solution space? in most cases lack of constraints provide a space of solutions what can we do with...
TRANSCRIPT
![Page 1: 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)](https://reader030.vdocuments.site/reader030/viewer/2022032607/56649eca5503460f94bd82aa/html5/thumbnails/1.jpg)
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
![Page 2: 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)](https://reader030.vdocuments.site/reader030/viewer/2022032607/56649eca5503460f94bd82aa/html5/thumbnails/2.jpg)
Lecture Outline
1. LP and MILP basic solution enumeration2. Flux variability analysis (FVA)3. Flux coupling
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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)
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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
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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
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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
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Types of coupling
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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
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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
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Biological Network Analysis:Regulation of Metabolism
Tomer ShlomiWinter 2008
![Page 12: 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)](https://reader030.vdocuments.site/reader030/viewer/2022032607/56649eca5503460f94bd82aa/html5/thumbnails/12.jpg)
Lecture Outline
1. Transcriptional regulation2. Steady-state Regulatory FBA (SR-FBA)3. Regulatory FBA
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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
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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
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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
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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
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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
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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
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4. Regulatory FBA (rFBA)
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Regulatory Feedback• Many regulatory mechanisms cannot be described via
steady-state description• Depends on
– E synthesis rate– E degradation rate
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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
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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
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Acetate re-utilization experiments