Explorations ofMultidimensional Sequence Space
one symbol -> 1D
coordinate of dimension = pattern length
Two symbols -> Dimension = length of pattern
length 1 = 1D:
Two symbols -> Dimension = length of pattern
length 2 = 2D:
dimensions correspond to positionFor each dimension two possibiities
Note: Here is a possible bifurcation: a larger alphabet could be represented as more choices along the axis of position!
Two symbols -> Dimension = length of pattern
length 3 = 3D:
Two symbols -> Dimension = length of pattern
length 4 = 4D:
aka Hypercube
Two symbols -> Dimension = length of pattern
Three Symbols (another solution is to use more values for each dimension)
Four Symbols:
I.e.: with an alphabet of 4, we have a hypercube (4D) already with a pattern size of 2, provided we stick to a binary pattern in each dimension.
hypercubes at 2 and 4 alphabets
2 character alphabet, pattern size 4
4 character alphabet, pattern size 2
Three Symbols Alphabet suggests fractal representation
3 fractal
enlarge fill in
outer pattern repeats inner pattern= self similar= fractal
3 character alphapet3 pattern fractal
3 character alphapet4 pattern fractal Conjecture:
For n -> infinity, the fractal midght fill a 2D triangle
Note: check Mandelbrot
Same for 4 character alphabet
1 position
2 positions
3 positions
4 character alphabet continued
(with cheating I didn’t actually add beads)
4 positions
4 character alphabet continued
(with cheating I didn’t actually add beads)
5 positions
4 character alphabet continued
(with cheating I didn’t actually add beads)
6 positions
4 character alphabet continued
(with cheating I didn’t actually add beads)
7 positions
Animated GIf 1-12 positions
Protein Space in JalView
Alignment of V F A ATPase ATP binding SU(catalytic and non-catalytic SU)
UPGMA tree of V F A ATPase ATP binding SU with line dropped to partition (and colour) the 4 SU types (VA cat and non cat, F cat and non cat). Note that details of the tree $%#&@.
PCA analysis of V F A ATPase ATP binding SU using colours from the UPGMA tree
Same PCA analysis of V F A ATPase ATP binding SU using colours from the UPGMA tree, but turned slightly. (Giardia A SU selected in grey.)
Same PCA analysis of V F A ATPase ATP binding SU Using colours from the UPGMA tree, but replacing the 1st with the 5th axis. (Eukaryotic A SU selected in grey.)
Same PCA analysis of V F A ATPase ATP binding SU Using colours from the UPGMA tree, but replacing the 1st with the 6th axis. (Eukaryotic B SU selected in grey - forgot rice.)
Problems• Jalview’s approach requires an alignment.• Solution: Use pattern absence / presence as coordinate• Which patterns?
– GBLOCKS (new additions use PSSMs)– CDD PSSM profiles– It would be nice to stick to small words.
• One could screen for words/motifs/PSSMs that have a good power of resolution:– PCA with all, choose only the ones that contribute to the main axis– probably better to do data bank search and find how often it is
present. One could generate random motifs (or all possible motifs) and check them out (Criterion needs work).
– Empirical orthogonality– Exhaustive vs random – How to judge discriminatory power (maybe 5% significance value)– Present absence - optimal discriminatory power?