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Parikh vectors
4DSZ_1 9AUQ_1 8RFN_1 Letter Amino acid
47 6 19 I Isoleucine
43 6 12 D Aspartic acid
40 10 17 E Glutamic acid
19 1 7 H Histidine
27 4 13 P Proline
64 10 17 S Serine
45 9 9 T Threonine
12 0 1 C Cysteine
37 1 24 K Lycine
31 2 17 F Phenylalanine
30 7 12 Q Glutamine
40 24 21 G Glycine
14 2 7 M Methionine
20 6 6 W Tryptophan
44 8 14 V Valine
35 8 25 A Alanine
30 3 9 R Arginine
39 14 9 N Asparagine
55 7 25 L Leucine
56 4 10 Y Tyrosine 

4DSZ_1|Chains A, B|Dipeptidyl peptidase 4|Homo sapiens (9606)
>9AUQ_1|Chains A, B|Lectin|Planktothrix agardhii (1160)
>8RFN_1|Chains A, B, C, D|NAD(P)H dehydrogenase [quinone] 1|Homo sapiens (9606)
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 \(w=f(c)\)
4DSZ , Knot 280 728 0.84 40 306 686 
SRKTYTLTDYLKNTYRLKLYSLRWISDHEYLYKQENNILVFNAEYGNSSVFLENSTFDEFGHSINDYSISPDGQFILLEYNYVKQWRHSYTASYDIYDLNKRQLITEERIPNNTQWVTWSPVGHKLAYVWNNDIYVKIEPNLPSYRITWTGKEDIIYNGITDWVYEEEVFSAYSALWWSPNGTFLAYAQFNDTEVPLIEYSFYSDESLQYPKTVRVPYPKAGAVNPTVKFFVVNTDSLSSVTNATSIQITAPASMLIGDHYLCDVTWATQERISLQWLRRIQNYSVMDICDYDESSGRWNCLVARQHIEMSTTGWVGRFRPSEPHFTLDGNSFYKIISNEEGYRHICYFQIDKKDCTFITKGTWEVIGIEALTSDYLYYISNEYKGMPGGRNLYKIQLSDYTKVTCLSCELNPERCQYYSVSFSKEAKYYQLRCSGPGLPLYTLHSSVNDKGLRVLEDNSALDKMLQNVQMPSKKLDFIILNETKFWYQMILPPHFDKSKKYPLLLDVYAGPCSQKADTVFRLNWATYLASTENIIVASFDGRGSGYQGDKIMHAINRRLGTFEVEDQIEAARQFSKMGFVDNKRIAIWGWSYGGYVTSMVLGSGSGVFKCGIAVAPVSRWEYYDSVYTERYMGLPTPEDNLDHYRNSTVMSRAENFKQVEYLLIHGTADDNVHFQQSAQISKALVDVGVDFQAMWYTDEDHGIASSTAHQHIYTHMSHFIKQCFSLP
9AUQ , Knot 48 132 0.59 38 71 91 
ALYNVENQWGGSSAPWNEGGQWEIGSRSDQNVVAINVESGDDGQTLNGTMTYAGEGPIGFRATLLGNNSYEVENQWGGDSAPWHSGGNWILGSRENQNVVAINVESGDDGQTLNGTMTYAGEGPIGFKGTLT
8RFN , Knot 123 274 0.84 40 178 265 
MVGRRALIVLAHSERTSFNYAMKEAAAAALKKKGWEVVESDLYAMNFNPIISRKDITGKLKDPANFQYPAESVLAYKEGHLSPDIVAEQKKLEAADLVIFQFPLQWFGVPAILKGWFERVFIGEFAYTYAAMYDKGPFRSKKAVLSITTGGSGSMYSLQGIHGDMNVILWPIQSGILHFCGFQVLEPQLTYSIGHTPADARIQILEGWKKRLENIWDETPLYFAPSSLFDLNFQAGFLMKKEVQDEEKNKKFGLSVGHHLGKSIPTDNQIKARK

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 \(c\).

\(|P_{f(4DSZ_1)}(2) \setminus P_{f(9AUQ_1)}(2)|=246\), \(|P_{f(9AUQ_1)}(2) \setminus P_{f(4DSZ_1)}(2)|=11\). 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 1:00000010001000001010010110000010000001111010010001110000100110010000101010111100001001000001000100100001100001100001101011100110110001010101011000101010001100110011000011010011110101011101010000111100010000010010010110101111010101111000010010010010101110111100010010110000101011001000011010000000101001110001010001111010100101010100100110000100010010100000011001010111101100001001000001111100100101000001001000101000000010100010000100011111100100010001101100001100110010110001011110000110011111010000001111010111000010011010110011000011110101010100100110110001101010001011001001111000011111100110100111101011100111111100100000100000111101000100000001100100100100111010100010100010100111011101011100000011100010001000100110001011
Pair \(Z_2\) Length of longest common subsequence
4DSZ_1,9AUQ_1 257 3
4DSZ_1,8RFN_1 188 4
9AUQ_1,8RFN_1 171 3

Newick tree

 
[
	4DSZ_1:12.25,
	[
		8RFN_1:85.5,9AUQ_1:85.5
	]:34.75
]

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 \((0.85)(0.8)(\frac{860 }{\log_{20} 860}-\frac{132}{\log_{20}132})=204.\)
Status Protein1 Protein2 d d1/2
Query variables 4DSZ_1 9AUQ_1 264 149
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]