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Parikh vectors
8AEM_1 6PDV_1 4UZD_1 Letter Amino acid
15 13 20 A Alanine
19 9 13 I Isoleucine
26 17 14 V Valine
26 19 10 D Aspartic acid
9 11 10 Q Glutamine
14 22 15 G Glycine
30 26 35 L Leucine
9 2 4 M Methionine
14 17 14 P Proline
15 18 17 S Serine
5 7 4 W Tryptophan
23 8 12 Y Tyrosine
24 7 21 R Arginine
15 10 9 N Asparagine
2 1 4 C Cysteine
16 12 16 H Histidine
20 24 19 K Lycine
11 12 13 T Threonine
22 13 25 E Glutamic acid
13 12 12 F Phenylalanine 

8AEM_1|Chain A|Casein kinase II subunit alpha|Homo sapiens (9606)
>6PDV_1|Chain A|Carbonic anhydrase 2|Homo sapiens (9606)
>4UZD_1|Chains A, B|AURORA KINASE A|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 \(w=f(c)\)
8AEM , Knot 147 328 0.86 40 194 315 
SGPVPSRARVYTDVNTHRPSEYWDYESHVVEWGNQDDYQLVRKLGRGKYSEVFEAINITNNEKVVVKILKPVAAAKIKREIKILENLRGGPNIITLADIVKDPVSRTPALVFEHVNNTDFKQLYQTLTDYDIRFYMYEILKALDYCHSMGIMHRDVKPHNVMIDHEHRKLRLIDWGLAEFYHPGQEYNVRVASRYFKGPELLVDYQMYDYSLDMWSLGCMLASMIFRKEPFFHGHDNYDQLVRIAKVLGTEDLYDYIDKYNIELDPRFNDILGRHSRKRWERFVHSENQHLVSPEALDFLDKLLRYDHQSRLTAREAMEHPYFYTVVK
6PDV , Knot 112 260 0.79 40 176 249 
MSHHWGYGKHNGPEHWHKDFPIAKGERQSPVDIDTHTAKYDPSLKPLSVSYDQATSLRILNNGHAFNVEFDDSQDKAVLKGGPLDGTYRLIQFHFHWGSLDGQGSEHTVDKKKYAAELHLVHWNTKYGDFGKAVQQPDGLAVLGIFLKVGSAKPGLQKVVDVLDSIKTKGKSADFTNFDPRGLLPESLDYWTYPGSLTTPPLLECVTWIVLKEPISVSSEQVLKFRKLNFNGEGEPEELMVDNWRPAQPLKNRQIKASFK
4UZD , Knot 129 287 0.84 40 178 274 
MKRQWALEDFEIGRPLGKGKFGNVYLAREKQSKFILALKVLFKAQLEKAGVEHQLRREVEIQSHLRHPNILRLYGYFHDATRVYLILEYAPLGTVYRELQKLSKFDEQRTATYITELANALSYCHSKRVIHRDIKPENLLLGSAGELKIADFGWSVHAPSSRRTTLCGTLDYLPPEMIEGRMHDEKVDLWSLGVLCYEFLVGKPPFEANTYQETYKRISRVEFTFPDFVTEGARDLISRLLKHNPSQRPMLREVLEHPWITANSSKPSNCQNKESAAAALEHHHHHH

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 \(c\).

\(|P_{f(8AEM_1)}(2) \setminus P_{f(6PDV_1)}(2)|=101\), \(|P_{f(6PDV_1)}(2) \setminus P_{f(8AEM_1)}(2)|=83\). 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 1:0111100101000100001000100000110110000001100110100001101101000001110110111110100010110010111011011011001100011111001000010010001000010101001101100000111100010100111000000101101111010011000010110001011011100010000101101101110111000111010000001101101110001000100001010101001110000001001100000011010110110011000000010100110010100110
Pair \(Z_2\) Length of longest common subsequence
8AEM_1,6PDV_1 184 3
8AEM_1,4UZD_1 172 4
6PDV_1,4UZD_1 178 3

Newick tree

 
[
	6PDV_1:91.96,
	[
		8AEM_1:86,4UZD_1:86
	]:5.96
]

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 \((0.85)(0.8)(\frac{588 }{\log_{20} 588}-\frac{260}{\log_{20}260})=92.5\)
Status Protein1 Protein2 d d1/2
Query variables 8AEM_1 6PDV_1 121 106.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]