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Parikh vectors
5OTR_1 6OGL_1 1LXF_1 Letter Amino acid
9 7 4 Q Glutamine
14 13 6 G Glycine
21 4 6 K Lycine
5 2 0 W Tryptophan
25 1 1 Y Tyrosine
2 2 2 C Cysteine
17 1 0 H Histidine
19 15 4 I Isoleucine
10 2 7 M Methionine
15 0 3 S Serine
28 6 7 V Valine
25 5 2 R Arginine
15 4 2 N Asparagine
26 4 11 D Aspartic acid
24 4 12 E Glutamic acid
14 3 5 F Phenylalanine
15 7 2 P Proline
16 3 5 A Alanine
12 8 4 T Threonine
30 8 6 L Leucine 

5OTR_1|Chain A|Casein kinase II subunit alpha|Homo sapiens (9606)
>6OGL_1|Chain A|Protease|Human immunodeficiency virus 1 (11676)
>1LXF_1|Chain A[auth C]|TROPONIN C, SLOW SKELETAL AND CARDIAC MUSCLES|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 \(w=f(c)\)
5OTR , Knot 148 342 0.84 40 194 316 
SGPVPSRARVYTDVNTHRPSEYWDYESHVVEWGNQDDYQLVRKLGRGKYSEVFEAINITNNEKVVVKILKPVAAAKIKREIKILENLRGGPNIITLADIVKDPVSRTPALVFEHVNNTDFKQLYQTLTDYDIRFYMYEILKALDYCHSMGIMHRDVKPHNVMIDHEHRKLRLIDWGLAEFYHPGQEYNVRVASRYFKGPELLVDYQMYDYSLDMWSLGCMLASMIFRKEPFFHGHDNYDQLVRIAKVLGTEDLYDYIDKYNIELDPRFNDILGRHSRKRWERFVHSENQHLVSPEALDFLDKLLRYDHQSRLTAREAMEHPYFYTVVKREAMEHPYFYTVVK
6OGL , Knot 52 99 0.80 38 79 96 
PQITLWQRPIVTIKVGGQLREALIDTGADDTIFEEINLPGRWKPKLIGGIGGFMKVRQYDQIPIEICGHQAIGTVLVGPTPINVIGRNMLTQIGCTLNF
1LXF , Knot 47 89 0.79 36 71 85 
MDDIYKAAVEQLTEEQKNEFKAAFDIFVLGAEDGCISTKELGKVMRMLGQNPTPEELQEMIDEVDEDGSGTVDFDEFLVMMVRCMKDDS

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 \(c\).

\(|P_{f(5OTR_1)}(2) \setminus P_{f(6OGL_1)}(2)|=149\), \(|P_{f(6OGL_1)}(2) \setminus P_{f(5OTR_1)}(2)|=34\). 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 1:011110010100010000100010000011011000000110011010000110110100000111011011111010001011001011101101101100110001111100100001001000100001010100110110000011110001010011100000010110111101001100001011000101101110001000010110110111011100011101000000110110111000100010000101010100111000000100110000001101011011001100000001010011001010011000110010100110
Pair \(Z_2\) Length of longest common subsequence
5OTR_1,6OGL_1 183 3
5OTR_1,1LXF_1 183 3
6OGL_1,1LXF_1 112 3

Newick tree

 
[
	5OTR_1:10.58,
	[
		6OGL_1:56,1LXF_1:56
	]:44.58
]

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 \((0.85)(0.8)(\frac{441 }{\log_{20} 441}-\frac{99}{\log_{20}99})=103.\)
Status Protein1 Protein2 d d1/2
Query variables 5OTR_1 6OGL_1 131 84
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]