On the time dependent diffusion of macromolecules through transient open junctions and their subendothelial spread. 2. Long time model for interaction between leakage sites

In Part 1 of this study (Weinbaum et al., 1988) a short time model has been proposed to describe the initial time dependent leakage of macromolecules at short distances (5 microns or less) from the exit of a transient open junction which the authors have hypothesized as a characteristic feature of endothelial cells in the process of turnover (Weinbaum et al., 1985). This open junction pathway has also been proposed (Weinbaum et al., 1988) to be the primary ultrastructural correlate of the 20 nm diameter large pore suggested by Renkin et al. (1977) using the predictions of cylindrical pore theory. The short time model in (Weinbaum et al., 1988), however, has major limitations in that it neglects the interaction between leakage sites, macromolecular entry through other pathways, the finite thickness of the vessel wall and the curvature of the cell perimeter. The longer time model developed herein will attempt to describe each of these features and also present an improved model and analytic solution for the steady state flux and uptake. In the previous steady state model developed by Weinbaum et al. (1985) the effect of the resistance of the transient open junctions and the non-isotropic diffusion in the underlying tissue due to the internal elastic lamina (IEL) were both neglected. New solutions are first presented which describe the effect of these important model refinements on the steady state macromolecular permeability of the major arteries. Time dependent solutions are then presented to predict the transient longer time labeling following the introduction of tracer macromolecules of varying size. These solutions and the corresponding short time solutions in Weinbaum et al. (1988) are the first solutions to our knowledge to describe the difficult time-dependent boundary value problem to determine how the channel exit concentration and flux at a leaky junction vary with time. This is accomplished by casting the boundary value problem in the form of an integral equation for the unknown flux at the cleft exit and then solving this problem using a specially designed numerical technique. The theoretical predictions are used to interpret the behavior of the localized leaks to HRP and albumin that have been reported in Stemerman et al. (1986) and our own recent experiments (Lin et al., 1988).

On the time-dependent diffusion of macromolecules through transient open junctions and their subendothelial spread. I. Short-time model for cleft exit region

In this two-part study we shall quantitatively study, using time-dependent models, the hypothesis that transient open junctions associated with widely scattered endothelial cells undergoing mitosis are the structural equivalent for the large pore pathway via which macromolecules the size of albumin or larger cross the vascular endothelium. In an earlier steady-state model [Am. J. Physiol. 248, H945-960 (1985)], the authors demonstrated that such an open-junction pathway could quantitatively account for the regional differences in macromolecular permeability observed in various mammalian arteries in regions of enhanced cell turnover as indicated by 3H-thymidine although these cells were less than 1% of the population and the open junctions occupied less than 10(-5) of the endothelial surface. The time-dependent models described herein have been used to identify a time window and size of probe molecule wherein this hypothesis could be tested experimentally in the larger blood vessels. The first stages of these experiments have now been completed and provide convincing evidence that the junctions of virtually all endothelial cells in the M phase of the cell cycle are leaky to macromolecules (Lin et al., 1988). The statistical frequency of such leakage sites has also been determined. The time-dependent models developed herein contain two important refinements that were not contained in the earlier steady state model. First the finite resistance of the open cleft as a function of molecular size is accounted for by introducing a diffusion coefficient ratio Dj/Dz describing the relative resistance of the open cleft compared to the subendothelial tissue in the direction normal to the endothelial surface. Second the non-isotropy of the vessel wall due to the elastic lamina is considered by introducing a second diffusion coefficient ratio Dx/Dz describing the relative resistance in the lateral as compared to the normal direction. This second ratio can be as large as 100 for the arterial intima, but is of order unity for capillaries. In Part I a short time model is presented to describe the initial labeling of the open cleft and the subendothelial space in the vicinity of the cleft exit following the introduction of a tracer macromolecule. This model is valid for both larger vessels and capillaries since wall thickness and curvature and the interaction between leakage sites does not enter into the model description. In Part II (Wen et al., 1988) a long-time model is developed for larger vessels only which is valid for greater times including steady-state labeling.