We consider the situation of integral membrane proteins in a lipid bilayer matrix where the size of the polar group of the protein is important in determining the lateral packing of the proteins. We represent the cross-section of the protein hydrophobic core as a hexagon moving on a lattice, and represent the projection of the polar group onto the plane of the bilayer as a shape, parts of which overlap the hexagon. Lattice sites represent lipid molecules. We calculate the fraction of lipid molecules which are adjacent to the hydrophobic core of at least one protein. We use this data to consider the "motion restricted" spectrum observed in electron spin resonance (ESR) probe studies, and compute the dependence of the "motion restricted" fraction upon protein concentration. The resulting curves can be used to analyse ESR data in order to deduce the size and shape of the proteins' polar segment. We have used the range of models examined to study the dependence upon protein concentration of the particular case of the "motion restricted" spectrum of a spin-labelled lipid freely diffusing or, alternatively, covalently bound to cytochrome c oxidase. We find that our calculations are in accord with a model where approximately 60 lipid molecules can fit around an isolated such protein in both halves of the bilayer, and where the polar segment is substantially anisotropic and extends laterally beyond the limits of the hydrophobic core. The latter is in accord with what is known about the structure of cytochrome c oxidase. We indicate further measurements that should be performed in order to establish more definitively the dependence of the "motion restricted" component upon protein concentration, giving the lipid protein ratios at which they should be performed, and we make predictions concerning the results. Finally we argue for a particular unified way of plotting experimental data.