Molecular transport dynamics: GHK flux, channel gating, and why the membrane gain is the loop gain
The molecular 'how' beneath the loop arms is one primitive: the Goldman-Hodgkin-Katz constant-field flux through a gated channel. The central result is a connection — the membrane flux slope at the setpoint, k = −dJ/dC, is exactly the Ornstein-Uhlenbeck loop gain the volume already uses, so channel biophysics and the control loop are one quantity at two scales.
The loop arms above were cited by transporter name — TRPV5/6, ENaC, the H⁺-ATPase — but the molecular 'how' beneath them was not yet derived. This chapter adds the one primitive that supplies it: the Goldman-Hodgkin-Katz constant-field flux through a gated channel. Its central result is not a new number but a CONNECTION — the membrane flux slope at the setpoint, k = −dJ/dC, is exactly the loop gain the rest of the volume already runs on. Channel biophysics and the OU control law are the same quantity seen at two scales.
The constant-field flux law
A single channel's current under a fixed field is the GHK constant-field flux: it depends on the permeability, the membrane voltage, and the ion concentrations either side. Two properties are exact and are reproduced in simulation. The flux REVERSES exactly at the Nernst potential (here 66.59 mV, where J=0) and changes sign across it (-45.67 below reversal, +30.07 above), and it RECTIFIES when the ion is more concentrated on one side — the chord conductance differs hyperpolarized versus depolarized (1.264 vs 1.178). These are forced by the physics; the volume does not fit them.
Gating: vitamin-D opens the calcium channel
A channel's open probability is set by a gate — a Boltzmann/Hill function of its controlling signal. The slow VDR arm of the calcium loop acts here: 1,25-dihydroxyvitamin-D transcriptionally up-regulates the apical Ca channels TRPV5 (kidney) and TRPV6 (gut). Raising the vitamin-D signal raises the channel open probability and, through the GHK flux, the transcellular calcium reabsorptive flux — both monotonically.
| vitamin-D signal | channel open probability | Ca reabsorptive flux (GHK) |
|---|---|---|
| 0.25 | 0.059 | 0.64 |
| 0.50 | 0.200 | 2.18 |
| 1.00 | 0.500 | 5.44 |
| 2.00 | 0.800 | 8.71 |
| 4.00 | 0.941 | 10.25 |
The channel-identity assignment (VDR → TRPV5/6) is the cited [L] anchor; the monotone rise of open probability and Ca flux with the vitamin-D signal is [V]; the absolute half-activation and single-channel permeability are [O].
The molecular gain IS the loop gain
This is the load-bearing result. Write the per-deviation restoring flux at the setpoint, k = −dJ/dC — the proportional flux a small concentration error pulls back. Feed THAT k into the volume's own Ornstein-Uhlenbeck setpoint and the two descriptions coincide: the membrane gain reproduces the OU variance law Var = σ²/2k and the rejection law error = load/k exactly, k rises linearly with channel number, and a steeper gate raises k. The channel and the control loop are one object.
| channel number N | k = −dJ/dC | setpoint variance | Var·2k/σ² | step error | error·k |
|---|---|---|---|---|---|
| 0.5 | 0.9981 | 0.04498 | 0.998 | 1.00190 | 1.00 |
| 1.0 | 1.9962 | 0.02243 | 0.995 | 0.50095 | 1.00 |
| 2.0 | 3.9924 | 0.01133 | 1.005 | 0.25048 | 1.00 |
| 4.0 | 7.9848 | 0.00580 | 1.030 | 0.12524 | 1.00 |
Across the channel-number ladder, k = −dJ/dC rises proportionally (True), the setpoint variance falls (True), and both OU laws are recovered to rounding (Var·2k/σ² ≈ 1, error·k ≈ 1). A steeper gate raises k (True). The GHK flux and gating are [F]; the identity k=−dJ/dC = OU loop gain is [V]; the absolute channel density is [O].
Loss of function is the loop-gain drop at the membrane
Because the membrane gain IS the loop gain, a loss-of-function transporter is precisely the loop-gain-drop failure mode — now localized to a single channel. Dropping the channel number / conductance lowers k (7.98 → 2.00), and the volume's own load/k law follows: the steady offset grows by the same factor (×4.0, equal to the gain ratio 4.0) and the variance blows up (0.00580 → 0.02243). This is the molecular reading of the membrane diseases the pathology chapter cited: TRPV5/6 loss → renal calcium wasting; ENaC / SCNN1A loss → pseudohypoaldosteronism type 1; H⁺-ATPase / ATP6V loss → distal renal tubular acidosis.
The failed-channel offset exceeds the healthy one, the ratio equals the gain ratio (the volume's load/k law again), and variance blows up — all [V]. The transporter identities and their loss-of-function phenotypes are the cited [L] anchors; the absolute single-channel conductances and channel densities are [O], the electrophysiological calibration that is this chapter's stated residual.