Flux Analysis

An algorithm to assess the flow of metabolites occuring in a metabolic network. It answers the question Given all the reactions in a metabolic network, how much of each metabolite is being produced or consumed at steady state?

Analysis Workflow

Flux Analysis

Select the "Flux Analysis" button under the Analysis Options dropdown.
Set bounds for model associated reactions
Select an objective function.

Selecting an objective function

A reaction that you want the model to maximize or minimize. If you pick the Biomass reaction as the objective, the model will find a flux distribution where this reactions as strongly as possible subject to all constraints.
Choosing a Flux algorithm
Once the analysis is done, the flux values acts as weights and are applied to the model that is reflected in the color gradient.

Recomended

To learn more about Flux Weight File. Check out Uploading a Flux weight File

Types of Flux Analysis Algorithms

Standard Flux-Balanace Analysis (FBA)

Standard FBA predicts the steady-state flux distribution in a metabolic network by optimizing an objective.

Maximize or minimize an objective function \(Z\) (e.g., biomass flux) using Flux Balance Analysis (FBA):

\[ \text{maximize } Z = c^T v \]

subject to:

\[ S \cdot v = 0 \quad \text{(steady-state mass balance)} \]

\[ v_\text{min} \le v \le v_\text{max} \quad \text{(flux bounds)} \]

Where:

\(S\) = stoichiometric matrix
\(v\) = vector of reaction fluxes
\(c\) = vector defining the objective (1 for biomass reaction, 0 for others)

Cycle Free Flux (Loopless solution)

Why was Loopless solution developed

Loopless solution was developed to solved the problem of standard FBA: Possibility of predicting thermodynamically infeasible cycles. These cycles can produce flux distributions where metabolites appear to be produced without the substrate being consumed.

Loopless FBA considers additional constraints to prevent such cycles and this algorithm becomes a Mixed Integer Linear Programming (MILP) problem that is computationally expensive.

Adds constraints to remove internal loops:

\[ v_j \cdot (v_j - M y_j) = 0 \quad \forall j \in \text{reversible reactions} \]

Where:

\(y_j\) = binary variable indicating reaction direction
\(M\) = a large constant (Big-M method)

Single-Reaction Deletion

An approach where each reaction is deleted (knock-out) one at a time and the resulting flux distribution is calculated.

This methods helps you to identify which reactions are essential for product formation as deletion of one reaction results in a loss of objective value.

For each reaction \(r_i\):

\[ v_i = 0 \]

Then solve Standard FBA or Loopless FBA to see how the objective \(Z\) changes.

Parsimonous Flux Balance Analysis (pFBA)

Why was pFBA developed

pFBA was developed to solved the problem of standard FBA: Flux Redundancy.

In some cases, multiple flux distributions achieve the same optimal objective value, but biologically not all are equally likely. pFBA includes a second optimization step where the total flux is minimized, subject to achieving the same maximum objective. This tends to identify the most efficient flux distribution

Two-step Linear Program (LP):

Solve standard FBA to get optimal objective \(Z^*\)
Minimize total flux while keeping \(Z = Z^*\):

\[ \begin{aligned} \text{minimize} \quad & \sum_j |v_j| \\ \text{subject to} \quad & S \cdot v = 0 \\ & v_{\min} \leq v \leq v_{\max} \\ & c^T v = Z^* \end{aligned} \]

Flux Variability Analysis (FVA)

Flux Variability Analysis quantifies the range of possible flux values for each reaction while still achieving the same optimal objective value. Unlike FBA, which returns a single flux distribution, FVA reveals alternative pathways and network flexibility.

Why was FVA developed

Standard FBA provides only one optimal solution, even though many equally optimal solutions may exist. FVA helps identify reactions that are rigid (fixed flux) versus flexible (variable flux) under the same biological objective.

For each reaction ( v_i ), FVA computes:

Minimum possible flux
Maximum possible flux

while keeping the objective fixed at its optimal value ( Z^* ).

For each reaction ( i ):

\[ \begin{aligned} \text{minimize / maximize} \quad & v_i \\ \text{subject to} \quad & S \cdot v = 0 \\ & v_{\min} \le v \le v_{\max} \\ & c^T v = Z^* \end{aligned} \]

Where:

( Z^* ) is the optimal objective value obtained from standard FBA
Each reaction is optimized independently

Interpretation:

Narrow flux range → reaction is essential or tightly regulated
Wide flux range → reaction is flexible or part of alternative pathways

Single-Gene Deletion

Single-Gene Deletion simulates the knockout of individual genes to assess their impact on network functionality and the objective value.

Why Single-Gene Deletion is useful

Genes often control multiple reactions through gene–protein–reaction (GPR) associations. Deleting a gene allows identification of essential genes and genetic robustness in the metabolic network.

For a gene ( g ):

All reactions associated with ( g ) (based on GPR rules) are constrained to zero flux:

\[ v_j = 0 \quad \forall j \in \text{reactions associated with } g \]

Then solve Standard FBA or Loopless FBA to compute the new objective value ( Z_g ).

Interpretation:

( Z_g = 0 ) → gene is essential
( Z_g < Z^ ) → gene is important but non-essential*
( Z_g = Z^ ) → gene deletion has no effect*

Blocked Reactions

Blocked reactions are reactions that cannot carry any flux under any feasible steady-state condition, given the network topology and flux bounds. Identifying these reactions helps reveal structural limitations, dead-end pathways, and gaps in the metabolic model.

Why Blocked Reactions analysis is useful

A metabolic model may contain reactions that are never active due to missing connections, incomplete pathways, or restrictive bounds. Detecting blocked reactions highlights network gaps and guides model curation by pinpointing areas that need additional reactions or corrected constraints.

A reaction \(v_j\) is considered blocked if:

\[ v_j^{\min} = v_j^{\max} = 0 \]

This is determined by solving two optimization problems for each reaction:

\[ \begin{aligned} v_j^{\min} &= \min \, v_j \\ v_j^{\max} &= \max \, v_j \\ \text{subject to} \quad & S \cdot v = 0 \\ & v_{\min} \le v \le v_{\max} \end{aligned} \]

If both the minimum and maximum flux are zero, the reaction is blocked.

Interpretation:

Blocked reaction → reaction is structurally unable to carry flux
Non-blocked reaction → reaction can participate in at least one feasible flux distribution