Module 7 · Page 3 of 3 — The Quantum Frontier

III. Quantum coherence at the pump

Pages 1 and 2 established that the leaf is a distributed cellular automaton whose operating point is geometrically constrained by the atmosphere. Both results are classical. The remaining question — the one that connects this module to quantum-proteins.ai — is whether quantum mechanics enters at the mechanism that drives the whole system.

H⁺-ATPase is the engine of guard-cell turgor change. Each aperture transition requires protons to cross the plasma membrane against an electrochemical gradient of ~200 mV. The proton mass is \(m_p = 1.67 \times 10^{-27}\) kg and the membrane presents a potential barrier whose width is comparable to the proton de Broglie wavelength at physiological energies. The quantum mechanical question is not exotic: it is the same question physically demonstrated in conformationally-constrained bicyclic systems in 1985.

The proton is not too small to be classical. It is too light to ignore the barrier geometry.
Eq. 8a — Temperature-dependent proton transfer rate
\[ k(T) = \underbrace{\nu_0\,e^{-E_a/k_BT}}_{\text{classical}} + \underbrace{k_\mathrm{tunn}(T)}_{\text{quantum}} \]

At T ≫ T*, classical term dominates. At T ≪ T*, tunnelling dominates and k(T) → const.

Eq. 8b — Quantum tunnelling contribution (WKB)
\[ k_\mathrm{tunn}(T) = \nu_0\!\int_0^{E_a}\!T(E)\,e^{-E/k_BT}\,\frac{dE}{k_BT} \]

Boltzmann-weighted integral over all energies below the barrier. T(E) is the WKB transmission coefficient from Eq. 7.

Eq. 8c — Crossover temperature T*
\[ T^* = \frac{\hbar\,\omega_b}{2\pi\,k_B}, \qquad \omega_b = \sqrt{\frac{|V''(x_\mathrm{top})|}{m_p}} \]

ω_b — imaginary frequency at barrier top. For a = 1 Å and V₀ = 0.2 eV, T* ≈ 150 K. Stiffer barriers push T* into the physiological range.

Eq. 9 — Kinetic Isotope Effect (KIE) H⁺/D⁺
\[ \mathrm{KIE} = \frac{k_H}{k_D} \qquad k_\mathrm{tunn}^{(D)} \approx T(E)^{\sqrt{2}} \]

Since m_D = 2m_H, the deuterium WKB exponent is √2 ≈ 1.41× larger. Classical KIE ≈ 6–7. Tunnelling signature: KIE ≫ 7, temperature-independent.

H⁺-ATPase barrier — live parameter model

Rate and KIE as functions of temperature

Historical connection · IFP 1985 → quantum-proteins.ai 2025

From bicyclic systems to guard cells — the geometry principle

The 1985 IFP PhD research on hydrogen transfer in conformationally-constrained bicyclic systems demonstrated that rigid molecular scaffolds suppressed hydrogen transfer despite thermodynamic favorability. In retrospect, this was a clean experimental demonstration of the quantum tunnelling geometry requirement: the tunnelling probability \(T(E)\) decreases exponentially with the donor–acceptor distance \(2a\).

The same principle applies to the H⁺-ATPase transmembrane domain. Conformational changes induced by ATP hydrolysis alter the effective barrier geometry. The guard cell may modulate turgor not just by changing the thermodynamic driving force, but by modulating the quantum mechanical efficiency of proton transfer at the pump. This is the claim that quantum-proteins.ai Problem 11 asks to be quantified. The KIE calculation in Cell 10 of the Python Runtime is its sharpest experimental test.