6 | FCA & omics data Flashcards
What is the goal of FCA?
Main goal: Classify reaction pairs based on their dependencies (couplings)
i.e. to answer these questions:
- more efficient way to determine reactions whose fluxes show correlation of 1 – do we have to sample, or one LP sufficient?
- Can we specify how structure of metabolic network leads to dependencies between reaction fluxes?
- What are the biological implications of such dependencies?
What is directional coupling?
Reaction Ri is directionally coupled to Rj if:
* Whenever vi ≠ 0, then vj ≠ 0 in all steady states.
* The reverse is not necessarily true.
Notation: Ri → Rj or i → j
What is partial coupling?
Reactions Ri and Rj are partially coupled if:
* vi ≠ 0 ⇔ vj ≠ 0, but
* The ratio vi/vj is not constant.
Notation: Ri ↔ Rj or i ↔ j
What is an emergent property? Where do we see this in coupling?
Partial coupling is an emergent property – arises as result of network as a whole
You can’t detect it by only examining individual reactions or pairwise stoichiometry — it emerges from the interplay of multiple reactions across the whole network.
It is not a trivial task to arrive at an example for partial coupling
What is full coupling?
Reactions Ri and Rj are fully coupled if:
* vi ≠ 0 ⇔ vj ≠ 0, and
* vi/vj = constant across all steady states.
Notation: Ri ⟺ Rj or i ⟺ j
What are uncoupled reactions?
- Reactions that are not directionally, partially, or fully coupled.
- No flux dependence between them.
What can often contribute to emergent partial coupling, though not a strict requirement?
Partial coupling is emergent: it can’t be deduced by just looking at two reactions.
Multisubstrate reactions contribute to such complex dependencies, though they aren’t the only way partial coupling can arise.
What kind of LP do we need to solve in order to determine reaction couplings computationally?
Solve a linear fractional program (LFP):
max/min (vi / vj)
Subject to:
* Nv = 0 (steady state)
* vmin ≤ v ≤ vmax
What are the five possible outcomes of max(min)v vi/vj, assuming that all reactions are irreversible?
For some constant c > 0:
1. 0 ≤ vi/vj ≤ c → directional: i → j
2. c1 ≤ vi/vj ≤ c2 → partial coupling: i ↔ j
3. vi/vj = c → full coupling: i ⟺ j
4. vj/vi ≥ c, no upper bound → directional: j → i
5. 0 ≤ vi/vj, no upper bound → uncoupled
Why is solving LFPs hard?
- LFPs are not linear
- But: can be converted using Charnes-Cooper transformation into LPs
What is an LFP?
Linear fractional program
Ie a fraction of linear functions
What is the Charnes-Cooper transformation?
A method to convert an LFP: max (cTx + α) / (dTx + β) Into a standard LP form using substitutions:
* y = x / (dTx + β)
* T = 1 / (dTx + β)
Results in a new LP with linear constraints and objective.
Complexity:
How many LPs must be solved to classify all couplings?
If there are m reactions:
For one reaction, we need to solve 𝟐(𝒎 − 𝟏) linear programs to determine the couplings to the other (𝒎−𝟏) reactions
→ For all reactions – we need to solve 𝟐𝒎(𝒎 − 𝟏) LPs
i.e one min and one max for each ordered pair of reactions (i, j)
For instance, calculation of coupling in the large-scale network of E. coli (iJR904) would take more than 24 hours
Recent methods and tools are based on assessing feasibility programs, rather than linear programs
This allows calculations in a matter of few hours for the same network
Why can’t we solve vi / vj directly in FCA?
- Because it’s a linear fractional program (LFP) — not a linear function.
- LP solvers require a linear objective.
Consider the following LFP: max/min (vi / vj).
What would be the result of applying the Charnes-Cooper transformation?
- It converts the LFP into a linear program (LP) by substituting:
- t = 1 / vj
- v’ = v · t
- The new variable v’i becomes the objective function.
- v’j = 1 is added as a constraint.
- All constraints are rewritten in terms of v’ and t.
Consider the following LFP:
max/min (vi / vj)
st:
Nv = 0 (steady state)
vmin ≤ v ≤ vmax
What is the final form of the LP after Charnes-Cooper transformation?
max v’i
s.t:
N v’ = 0
0 ≤ v’k ≤ vk,max · t
v’j = 1
t ≥ 0
What is the general form of a Linear Fractional Program (LFP)?
max (cTx + α) / (dTx + β)
s.t:
A x ≤ b
Consider the LFP max (cTx + α) / (dTx + β)
s.t:
A x ≤ b
What substitution does Charnes-Cooper use to linearize such an LFP?
- Let t = 1 / (dTx + β)
- Let y = x · t = x / (dTx + β)
Consider the general form of an LFP:
max (cTx + α) / (dTx + β)
s.t:
A x ≤ b
What is the transformed objective in Charnes-Cooper form?
max cTy + αt
Consider the general form of an LFP:
max (cTx + α) / (dTx + β)
s.t:
A x ≤ b
What are the constraints in Charnes-Cooper LP form?
- A y ≤ b t
- dT y + β t = 1
- t ≥ 0
Consider the LFP max (cTx + α) / (dTx + β)
s.t:
A x ≤ b
After transformation as follows:
* Let t = 1 / (dTx + β)
* Let y = x · t = x / (dTx + β)
How do we recover the original variable x from the transformed variables?
x = y / t
How does the general form of an LFP relate to FCA?
- FCA optimizes vi / vj, which is a special case of an LFP.
- In this case:
- x = v,
- c = ei
- d = ej
- α = β = 0
Why does FCA solve both max and min of vi/vj?
- To determine how much the flux ratio can vary.
- A narrow or fixed range indicates strong coupling; a wide or unbounded range suggests weak or no coupling.
What does a fixed value of vi / vj indicate?
the fluxes are always in a fixed proportion → full coupling
