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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.

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 [

Can the finding of relatively simple dynamic properties of transport processes in epithelia be compatible with the complex structural conception of the cell interior? The general objective of this work is to contribute with the basic aspects of one formal theoretical approach to answer this question. In particular, this study employs mathematical modeling to uncover properties that could be employed to measure structural cellular complexity. Since a detailed computer simulation of the solute movement throughout the intracellular medium would, although more realistic, not be easy to incorporate in a representation of the overall transport process, this study adopts a simpler approach which, nevertheless, may provide some basic conclusions. In this way, as an alternative to explicit computational simulations of the intracellular media, this study assumes that the unidirectional solute movement can approximately be represented by a discrete, multicompartment model. To be noted, discrete approaches to describe flow through nonhomogeneous media have been employed to understand the basic aspects of percolation [

In essence, the findings of this theoretical study suggest that the basic processes of transcellular transport across an epithelial cell between the two extracellular compartments may be reduced to an equivalent two-state linear model. The strategy of model reduction represents an alternative to study discrete systems with a high degree of complexity, such as biochemical networks, and permits to derive models that retain some of the relevant system properties under specific conditions. In this respect, linear systems of a relatively large number of components can be handled in a rather straightforward fashion. Thus, in macromolecular systems the reduction of linear intermediate transitions of multistate diagrams to yield simpler models has provided a tool, for instance, to understand the finding of simple kinetic behaviors in complex membrane transport systems [

One of the simplest models of transcellular transport of a solute across an epithelial cell (e.g., an intestinal cell) is depicted in Figure

Scheme of an epithelial cell performing transcellular transport of a solute. The solute enters the cell at the apical membrane (ap) via active transport at rate

Different models of transcellular solute transport across an epithelial cell (cf. Figure

A somewhat more complex model can be obtained if one assumes, for instance, that an unstirred layer additionally exists at the intracellular surface of the apical membrane (Figure

As mentioned above (Section

In order to illustrate the concepts introduced here, the next section contains numerical studies of some dynamic properties of the models. Of particular interest is the finding that, under conditions not permitting strict reduction, the models nevertheless exhibit a dynamic behavior approximately equivalent to a two-state dynamic model governed by the general equations

In this section, numerical studies are performed to compare the dynamic behaviors between the original and the reduced models, for the different models considered and for different values of some of the parameters. In essence, the procedure followed here consists in simulating the time courses of the model dynamics in response to perturbations from the steady state. The results shown are not exhaustive and only intended to illustrate some basic properties of the models. For this reason, the numerical values employed here for the rate constants are arbitrary and only results of the relative variations of the variable

The increasing complexity of the models (i.e., from Model I towards Model V, Figure

Plots of the time delay

Figures

(a)–(c) Plots of

Similar to Figure

Similar results to the ones displayed in Figures

The results of this theoretical study permit to suggest that complex models of transepithelial transport of solutes may nevertheless exhibit dynamic properties undistinguishable from those of simple models. At least in the realm of linear models of transport, it was shown here that models incorporating several intermediate states of the solute in the intracellular compartment may, under the proper conditions, be reduced to simple two-state models that assume the existence of a unique concentration value for the intracellular solute. Even if those reduction conditions are not accomplished, the numerical studies also permitted to obtain approximate two-state dynamic models to mimic the original complex ones. Taken together, the results of this work permit to ascertain that, at least for the case of the elementary processes of epithelial transport of solutes, it may be possible to reconcile the finding of relatively simple transcellular transport dynamics with the current conception of the cell interior as a highly complex structural media.

This work has, therefore, focused on the case of the transport of solutes across epithelial cells to illustrate that complex models of transport can exhibit dynamic behaviors undistinguishable from those of simple ones. The occasional emergence of relatively simple dynamic properties may be a property encountered for the case of many other complex biological processes, such as transitions between macromolecular states [

As a reference to this study, this section resumes the basic properties of the general two-state model of transepithelial solute transport. This model is given by

The solution of the system given by (

Since

For the case that

In the numerical studies, an approximate solution to the dynamics of a complete (i.e., nonreduced) model can be obtained by introducing an adjust factor

The approximate solution reads

This section summarizes the procedure to reduce dynamic models of transcellular transport, for the case that some states of the transported species are transient intermediates. The method described in this section is based upon the techniques originally developed by Hill [

For the case of Model II, if

Analogously, the more complex models (Models III to VI) can be reduced to the general two-state model given by (

This work was supported by Grants from the Programa para el Desarrollo de las Ciencias Básicas (PEDECIBA) and from the Comisión Sectorial de Investigación Científica (CSIC), Universidad de la República, Uruguay.

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^{2+}binding and diffusion, during activation of frog skeletal muscle