5 | FVA & flux sampling Flashcards by Stevie Davies

The objective 𝑧 = 𝑐^𝑇𝑣 is a ______ combination of ______.
FBA uses maximization of the flux through the ______.

The objective 𝑧 = 𝑐_𝑇𝑣 is a linear combination of fluxes.
FBA uses maximization of the flux through the biomass reaction.

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What makes constraints constraints in an FBA LP?

They stay the same when simulating and analysing
Only the objective function can change

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What are different types of FBA objectives?

Probing metabolic capabilities
Physiological objectives
Biotechnological objectives

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Different types of FBA objectives:
Give examples of objectives probing metabolic capabilities.

Objectives that probe the generating capabilities of a network:
* Maximize ATP production from a given substrate ( → maximise ATP export)
* Maximize amino acid production from a given substrate

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Different types of FBA objectives:
Give examples of physiological objectives.

Objectives that represent likely physiological tasks:
* Maximize biomass production (v_bio)
* Minimize total flux (assumes minimal (parsimonious) resource usage)

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Different types of FBA objectives:
Give examples of biotechnological objectives.

Objectives that represent biotechnological goals
* Optimize production of a target compound (e.g., an amino acid) → metabolic engineering

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Maximisation of flux through an artificial biomass reaction (a ______ output from a network of ______ needed for the synthesis of cellular mass) captures the belief that a microorganism has evolved to ______ ______.

Maximisation of flux through an artificial biomass reaction (a balanced output from a network of precursors needed for the synthesis of cellular mass) captures the belief that a microorganism has evolved to optimise growth.

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What hypothesis does minimization of total flux capture?

Hypothesis = biological system has evolved to economize resource usage

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How is minimizing total flux implemented in FBA?

Set c = [1, 1, …, 1], so all fluxes contribute equally to the objective.
Trivial solution: v = 0 for all reactions → Not useful.
Constraint added: Biomass flux (v_bio) must be above a certain threshold.

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Commonly applied objectives in FBA like maximizing ATP production, minimizing nutrient uptake, maximizing metabolite production, minimizing the total flux, are all ______ functions, and hence are ______, but not necessarily ______ ______.

Commonly applied objectives in FBA like maximizing ATP production, minimizing nutrient uptake, maximizing metabolite production, minimizing the total flux, are all linear functions, and hence are convex, but not necessarily strictly convex

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What would be a possible objective if we want to explore the trade-off between cell growth and production of particular metabolite (flux 𝑣_p)?

i.e maximize a linear combination of biomass flux and metabolite production
𝑐₁𝑣_𝑏𝑖𝑜 + 𝑐₂𝑣₂
eg give more weight to 𝑐₁ if we want to prioritise biomass production… eg 3𝑣_𝑏𝑖𝑜 + v_alanine …

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What is a convex function - definition in english?

If we take any two points, the function evaluated at a convex combination of these points should not be larger than the same convex combination of the 𝑓(𝑥) and 𝑓(𝑦)
convex region: lines from two points are always within the region - The segment must lie above the graph of 𝑓
In simple terms, a convex function is a type of function where, if you pick any two points on its curve, the line connecting those points will always stay on or above the curve. This means that the function is “bowl-shaped” and doesn’t have any dips or sharp turns.

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What is a convex function - mathematical definition?

A function f(x) or 𝑓: ℝ^𝑛 → ℝ is convex if its domain is a convex set and for all 𝑥 and 𝑦 in its domain, and all 𝜆 ∈ [0,1], it holds that:

f(λx + (1 − λ)y) ≤ λ f(x) + (1 − λ) f(y)

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What do convex functions ensure in FBA?

a well-defined solution space
at least one best solution (optimum).

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Why is convexity important in FBA?

If the function is strictly convex, there is a unique optimal solution.
Linear functions (like FBA objectives) are convex, but not strictly convex → can lead to alternative optima.

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Sketch the mathematical definition of convexity.

https://drive.google.com/file/d/1ybrY03_boaEsZKRM3yzxjJza2d8ZkY-H/view?usp=sharing

https://drive.google.com/file/d/1QQu6WO_0zHjeJpfr4NdORZRmSPnZmJDI/view?usp=sharing

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What is the definition of strict convexity?

A function f(x) is strictly convex if for all x ≠ y and λ ∈ (0,1):
f(λx+(1−λ)y) < λf(x)+(1−λ)f(y)
This means the function curves inward more sharply, ensuring a unique optimum.

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What are the key properties of convex and strictly convex functions?

Convex: line between two points stays on or above the function.
Strictly convex: line stays strictly above the function, except at the endpoints.

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What is the difference in the curves of convex and strictly convex functions?

Convex: The function can be a straight line in some regions.
Strictly convex: The function must curve in all regions, meaning no straight-line sections.

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True or false: no linear function is strictly convex!
What does this mean for FBA?

True: not strictly convex, only convex
→ no unique optimal solution, but rather several solutions
(some functions applied in metabolic modeling are strictly convex

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What are alternative optima in FBA?

Multiple flux distributions that achieve the same optimal objective value.

Implications:
* The network is robust (can use different pathways).
* Alternative flux distributions may correspond to silent phenotypes.

How are alternative optima represented? What can this be used to compare? What does this show?

Represented as a boxplot or barplot showing the range of fluxes for each reaction.
Represented as a fraction of reactions with given percentage of feasible range used to achieve the optimum of an objective [𝑣_𝑖^{𝑚𝑎𝑥,𝑜} − 𝑣_𝑖^{𝑚𝑖𝑛,𝑜}] / [𝑣_𝑖^{𝑚𝑎𝑥,𝑓} − 𝑣_𝑖^{𝑚𝑖𝑛,𝑓}]
Shows the minimum and maximum possible flux values while maintaining the same objective function value.
Can be used to compare two different scenarios
- conditions (e.g. carbon sources)
- objectives
- focus on reactions with non-overlapping ranges

(see doc for graphic)

How can FVA results be expressed as a fraction of the feasible range?

The fraction of the range used to achieve the optimum of an objective:
( v_i^max,o - v_i^min,o ) / ( v_i^max,f - v_i^min,f )
Helps compare reactions across different conditions or objectives.

How can FVA be used to compare different scenarios?

Can compare different carbon sources (e.g., glucose vs. succinate).
Can compare different objectives (e.g., growth vs. metabolite production).
Focus on reactions with non-overlapping flux ranges, as they highlight metabolic differences.

How can FVA be used to identify alternative flux distributions?

* Given the min/max fluxes v_j^min,o and v_j^max,o, there must be a flux distribution where: * v_j = v_j^min,o * v_j = v_j^max,o * Different reactions can take different values while still achieving the same objective.

How can we obtain alternative optima computationally?

One way is to solve an LP by fixing some fluxes while maximizing the objective: * max_v c^T v * Subject to: * N v = 0 (steady state) * v_i^min ≤ v_i ≤ v_i^max (capacity constraints) * Fix some fluxes: * v_j = v_j^min,o or v_j^max,o

Key takeaways from FVA and alternative optima: FVA identifies ______ ______ that maintain the same ______. Reactions with large ______ ranges indicate ______ in metabolism. Comparing different conditions (e.g., carbon sources) helps identify metabolic shifts. Alternative ______ solutions highlight multiple valid flux distributions.

Key takeaways from FVA and alternative optima: FVA identifies _flux ranges_ that maintain the same _objective_. Reactions with large _feasible_ ranges indicate _flexibility_ in metabolism. Comparing different conditions (e.g., carbon sources) helps identify metabolic shifts. Alternative _optimal_ solutions highlight multiple valid flux distributions.

What is Flux Variability Analysis (FVA)?

* Determines the feasible ranges, i.e range of possible flux values for each reaction at steady-state that still satisfy the optimum growth rate * Objective: Find v_min and v_max for every flux in the network. ## Footnote 🔵 If you DON'T enforce the optimum growth rate, then FVA gives you the: Feasible range — all steady-state fluxes that satisfy the model’s constraints, regardless of objective value. This is broader and includes sub-optimal solutions.

What are the key constraints in FVA?

* N v = 0 (steady state) * v_min ≤ v ≤ v_max (capacity constraints) * z = z* (fix the optimal objective value)

What are blocked reactions?

* Reactions that can never carry flux in any steady state. * Condition: v_i,min = v_i,max = 0.

What is the operational range in FVA?

* The range of flux values in steady states that achieve the optimal objective value (z^*). * Different from the feasible range, which considers all steady states.

Why is FVA computationally expensive? How many LP problems does it solve?

* Requires solving 2n LP problems (n = number of reactions). * Large-scale metabolic models may require long computation times.

Which is narrower - operational or feasible range?

operational

How can FVA be used to classify reactions?

Assume that 𝑧∗ is the optimum of the objective 𝑧, and 𝛼 is a percentage of 𝑧∗ which we are willing to sacrifice. min(max)_𝑣 𝑣_𝑖 s.t. 𝑁𝑣 = 0 ∀𝑖, 1 ≤ 𝑖 ≤ 𝑚, 𝑣_𝑖^𝑚𝑖𝑛 ≤ 𝑣_𝑖 ≤ 𝑣_𝑖^𝑚𝑎𝑥 𝑧 ≥𝛼𝑧^∗ ## Footnote (see graphic on google doc)

How can reactions be classified based on FVA?

* Hard-coupled * Partially-coupled * Not-coupled * Blocked

Classification based on FVA: hard-coupled reactions?

→ Flux scales directly with the objective.

Classification based on FVA: Partially-coupled reactions

→ Flux is nonzero, but varies.

Classification based on FVA: Not-coupled reactions

→ Flux can be zero.

Classification based on FVA: Blocked reactions

→ Flux is always zero.

If a reaction flux has a feasible range in FVA, does this mean all flux values in the range are equally likely?

* No! The probability of observing different flux values may not be uniform.

What is the goal of flux sampling?

* Uses uniform random sampling to characterise the feasible or optimal space of an LP * → unbiased assessment of the network properties * → find out which point/area is the most probable of the feasible region

What is flux sampling?

* A method to explore the distribution of flux values across the feasible space. * Ensures that all feasible solutions are sampled without bias.

What are two common methods for flux sampling?

* Walk with Ball Steps * Hit-and-Run Sampling

Describe the walk with Ball Steps method for flux sampling, what issue is there?

* Selects random points inside a ball of radius δ. * Problems: how to select δ? Can be slow and inefficient.

Describe the Hit-and-Run Sampling method for flux sampling

* Selects a random direction, then moves within feasible limits. * A more involved way to sample – Hit-and-run walk Given a point 𝑣 in the feasbile space - the point is a flux distribution * Select a line 𝐿 through 𝑣 uniformly at random * Choose the next point 𝑢 uniformly at random from the segment determined by the points in which 𝐿 crosses the boundary of the feasible region * Update 𝑣 to be 𝑢 and repeat the above step

What does the Hit-and-Run flux sampling method ensure?

Ensures each step stays in the feasible region.

Hit and run flux sampling - disadvantage?

More difficult to implement

What classification is possible with flux sampling?

Reactions can be classified based on the shape of the probability distribution: * Left peak - Lowest value is the most probable * Right peak - Highest value is the most probable * Central peak- A value between the lower and upper boundary is the most likely * Broad peak - Plateau of equally probable flux values exist

How can correlation between reaction pairs be determined using flux sampling?

* Correlation chart of pairwise Scatter plots show how fluxes of two reactions change together. * Pearson correlation coefficient: r(𝑋,𝑌) = 𝑐𝑜𝑣(𝑋,𝑌) / 𝜎_𝑋𝜎_𝑌

How / for what can statistical tests be used to compare reaction fluxes?

Reactions can be compared between conditions Given two probability distribtuons (and respective samples), one can ask: - Difference in means? → tests for difference in means eg t-test. - Difference in variance? → statistical tests for variance eg F-test. - Overall distribution difference? → Kolmogorov-Smirnov test.