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Parikh vectors
2WAO_1 8QSB_1 3LKB_1 Letter Amino acid
27 19 14 S Serine
8 2 3 W Tryptophan
21 12 32 V Valine
11 13 22 R Arginine
17 14 15 D Aspartic acid
15 10 14 Y Tyrosine
22 7 18 N Asparagine
26 24 30 L Leucine
11 3 14 H Histidine
16 17 20 K Lycine
17 5 19 F Phenylalanine
19 5 21 P Proline
7 7 16 Q Glutamine
19 29 28 E Glutamic acid
32 14 34 G Glycine
15 10 21 I Isoleucine
7 9 6 M Methionine
21 10 20 T Threonine
24 24 43 A Alanine
6 2 2 C Cysteine 

2WAO_1|Chain A|ENDOGLUCANASE E|CLOSTRIDIUM THERMOCELLUM (1515)
>8QSB_1|Chains A, C[auth J]|14-3-3 protein sigma|Homo sapiens (9606)
>3LKB_1|Chains A, B|Probable branched-chain amino acid ABC transporter, amino acid binding protein|Thermus thermophilus (300852)
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 \(w=f(c)\)
2WAO , Knot 152 341 0.86 40 216 329 
MASPDEDNPGILYNGRFDFSDPNGPKCAWSGSNVELNFYGTEASVTIKSGGENWFQAIVDGNPLPPFSVNATTSTVKLVSGLAEGAHHLVLWKRTEASLGEVQFLGFDFGSGKLLAAPKPLERKIEFIGDSITCAYGNEGTSKEQSFTPKNENSYMSYAAITARNLNASANMIAWSGIGLTMNYGGAPGPLIMDRYPYTLPYSGVRWDFSKYVPQVVVINLGTNDFSTSFADKTKFVTAYKNLISEVRRNYPDAHIFCCVGPMLWGTGLDLCRSYVTEVVNDCNRSGDLKVYFVEFPQQDGSTGYGEDWHPSIATHQLMAERLTAEIKNKLGWLEHHHHHH
8QSB , Knot 108 236 0.83 40 164 229 
GAMGSMERASLIQKAKLAEQAERYEDMAAFMKGAVEKGEELSCEERNLLSVAYKNVVGGQRAAWRVLSSIEQKSNEEGSEEKGPEVREYREKVETELQGVCDTVLGLLDSHLIKEAGDAESRVFYLKMKGDYYRYLAEVATGDDKKRIIDSARSAYQEAMDISKKEMPPTNPIRLGLALNFSVFHYEIANSPEEAISLAKTTFDEAMADLHTLSEDSYKDSTLIMQLLRDNLTLWT
3LKB , Knot 169 392 0.85 40 220 375 
MSLGQQQVTLFWSGAITGPTSDAGAPYGAAVEDYCKWANERKLVPGVVFNCVVRDDQYNNANTQRFFEEAVDRFKIPVFLSYATGANLQLKPLIQELRIPTIPASMHIELIDPPNNDYIFLPTTSYSEQVVALLEYIAREKKGAKVALVVHPSPFGRAPVEDARKAARELGLQIVDVQEVGSGNLDNTALLKRFEQAGVEYVVHQNVAGPVANILKDAKRLGLKMRHLGAHYTGGPDLIALAGDAAEGFLWATSFYMAHEDTPGIRLQKEIGRKYGRPENFIESVNYTNGMLAAAIAVEAIRRAQERFKRITNETVYQAIVGMNGPNAFKPGFAVSTKQGVEIDFTKSEHTGAEGLRILEAKGGRFVPVTEPFTSALFRKVHYGEGHHHHHH

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 \(c\).

\(|P_{f(2WAO_1)}(2) \setminus P_{f(8QSB_1)}(2)|=122\), \(|P_{f(8QSB_1)}(2) \setminus P_{f(2WAO_1)}(2)|=70\). 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 1:11010000111100101010010110011010010101010010101001100110111010111110101000010110111011001111000010110101111011010111110110001011100100101001000000101000000100111010010101011110111101001111111110001001100110101000110111101100010001100001101000110010000101011001111111011010000100110000001010101101100010010100101011000111001010100011110000000
Pair \(Z_2\) Length of longest common subsequence
2WAO_1,8QSB_1 192 4
2WAO_1,3LKB_1 150 6
8QSB_1,3LKB_1 166 4

Newick tree

 
[
	8QSB_1:94.13,
	[
		2WAO_1:75,3LKB_1:75
	]:19.13
]

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 \((0.85)(0.8)(\frac{577 }{\log_{20} 577}-\frac{236}{\log_{20}236})=96.8\)
Status Protein1 Protein2 d d1/2
Query variables 2WAO_1 8QSB_1 126 106
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]