Hemodynamic Homeostasis · §5 · dV/dt = intake − k(P − P_set)

Pressure-natriuresis as an integral controller (Guyton)

The slow loop is an integral controller, and the kidney is the integrator. Renal pressure-natriuresis obeys dV/dt = intake − k(P − P_set), so any constant disturbance is driven to zero steady error. Two transient volume loads both correct to the same defended pressure, 93 mmHg (spread 0, Guyton’s infinite gain). Shape sim-reproduced [V], anchor cited [L], setpoint open [O].

On the hours-to-days track the kidney defends the long-run pressure by excreting volume in proportion to the pressure error. Modelled as dV/dt = intake − k(P − P_set), the controller shows perfect adaptation: transient salt/volume loads return to the same 93 mmHg (spread 0), the signature of infinite steady-state gain [V]; the absolute setpoint is open [O].

The kidney is the integrator of the pressure loop

The long-run pressure is held by an integral, not a proportional, law — and that is the whole reason blood pressure is stable over a lifetime. In Guyton’s renal–body-fluid feedback the kidney excretes volume in proportion to the pressure error, dV/dt = intake − k(P − P_set), so the steady error of any constant disturbance is driven to zero (research target RP3). Where the fast baroreflex only attenuates, the renal integrator eliminates steady error.

The integrator node is the kidney, master SIX2 (γ = 1.5556, DNA-measured [V]); the renin and aldosterone arm is the RAAS endocrine node, master REN (γ = 1.3634, measured on the same nearest-neighbour stacking pipeline [V]). Both identities are grounded in genomic thermodynamics and never fitted (see the DNA-grounding chapter), so the slow loop is built on two real master genes rather than two labels.

Perfect adaptation = infinite steady-state gain

The integral law makes the defended pressure load-independent, which is the precise, testable signature of an integral controller. In the model two distinct transient volume loads both correct to 93 mmHg, with a load-independent spread of 0 mmHg — perfect adaptation, the property Guyton called “infinite gain.” A proportional controller would leave a load-dependent residual; an integral one does not, and the reproduced zero spread is the discriminant.

This is the structural reason sustained hypertension cannot be a pure operating-point disturbance. An infinite-gain integrator rejects any fixed push back toward its reference, so a durable pressure change requires moving the reference itself — the reset that the next section formalises. The controller shape (perfect adaptation) is reproduced [V]; the cited renal handling is an anchor [L]; the absolute setpoint in mmHg is open [O], with the obstacle stated in the ledger.