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): 5913-7 Casari, G., C. Sander and A. Valencia (1995). "A method to predict functional residues in proteins." Nat Struct Biol 2(2): 171-8 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 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 (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 alphabet3 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 - only

homologous sequences can be depicted in the same space

• Solution: One could use pattern absence / presence as coordinates