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Parikh vectors
9VXS_1 6BRF_1 1TZS_1 Letter Amino acid
7 16 23 Q Glutamine
12 37 13 E Glutamic acid
18 27 20 I Isoleucine
24 19 6 K Lycine
9 20 22 F Phenylalanine
12 34 29 V Valine
15 16 14 N Asparagine
21 27 19 D Aspartic acid
2 12 7 C Cysteine
2 13 5 H Histidine
5 10 9 M Methionine
9 18 13 Y Tyrosine
4 4 5 W Tryptophan
28 36 19 A Alanine
15 21 4 R Arginine
22 36 38 G Glycine
18 32 23 L Leucine
8 20 24 P Proline
19 23 36 S Serine
17 29 22 T Threonine 

9VXS_1|Chains A, B|Beta-lactamase SME-1|Serratia marcescens (615)
>6BRF_1|Chains A, C|Tubulin alpha-1B chain|Sus scrofa (9823)
>1TZS_1|Chain A|Cathepsin E|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 \(w=f(c)\)
9VXS , Knot 122 267 0.85 40 180 254 
NKSDAAAKQIKKLEEDFDGRIGVFAIDTGSGNTFGYRSDERFPLCSSFKGFLAAAVLERVQQKKLDINQKVKYESRDLEYHSPITTKYKGSGMTLGDMASAALQYSDNGATNIIMERFLGGPEGMTKFMRSIGDNEFRLDRWELELNTAIPGDKRDTSTPKAVANSLNKLALGNVLNAKVKAIYQNWLKGNTTGDARIRASVPADWVVGDKTGSCGAYGTANDYAVIWPKNRAPLIVSIYTTRKSKDDKHSDKTIAEASRIAIQAID
6BRF , Knot 192 450 0.87 40 257 425 
MRECISIHVGQAGVQIGNACWELYCLEHGIQPDGQMPSDKTIGGGDDSFNTFFSETGAGKHVPRAVFVDLEPTVIDEVRTGTYRQLFHPEQLITGKEDAANNYARGHYTIGKEIIDLVLDRIRKLADQCTGLQGFLVFHSFGGGTGSGFTSLLMERLSVDYGKKSKLEFSIYPAPQVSTAVVEPYNSILTTHTTLEHSDCAFMVDNEAIYDICRRNLDIERPTYTNLNRLISQIVSSITASLRFDGALNVDLTEFQTNLVPYPRIHFPLATYAPVISAEKAYHEQLSVAEITNACFEPANQMVKCDPRHGKYMACCLLYRGDVVPKDVNAAIATIKTKRSIQFVDWCPTGFKVGINYQPPTVVPGGDLAKVQRAVCMLSNTTAIAEAWARLDHKFDLMYAKRAFVHWYVGEGMEEGEFSEAREDMAALEKDYEEVGVDSVEGEGEEEGEE
1TZS , Knot 148 351 0.82 40 201 334 
IQFTESCSMDQSAKEPLINYLDMEYFGTISIGSPPQNFTVIFDTGSSNLWVPSVYCTSPACKTHSRFQPSQSSTYSQPGQSFSIQYGTGSLSGIIGADQVSVEGLTVVGQQFGESVTEPGQTFVDAEFDGILGLGYPSLAVGGVTPVFDNMMAQNLVDLPMFSVYMSSNPEGGAGSELIFGGYDHSHFSGSLNWVPVTKQAYWQIALDNIQVGGTVMFCSEGCQAIVDTGTSLITGPSDKIKQLQNAIGAAPVDGEYAVECANLNVMPDVTFTINGVPYTLSPTAYTLLDFVDGMQFCSSGFQGLDIHPPAGPLWILGDVFIRQFYSVFDRGNNRVGLAPAVPWSHPQFEK

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 \(c\).

\(|P_{f(9VXS_1)}(2) \setminus P_{f(6BRF_1)}(2)|=44\), \(|P_{f(6BRF_1)}(2) \setminus P_{f(9VXS_1)}(2)|=121\). 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 1:000011100100100010101111110010100110000001110001011111111001000010100010000001000011000001011011011011100000110011100111110110011001100010100101010011110000000101110010011110110101011000110100010101010111011110001001101010001111100011111010000000000000001101001110110
Pair \(Z_2\) Length of longest common subsequence
9VXS_1,6BRF_1 165 5
9VXS_1,1TZS_1 167 4
6BRF_1,1TZS_1 166 4

Newick tree

 
[
	1TZS_1:83.49,
	[
		9VXS_1:82.5,6BRF_1:82.5
	]:0.99
]

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 \((0.85)(0.8)(\frac{717 }{\log_{20} 717}-\frac{267}{\log_{20}267})=124.\)
Status Protein1 Protein2 d d1/2
Query variables 9VXS_1 6BRF_1 158 122
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]