## Modeling proximal tubule cell homeostasis: tracking changes in luminal flow.

Bulletin of Mathematical Biology 71(6):1285 (2009) PMID 19280266 PMCID PMC2793416

During normal kidney function, there are routinely wide swings in proximal tubule fluid flow and proportional changes in Na(+) reabsorption across tubule epithelial cells. This "glomerulotubular balance" occurs in the absence of any substantial change in cell volume, and is thus a challenge to coordinate luminal membrane solute entry with peritubular membrane solute exit. In this work, linear optimal control theory is applied to generate a configuration of regulated transporters that could achieve this result. A previously developed model of rat proximal tubule epithelium is linearized about a physiologic reference condition; the approximate linear system is recast as a dynamical system; and a Riccati equation is solved to yield the optimal linear feedback that stabilizes Na(+) flux, cell volume, and cell pH. The first observation is that optimal feedback control is largely consigned to three physiologic variables, cell volume, cell electrical potential, and lateral intercellular hydrostatic pressure. Parameter modulation by cell volume stabilizes cell volume; parameter modulation by electrical potential or interspace pressure act to stabilize Na(+) flux and cell pH. This feedback control is utilized in a tracking problem, in which reabsorptive Na(+) flux varies over a factor of two, in order to represent a substantial excursion of glomerulotubular balance. The resulting control parameters consist of two terms, an autonomous term and a feedback term, and both terms include transporters on both luminal and peritubular cell membranes. Overall, the increase in Na(+) flux is achieved with upregulation of luminal Na(+)/H(+) exchange and Na(+)-glucose cotransport, with increased peritubular Na(+)-3HCO(3)(-) and K(+)-Cl(-) cotransport, and with increased Na(+), K(+)-ATPase activity. The configuration of activated transporters emerges as a testable hypothesis of the molecular basis for glomerulotubular balance. It is suggested that the autonomous control component at each cell membrane could represent the cytoskeletal effects of luminal flow.

DOI: 10.1007/s11538-009-9402-1