redundancies

Contradict each other .. Can be eliminated.
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Compressing the network

• Removing redundancies

Active in any stead state.
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Compressing the network

• Removing redundancies
• Some redundancies can be detected as dependent
  linear rows in the kernel matrix.
• Iterative approach, remove redundancies until non
  detected.
• Produce better results.

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General framework

• Preprocessing -
• Metabolic networks yield deeper insight of
  organisms metabolism.
• Failure modes analysis will provide
• Crucial parts identification.
• Suitable targets for repressing undesired
  metabolic functions.
• Apply NSA algorithm.
• Post process.

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One more tweak

• The authors offers an efficient implementation to
  the NSA and CBA (Combined basis Schuster et.
  al.) algorithms.
• Using binary representation for vectors.
• Fast bit operators.
• Efficient memory usage (up to 1.6 of original!)

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Seminar in bioinformatics

• Minimal cut sets in biochemical reaction
• networks

Steffen Klamt and Ernst Dieter Gilles
Bioinformatics Vol. 20 no. 2 2004, pages 226234
Elad Gerson, Spring 2006, Technion.
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Abstract

• Motivation
• Metabolic networks yield deeper insight of
  organisms metabolism.
• Failure modes analysis will provide
• Crucial parts identification.
• Suitable targets for repressing undesired
  metabolic functions.
• Results
• The biochemical networks minimal cut sets
  concept.
• Algorithm which computes MCS with respect to an
  objective reaction.
• Potential applications includes
• phenotype predictions.
• Network verifications.
• Structural robustness and fragility assessment.
• Metabolic flux analysis.
• Target identification in drug discovery.

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Introduction

• Assume we wish to prevent the production of
  metabolite X.
• i.e. there is no balanced flux distribution
  possible which involves obR.
• Can be done by gene deletion or enzyme inhibition.

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Introduction

• Definition - We call a set of reactions a cut set
  (with
• respect to a defined objective reaction) if after
  the removal
• of these reactions from the network no feasible
  balanced flux
• distribution involves the objective reaction.

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Introduction

• Thats easy .. Consider C0 obR
• One might wish to cut the reaction at the
  beginning.
• What if there are numerous obRs ?
• Simultaneous failure might be achieved more
  efficiently.

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Introduction

• Take two Remove all reactions except for oBR.
• Not efficient.
• Not intelligent.

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Introduction

• Consider C1 R5, R8
• Sufficient.
• Neither the removal of R5 nor R8 is sufficient.
• No subset of C1 is a valid cut set ? C1 is
  minimal.

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Introduction

• Definition - A cut set C (related to a defined
  objective reaction)
• is a minimal cut set (MCS) if no proper subset of
  C is a
• cut set.
• Can you spot all the MCS in the network ?

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Introduction
Is C2 R2, R4, R6 minimal ?
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Introduction
Is C3 R2, R5, R7 ?
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Introduction
How about C1 R1 ?
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Introduction

• OK, what about Graph disconnectivity algorithms ?
• No good, They dont take the hypergraph nature of
  metabolic pathways into account.

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The algorithm

• Initialization
• Calculate the EMs in the given network
• Define the objective reaction obR
• (3) Choose all EMs where reaction obR is
  non-zero and
• store it in the binary array em_obR
  (em_obRij1
• means that reaction j is involved in EM i)
• (4) Initialize arrays mcs and precutsets as
  follows (each
• array contains sets of reaction indices)
  append j to mcs if reaction j is
• essential (em_obRij1 for each EM i),
  otherwise to precutsets

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The algorithm

• (5) FOR i2 TO MAX_CUTSETSIZE
• (5.1) new_precutsets
• (5.2) FOR j 1 TO q (q number of reactions)
• (5.2.1) Remove all sets from precutsets where
  reaction j participates
• (5.2.2) Find all sets of reactions in
  precutsets that do not cover at least one EM in
  em_obR where reaction j participates combine
  each of these sets
• with reaction j and store the new preliminary
  cut sets in temp_precutsets
• (5.2.3) Drop all temp_precutsets which are a
  superset of any of the already determined
  minimal cut sets stored in mcs
• (5.2.4) Find all retained temp_precutsets which
  do nowcover all EMs and
• append them to mcs append all others to
  new_precutsets
• ENDFOR
• (5.3) If isempty(new_precutsets)
• (5.3.1) Break
• ELSE
• (5.3.2) precutsetsnew_precutsets
• ENDIF
• ENDFOR
• (6) result mcs contains the MCSs

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Running example

• Initialization Calculate EM

We are only interested in EM containing obR
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Running example

• Initialization
• mcs 1, precutsets 2,3,4,5,6,7
  ,8

R8 R7 R6 R5 R4 R3 R2 R1 em_obR
1 1 1 0 0 0 0 1 1 (EM2)
0 0 0 1 0 1 1 1 2 (EM3)
0 0 0 1 1 0 0 1 3 (EM4)
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Running example

• I 2, j 1
• mcs 1, precutsets 2,3,4,5,6,7
  ,8
• new_precutsets
• temp_precutsets

R8 R7 R6 R5 R4 R3 R2 R1 em_obR
1 1 1 0 0 0 0 1 1 (EM2)
0 0 0 1 0 1 1 1 2 (EM3)
0 0 0 1 1 0 0 1 3 (EM4)
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Running example

• I 2, j 1
• mcs 1, precutsets 2,3,4,5,6,7
  ,8
• new_precutsets
• temp_precutsets 1 2

R8 R7 R6 R5 R4 R3 R2 R1 em_obR
1 1 1 0 0 0 0 1 1 (EM2)
0 0 0 1 0 1 1 1 2 (EM3)
0 0 0 1 1 0 0 1 3 (EM4)
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Running example

• I 2, j 1
• mcs 1, precutsets 2,3,4,5,6,7
  ,8
• new_precutsets
• temp_precutsets 1 2, 1 3, 1 4, 1 5 1
  6

R8 R7 R6 R5 R4 R3 R2 R1 em_obR
1 1 1 0 0 0 0 1 1 (EM2)
0 0 0 1 0 1 1 1 2 (EM3)
0 0 0 1 1 0 0 1 3 (EM4)
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Running example

• I 2, j 1
• mcs 1, precutsets 2,3,4,5,6,7
  ,8
• new_precutsets
• temp_precutsets 1 2, 1 3, 1 4, 1 5 1
  6 1 7 1 8

R8 R7 R6 R5 R4 R3 R2 R1 em_obR
1 1 1 0 0 0 0 1 1 (EM2)
0 0 0 1 0 1 1 1 2 (EM3)
0 0 0 1 1 0 0 1 3 (EM4)
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Running example

• I 2, j 1
• mcs 1, precutsets 2,3,4,5,6,7
  ,8
• new_precutsets
• temp_precutsets 1 2, 1 3, 1 4, 1 5, 1
  6, 1 7, 1 8

R8 R7 R6 R5 R4 R3 R2 R1 em_obR
1 1 1 0 0 0 0 1 1 (EM2)
0 0 0 1 0 1 1 1 2 (EM3)
0 0 0 1 1 0 0 1 3 (EM4)
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Running example

• I 2, j 2
• mcs 1, precutsets 2,3,4,5,6,7
  ,8
• new_precutsets
• temp_precutsets

R8 R7 R6 R5 R4 R3 R2 R1 em_obR
1 1 1 0 0 0 0 1 1 (EM2)
0 0 0 1 0 1 1 1 2 (EM3)
0 0 0 1 1 0 0 1 3 (EM4)
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Running example

• I 2, j 2
• mcs 1, precutsets 3,4,5,6,7,8
  
• new_precutsets
• temp_precutsets

R8 R7 R6 R5 R4 R3 R2 R1 em_obR
1 1 1 0 0 0 0 1 1 (EM2)
0 0 0 1 0 1 1 1 2 (EM3)
0 0 0 1 1 0 0 1 3 (EM4)
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Running example

• I 2, j 2
• mcs 1, precutsets 3,4,5,6,7,8
  
• new_precutsets
• temp_precutsets 2 4

R8 R7 R6 R5 R4 R3 R2 R1 em_obR
1 1 1 0 0 0 0 1 1 (EM2)
0 0 0 1 0 1 1 1 2 (EM3)
0 0 0 1 1 0 0 1 3 (EM4)
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Running example

• I 2, j 2
• mcs 1, precutsets 3,4,5,6,7,8
  
• new_precutsets
• temp_precutsets 2 4,2 6,2 7,2 8

R8 R7 R6 R5 R4 R3 R2 R1 em_obR
1 1 1 0 0 0 0 1 1 (EM2)
0 0 0 1 0 1 1 1 2 (EM3)
0 0 0 1 1 0 0 1 3 (EM4)
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Running example

• I 2, j 2
• mcs 1, precutsets 3,4,5,6,7,8
  
• new_precutsets 2 4
• temp_precutsets 2 6,2 7,2 8

R8 R7 R6 R5 R4 R3 R2 R1 em_obR
1 1 1 0 0 0 0 1 1 (EM2)
0 0 0 1 0 1 1 1 2 (EM3)
0 0 0 1 1 0 0 1 3 (EM4)
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Running example

• I 2, j 2
• mcs 1, precutsets 3,4,5,6,7,8
  
• new_precutsets 2 4,2 6,2 7,2 8
• temp_precutsets

R8 R7 R6 R5 R4 R3 R2 R1 em_obR
1 1 1 0 0 0 0 1 1 (EM2)
0 0 0 1 0 1 1 1 2 (EM3)
0 0 0 1 1 0 0 1 3 (EM4)
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Running example

• I 2, j 5
• mcs 1, precutsets 5,6,7,8
• new_precutsets 2 4,2 6,2 7,2 8, ..
• temp_precutsets

R8 R7 R6 R5 R4 R3 R2 R1 em_obR
1 1 1 0 0 0 0 1 1 (EM2)
0 0 0 1 0 1 1 1 2 (EM3)
0 0 0 1 1 0 0 1 3 (EM4)
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Running example

• I 2, j 5
• mcs 1, precutsets 6,7,8
• new_precutsets 2 4,2 6,2 7,2 8, ..
• temp_precutsets 5 6,5 7,5 8

R8 R7 R6 R5 R4 R3 R2 R1 em_obR
1 1 1 0 0 0 0 1 1 (EM2)
0 0 0 1 0 1 1 1 2 (EM3)
0 0 0 1 1 0 0 1 3 (EM4)
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Running example

• I 2, j 5
• mcs 1, precutsets 6,7,8
• new_precutsets 2 4,2 6,2 7,2 8, ..
• temp_precutsets 5 6,5 7,5 8

R8 R7 R6 R5 R4 R3 R2 R1 em_obR
1 1 1 0 0 0 0 1 1 (EM2)
0 0 0 1 0 1 1 1 2 (EM3)
0 0 0 1 1 0 0 1 3 (EM4)
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Running example

• I 2, j 5
• mcs 1, 5 6, precutsets 6,7,8
• new_precutsets 2 4,2 6,2 7,2 8, ..
• temp_precutsets 5 7,5 8

R8 R7 R6 R5 R4 R3 R2 R1 em_obR
1 1 1 0 0 0 0 1 1 (EM2)
0 0 0 1 0 1 1 1 2 (EM3)
0 0 0 1 1 0 0 1 3 (EM4)
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Running example

• I 2, j 5
• mcs 1, 5 6, 5 7, precutsets
  6,7,8
• new_precutsets 2 4,2 6,2 7,2 8, ..
• temp_precutsets 5 8

R8 R7 R6 R5 R4 R3 R2 R1 em_obR
1 1 1 0 0 0 0 1 1 (EM2)
0 0 0 1 0 1 1 1 2 (EM3)
0 0 0 1 1 0 0 1 3 (EM4)
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Running example

• I 2, j 8
• mcs 1, 5 6, 5 7, 5 8, precutsets
• new_precutsets 2 4,2 6,2 7,2 8, ..
• temp_precutsets

R8 R7 R6 R5 R4 R3 R2 R1 em_obR
1 1 1 0 0 0 0 1 1 (EM2)
0 0 0 1 0 1 1 1 2 (EM3)
0 0 0 1 1 0 0 1 3 (EM4)
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Running example

• I 3, j 2
• mcs 1, 5 6, 5 7, 5 8, precutsets
  2 4,2 6,2 7,2 8,4 6,
• new_precutsets
• temp_precutsets

R8 R7 R6 R5 R4 R3 R2 R1 em_obR
1 1 1 0 0 0 0 1 1 (EM2)
0 0 0 1 0 1 1 1 2 (EM3)
0 0 0 1 1 0 0 1 3 (EM4)
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Running example

• I 3, j 2
• mcs 1, 5 6, 5 7, 5 8, precutsets
  4 6,
• new_precutsets
• temp_precutsets

R8 R7 R6 R5 R4 R3 R2 R1 em_obR
1 1 1 0 0 0 0 1 1 (EM2)
0 0 0 1 0 1 1 1 2 (EM3)
0 0 0 1 1 0 0 1 3 (EM4)
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Running example

• I 3, j 2
• mcs 1, 5 6, 5 7, 5 8, precutsets
  4 6,
• new_precutsets
• temp_precutsets 2 4 6,

R8 R7 R6 R5 R4 R3 R2 R1 em_obR
1 1 1 0 0 0 0 1 1 (EM2)
0 0 0 1 0 1 1 1 2 (EM3)
0 0 0 1 1 0 0 1 3 (EM4)
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Running example

• I 3, j 2
• mcs 1, 5 6, 5 7, 5 8, 2 4 6,
  precutsets 4 6,
• new_precutsets
• temp_precutsets

R8 R7 R6 R5 R4 R3 R2 R1 em_obR
1 1 1 0 0 0 0 1 1 (EM2)
0 0 0 1 0 1 1 1 2 (EM3)
0 0 0 1 1 0 0 1 3 (EM4)
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Running example

• I 3, j 8
• mcs 1, 5 6, 5 7, 5 8, 2 4 6,,
  precutsets
• new_precutsets
• temp_precutsets

R8 R7 R6 R5 R4 R3 R2 R1 em_obR
1 1 1 0 0 0 0 1 1 (EM2)
0 0 0 1 0 1 1 1 2 (EM3)
0 0 0 1 1 0 0 1 3 (EM4)
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Complexity

• Let q be the number of reactions.
• Assuming EM ltlt q.
• In initialization q singletons are generated and
  tested.
• In the i-th iteration
• Overall number of temp_precutsets generated
• O(p) comparisons are made.
• Hence, (All subsets of q items)
• Yes .. exponential..
• Maximal MCS size ltlt q bounds polynomial
  approximation.

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MCS in central metabolism of E. coli

• MCS calculated with biomass synthesisas
  objective reaction (growth).
• Network comprises 110 reactions and 89
  metabolites.
• Catabolic (material breakdown) part modeled in
  details.
• Enables excretion of 5 metabolites.
• Uptake of glucose, acetate, glycerol and
  succinate.
• Growth on each substrate was tested separately.

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MCS in central metabolism of E. coli
57
Possible applications

• Structural fragility and robustness
• MCS can be used for risk assessment in
  metabolic pathways.
• More EMs suggested a more robust and less fragile
  pathway.
• EMs number and MCSs size are strongly correlated.
  (More elements must fail).
• We seek a better criteria.

Glucose is known to be the least fragile growth
substrate having most EMs and apparently longest
MCSs
Dangerous MCSs
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Possible applications
Structural fragility and robustness
Definition Reaction fragility factor
Fi is the reciprocal of the average size of all
the MCSs the reaction i participates.
59
Possible applications
Structural fragility and robustness
Definition Reaction fragility factor
Fi is the reciprocal of the average size of all
the MCSs the reaction i participates.
May suggest reactions importance.
60
Possible applications
Structural fragility and robustness
Definition Reaction fragility factor
Fi is the reciprocal of the average size of all
the MCSs the reaction i participates.
Is there a correlation between Fi and the number
of EMs the reaction participates?
61
Possible applications
Structural fragility and robustness

62
Possible applications
Structural fragility and robustness D
efinition Network fragility F is defined
as where q is then number of reactions.
63
Possible applications

• Network verification and mutant phenotype
  predictions.
• Cutting an MCS is predicted to leave a metabolic
  pathway dysfunctional.
• Apply the algorithm with growth as obR.
• If a set of gene deletions (or mutants) contains
  an MCS a non-viable phenotype is expected.
• Viable phenotype would be a false negative.
• Proof for incorrect or incomplete network.
• Otherwise growth is possible.
• Non-viable phenotype would be a false positive.
• May suggest a false assumption in the network
  structure.
• One of the reactions in the MCS might be of
  regulatory nature.

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Possible applications

• Target identification and repressing cellular
  functions.
• MCS offers a theoretical tool for target
  identification in drug discovery.
• An irreducible set of interventions needed for
  pathway dysfunction.
• Usually we will look for minimal size of MCS.
• Other pathways should be weakly affected.
• Can be checked easily set of untouched EMs.
• MCS 0, 2, 3, 4 will not affect EM1