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Parikh vectors
8SHA_1 2JJZ_1 1HBP_1 Letter Amino acid
27 7 17 D Aspartic acid
11 6 6 I Isoleucine
24 15 11 K Lycine
17 7 12 F Phenylalanine
18 9 13 R Arginine
14 7 9 N Asparagine
11 1 3 H Histidine
11 11 4 M Methionine
40 11 11 S Serine
29 7 12 V Valine
33 7 14 A Alanine
28 11 11 G Glycine
32 15 12 L Leucine
23 4 8 T Threonine
8 0 4 W Tryptophan
19 1 6 C Cysteine
21 17 11 E Glutamic acid
10 3 7 Y Tyrosine
12 3 7 Q Glutamine
7 8 5 P Proline 

8SHA_1|Chain A[auth N]|Guanine nucleotide-binding protein subunit beta-5|Homo sapiens (9606)
>2JJZ_1|Chain A|IONIZED CALCIUM-BINDING ADAPTER MOLECULE 2|HOMO SAPIENS (9606)
>1HBP_1|Chain A|RETINOL BINDING PROTEIN|Bos taurus (9913)
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 \(w=f(c)\)
8SHA , Knot 169 395 0.85 40 221 367 
MCDQTFLVNVFGSCDKCFKQRALRPVFKKSQQLSYCSTCAEIMATEGLHENETLASLKSEAESLKGKLEEERAKLHDVELHQVAERVEALGQFVMKTRRTLKGHGNKVLCMDWCKDKRRIVSSSQDGKVIVWDSFTTNKEHAVTMPCTWVMACAYAPSGCAIACGGLDNKCSVYPLTFDKNENMAAKKKSVAMHTNYLSACSFTNSDMQILTASGDGTCALWDVESGQLLQSFHGHGADVLCLDLAPSETGNTFVSGGCDKKAMVWDMRSGQCVQAFETHESDINSVRYYPSGDAFASGSDDATCRLYDLRADREVAIYSKESIIFGASSVDFSLSGRLLFAGYNDYTINVWDVLKGSRVSILFGHENRVSTLRVSPDGTAFCSGSWDHTLRVWA
2JJZ , Knot 76 150 0.84 38 113 146 
MSGELSNRFQGGKAFGILKARQERRLAEINREFLCDQKYSDEENLPEKLTAFKEKYMEFDLNNEGEIDLMSLKRMMEKLGVPKTHLEMKKMISEVTGGVSDTISYRDFVNMMLGKRSAVLKLVMMFEGKANESSPKPVGPPPERDIASLP
1HBP , Knot 88 183 0.83 40 138 177 
ERDCRVSSFRVKENFDKARFAGTWYAMAKKDPEGLFLQDNIVAEFSVDENGHMSATAKGRVRLLNNWDVCADMVGTFTDTEDPAKFKMKYWGVASFLQKGNDDHWIIDTDYETFAVQYSCRLLNLDGTCADSYSFVFARDPSGFSPEVQKIVRQRQEELCLARQYRLIPHNGYCDGKSERNIL

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 \(c\).

\(|P_{f(8SHA_1)}(2) \setminus P_{f(2JJZ_1)}(2)|=152\), \(|P_{f(2JJZ_1)}(2) \setminus P_{f(8SHA_1)}(2)|=44\). 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 1:10000111011100000100011011100000100000010111001100000110100010010101000010100101001100101110111000001010100110101000000110000010111100100000011011001111010110101110111000001011010000011100001110000101001000010110101010011101001011001010110110101110001001101100001111010010010110000001001000101011101000100010010100011100000111110010101010111110000010110110100101111000010010101010110010100010111
Pair \(Z_2\) Length of longest common subsequence
8SHA_1,2JJZ_1 196 3
8SHA_1,1HBP_1 179 5
2JJZ_1,1HBP_1 147 3

Newick tree

 
[
	8SHA_1:99.70,
	[
		1HBP_1:73.5,2JJZ_1:73.5
	]:26.20
]

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 \((0.85)(0.8)(\frac{545 }{\log_{20} 545}-\frac{150}{\log_{20}150})=115.\)
Status Protein1 Protein2 d d1/2
Query variables 8SHA_1 2JJZ_1 146 100
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]