Most models developed to represent transport across epithelia assume that the cell interior constitutes a homogeneous compartment, characterized by a single concentration value of the transported species. This conception differs significantly from the current view, in which the cellular compartment is regarded as a highly crowded media of marked structural heterogeneity. Can the finding of relatively simple dynamic properties of transport processes in epithelia be compatible with this complex structural conception of the cell interior? The purpose of this work is to contribute with one simple theoretical approach to answer this question. For this, the techniques of model reduction are utilized to obtain a two-state reduced model from more complex linear models of transcellular transport with a larger number of intermediate states. In these complex models, each state corresponds to the solute concentration in an intermediate intracellular compartment. In addition, the numerical studies reveal that it is possible to approximate a general two-state model under conditions where strict reduction of the complex models cannot be performed. These results contribute with arguments to reconcile the current conception of the cell interior as a highly complex medium with the finding of relatively simple dynamic properties of transport across epithelial cells. 1. Introduction The transport of water and solutes across epithelia is a relevant physiological property of higher organisms. To perform transport, epithelial cells develop a polarized distribution of membrane molecules, which localize at distinct apical and basolateral domains of the plasma membrane [1, 2]. The analysis and interpretation of quantitative data about solute and water transport across epithelia have constituted a major objective of cell physiologists. The majority of the models classically developed to represent solute transport across epithelia have considered that the interior of the epithelial cells constitutes a well-stirred, homogeneous compartment, characterized by a single value of concentration of the transported species [3–5]. This view implicitly assumes that the intracellular diffusion coefficient of the species remains constant and that diffusion occurs freely and rapidly, so that the intracellular solute concentrations attain a single equilibrium value at a faster time scale than the overall process. This conception differs markedly from the current view about the structural and functional characteristics of the cell interior. In this conception, the intracellular compartment is
References
[1]
C. Yeaman, K. K. Grindstaff, and W. J. Nelson, “New perspectives on mechanisms involved in generating epithelial cell polarity,” Physiological Reviews, vol. 79, no. 1, pp. 73–98, 1999.
[2]
W. J. Nelson, “Epithelial cell polarity from the outside looking in,” News in Physiological Sciences, vol. 18, no. 4, pp. 143–146, 2003.
[3]
G. Whittembury and L. Reuss, “Mechanisms of coupling of solute and solvent transport in epithelia,” in The Kidney: Physiology and Pathophysiology, D. W. Seldin and G. Giebishch, Eds., pp. 317–360, Raven Press, New York, NY, USA, 2nd edition, 1992.
[4]
S. G. Schultz, “A century of (epithelial) transport physiology: from vitalism to molecular cloning,” American Journal of Physiology, vol. 274, no. 1, pp. C13–C23, 1998.
[5]
L. G. Palmer and O. S. Andersen, “The two-membrane model of epithelial transport: Koefoed-Johnsen and Ussing (1958),” Journal of General Physiology, vol. 132, no. 6, pp. 607–612, 2008.
[6]
D. S. Goodsell, “Inside a living cell,” Trends in Biochemical Sciences, vol. 16, pp. 203–206, 1991.
[7]
K. Luby-Phelps, “Cytoarchitecture and physical properties of cytoplasm: volume, viscosity, diffusion, intracellular surface area,” International Review of Cytology, vol. 192, pp. 189–221, 2000.
[8]
J. A. Dix and A. S. Verkman, “Crowding effects on diffusion in solutions and cells,” Annual Review of Biophysics, vol. 37, pp. 247–263, 2008.
[9]
H. X. Zhou, G. Rivas, and A. P. Minton, “Macromolecular crowding and confinement: biochemical, biophysical, and potential physiological consequences,” Annual Review of Biophysics, vol. 37, pp. 375–397, 2008.
[10]
D. L. Ermak and J. A. McCammon, “Brownian dynamics with hydrodynamic interactions,” The Journal of Chemical Physics, vol. 69, no. 4, pp. 1352–1360, 1978.
[11]
M. Dlugosz and J. Trylska, “Diffusion in crowded biological environments: applications of Brownian dynamics,” BMC Biophysics, vol. 4, no. 1, article 3, 2011.
[12]
C. C. W. Hsia, C. J. C. Chuong, and R. L. Johnson, “Red cell distortion and conceptual basis of diffusing capacity estimates: finite element analysis,” Journal of Applied Physiology, vol. 83, no. 4, pp. 1397–1404, 1997.
[13]
P. Bauler, G. A. Huber, and J. A. McCammon, “Hybrid finite element and Brownian dynamics method for diffusion-controlled reactions,” Journal of Chemical Physics, vol. 136, no. 16, Article ID 164107, 2012.
[14]
I. I. Moraru, J. C. Schaff, B. M. Slepchenko et al., “Virtual Cell modelling and simulation software environment,” IET Systems Biology, vol. 2, no. 5, pp. 352–362, 2008.
[15]
I. L. Novak, P. Kraikivski, and B. M. Slepchenko, “Diffusion in cytoplasm: effects of excluded volume due to internal membranes and cytoskeletal structures,” Biophysical Journal, vol. 97, no. 3, pp. 758–767, 2009.
[16]
B. Berkowitz and R. P. Ewing, “Percolation theory and network modeling applications in soil physics,” Surveys in Geophysics, vol. 19, no. 1, pp. 23–72, 1998.
[17]
C. S. Patlak, F. E. Hospod, S. D. Trowbridge, and G. C. Newman, “Diffusion of radiotracers in normal and ischemic brain slices,” Journal of Cerebral Blood Flow and Metabolism, vol. 18, no. 7, pp. 776–802, 1998.
[18]
S. M. Baylor and S. Hollingworth, “Model of sarcomeric Ca2+ movements, including ATP Ca2+ binding and diffusion, during activation of frog skeletal muscle,” Journal of General Physiology, vol. 112, no. 3, pp. 297–316, 1998.
[19]
J. A. Hernández and J. C. Valle Lisboa, “Reduced kinetic models of facilitative transport,” Biochimica Et Biophysica Acta, vol. 1665, pp. 65–74, 2004.
[20]
S. G. Schultz S. G, “Homocellular regulatory mechanisms in sodium-transporting epithelia: avoidance of extinction by "flush-through",” American Journal of Physiology, vol. 242, no. 6, pp. F579–F590, 1981.
[21]
J. M. Diamond, “Transcellular cross-talk between epithelial cell membranes,” Nature, vol. 300, no. 5894, pp. 683–685, 1982.
[22]
L. Reuss and C. U. Cotton, “Volume regulation in epithelia: transcellular transport and cross-talk,” in Cellular and Molecular Physiology of Cell Volume Regulation, K. Strange, Ed., pp. 31–47, CRC Press, Boca Raton, Fla, USA, 1994.
[23]
M. R. Maurya, S. J. Bornheimer, V. Venkatasubramanian, and S. Subramaniam, “Reduced-order modelling of biochemical networks: application to the GTPase-cycle signalling module,” IEE Proceedings Systems Biology, vol. 152, no. 4, pp. 229–242, 2005.
[24]
T. J. Pedley and J. Fischbarg, “Unstirred layer effects on osmotic water flow across gallbladder epithelium,” Journal of Membrane Biology, vol. 54, no. 2, pp. 89–102, 1980.
[25]
K. R. Spring, “Routes and mechanism of fluid transport by epithelia,” Annual Review of Physiology, vol. 60, pp. 105–119, 1998.
[26]
A. M. Weinstein, “Dynamics of cellular homeostasis: recovery time for a perturbation from equilibrium,” Bulletin of Mathematical Biology, vol. 59, no. 3, pp. 451–481, 1997.
[27]
A. M. Weinstein, “Modeling epithelial cell homeostasis: assessing recovery and control mechanisms,” Bulletin of Mathematical Biology, vol. 66, no. 5, pp. 1201–1240, 2004.
[28]
J. A. Hernández, “Stability properties of elementary dynamic models of membrane transport,” in Bulletin of Mathematical Biology, vol. 65, pp. 175–197, 2003.
[29]
J. A. Hernández, “A general model for the dynamics of the cell volume,” Bulletin of Mathematical Biology, vol. 69, no. 5, pp. 1631–1648, 2007.
[30]
T. L. Hill, Free Energy Transduction in Biology, Academic Press, New York, NY, USA, 1977.
