Solution Space?• In most cases lack of constraints provide a space

of solutions

• What can we do with this space?

1. Optimization methods (previous lesson)– May result in a single, unique solution– May still result in a (smaller) convex solution

space

2. Explore alternative solutions in this space

Lecture Outline

1. LP and MILP basic solution enumeration2. Flux variability analysis (FVA)3. Flux coupling

Flux Variability Analysis• Determine for each reaction its range of

possible flux (within feasible solutions)• Computed via 2 LP problems for each reaction

(to find the lower and upper bounds)

Flux Variability Analysis• For the E. coli metabolic network, 3% of the

metabolic fluxes can vary and still allow for optimal biomass production on glucose

• Assuming sub-optimal growth rate of above 95% of the maximal rate – up to 50% of the fluxes can vary!

• This is a major issue with constraint-based modeling!

• Various studies still ignore this and simply choose a single arbitrary FBA solution for their analysis

Alternative MILP Solutions

• Identify solutions with different integer values

• The integer variables denoted yi and the number of reactions is M

• Each “integer cut” excludes one previously found solution yj*

• Which is equivalent to |yj* - yj|>0

Flux Coupling Analysis (FCA)

• Used to check how pairs of fluxes affect one another

• Done by calculating the minimum and maximum ratio between two fluxes

• Transformation needed to make it a linear problem

Types of coupling

Identifying coupled reaction sets

• A much higher percentage of reactions that are member of coupled sets in H. pylori (with the smaller network) compared to S. cerevisiae and E.coli

• If the biomass production rate is fixed to its maximal rate, we get ~40% of the reactions coupled to the biomass production rate

9

Alternative Optima: Hit and Run Sampling

• Almaas, et. al, 2004• Based on a random walk inside the solution space

polytope• Choose an arbitrary solution • Iteratively make a step in a random direction• Bounce off the walls of the polytope in random

directions

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Alternative Optima: Uniform Random Sampling

• Wiback, et. al, 2004• The problem of uniform sampling a high-dimensional

polytope is NP-Hard• Find a tight parallelepiped object that binds the

polytope• Randomly sample solutions from the parallelepiped• Can be used to estimate the volume of the polytope

Biological Network Analysis:Regulation of Metabolism

Tomer ShlomiWinter 2008

Lecture Outline

1. Transcriptional regulation2. Steady-state Regulatory FBA (SR-FBA)3. Regulatory FBA

Transcriptional Regulation• RNA polymerase – proteinmachinery that transcribes genes

• Transcription factors (TFs) bind to specific binding sites in the promoter region of a gene

• After binding to DNA TFs either enhance (activator) or disrupt (repressor) RNApolymerase bindingto DNA

Transcriptional Regulatory Network

• Nodes – transcription factors (TFs) and genes;• Edges – directed from transcription factor to the

genes it regulates • Reflect the cell’s genetic regulatory circuitry• Derived through:

1062 TFs, X genes 1149 interactions

S. cerevisiae

▲ Chromatin IP ▲ Microarrays

3. Steady-state Regulatory FBA (SR-FBA)

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Integrated Metabolic/Regulatory Models

• Genome-scale integrated model for E. coli (Covert 2004)• 1010 genes (104 TFs, 906 genes)• 817 proteins• 1083 reactions

• The Extreme Pathways approach can’t work on such large-scale models

The Steady-state Regulatory FBA Method

• SR-FBA is an optimization method that finds a consistent pair of metabolic and regulatory steady-states

• Based on Mixed Integer Linear Programming• Formulate the inter-dependency between the metabolic and

regulatory state using linear equations

Regulatory

state

Metabolic state

v

v1

v2

v3

…

g

0

1

1

…

g1 = g2 AND NOT (g3)

g3 = NOT g4

…

S·v = 0vmin < v <

vmax

Stoichiometric matrix

SR-FBA: Regulation → Metabolism

• The activity of each reaction depends on the presence specific catalyzing enzymes

• For each reaction define a Boolean variable ri specifying whether the reaction can be catalyzed by enzymes available from the expressed genes

• Formulate the relation between the Boolean variable ri and the flux through reaction i

Met1 Met3

Met2

Gene2Gene1 Gene3

Protein2 Protein3

Enzyme1Enzyme

complex2

AND

ORiiii rv )1(

iiii rv )1(

)0( iriii v

if then

else

0iv

r1

r1 = g1 OR (g2 AND g3)

g1 g2 g3

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SR-FBA: Metabolism → Regulation

• The presence of certain metabolites activates/represses the activity of specific TFs

• For each such metabolite we define a Boolean variable mj specifying whether it is actively synthesized, which is used to formulate TF regulation equations

Me1

Met2 Met4

Met3

TF2 TF3TF1

TF2 = NOT(TF1) AND (MET3 OR TF3)

)0( ivif then 1jm0jmelse

iij vm )(

iiij vm )(mj

SR-FBA Formulation• Boolean variables

– Regulatory state – g– Protein state – p– Reaction state – r– Reaction predicate - b

Recursive formulation of regulatory logic as linear equations

Formulation of Boolean G2R mapping

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Results: Validation of Expression and Flux

Predictions• Prediction of expression state changes between aerobic

and anaerobic conditions are in agreement with experimental data (p-value = 10-300)

• Prediction of metabolic flux values in glucose medium are significantly correlated with measurements via NMR spectroscopy (spearman correlation 0.942)

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The Functional Effects of Regulation on Metabolism

• Metabolic constraints determine the activity of 45-51% of the genes depending of growth media (covering 57% of all genes)

• The integrated model determines the activity of additional 13-20% of the genes (covering 36% of all genes)– 13-17% are directly regulated (via a TF)– 2-3% are indirectly regulated

• The activity of the remaining 30% of the genes is undetermined

4. Regulatory FBA (rFBA)

Regulatory Feedback• Many regulatory mechanisms cannot be described via

steady-state description• Depends on

– E synthesis rate– E degradation rate

Dynamic FBA Profiles• Separation of time-scales

– Transcriptional regulation: minutes– Metabolism: seconds

• Divide experimental time to small steps• Regulatory changes are continuous across time intervals• Metabolic behavior is in steady-state within each time-interval

Δt0

Metabolic state

Δt1

Regulatory

state

Metabolic state

Δt2

Regulatory

state

Regulatory FBA• Input:

– Initial biomass, X0

– Initial extra-cellular concentrations So

• Method– Compute maximal metabolite uptake rates

– Apply FBA to compute a flux distribution, v, with maximal growth rate, µ (considering regulatory constraints, derived from protein exp. state p)

– Compute new biomass:

– Compute new extra-cellular concentrations:– Update gene expression state, g– Update protein expression state, p, based on protein synthesis and

degradation constant

– Extra-cellular metabolite concentrations, Sc

– Cell density (biomass), X– Growth rate, µ– Flux distribution, v– Gene expression state, g– Protein expression state,

p

Acetate re-utilization experiments