4 | FBA Flashcards
What is a metabolic network?
A collection of biochemical reactions transforming metabolites
A metabolic network can be represented by a _____________.
Rows correspond to ________, columns correspond to _______.
Entries are _________.
Negative → _______
Positive → _______
.. represented by a Stoichiometric matrix.
Rows - metabolites
Columns - reactions
Negative - Substrates
Positive - Products
What is our aim in constraint based modeling? cf ACN?
- making statements/predictions
- about underlying properties of biochemical reactions
- that make up a metabolic network
What is a reaction rate (flux)?
- Represents throughput of reaction rj (=conversion rate)
- Describes how much matter is transformed per unit time.
- A physical quantity
- Denoted as vj
What is the unit of a reaction rate (flux)?
- Units: mol / gDW∙h
- Content / over time - cf concentration over time in TSB - but the unit of flux must match the unit of concentration of time
The degree of temporal change in concentration of 𝑋𝑖 imposed by a reaction equals the product of the ______ and the ______ with which 𝑋𝑖 enters the reaction
The degree of temporal change in concentration of 𝑋𝑖 imposed by a reaction equals the product of the reaction rate and the molarity with which 𝑋𝑖 enters the reaction
What does Xi denote? How do reaction rates affect this?
- Metabolite concentration Xi
- Changes based on:
- Reactions that produce/synthesize Xi → +ve contribution
- Reactions that consume/degrade Xi → -ve contribution
What is v / v(t)?
- Reaction rate = flux
- It’s a vector: ( v1(t), v2(t), …, vn(t) ) → rate for each reaction
What is the equation in CBM for the rate of change of a metabolite concentration? Also give the equation that this is derived from.
- dxi/dt = Ni. * v(t) (a differential equation)
- Ni. = the i-th row of the stoichiometric model
- v(t) = vector of fluxes (reaction rates) for all reactions in a network
- Derived by taking the limit of (instantaneous change): [ xi(t + ∆t) - xi(t) ] / ∆t
Consider a closed network with 2 reactions and 2 metabolites.
How can we relate a vector of the rate of change in concentration of all metabolites to a vector of the fluxes of all the reactions?
- [dxA/dt, dxB/dt ] = [ -1 1, 1 -1] [ v1(t), v2(t) ]
- A system of linear equations!
Consider a metabolic network consisting of two metabolites A and B and two reactions r1 and r2 with the stoichiometric matrix [ -1 1, 1 -1].
Write the differential equations for this network.
dxA/dt = - v1(t) + v2(t)
dxB/dt = v1(t) - v1(t)
Reaction rates are a function of … ?
Which factors do we often not know?
- concentration of metabolites, 𝑥
- concentration of enzymes, 𝐸, that catalyze the reaction
- concentration of effectors (activators / inhibitors - also metabolites)
- kinetic parameters of the enzyme, e.g. catalytic rate, 𝑘𝑐𝑎𝑡
More specifically, for the reaction 𝑟𝑗:
* 𝑣𝑗 = 𝑓𝑗(𝑥,𝐸,𝑘)
Often we do not know:
* The effectors
* The kinetic parameters, species and condition-dependent
What are two different ways to mathematically represent reaction kinetics, and when are they used?
Fine-grained → Mass Action kinetics
Assumes all reactions are elementary
Example: A → : v = kr · xA
Example: 2A → : v = kr · xA2
Coarse-grained → Michaelis-Menten kinetics
Used for enzyme-catalyzed reactions
General form: v = k · E · ( xA / (xA + kMA ) )
What problem arises when using mass-action kinetics in Constraint-Based Modeling (CBM)?
- Stoichiometry > 1 leads to nonlinear terms
- Makes system of equations much more difficult mathematically and computationally.
- We lack kinetic parameters, such as enzyme concentrations and reaction rates.
When using mass-action kinetics, a stoichiometry > 1 leads to nonlinear terms.
This makes system of equations much more difficult mathematically and computationally, we also lack kinetic parameters, such as enzyme concentrations and reaction rates.
How does CBM solve this issue?
- Shortcut: Replace reaction rate equations with a flux variable (vj) that is independent of concentrations, effectors, and kinetics.
- This requires an assumption: steady state (N v = 0), ensuring mass balance without explicitly modeling reaction kinetics.
What does it mean for a system to be at steady state?
- Steady state assumption
- No net change in metabolite concentrations over time
- what goes out in to a metabolic pool = what comes out
In what state do we choose to model our system? What can we say about the environment?
- Steady state
- Environment does not change
- There are no triggers that would induce change of gene expression
Steady state in CBM: is this a thermodynamic equilibrium?
No.
Thermodynamic equilibrium:
* Each reaction has a net rate of 0
* Reversible reactions may have flux in both directions but they cancel each other out
* Irreversible reactions have flux of 0
* Net rate of 0 (ie no flux) → cell death!
* Is a form of steady state because metabolite concentrations do not change!
Steady state:
* Metabolites have approximately constant concentrations, but there can be flux!
What assumptions is a steady state of metabolite concentrations based on?
Assumes relatively constant enzyme levels and gene expression
How can a steady state be represented mathematically?
- dXi/dt = 0
- → N * v = 0
- (dependence on time dropped, since once system is in steady state, does not move out unless perturbation in concentrations)
What does the equation N * v = 0 represent?
- A homogeneous system of linear equations
- Unknowns are the reaction rates v
- Coefficients are given by the stoichiometric matrix
- The null space of N contains all steady-state flux distributions
- Solutions define the set of all possible metabolic states
Why are metabolic networks underdetermined?
- Typically have more reactions than metabolites
- Leads to infinitely many possible steady-state solutions
- Requires additional constraints to identify a biologically relevant solution
What form of a linear program is a simplex method designed for?
- Canonical form: all of decision variables are non negative
- Ie all reactions are irreversible
Not interested in entire ______, but rather on a null space that imposes additional ______ on the ______ reaction → ______for a stoichiometric matrix C(N) = {v | Nv = 0 and v >= 0 for irreversible} (see second to last lecture)
Not interested in entire null space, but rather in one that imposes additional constraints on the reversible reaction → flux cone for a stoichiometric matrix C(N) = {v | Nv = 0 and v >= 0 for irreversible} (see second to last lecture)
