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Parikh vectors
8CJI_1 2NMZ_1 1ZMU_1 Letter Amino acid
18 6 13 P Proline
16 8 14 T Threonine
17 5 23 V Valine
21 1 16 S Serine
20 6 16 A Alanine
8 3 15 N Asparagine
16 4 18 D Aspartic acid
11 5 11 Q Glutamine
24 4 25 E Glutamic acid
24 2 15 F Phenylalanine
14 4 18 R Arginine
9 0 3 C Cysteine
31 10 37 L Leucine
4 2 10 M Methionine
2 2 2 W Tryptophan
21 13 22 G Glycine
17 1 7 H Histidine
10 15 17 I Isoleucine
27 7 29 K Lycine
16 1 16 Y Tyrosine 

8CJI_1|Chain A|Tryptophan 5-hydroxylase 1|Homo sapiens (9606)
>2NMZ_1|Chains A, B|PROTEASE|Human immunodeficiency virus 1 (11676)
>1ZMU_1|Chains A, B|MAP/Microtubule affinity regulating kinase 2|Rattus norvegicus (10116)
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 \(w=f(c)\)
8CJI , Knot 142 326 0.84 40 198 311 
MKHHHHHHHGAAGTSLYKKAGENLYFQGSVPWFPKKISDLDHCANRVLMYGSELDADHPGFKDNVYRKRRKYFADLAMNYKHGDPIPKVEFTEEEIKTWGTVFQELNKLYPTHACREYLKNLPLLSKYCGYREDNIPQLEDVSNFLKERTGFSIRPVAGYLSPRDFLSGLAFRVFHCTQYVRHSSDPFYTPEPDTCHELLGHVPLLAEPSFAQFSQEIGLASLGASEEAVQKLATCYFFTVEFGLCKQDGQLRVFGAGLLSSISELKHALSGHAKVKPFDPKITCKQECLITTFQDVYFVSESFEDAKEKMREFTKTIKRPFGVKY
2NMZ , Knot 53 99 0.82 38 77 95 
PQITLWKRPLVTIKIGGQLKEALLDTGADDTVIEEMSLPGRWKPKMIGGIGGFIKVRQYDQIIIEIAGHKAIGTVLVGPTPANIIGRNLLTQIGATLNF
1ZMU , Knot 146 327 0.86 40 208 317 
GNSATSADEQPHIGNYRLLKTIGKGNFAKVKLARHILTGKEVAVKIIDKTQLNSSSLQKLFREVRIMKVLNHPNIVKLFEVIETEKTLYLVMEYASGGEVFDYLVAHGRMKEKEARAKFRQIVSAVQYCHQKFIVHRDLKAENLLLDADMNIKIADFGFSNEFTFGNKLDTFCGSPPYAAPELFQGKKYDGPEVDVWSLGVILYTLVSGSLPFDGQNLKELRERVLRGKYRIPFYMSTDCENLLKKFLILNPSKRGTLEQIMKDRWMNVGHEDDELKPYVEPLPDYKDPRRTELMVSMGYTREEIQDSLVGQRYNEVMATYLLLGYK

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 \(c\).

\(|P_{f(8CJI_1)}(2) \setminus P_{f(2NMZ_1)}(2)|=150\), \(|P_{f(2NMZ_1)}(2) \setminus P_{f(8CJI_1)}(2)|=29\). 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 1:10000000011110010001100101010111110010010001001110100101001110001000000011011100001011101010000100110110010010100100001001111000010000011010010011000011010111101010011011110110000010000011001010000011101111101011010001111011100011001100011010111000010101111111001001001101010101101010000001100100101100010010001001000100111100
Pair \(Z_2\) Length of longest common subsequence
8CJI_1,2NMZ_1 179 3
8CJI_1,1ZMU_1 156 4
2NMZ_1,1ZMU_1 179 3

Newick tree

 
[
	2NMZ_1:93.01,
	[
		8CJI_1:78,1ZMU_1:78
	]:15.01
]

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 \((0.85)(0.8)(\frac{425 }{\log_{20} 425}-\frac{99}{\log_{20}99})=99.1\)
Status Protein1 Protein2 d d1/2
Query variables 8CJI_1 2NMZ_1 127 81.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]