Protocell Division: Membrane Growth & Excess-Area Fission

Growth Rates

Membrane synthesis k_A 0.22

Content synthesis k_C 0.18

Fission threshold ν_c 1.30

Population

Protocells

Total divisions

Nutrient pool

100%

How it works

Each protocell is a lipid vesicle with membrane area A and encapsulated macromolecule content C (a stand-in for a cell-free TX-TL expression system). Both grow under Monod-style nutrient limitation f(N) = N / (N + K):

dA/dt = k_A · f(N)
dC/dt = k_C · f(N)
V → C / c₀   (volume relaxes to keep osmotic concentration c₀ constant)

A sphere of volume V has the smallest possible area A_min = 4π(3V/4π)^(2/3). The reduced excess area ν = A / A_min measures how much membrane the vesicle carries beyond that minimum — real giant unilamellar vesicles stay near ν ≈ 1, but membrane synthesis outpaces volume growth (osmotic water influx is diffusion-limited), so ν climbs.

ν = A / A_min(V),  ν ≥ 1
fission when ν ≥ ν_c

k_A — how fast the membrane grows; higher values push ν up faster, triggering more frequent division.
k_C — how fast encapsulated content accumulates, which sets the osmotic target volume and slows ν's rise.
ν_c — the excess-area ratio at which the vesicle becomes shape-unstable and undergoes fission into two daughters; content and area split with a stochastic partition (binomial-like noise), so daughters are rarely identical.
A shared nutrient pool depletes as the population's total membrane and content synthesis consumes it, so growth slows and the population saturates — a finite-resource limit real synthetic-cell reactors face too.

This is the same excess-area-driven division mechanism proposed for early protocells (Zhu & Szostak-style vesicle models): membrane growth without matching osmotic volume growth is, on its own, enough to make a minimal cell divide — no genetic division machinery required.

Protocell (color = ν, excess area) Near fission threshold

Population cap reached — divisions paused

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