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Parikh vectors
5SWU_1 4MQW_1 4KWW_1 Letter Amino acid
12 3 11 F Phenylalanine
17 8 16 S Serine
12 7 19 P Proline
15 3 11 R Arginine
10 4 6 N Asparagine
2 10 6 C Cysteine
8 5 10 Q Glutamine
14 4 16 E Glutamic acid
6 10 7 H Histidine
17 1 6 I Isoleucine
17 7 30 V Valine
24 8 50 A Alanine
27 5 37 L Leucine
8 3 4 M Methionine
11 8 12 T Threonine
3 0 4 W Tryptophan
6 4 3 Y Tyrosine
19 2 12 D Aspartic acid
21 4 29 G Glycine
14 6 12 K Lycine 

5SWU_1|Chains A, B|Pentafunctional AROM polypeptide|Neosartorya fumigata (strain CEA10 / CBS 144.89 / FGSC A1163) (451804)
>4MQW_1|Chains A, D, G|Glycoprotein hormones, alpha polypeptide|Homo sapiens (9606)
>4KWW_1|Chains A, B, C, D, E, F|Nicotinate-nucleotide pyrophosphorylase [carboxylating]|Homo sapiens (9606)
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 \(w=f(c)\)
5SWU , Knot 119 263 0.84 40 170 251 
FIKFVTGLEDSLGTIKKKQHSFFVSLTLPDVRGADQILEQACVGSDAVELRVDLLEDPDSSNGIPTVDFVADQISYLRSRITLPVIFTIRTKGQGGRFPDDAHAEAMQLYRLAVRSGCEFVDLEIAFPDEMLRAVTEMKGYSKIIASHHDPNGELSWANMSWMKYYNRALEYGDVIKLVGVARNLDDNTALRKFKNWAEEAHDVPLIAINMGGNGQLSRILNGFMTPVSHPALPFRAAPGQLSATDIRKGLSLMGEIKKGAAL
4MQW , Knot 50 102 0.75 38 83 95 
APDVQDCPECTLQENPLFSQPGAPILQCMGCCFSRAYPTPLRSKKTMLVQKNVTSESTCCVAKSYNRVTVMGGFKVENHTACHCSTCYYHKSAAAHHHHHHH
4KWW , Knot 129 301 0.81 40 170 282 
GPGSMDAEGLALLLPPVTLAALVDSWLREDCPGLNYAALVSGAGPSQAALWAKSPGVLAGQPFFDAIFTQLNCQVSWFLPEGSKLVPVARVAEVRGPAHCLLLGERVALNTLARCSGIASAAAAAVEAARGAGWTGHVAGTRKTTPGFRLVEKYGLLVGGAASHRYDLGGLVMVKDNHVVAAGGVEKAVRAARQAADFTLKVEVECSSLQEAVQAAEAGADLVLLDNFKPEELHPTATVLKAQFPSVAVEASGGITLDNLPQFCGPHIDVISMGMLTQAAPALDFSLKLFAKEVAPVPKIH

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 \(c\).

\(|P_{f(5SWU_1)}(2) \setminus P_{f(4MQW_1)}(2)|=130\), \(|P_{f(4MQW_1)}(2) \setminus P_{f(5SWU_1)}(2)|=43\). 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 1:11011011000110100000011101011010110011001011001101010110010000111010111001001000101111101000101101100101011010011100100110101111001101100101000111000010101011010110000011001011011111001000011001001100100111111011101010011011101100111110111101010010011011101001111
Pair \(Z_2\) Length of longest common subsequence
5SWU_1,4MQW_1 173 3
5SWU_1,4KWW_1 156 3
4MQW_1,4KWW_1 165 4

Newick tree

 
[
	4MQW_1:86.58,
	[
		5SWU_1:78,4KWW_1:78
	]:8.58
]

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 \((0.85)(0.8)(\frac{365 }{\log_{20} 365}-\frac{102}{\log_{20}102})=81.0\)
Status Protein1 Protein2 d d1/2
Query variables 5SWU_1 4MQW_1 103 71
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]