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Parikh vectors
1EYY_1 7ZIG_1 5OYD_1 Letter Amino acid
45 31 35 L Leucine
7 4 12 M Methionine
32 16 27 T Threonine
4 2 12 W Tryptophan
18 16 24 D Aspartic acid
33 24 23 E Glutamic acid
11 27 14 K Lycine
22 24 16 F Phenylalanine
36 18 14 P Proline
38 21 21 S Serine
35 17 23 V Valine
26 14 15 R Arginine
27 11 22 Q Glutamine
35 21 32 G Glycine
13 17 14 H Histidine
8 16 14 Y Tyrosine
70 20 37 A Alanine
18 8 27 N Asparagine
6 9 0 C Cysteine
26 10 14 I Isoleucine 

1EYY_1|Chains A, B, C, D|ALDEHYDE DEHYDROGENASE|Vibrio harveyi (669)
>7ZIG_1|Chain A|Tryptophan 5-hydroxylase 1|Homo sapiens (9606)
>5OYD_1|Chains A, B|Cellulase, putative, cel5D|Cellvibrio japonicus (498211)
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 \(w=f(c)\)
1EYY , Knot 203 510 0.82 40 232 473 
MNPQTDNVFYATNAFTGEALPLAFPVHTEVEVNQAATAAAKVARDFRRLNNSKRASLLRTIASELEARSDDIIARAHLETALPEVRLTGEIARTANQLRLFADVVNSGSYHQAILDTPNPTRAPLPKPDIRRQQIALGPVAVFGASNFPLAFSAAGGDTASALAAGCPVIVKGHTAHPGTSQIVAECIEQALKQEQLPQAIFTLLQGNQRALGQALVSHPEIKAVGFTGSVGGGRALFNLAHERPEPIPFYGELGAINPTFIFPSAMRAKADLADQFVASMTMGCGQFCTKPGVVFALNTPETQAFIETAQSLIRQQSPSTLLTPGIRDSYQSQVVSRGSDDGIDVTFSQAESPCVASALFVTSSENWRKHPAWEEEIFGPQSLIVVCENVADMLSLSEMLAGSLTATIHATEEDYPQVSQLIPRLEEIAGRLVFNGWPTGVEVGYAMVHGGPYPASTHSASTSVGAEAIHRWLRPVAYQALPESLLPDSLKAENPLEIARAVDGKAAHS
7ZIG , Knot 142 326 0.84 40 198 311 
MKHHHHHHHGAAGTSLYKKAGENLYFQGSVPWFPKKISDLDHCANRVLMYGSELDADHPGFKDNVYRKRRKYFADLAMNYKHGDPIPKVEFTEEEIKTWGTVFQELNKLYPTHACREYLKNLPLLSKYCGYREDNIPQLEDVSNFLKERTGFSIRPVAGYLSPRDFLSGLAFRVFHCTQYVRHSSDPFYTPEPDTCHELLGHVPLLAEPSFAQFSQEIGLASLGASEEAVQKLATCYFFTVEFGLCKQDGQLRVFGAGLLSSISELKHALSGHAKVKPFDPKITCKQECLITTFQDVYFVSESFEDAKEKMREFTKTIKRPFGVKY
5OYD , Knot 164 396 0.82 38 227 376 
MGSSHHHHHHSSGLVPRGSHMASGLYPSYNTSPAAPDSTGMQSTAVQLAGKIRLGWNIGNTMEAIGGETAWGNPMVSNELLKLVKDSGFDAVRIPVAWDQYANQESAEISAAWLNRVKQVVQMAIDNELYVLINIHWDGGWLENNITPAKKDENNAKQKAFWEQIATHLRDFDEHLLFAGTNEPNAENAEQMDVLNSYLQTFVDAVRSTGGKNAYRVLVLQGPVTDIEKTNELWTHMPADTATDRLMAEVHFYTPYNFALMRQDESWGKQFYYWGEGFLSTTDTERNPTWGEEATIDQLFDLMKTKFVDQGIPVVLGEFSAMRRTNLTGDALTLHLAGRAYYHKYVTQQALARGLLPFYWDNGGNDNFSSGIFNRQQNTVFDQQVLDALLEGAGAQ

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 \(c\).

\(|P_{f(1EYY_1)}(2) \setminus P_{f(7ZIG_1)}(2)|=105\), \(|P_{f(7ZIG_1)}(2) \setminus P_{f(1EYY_1)}(2)|=71\). 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 1:101000011010011010111111110001010011011101100100100000101100110010100001110101001110101010110010010111011001000011100101001111010100001111111111100111110111100101111101111010010110001110010011000011011101101000111011100101011110101111011101100010111101011110101111011010101100111010110101000111111100100011100100110000100110111000000011001000110101001001011011110000010001110001111001111000110110100111101010101000001010011101001110111011101101101110111011000010001110110011011100111001110010100110110110101100
Pair \(Z_2\) Length of longest common subsequence
1EYY_1,7ZIG_1 176 4
1EYY_1,5OYD_1 143 4
7ZIG_1,5OYD_1 175 6

Newick tree

 
[
	7ZIG_1:92.53,
	[
		1EYY_1:71.5,5OYD_1:71.5
	]:21.03
]

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 \((0.85)(0.8)(\frac{836 }{\log_{20} 836}-\frac{326}{\log_{20}326})=138.\)
Status Protein1 Protein2 d d1/2
Query variables 1EYY_1 7ZIG_1 171 141
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]