Modeling proximal tubule cell homeostasis: Tracking changes in luminal flow. Weinstein, A. & Sontag, E. Bulletin of Mathematical Biology, 71:1285-1322, 2009.
abstract   bibtex   
During normal kidney function, there are 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 optimal linear feedback that stabilizes Na+ flux, cell volume, and cell pH. This optimal feedback control is largely consigned to three physiologic variables, cell volume, cell electrical potential, and lateral intercellular hydrostatic pressure. Transport modulation by cell volume stabilizes cell volume; transport 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. 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 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.
@ARTICLE{weinstein07,
   AUTHOR       = {A.M. Weinstein and E.D. Sontag},
   JOURNAL      = {Bulletin of Mathematical Biology},
   TITLE        = {Modeling proximal tubule cell homeostasis: Tracking 
      changes in luminal flow},
   YEAR         = {2009},
   OPTMONTH     = {},
   OPTNOTE      = {},
   OPTNUMBER    = {},
   PAGES        = {1285-1322},
   VOLUME       = {71},
   PDF          = {../../FTPDIR/weinstein_sontag_bulletin_math_biology_proximal_tubule_2009.pdf},
   ABSTRACT     = {During normal kidney function, there are 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 optimal linear feedback that 
      stabilizes Na+ flux, cell volume, and cell pH. This optimal feedback 
      control is largely consigned to three physiologic variables, cell 
      volume, cell electrical potential, and lateral intercellular 
      hydrostatic pressure. Transport modulation by cell volume stabilizes 
      cell volume; transport 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. 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 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. }
}

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