8 | metabolic engineering strategies I Flashcards

1
Q

What is the motivation behind coupling product formation to growth?

A
  • force cell to produce desired compound as by-product of growth
  • β†’ ensures production is evolutionarily stable.
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2
Q

What is strong coupling in metabolic engineering?

A

KO strategy: product formation strongly coupled to growth (biomass production)
ensure any growth-supporting flux dist also produces product > threshold.

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3
Q

What is weak coupling?

A

Product is only formed at the optimal growth flux, not necessarily under all conditions.

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4
Q

What is the exhaustive KO strategy covered in lecture 8 on metabolic engineering?

A
  • Try all possible subsets of knock-outs from the list of genes.
  • Evaluate product yield or other objectives, solution = maximum of all maxima
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5
Q

Write the LP for an exhaustive knock out enumeration.

A

For every subset 𝑇 βŠ† 𝐿:
max π‘£π‘β„Žπ‘’π‘šπ‘–π‘π‘Žπ‘™
s.t.
𝑁𝑣 =0
βˆ€π‘— ∈ 𝑇, 𝑣𝑗= 0,
βˆ€π‘–, 𝑖 βˆ‰ 𝑇,π‘£π‘–π‘šπ‘–π‘› ≀ 𝑣𝑖 ≀ π‘£π‘–π‘šπ‘Žπ‘₯
π‘£π‘π‘–π‘œ β‰₯ π›Όπ‘§βˆ—
If we want sufficient high product yield to be at some level of biomass yield:
where π‘§βˆ— is the optimal growth.

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6
Q

What additional constraint ensures viability in exhaustive knock-out tests?

A

vbio β‰₯ Ξ± Β· z, where z is wild-type biomass flux

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7
Q

What are the limitations of exhaustive knock-out search?

A
  • Combinatorial explosion: infeasible for large gene lists
  • Cannot scale to full genome-wide searches
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8
Q

What is a bi-level optimization problem?

A
  • One optimization problem (outer) depends on the solution of another (inner)
  • I.e two nested optimizations
  • The outer objective uses the result of the inner optimization
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9
Q

Why do bi-level problems appear in metabolic engineering?

A
  • The outer problem wants to maximize product formation (engineer)
  • The inner problem responds by optimizing its own growth (cell)
  • The engineering strategy must account for the internal optimization
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10
Q

In OptKnock, what are the outer and inner problems?

A
  • Outer: maximize vtarget (e.g. product formation) by choosing KOs
  • Inner: maximize vbio (growth) given knock-outs
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11
Q

How are knock-outs encoded in OptKnock?

A
  • Binary variable yi ∈ {0, 1}
  • yi = 1 β†’ reaction is active
  • yi = 0 β†’ reaction is knocked out β†’ vi = 0
  • Use bounds: yi * vimin ≀ vi ≀ yi * vimax
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12
Q

What does the inner problem in a bi-level formulation depend on?

A
  • The outer decision variables (e.g. knock-outs y)
  • These change the feasible region for the inner problem
  • The inner LP is parameterized by y
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13
Q

What makes OptKnock a mixed-integer bi-level program?

A
  • Outer layer includes binary variables yi
  • Inner problem is a linear program over flux variables v
  • The feasible set for v depends on y
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14
Q

What is OptKnock?

A
  • A bi-level MILP to find knock-outs that couple growth and production
  • Automates the search for knock-out designs
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15
Q

How is the number of knock-outs restricted in OptKnock?

A

Add a constraint for K, the max allowed deletions: βˆ‘(1βˆ’yi) ≀ K

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16
Q

What is the full program for OptKnock (Burgard et al. (2003) Biotech. & Bioeng.)

A

max over y: vchemical
st:
max vbio
st:
N v = 0
for all i ≀ 1 ≀ m, yi Β· vimin ≀ vi ≀ yi Β· vimax
yi ∈ {0, 1}
βˆ‘i=1m(1βˆ’yi) ≀ K

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17
Q

What does the inner problem in OptKnock represent?

A

The cell’s behavior: maximize biomass production given knock-outs

18
Q

What does the outer problem in OptKnock represent?

A

The engineer’s goal: maximize product formation by choosing which knock-outs to apply

19
Q

What assumptions does OptKnock make about the cell?

A
  • The cell optimizes growth
  • Product formation is only induced indirectly via knock-outs
20
Q

Why are bi-level problems difficult to solve?

A
  • The inner problem changes with each outer decision
  • You can’t just solve them separately
  • to nesting, discrete nature of int vars β†’ problem non-convex, combinatorial
21
Q

What does β€œcombinatorial” mean in the context of bi-level optimization?

A
  • refers to having to choose among many discrete combinations, eg KO sets
  • Each combination may lead to a different inner solution
  • possible combinations can grow very large - β€˜combinatorial explosion’
  • Can make problem hard to solve efficiently
22
Q

Why can’t bi-level problems be solved directly?

A

Inner LP depends on outer variables β†’ standard MILP solvers unable to handle it directly.

23
Q

How can bi-level problems be solved in OptKnock?

A

Two approaches:
* By replacing the inner LP with its dual problem
* Apply Karush-Kuhn-Tucker (KKT) conditions (optimality) to connect primal, dual

24
Q

What are the KKT conditions used to solve bi-level problems?

(not so relevant ! )

A
  • Stationarity
  • Primal feasibility
  • Dual feasibility
  • Complementary slackness
25
What is complementary slackness?
For each constraint: * If it's active, dual variable may be positive * If it's inactive, dual variable is zero
26
What are mixed-integer linear programs (MILPs)?
LPs with some variables required to be integers (e.g. binary knock-out indicators)
27
What is LP duality?
* Every LP has a dual problem * The dual variables represent the shadow prices of the constraints
28
What is a shadow price in LP duality?
Each constraint in the primal has a shadow price. * = The value of a dual variable corresponding to a primal constraint * = how much objective function would improve if that constraint relaxed by 1 unit * AKA marginal value of a resource
29
If you have n variables and m constraints in an LP, how many constraints will you have in its dual?
* m variables * n constraints
30
What is the optimality property in LP duality?
Let π‘₯ be a feasible solution to a primal LP, and πœ‡ be a feasible solution to the dual LP. If 𝑐𝑇π‘₯ = π‘π‘‡πœ‡ then π‘₯ and πœ‡ are optimal solutions to the primal and dual LPs, respectively.
31
What is the strong duality property in LP duality?
* If primal has finite optimal solution, then so does its dual, and these values are equal. * ie maxπ‘₯ 𝑐𝑇π‘₯ = minπœ‡ π‘π‘‡πœ‡ That means: * Solve the primal β†’ get a value; Solve the dual β†’ get a value. βœ… Both values are equal This only holds if: * The primal or dual has a feasible solution (satisfies all constraints) * The problem is bounded (no infinite solutions)
32
How do you convert a primal LP into its dual (without Lagrangian)?
* Each constraint in the primal becomes a variable in the dual * Each variable in the primal becomes a constraint in the dual * The primal objective coefficients become RHS values in the dual constraints * The primal constraint matrix A is transposed in the dual
33
In LP duality, what happens to β‰₯ constraints in the primal?
* They become ≀ constraints in the dual * Their corresponding dual variables must be β‰₯ 0
34
What is the dual of this primal LP? min cT x s.t. Aineq x β‰₯ bineq x β‰₯ 0 Aeq x = beq
max bineqT y1 + beqT y2 s.t. AineqT y1 + AeqT y2 ≀ c y1 β‰₯ 0
35
Why is LP duality useful in OptKnock?
* Allows removing inner LP, replacing with constraints involving dual variables * Converts the bi-level problem to a single-level MILP
36
What are two approaches to flattening bi-level optimization problems?
* LP duality: replace the inner LP with its dual and enforce strong duality * KKT conditions: use stationarity, feasibility, and complementary slackness via the Lagrangian
37
Do LP duality and KKT methods give the same solution for bi-level LPs?
* Yes, they both produce the same optimal result * They enforce optimality of the inner problem in different ways
38
Do LP duality and KKT produce the same flattened single-level program?
No, the formulations differ * LP duality leads to a more compact and structured MILP * KKT-based reformulations are often larger and more complex
39
What does it mean to "not use the Lagrangian approach" for duality?
* You should derive the dual using the standard LP rulebook, not calculus or KKT * Don’t introduce Lagrange multipliers * Just work structurally from constraint-to-variable transformation
40
For a primal minimization problem: min cTx s.t. A x β‰₯ b x β‰₯ 0 What would be the key steps to formulate the dual?
Follow the standard duality principles in optimization. 1. Identify Primal Components: 2. Assign Dual Variables: 3. Formulate Dual Objective: 4. Formulate Dual Constraints:
41
For a primal minimization problem: min cTx s.t. A x β‰₯ b x β‰₯ 0 What is the dual?
maxπœ‡ 4πœ‡ s.t πœ‡ = 0 ------- Perplexity explanation: Original (primal) problem: Minimize y st y β‰₯ 0, x + y β‰₯ 4. **1. Identify Primal Components:** Variables: yβ‰₯0, x (unrestricted). Objective: Minimize y Constraints: One explicit constraint x+yβ‰₯4. **Assign Dual Variables:** Each primal constraint becomes a dual variable. Here, the constraint x+yβ‰₯4 corresponds to a dual variable Ξ»β‰₯0. **Formulate Dual Objective:** The dual objective is derived from the right-hand side of the primal constraint multiplied by Ξ»: 4Ξ». **Formulate Dual Constraints:** For each primal variable, create a dual constraint based on its sign: For x (unrestricted): The coefficient of x in the primal constraint (1) forms an equality constraint in the dual: Ξ»=0. For y (non-negative): The coefficient of y in the primal constraint (1) forms an inequality constraint: λ≀1. Combine Constraints: The dual constraints are Ξ»=0, λ≀1, and Ξ»β‰₯0. **Dual Problem:** Maximize 4Ξ» subject to Ξ»=0, Ξ»β‰₯0, λ≀1. **Interpretation:** The only feasible solution for the dual is Ξ»=0, yielding an optimal value of 0 This matches the primal optimal value y=0 (achieved when xβ‰₯4), demonstrating strong duality. **Summary:** The dual LP is trivial in this case because the unrestricted variable x forces Ξ»=0. This ensures the primal and dual optimal values align, satisfying the duality theorems in linear programming.
42
What is the simplified OptKnock reformulation/approximation from the exercise, often used computationally, which: * Turns the bilevel OptKnock problem into a single-level MILP * Focuses on minimizing the number of knockouts * Adds a small constant Ξ΅ to soften knockouts ?
min βˆ‘ yi s.t. N v = 0 vbio β‰₯ Ξ± Β· z* vtarget β‰₯ Ξ² Β· wtarget v β‰₯ 0 βˆ€ i:  (1 βˆ’ yi) Β· Ξ΅ ≀ vi  vi ≀ (1 βˆ’ yi) Β· vimax + yi Β· Ξ΅ y ∈ {0,1}