Heparan sulphate (HS) is a glycosaminoglycan present in all metazoan organisms. It is an unbranched chain made up of repeating disaccharide units of uronic acid and glucosamine sugars, and is present in both cells and the extracellular matrix. It is one of the most structurally diverse biological molecules and its biosynthesis involves a variety of enzymic modification steps. Unlike the genome and the transcriptome, HS synthesis is not template driven. Nevertheless, the HS structure and function are highly regulated with modification steps occurring in discrete regions of the polysaccharide chain to give rise to diverse structures interacting with, and regulating, many different proteins. The resulting variation leads to diverse biological roles of HS. To study this structural diversity, rapid isolation and characterization of HS from small amounts of tissues, followed by digestion with bacterially derived enzymes (heparitinases) and chromatography techniques can be used to separate HS oligosaccharides of different size and charge. However, this leads to complex datasets where comparison of just a few samples leads to difficulties in data analysis. Using automatically integrated peak data obtained from chromatographic software, one can apply the effective disc technique to the data points to obtain the centre of mass in each dataset, for example from different murine tissues. This allows facile comparative analysis of different datasets. When the cloud of points displays some preferential direction (anisotropy), it is preferable to compute its effective ellipse. Analysis of the dynamics of the cloud of points for repeated experiments allows the quantification of their reproducibility through evaluation of an average Lyapunov exponent characterizing the area-preserving nature of a sequence of effective ellipses. These basic mathematical approaches allow a more systematic comparison of datasets derived from structural analysis using basic spreadsheet software calculations and contribute to the development of system biology strategies for tackling biocomplexity of HS polysaccharides.
Josephson's junction, annulus maps, Birkhoff attractors, horseshoes and rotation sets
