7 | KO mutants (MOMA, ROOM)

Question 1

Is FBA suitable for simulating knock-out mutants? Why/ why not?

Answer 1

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.

Question 2

How is a gene knock-out simulated in FBA?

Answer 2

• The flux of the associated reaction is set to zero (v_r = 0).
• FBA is solved again with this constraint.

Question 3

What are isoenzymes and enzyme complexes, and how can they be modelled in FBA?

Answer 3

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

Question 4

How are GPR associations handled in knock-out simulations?

Answer 4

• GPRs are represented using Boolean logic.
• A reaction is knocked out if its GPR evaluates to FALSE.

Question 5

How are the feasibility spaces of mutants and wild type related?

Answer 5

• Feasibility space of mutant properly contained in feasibility space of WT!
• assuming all lower and upper boundaries on flux capacities are the same

Question 6

What are relevant formulae for eigenvalues / vectors? Example with 𝑄 = [ 2 1, 1 2 ] ?

Answer 6

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 !

Question 7

MOMA relaxes the assumption of ______ ______ ______ ______for gene ______. A mutant is likely to initially display a ______ ______distribution.

Answer 7

MOMA relaxes the assumption of optimal growth flux states for gene deletions. A mutant is likely to initially display a suboptimal flux distribution.

Question 8

What does MOMA stand for? What is the hypothesis behind MOMA?

Answer 8

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.

Question 9

The hypothesis behind MOMA is to ______ ______. The maths behind it differs from the ______ used in FBA. It involves ______ minimization in ______ ______.

Answer 9

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.

Question 10

How does MOMA formulate the problem?

Answer 10

• 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.

Question 11

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 (w_i - v_i)²]? I.e the next step in being able to solve this

Answer 11

Clearly its minimized when Σ (w_i - v_i)² is minimized

Question 12

In MOMA, why can the term v² be ignored in the objective function?

Answer 12

• 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.

Question 13

What is the starting MOMA objective function based on the hypothesis?

Answer 13

• min ∑_i(w_i − v_i)²
• This expands to: ∑_iw_i² - 2∑_iv_iw_i + ∑_iv_i²
• the last term is constant and can be removed

✅

Question 14

What is the final MOMA optimization problem?

Answer 14

min_w w^TI_m w + (−2v^T)w

s.t:

N w = 0
w_j = 0 (for knocked-out reaction)
∀i, 1 ≤ i ≠ j ≤ m, w_i^min ≤ w_i ≤ w_i^max

✅

Question 15

What type of optimization problem is MOMA?

Answer 15

A quadratic program (QP) with linear constraints.

Question 16

What is a Quadratic Programming (QP) problem?

Answer 16

• Optimization with quadratic objective: eg min_x ½x^TQx + c^Tx
• And linear constraints.

✅

Question 17

In MOMA, how can Σ w_i² be written using matrix notation?

Answer 17

• Σ w_i² = w^TI_mw, where I is the identity matrix.
• This is the squared Euclidean norm of vector w.

Question 18

What are the constraints in the final MOMA?

Answer 18

• N w = 0 (steady state)
• w_j = 0 (if reaction j is knocked out)
• ∀i, 1 ≤ i ≠ j ≤ m, w_i^min ≤ w_i ≤ w_i^max

✅

Question 19

What is the effect of the Euclidean norm in MOMA?

Answer 19

• It favors many small changes over few large ones.
• This may not reflect true biological behavior.

Question 20

Why does MOMA result in convex optimization?

Answer 20

• Because Q = I, which is positive definite.
• The objective is strictly convex → unique solution.

Question 21

Why are eigenvalues of Q important in QP?

Answer 21

• They tell you if the function is convex.
• All positive eigenvalues → strictly convex → unique minimum.

Question 22

What does it mean for x^TQx if all eigenvalues of Q are positive?

Answer 22

• Q is positive definite → the function is strictly convex.
• Ensures a unique global minimum.

Question 23

In QP, if the matrix Q is not positive semidefinite it can be…

Answer 23

• negative semidefinite (some zero, all other negative) or
• indefinite (eigenvalues of mixed signs)

Question 24

What does it mean for for x^TQx if Q is positive semidefinite?

Answer 24

The function is convex but may have multiple minima.

Question 25

Strict convexity in a QP: x^TQx is strictly convex if Q is ______ ______ , and this means that the ______ ______ coincides with the ______ ______ and 𝑥 is unique ______ .

Answer 25

x^TQx 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_..

Question 26

What methods can solve a QP?

Answer 26

* Interior point * active set * conjugate gradient * augmented Lagrangian * simplex extensions.

Question 27

For QPs for which 𝑄 is ______ ______ and contain only ______ constraints, the problem reduces to solving a system of ______ ______ .

Answer 27

For QPs for which 𝑄 is _positive definite_ and contain only _equality constraints_, the problem reduces to solving a system of _linear equations_.

Question 28

Does MOMA optimize biomass?

Answer 28

No, it only minimizes the flux difference from the wild type.

Question 29

What norm does MOMA use to measure flux changes?

Answer 29

The Euclidean norm (L2): Minimizes squared diff between WT and mutant fluxes.

Question 30

What kind of changes does MOMA favor?

Answer 30

* Many small changes across multiple fluxes. * This leads to smoother, less extreme metabolic adjustments.

Question 31

What is the QP form of the MOMA objective function?

Answer 31

min_w w^TI_mw + (−2v^T)w ## Footnote ✅

Question 32

Why is there a 1/2 in the QP objective?

Answer 32

* To simplify the derivative of the quadratic term. * It doesn’t affect the solution.

Question 33

Is MOMA always strictly convex?

Answer 33

* Yes, because Q = I, which is positive definite. * The solution is unique and stable.

Question 34

Where does the wild-type flux vector come from in MOMA?

Answer 34

* It must be provided as input. * Typically obtained experimentally or via FBA

Question 35

What is the L2 norm used in MOMA?

Answer 35

* The Euclidean distance between mutant and wild-type fluxes. * L2 norm = √Σ(w_i − v_i)²

Question 36

What is the difference between L2 and L0 norms?

Answer 36

* L2 norm (MOMA): favors many small changes * L0 norm (ROOM-like): counts number of large changes

Question 37

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 ______ ______!

Answer 37

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_!

Question 38

What is the hypothesis behind ROOM?

Answer 38

* After a knock-out, cells minimize the number of large flux changes. * Focus is on minimizing regulatory changes, not total flux difference.

Question 39

How does ROOM formulate the problem?

Answer 39

* It minimizes the number of reactions with significant flux changes * This is formulated as a Mixed Integer Linear Program (MILP).

Question 40

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.

Answer 40

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.

Question 41

In ROOM, the regulatory changes are parsimoniously described by ______ ______ dynamics. This assigns ______ ______ to each regulatory change, regardless of its ______.

Answer 41

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_.

Question 42

What is the ROOM objective function?

Answer 42

* min Σ y_i, * where y_i = 1 if reaction i has a large flux change, 0 otherwise.

Question 43

How are significant changes defined in ROOM?

Answer 43

Using the wild-type flux v_i, define bounds: * v_i^u = v_i + δ |v_i| + ε * v_i^l = v_i − δ |v_i| − ε ## Footnote ✅

Question 44

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?

Answer 44

**bounds on fluxes:** For all i, w_i^min ≤ w_i ≤ w_i^max **binary variables:** For all i, y_i ∈ {0,1} y_i = 1 if reaction i has a large flux change **constraints linking fluxes and binary indicators:** w_i − y_i(w_i^max − v_i^u) ≤ v_i^u w_i − y_i(w_i^min − v_i^l) ≥ v_i^l ## Footnote ✅

Question 45

How are the upper and lower bounds v_i^u and v_i^l defined in ROOM?

Answer 45

v_i^u = v_i + δ|v_i| + ε v_i^l = v_i − δ|v_i| − ε These bounds are set based on a given wild type flux distribution v and two parameters: * δ – determining relative changes * ε – sensitivity bound ## Footnote ✅

Question 46

What are all of the constraints in ROOM?

Answer 46

Nw = 0 w_j = 0 ∀i, 1 ≤ i ≤ m, w_i^min ≤ w_i ≤ w_i^max ∀i, 1 ≤ i ≤ m, y_i ∈ {0,1} [*y_i = 1 if reaction i has a large flux change*] w_i − y_i(w_i^max − v_i^u) ≤ v_i^u w_i − y_i(w_i^min − v_i^l) ≥ v_i^l v_i^u = v_i + δ|v_i| + ε v_i^l = v_i − δ|v_i| − ε ## Footnote ✅

Question 47

How does the value of y_i affect the other ROOM constraints?

Answer 47

For each flux w_i, * If y_i = 0, flux stays near WT (between v^l, v^u). * If y_i = 1, flux is unconstrained (allowed to differ more).

Question 48

What type of optimization problem is ROOM?

Answer 48

* A Mixed Integer Linear Program (MILP). * It uses binary variables to mark large flux changes.

Question 49

What is a branch point in a metabolic network?

Answer 49

A point where flux can go down two or more alternate paths.

Question 50

What is the "linearity hypothesis" in ROOM?

Answer 50

The idea that ROOM will favor a single linear path through a branch point. I.e. it avoids spreading small changes across multiple paths

Question 51

What’s the key difference between MOMA and ROOM?

Answer 51

* MOMA minimizes the total flux difference (Euclidean distance). * ROOM minimizes the number of reactions with significant changes.

Question 52

Do MOMA and ROOM explicitly optimize biomass?

Answer 52

No, both assume that flux adjustments occur without prioritizing growth.

Question 53

How does MOMA behave at a branch point? How does ROOM behave at a branch point?

Answer 53

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.

Question 54

What are the advantages of ROOM over MOMA?

Answer 54

ROOM: * leads to sparser changes (fewer large flux shifts). * Produces better correlation with experimental fluxes in many cases.

Question 55

What is a key difference in flux distribution predicted by ROOM vs MOMA at branch points?

Answer 55

ROOM prefers to turn one path on and the other off (linearity). MOMA may split fluxes to minimize deviation.

Question 56

How does ROOM compare to MOMA and FBA in predicting mutant fluxes?

Answer 56

ROOM shows higher correlation with experimental fluxes in knock-out mutants.

Question 57

What is the advantage of ROOM’s sparsity assumption?

Answer 57

It predicts fewer, more significant changes, which aligns better with real biological responses to gene knockouts.

Question 58

What does MOMA imply about mutant adaptation?

Answer 58

* Adaptation leads to small shifts across many fluxes, which may be unrealistic. * Reflects the Euclidean norm used in the objective.

Question 59

What is an open problem in mutant simulation?

Answer 59

Finding better distance functions between mutant and wild-type flux states. Needs to consider: * Thermodynamics * Metabolite concentrations * Biomass composition changes