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Parikh vectors
4WSX_1 8PAM_1 2BWM_1 Letter Amino acid
5 3 9 W Tryptophan
19 10 21 R Arginine
26 9 34 N Asparagine
14 6 16 K Lycine
8 3 5 M Methionine
17 32 30 A Alanine
25 23 14 S Serine
17 30 47 V Valine
10 11 6 H Histidine
8 4 22 F Phenylalanine
15 15 11 P Proline
28 14 19 T Threonine
9 1 4 C Cysteine
13 5 11 Q Glutamine
14 18 6 E Glutamic acid
35 24 60 G Glycine
8 9 11 Y Tyrosine
11 13 27 D Aspartic acid
22 11 21 I Isoleucine
23 25 27 L Leucine 

4WSX_1|Chains A, C, E, G, I, K, M, O, Q, S, U, W|Hemagglutinin HA1 chain|Influenza A virus (11320)
>8PAM_1|Chain A|Beta-lactamase VIM-1|Pseudomonas aeruginosa (287)
>2BWM_1|Chain A|PSATHYRELLA VELUTINA LECTIN PVL|PSATHYRELLA VELUTINA (71681)
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 \(w=f(c)\)
4WSX , Knot 144 327 0.85 40 201 312 
ADPGDKICLGHHAVANGTIVKTLTNEQEEVTNATETVESTGINRLCMKGRKHKDLGNCHPIGMLIGTPACDLHLTGMWDTLIERENAIAYCYPGATVNVEALRQKIMESGGINKISTGFTYGSSINSAGTTRACMRNGGNSFYAELKWLVSKSKGQNFPQTTNTYRNTDTAEHLIMWGIHHPSSTQEKNDLYGTQSLSISVGSSTYRNNFVPVVGARPQVNGQSGRIDFHWTLVQPGDNITFSHNGGLIAPSRVSKLIGRGLGIQSDAPIDNNCESKCFWRGGSINTRLPFQNLSPRTVGQCPKYVNRRSLMLATGMRNVPELIQGR
8PAM , Knot 117 266 0.81 40 152 253 
MLKVISSLLVYMTASVMAVASPLAHSGEPSGEYPTVNEIPVGEVRLYQIADGVWSHIATQSFDGAVYPSNGLIVRDGDELLLIDTAWGAKNTAALLAEIEKQIGLPVTRAVSTHFHDDRVGGVDVLRAAGVATYASPSTRRLAEAEGNEIPTHSLEGLSSSGDAVRFGPVELFYPGAAHSTDNLVVYVPSANVLYGGCAVHELSSTSAGNVADADLAEWPTSVERIQKHYPEAEVVIPGHGLPGGLDLLQHTANVVKAHKNRSVAE
2BWM , Knot 144 401 0.71 40 178 307 
SVVVISQALPVPTRIPGVADLVGFGNGGVYIIRNSLLIQVVKVINNFGYDAGGWRVEKHVRLLADTTGDNQSDVVGFGENGVWISTNNGNNTFVDPPKMVLANFAYAAGGWRVEKHIRFMADLRKTGRADIVGFGDGGIYISRNNGGGQFAPAQLALNNFGYAQGWRLDRHLRFLADVTGDGLLDVVGFGENQVYIARNSGNGTFQPAQAVVNNFCIGAGGWTISAHPRVVADLTGDRKADILGFGVAGVYTSLNNGNGTFGAVNLVLKDFGVNSGWRVEKHVRCVSSLTNKKVGDIIGFGDAGVYVALNNGNGTFGPVKRVIDNFGYNQGWRVDKHPRFVVDLTGDGCADIVGFGENSVWACMNKGDGTFGPIMKLIDDMTVSKGWTLQKTVRYAANLYL

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 \(c\).

\(|P_{f(4WSX_1)}(2) \setminus P_{f(8PAM_1)}(2)|=111\), \(|P_{f(8PAM_1)}(2) \setminus P_{f(4WSX_1)}(2)|=62\). 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 1:101100101100111010110010000001001000100011001010100000110001111111011001010111001100001110001110101011000110011100100110010010011000101001100101010111000010011000000000001001111110010000000010100010101100000001111111010101001010101011011001010001111110010011101111000111000000001101101000111001010011001001000011110110011011010
Pair \(Z_2\) Length of longest common subsequence
4WSX_1,8PAM_1 173 4
4WSX_1,2BWM_1 159 3
8PAM_1,2BWM_1 152 4

Newick tree

 
[
	4WSX_1:85.30,
	[
		2BWM_1:76,8PAM_1:76
	]:9.30
]

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 \((0.85)(0.8)(\frac{593 }{\log_{20} 593}-\frac{266}{\log_{20}266})=92.1\)
Status Protein1 Protein2 d d1/2
Query variables 4WSX_1 8PAM_1 120 107.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]