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Parikh vectors
2ELO_1 9ETZ_1 7ZAR_1 Letter Amino acid
1 8 7 Y Tyrosine
2 9 2 H Histidine
1 51 13 L Leucine
0 5 5 M Methionine
0 7 3 W Tryptophan
0 18 9 Q Glutamine
3 25 8 E Glutamic acid
2 21 7 I Isoleucine
1 22 8 T Threonine
2 13 3 R Arginine
1 35 12 N Asparagine
1 22 6 D Aspartic acid
2 1 2 C Cysteine
2 30 5 V Valine
3 26 3 G Glycine
1 19 5 F Phenylalanine
0 35 5 A Alanine
3 28 12 K Lycine
2 12 12 P Proline
10 44 2 S Serine 

2ELO_1|Chain A|Zinc finger protein 406|Homo sapiens (9606)
>9ETZ_1|Chains A, K[auth L]|Cytochrome b-c1 complex subunit 1, mitochondrial|Saccharomyces cerevisiae (4932)
>7ZAR_1|Chain A[auth AAA]|Isoform C of Bromodomain-containing protein 4|Homo sapiens (9606)
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 \(w=f(c)\)
2ELO , Knot 21 37 0.68 32 31 33 
GSSGSSGRSYSCPVCEKSFSEDRLIKSHIKTNHPEVS
9ETZ , Knot 177 431 0.83 40 215 406 
AEVTQLSNGIVVATEHNPSAHTASVGVVFGSGAANENPYNNGVSNLWKNIFLSKENSAVAAKEGLALSSNISRDFQSYIVSSLPGSTDKSLDFLNQSFIQQKANLLSSSNFEATKKSVLKQVQDFEENDHPNRVLEHLHSTAFQNTPLSLPTRGTLESLENLVVADLESFANNHFLNSNAVVVGTGNIKHEDLVNSIESKNLSLQTGTKPVLKKKAAFLGSEVRLRDDTLPKAWISLAVEGEPVNSPNYFVAKLAAQIFGSYNAFEPASRLQGIKLLDNIQEYQLCDNFNHFSLSYKDSGLWGFSTATRNVTMIDDLIHFTLKQWNRLTISVTDTEVERAKSLLKLQLGQLYESGNPVNDANLLGAEVLIKGSKLSLGEAFKKIDAITVKDVKAWAGKRLWDQDIAIAGTGQIEGLLDYMRIRSDMSMMRW
7ZAR , Knot 68 129 0.85 40 112 125 
GSHMNPPPPETSNPNKPKRQTNQLQYLLRVVLKTLWKHQFAWPFQQPVDAVKLNLPDYYKIIKTPMDMGTIKKRLENNYYWNAQECIQDFNTMFTNCYIYNKPGDDIVLMAEALEKLFLQKINELPTEE

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 \(c\).

\(|P_{f(2ELO_1)}(2) \setminus P_{f(9ETZ_1)}(2)|=13\), \(|P_{f(9ETZ_1)}(2) \setminus P_{f(2ELO_1)}(2)|=197\). 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 1:1001001000001100001000011000100001010
Pair \(Z_2\) Length of longest common subsequence
2ELO_1,9ETZ_1 210 3
2ELO_1,7ZAR_1 125 3
9ETZ_1,7ZAR_1 181 3

Newick tree

 
[
	9ETZ_1:10.27,
	[
		2ELO_1:62.5,7ZAR_1:62.5
	]:44.77
]

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 \((0.85)(0.8)(\frac{468 }{\log_{20} 468}-\frac{37}{\log_{20}37})=134.\)
Status Protein1 Protein2 d d1/2
Query variables 2ELO_1 9ETZ_1 167 90
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]

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