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Parikh vectors
1PZT_1 7QEW_1 8GTV_1 Letter Amino acid
16 7 17 D Aspartic acid
11 11 9 E Glutamic acid
25 18 29 L Leucine
15 17 11 K Lycine
17 22 27 V Valine
12 11 17 A Alanine
17 6 21 N Asparagine
21 11 26 G Glycine
18 20 11 I Isoleucine
14 16 17 F Phenylalanine
4 9 12 C Cysteine
11 7 10 M Methionine
18 7 13 P Proline
11 14 24 T Threonine
16 9 11 Y Tyrosine
17 10 11 R Arginine
15 6 14 Q Glutamine
7 5 7 H Histidine
19 13 16 S Serine
2 7 3 W Tryptophan 

1PZT_1|Chain A|Beta-1,4-galactosyltransferase 1|Bos taurus (9913)
>7QEW_1|Chains A[auth G], B[auth H], C[auth I], D[auth J], E[auth K], F[auth L]|Gap junction beta-2 protein|Homo sapiens (9606)
>8GTV_1|Chains A, B|3C-like proteinase|Severe acute respiratory syndrome coronavirus 2 (2697049)
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 \(w=f(c)\)
1PZT , Knot 130 286 0.85 40 202 282 
ASMTGGQQMGRGSSLTACPEESPLLVGPMLIEFNIPVDLKLVEQQNPKVKLGGRYTPMDCISPHKVAIIIPFRNRQEHLKYWLYYLHPILQRQQLDYGIYVINQAGESMFNRAKLLNVGFKEALKDYDYNCFVFSDVDLIPMNDHNTYRCFSQPRHISVAMDKFGFSLPYVQYFGGVSALSKQQFLSINGFPNNYWGAGGEDDDIYNRLAFRGMSVSRPNAVIGKTRMIRHSRDKKNEPNPQRFDRIAHTKETMLSDGLNSLTYMVLEVQRYPLYTKITVDIGTPS
7QEW , Knot 105 226 0.84 40 168 220 
MDWGTLQTILGGVNKHSTSIGKIWLTVLFIFRIMILVVAAKEVWGDEQADFVCNTLQPGCKNVCYDHYFPISHIRLWALQLIFVSTPALLVAMHVAYRRHEKKRKFIKGEIKSEFKDIEEIKTQKVRIEGSLWWTYTSSIFFRVIFEAAFMYVFYVMYDGFSMQRLVKCNAWPCPNTVDCFVSRPTEKTVFTVFMIAVSGICILLNVTELCYLLIRYCSGKSKKPV
8GTV , Knot 134 306 0.83 40 196 290 
SGFRKMAFPSGKVEGCMVQVTCGTTTLNGLWLDDVVYCPRHVICTSEDMLNPNYEDLLIRKSNHNFLVQAGNVQLRVIGHSMQNCVLKLKVDTANPKTPKYKFVRIQPGQTFSVLACYNGSPSGVYQCAMRPNFTIKGSFLNGSCGSVGFNIDYDCVSFCYMHHMELPTGVHAGTDLEGNFYGPFVDRQTAQAAGTDTTITVNVLAWLYAAVINGDRWFLNRFTTTLNDFNLVAMKYNYEPLTQDHVDILGPLSAQTGIAVLDMCASLKELLQNGMNGRTILGSALLEDEFTPFDVVRQCSGVTFQ

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 \(c\).

\(|P_{f(1PZT_1)}(2) \setminus P_{f(7QEW_1)}(2)|=108\), \(|P_{f(7QEW_1)}(2) \setminus P_{f(1PZT_1)}(2)|=74\). 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 1:1010110011010010101000111111111010111010110000101011100011001010011111110000001001100101110000100110110011001100101101110011000000011100101111000000001001001011100111011010011110110000110101110001111100001000111011010010111100011000000000101001001100000110011001001110100011000101011010
Pair \(Z_2\) Length of longest common subsequence
1PZT_1,7QEW_1 182 4
1PZT_1,8GTV_1 172 3
7QEW_1,8GTV_1 184 4

Newick tree

 
[
	7QEW_1:93.26,
	[
		1PZT_1:86,8GTV_1:86
	]:7.26
]

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 \((0.85)(0.8)(\frac{512 }{\log_{20} 512}-\frac{226}{\log_{20}226})=82.2\)
Status Protein1 Protein2 d d1/2
Query variables 1PZT_1 7QEW_1 106 94.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]