Abstract
We perform an analytical study of the dynamics of a multi-solute model for water transport across a cell membrane under periodic fluctuations of the extracellular solute molalities. Under the presence of non-permeating intracellular solute, water volume experiences periodic oscillations if and only if the extracellular non-permeating solute molality is positive in the average. On the other hand, in the absence of non-permeating intracellular solute, a sufficient condition for the existence of an infinite number of periodic solutions of the model is provided. Such sufficient condition holds automatically in the case of only one permeating solute. The proofs are based on classical tools from the qualitative theory of differential equations, namely Brouwer degree, upper and lower solutions and comparison arguments.
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References
Bernardi M, DePalma R, Trevisani F, Capani F, Santini C, Baraldini M, Gasbarrini G (1985) Serum potassium circadian rhythm. Relationship with aldosterone. Horm Metab Res 17(12):695
Benson JD, Chicone CC, Critser JK (2005) Exact solutions of a two parameter flux model and cryobiological applications. Cryobiology 50(3):308–316
Benson JD, Chicone CC, Critser JK (2011) A general model for the dynamics of cell volume, global stability and optimal control. J Math Biol 63(2):339–359
Capietto A, Mawhin J, Zanolin F (1992) Continuation theorems for periodic perturbations of autonomous systems. Trans Am Math Soc 329:41–72
Deimling D (1985) Nonlinear functional analysis. Springer, Berlin
Fijorek K, Puskulluoglu M, Polak S (2013) Circadian models of serum potassium, sodium, and calcium concentrations in healthy individuals and their application to cardiac electrophysiology simulations at individual level. Comput Math Methods Med1–8. Article ID 429037
Haydon MJ, Bell LJ, Webb AAR (2011) Interactions between plant circadian clocks and solute transport. J Exp Bot 62(7):2333–2348
Hernández JA (2007) A general model for the dynamics of the cell volume. Bull Math Biol 69(5):1631–1648
Kanabrocki EL, Scheving LE, Halberg F (1973) Circadian variations in presumably healthy men under conditions of peace time army reserve unit training. Space Life Sci 4(2):258–270
Katkov I (2000) A two-parameter model of cell membrane permeability for multisolute systems. Cryobiology 40(1):64–83
Katkov I (2002) The point of maximum cell water volume excursion in case of presence of an impermeable solute. Cryobiology 44(3):193–203
Kleinhans FW (1998) Membrane permeability modeling: Kedem–Katchalsky vs a two-parameter formalism. Cryobiology 37(4):271–289
Krasnoselskii MA, Zabreiko PP (1984) Geometrical methods of nonlinear analysis. Springer, Berlin
Nkashama MN (1989) A generalized upper and lower solutions method and multiplicity results for nonlinear first-order ordinary differential equations. J Math Anal Appl 140(2):381–395
Sennels HP, Jørgensen HL, Goetze JP, Fahrenkrug J (2012) Rhythmic 24-hour variations of frequently used clinical biochemical parameters in healthy young males-the Bispebjerg study of diurnal variations. Scand J Clin Lab Investig 72:287–295
Sothern RB, Vesely DL, Kanabrocki EL et al (1996) Circadian relationships between circulating atrial natriuretic peptides and serum sodium and chloride in healthy humans. Am J Nephrol 16(6):462–470
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I wish to thank two anonymous reviewers for their valuable comments and suggestions. Partially supported by Spanish MICINN Grant with FEDER funds MTM2011-23652 and project FQM-1861, Junta de Andalucía
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Torres, P.J. Periodic oscillations of a model for membrane permeability with fluctuating environmental conditions. J. Math. Biol. 71, 57–68 (2015). https://doi.org/10.1007/s00285-014-0815-6
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DOI: https://doi.org/10.1007/s00285-014-0815-6