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Parikh vectors
3AJG_1 2OLH_1 7LII_1 Letter Amino acid
14 26 12 S Serine
3 8 5 W Tryptophan
6 5 3 C Cysteine
30 34 31 L Leucine
8 16 5 Y Tyrosine
18 32 11 A Alanine
6 4 1 M Methionine
10 24 17 D Aspartic acid
10 21 15 F Phenylalanine
17 15 9 P Proline
8 24 13 T Threonine
14 17 19 V Valine
12 25 18 R Arginine
5 15 7 N Asparagine
14 27 18 G Glycine
6 9 1 H Histidine
12 17 17 I Isoleucine
9 20 18 K Lycine
24 11 5 Q Glutamine
22 11 33 E Glutamic acid 

3AJG_1|Chains A, B|Phycocyanobilin:ferredoxin oxidoreductase|Synechocystis (1148)
>2OLH_1|Chain A|Chitinase-3-like protein 1|Capra hircus (9925)
>7LII_1|Chain A|Maritimacin|Thermotoga maritima (strain ATCC 43589 / MSB8 / DSM 3109 / JCM 10099) (243274)
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 \(w=f(c)\)
3AJG , Knot 115 248 0.85 40 175 238 
MAVTDLSLTNSSLMPTLNPMIQQLALAIAASWQSLPLKPYQLPEDLGYVEGRLEGEKLVIENRCYQTPQFRKMHLELAKVGKGLDILHCVMFPEPLYGLPLFGCDIVAGPGGVSAAIADLSPTQSDRQLPAAYQKSLAELGQPEFEQQRELPPWGEIFSEYCLFIRPSNVTEEERFVQRVVDFLQIHCHQSIVAEPLSEAQTLEHRQGQIHYCQQQQKNDKTRRDLEKAFGEAWAERYMSQVLFDVIQ
2OLH , Knot 155 361 0.84 40 217 342 
YKLICYYTSWSQYREGDGSCFPDAIDPFLCTHVIYSFANISNNEIDTWEWNDVTLYDTLNTLKNRNPKLKTLLSVGGWNFGPERFSKIASKTQSRRTFIKSVPPFLRTHGFDGLDLAWLYPGRRDKRHLTALVKEMKAEFAREAQAGTERLLLSAAVSAGKIAIDRGYDIAQISRHLDFISLLTYDFHGAWRQTVGHHSPLFRGNSDASSRFSNADYAVSYMLRLGAPANKLVMGIPTFGRSFTLASSKTDVGAPISGPGIPGRFTKEKGILAYYEICDFLHGATTHRFRDQQVPYATKGNQWVAYDDQESVKNKARYLKNRQLAGAMVWALDLDDFRGTFCGQNLTFPLTSAVKDVLARV
7LII , Knot 118 258 0.84 40 158 248 
MEFLKRSFAPLTEKQWQEIDNRAREIFKTQLYGRKFVDVEGPYGWEYAAHPLGEVEVLSDENEVVKWGLRKSLPLIELRATFTLDLWELDNLERGKPNVDLSSLEETVRKVAEFEDEVIFRGCEKSGVKGLLSFEERKIECGSTPKDLLEAIVRALSIFSKDGIEGPYTLVINTDRWINFLDDEKRVEECLRGGKIITTPRIEDALVVSERGGDFKLILGQDLSIGYEDREKDAVRLFITETFTFQVVNPEALILLKF

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 \(c\).

\(|P_{f(3AJG_1)}(2) \setminus P_{f(2OLH_1)}(2)|=69\), \(|P_{f(2OLH_1)}(2) \setminus P_{f(3AJG_1)}(2)|=111\). 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 1:11100101000011101011100111111101001110100110011010101010011100000001010010101101101101100111101101111110011111111011110101000000111100001101101010000011111011000011101001000001100110110100000111011001001000010100000000000000010011101110001001110110
Pair \(Z_2\) Length of longest common subsequence
3AJG_1,2OLH_1 180 3
3AJG_1,7LII_1 155 4
2OLH_1,7LII_1 175 3

Newick tree

 
[
	2OLH_1:92.20,
	[
		3AJG_1:77.5,7LII_1:77.5
	]:14.70
]

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 \((0.85)(0.8)(\frac{609 }{\log_{20} 609}-\frac{248}{\log_{20}248})=101.\)
Status Protein1 Protein2 d d1/2
Query variables 3AJG_1 2OLH_1 133 110.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]