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Parikh vectors
2QUN_1 1WKG_1 5VIX_1 Letter Amino acid
7 1 0 C Cysteine
18 18 14 K Lycine
10 22 14 T Threonine
20 34 8 V Valine
25 10 10 D Aspartic acid
12 26 4 P Proline
15 14 4 F Phenylalanine
6 5 0 W Tryptophan
7 9 6 N Asparagine
12 13 3 M Methionine
4 7 12 Q Glutamine
21 42 12 E Glutamic acid
22 38 12 G Glycine
5 6 8 H Histidine
13 16 14 I Isoleucine
24 41 18 L Leucine
28 45 3 A Alanine
19 33 8 R Arginine
14 7 8 S Serine
8 8 2 Y Tyrosine 

2QUN_1|Chains A, B, C, D|D-tagatose 3-epimerase|Pseudomonas cichorii (36746)
>1WKG_1|Chains A, B|Acetylornithine/acetyl-lysine aminotransferase|Thermus thermophilus (274)
>5VIX_1|Chain A|Polyubiquitin|Naegleria fowleri (5763)
Protein code \(c\) LZ-complexity \(\mathrm{LZ}(w)\) Length \(n=|w|\) \(\frac{\mathrm{LZ}(w)}{n /\log_{20} n}\) \(p_w(1)\) \(p_w(2)\) \(p_w(3)\) Sequence \(w=f(c)\)
2QUN , Knot 130 290 0.84 40 194 279 
MNKVGMFYTYWSTEWMVDFPATAKRIAGLGFDLMEISLGEFHNLSDAKKRELKAVADDLGLTVMCCIGLKSEYDFASPDKSVRDAGTEYVKRLLDDCHLLGAPVFAGLTFCAWPQSPPLDMKDKRPYVDRAIESVRRVIKVAEDYGIIYALEVVNRFEQWLCNDAKEAIAFADAVDSPACKVQLDTFHMNIEETSFRDAILACKGKMGHFHLGEANRLPPGEGRLPWDEIFGALKEIGYDGTIVMEPFMRKGGSVSRAVGVWRDMSNGATDEEMDERARRSLQFVRDKLA
1WKG , Knot 162 395 0.81 40 191 365 
METRTLEDWRALLEAEKTLDSGVYNKHDLLIVRGQGARVWDAEGNEYIDCVGGYGVANLGHGNPEVVEAVKRQAETLMAMPQTLPTPMRGEFYRTLTAILPPELNRVFPVNSGTEANEAALKFARAHTGRKKFVAAMRGFSGRTMGSLSVTWEPKYREPFLPLVEPVEFIPYNDVEALKRAVDEETAAVILEPVQGEGGVRPATPEFLRAAREITQEKGALLILDEIQTGMGRTGKRFAFEHFGIVPDILTLAKALGGGVPLGVAVMREEVARSMPKGGHGTTFGGNPLAMAAGVAAIRYLERTRLWERAAELGPWFMEKLRAIPSPKIREVRGMGLMVGLELKEKAAPYIARLEKEHRVLALQAGPTVIRFLPPLVIEKEDLERVVEAVRAVLA
5VIX , Knot 45 160 0.47 36 66 81 
MAHHHHHHMQIFVKTLTGKTITLEVESNDTIENVKSKIQDKEGIPPDQQRLIFAGKQLEDGRTLSDYNIQKESTLHLVLRLRGGMQIFVKTLTGKTITLEVESNDTIENVKSKIQDKEGIPPDQQRLIFAGKQLEDGRTLSDYNIQKESTLHLVLRLRGG

Let \(P_w(n)\) be the set of distinct subwords (intervals) in a word \(w\). Let \(p_w(n)\) be the cardinality of \(P_w(n)\). Let \(f(c)\) be the sequence in FASTA with 4-symbol Protein Data Bank code \(c\).

\(|P_{f(2QUN_1)}(2) \setminus P_{f(1WKG_1)}(2)|=76\), \(|P_{f(1WKG_1)}(2) \setminus P_{f(2QUN_1)}(2)|=73\). Let \( Z_k(x,y)=|P_x(k)\setminus P_y(k)|+|P_y(k)\setminus P_x(k)| \) be a LZ76 style (set of subwords) Jaccard distance numerator for \(P(k)\).Hydrophobic-polar version of Sequence 1:10011110001000111011101001111110110101101001001000010111001110110011100000110100010011000100110000111111111101011100111010000101001100100110110001110110110010011000100111110110011001010010101000010011110010110101101001111010111001111100110010111011100110100111110010011000010001000101100011
Pair \(Z_2\) Length of longest common subsequence
2QUN_1,1WKG_1 149 3
2QUN_1,5VIX_1 188 3
1WKG_1,5VIX_1 181 3

Newick tree

 
[
	5VIX_1:97.47,
	[
		2QUN_1:74.5,1WKG_1:74.5
	]:22.97
]

Let d be the Otu--Sayood distance d.
Let d1 be the Otu--Sayood distance d1. (This makes the 4TYN sequence AAAAAA a close match...)
A roughly speaking expected distance is \((0.85)(0.8)(\frac{685 }{\log_{20} 685}-\frac{290}{\log_{20}290})=109.\)
Status Protein1 Protein2 d d1/2
Query variables 2QUN_1 1WKG_1 137 120
Was not able to put for d
Was not able to put for d1

In notation analogous to [Theorem 16, Kjos-Hanssen, Niraula and Yoon (2022)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]