Explorations of Multidimensional 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 position For 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 alphapet 3 pattern fractal

3 character alphapet 4 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?