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Parikh vectors
5MUA_1 1PKZ_1 9LAE_1 Letter Amino acid
11 5 7 Q Glutamine
4 3 1 H Histidine
15 15 9 I Isoleucine
7 12 13 P Proline
10 1 2 W Tryptophan
20 7 21 N Asparagine
17 12 9 D Aspartic acid
17 10 16 F Phenylalanine
3 1 9 C Cysteine
29 10 15 G Glycine
21 30 14 L Leucine
12 24 12 K Lycine
3 8 0 M Methionine
12 10 15 Y Tyrosine
28 16 12 A Alanine
7 12 11 R Arginine
22 4 13 T Threonine
15 10 20 V Valine
8 20 7 E Glutamic acid
25 12 17 S Serine 

5MUA_1|Chains A, B|Ricin B-related lectin|Polyporus squamosus (2829415)
>1PKZ_1|Chains A, B|Glutathione S-transferase A1|Homo sapiens (9606)
>9LAE_1|Chain A[auth G]|Spike protein S1|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 \(w=f(c)\)
5MUA , Knot 131 286 0.86 40 186 279 
MSFQGHGIYYIASAYVANTRLALSEDSSANKSPDVIISSDAVDPLNNLWLIEPVGEADTYTVRNAFAGSYMDLAGHAATDGTAIIGYRPTGGDNQKWIISQINDVWKIKSKETGTFVTLLNGDGGGTGTVVGWQNITNNTSQNWTFQKLSQTGANVHATLLACPALRQDFKSYLSDGLYLVLTRDQISSIWQASGLGSTPWRSEIFDCDDFATVFKGAVAKWGNENFKANGFALLCGLMFGSKSSGAHAYNWFVERGNFSTVTFFEPQNGTYSANAWDYKAYFGLF
1PKZ , Knot 104 222 0.84 40 153 217 
MAEKPKLHYFNARGRMESTRWLLAAAGVEFEEKFIKSAEDLDKLRNDGYLMFQQVPMVEIDGMKLVQTRAILNYIASKYNLYGKDIKERALIDMYIEGIADLGEMILLLPVCPPEEKDAKLALIKEKIKNRYFPAFEKVLKSHGQDYLVGNKLSRADIHLVELLYYVEELDSSLISSFPLLKALKTRISNLPTVKKFLQPGSPRKPPMDEKSLEEARKIFRF
9LAE , Knot 103 223 0.83 38 153 214 
RVQPTESIVRFPNITNLCPFGEVFNATRFASVYAWNRKRISNCVADYSVLYNSASFSTFKCYGVSPTKLNDLCFTNVYADSFVIRGDEVRQIAPGQTGKIADYNYKLPDDFTGCVIAWNSNNLDSKVGGNYNYLYRLFRKSNLKPFERDISTEIYQAGSTPCNGVEGFNCYFPLQSYGFQPTNGVGYQPYRVVVLSFELLHAPATVCGPKKSTNLVKNKCVNF

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 \(c\).

\(|P_{f(5MUA_1)}(2) \setminus P_{f(1PKZ_1)}(2)|=108\), \(|P_{f(1PKZ_1)}(2) \setminus P_{f(5MUA_1)}(2)|=75\). 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 1:1010101100110101100011100000100010111000110110011110111010000100111100101110110010111100101100001110010011010000010110110101110101111001000000010100100011010101110111000100010011011100001001101011100110001100001101101111011000101011111011111000011010011100101001011010010001011000101111
Pair \(Z_2\) Length of longest common subsequence
5MUA_1,1PKZ_1 183 4
5MUA_1,9LAE_1 159 4
1PKZ_1,9LAE_1 146 4

Newick tree

 
[
	5MUA_1:89.54,
	[
		9LAE_1:73,1PKZ_1:73
	]:16.54
]

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 \((0.85)(0.8)(\frac{508 }{\log_{20} 508}-\frac{222}{\log_{20}222})=82.3\)
Status Protein1 Protein2 d d1/2
Query variables 5MUA_1 1PKZ_1 105 95
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]