ways to construct protein space construction of sequence space from (eigen et al. 1988) illustrating...

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