Explain metabolic network modeling

WHAT is a metabolic network?

• Nodes = metabolites (e.g. glucose, ATP, pyruvate).
• Edges/reactions = enzyme-catalysed conversions.
• Each reaction has a flux $v_j$ = rate at which it runs (units: mmol · gDW⁻¹ · h⁻¹, "per gram dry weight per hour").

The stoichiometric matrix $S$ (first principles)

We want to bookkeep how each reaction changes each metabolite.

Deriving the mass-balance equation from scratch.

Consider metabolite $i$. Its concentration $x_i$ changes because reactions produce or consume it. The rate of change is the sum over all reactions of (how much reaction $j$ makes of $i$) × (how fast $j$ runs):

$\frac{dx_i}{dt} = \sum_{j=1}^{n} S_{ij}\, v_j$

Stack all metabolites into a vector $\mathbf{x}$ and all fluxes into $\mathbf{v}$:

The steady-state assumption

This is a set of linear equations. Because there are usually far more reactions than metabolites ($n \gg m$), the system is underdetermined: infinitely many flux vectors $\mathbf v$ satisfy it. That set of solutions is the null space of $S$, called the flux cone.

Flux Balance Analysis (FBA) — picking one solution

We can't solve $S\mathbf v = 0$ uniquely, so we add:

1. Bounds on each flux (thermodynamics/enzyme limits): $\alpha_j \le v_j \le \beta_j$. Irreversible reactions have $\alpha_j = 0$.
2. An objective to maximise, usually the biomass reaction $v_{\text{biomass}}$ (a pseudo-reaction draining precursors in the ratios needed to build one new cell).

Why a linear program? All constraints and the objective are linear in the fluxes, so we use linear programming (LP), which is fast even for genome-scale models (thousands of reactions).

Worked example 1 — a tiny 3-reaction network

Reactions:

• $v_1$: $\;\to A$ (uptake of A)
• $v_2$: $A \to B$
• $v_3$: $B \to$ (biomass/export)

Metabolites $A, B$. Build $S$:

$S=\begin{pmatrix} +1 & -1 & 0 \\ 0 & +1 & -1 \end{pmatrix}$

Why these signs? Row A: $v_1$ makes A ($+1$), $v_2$ eats A ($-1$). Row B: $v_2$ makes B ($+1$), $v_3$ eats B ($-1$).

Steady state $S\mathbf v = 0$:

• Row A: $v_1 - v_2 = 0 \Rightarrow v_1 = v_2$.
• Row B: $v_2 - v_3 = 0 \Rightarrow v_2 = v_3$.

Why this step? Steady state means each internal pool is balanced, forcing all three fluxes equal — the classic conservation of throughput in a linear pathway.

If uptake is capped at $v_1 \le 10$ and we maximise $v_3$: answer $v_1=v_2=v_3=10$. The bottleneck is the intake.

Worked example 2 — a branch point

Add $v_4$: $A \to C$ (competing branch). Now maximise biomass that needs only B (i.e. maximise $v_3$).

Row A balance: $v_1 = v_2 + v_4$.

• Why? A is now consumed by two reactions, so its production must equal the sum of both consumers.

To maximise $v_3$ (=$v_2$), set $v_4 = 0$ so all A flows toward B.

Why this step? Any flux into the side branch $v_4$ steals substrate from the biomass route. The optimiser learns to shut off wasteful branches — this is how FBA predicts which pathways a cell "chooses" under an objective.

Worked example 3 — reading a knockout

Delete the enzyme for $v_2$: set $v_2 = 0$. Then row B: $0 - v_3 = 0 \Rightarrow v_3 = 0$. Biomass = 0 → lethal knockout.

Why this step? With the only producer of B removed, B can't be made, so nothing downstream runs. This is exactly how FBA predicts gene essentiality in silico.

Common mistakes (Steel-manned)

Flashcards

Connections

• Flux Balance Analysis
• Linear Programming
• Stoichiometry and Mass Conservation
• Genome-scale Metabolic Models (GEMs)
• Null Space and Linear Algebra
• Enzyme Kinetics (source of the flux bounds)
• Gene Essentiality and Knockout Screens
• Systems Biology & Frontiers

Concept Map

Hinglish (regional understanding)