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Parikh vectors
5XHO_1 4RXX_1 5DSV_1 Letter Amino acid
12 25 24 A Alanine
8 23 5 H Histidine
4 25 14 I Isoleucine
9 9 9 N Asparagine
13 19 17 D Aspartic acid
18 25 23 E Glutamic acid
19 64 18 L Leucine
8 21 9 F Phenylalanine
3 23 5 P Proline
9 34 21 S Serine
7 12 19 G Glycine
15 19 18 K Lycine
5 10 9 M Methionine
5 20 7 T Threonine
8 8 13 Y Tyrosine
11 36 21 V Valine
9 17 13 R Arginine
2 9 4 C Cysteine
8 24 4 Q Glutamine
1 7 2 W Tryptophan 

5XHO_1|Chain A|Ferritin, middle subunit|Rana catesbeiana (8400)
>4RXX_1|Chain A|Ubiquitin carboxyl-terminal hydrolase 38|Homo sapiens (9606)
>5DSV_1|Chains A, B, C, D, E, F, G, H, I, J, K, L, M, N|Proteasome subunit alpha type-3|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 \(w=f(c)\)
5XHO , Knot 85 174 0.84 40 134 170 
VSQVRQNYHSDCEAAVNRMLNLELYASYTYSSMYAFFDRDDVALHNVAEFFKEHSHEEREHAEKFMKYQNKRGGRVVLQDIKKPERDEWGNTLEAMQAALQLEKTVNQALLDLHKLATDKVDPHLCDFLESEYLKEQVKDIKRIGDFITNLKRLGLPENGMGEYLFDKHSVKES
4RXX , Knot 181 430 0.85 40 236 407 
MDKILEGLVSSSHPLPLKRVIVRKVVESAEHWLDEAQCEAMFDLTTRLILEGQDPFQRQVGHQVLEAYARYHRPEFESFFNKTFVLGLLHQGYHSLDRKDVAILDYIHNGLKLIMSCPSVLDLFSLLQVEVLRMVCERPEPQLCARLSDLLTDFVQCIPKGKLSITFCQQLVRTIGHFQCVSTQERELREYVSQVTKVSNLLQNIWKAEPATLLPSLQEVFASISSTDASFEPSVALASLVQHIPLQMITVLIRSLTTDPNVKDASMTQALCRMIDWLSWPLAQHVDTWVIALLKGLAAVQKFTILIDVTLLKIELVFNRLWFPLVRPGALAVLSHMLLSFQHSPEAFHLIVPHVVNLVHSFKNDGLPSSTAFLVQLTELIHCMMYHYSGFPDLYEPILEAIKDFPKPSEEKIKLILNQSAWTSHHHHHH
5DSV , Knot 112 255 0.81 40 172 246 
MSSIGTGYDLSASTFSPDGRVFQVEYAMKAVENSSTAIGIRCKDGVVFGVEKLVLSKLYEEGSNKRLFNVDRHVGMAVAGLLADARSLADIAREEASNFRSNFGYNIPLKHLADRVAMYVHAYTLYSAVRPFGCSFMLGSYSVNDGAQLYMIDPSGVSYGYWGCAIGKARQAAKTEIEKLQMKEMTCRDIVKEVAKIIYIVHDEVKDKAFELELSWVGELTNGRHEIVPKDIREEAEKYAKESLKEEDESDDDNM

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 \(c\).

\(|P_{f(5XHO_1)}(2) \setminus P_{f(4RXX_1)}(2)|=37\), \(|P_{f(4RXX_1)}(2) \setminus P_{f(5XHO_1)}(2)|=139\). 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 1:100100000000011100110101010000001011100001110011011000000000010011000000110111001001000011001011011101000100111010011000101010011000010001001001101100100111100111001100001000
Pair \(Z_2\) Length of longest common subsequence
5XHO_1,4RXX_1 176 4
5XHO_1,5DSV_1 154 3
4RXX_1,5DSV_1 166 4

Newick tree

 
[
	4RXX_1:88.19,
	[
		5XHO_1:77,5DSV_1:77
	]:11.19
]

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 \((0.85)(0.8)(\frac{604 }{\log_{20} 604}-\frac{174}{\log_{20}174})=123.\)
Status Protein1 Protein2 d d1/2
Query variables 5XHO_1 4RXX_1 156 107.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]