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Parikh vectors
4WOL_1 8XGW_1 3WQH_1 Letter Amino acid
2 1 45 T Threonine
0 0 39 N Asparagine
0 0 19 H Histidine
2 0 47 I Isoleucine
0 0 14 M Methionine
1 0 31 F Phenylalanine
1 0 28 P Proline
3 1 35 A Alanine
1 0 30 R Arginine
0 0 40 E Glutamic acid
7 0 55 L Leucine
1 0 56 Y Tyrosine
7 0 44 V Valine
1 0 43 D Aspartic acid
0 0 30 Q Glutamine
4 2 41 G Glycine
0 0 37 K Lycine
1 2 12 C Cysteine
2 0 64 S Serine
0 0 20 W Tryptophan 

4WOL_1|Chains A, B, C|TYRO protein tyrosine kinase-binding protein|Homo sapiens (9606)
>8XGW_1|Chains A, B|DNA (5'-D(*CP*AP*CP*GP*TP*G)-3')|synthetic construct (32630)
>3WQH_1|Chains A, B|Dipeptidyl peptidase 4|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 \(w=f(c)\)
4WOL , Knot 21 33 0.74 26 28 31 
CSTVSPGVLAGIVVGDLVLTVLIALAVYFLGRL
8XGW , Knot 5 6 0.49 8 5 4 
CACGTG
3WQH , Knot 282 730 0.85 40 306 688 
GPSRKTYTLTDYLKNTYRLKLYSLRWISDHEYLYKQENNILVFNAEYGNSSVFLENSTFDEFGHSINDYSISPDGQFILLEYNYVKQWRHSYTASYDIYDLNKRQLITEERIPNNTQWVTWSPVGHKLAYVWNNDIYVKIEPNLPSYRITWTGKEDIIYNGITDWVYEEEVFSAYSALWWSPNGTFLAYAQFNDTEVPLIEYSFYSDESLQYPKTVRVPYPKAGAVNPTVKFFVVNTDSLSSVTNATSIQITAPASMLIGDHYLCDVTWATQERISLQWLRRIQNYSVMDICDYDESSGRWNCLVARQHIEMSTTGWVGRFRPSEPHFTLDGNSFYKIISNEEGYRHICYFQIDKKDCTFITKGTWEVIGIEALTSDYLYYISNEYKGMPGGRNLYKIQLSDYTKVTCLSCELNPERCQYYSVSFSKEAKYYQLRCSGPGLPLYTLHSSVNDKGLRVLEDNSALDKMLQNVQMPSKKLDFIILNETKFWYQMILPPHFDKSKKYPLLLDVYAGPCSQKADTVFRLNWATYLASTENIIVASFDGRGSGYQGDKIMHAINRRLGTFEVEDQIEAARQFSKMGFVDNKRIAIWGWSYGGYVTSMVLGSGSGVFKCGIAVAPVSRWEYYDSVYTERYMGLPTPEDNLDHYRNSTVMSRAENFKQVEYLLIHGTADDNVHFQQSAQISKALVDVGVDFQAMWYTDEDHGIASSTAHQHIYTHMSHFIKQCFSLP

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 \(c\).

\(|P_{f(4WOL_1)}(2) \setminus P_{f(8XGW_1)}(2)|=28\), \(|P_{f(8XGW_1)}(2) \setminus P_{f(4WOL_1)}(2)|=5\). 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 1:000101111111111011101111111011101
Pair \(Z_2\) Length of longest common subsequence
4WOL_1,8XGW_1 33 1
4WOL_1,3WQH_1 278 3
8XGW_1,3WQH_1 305 2

Newick tree

 
[
	3WQH_1:16.20,
	[
		4WOL_1:16.5,8XGW_1:16.5
	]:15.70
]

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 \((0.85)(0.8)(\frac{39 }{\log_{20} 39}-\frac{6}{\log_{20}6})=14.8\)
Status Protein1 Protein2 d d1/2
Query variables 4WOL_1 8XGW_1 19 11
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]

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