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Parikh vectors
8WCI_1 7PXW_1 1TBG_1 Letter Amino acid
20 18 27 G Glycine
1 8 8 H Histidine
15 16 18 I Isoleucine
12 6 11 F Phenylalanine
8 8 12 Q Glutamine
0 2 14 C Cysteine
7 1 8 M Methionine
10 23 17 V Valine
4 17 17 N Asparagine
4 20 5 P Proline
0 6 8 W Tryptophan
3 10 8 Y Tyrosine
2 16 30 D Aspartic acid
0 7 22 R Arginine
4 5 9 E Glutamic acid
19 11 30 L Leucine
6 3 10 K Lycine
8 20 28 S Serine
10 16 28 T Threonine
23 22 30 A Alanine 

8WCI_1|Chains A, B, C, D, E, F, G, H, I, J|V-type sodium ATPase subunit K|Enterococcus hirae ATCC 9790 (768486)
>7PXW_1|Chain A|Auxiliary activity 9|Lentinus similis (292560)
>1TBG_1|Chains A, C[auth B], E[auth C], G[auth D]|TRANSDUCIN|Bos taurus (9913)
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 \(w=f(c)\)
8WCI , Knot 72 156 0.77 34 107 146 
MMDYLITQNGGMVFAVLAMATATIFSGIGSAKGVGMTGEAAAALTTSQPEKFGQALILQLLPGTQGLYGFVIAFLIFINLGSDMSVVQGLNFLGASLPIAFTGLFSGIAQGKVAAAGIQILAKKPEHATKGIIFAAMVETYAILGFVISFLLVLNA
7PXW , Knot 105 235 0.81 40 154 226 
HTLVWGVWVNGVDQGDGRNIYIRSPPNNNPVKNLTSPDMTCNVDNRVVPKSVPVNAGDTLTFEWYHNTRDDDIIASSHHGPIAVYIAPAASNGQGNVWVKLFEDAYNVTNSTWAVDRLITAHGQHSVVVPHVAPGDYLFRAEIIALHEADSLYSQNPIRGAQFYISCAQITINSSDDSTPLPAGVPFPGAYTDSTPGIQFNIYTTPATSYVAPPPSVWSGALGGSIAQVGDASLE
1TBG , Knot 146 340 0.83 40 192 317 
MSELDQLRQEAEQLKNQIRDARKACADATLSQITNNIDPVGRIQMRTRRTLRGHLAKIYAMHWGTDSRLLVSASQDGKLIIWDSYTTNKVHAIPLRSSWVMTCAYAPSGNYVACGGLDNICSIYNLKTREGNVRVSRELAGHTGYLSCCRFLDDNQIVTSSGDTTCALWDIETGQQTTTFTGHTGDVMSLSLAPDTRLFVSGACDASAKLWDVREGMCRQTFTGHESDINAICFFPNGNAFATGSDDATCRLFDLRADQELMTYSHDNIICGITSVSFSKSGRLLLAGYDDFNCNVWDALKADRAGVLAGHDNRVSCLGVTDDGMAVATGSWDSFLKIWN

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 \(c\).

\(|P_{f(8WCI_1)}(2) \setminus P_{f(7PXW_1)}(2)|=56\), \(|P_{f(7PXW_1)}(2) \setminus P_{f(8WCI_1)}(2)|=103\). 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 1:110011000111111111110101101110101111010111110000100110111101111001101111111111011001011011011110111110111011101011111101110010010011111111000111111101111101
Pair \(Z_2\) Length of longest common subsequence
8WCI_1,7PXW_1 159 3
8WCI_1,1TBG_1 175 3
7PXW_1,1TBG_1 154 4

Newick tree

 
[
	8WCI_1:85.68,
	[
		7PXW_1:77,1TBG_1:77
	]:8.68
]

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 \((0.85)(0.8)(\frac{391 }{\log_{20} 391}-\frac{156}{\log_{20}156})=70.5\)
Status Protein1 Protein2 d d1/2
Query variables 8WCI_1 7PXW_1 88 72.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]