Fluid–structure interactions in the xylem: A nonlinear reaction–diffusion model of hydraulic segmentation and pressure-actuated microvalves

Abstract

While the Cohesion-Tension theory explains primary advective sap ascent, standard continuous models often neglect the mechanical regulatory role of bordered pits during secondary transport, such as capillary refilling. To address this, we model the conifer xylem as a network of pressure-actuated elastic microvalves governed by fluid–structure interactions. By coupling Darcy flow with state-dependent hydraulic capacitance and aquaporin-mediated transcellular refilling, we derive a macroscopic, asymmetric Fisher–Kolmogorov reaction–diffusion equation. Analytically, unimpeded fluid recovery propagates as coherent saturation wavefronts bounded by a Kolmogorov–Petrovsky–Piskunov critical velocity, which places the threshold on the same scale as overnight hydraulic recovery, although a front velocity for conifer stems is not at present well constrained empirically. If macroscopic diffusivity or elastic buffering rates drop below this structurally invariant limit, the wavefront collapses, providing a theoretical analog for catastrophic hydraulic failure. Conversely, once a refilling front separates a continuous liquid column from an embolized compartment held near atmospheric pressure by its gas phase, the membrane between them sustains the full head of that column — as tension developed at the column’s own top, not as a weight resting upon it — and the microvalves actuate when this differential exceeds the elastic restitution limit of the margo. Crucially, our framework evaluates this topological shift via a piecewise integral: valve closures generate structural septa that physically isolate the lower fluid column and mathematically reset the pressure field for subsequent ascent. This provides a formal mathematical demonstration of hydraulic segmentation—the iterative stacking of mechanically decoupled fluid reservoirs, each of which need only sustain the tension generated over its own height rather than over the whole tree. Furthermore, by deducing a state-dependent capacitance factor to enforce absolute fluid-volume conservation, the closed-valve asymptotic limit yields an exact localized profile that functions as a threshold rather than an attractor: it sets a minimum sequestered volume, $4\sqrt{D/\gamma }$, below which an isolated reserve is reabsorbed by the surrounding living tissue. These profiles represent hydraulically sequestered capacitive buffers, mathematically mirroring how trees sequence drought responses to prioritize stem elastic water storage. Because the hydrostatic multiplier is fixed by the specific weight of water and the margo threshold has been measured directly, the predicted spacing of these septa follows with no adjustable parameter, and places segmentation in the range 33–97 m for the Pinaceae, its onset governed by the weakest pits of the population—within the range of heights attained by the tallest conifers. We emphasize that this remains a mechanism permitted by the physics and compatible with the known biology rather than one yet demonstrated: its distinctive signature, a discrete vertical periodicity in the aspirated state of the pit population, has not been looked for. Ultimately, this perspective on vascular fault tolerance offers scalable blueprints for resilient, biomimetic microfluidic technologies.