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Parikh vectors
3FGE_1 8CCW_1 1BPN_1 Letter Amino acid
14 17 53 A Alanine
11 6 22 N Asparagine
2 5 7 C Cysteine
12 16 37 E Glutamic acid
13 21 42 G Glycine
15 16 29 S Serine
13 27 33 V Valine
10 7 8 H Histidine
16 10 29 I Isoleucine
9 10 33 K Lycine
4 4 11 M Methionine
0 2 7 W Tryptophan
8 19 21 R Arginine
16 37 39 L Leucine
14 12 25 T Threonine
9 7 9 Y Tyrosine
12 18 25 D Aspartic acid
9 9 14 Q Glutamine
9 15 20 F Phenylalanine
7 24 23 P Proline 

3FGE_1|Chain A|putative flavin reductase with split barrel domain|Shewanella frigidimarina NCIMB 400 (318167)
>8CCW_1|Chain A|NAD-dependent protein deacetylase sirtuin-3, mitochondrial|Homo sapiens (9606)
>1BPN_1|Chain A|LEUCINE AMINOPEPTIDASE|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 \(w=f(c)\)
3FGE , Knot 98 203 0.85 38 153 200 
GMHFSKEHINALETRTRAHFINSLSGFKSANLIGTQDRQGNTNLSIVSSVIHLGANPPLMGMIIRPHSVPRHTFENIMQTGLYTINHVNQSIYEQAHQTSARYDKDESEFEATGLTPEYLSDFCAPFVKESRLKYSVKLVEHQHLAINGTEFVIGEIVDVYVDDNAVQTDGFIDLQAIDTVAISGLDCYYTGDKLARLPYAKK
8CCW , Knot 123 282 0.82 40 168 261 
SDKGKLSLQDVAELIRARACQRVVVMVGAGISTPSGIPDFRSPGSGLYSNLQQYDLPYPEAIFELPFFFHNPKPFFTLAKELYPGNYKPNVTHYFLRLLHDKGLLLRLYTQNIDGLERVSGIPASKLVEAHGTFASATCTVCQRPFPGEDIRADVMADRVPRCPVCTGVVKPDIVFFGEPLPQRFLLHVVDFPMADLLLILGTSLEVEPFASLTEAVRSSVPRLLINRDLVGPLAWHPRSRDVAQLGDVVHGVESLVELLGWTEEMRDLVQRETGKLDGPDK
1BPN , Knot 203 487 0.86 40 248 458 
TKGLVLGIYSKEKEEDEPQFTSAGENFNKLVSGKLREILNISGPPLKAGKTRTFYGLHEDFPSVVVVGLGKKTAGIDEQENWHEGKENIRAAVAAGCRQIQDLEIPSVEVDPCGDAQAAAEGAVLGLYEYDDLKQKRKVVVSAKLHGSEDQEAWQRGVLFASGQNLARRLMETPANEMTPTKFAEIVEENLKSASIKTDVFIRPKSWIEEQEMGSFLSVAKGSEEPPVFLEIHYKGSPNASEPPLVFVGKGITFDSGGISIKAAANMDLMRADMGGAATICSAIVSAAKLDLPINIVGLAPLCENMPSGKANKPGDVVRARNGKTIQVDNTDAEGRLILADALCYAHTFNPKVIINAATLTGAMDIALGSGATGVFTNSSWLWNKLFEASIETGDRVWRMPLFEHYTRQVIDCQLADVNNIGKYRSAGACTAAAFLKEFVTHPKWAHLDIAGVMTNKDEVPYLRKGMAGRPTRTLIEFLFRFSQDSA

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 \(c\).

\(|P_{f(3FGE_1)}(2) \setminus P_{f(8CCW_1)}(2)|=75\), \(|P_{f(8CCW_1)}(2) \setminus P_{f(3FGE_1)}(2)|=90\). 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 1:11010000101100000101100101100101110000010001011001101110111111110100110001001100110010010001000100001000000001010110100100101111000010001011000011101001111011010100011000111010110011101100000100110110100
Pair \(Z_2\) Length of longest common subsequence
3FGE_1,8CCW_1 165 4
3FGE_1,1BPN_1 181 3
8CCW_1,1BPN_1 174 4

Newick tree

 
[
	1BPN_1:90.76,
	[
		3FGE_1:82.5,8CCW_1:82.5
	]:8.26
]

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 \((0.85)(0.8)(\frac{485 }{\log_{20} 485}-\frac{203}{\log_{20}203})=81.9\)
Status Protein1 Protein2 d d1/2
Query variables 3FGE_1 8CCW_1 101 88
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]