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Parikh vectors
4PCA_1 8UKA_1 6GVC_1 Letter Amino acid
15 18 28 E Glutamic acid
22 4 26 L Leucine
16 15 19 K Lycine
13 3 12 N Asparagine
3 0 6 C Cysteine
6 6 11 Q Glutamine
15 4 23 S Serine
22 19 29 A Alanine
9 6 22 D Aspartic acid
12 4 17 M Methionine
15 2 30 I Isoleucine
8 2 12 F Phenylalanine
13 10 27 T Threonine
14 19 21 V Valine
5 0 18 R Arginine
12 18 28 G Glycine
13 1 9 H Histidine
5 5 19 P Proline
1 0 4 W Tryptophan
7 4 16 Y Tyrosine 

4PCA_1|Chains A, B, C, D|O-methyltransferase family protein|Anaplasma phagocytophilum str. HGE1 (1217107)
>8UKA_1|Chains A, B, C, D, E, F[auth a], G[auth b], H[auth c], I[auth d], J[auth e]|Alpha-synuclein|Homo sapiens (9606)
>6GVC_1|Chains A, B, C, D|Actin, alpha skeletal muscle|Oryctolagus cuniculus (9986)
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 \(w=f(c)\)
4PCA , Knot 101 226 0.80 40 158 215 
MAHHHHHHMRNVSLSKQDEYLNKLFAVDTEGALKAHKTAPSELRMAQLGTVEGQMLQLLIRMAGIHSIVEVGTCVGFSAICMAHALPSKGHIYTIEKDYENVVTANQNIVNCKLEDKITVLHGEALAQLNTLKEMAPFDMIFIDANKSSYLAYLNWAKMYIRKGGLIVADNTFLFGSVFDEHPTEKVSSNAHASMRAFNDELANKEKYLSTIIPTSEGMMVSIKLT
8UKA , Knot 59 140 0.69 34 86 121 
MDVFMKGLSKAKEGVVAAAEKTKQGVAEAAGKTKEGVLYVGSKTKEGVVHGVATVAEKTKEQVTNVGGAVVTGVTAVAQKTVEGAGSIAAATGFVKKDQLGKNEEGAPQEGILEDMPVDPDNEAYEMPSEEGYQDYEPEA
6GVC , Knot 160 377 0.84 40 220 362 
MCDEDETTALVCDNGSGLVKAGFAGDDAPRAVFPSIVGRPRHQGVMVGMGQKDSYVGDEAQSKRGILTLKYPIEHGIITNWDDMEKIWHHTFYNELRVAPEEHPTLLTEAPLNPKANREKMTQIMFETFNVPAMYVAIQAVLSLYASGRTTGIVLDSGDGVTHNVPIYEGYALPHAIMRLDLAGRDLTDYLMKILTERGYSFVTTAEREIVRDIKEKLCYVALDFENEMATAASSSSLEKSYELPDGQVITIGNERFRCPETLFQPSFIGMESAGIHETTYNSIMKCDIDIRKDLYANNVMSGGTTMYPGIADRMQKEITALAPSTMKIKIIAPPERKYSVWIGGSILASLSTFQQMWITKQEYDEAGPSIVHRKCF

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 \(c\).

\(|P_{f(4PCA_1)}(2) \setminus P_{f(8UKA_1)}(2)|=110\), \(|P_{f(8UKA_1)}(2) \setminus P_{f(4PCA_1)}(2)|=38\). 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 1:1100000010010100000010011110001110100011001011011010101101110111100110110011101101101110010100100000011010001100010001011010111010010011110111101000001101011010100111111000111101100010001000101010110001100000100111000111101010
Pair \(Z_2\) Length of longest common subsequence
4PCA_1,8UKA_1 148 4
4PCA_1,6GVC_1 182 4
8UKA_1,6GVC_1 188 3

Newick tree

 
[
	6GVC_1:97.90,
	[
		4PCA_1:74,8UKA_1:74
	]:23.90
]

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 \((0.85)(0.8)(\frac{366 }{\log_{20} 366}-\frac{140}{\log_{20}140})=68.6\)
Status Protein1 Protein2 d d1/2
Query variables 4PCA_1 8UKA_1 86 66.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]