Introduction to Steady State Metabolic Modeling

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Introduction to Steady State Metabolic Modeling

• Concepts
• What is a metabolic network?
• Modeling chemical reactions as flows
• Enzyme Kinetics, Michaelis-Menten equation
• Transcription (Repression and Induction)
• Mathematical notation and solving systems of ODEs
  computationally using numerical integration
• Applications
• Predicting knockout phenotypes
• Quantitative Flux Prediction

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Introduction to Steady State Metabolic Modeling

• Reading (this week)
• chapter 17 in text
• Sources
• Notes David Fell talk The Metabolic Network

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Why Model Metabolism?

• Predict the effects of drugs on metabolism
• e.g. what genes should be disrupted to prevent
  mycolic acid synthesis
• Many infectious disease processes involve
  microbial metabolic changes
• e.g. switch from sugar to fatty acid metabolism
  in TB in macrophages

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Genome Wide View of Metabolism
Streptococcus pneumoniae

• Explore capabilities of global network
• How do we go from a pretty picture to a model we
  can manipulate?

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Online Metabolic Databases
Kegg

• Pathlogic/BioCyc

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Metabolic Pathways
hexokinase

• Metabolites
• glucose
• Enzymes
• phosphofructokinase
• Reactions Stoichiometry
• 1 F6P gt 1 FBP
• Kinetics
• Regulation
• gene regulation
• metabolite regulation

phosphoglucoisomerase
phosphofructokinase
aldolase
triosephosphate isomerase
G3P dehydrogenase
phosphoglycerate kinase
phosphoglycerate mutase
enolase
pyruvate kinase
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Metabolic Modeling The Dream
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Steady State Assumptions

• Dynamics are transient
• At appropriate time-scales and conditions,
  metabolism is in steady state

uptake
conversion
secretion
A
B
uptake
Conversion
t
t

• Two key implications
• Fluxes are roughly constant
• Internal metabolite concentrations are constant

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Metabolic Flux
Input fluxes
Volume of pool of water metabolite concentration
Output fluxes
Slide Credit Jeremy Zucker
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Reaction Stoichiometries Are Universal

• The conversion of glucose to glucose 6-phosphate
  always follows this stoichiometry
• 1ATP 1glucose 1ADP 1glucose 6-phosphate
• This is chemistry not biology.
• Biology gt the enzymes catalyzing the reaction

• Enzymes influence rates and kinetics
• Activation energy
• Substrate affinity
• Rate constants

Not required for steady state modeling!
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Metabolic Flux Analysis
Use universal reaction stoichiometries to predict
network metabolic capabilities at steady state
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Notation

• Capital letter Metabolite or protein
  concentration A, B, or F
• v is a flux, or rate of change between two
  concentration nodes.
• Generally
• S is a substrate
• P is a product
• E is an enzyme

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Stoichiometry As Vectors

• We can denote the stoichiometry of a reaction by
  a vector of coefficients
• One coefficient per metabolite (convention)
• Positive if metabolite is produced
• Negative if metabolite is consumed

Example
Metabolites A B C D T
Reactions 2A B -gt C C -gt D
Stoichiometry Vectors -2 -1 1 0 T 0
0 -1 1 T
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Approach

• Step 1 Number fluxes
• Step 2 Set up equations for rate of concentration
  change vs. time
• Step 3 model the individual fluxes depending on
  reaction type

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A (Very) Simple System

• Reversible reactions are represented by two
  reactions that proceed in each direction (e.g.
  v4, v5)
• Flow conventions positive is flow into a node,
  negative flow out
• Exchange reactions allow for fluxes from/into
  an infinite pool outside the system (e.g. vin and
  vout). These are frequently the only fluxes
  experimentally measured.

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A (Very) Simple System
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Flux Model Activity

• Step 1 Number fluxes
• Step 2 Set up equations for rate of concentration
  vs. time
• What is flux for Fructose-6P?
• Glucose-6P

Glucose
Trehalose
v1
v2
v4
UDPG
Glucose-6P
v3
v7
v5
v6
Glycogen
Fructose-6P
ATP
v11
v8
ADP
Fructose-1,6-DP
2 ADP
v12
v9
2 ATP
2 Phosphenolpyruvate
v10
Ethanol
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The Stoichiometric Matrix (S)
Reactions
Metabolites
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A Simple System
v1 v2 v3 v4 v5 vin vout
A B C D
-1 0 0 1
0 0 -1 0
1 0 0 0
-1 1 0 0
0 -1 1 0
0 0 1 -1
0 0 -1 1
Exchange Reactions
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The Stoichiometric Matrix
V is a vector of fluxes through each
reaction Then SV is a vector describing the
change in concentration of each metabolite per
unit time
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Calculating changes in concentration
What happens if vin is 1 unit per second
We can calculate this with S
dA/dt dB/dt dC/dt dD/dt
A grows by 1 unit per second

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Some advantages of S

• Chemistry not Biology the stoichiometry of a
  given reaction is preserved across organisms,
  while the reaction rates may not be preserved
• Does NOT depend on kinetics or reaction rates
• Depends on gene annotations and mapping from gene
  to reactions

Depends on information we frequently already have
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Genes to Reactions

• Expasy enzyme database
• Indexed by EC number
• EC numbers can be assigned to genes by
• Blast to known genes
• PFAM domains

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Online Metabolic Databases
There are several online databases with curated
and/or automated EC number assignments for
sequenced genomes
KEGG

• Pathlogic/BioCyc

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From Genomes to the S Matrix
Examples

• Columns encode reactions
• Relationships btw genes and rxns
• 1 gene 1 rxn
• 1 gene 1 rxns
• 1 genes 1 rxn
• The same reaction can be included as multiple
  roles (paralogs)

Gene A
Gene B Gene C
Gene D
Gene E Gene E
Enzyme A
Enzyme B/C
Enyzme D
Enzyme E Enzyme E
R1 R2 R3 R4 R5 R6 R7 R8 R9 R10
-1 0 0 -2 0 0 1 0 1
0 -1 0 -1 0 0 0 1 0
A B C D E F G H I
Same rxn
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What Can We Use S For?
From S we can investigate the metabolic
capabilities of the system. We can determine
what combination of fluxes (flux configurations)
are possible at steady state
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The Steady State Constraint

• We have
• But also recall that at steady state, metabolite
  concentrations are constant dx/dt0

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Activity

• How does an enzyme physically work?
• Reaction sequence/picture
• Kinetics
• Biological example

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Enzyme Activity

• From Wikipedia, the free encyclopedia
• Enzymes are biomolecules that catalyze (i.e.,
  increase the rates of) chemical reactions. Almost
  all enzymes are proteins.
• In enzymatic reactions, the enzyme converts
  substrates into different molecules, the
  products.
• Almost all processes in a biological cell need
  enzymes to occur at significant rates.
• Like all catalysts, enzymes work by lowering the
  activation energy (Ea or ?G) for a reaction, thus
  dramatically increasing the rate of the reaction.

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Enzyme Activity
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Enzyme activation methods

• Lowering the activation energy by creating an
  environment in which the transition state is
  stabilized (e.g. straining the shape of a
  substrateby binding the transition-state
  conformation of the substrate/product molecules,
  the enzyme distorts the bound substrate(s) into
  their transition state form).
• Lowering the energy of the transition state, but
  without distorting the substrate, by creating an
  environment with the opposite charge distribution
  to that of the transition state.
• Providing an alternative pathway. E.g.,
  temporarily reacting with the substrate to form
  an intermediate ES complex.
• Reducing the reaction entropy change by bringing
  substrates together in the correct orientation to
  react.

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Biological Example

• Reaction catalyzed by orotidine 5'-phosphate
  decarboxylase (converts orotidine monophosphate
  (OMP) to uridine monophosphate (UMP) by
  liberating carbon dioxide)
• If no decarboxylase is present, consumes half of
  its substrate in 78 million years.
• When decarboxylase present, the same process
  takes just 25 milliseconds

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Enzyme Kinetics

• Based on chemistry and physics
• Mass-action law
• Thermodynamics
• Use established modeling methods

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Key Variables

• Concentration, S, of a substance S (number of
  molecules, n, of the substance per volume, V)
• Rate, v, of a reaction (change in concentration S
  per time, t)
• Assume a well-stirred homogeneous bioreactor
  (well-shaken-up bag of goo)

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Law of Mass Action

• When two reactants, A and B, react together at a
  given temperature in a "substitution reaction,"
  the affinity, or chemical force between them, is
  proportional to the active masses, A and B,
  each raised to a particular power

Simplified
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Example

• Reaction
• Rate Equation
• Net rates n , units M.s-1
• Kinetic or rate constants, k, units M.s-1
• Concentrations, moles/liter

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General Mass Action Law

• Substrate concentrations, Si
• Product concentrations, Pj
• Molecularities for the reaction, mi and mj

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Equilibrium

• Forward and backward rates are the same

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Concentration dynamics
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Thermodynamics

• Looks at the energy exchange and balance in
  systems and entropy (a measure of the
  unavailability of a systems energy to do work
  and the disorder of molecules in a system
• Three laws
• Energy conservation
• Process occurs only if increases entropy of a
  system
• As temperature approaches absolute zero, the
  entropy of a system approaches a constant
  minimum.

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Entropy and Free Energy

• Entropy hard to measure
• Gibbs Free energy energy capable of carrying out
  work under constant pressure and temperature
  conditions.
• Change in free energy
• For a reaction

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Thermally unstable, Equilibrium, Stable
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Enzyme Activity
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Michaelis-Menten Kinetics
Image Tom Vickers

• Enzyme not used up
• Based on the law of mass action

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Enzyme Rates

• Depend on solution conditions and substrate
  concentration.
• Conditions that denature the protein abolish
  enzyme activity, such as high temperatures,
  extremes of pH or high salt concentrations
• Raising substrate concentration tends to increase
  activity.

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Michaelis-Menten Saturation Curve

• To find the maximum speed of an enzymatic
  reaction, the substrate concentration is
  increased until a constant rate of product
  formation is seen.
• As substrate concentration increases, more and
  more of the free enzyme is converted into the
  substrate-bound ES form.
• At the maximum velocity (Vmax), all the enzyme
  active sites are bound to substrate, and the
  amount of ES complex total amount of enzyme
• The amount of substrate needed to achieve a given
  rate of reaction is given by the Michaelis-Menten
  constant (Km), the substrate concentration
  required for an enzyme to reach one-half its
  maximum velocity.
• kcat, is the number of substrate molecules
  handled by one active site per second.

V
S
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Michaelis-Menten Kinetics

• Start with the flows

Overall flow
v
v2
Image Tom Vickers
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Michaelis-Menten Kinetics

• From the Law of Mass Action

k1
k-1
k2
Image Tom Vickers
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Michaelis-Menten Kinetics

• From the Law of Mass Action

k1
k-1
k2
Image Tom Vickers
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Assumptions

• Quasi-equilibrium assumption
• Time to create ES (complex) is much faster than
  product creation
• Quasi-Steady State (Briggs and Haldane, 1925)
• Period in reaction where ES concentration is
  constant
• Only makes sense when SgtgtE

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Simplify equations

• Use quasi-steady state
• Also, recall that the total amount of enzyme, E,
  is not used up

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Apply total enzyme constraint and quasi steady
state
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After, Applying enzyme constraint and quasi
steady state
This gives the overall reaction rate, v
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Simpler form
Michaelis Constant, For quasi-equilibrium
Maximum velocity for the reaction
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Steps for finding rate equations

• Draw a diagram with concentrations of key
  substrates, products and intermediate complexes
• Write down the concentration rates of change
  using flows
• Use the law of mass action to get flows with
  respect to the kinetic rate constants

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Simplifying assumptions and constraints

• Sum of enzyme (both free and in complexes) is a
  constant, Etotal
• Apply quasi-steady state assumption for n-1
  enzyme species (12 give n algebraic equations)
• Overall reaction rate is equal to rate of product
  formation

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in silico Deletion Analysis
wild-type
mutant
Gene knockouts modeled by removing a reaction
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Regulation by protein interaction
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Competitive Inhibition
Uncompetitive Inhibition
Partial Inhibition
Noncompetitive Inhibition
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Analysis

• Enzyme is conserved
• Overall reaction rate is equal to rate of product
  formation

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Other types of Inhibition

• Irreversible binding of inhibitor to enzyme
  active site
• Substrate Inhibition after a certain
  concentration, S is reached, ESS is formed which
  prevents product formation
• Reversible if second substrate can be released
• Inhibition by binding inhibitor to substrate

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Ligands to Proteins

• Ligand molecule that binds to a protein
• Single binding site

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Cooperativity

• If protein has several binding sites, then
  interactions get more complex
• Positive cooperativity can make binding easier
  for the substrate
• Negative decreases affinity of protein to other
  ligands

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Positive Homotrophic cooperativity

• protein with multiple identical binding sites
• Look at dimerized protein, E2.

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Overall rate
n3

• n is the number of binding sites

n2
n1
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Interpreting Array Data in Metabolic Context
Clustering, GSEA
Kegg, PathwayExplorer