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Parikh vectors
6WNF_1 9EWO_1 2YDR_1 Letter Amino acid
14 17 46 D Aspartic acid
11 9 59 E Glutamic acid
9 29 44 L Leucine
2 10 15 M Methionine
0 13 25 P Proline
5 19 51 A Alanine
6 9 18 R Arginine
3 12 2 C Cysteine
4 11 23 Y Tyrosine
12 26 35 V Valine
7 21 43 N Asparagine
4 13 18 Q Glutamine
9 25 37 G Glycine
2 7 8 H Histidine
4 11 36 I Isoleucine
14 11 34 K Lycine
4 3 10 W Tryptophan
8 16 21 F Phenylalanine
3 15 34 S Serine
12 23 31 T Threonine 

6WNF_1|Chains A, B|Retinol-binding protein 2|Homo sapiens (9606)
>9EWO_1|Chain A|Non-structural protein 11|Homo sapiens (9606)
>2YDR_1|Chain A|O-GLCNACASE NAGJ|CLOSTRIDIUM PERFRINGENS (1502)
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 \(w=f(c)\)
6WNF , Knot 64 133 0.78 38 100 128 
TRDFNGTWEMESNENFEGYMKALDIDFATRKIAVRLTFTDVIDQDGDNFKTKATSTHLNYDVDFTVGVEFDEYTKSLDNRHVKALVTWEGDVLVCVQKGEKENRGWKKWIEGDKLYLELTCGDQVCRQVFKKK
9EWO , Knot 133 300 0.84 40 195 285 
SGFAKMAFPSGKVEGCMVQVTCGTTTLNGLWLDDVVYCPRHVICTSEDMLNPNYEDLLIRKSNHNFLVQAGNVQLRVIGHSMQNCVLKLKVDTANPKTPKYKFVRIQPGQTFSVLACYNGSPSGVYQCAMRPNFTIKGSFLNGSCGSVGFNIDYDCVSFCYMHHMELPTGVHAGTDLEGNFYGPFVDRQTAQAAGTDTTITVNVLAWLYAAVINGDRWFLNRFTTTLNDFNLVAMKYNYEPLTQDHVDILGPLSAQTGIAVLDMCASLKELLQNGMNGRTILGSALLEDEFTPFDVVAQC
2YDR , Knot 230 590 0.83 40 256 549 
GSVGPKTGEENQVLVPNLNPTPENLEVVGDGFKITSSINLVGEEEADENAVNALREFLTANNIEINSENDPNSTTLIIGEVDDDIPELDEALNGTTAENLKEEGYALVSNDGKIAIEGKDGDGTFYGVQTFKQLVKESNIPEVNITDYPTVSARGIVEGFYGTPWTHQDRLDQIKFYGENKLNTYIYAPKDDPYHREKWREPYPESEMQRMQELINASAENKVDFVFGISPGIDIRFDGDAGEEDFNHLITKAESLYDMGVRSFAIYWDNIQDKSAAKHAQVLNRFNEEFVKAKGDVKPLITVPTEYDTGAMVSNGQPRAYTRIFAETVDPSIEVMWTGPGVVTNEIPLSDAQLISGIYDRNMAVWWNYPVTDYFKGKLALGPMHGLDKGLNQYVDFFTVNPMEHAELSKISIHTAADYSWNMDNYDYDKAWNRAIDMLYGDLAEDMKVFANHSTRMDNKTWAKSGREDAPELRAKMDELWNKLSSKEDASALIEELYGEFARMEEACNNLKANLPEVALEECSRQLDELITLAQGDKASLDMIVAQLNEDTEAYESAKEIAQNKLNTALSSFAVISEKVAQSFIQEALS

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 \(c\).

\(|P_{f(6WNF_1)}(2) \setminus P_{f(9EWO_1)}(2)|=40\), \(|P_{f(9EWO_1)}(2) \setminus P_{f(6WNF_1)}(2)|=135\). 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 1:0001010101000001010101101011000111010100110001001000100001000101011101000000100001011101010111010010000011001101001010100100100011000
Pair \(Z_2\) Length of longest common subsequence
6WNF_1,9EWO_1 175 3
6WNF_1,2YDR_1 196 3
9EWO_1,2YDR_1 171 4

Newick tree

 
[
	6WNF_1:95.23,
	[
		9EWO_1:85.5,2YDR_1:85.5
	]:9.73
]

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 \((0.85)(0.8)(\frac{433 }{\log_{20} 433}-\frac{133}{\log_{20}133})=89.8\)
Status Protein1 Protein2 d d1/2
Query variables 6WNF_1 9EWO_1 114 82
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]