Ways to construct Protein Space Construction of sequence space from (Eigen et al. 1988) illustrating the construction of a high dimensional sequence space. Each additional sequence position adds another dimension, doubling the diagram for the shorter sequence. Shown is the progression from a single sequence position (line) to a tetramer (hypercube). A four (or twenty) letter code can be accommodated either through allowing four (or twenty) values for each dimension (Rechenberg 1973; Casari et al. 1995), or through additional dimensions (Eigen and Winkler-Oswatitsch 1992). Eigen, M. and R. Winkler-Oswatitsch (1992). Steps Towards Life: A Perspective on Evolution. Oxford; New York, Oxford University Press. Eigen, M., R. Winkler-Oswatitsch and A. Dress (1988). "Statistical geometry in sequence space: a method of quantitative comparative sequence analysis." Proc Natl Acad Sci U S A 85(16): Casari, G., C. Sander and A. Valencia (1995). "A method to predict functional residues in proteins." Nat Struct Biol 2(2): Rechenberg, I. (1973). Evolutionsstrategie; Optimierung technischer Systeme nach Prinzipien der biologischen Evolution. Stuttgart-Bad Cannstatt, Frommann-Holzboog.

Diversion: From Multidimensional Sequence Space to Fractals

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 (the other fork)

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 alphabet 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 - only homologous sequences can be depicted in the same space Solution: One could use pattern absence / presence as coordinates