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Parikh vectors
7TOK_1 8UVL_1 1YKF_1 Letter Amino acid
8 25 20 D Aspartic acid
0 11 4 C Cysteine
12 23 26 I Isoleucine
8 25 24 K Lycine
8 27 21 P Proline
8 17 13 T Threonine
12 27 37 V Valine
4 28 14 R Arginine
7 12 10 N Asparagine
12 27 43 G Glycine
4 26 10 H Histidine
11 43 23 L Leucine
6 35 21 E Glutamic acid
6 23 14 F Phenylalanine
4 14 6 Y Tyrosine
10 24 35 A Alanine
9 19 3 Q Glutamine
3 10 15 M Methionine
17 29 9 S Serine
1 3 4 W Tryptophan 

7TOK_1|Chains A, B|Acetylxylan esterase I|Flavobacterium johnsoniae (986)
>8UVL_1|Chain A|Serine/threonine-protein kinase/endoribonuclease IRE1|Homo sapiens (9606)
>1YKF_1|Chains A, B, C, D|NADP-DEPENDENT ALCOHOL DEHYDROGENASE|Thermoanaerobacter brockii (29323)
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 \(w=f(c)\)
7TOK , Knot 72 150 0.80 38 116 145 
PPEPGLAQNTLRQIIKVSLGGKQIRMRFSNLFSDQPAVLKSVSVANVTEAPAVDIKTQKILSFKGSPQVTLGADEVMYSDAFDFELQPGQLLAITIHYGEISSNVSGHPGSRTTSYILQGDHINNESFAGAVKTDHWYSIMGVDISSVKN
8UVL , Knot 184 448 0.83 40 241 430 
MHHHHHHGENLYFQGSSPSLEQDDGDEETSVVIVGKISFCPKDVLGHGAEGTIVYRGMFDNRDVAVKRILPECFSFADREVQLLRESDEHPNVIRYFCTEKDRQFQYIAIELCAATLQEYVEQKDFAHLGLEPITLLQQTTSGLAHLHSLNIVHRNLKPHNILISMPNAHGKIKAMISDFGLCKKLAVGRHSFSRRSGVPGTEGWIAPEMLSEDCKENPTYTVDIFSAGCVFYYVISEGSHPFGKSLQRQANILLGACSLDCLHPEKHEDVIARELIEKMIAMDPQKRPSAKHVLKHPFFWSLEKQLQFFQDVSDRIEKESLDGPIVKQLERGGRAVVKMDWRENITVPLQTDLRKFRTYKGGSVRDLLRAMRNKKHHYRELPAEVRETLGSLPDDFVCYFTSRFPHLLAHTYRAMELCSHERLFQPYYFHEPPEPQPPVTPDALGNS
1YKF , Knot 145 352 0.80 40 192 332 
MKGFAMLSIGKVGWIEKEKPAPGPFDAIVRPLAVAPCTSDIHTVFEGAIGERHNMILGHEAVGEVVEVGSEVKDFKPGDRVVVPAITPDWRTSEVQRGYHQHSGGMLAGWKFSNVKDGVFGEFFHVNDADMNLAHLPKEIPLEAAVMIPDMMTTGFHGAELADIELGATVAVLGIGPVGLMAVAGAKLRGAGRIIAVGSRPVCVDAAKYYGATDIVNYKDGPIESQIMNLTEGKGVDAAIIAGGNADIMATAVKIVKPGGTIANVNYFGEGEVLPVPRLEWGCGMAHKTIKGGLCPGGRLRMERLIDLVFYKRVDPSKLVTHVFRGFDNIEKAFMLMKDKPKDLIKPVVILA

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 \(c\).

\(|P_{f(7TOK_1)}(2) \setminus P_{f(8UVL_1)}(2)|=35\), \(|P_{f(8UVL_1)}(2) \setminus P_{f(7TOK_1)}(2)|=160\). 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 1:110111100010011010111001010100110001111001011010011110100001101010101011100110001101010110111101001010001010110000001101001000011111000010011110100100
Pair \(Z_2\) Length of longest common subsequence
7TOK_1,8UVL_1 195 4
7TOK_1,1YKF_1 174 3
8UVL_1,1YKF_1 159 4

Newick tree

 
[
	7TOK_1:96.31,
	[
		1YKF_1:79.5,8UVL_1:79.5
	]:16.81
]

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 \((0.85)(0.8)(\frac{598 }{\log_{20} 598}-\frac{150}{\log_{20}150})=129.\)
Status Protein1 Protein2 d d1/2
Query variables 7TOK_1 8UVL_1 168 109.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]