## 3 Results and Discussion  
### Whole Genome Transforms  
Figure 1 illustrates the process of comparing difference spectra for chromosomes 21 and 22, for two properties, roll3 and sequence. Chromosomes 21 and 22 were each divided into 1031 bp blocks and their mean power spectra obtained (panels (c), (d), and (h), (i) respectively). Their shuffled power spectra (as described in the Approach section) were also obtained (panels (a), (b), and (f), (g)). Their difference spectra are shown in Figure 1(e) and (j). For example, panel(e) shows the difference spectra  
of chromosome 21 (obtained by subtracting panel (a) from panel (c)) in blue with that of chromosome 22 (obtained by subtracting panel (b) from panel (d)) in red. Notice that the roll3 spectra of both the randomised and shuffled sequences have the same overall sine wave shape. This is due to the structure imposed on the spectra by considering the sequences as a series of overlapping octamers. Since an octamer differs in only one nucleotide from its neighbour octamer, this imposes a relationship on the neighbouring parameter values which are generally similar from octamer to octamer. Although the particular shape of the spectra is parameter-dependent, the same effect is seen in all structural parameter spectra plots. This is in sharp contrast to the sequence plot [Figures 1(f) - (i)], where, since there is no relationship between the neighbouring bases in the shuffled sequence, the shuffled plots are straight. Figure 1: Comparing Spectra. Left-hand panels are roll3, right-hand are sequence. (a) roll3 spectra of shuffled chromosome 21; (b) roll3 spectra of shuffled chromosome 22; (c) roll3 spectra of chromosome 21; (d) roll3 spectra of chromosome 22; (e) roll3 difference spectra of chromosome 21 (blue line) and chromosome 22 (red line). (f) sequence spectra of shuffled chromosome 21; (g) sequence spectra of shuffled chromosome 22; (h) sequence spectra of chromosome 21; (i) sequence spectra of chromosome 22; (j) sequence difference spectra of chromosome 21 (blue line) and chromosome 22 (red line). In each case the y-axis is the spectral value divided by 10,000. All figures were drawn in Matlab (www.mathworks.com). Although the shape of the shuffled and genomic roll3 plots are similar, the genomic plots have many sharp peaks, indicating the non-random organization of structural features, clearly absent from the shuffled sequences. In Figure 2 we plot the difference spectra of human chromosomes 21 and 22 (in blue and red respectively), as per Figure 1(e), for each of the structural parameters of Table 1 and also for sequence. All the spectra increase in intensity as the periodicity decreases. This tends to obscure meaningful peaks - thus we have plotted only the region between 55 bp and 2.5 bp. The spectra of chromosomes 21 and 22 are strikingly similar. The only differences are in the relative intensity of the peaks rather than in their positions. The sequence plot has many peaks, clearly present in both chromosomes, representing the non-random organization of genomic sequence. Both chromosomes show a clear peak at 3 bp which has previously been noted by many other workers [5,10,13] and is indicative of the presence of coding sequences. The peaks at 6.1 and 12 bp are overtones of this primary peak. Some parameters are related - for example decreasing roll flexibility, decreasing roll3 flexibility, Q\({}^{\prime}\) Roll, and 3 Q\({}_{\textrm{ Roll}}\) all describe in some measure the ability of DNA to move to a lower value of roll. Such sets of parameters commonly exhibit at least some of the same peaks. Peaks are found at 6.4 bp for all decreasing roll flexibility parameters, and at 23 bp for all increasing roll flexibility parameters. Neither of these peaks are present in the sequence spectra. Figure 2: Whole Chromosome Difference Spectra.