Resolution effect on the stereological estimation of surface and volume and its interpretation in terms of fractal dimensions

TLDR

Estimating surface and volume density of subcellular membrane systems at different magnifications yields inconsistent results. The study interprets these magnification‑dependent estimates using Mandelbrot’s fractal concept, estimating membrane fractal dimensions and deriving resolution‑correction factors to convert measurements to the critical magnification. The authors applied Mandelbrot’s fractal framework to estimate the fractal dimensions of endoplasmic reticulum, inner and outer mitochondrial membranes from stereological data at varying magnifications. Increasing magnification from 18,000× to 130,000× raised surface density estimates of endoplasmic reticulum and inner mitochondrial membranes threefold and volume density of ER threefold, while outer mitochondrial membranes increased only 20%; beyond 130,000× no further change, defining a critical magnification that may explain large literature discrepancies.

Abstract

SUMMARY Estimating surface and volume density of subcellular membrane systems at different magnifications yield different results. As the magnification is increased from × 18,000 to × 130,000 the estimates of surface density of endoplasmic reticulum and inner mitochondrial membranes increase by a factor of 3, whereas that for outer mitochondrial membranes increases only by 20%. The estimate of volume density of endoplasmic reticulum also increases by a factor of 3. No further increase is observed at magnifications above × 130,000 which is therefore called critical magnification. The findings are interpreted on the basis of the concept of fractals proposed by Mandelbrot, and the fractal dimensions of the membrane systems considered are estimated. This can lead to the derivation of resolution correction factors which permit measurements obtained at any magnification to be converted to estimates at critical magnification. These findings may explain, at least in part, the large discrepancy in the estimates of the surface of cytomembranes found in the literature.