Metabolic flux distributions

Economic Principles in Cell Biology · Ch. 4 · 10.5281/zenodo.8156504

Zoom out from single reactions to the whole network and one assumption does most of the work: in steady growth, internal metabolites hold constant. Whatever is made is used. That single equation, Nv = 0, carves the space of all possible flux patterns down to a cone. This chapter is the geometry the next one optimizes over.

Steady state as a balance sheet

Collect the network's stoichiometry into a matrix N: one row per internal metabolite, one column per reaction, entries counting how many molecules each reaction makes or consumes. A flux vector v lists the rate of every reaction. The product Nv is the net rate of change of every metabolite pool. Steady state means none of them drift, so Nv = 0. Every metabolite's ledger balances: total in equals total out.

The linear chain above makes A, converts A to B, and consumes B. Steady state forces v₁ = v₂ = v₃. Nothing can accumulate in a pipe. Let's confirm it and find the whole solution space.

Python

import numpy as np
from scipy.linalg import null_space

# rows = internal metabolites A, B ; columns = reactions v1, v2, v3
#   v1: -> A      (+1 to A)
#   v2: A -> B    (-1 A, +1 B)
#   v3: B ->      (-1 B)
N = np.array([[ 1, -1,  0],
              [ 0,  1, -1]])

v = np.array([5, 5, 5])
print("N @ v =", N @ v, "->", "steady state" if np.allclose(N@v, 0) else "drifting")

# the space of ALL steady-state flux vectors is the null space of N
ns = null_space(N)
print("degrees of freedom:", ns.shape[1])
print("basis flux mode:", np.round(ns[:,0] / ns[0,0], 3))  # scaled so v1 = 1

The null space has dimension one: every steady-state flux through this network is a multiple of the same pattern (1, 1, 1). Set the input rate and the rest follows. Real networks have branches and cycles, so the solution space grows into a whole cone of allowed flux patterns.

The flux cone

Add one more rule: some reactions run only forward, their flux can't go negative. The solution set was a subspace when reactions could run either direction. Restricted to non-negative flux, it becomes a cone, closed under adding allowed patterns and scaling them up, but with walls where irreversible reactions hit zero. The cone's edges are the elementary flux modes, the minimal self-contained routes through the network. Any steady flux the cell can run is a positive blend of these routes. The source develops this geometry in full. What we need next is only that the feasible region is a convex cone, because that's exactly the shape a linear program knows how to optimize.

Neighbors

Related chapters

🦠 Ch.5 Optimizing fluxes — pick the best point in this cone: flux balance analysis
📐 Linear Algebra — null spaces, the machinery under Nv = 0

Foundations (Wikipedia)

Adaptation notes

The source chapter covers the flux cone, elementary flux modes, flux polyhedra, and several methods for exploring flux space. We keep the core, steady state as Nv = 0 and the feasible set as a convex cone, and make the null-space claim runnable on the smallest network that shows it. Elementary modes, cofactor balancing, and the full polyhedral picture are in the source.

Want the full treatment? Read Economic Principles in Cell Biology, Ch. 4.

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