Maximum density of random placing of membrane particles

  • An update to this article was published on 01 April 1979


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 simulation2,3 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.

Access options

Rent or Buy article

Get time limited or full article access on ReadCube.


All prices are NET prices.


  1. 1

    Solomon, H. Proc. 5th Berkeley Symp. Math. Statist. Probability 3, 119–134 (1967).

  2. 2

    Finegold, L. Biochem. biophys. Acta 448, 393–398 (1976); Proc. Workshop on Physical Chemical Aspects of Cell Surface Events in Cellular Regulation (ed. Delisi, C.) (Elsevier, New York, 1979).

  3. 3

    Donnell, J. T. thesis, Drexel Univ. (1979).

  4. 4

    Sleytr, U. B. & Robards, A. W. J. Microsc. 111, 77–100 (1977).

  5. 5

    Palásti, I. Magy. tudom. Akad. Mat. Kut. Intéz. Közl. 5, 353–360 (1960).

  6. 6

    Akeda, Y. & Hori, M. Nature 254, 318–319 (1975).

  7. 7

    Akeda, Y. & Hori, M. Biometrica 63, 361–366 (1976).

  8. 8

    Rotham, J. E. & Lenard, J. Science 195, 743–753 (1977).

  9. 9

    Singer, S. J. J. supramolec. Struct. 6, 313–323 (1977).

Download references

Author information



Rights and permissions

Reprints and Permissions

About this article

Cite this article

FINEGOLD, L., DONNELL, J. Maximum density of random placing of membrane particles. Nature 278, 443–445 (1979).

Download citation

Further reading


By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.