7 | KO mutants (MOMA, ROOM) Flashcards
Is FBA suitable for simulating knock-out mutants? Why/ why not?
No. Knock outs can be easily implemented, however:
- FBA assumes cells optimize biomass, but mutants may not do so.
- Lab-generated mutants aren’t under long-term evolutionary pressure.
- Knock-out mutants lack evolved regulatory mechanisms to redirect flux optimally.
How is a gene knock-out simulated in FBA?
- The flux of the associated reaction is set to zero (vr = 0).
- FBA is solved again with this constraint.
What are isoenzymes and enzyme complexes, and how can they be modelled in FBA?
Isoenzymes: different but catalyzing same reaction — all must be knocked out.
Enzyme complexes: multiple subunits — removal of any subunit inactivates complex
Modelled with the help of gene-protein-reaction (GPR) associations represented in boolean format
How are GPR associations handled in knock-out simulations?
- GPRs are represented using Boolean logic.
- A reaction is knocked out if its GPR evaluates to FALSE.
How are the feasibility spaces of mutants and wild type related?
- Feasibility space of mutant properly contained in feasibility space of WT!
- assuming all lower and upper boundaries on flux capacities are the same
What are relevant formulae for eigenvalues / vectors? Example with 𝑄 = [ 2 1, 1 2 ] ?
𝑄𝑥 = 𝜆𝑥 has a non-zero solution for 𝑥 if and only if:
det(𝑄−𝜆𝐼) = 0
Example:
𝑄 = [ 2 1, 1 2 ]
det(𝑄−𝜆𝐼) = [2−𝜆 1, 1 2−𝜆 ] = 3 − 4𝜆 + 𝜆<sup2</sup>
→ 𝜆1 = 1, 𝜆2 = 3
Solve for 𝑥 by substitution in (𝑄−𝜆𝐼)𝑥 = 0 !
MOMA relaxes the assumption of ______ ______ ______ ______for gene ______. A mutant is likely to initially display a ______ ______distribution.
MOMA relaxes the assumption of optimal growth flux states for gene deletions. A mutant is likely to initially display a suboptimal flux distribution.
What does MOMA stand for? What is the hypothesis behind MOMA?
Minimization Of Metabolic Adjustment
Hypothesis:
- After a gene deletion, the cell adopts a flux distribution close to the wild type.
- It does not immediately re-optimize for growth.
The hypothesis behind MOMA is to ______ ______. The maths behind it differs from the ______ used in FBA. It involves ______ minimization in ______ ______.
The hypothesis behind MOMA is to be tested. The maths behind it differs from the LP used in FBA. It involves distance minimization in flux space.
How does MOMA formulate the problem?
- MOMA finds the flux vector w (mutant) that is closest to the wild-type vector v, under constraints that represent the mutant network.
- i.e minimizes Euclidean distance between WT flux vector v and mutant flux vector w.
- results in a quadratic programming (QP) problem.
in MOMA, when considering the WT flux vector v and the mutant flux vector w, what can be said about the euclidean distance sqrt [ Σ (wi - vi)2]? I.e the next step in being able to solve this
Clearly its minimized when Σ (wi - vi)2 is minimized
In MOMA, why can the term v2 be ignored in the objective function?
- Because v is constant (wild-type flux is known).
- w is the variable, so only terms involving w affect the optimization.
- Constants don’t change the position of the minimum.
What is the starting MOMA objective function based on the hypothesis?
- min ∑i(wi − vi)2
- This expands to: ∑iwi2 - 2∑iviwi + ∑ivi2
- the last term is constant and can be removed
✅
What is the final MOMA optimization problem?
minw wTIm w + (−2vT)w
s.t:
N w = 0
wj = 0 (for knocked-out reaction)
∀i, 1 ≤ i ≠ j ≤ m, wimin ≤ wi ≤ wimax
✅
What type of optimization problem is MOMA?
A quadratic program (QP) with linear constraints.
What is a Quadratic Programming (QP) problem?
- Optimization with quadratic objective: eg minx ½xTQx + cTx
- And linear constraints.
✅
In MOMA, how can Σ wi2 be written using matrix notation?
- Σ wi² = wTImw, where I is the identity matrix.
- This is the squared Euclidean norm of vector w.
What are the constraints in the final MOMA?
- N w = 0 (steady state)
- wj = 0 (if reaction j is knocked out)
- ∀i, 1 ≤ i ≠ j ≤ m, wimin ≤ wi ≤ wimax
✅
What is the effect of the Euclidean norm in MOMA?
- It favors many small changes over few large ones.
- This may not reflect true biological behavior.
Why does MOMA result in convex optimization?
- Because Q = I, which is positive definite.
- The objective is strictly convex → unique solution.
Why are eigenvalues of Q important in QP?
- They tell you if the function is convex.
- All positive eigenvalues → strictly convex → unique minimum.
What does it mean for xTQx if all eigenvalues of Q are positive?
- Q is positive definite → the function is strictly convex.
- Ensures a unique global minimum.
In QP, if the matrix Q is not positive semidefinite it can be…
- negative semidefinite (some zero, all other negative) or
- indefinite (eigenvalues of mixed signs)
What does it mean for for xTQx if Q is positive semidefinite?
The function is convex but may have multiple minima.
Strict convexity in a QP: xTQx is strictly convex if Q is ______ ______ , and this means that the ______ ______ coincides with the ______ ______ and 𝑥 is unique ______ .
xTQx is strictly convex if Q is positive definite, and this means that the local optimum coincides with the global optimum and x is unique optimizer..
What methods can solve a QP?
- Interior point
- active set
- conjugate gradient
- augmented Lagrangian
- simplex extensions.
For QPs for which 𝑄 is ______ ______ and contain only ______ constraints, the problem reduces to solving a system of ______ ______ .
For QPs for which 𝑄 is positive definite and contain only equality constraints, the problem reduces to solving a system of linear equations.
Does MOMA optimize biomass?
No, it only minimizes the flux difference from the wild type.
What norm does MOMA use to measure flux changes?
The Euclidean norm (L2): Minimizes squared diff between WT and mutant fluxes.
What kind of changes does MOMA favor?
- Many small changes across multiple fluxes.
- This leads to smoother, less extreme metabolic adjustments.
What is the QP form of the MOMA objective function?
minw wTImw + (−2vT)w
✅
Why is there a 1/2 in the QP objective?
- To simplify the derivative of the quadratic term.
- It doesn’t affect the solution.
Is MOMA always strictly convex?
- Yes, because Q = I, which is positive definite.
- The solution is unique and stable.
Where does the wild-type flux vector come from in MOMA?
- It must be provided as input.
- Typically obtained experimentally or via FBA
What is the L2 norm used in MOMA?
- The Euclidean distance between mutant and wild-type fluxes.
- L2 norm = √Σ(wi − vi)²
What is the difference between L2 and L0 norms?
- L2 norm (MOMA): favors many small changes
- L0 norm (ROOM-like): counts number of large changes
Is FBA suitable for simulation of flux distributions for KO mutants?
Experimental evidence indicates that, in many cases, growth rate after deletion drops in comparison to that of WT, but then gradually increases and reaches nearly that of WT.
MOMA provides one way to explain these changes; however it implies ______ ______ ______ in all fluxes rather than ______ ______ ______ in few fluxes (of same value).
These characteristics reflect the used ______ ______!
MOMA provides one way to explain these changes; however it implies numerous small changes in all fluxes rather than few large changes in few fluxes (of same value).
These characteristics reflect the used Euclidean norm!
What is the hypothesis behind ROOM?
- After a knock-out, cells minimize the number of large flux changes.
- Focus is on minimizing regulatory changes, not total flux difference.
How does ROOM formulate the problem?
- It minimizes the number of reactions with significant flux changes
- This is formulated as a Mixed Integer Linear Program (MILP).
ROOM is based on the hypothesis that the ______ ______ ______ required for realizing flux changes after gene knock-outs are ______ by the cell, i.e it minimizes ______ costs.
ROOM is based on the hypothesis that the genetic regulatory changes required for realizing flux changes after gene knock-outs are minimized by the cell, i.e it minimizes adaptation costs.
In ROOM, the regulatory changes are parsimoniously described by ______ ______ dynamics. This assigns ______ ______ to each regulatory change, regardless of its ______.
In ROOM, the regulatory changes are parsimoniously described by Boolean on/off dynamics. This assigns fixed cost to each regulatory change, regardless of its magnitude.
What is the ROOM objective function?
- min Σ yi,
- where yi = 1 if reaction i has a large flux change, 0 otherwise.
How are significant changes defined in ROOM?
Using the wild-type flux vi, define bounds:
- viu = vi + δ |vi| + ε
- vil = vi − δ |vi| − ε
✅
What are the bounds on fluxes in ROOM?
What are the binary variables in ROOM and what values can they take?
What are the ROOM constraints linking fluxes and binary indicators?
bounds on fluxes:
For all i, wimin ≤ wi ≤ wimax
binary variables:
For all i, yi ∈ {0,1}
yi = 1 if reaction i has a large flux change
constraints linking fluxes and binary indicators:
wi − yi(wimax − viu) ≤ viu
wi − yi(wimin − vil) ≥ vil
✅
How are the upper and lower bounds viu and vil defined in ROOM?
viu = vi + δ|vi| + ε
vil = vi − δ|vi| − ε
These bounds are set based on a given wild type flux distribution v and two parameters:
- δ – determining relative changes
- ε – sensitivity bound
✅
What are all of the constraints in ROOM?
Nw = 0
wj = 0
∀i, 1 ≤ i ≤ m, wimin ≤ wi ≤ wimax
∀i, 1 ≤ i ≤ m, yi ∈ {0,1} [yi = 1 if reaction i has a large flux change]
wi − yi(wimax − viu) ≤ viu
wi − yi(wimin − vil) ≥ vil
viu = vi + δ|vi| + ε
vil = vi − δ|vi| − ε
✅
How does the value of yi affect the other ROOM constraints?
For each flux wi,
- If yi = 0, flux stays near WT (between vl, vu).
- If yi = 1, flux is unconstrained (allowed to differ more).
What type of optimization problem is ROOM?
- A Mixed Integer Linear Program (MILP).
- It uses binary variables to mark large flux changes.
What is a branch point in a metabolic network?
A point where flux can go down two or more alternate paths.
What is the “linearity hypothesis” in ROOM?
The idea that ROOM will favor a single linear path through a branch point.
I.e. it avoids spreading small changes across multiple paths
What’s the key difference between MOMA and ROOM?
- MOMA minimizes the total flux difference (Euclidean distance).
- ROOM minimizes the number of reactions with significant changes.
Do MOMA and ROOM explicitly optimize biomass?
No, both assume that flux adjustments occur without prioritizing growth.
How does MOMA behave at a branch point? How does ROOM behave at a branch point?
MOMA: may split flux between branches, to minimize total deviation from WT.
ROOM: tends to send flux through only 1 branch, minimizing number of changed reactions.
What are the advantages of ROOM over MOMA?
ROOM:
- leads to sparser changes (fewer large flux shifts).
- Produces better correlation with experimental fluxes in many cases.
What is a key difference in flux distribution predicted by ROOM vs MOMA at branch points?
ROOM prefers to turn one path on and the other off (linearity).
MOMA may split fluxes to minimize deviation.
How does ROOM compare to MOMA and FBA in predicting mutant fluxes?
ROOM shows higher correlation with experimental fluxes in knock-out mutants.
What is the advantage of ROOM’s sparsity assumption?
It predicts fewer, more significant changes, which aligns better with real biological responses to gene knockouts.
What does MOMA imply about mutant adaptation?
- Adaptation leads to small shifts across many fluxes, which may be unrealistic.
- Reflects the Euclidean norm used in the objective.
What is an open problem in mutant simulation?
Finding better distance functions between mutant and wild-type flux states.
Needs to consider:
* Thermodynamics
* Metabolite concentrations
* Biomass composition changes
