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Parikh vectors
3CNW_1 6YPG_1 2DNE_1 Letter Amino acid
4 24 2 R Arginine
8 15 2 N Asparagine
12 11 7 T Threonine
8 14 6 P Proline
7 26 6 D Aspartic acid
0 2 2 C Cysteine
9 19 9 I Isoleucine
4 20 8 K Lycine
4 9 2 M Methionine
7 13 2 F Phenylalanine
6 15 10 A Alanine
5 9 2 Q Glutamine
13 22 11 E Glutamic acid
10 14 12 G Glycine
9 16 0 H Histidine
8 26 6 V Valine
13 30 6 L Leucine
12 15 12 S Serine
3 5 1 W Tryptophan
6 23 2 Y Tyrosine 

3CNW_1|Chains A, B|Protein XoxI|Bacillus cereus ATCC 14579 (226900)
>6YPG_1|Chain A|Casein kinase II subunit alpha|Homo sapiens (9606)
>2DNE_1|Chain A|Dihydrolipoyllysine-residue acetyltransferase component of pyruvate dehydrogenase complex|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 \(w=f(c)\)
3CNW , Knot 75 148 0.84 38 121 143 
MNMAHTTTSMEIFGSPEQVWQLIGGFNSLPDWLPYIPSSKLTEGGRVRHLANPDGDTIIERLEVFNDKERYYTYSIMNAPFPVTNYLSTIQVKEGTESNTSLVEWSGTFTPVEVSDEEAINLFHGIYSDGLKALQQAFLDLEHHHHHH
6YPG , Knot 147 328 0.86 40 194 315 
SGPVPSRARVYTDVNTHRPSEYWDYESHVVEWGNQDDYQLVRKLGRGKYSEVFEAINITNNEKVVVKILKPVAAAKIKREIKILENLRGGPNIITLADIVKDPVSRTPALVFEHVNNTDFKQLYQTLTDYDIRFYMYEILKALDYCHSMGIMHRDVKPHNVMIDHEHRKLRLIDWGLAEFYHPGQEYNVRVASRYFKGPELLVDYQMYDYSLDMWSLGCMLASMIFRKEPFFHGHDNYDQLVRIAKVLGTEDLYDYIDKYNIELDPRFNDILGRHSRKRWERFVHSENQHLVSPEALDFLDKLLRYDHQSRLTAREAMEHPYFYTVVK
2DNE , Knot 55 108 0.79 38 87 101 
GSSGSSGQKVPLPSLSPTMQAGTIARWEKKEGDKINEGDLIAEVETDKATVGFESLEECYMAKILVAEGTRDVPIGAIICITVGKPEDIEAFKNYTLDSSAASGPSSG

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 \(c\).

\(|P_{f(3CNW_1)}(2) \setminus P_{f(6YPG_1)}(2)|=53\), \(|P_{f(6YPG_1)}(2) \setminus P_{f(3CNW_1)}(2)|=126\). 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 1:1011000001011101001101111100110111011000100110100110101001100101100000000001101111100010010100100000011010101011010000110110110001101100111010000000
Pair \(Z_2\) Length of longest common subsequence
3CNW_1,6YPG_1 179 4
3CNW_1,2DNE_1 134 3
6YPG_1,2DNE_1 173 3

Newick tree

 
[
	6YPG_1:93.97,
	[
		3CNW_1:67,2DNE_1:67
	]:26.97
]

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 \((0.85)(0.8)(\frac{476 }{\log_{20} 476}-\frac{148}{\log_{20}148})=96.9\)
Status Protein1 Protein2 d d1/2
Query variables 3CNW_1 6YPG_1 128 90.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]