Download Gene Expression Analysis, DNA Chips and Genetic Networks

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 enumeration
2. 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
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
10
Biological Network Analysis:
Regulation of Metabolism
Tomer Shlomi
Winter 2008
Lecture Outline
1. Transcriptional regulation
2. Steady-state Regulatory FBA (SR-FBA)
3. Regulatory FBA
Transcriptional Regulation
• RNA polymerase – protein
machinery 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) RNA
polymerase binding
to 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:
▲ Chromatin IP
▲ Microarrays
S. cerevisiae
1062 TFs, X genes
1149 interactions
3. Steady-state Regulatory FBA (SR-FBA)
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
g
0
1
1
v
Regulatory
state
Metabolic
state
v2
v3
…
…
g1 = g2 AND NOT (g3)
g3 = NOT g4
…
v1
Stoichiometric
matrix
S·v = 0
vmin < v < vmax
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
g1
g2
g3
if (ri  0)
then vi  0
Gene1
else  i  vi   i
Enzyme1
vi  (1  ri )  i   i
 i  vi  (1  ri ) i
r1
Gene2
Gene3
Protein2
Protein3
OR
Met1
Enzyme
complex2
AND
Met3
Met2
r1 = g1 OR (g2 AND g3)
•
•
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
TF2 = NOT(TF1) AND (MET3 OR TF3)
if
(vi  0)
then m j  1
else m j  0
TF1
m j (   i )  vi  
TF2
TF3
Me1
Met3
Met2
Met4
m j ( i   )  vi   i
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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
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)
Regulatory Feedback
• Many regulatory mechanisms cannot be described via steadystate 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 timeinterval
Δt1
Δt0
Regulatory
state
Metabolic
state
Metabolic
state
Δt2
Regulatory
state
Regulatory FBA
•
•
Input:
– Initial biomass, X0
– Initial extra-cellular concentrations So
Method
– Compute maximal metabolite uptake rates
– Extra-cellular metabolite
concentrations, Sc
– Cell density (biomass), X
– Growth rate, µ
– Flux distribution, v
– Gene expression state, g
– Protein expression state, p
– 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
Acetate re-utilization experiments