Module 7 · Biological Information Systems · 3 pages + Python runtime

The
GreenNetwork

From guard-cell logic to quantum coherence — the leaf as a distributed computing system operating across four scales of physics

3 pagesCore module
9 equationsTypeset MathJax
10 cellsRunnable Python
3 platformsCross-linked
~90 minReading + running

Module arc — four movements

1

Biology → Mathematics

The Cellular Automaton

Guard-cell logic. Wolfram-class CA. Ball–Woodrow–Berry steady state. Turing patterning. No central coordinator — only local rules.

\( \dot g_s = \tfrac{1}{\tau}[g_\mathrm{eq}(\xi)-g_s] \)
Open Page 1 →
2

Geometry

The Constraint Manifold

WUE as a topological invariant. The {A, E, g_s} constraint surface in WebGL. Diurnal trajectory animation.

\( \mathrm{WUE} = P(c_a-c_i)/1.6\,\mathrm{VPD} \)
Open Page 2 →
3

Quantum Frontier

Quantum Coherence at the Pump

H⁺-ATPase proton tunnelling. Temperature-dependent rates. KIE H⁺/D⁺. Crossover temperature. Connection to Problem 11 and the 1985 bicyclic work.

\( k(T) = \nu_0 e^{-E_a/k_BT} + k_\mathrm{tunn}(T) \)
Open Page 3 →
★

Python Runtime · Pyodide

In-Browser Simulation

10 executable cells. NumPy · Matplotlib · SciPy. Stale-cell indicator. Narrative mode. Variable inspector. Dependency graph. Export .ipynb / .py / .csv.

Shift+Enter to run · Auto-run on inject
Open Runtime →

Equations in this module

Eq. 1 — Ball–Woodrow–Berry
\( g_s = g_0 + m\,Ah_s/c_s \)
Eq. 2 — Guard-cell ODE
\( \dot g_s = \tfrac{1}{\tau}[g_{s,\mathrm{eq}}-g_s] \)
Eq. 3 — Sigmoid equilibrium
\( g_{s,\mathrm{eq}} = g_{\max}\sigma(\xi) \)
Eq. 4 — Fick's law fluxes
\( J_{\mathrm{CO_2}}=g_s(c_a-c_i),\;J_{H_2O}=1.6g_s\,\mathrm{VPD}/P \)
Eq. 5 — WUE invariant
\( \mathrm{WUE}=P(c_a-c_i)/1.6\,\mathrm{VPD} \)
Eq. 6 — Turing patterning
\( \partial_t u = D_u\nabla^2u + f(u,v) \)
Eq. 7 — WKB tunnelling
\( T \approx \exp(-\tfrac{2}{\hbar}\int\sqrt{2m(V-E)}\,dx) \)
Eq. 8 — Rate k(T) + crossover T*
\( k(T) = \nu_0 e^{-E_a/k_BT} + k_\mathrm{tunn}(T) \)
Eq. 9 — Kinetic Isotope Effect
\( \mathrm{KIE} = k_H/k_D \)

Prerequisites

First-order linear ODE
Sigmoid / logistic dynamics
Fick's first law of diffusion
Reaction-diffusion (Turing 1952)
Wolfram CA classification
Python 3 / NumPy
WKB approximation in QM (Page 3)
Kinetic isotope effects (Page 3)
Arrhenius kinetics + crossover temperature
● Accessible● Intermediate● Advanced (Page 3)

Begin with Page 1

The cellular automaton model. Guard-cell logic. Phase portrait and bifurcation. Animated step-by-step computation. The same mathematics as seismicshift.earth, eight orders of magnitude away.

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