Kinetics of extracellular ATP from goldfish hepatocytes: a lesson from mathematical modeling

In goldfish hepatocytes, hypotonic exposure leads to cell swelling, followed by a compensatory shrinkage termed RVD. It has been previously shown that ATP is accumulated in the extracellular medium of swollen cells in a non-linear fashion, and that extracellular ATP (ATPe) is an essential intermediate to trigger RVD. Thus, to understand how RVD proceeds in goldfish hepatocytes, we developed two mathematical models accounting for the experimental ATPe kinetics reported recently by Pafundo et al. in Am. J. Physiol. 294, R220-R233, 2008. Four different equations for ATPe fluxes were built to account for the release of ATP by lytic (J(L)) and nonlytic mechanisms (J(NL)), ATPe diffusion (J(D)), and ATPe consumption by ectonucleotidases (J(V)). Particular focus was given to J(NL), defined as the product of a time function (J(R)) and a positive feedback mechanism whereby ATPe amplifies J(NL). Several J (R) functions (Constant, Step, Impulse, Gaussian, and Lognormal) were studied. Models were tested without (model 1) or with (model 2) diffusion of ATPe. Mathematical analysis allowed us to get a general expression for each of the models. Subsequently, by using model dependent fit (simulations) as well as model analysis at infinite time, we observed that: - use of J(D) does not lead to improvements of the models. - Constant and Step time functions are only applicable when J(R)=0 (and thus, J(NL)=0), so that the only source of ATPe would be J(L), a result incompatible with experimental data. - use of impulse, Gaussian, and lognormal J(R)s in the models led to reasonable good fits to experimental data, with the lognormal function in model 1 providing the best option. Finally, the predictive nature of model 1 loaded with a lognormal J(R) was tested by simulating different putative in vivo scenarios where J(V) and J(NL) were varied over ample ranges.