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Parikh vectors
5XKO_1 6BSK_1 1EIX_1 Letter Amino acid
4 14 11 F Phenylalanine
4 7 3 Y Tyrosine
13 21 15 R Arginine
12 19 18 D Aspartic acid
9 24 9 E Glutamic acid
3 8 9 K Lycine
5 18 15 S Serine
3 6 5 N Asparagine
2 11 13 Q Glutamine
4 18 8 I Isoleucine
13 31 23 L Leucine
4 7 1 W Tryptophan
12 22 24 V Valine
25 9 33 A Alanine
13 20 16 G Glycine
6 11 4 H Histidine
12 7 14 T Threonine
3 5 3 C Cysteine
2 3 9 M Methionine
9 15 12 P Proline 

5XKO_1|Chains A, B|Cytidine/deoxycytidylate deaminase, zinc-binding region|Mycobacterium smegmatis (strain ATCC 700084 / mc(2)155) (246196)
>6BSK_1|Chain A|Serine/threonine-protein kinase pim-1|Homo sapiens (9606)
>1EIX_1|Chains A, B, C, D|OROTIDINE 5'-MONOPHOSPHATE DECARBOXYLASE|Escherichia coli (562)
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 \(w=f(c)\)
5XKO , Knot 77 158 0.82 40 122 155 
AISDADLKYLRRCVDLAREALDDGDEPFGSVLVDHTGTTLFEDRNRVKDGDATAHPEFAIARWAARHLTPDRRARATVYTSGEHCPMCAAAHAWVGLGRIVYATSSAQLGGWLTEWGAQAPPVATLPINTVAPGVVVDGPAEELAETMHNLYRAKFGR
6BSK , Knot 122 276 0.82 40 176 259 
EKEPLESQYQVGPLLGSGGFGSVYSGIRVSDNLPVAIKHVEKDRISDWGELPNGTRVPMEVVLLKKVSSGFSGVIRLLDWFERPDSFVLILERPEPVQDLFDFITERGALQEELARSFFWQVLEAVRHCHNCGVLHRDIKDENILIDLNRGELKLIDFGSGALLKDTVYTDFDGTRVYSPPEWIRYHRYHGRSAAVWSLGILLYDMVCGDIPFEHDEEIIRGQVFFRQRVSSECQHLIRWCLALRPSDRPTFEEIQNHPWMQDVLLPQETAEIHLH
1EIX , Knot 111 245 0.83 40 158 237 
MTLTASSSSRAVTNSPVVVALDYHNRDDALAFVDKIDPRDCRLKVGKEMFTLFGPQFVRELQQRGFDIFLDLKFHDIPNTAAHAVAAAADLGVWMVNVHASGGARMMTAAREALVPFGKDAPLLIAVTVLTSMEASDLVDLGMTLSPADYAERLAALTQKCGLDGVVCSAQEAVRFKQVFGQEFKLVTPGIRPQGSEAGDQRRIMTPEQALSAGVDYMVIGRPVTQSVDPAQTLKAINASLQRSA

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 \(c\).

\(|P_{f(5XKO_1)}(2) \setminus P_{f(6BSK_1)}(2)|=54\), \(|P_{f(6BSK_1)}(2) \setminus P_{f(5XKO_1)}(2)|=108\). 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 1:11001010010001011001100100111011100010011000001001010101011110111001010001010100010001101110111111011010001011111001110111110111001111111011100110010010010110
Pair \(Z_2\) Length of longest common subsequence
5XKO_1,6BSK_1 162 4
5XKO_1,1EIX_1 148 5
6BSK_1,1EIX_1 168 4

Newick tree

 
[
	6BSK_1:85.16,
	[
		5XKO_1:74,1EIX_1:74
	]:11.16
]

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 \((0.85)(0.8)(\frac{434 }{\log_{20} 434}-\frac{158}{\log_{20}158})=82.0\)
Status Protein1 Protein2 d d1/2
Query variables 5XKO_1 6BSK_1 101 78.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]