Role of topology in bioenergetics of sodium transport in complex epithelia.

There has been some controversy as to whether in sodium-transporting epithelia (e.g., frog skin, toad urinary bladder) the Na-O2 ratio is independent of the net rate of transepithelial sodium transport or remains constant over a wide range of transport rates. This computer simulation study shows that both views are defensible, depending on whether one wishes to exclude or include "static head" energy in the calculations. This energy arises from intramembrane sodium recirculation, as shown here when applying a multicompartment epithelial membrane model. Assuming reasonable kinetic parameters of a transport model whose sodium transport rate is varied over a wide range by a step-by-step simulated amiloride action, the computations have shown the following. When static head energy (O2 consumption) is excluded from the calculations, the Na-O2 ratio is constant over a wide range of transepithelial sodium flux, up to the tested value of 20 neq X cm-2 X min-1, a "normal" value found in frog skin. The Na-O2 ratios were 19.4 and 28.7 in low- and high-sodium models, respectively. If the static head energy is included in the calculations, the Na-O2 ratios increase with increasing transport rate from zero values to values up to 15.7. These data are in good agreement with laboratory results, as are derived data on phenomenological coefficients and thermodynamic coupling coefficients (LNa = 80, 128; LNa,r = 4.4; Lr = 0.27, 0.58; q = 0.50, 0.90, depending on the chosen model parameters).

Compartmental analysis of the Na+ flux ratio with application to data on frog skin epidermis.

In this computer simulation study, the role of the topological factor on the Na+ influx/backflux (efflux) ratio in multicompartmental model membranes with active Na+ transport has been investigated. As in the classical "three compartment model", so also in multicompartment models with series order of compartments (series topology), the flux ratios are time-independent. By contrast, in models with series-parallel order of compartments (series-parallel topology), inclusive shunt pathways, the flux ratios are time-dependent. The values of the ratios can increase, or decrease with time, reaching steady state values, depending on the nature of the chosen topology. In a similar manner, the apparent value of the driving force, ENa, of the Na+-pumps, calculated from the Ussing-Teorell flux ratio equation and using global flux ratios, can vary in models with series-parallel topology. This is not the case in models with series topology. On the other hand, the true value of the driving forces of the Na+ pumps, calculated from local flux ratios, are higher, and time-independent. In the absence of Na+ pumps (simulated ouabain effect) the flux ratios have in all cases the values of 1.0. These theoretical results are in good agreement with the theoretical results recently published by Sten-Knudsen & Ussing (1981) whose analysis utilized principles differing from those used here. In the design of the multicompartment model and the choice of kinetic parameters, frog skin epidermis served as a guide, such that simulated outputs closely agreed with experimental data in the literature. This includes the realization of a "fast" paracellular, and a "slow" cellular pathway for transepidermal flow of Na+.

Sodium transport and distribution of electrolytes in frog skin.

The objective of this study on frog skin was to examine correlations between transepidermal active Na-transport and intracellular [Na]c, [K]c, [Cl]c homeostasis. Isolated, whole skins, and "split skins" were used in measurements of short-circuit current (SCC) and open skin potential (PD). Water and ion contents were estimated on split skins. Absolute [Na]c and [K]c varied over the range of 18 to 46, and 113 to 80 mM, respectively (Figure 7), but a complementary relationship existed between Na and K, such that [Na]c + [K]c remained approximately equal to 129 mM. Average values for [Na]c and [K]c were approximately equal to 31 and approximately equal to 96 mM, respectively. [Cl]c remained constant at approximately equal to 38 mM. This complementary relationship does not seem to be an artifact, caused by collagenase, used in the preparation of split skins. Whole skins and split skins in Ringer's solution, when treated with fluoroacetate (FAc), ouabain (Ou), or vanadate (Va) over wide ranges of concentrations, showed that FAc greatly depressed the SCC and the PD, without changing [Na]c, [K]c, [Cl]c. FAc acted only from the corium side of the skin. The decreasing SCC remained a Na-current, as in control skins. By comparison, such a separation of cellular functions could not be established with Ou, or Va. These inhibitors either affected SCC, PD, and cellular ion concentration, or they had no effect on any of these parameters. The complementary relationship between [Na]c and [K]c, with [Cl]c remaining again at approximately equal to 38 mM, was also found in tissues exposed to inhibitors. These results indicate that transcellular active Na transport and electrolyte homeostasis are not always rigidly coupled, suggesting that these processes may not be uniformly distributed within the epithelial cells, or among the interconnected cell layers of the frog skin epidermis.

Regulatory aspects of Na+ transport in complex epithelia (frog skin).

In the multilayered epithelial membranes, topological factors, in addition to factors operating in cell units, must be considered in the regulation of ion transport. One such factor, i.e., the role of cell-to-cell junctions as regulators, is considered in the present computer simulation study. The primary aim was to correlate electron microprobe data on cellular Na+ pool sizes of frog skin with well-known rates of transepidermal Na+ influx and efflux under several laboratory conditions, including single or combined application of ouabain and amiloride. A complete kinetic analysis of data obtained on a simplified multicompartmental model of the epidermis suggests, on purely topological grounds, the occurrence in the epidermis of multiple physiological steady states in which Na+ is pumped transepidermally at equal rates but associated with very different states of intercellular Na+ flow dynamics. The possible significance of these multiple states for the understanding of the responses of the epidermis to perturbing influences is discussed.

Kinetic studies on the effects of ouabain on Na+ fluxes in frog skin.

Among 48 pieces of paired frog skins of Rana pipiens in Ringer's solution, 10 pieces showed a strictly monotone decrease in the short circuit current (SCC) following ouabain treatment (10(-4) M). In 9 cases a transient attenuation, and in 27 cases a distinct wave in the ebb of the SCC, was seen. In 2 instances, two waves were seen. Associated with the not-monotone events was a transient rise in electrical skin conductance. The reasons for these mixed skin responses are unknown. One possible reason is considered here: Early during the ouabain action, some of the Na+ entering from the mucosal side is trapped in the skin by electroneutral processes, in keeping with the already known fact that ultimately cellular KCl is partly replaced by NaCl. Computer assisted model studies show how monotone, and not-monotone "transepithelial" net Na+ flux curves can be generated. Essential conditions for the generation of not-monotone Na+ flux curves are: 1. Presence of two distinct "cellular", active Na+ pools in the model. 2. Presence of a loop pathway in which a principal "transepithelial Na+ transport compartment", and a constituent "Na+/K+ maintenance compartment", are connected to each other and to the "extracellular" compartment. The model, then, predicts under which kinetic conditions monotone and not-monotone transepithelial Na+ flux curves will be seen.

Topological aspects of ion transport in complex epithelia (frog skin).

A weak point of the current concept of the kinetics of ion flow in complex epithelial tissue membranes, such as frog skin, is the supposition that these tissue membranes and their exterior environments can be looked upon as a "three compartment system." In the present study a new and more realistic conceptual framework, a "multicompartment system," is applied to a computer assisted kinetic analysis of experimental data. These deal with Na+ flows in "tight" and "leaky" frog skins, prior to and after the treatment with the Na+-blocking drug amiloride. It is shown by numerical examples that unpredictable Na+ flux patterns in frog skin arise from two diverse contributing factors: 1) The constitutive physical relationships which govern the local events at the level of the plasma membranes, and 2) the much neglected topology, i.e., the "connectedness" of the heterogeneous compartments.

Network thermodynamic approach compartmental analysis. Na+ transients in frog skin.

We introduce a general network thermodynamic method for compartmental analysis which uses a compartmental model of sodium flows through frog skin as an illustrative example (Huf and Howell, 1974a). We use network thermodynamics (Mikulecky et al., 1977b) to formulate the problem, and a circuit simulation program (ASTEC 2, SPICE2, or PCAP) for computation. In this way, the compartment concentrations and net fluxes between compartments are readily obtained for a set of experimental conditions involving a square-wave pulse of labeled sodium at the outer surface of the skin. Qualitative features of the influx at the outer surface correlate very well with those observed for the short circuit current under another similar set of conditions by Morel and LeBlanc (1975). In related work, the compartmental model is used as a basis for simulation of the short circuit current and sodium flows simultaneously using a two-port network (Mikulecky et al., 1977a, and Mikulecky et al., A network thermodynamic model for short circuit current transients in frog skin. Manuscript in preparation; Gary-Bobo et al., 1978). The network approach lends itself to computation of classic compartmental problems in a simple manner using circuit simulation programs (Chua and Lin, 1975), and it further extends the compartmental models to more complicated situations involving coupled flows and non-linearities such as concentration dependencies, chemical reaction kinetics, etc.