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Parikh vectors
3WFU_1 8CAU_1 6ICS_1 Letter Amino acid
11 12 1 H Histidine
2 13 2 M Methionine
8 27 17 D Aspartic acid
6 18 3 Q Glutamine
13 24 17 E Glutamic acid
2 18 4 Y Tyrosine
5 36 33 S Serine
7 23 30 T Threonine
2 12 1 W Tryptophan
19 20 32 K Lycine
7 22 3 F Phenylalanine
4 32 1 P Proline
0 17 0 C Cysteine
15 29 21 G Glycine
9 26 10 I Isoleucine
17 52 22 L Leucine
7 43 23 V Valine
15 30 13 A Alanine
2 22 2 R Arginine
2 16 12 N Asparagine 

3WFU_1|Chain A|Myoglobin|Equus caballus (9796)
>8CAU_1|Chains A, B, C, D, E|Neuronal acetylcholine receptor subunit alpha-7|Homo sapiens (9606)
>6ICS_1|Chain A[auth O]|Outer surface protein A|Borrelia burgdorferi (139)
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 \(w=f(c)\)
3WFU , Knot 75 153 0.82 38 115 148 
GLSDGEWQQVLNVWGKVEADIAGHGQEVLIRLFTGHPETLEKFDKFKHLKTEAEMKASEDLKKHGTVVLTALGGILKKKGHHEAELKPLAQSHATKHKIPIKYLEFISDAIIHVLHSKHPGDFGADAQGAMTKALELFRNDIAAKYKELGFQG
8CAU , Knot 206 492 0.86 40 258 475 
EFQRKLYKELVKNYNPLERPVANDSQPLTVYFSLSLLQIMDVDEKNQVLTTNIWLQMSWTDHYLQWNVSEYPGVKTVRFPDGQIWKPDILLYNSADERFDATFHTNVLVNSSGHCQYLPPGIFKSSCYIDVRWFPFDVQHCKLKFGSWSYGGWSLDLQMQEADISGYIPNGEWDLVGIPGKRSERFYECCKEPYPDVTFTVTMRRRTLYYGLNLLIPCVLISALALLVFLLPADSGEKISLGITVLLSLTVFMLLVAEIMPATSDSVPLIAQYFASTMIIVGLSVVVTVIVLQYHHHDPDGGKMPKWTRVILLNWCAWFLRMKRPGEDKVRPACQHKQRRCSLASVEMSAVAPPPASNGNLLYIGFRGLDGVHCVPTPDSGVVCGRMACSPTHDEHLLHGGQPPEGDPDLAKILEEVRYIANRFRCQDESEAVCSEWKFAACVVDRLCLMAFSVFTIICTIGILMSAPNFVEAVSKDFASAGLTETSQVAPA
6ICS , Knot 98 247 0.72 38 117 218 
GSHMKNSVSVDLPGSMKVLVSKSSNADGKYDLIATVDALELSGTSDKNNGSGVLEGVKADASKVKLTISDDLGQTTLEVFKSDGSTLVSKKVTSKDKSSTEEKFNEKGEVSEKIITRADGTRLEYTGIKSDGSGKAKEVLKGYVLEGTLTAEKTTLVVKEGTVTLSKNISKSGAVSVELNDTDTKKTAAWNSGTSTLTITVNSKKTKDLVFTSSNTITVQQYDSNGTSLEGSAVEITKLDEIKNALK

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 \(c\).

\(|P_{f(3WFU_1)}(2) \setminus P_{f(8CAU_1)}(2)|=32\), \(|P_{f(8CAU_1)}(2) \setminus P_{f(3WFU_1)}(2)|=175\). 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 1:110010100110111010101110100111011010100100100100100010101000100010111011111100010001010111000100001110010110011101100001101110101110011011000111000011101
Pair \(Z_2\) Length of longest common subsequence
3WFU_1,8CAU_1 207 4
3WFU_1,6ICS_1 112 4
8CAU_1,6ICS_1 193 3

Newick tree

 
[
	8CAU_1:11.92,
	[
		3WFU_1:56,6ICS_1:56
	]:54.92
]

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 \((0.85)(0.8)(\frac{645 }{\log_{20} 645}-\frac{153}{\log_{20}153})=141.\)
Status Protein1 Protein2 d d1/2
Query variables 3WFU_1 8CAU_1 183 118
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]