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Flux Balance Analysis Gapless metabolic reconstruction Esa Pitk anen 27.3.2009 Metabolic Modeling, spring 2009 MBI Programme Department of Computer Science University of Helsinki FBA and gapless reconstruction p. 1/48 Topics today

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SLIDE 1

Flux Balance Analysis Gapless metabolic reconstruction

Esa Pitk¨ anen 27.3.2009 Metabolic Modeling, spring 2009 MBI Programme Department of Computer Science University of Helsinki

FBA and gapless reconstruction – p. 1/48

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SLIDE 2

Topics today

Metabolic reconstruction (revisited) In silico validation of reconstructed models Flux Balance Analysis (FBA) Gapless metabolic reconstruction

FBA and gapless reconstruction – p. 2/48

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Goals of this lecture

Introduce two methods for metabolic network analysis FBA (established) Gapless reconstruction (recent work) Discuss (integer) linear programming (on a brief level only) Discuss some of the many challenges of metabolic modeling Is it possible to achieve useful results with simple models such as stoichiometric models

FBA and gapless reconstruction – p. 3/48

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Metabolic reconstruction (revisited)

FBA and gapless reconstruction – p. 4/48

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Reconstruction process

FBA and gapless reconstruction – p. 5/48

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In silico validation of metabolic mod- els

Reconstructed genome-scale metabolic networks are very large: hundreds or thousands of reactions and metabolites Manual curation is often necessary Amount of manual work needed can be reduced with computational methods Aims to provide a good basis for further analysis and experiments Does not remove the need for experimental verification

FBA and gapless reconstruction – p. 6/48

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Flux Balance Analysis: preliminaries

Recall that in a steady state, metabolite concentrations are constant over time, dXi dt =

r

  • j=1

sijvj = 0, for i = 1, . . . , n, and that a stoichiometric model is given by S = [SII SIE] where SII describes internal metabolites - internal reactions, and SIE internal metabolites - exchange reactions.

FBA and gapless reconstruction – p. 7/48

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Flux Balance Analysis (FBA)

FBA is a framework for investigating the theoretical capabilities of a stoichiometric metabolic model S Analysis is constrained by

  • 1. Steady state assumption Sv = 0
  • 2. Thermodynamic constraints: (ir)reversibility of

reactions

  • 3. Limited reaction rates of enzymes:

Vmin ≤ v ≤ Vmax Note that constraints (2) can be included in Vmin and Vmax.

FBA and gapless reconstruction – p. 8/48

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Flux Balance Analysis (FBA)

In FBA, we are interested in determining the theoretical maximum (minimum) yield of some metabolite, given model For instance, we may be interested in finding how efficiently yeast is able to convert sugar into ethanol Figure: glycolysis in KEGG

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Flux Balance Analysis (FBA)

FBA has applications both in metabolic engineering and metabolic reconstruction Metabolic engineering: find out possible reactions (pathways) to insert or delete Metabolic reconstruction: validate the reconstruction given observed metabolic phenotype

FBA and gapless reconstruction – p. 10/48

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Formulating an FBA problem

We formulate an FBA problem by specifying parameters c in the optimization function Z, Z =

r

  • i=1

civi. Examples: Set ci = 1 if reaction i produces “target” metabolite, and ci = 0 otherwise Growth function: maximize production of biomass constituents Energy: maximize ATP (net) production

FBA and gapless reconstruction – p. 11/48

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Solving an FBA problem

Given a model S, we then seek to find the maximum

  • f Z while respecting the FBA constraints,

max

v

Z = max

v r

  • i=1

civi such that (1) Sv = 0 (2) Vmin ≤ v ≤ Vmax (3) (We could also replace max with min.) This is a linear program, having a linear objective function and linear constraints

FBA and gapless reconstruction – p. 12/48

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Solving a linear program

General linear program formulation: max

xi

  • i

cixi such that Ax ≤ b Algorithms: simplex (worst-case exponential time), interior point methods (polynomial) Matlab solver: linprog (Statistical Toolbox) Many solvers around, efficiency with (very) large models varies

FBA and gapless reconstruction – p. 13/48

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Linear programs

Linear constraints define a convex polyhedron (feasible region) If the feasible region is empty, the problem is infeasible. Unbounded feasible region (in direction of objective function): no optimal solution Given a linear objective func- tion, where can you find the maximum value?

FBA and gapless reconstruction – p. 14/48

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Flux Balance Analysis: example

r1 6PGL NADPH r2 6PG r3 R5P r4 X5P r5 bG6P r6 bF6P r7 r8 aG6P r9 r10 NADPP H2O

Let’s take the course’s running example... Unconstrained uptake (exchange) reactions for NADP+ (r10), NADPH and H2O (not drawn) Constrained uptake for αG6P, 0 ≤ v8 ≤ 1 Objective: production of X5P (v9) c = (0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0)

FBA and gapless reconstruction – p. 15/48

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Flux Balance Analysis: example

r1 6PGL NADPH r2 6PG r3 R5P r4 X5P r5 bG6P r6 bF6P r7 r8 aG6P r9 r10 NADPP H2O

r1 r2 r3 r4 r5 r6 r7 r8 r9 r10 r11 r12 βG6P

  • 1

1

  • 1

αG6P

  • 1
  • 1

1 βF6P 1 1 6PGL 1

  • 1

6PG 1

  • 1

R5P 1

  • 1

X5P 1

  • 1

NADP+

  • 1
  • 1

1 NADPH 1 1 1 H2O

  • 1

1

FBA and gapless reconstruction – p. 16/48

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Flux Balance Analysis: example

Solve the linear program max

v r

  • i

civi = max v9 subject to

r

  • i=1

sijvi = 0 for all j = 1, . . . , 10 0 ≤ v8 ≤ 1 Hint: Matlab’s linprog offers nice convenience functions for specifying equality constraints and bounds

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Flux Balance Analysis: example

r1 1.00 6PGL NADPH r2 1.00 6PG r3 1.00 R5P r4 1.00 X5P r5 0.57 bG6P r6 0.43 bF6P r7 -0.43 r8 1.00 aG6P r9 1.00 r10 2.00 NADPP H2O

Figure gives one possible solution (flux assignment v) Reaction r7 (red) operates in backward direction Uptake of NADP+ v10 = 2v8 = 2 How many solutions (different flux assignments) are there for this problem?

FBA and gapless reconstruction – p. 18/48

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FBA validation of a reconstruction

Check if it is possible to produce metabolites that the organism is known to produce Maximize production of each such metabolite at time Make sure max. production is above zero To check biomass production (growth), add a reaction to the model with stoichiometry corresponding to biomass composition

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FBA validation of a reconstruction

If a maximum yield of some metabolite is lower than measured → missing pathway Iterative process: find metabolite that cannot be produced, fix the problem by changing the model, try again

r1 0.00 6PGL NADPH r2 0.00 6PG r3 0.00 R5P r4 0.00 X5P r5 0.00 bG6P r6 0.00 bF6P r7 0.00 r8 0.00 aG6P r9 0.00 NADPP H2O r1 1.00 6PGL NADPH r2 1.00 6PG r3 1.00 R5P r4 1.00 X5P r5 0.57 bG6P r6 0.43 bF6P r7 -0.43 r8 1.00 aG6P r9 1.00 r10 2.00 NADPP H2O

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FBA validation of a reconstruction

FBA gives the maximum flux given stoichiometry only, i.e., not constrained by regulation or kinetics In particular, assignment of internal fluxes on alternative pathways can be arbitrary (of course subject to problem constraints)

r1 1.00 6PGL NADPH r2 1.00 6PG r3 1.00 R5P r4 1.00 X5P r5 0.57 bG6P r6 0.43 bF6P r7 -0.43 r8 1.00 aG6P r9 1.00 r10 2.00 NADPP H2O r1 1.00 6PGL NADPH r2 1.00 6PG r3 1.00 R5P r4 1.00 X5P r5 0.00 bG6P r6 1.00 bF6P r7 -1.00 r8 1.00 aG6P r9 1.00 r10 2.00 NADPP H2O

FBA and gapless reconstruction – p. 21/48

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Gapless metabolic reconstruction

Motivation: Current workflows choose “good” reactions by sequence evidence, fix problems later manually or automatically

FBA and gapless reconstruction – p. 22/48

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What is a (reaction) gap?

A reaction in the metabolic network that “should be there” but is not Sequencing failure Correct ortholog not found Correct ortholog found but misannotated Correct reaction not in metabolic database(s) (previously unknown function) In the prediction context, a gap is a false negative prediction

FBA and gapless reconstruction – p. 23/48

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Gaps in metabolic models

Central metabolism usually well covered (well conserved!) Glycolysis TCA cycle Pentose phosphate pathway Amino acid pathways Lots of problems with other parts even in commonly used models

FBA and gapless reconstruction – p. 24/48

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Why bother with gaps?

Gaps cause problems with both qualitative and quantitive analyses Consider FBA for example A single reaction gap can block flux through multiple reactions Particularly problematic with branching pathways Ultimately, gaps can lead into false predictions, leading in the worst case to unnecessary experiments (Same applies to false positives, i.e., extra reactions)

FBA and gapless reconstruction – p. 25/48

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Effect of gaps

Red: gap, Orange: cannot carry flux, Green: can carry flux

r1 6PGL NADPH r2 6PG r3 R5P r4 X5P r5 bG6P r6 bF6P r7 r8 aG6P r9 r10 NADPP H2O r1 6PGL NADPH r2 6PG r3 R5P r4 X5P r5 bG6P r6 bF6P r7 r8 aG6P r9 r10 NADPP H2O r1 6PGL NADPH r2 6PG r3 R5P r4 X5P r5 bG6P r6 bF6P r7 r8 aG6P r9 r10 NADPP

FBA and gapless reconstruction – p. 26/48

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Modeling gaps with AND-OR graphs

Let A be the set of input metabolites and reactions Reaction r is reachable, iff all its substrates are reachable, or r ∈ A (AND node) Metabolite m is reachable, iff at least one of its producing reactions is reachable, or m ∈ A (OR node)

r1 6PGL NADPH r2 6PG r3 R5P r4 X5P r5 bG6P r6 bF6P r7 r8 aG6P r9 r10 NADPP H2O

For example, reaction r1 is reachable only if both NADPP and βG6P are reachable.

FBA and gapless reconstruction – p. 27/48

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Modeling gaps with AND-OR graph

Reachable reaction: has substrates that the network is able to produce (under this simple model) Reachable metabolite: can be produced by one or more reactions that are able to operate A gap in the model: reaction or metabolite not reachable given inputs A. Let’s take an example...

FBA and gapless reconstruction – p. 28/48

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Reachability in AND-OR graphs 1/7

r1 m4 r2 m7 r3 m5 m6 r4 r5 m8 r6 m1 r7 m2 r8 m3

Set A = {r6, r7}. Inputs A used to model system boundaries.

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Reachability in AND-OR graphs 2/7

r1 m4 r2 m7 r3 m5 m6 r4 r5 m8 r6 m1 r7 m2 r8 m3 r1 m4 r2 m7 r3 m5 m6 r4 r5 m8 r6 m1 r7 m2 r8 m3

r6 produces m1, r7 produces m2.

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Reachability in AND-OR graphs 3/7

r1 m4 r2 m7 r3 m5 m6 r4 r5 m8 r6 m1 r7 m2 r8 m3 r1 m4 r2 m7 r3 m5 m6 r4 r5 m8 r6 m1 r7 m2 r8 m3

Both r1 and r2 have all substrates reachable.

FBA and gapless reconstruction – p. 31/48

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Reachability in AND-OR graphs 4/7

r1 m4 r2 m7 r3 m5 m6 r4 r5 m8 r6 m1 r7 m2 r8 m3 r1 m4 r2 m7 r3 m5 m6 r4 r5 m8 r6 m1 r7 m2 r8 m3

r4 is reachable. r3 and r5 remain unreachable.

FBA and gapless reconstruction – p. 32/48

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Reachability in AND-OR graphs 5/7

r1 m4 r2 m7 r3 m5 m6 r4 r5 m8 r6 m1 r7 m2 r8 m3 r1 m4 r2 m7 r3 m5 m6 r4 r5 m8 r6 m1 r7 m2 r8 m3

If r8 is added to the initially reachable reactions, A = {r6, r7, r8}, m3 becomes reachable.

FBA and gapless reconstruction – p. 33/48

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Reachability in AND-OR graphs 6/7

r1 m4 r2 m7 r3 m5 m6 r4 r5 m8 r6 m1 r7 m2 r8 m3 r1 m4 r2 m7 r3 m5 m6 r4 r5 m8 r6 m1 r7 m2 r8 m3

r3, m5 and m6 are reached...

FBA and gapless reconstruction – p. 34/48

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SLIDE 35

Reachability in AND-OR graphs 7/7

r1 m4 r2 m7 r3 m5 m6 r4 r5 m8 r6 m1 r7 m2 r8 m3 r1 m4 r2 m7 r3 m5 m6 r4 r5 m8 r6 m1 r7 m2 r8 m3

...and finally r5 and m8.

FBA and gapless reconstruction – p. 35/48

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Gapless metabolic reconstruction

Input A set of reactions R (e.g., all reactions in KEGG) Inputs A Score function f : R → R. Task: find a subset R ⊆ R such that F(R) =

r∈R f(r) is maximized

All reactions r ∈ R are reachable given A in the AND-OR graph induced by R

FBA and gapless reconstruction – p. 36/48

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Score function f

If f > 0, the solution contains all reachable reactions If f < 0, the solution is empty Interesting case: reactions have both positive and negative scores

r1, f(r1)=1 m4 r2, f(r2)=-3 m7 r3, f(r3)=-2 m5 m6 r4, f(r4)=-1 r5, f(r5)=4 m8 r6 m1 r7 m2 r8 m3

FBA and gapless reconstruction – p. 37/48

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Example sets R

r1, f(r1)=1 m4 r2, f(r2)=-3 m7 r3, f(r3)=-2 m5 m6 r4, f(r4)=-1 r5, f(r5)=4 m8 r6 m1 r7 m2 r8 m3

R1 = {r1, r2, r3, r4, r5}, F(R1) = 1 − 3 − 2 − 1 + 4 = −1 R2 = {r1, r2, r3, r5}, F(R2) = 1 − 3 − 2 + 4 = 0 R3 = {r1, r3, r4, r5}, F(R3) = 1 − 2 − 1 + 4 = 2 R4 = {r1}, F(R4) = 1 Reactions in R1, . . . , R4 are reachable in the induced graphs.

FBA and gapless reconstruction – p. 38/48

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Establishing connection between genome and reaction scoring

We would like scores f(r) to reflect the degree of confidence to that the genome codes for an enzyme catalyzing reaction r Assume that we have A set of protein sequences from the genome under study An annotated protein sequence database (such as UniProt) A reaction database (such as KEGG or BioCyc)

FBA and gapless reconstruction – p. 39/48

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Establishing connection between genome and reaction scoring

We assign each reaction r score f(r) = max

s∈G max t∈C(r) B(s, t) − b,

where G is the set of protein sequences from genome, C(r) are the sequences in the database annotated with reaction r, B(s, t) is the BLAST score of alignment of s and t and b ∈ R

FBA and gapless reconstruction – p. 40/48

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Establishing connection between genome and reaction scoring

A reaction with a negative score only appear in the solution when it fills a gap!

FBA and gapless reconstruction – p. 41/48

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Solving gapless reconstruction

Gapless reconstruction can be formulated as a mixed integer linear program (MILP) Some variables allowed to only take integer values Formulation resembles Flux Balance Analysis We add binary decision variables for each reaction Instead of pure steady state, we allow metabolite net production Futile cycles disallowed

FBA and gapless reconstruction – p. 42/48

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Gapless reconstruction as ILP

max

x

  • ri

f(ri)xi such that 1 N xi ≤ vi vi ≤ Mxi

  • i

sijvi − tj ≥ tj ≥ α

  • ri∈P(mj)

vi xi ∈ {0, 1} N, M: large numbers First two constraints ensure xi = 0 ⇔ vi = 0 tj: removes a fraction (α) of flux from the system to disallow futile cycles P(mj): producers of mj

FBA and gapless reconstruction – p. 43/48

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Gapless reconstruction as an Integer Linear Program

max

x

  • ri

f(ri)xi such that 1 N xi ≤ vi vi ≤ Mxi

  • i

sijvi − tj ≥ tj ≥ α

  • ri∈P(mj)

vi xi ∈ {0, 1}

FBA and gapless reconstruction – p. 44/48

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Complexity of ILP

Unfortunately, integer linear programming is in general NP-hard NP-hard even with 0-1 (binary) variables (one of Karp’s famous 21 NP-hard problems) Solvers typically resort to branch-and-bound or cutting plane methods (such as GLPK or lp_solve) We are unable to solve genome-scale gapless reconstruction with previous formulation Divide-and-conquer heuristic applied

FBA and gapless reconstruction – p. 45/48

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Gapless reconstruction

Gapless reconstruction combines two steps in the reconstruction workflow Selection of the initial reaction set Curation of the initial reaction set

FBA and gapless reconstruction – p. 46/48

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Gapless reconstruction

No previous knowledge on metabolic pathways needed! However, a priori knowledge on metabolites and reactions can be plugged in Possible to discover pathways that are not previously known Article:

  • E. Pitkänen, A. Rantanen, J. Rousu, E. Ukkonen: A

computational method for reconstructing gapless metabolic

  • networks. 2nd International Conference on Bioinformatics

Research and Development (BIRD’08), Communications in Computer and Information Science, Vol. 13, Springer, 2008.

FBA and gapless reconstruction – p. 47/48

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FBA and gapless reconstruction – p. 48/48