Fluorescence energy transfer can be used to investigate spatial distributions of acceptor molecules on surfaces or in membranes. In these applications energy transfer is observed from multiple donors to multiple acceptors. The fluorescence decay in such instances is non- exponential and is described by a 'stretched' exponential. This behavior is sometimes referred to as being fractal in time. A theoretical analysis of these decays is presented. Using this analysis the time-resolved fluorescence of the donor can be used to determine the fractal dimension of the surface distribution as well as the size of the acceptor domains. Application of these theoretical results to experimental data on the structure of membrane protein aggregates is described. A key problem is to determine the functional unit of membrane proteins. Two different proteins, bacteriorhodopsin and the calcium ATPase, have been investigated in model reconstituted vesicle systems. Energy transfer experiments can be devised that will measure two different fractal dimensions characterizing these aggregates. They are the fractal dimension associated with the density distribution and the one associated with the length of the 'coastline'. This information can be interpreted in terms of the number of proteins in the unit structure. In bacteriorhodopsin both hexagonal-packed and monomeric units are stable and the monomer is functional. The calcium ATPase results are consistent with a tetrameric functional unit.
T. Gregory Dewey,
"Fractal dynamics of fluorescence energy transfer in biomembranes", Proc. SPIE 1432, Biomolecular Spectroscopy II, (1 June 1991); doi: 10.1117/12.44209; https://doi.org/10.1117/12.44209