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Parikh vectors
5XAO_1 5VYV_1 4TTB_1 Letter Amino acid
28 8 20 K Lycine
9 6 6 M Methionine
24 7 16 P Proline
37 12 11 G Glycine
18 3 16 I Isoleucine
19 12 14 T Threonine
10 6 7 Y Tyrosine
17 9 11 N Asparagine
30 10 13 D Aspartic acid
11 12 7 Q Glutamine
19 6 6 F Phenylalanine
11 0 5 W Tryptophan
48 18 16 A Alanine
20 3 16 H Histidine
38 28 23 L Leucine
27 12 12 S Serine
29 19 22 V Valine
21 17 16 R Arginine
7 3 2 C Cysteine
22 24 26 E Glutamic acid 

5XAO_1|Chains A, B[auth C]|Uncharacterized protein|Phaeosphaeria nodorum (strain SN15 / ATCC MYA-4574 / FGSC 10173) (321614)
>5VYV_1|Chains A, B|UPF0502 protein YceH|Escherichia coli O157:H7 (83334)
>4TTB_1|Chains A, B|Iodotyrosine dehalogenase 1|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 \(w=f(c)\)
5XAO , Knot 183 445 0.83 40 243 420 
MAPSRANTSVIVVGGGGTIGSSTALHLVRSGYTPSNVTVLDAYPIPSSQSAGNDLAKIMGVSLRNPVDLQLALEARQMWNEDELFKKFFHNTGRLDCAHGEKDIADLKSGYQALVDAGLDATNEWLDSEDEILKRMPLLSRDQIKGWKAIFSKDGGWLAAAKAINAVGEYLRDQGVRFGFYGAGSFKAPLLAEGVCIGVETVDGTRYYADKVVLAAGAWSPTLVELHEQCVSKAWVYGHIQLTPEEAARYKNSPVVYNGDVGFFFEPNEHGVIKVCDEFPGFTRFKMHQPFGAKAPKRISVPRSHAKHPTDTIPDASDVSIRRAIATFMPQFKNKKMFNQAMCWCTDTADAALLICEHPEWKNFVLATGDSGHSFKLLPNIGKHVVELLEGTLADDLAHAWRWRPGSGDALKSRRSAPAKDLADMPGWNHDKPRANLLEHHHHHH
5VYV , Knot 97 215 0.80 38 134 204 
MKYQLTALEARVIGCLLEKQVTTPEQYPLSVNGVVTACNQKTNREPVMNLSESEVQEQLDNLVKRHYLRTVSGFGNRVTKYEQRFCNSEFGDLKLSAAEVALITTLLLRGAQTPGELRSRAARMYEFSDMAEVELTLEQLANREDGPFVVRLAREPGKRESRYMHLFSGEVEDQPAVTDMSNAVDGDLQARVEALEIEVAELKQRLDSLLAHLGD
4TTB , Knot 121 265 0.85 40 181 253 
SGEPRTRAEARPWVDEDLKDSSDLHQAEEDADEWQESEENVEHIPFSHNHYPEKEMVKRSQEFYELLNKRRSVRFISNEQVPMEVIDNVIRTAGTAPSGAHTEPWTFVVVKDPDVKHKIRKIIEEEEEINYMKRMGHRWVTDLKKLRTNWIKEYLDTAPILILIFKQVHGFAANGKKKVHYYNEISVSIACGILLAALQNAGLVTVTTTPLNCGPRLRVLLGRPAHEKLLMLLPVGYPSKEATVPDLKRKPLDQIMVTVHHHHHH

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 \(c\).

\(|P_{f(5XAO_1)}(2) \setminus P_{f(5VYV_1)}(2)|=151\), \(|P_{f(5VYV_1)}(2) \setminus P_{f(5XAO_1)}(2)|=42\). 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 1:1110010001111111101100011011001001001011010111000011001101111010011010111010011000011001100010100101000110100100111011101000110000011001111000010110111000111111101101110010001101110111010111110110111001010000100111111110101101000010011101010101001100000111001011111010001110100011110010100111101100101100010010001101001010011101110100001100110100001011111000101001111010010010111011001101101011001101101011010110000011100110111100001010110000000
Pair \(Z_2\) Length of longest common subsequence
5XAO_1,5VYV_1 193 5
5XAO_1,4TTB_1 158 6
5VYV_1,4TTB_1 149 3

Newick tree

 
[
	5XAO_1:92.29,
	[
		4TTB_1:74.5,5VYV_1:74.5
	]:17.79
]

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 \((0.85)(0.8)(\frac{660 }{\log_{20} 660}-\frac{215}{\log_{20}215})=125.\)
Status Protein1 Protein2 d d1/2
Query variables 5XAO_1 5VYV_1 156 113
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]