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Parikh vectors
7NML_1 6YUL_1 6OSW_1 Letter Amino acid
2 16 1 H Histidine
10 13 5 F Phenylalanine
9 27 5 D Aspartic acid
6 2 0 C Cysteine
7 22 6 E Glutamic acid
7 24 8 P Proline
5 25 8 S Serine
1 5 0 W Tryptophan
13 22 6 A Alanine
13 17 2 N Asparagine
8 23 7 K Lycine
3 12 4 Q Glutamine
1 15 3 M Methionine
4 13 5 T Threonine
12 33 7 L Leucine
2 23 1 Y Tyrosine
10 30 6 V Valine
5 26 8 R Arginine
11 22 6 G Glycine
4 21 6 I Isoleucine 

7NML_1|Chains A, B|Galectin-1|Homo sapiens (9606)
>6YUL_1|Chains A[auth AAA], B[auth GGG]|Casein kinase II subunit alpha|Homo sapiens (9606)
>6OSW_1|Chain A|Forkhead box M1|Danio rerio (7955)
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 \(w=f(c)\)
7NML , Knot 67 133 0.82 40 104 129 
CGLVASNLNLKPGECLRVRGEVAPDAKSFVLNLGKDSNNLCLHFNPRFNAHGDANTIVCNSKDGGAWGTEQREAVFPFQPGSVAEVCITFDQANLTVKLPDGYEFKFPNRLNLEAINYMAADGDFKIKCVAFD
6YUL , Knot 170 391 0.86 40 217 371 
MSGPVPSRARVYTDVNTHRPREYWDYESHVVEWGNQDDYQLVRKLGRGKYSEVFEAINITNNEKVVVKILKPVKKKKIKREIKILENLRGGPNIITLADIVKDPVSRTPALVFEHVNNTDFKQLYQTLTDYDIRFYMYEILKALDYCHSMGIMHRDVKPHNVMIDHEHRKLRLIDWGLAEFYHPGQEYNVRVASRYFKGPELLVDYQMYDYSLDMWSLGCMLASMIFRKEPFFHGHDNYDQLVRIAKVLGTEDLYDYIDKYNIELDPRFNDILGRHSRKRWERFVHSENQHLVSPEALDFLDKLLRYDHQSRLTAREAMEHPYFYTVVKDQARMGSSSMPGGSTPVSSANMMSGISSVPTPSPLGPLAGSPVIAAANPLGMPVPAAAGAQQ
6OSW , Knot 49 94 0.79 36 81 92 
GEFMRESPRRPIILKRRKLPFAKSTARSFPDGIRVMDHPTMPDTQVVVIPKSADLQSVISVLTAKGKEAGPQGRNKFILLSGDTSAEEENLYFQ

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 \(c\).

\(|P_{f(7NML_1)}(2) \setminus P_{f(6YUL_1)}(2)|=42\), \(|P_{f(6YUL_1)}(2) \setminus P_{f(7NML_1)}(2)|=155\). 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 1:0111100101011001010101110100111011000001010101010101010011000001111100000111110110110101010010101011010010110010101100111010101001110
Pair \(Z_2\) Length of longest common subsequence
7NML_1,6YUL_1 197 4
7NML_1,6OSW_1 133 3
6YUL_1,6OSW_1 200 3

Newick tree

 
[
	6YUL_1:10.98,
	[
		7NML_1:66.5,6OSW_1:66.5
	]:41.48
]

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 \((0.85)(0.8)(\frac{524 }{\log_{20} 524}-\frac{133}{\log_{20}133})=115.\)
Status Protein1 Protein2 d d1/2
Query variables 7NML_1 6YUL_1 149 97.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]