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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, 5:175. Elad Gerson, Spring 2006, Technion. Agenda. Quick overview of last week’s lecture. Extension of the EP concept.

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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, 5:175

Elad Gerson, Spring 2006, Technion.

• Quick overview of last week’s lecture.

• Extension of the EP concept.

• Enter EM.

• General framework for EM computation.

• Reversible reactions split.

• Network compression.

• Post processing.

• Some implementation tweaks.

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.

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

Notice that we are interested only in solutions where

(sign suggests reaction’s 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.

Biology suggests some reaction are reversible.

Consider the following network for instance –

R5 can work in both directions (Not simultaneously!)

Remove the restriction, signs suggests direction ..

• Not all reactions are reversible.

• Solutions no longer take the form of a polyhedral cone.

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

Removing redundancies

Can be united..

Removing redundancies

R1 is null in any feasible

Removing redundancies

Contradict each other ..

Can be eliminated.

Removing redundancies

Active in any stead state.

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.

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.

• 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!)

### Seminar in bioinformatics

Minimal cut sets in biochemical reaction

networks

Steffen Klamt and Ernst Dieter Gilles

Bioinformatics Vol. 20 no. 2 2004, pages 226–234

Elad Gerson, Spring 2006, Technion.

• 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.

• 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.

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.

• That’s easy .. Consider C0 = {obR}

• One might wish to cut the reaction at the beginning.

• What if there are numerous obR’s ?

• Simultaneous failure might be achieved more efficiently.

• Take two – Remove all reactions except for oBR.

• Not efficient.

• Notintelligent.

• 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.

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 ?

Is C2 = {R2, R4, R6} minimal ?

Is C3 = {R2, R5, R7} ?

How about C1 = {R1} ?

• OK, what about Graph disconnectivity algorithms ?

• No good, They don’t take the hypergraph nature of metabolic pathways into account.

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_obR[i][j]==1

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_obR[i][j]=1 for each EM i), otherwise to precutsets

(5) FOR i=2 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) precutsets=new_precutsets

ENDIF

ENDFOR

(6) result: mcs contains the MCSs

Initialization – Calculate EM

We are only interested

in EM containing obR

Initialization

mcs = {{1}}, precutsets = {{2},{3},{4},{5},{6},{7},{8}}

I = 2, j = 1

mcs = {{1}}, precutsets = {{2},{3},{4},{5},{6},{7},{8}}

new_precutsets = {}

temp_precutsets = {}

I = 2, j = 1

mcs = {{1}}, precutsets = {{2},{3},{4},{5},{6},{7},{8}}

new_precutsets = {}

temp_precutsets = {{1 2}}

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}}

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}}

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}}

I = 2, j = 2

mcs = {{1}}, precutsets = {{2},{3},{4},{5},{6},{7},{8}}

new_precutsets = {}

temp_precutsets = {}

I = 2, j = 2

mcs = {{1}}, precutsets = {{3},{4},{5},{6},{7},{8}}

new_precutsets = {}

temp_precutsets = {}

I = 2, j = 2

mcs = {{1}}, precutsets = {{3},{4},{5},{6},{7},{8}}

new_precutsets = {}

temp_precutsets = {{2 4}}

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}}

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}}

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 = {}

I = 2, j = 5

mcs = {{1}}, precutsets = {{5},{6},{7},{8}}

new_precutsets = {{2 4},{2 6},{2 7},{2 8}, ..}

temp_precutsets = {}

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}}

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}}

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}}

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}}

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 = {}

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 = {}

I = 3, j = 2

mcs = {{1}, {5 6}, {5 7}, {5 8}}, precutsets = {…{4 6},…}

new_precutsets = {}

temp_precutsets = {…}

I = 3, j = 2

mcs = {{1}, {5 6}, {5 7}, {5 8}}, precutsets = {…{4 6},…}

new_precutsets = {…}

temp_precutsets = {{2 4 6},…}

I = 3, j = 2

mcs = {{1}, {5 6}, {5 7}, {5 8}, {2 4 6}}, precutsets = {…{4 6},…}

new_precutsets = {…}

temp_precutsets = {…}

I = 3, j = 8

mcs = {{1}, {5 6}, {5 7}, {5 8}, {2 4 6},…}, precutsets = {}

new_precutsets = {…}

temp_precutsets = {…}

• Let q be the number of reactions.

• Assuming |EM| << 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 << q bounds polynomial approximation.

• MCS calculated with ‘biomass synthesis’as 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, glyceroland succinate.

• Growth on each substrate was tested separately.

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

Structural fragility and robustness

Definition – Reaction fragility factor Fi is the reciprocal of the

average size of all the MCSs the reaction i participates.

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 reaction’s

importance.

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?

Structural fragility and robustness

Structural fragility and robustness

Definition – Network fragility F is defined as

where q is then number of reactions.

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.

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 EM’s.

• MCS 0, 2, 3, 4 will not affect EM1