Module 7 · Biological Information Systems · Page 1 of 3

The Green
Network

Stomata as cellular automata: distributed intelligence without a centre

Look at a leaf and see biology. Let information theory look and it sees a distributed network of microprocessors — ten thousand pores, each adjusting aperture on purely local signals, with no central controller. No conductor. No master clock. Yet the whole leaf breathes in concert.

I

Foundation

Guard Cell Logic

Each pore integrates four independent signals: blue light (phototropins), CO₂, ABA, and VPD — through competing kinase cascades. No signal is privileged. No hierarchy.

II

Mathematics

Wolfram-Class Automata

Guard-cell response rules formalised via the BWB model and the full ODE produce canopy-level gas exchange from entirely local computation — Wolfram Class III–IV.

III

Frontier

Quantum Tunnelling at the Pump

Turgor change is driven by H⁺-ATPase proton pumps. Proton translocation is a candidate site for WKB quantum tunnelling — connecting to quantum-proteins.ai Problem 11.

I. The Ball–Woodrow–Berry conductance model

The earliest quantitative account of stomatal behaviour at steady state is the Ball–Woodrow–Berry (BWB) model (1987). It expresses stomatal conductance \(g_s\) as a function of net CO₂ assimilation \(A\), relative humidity at the leaf surface \(h_s\), and CO₂ concentration at the surface \(c_s\):

Eq. 1 — Ball–Woodrow–Berry model
\[ g_s = g_0 + m\,\frac{A\cdot h_s}{c_s} \]

g₀ — residual conductance  ·  m — empirical slope ≈ 9 for C₃ plants  ·  A — net assimilation [µmol m⁻² s⁻¹]

II. Guard-cell dynamics — the full ODE

Eq. 2 — Guard-cell aperture ODE
\[ \frac{dg_s}{dt} = \frac{1}{\tau}\Bigl[g_{s,\mathrm{eq}}(\Phi,c_i,\mathrm{VPD},[\mathrm{ABA}]) - g_s\Bigr] \]

τ — stomatal time constant [s]  ·  Φ — blue-light flux  ·  VPD — vapour pressure deficit [kPa]

Eq. 3 — Sigmoid equilibrium conductance
\[ g_{s,\mathrm{eq}} = \frac{g_{\max}}{1+e^{-\xi}}, \quad \xi = \alpha\Phi - \beta c_a - \gamma\,\mathrm{VPD} - \delta[\mathrm{ABA}] \]

Each coefficient (α,β,γ,δ) is purely local — no global leaf-level term appears in ξ.

Eq. 4–5 — Fick's law fluxes and WUE
\[ J_{\mathrm{CO_2}} = g_s(c_a-c_i), \quad J_{\mathrm{H_2O}} = 1.6\,g_s\,\mathrm{VPD}/P \]\[ \mathrm{WUE} = \frac{P(c_a-c_i)}{1.6\,\mathrm{VPD}} \]

Factor 1.6 = ratio of diffusivities D(H₂O)/D(CO₂). Remarkably, g_s cancels in WUE — WUE is an atmospheric invariant.

Eq. 6 — Turing reaction-diffusion (stomatal patterning)
\[ \frac{\partial u}{\partial t} = D_u\nabla^2u + f(u,v), \quad \frac{\partial v}{\partial t} = D_v\nabla^2v + g(u,v) \]

Turing condition: D_v ≫ D_u — inhibitor diffuses faster than activator, producing minimum-distance rule.

Eq. 7 — WKB proton tunnelling probability
\[ T \approx \exp\!\left(-\frac{2}{\hbar}\int_{x_1}^{x_2}\sqrt{2m(V(x)-E)}\,dx\right) \]
The leaf has no strategy. It has only physics — and from that physics, something that looks very much like strategy emerges.

Dynamical systems analysis

Phase portrait — ġ_s vs g_s

Bifurcation — g_s* vs [ABA]

Animated CA step — computation made visible