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Parikh vectors
8ALW_1 8TWG_1 5QBQ_1 Letter Amino acid
24 28 29 A Alanine
14 25 21 D Aspartic acid
10 14 13 Y Tyrosine
12 21 23 V Valine
7 15 9 Q Glutamine
10 38 18 I Isoleucine
19 28 49 S Serine
10 13 47 T Threonine
13 23 1 R Arginine
7 21 6 N Asparagine
29 25 5 E Glutamic acid
17 23 11 K Lycine
9 11 0 M Methionine
5 9 13 P Proline
2 4 5 W Tryptophan
2 7 2 C Cysteine
14 30 38 G Glycine
3 15 1 H Histidine
24 31 19 L Leucine
5 21 20 F Phenylalanine 

8ALW_1|Chain A|14-3-3 protein sigma|Homo sapiens (9606)
>8TWG_1|Chain A[auth B]|6-hydroxynicotinate 3-monooxygenase|Legionella clemsonensis (1867846)
>5QBQ_1|Chain A|Endothiapepsin|Cryphonectria parasitica (5116)
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 \(w=f(c)\)
8ALW , Knot 108 236 0.83 40 164 229 
GAMGSMERASLIQKAKLAEQAERYEDMAAFMKGAVEKGEELSCEERNLLSVAYKNVVGGQRAAWRVLSSIEQKSNEEGSEEKGPEVREYREKVETELQGVCDTVLGLLDSHLIKEAGDAESRVFYLKMKGDYYRYLAEVATGDDKKRIIDSARSAYQEAMDISKKEMPPTNPIRLGLALNFSVFHYEIANSPEEAISLAKTTFDEAMADLHTLSEDSYKDSTLIMQLLRDNLTLWT
8TWG , Knot 173 402 0.86 40 232 383 
MGSSHHHHHHSSGKLMSKNIKILVVGAGIAGPAICYWLKRFGFSPVLIEKYASIRKGGQALDIRGIATHLARVMGIYDQICDRRTRIECGRFVDSAGKVLHEEHGEKFGFRQDDEVEILRGDLVEILMKTIADVPCYFNKSIISMQQNADKVIVKFKDGKIENYDIVIGADGIHSATRRMIFDKNEYQLIHLGSYLSTFTIPNYLGLSHTDLECEANHKLVSINSDNNPEIARAGFMFRSQHILKDIRDEHEQKQFLCDTFRDFGWETQNILNRMPESDDFYFDAITQVRMNSWTKGRIALIGDAGYCPSPLSGQGNNLAFVGAYILAGELKTANANYMQAFSRYNALLRSFVDTNQEFGVWVSKSFLVEDEVSKEIAEERSNRILAMIKLVSNAIILPQYE
5QBQ , Knot 132 330 0.77 38 160 297 
STGSATTTPIDSLDDAYITPVQIGTPAQTLNLDFDTGSSDLWVFSSETTASEVDGQTIYTPSKSTTAKLLSGATWSISYGDGSSSSGDVYTDTVSVGGLTVTGQAVESAKKVSSSFTEDSTIDGLLGLAFSTLNTVSPTQQKTFFDNAKASLDSPVFTADLGYHAPGTYNFGFIDTTAYTGSITYTAVSTKQGFWEWTSTGYAVGSGTFKSTSIDGIADTGTTLLYLPATVVSAYWAQVSGAKSSSSVGGYVFPCSATLPSFTFGVGSARIVIPGDYIDFGPISTGSSSCFGGIQSSAGIGINIFGDVALKAAFVVFNGATTPTLGFASK

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 \(c\).

\(|P_{f(8ALW_1)}(2) \setminus P_{f(8TWG_1)}(2)|=47\), \(|P_{f(8TWG_1)}(2) \setminus P_{f(8ALW_1)}(2)|=115\). 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 1:11110100101100101100100000111110111001001000000110110001111001110110010000000100001101000000100010110001111100011001101000110101010000011011010000011001001000110100001110011011111010110001100100110110001001110100100000000011101100010110
Pair \(Z_2\) Length of longest common subsequence
8ALW_1,8TWG_1 162 3
8ALW_1,5QBQ_1 158 4
8TWG_1,5QBQ_1 182 3

Newick tree

 
[
	8TWG_1:88.39,
	[
		8ALW_1:79,5QBQ_1:79
	]:9.39
]

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 \((0.85)(0.8)(\frac{638 }{\log_{20} 638}-\frac{236}{\log_{20}236})=113.\)
Status Protein1 Protein2 d d1/2
Query variables 8ALW_1 8TWG_1 147 115
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]