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Parikh vectors
8UTY_1 4OWF_1 7QSB_1 Letter Amino acid
32 12 13 L Leucine
10 2 0 M Methionine
23 0 9 S Serine
34 6 10 V Valine
36 9 7 A Alanine
27 3 4 I Isoleucine
20 1 6 P Proline
4 0 0 W Tryptophan
16 1 1 N Asparagine
36 2 11 G Glycine
13 2 3 H Histidine
19 11 1 K Lycine
12 0 1 C Cysteine
16 13 3 Q Glutamine
37 15 5 E Glutamic acid
20 2 2 F Phenylalanine
29 1 1 T Threonine
19 2 0 Y Tyrosine
21 4 10 R Arginine
27 4 5 D Aspartic acid 

8UTY_1|Chains A, C[auth E], F[auth S]|Tubulin alpha-1B chain|Sus scrofa (9823)
>4OWF_1|Chains A, B|NF-kappa-B essential modulator|Mus musculus (10090)
>7QSB_1|Chains A, B|Protein scribble homolog|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 \(w=f(c)\)
8UTY , Knot 192 451 0.86 40 258 426 
MRECISIHVGQAGVQIGNACWELYCLEHGIQPDGQMPSDKTIGGGDDSFNTFFSETGAGKHVPRAVFVDLEPTVIDEVRTGTYRQLFHPEQLITGKEDAANNYARGHYTIGKEIIDLVLDRIRKLADQCTGLQGFLVFHSFGGGTGSGFTSLLMERLSVDYGKKSKLEFSIYPAPQVSTAVVEPYNSILTTHTTLEHSDCAFMVDNEAIYDICRRNLDIERPTYTNLNRLISQIVSSITASLRFDGALNVDLTEFQTNLVPYPRIHFPLATYAPVISAEKAYHEQLSVAEITNACFEPANQMVKCDPRHGKYMACCLLYRGDVVPKDVNAAIATIKTKRSIQFVDWCPTGFKVGINYQPPTVVPGGDLAKVQRAVCMLSNTTAIAEAWARLDHKFDLMYAKRAFVHWYVGEGMEEGEFSEAREDMAALEKDYEEVGVDSVEGEGEEEGEEY
4OWF , Knot 43 90 0.71 34 63 82 
GMQLEDLRQQLQQAEEALVAKQELIDKLKEEAEQHKIVMETVPVLKAQADIYKADFQAERHAREKLVEKKEYLQEQLEQLQREFNKLKVG
7QSB , Knot 50 92 0.82 34 76 89 
SVEEIRLPRAGGPLGLSIVGGSDHSSHPFGVQEPGVFISKVLPRGLAARSGLRVGDRILAVNGQDVRDATHQEAVSALLRPCLELSLLVRRD

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 \(c\).

\(|P_{f(8UTY_1)}(2) \setminus P_{f(4OWF_1)}(2)|=209\), \(|P_{f(4OWF_1)}(2) \setminus P_{f(8UTY_1)}(2)|=14\). 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 1:1000101011011101101010100100110101011000011110001001100011100110111101010110010010000110100110100011000101000110011011100100110000110111110011110101100111001010010000101010111010011101000110000010000011110001100100001010010000100110011001010101011101010010001110101011110011110100100001011010010101100110001001001100110010111001011110100000101101010110111000110111110110100110110000111011101000101101001110101101100101001000111100000011100101010001000
Pair \(Z_2\) Length of longest common subsequence
8UTY_1,4OWF_1 223 3
8UTY_1,7QSB_1 208 3
4OWF_1,7QSB_1 111 2

Newick tree

 
[
	8UTY_1:12.29,
	[
		7QSB_1:55.5,4OWF_1:55.5
	]:64.79
]

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 \((0.85)(0.8)(\frac{541 }{\log_{20} 541}-\frac{90}{\log_{20}90})=134.\)
Status Protein1 Protein2 d d1/2
Query variables 8UTY_1 4OWF_1 176 104
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]