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Parikh vectors
7OIB_1 8GDW_1 5KHZ_1 Letter Amino acid
21 6 9 V Valine
5 1 0 C Cysteine
9 7 8 E Glutamic acid
29 5 11 G Glycine
5 3 5 F Phenylalanine
5 0 3 W Tryptophan
5 2 6 Y Tyrosine
31 13 16 A Alanine
7 8 10 D Aspartic acid
11 9 5 Q Glutamine
26 6 16 L Leucine
19 7 6 S Serine
15 3 12 N Asparagine
16 7 10 I Isoleucine
10 6 5 M Methionine
21 4 3 P Proline
31 3 13 R Arginine
7 3 7 H Histidine
17 5 13 K Lycine
15 5 12 T Threonine 

7OIB_1|Chain A[auth D]|39S ribosomal protein L2, mitochondrial|Homo sapiens (9606)
>8GDW_1|Chains A[auth AAA], B[auth BBB], C[auth CCC], D[auth DDD]|Ssr1698 protein|Synechocystis sp. PCC 6803 (1148)
>5KHZ_1|Chain A|Endolysin|Enterobacteria phage T4 (10665)
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 \(w=f(c)\)
7OIB , Knot 134 305 0.83 40 186 297 
MALCALTRALRSLNLAPPTVAAPAPSLFPAAQMMNNGLLQQPSALMLLPCRPVLTSVALNANFVSWKSRTKYTITPVKMRKSGGRDHTGRIRVHGIGGGHKQRYRMIDFLRFRPEETKSGPFEEKVIQVRYDPCRSADIALVAGGSRKRWIIATENMQAGDTILNSNHIGRMAVAAREGDAHPLGALPVGTLINNVESEPGRGAQYIRAAGTCGVLLRKVNGTAIIQLPSKRQMQVLETCVATVGRVSNVDHNKRVIGKAGRNRWLGKRPNSGRWHRKGGWAGRKIRPLPPMKSYVKLPSASAQS
8GDW , Knot 56 103 0.84 38 87 101 
MADPLTPAISDRICKHMNEDHASAIALYAQVFGQQTDVTMAQMQAIDPTGMDLVVESEGGSKTIRIEFEQPLKDSEDAHQVLIAMAKQARSVGKNSAENLYFQ
5KHZ , Knot 78 170 0.78 38 120 160 
MNIFEMLRIDEGLRLKIYKDTEGYYTIGIGHLLTKSPSLNAAKSELDKAIGRNTNGVITKDEAEKLFNQDVDAAVRGILRNAKLKPVYDSLDAVRRAALINMVFQMGETGVAGFTNSLRMLQQKRWDEAAVNLAKSRWYNQTPNRAKRVITTFRTGTWDAYKNLHHHHHH

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 \(c\).

\(|P_{f(7OIB_1)}(2) \setminus P_{f(8GDW_1)}(2)|=137\), \(|P_{f(8GDW_1)}(2) \setminus P_{f(7OIB_1)}(2)|=38\). 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 1:11101100110010111101111110111110110011100101111110011100111010110100000001011010001100001010101111100000011011010100000111000110100010001011111110000111100010110011000011011111001010111111110110010001101100101110011110010101110110000101100011011010010000011101100011100100101000111110010111110001011010100
Pair \(Z_2\) Length of longest common subsequence
7OIB_1,8GDW_1 175 4
7OIB_1,5KHZ_1 162 4
8GDW_1,5KHZ_1 151 2

Newick tree

 
[
	7OIB_1:87.05,
	[
		5KHZ_1:75.5,8GDW_1:75.5
	]:11.55
]

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 \((0.85)(0.8)(\frac{408 }{\log_{20} 408}-\frac{103}{\log_{20}103})=92.9\)
Status Protein1 Protein2 d d1/2
Query variables 7OIB_1 8GDW_1 117 77.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]