For smooth functions ${{a}_{1}}\,,\,{{a}_{2}}\,,\,{{a}_{3}}\,,\,{{a}_{4}}\,$ on a quaternion Heisenberg group, we characterize the existence of solutions of the partial differential operator system ${{X}_{1}}f\,=\,{{a}_{1}},\,{{X}_{2}}f=\,{{a}_{2}},\,{{X}_{3}}f\,=\,{{a}_{3}},\,\text{and}\,{{X}_{4}}f\,=\,{{a}_{4}}$. In addition, a formula for the solution function $f$ is deduced, assuming solvability of the system.

Crimp, a distinguishing feature of sheep fibres, significantly affects wool value, processing and final fabric attributes. Several explanations for fibre bending have been proposed. Most concentrate on relative differences in the physicochemical properties of the cortical cells, which comprise the bulk of the fibre. However, the associations between cortical properties and fibre crimp are not consistent and may not reflect the underlying causation of fibre curvature (FC). We have formulated a mechanistic model in which fibre shape is dictated primarily by the degree of asymmetry in cell supply from the follicle bulb, and the point at which keratinisation is completed within the follicle. If this hypothesis is correct, one would anticipate that most variations in fibre crimp would be accounted for by quantitative differences in both the degree of mitotic asymmetry in follicle bulbs and the distance from the bulb to the point at which keratinisation is completed. To test this hypothesis, we took skin biopsies from Merino sheep from sites producing wool differing widely in fibre crimp frequency and FC. Mitotic asymmetry in follicle bulbs was measured using a DNA-labelling technique and the site of final keratinisation was defined by picric acid staining of the fibre. The proportion of para- to ortho-cortical cell area was determined in the cross-sections of fibres within biopsy samples. Mitotic asymmetry in the follicle bulb accounted for 0.64 (P < 0.0001) of the total variance in objectively measured FC, while the point of final keratinisation of the fibre accounted for an additional 0.05 (P < 0.05) of the variance. There was no association between ortho- to para-cortical cell ratio and FC. FC was positively associated with a subjective follicle curvature score (P < 0.01). We conclude that fibre crimp is caused predominantly by asymmetric cell division in follicles that are highly curved. Differential pressures exerted by the subsequent asymmetric cell supply and cell hardening in the lower follicle cause fibre bending. The extent of bending is then modulated by the point at which keratinisation is completed; later hardening means the fibre remains pliable for longer, thereby reducing the pressure differential and reducing fibre bending. This means that even highly asymmetric follicles may produce a straight fibre if keratinisation is sufficiently delayed, as is the case in deficiencies of zinc and copper, or when keratinisation is perturbed by transgenesis. The model presented here can account for the many variations in fibre shape found in mammals.