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Parikh vectors
8XXZ_1 5REH_1 2QYA_1 Letter Amino acid
8 11 6 R Arginine
7 12 2 C Cysteine
6 14 2 Q Glutamine
14 17 7 D Aspartic acid
15 9 14 E Glutamic acid
15 17 2 F Phenylalanine
15 24 4 T Threonine
16 17 10 A Alanine
21 11 6 I Isoleucine
19 29 15 L Leucine
19 11 6 K Lycine
1 3 0 W Tryptophan
7 11 4 Y Tyrosine
15 21 2 N Asparagine
18 26 11 G Glycine
12 7 9 H Histidine
5 10 3 M Methionine
4 13 5 P Proline
15 16 5 S Serine
8 27 11 V Valine 

8XXZ_1|Chain A|Guanine nucleotide-binding protein G(o) subunit alpha|Homo sapiens (9606)
>5REH_1|Chain A|3C-like proteinase|Severe acute respiratory syndrome coronavirus 2 (2697049)
>2QYA_1|Chains A, B, C, D|Uncharacterized conserved protein|Methanopyrus kandleri AV19 (190192)
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 \(w=f(c)\)
8XXZ , Knot 106 240 0.80 40 162 229 
MGHHHHHHENLYFQGTLSAEERAALERSKAIEKNLKEDGISAAKDVKLLLLGADNSGKSTIVKQMKIIHGGSGGSGGTTGIVETHFTFKNLHFRLFDVGGQRSERKKWIHCFEDVTAIIFCVDLSDYNRMHESLMLFDSICNNKFFIDTSIILFLNKKDLFGEKIKKSPLTICFPEYTGPNTYEDAAAYIQAQFESKNRSPNKEIYCHMTCATDTNNAQVIFDAVTDIIIANNLRGCGLY
5REH , Knot 134 306 0.83 40 196 290 
SGFRKMAFPSGKVEGCMVQVTCGTTTLNGLWLDDVVYCPRHVICTSEDMLNPNYEDLLIRKSNHNFLVQAGNVQLRVIGHSMQNCVLKLKVDTANPKTPKYKFVRIQPGQTFSVLACYNGSPSGVYQCAMRPNFTIKGSFLNGSCGSVGFNIDYDCVSFCYMHHMELPTGVHAGTDLEGNFYGPFVDRQTAQAAGTDTTITVNVLAWLYAAVINGDRWFLNRFTTTLNDFNLVAMKYNYEPLTQDHVDILGPLSAQTGIAVLDMCASLKELLQNGMNGRTILGSALLEDEFTPFDVVRQCSGVTFQ
2QYA , Knot 62 124 0.80 38 93 115 
MSLARVLLNIHGTGDTVVLALCDEDLLGVELKYKGRTLHISEPFYSGKSMEPDRAAKKIREAVQEYEDEKTVAINALGELACSVVVDAGLAREDEIGELGGVPHVQMYILPREPFLEGHHHHHH

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 \(c\).

\(|P_{f(8XXZ_1)}(2) \setminus P_{f(5REH_1)}(2)|=79\), \(|P_{f(5REH_1)}(2) \setminus P_{f(8XXZ_1)}(2)|=113\). 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 1:110000000010101010100011100001100010001101100101111110001000110010110110110110011100010100101011011100000001100100101111010100000100011110010000111000111110000111001000110101100011000001110101010000001000100010010000010111011001111001010110
Pair \(Z_2\) Length of longest common subsequence
8XXZ_1,5REH_1 192 3
8XXZ_1,2QYA_1 165 7
5REH_1,2QYA_1 165 4

Newick tree

 
[
	5REH_1:91.72,
	[
		8XXZ_1:82.5,2QYA_1:82.5
	]:9.22
]

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 \((0.85)(0.8)(\frac{546 }{\log_{20} 546}-\frac{240}{\log_{20}240})=87.2\)
Status Protein1 Protein2 d d1/2
Query variables 8XXZ_1 5REH_1 112 99.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]