Maximum density of random placing of membrane particles

Abstract

THE apparently simple problems of (1) how many short lines (‘cars’) of the same length can be randomly placed (‘parked’) in non-overlapping fashion on a line (‘street’); (2) how many squares (Fig. 1) or disks (Fig. 2) of the same size can be randomly placed in an area without overlapping; (3) how many solid spheres (atoms) can be randomly packed in a given volume, are of interest from both physical and mathematical points of view, and occur in various situations¹. Our interest in the two-dimensional problem arises from a computer simulation^2,3 of particles in cell membranes. When the membrane bilayer is split into two leaflets by freeze-fracture method⁴ and an area of the split membrane is visualised by electron microscopy, then the embedded membrane particles (believed to be predominantly proteinaceous) are seen as circular shapes. In natural membranes the unaggregated particles seem to be random in position, and the areal density of the particles may be high. It seems that nature has provided us with a two-dimensional example of (nearly) maximum random parking. Hence we have estimated the maximum fractional area covered by circular disks on a two-dimensional membrane, using a computer simulation with periodic boundary conditions. The correspondence with particles in a biological membrane has implications for membrane biogenesis.