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Parikh vectors
8OPI_1 9IXZ_1 4NEE_1 Letter Amino acid
8 22 6 Q Glutamine
2 35 0 P Proline
16 53 2 S Serine
1 11 1 W Tryptophan
8 43 15 V Valine
17 47 8 A Alanine
6 26 8 D Aspartic acid
14 41 3 G Glycine
10 23 8 Y Tyrosine
4 12 4 M Methionine
6 31 11 F Phenylalanine
3 13 10 N Asparagine
2 10 3 C Cysteine
7 33 12 E Glutamic acid
0 10 4 H Histidine
2 31 9 I Isoleucine
6 58 15 L Leucine
8 25 6 T Threonine
6 48 8 R Arginine
3 39 9 K Lycine 

8OPI_1|Chain A|VHH Z70 mutant 1|Lama glama (9844)
>9IXZ_1|Chains A, B, C, D|Isoform 3 of Potassium voltage-gated channel subfamily KQT member 2|Homo sapiens (9606)
>4NEE_1|Chains A[auth D], B[auth F], C[auth I], D[auth L]|AP-2 complex subunit sigma|Rattus norvegicus (10116)
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 \(w=f(c)\)
8OPI , Knot 61 129 0.76 38 96 122 
GAMAEVQLQASGGVFVQSGGSLRLSCAASGATSTFDGMGWFRQAPGKEREFVSAISYEQGSYTYYADSVKGRFTISRDNSKNMVYLQMNSLRAEDTATYYCAPAYEGDLYAFDSYGEQGTQVTVSSAAA
9IXZ , Knot 245 611 0.85 40 278 578 
AGKPPKRNAFYRKLQNFLYNVLERPRGWAFIYHAYVFLLVFSCLVLSVFSTIKEYEKSSEGALYILEIVTIVVFGVEYFVRIWAAGCCCRYRGWRGRLKFARKPFCVIDIMVLIASIAVLAAGSQGNVFATSALRSLRFLQILRMIRMDRRGGTWKLLGSVVYAHSKELVTAWYIGFLCLILASFLVYLAEKGENDHFDTYADALWWGLITLTTIGYGDKYPQTWNGRLLAATFTLIGVSFFALPAGILGSGFALKVQEQHRQKHFEKRRNPAAGLIQSAWRFYATNLSRTDLHSTWQYYERTVTVPMYRLIPPLNQLELLRNLKSKSGLAFRKDPPPEPSPSQKVSLKDRVFSSPRGVAAKGKGSPQAQTVRRSPSADQSLEDSPSKVPKSWSFGDRSRARQAFRIKGAASRQNSEEASLPGEDIVDDKSCPCEFVTEDLTPGLKVSIRAVCVMRFLVSKRKFKESLRPYDVMDVIEQYSAGHLDMLSRIKSLQSRVDQIVGRGPAITDKDRTKGPAEAELPEDPSMMGRLGKVEKQVLSMEKKLDFLVNIYMQRMGIPPTETEAYFGAKEPEPAPPYHSPEDSREHVDRHGCIVKIVRSSSSTGQKNFS
4NEE , Knot 73 142 0.85 38 117 138 
MIRFILIQNRAGKTRLAKWYMQFDDDEKQKLIEEVHAVVTVRDAKHTNFVEFRNFKIIYRRYAGLYFCICVDVNDNNLAYLEAIHNFVEVLNEYFHNVCELDLVFNFYKVYTVVDEMFLAGEIRETSQTKVLKQLLMLQSLE

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 \(c\).

\(|P_{f(8OPI_1)}(2) \setminus P_{f(9IXZ_1)}(2)|=17\), \(|P_{f(9IXZ_1)}(2) \setminus P_{f(8OPI_1)}(2)|=199\). 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 1:111101010101111100110101001101100010111110011100001101100001000001001010101000000011010100101000100001110010101100010010010100111
Pair \(Z_2\) Length of longest common subsequence
8OPI_1,9IXZ_1 216 3
8OPI_1,4NEE_1 155 3
9IXZ_1,4NEE_1 209 4

Newick tree

 
[
	9IXZ_1:11.25,
	[
		8OPI_1:77.5,4NEE_1:77.5
	]:36.75
]

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 \((0.85)(0.8)(\frac{740 }{\log_{20} 740}-\frac{129}{\log_{20}129})=174.\)
Status Protein1 Protein2 d d1/2
Query variables 8OPI_1 9IXZ_1 224 133
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]