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Parikh vectors
5EHW_1 6LUI_1 1TYX_1 Letter Amino acid
12 0 23 F Phenylalanine
7 1 7 W Tryptophan
8 3 21 Y Tyrosine
13 6 39 A Alanine
7 16 21 R Arginine
1 1 8 C Cysteine
12 2 8 H Histidine
26 6 46 L Leucine
2 1 7 M Methionine
10 1 31 N Asparagine
13 6 19 E Glutamic acid
12 2 40 T Threonine
9 5 37 I Isoleucine
24 3 29 K Lycine
17 6 21 P Proline
18 7 44 S Serine
17 4 35 V Valine
19 3 40 D Aspartic acid
11 4 14 Q Glutamine
22 3 64 G Glycine 

5EHW_1|Chain A|Carbonic anhydrase 2|Homo sapiens (9606)
>6LUI_1|Chain A|Atherin|Homo sapiens (9606)
>1TYX_1|Chain A|TAILSPIKE PROTEIN|Enterobacteria phage P22 (10754)
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 \(w=f(c)\)
5EHW , Knot 112 260 0.79 40 176 249 
MSHHWGYGKHNGPEHWHKDFPIAKGERQSPVDIDTHTAKYDPSLKPLSVSYDQATSLRILNNGHAFNVEFDDSQDKAVLKGGPLDGTYRLIQFHFHWGSLDGQGSEHTVDKKKYAAELHLVHWNTKYGDFGKAVQQPDGLAVLGIFLKVGSAKPGLQKVVDVLDSIKTKGKSADFTNFDPRGLLPESLDYWTYPGSLTTPPLLECVTWIVLKEPISVSSEQVLKFRKLNFNGEGEPEELMVDNWRPAQPLKNRQIKASFK
6LUI , Knot 45 80 0.82 38 69 78 
SASPHYQEWILDTIDSLRSRKARPDLERICRMVRRRHGPEPERTRAELEKLIQQRAVLRVSYKGSISYRNAARVQPPRRG
1TYX , Knot 220 554 0.83 40 249 518 
YSIEADKKFKYSVKLSDYPTLQDAASAAVDGLLIDRDYNFYGGETVDFGGKVLTIECKAKFIGDGNLIFTKLGKGSRIAGVFMESTTTPWVIKPWTDDNQWLTDAAAVVATLKQSKTDGYQPTVSDYVKFPGIETLLPPNAKGQNITSTLEIRECIGVEVHRASGLMAGFLFRGCHFCKMVDANNPSGGKDGIITFENLSGDWGKGNYVIGGRTSYGSVSSAQFLRNNGGFERDGGVIGFTSYRAGESGVKTWQGTVGSTTSRNYNLQFRDSVVIYPVWDGFDLGADTDMNPELDRPGDYPITQYPLHQLPLNHLIDNLLVRGALGVGFGMDGKGMYVSNITVEDCAGSGAYLLTHESVFTNIAIIDTNTKDFQANQIYISGACRVNGLRLIGIRSTDGQGLTIDAPNSTVSGITGMVDPSRINVANLAEEGLGNIRANSFGYDSAAIKLRIHKLSKTLDSGALYSHINGGAGSGSAYTQLTAISGSTPDAVSLKVNHKDCRGAEIPFVPDIASDDFIKDSSCFLPYWENNSTSLKALVKKPNGELVRLTLATL

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 \(c\).

\(|P_{f(5EHW_1)}(2) \setminus P_{f(6LUI_1)}(2)|=133\), \(|P_{f(6LUI_1)}(2) \setminus P_{f(5EHW_1)}(2)|=26\). 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 1:10001101000110010001111010000110100001000101011010000100101100101101010000001110111101000110101011010101000010000011010110100001011011001011111111101101011100110110010001001010010101111001001001101001111001011110011010000110100101010101001110010110110000101010
Pair \(Z_2\) Length of longest common subsequence
5EHW_1,6LUI_1 159 3
5EHW_1,1TYX_1 163 4
6LUI_1,1TYX_1 228 3

Newick tree

 
[
	1TYX_1:10.81,
	[
		5EHW_1:79.5,6LUI_1:79.5
	]:25.31
]

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 \((0.85)(0.8)(\frac{340 }{\log_{20} 340}-\frac{80}{\log_{20}80})=81.6\)
Status Protein1 Protein2 d d1/2
Query variables 5EHW_1 6LUI_1 102 65.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]