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Parikh vectors
8EYL_1 2MXZ_1 7XPV_1 Letter Amino acid
12 8 15 F Phenylalanine
17 4 25 P Proline
11 6 12 Q Glutamine
22 12 29 G Glycine
12 3 23 H Histidine
9 9 11 I Isoleucine
26 9 38 L Leucine
24 14 1 K Lycine
7 3 6 W Tryptophan
17 7 36 V Valine
19 10 33 D Aspartic acid
12 6 17 T Threonine
10 6 13 N Asparagine
13 8 20 E Glutamic acid
13 14 39 A Alanine
7 5 33 R Arginine
1 0 7 C Cysteine
2 1 7 M Methionine
18 8 21 S Serine
8 4 9 Y Tyrosine 

8EYL_1|Chain A|Carbonic anhydrase 2|Homo sapiens (9606)
>2MXZ_1|Chain A|L-alanyl-D-glutamate peptidase|Enterobacteria phage T5 (10726)
>7XPV_1|Chains A, B|Transglycosylse|Marinactinospora thermotolerans (531310)
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 \(w=f(c)\)
8EYL , Knot 112 260 0.79 40 176 249 
MSHHWGYGKHNGPEHWHKDFPIAKGERQSPVDIDTHTAKYDPSLKPLSVSYDQATSLRILNNGHAFNVEFDDSQDKAVLKGGPLDGTYRLIQFHFHWGSLDGQGSEHTVDKKKYAAELHLVHWNTKYGDFGKAVQQPDGLAVLGIFLKVGSAKPGLQKVVDVLDSIKTKGKSADFTNFDPRGLLPESLDYWTYPGSLTTPPLLECVTWIVLKEPISVSSEQVLKFRKLNFNGEGEPEELMVDNWRPAQPLKNRQIKASFK
2MXZ , Knot 68 137 0.81 38 111 134 
MSFKFGKNSEKQLATVKPELQKVARRALELSPYDFTIVQGIRTVAQSAQNIANGTSFLKDPSKSKHITGDAIDFAPYINGKIDWNDLEAFWAVKKAFEQAGKELGIKLRFGADWNASGDYHDEIKRGTYDGGHVELV
7XPV , Knot 163 395 0.82 40 202 367 
GSSHHHHHHSSGENLYFQHMSGRDISTAVVVTTISDGGFLDRLAPALRDAGARLIVIPDRNTGPALFAACERHRRLGLDVVCPSVAEQQDLLERLAVPDLIPYHSDNRRNVGYLMAWMEGFDVIVSMDDDNLPTTDDFVERHQVVCQGPRTQPVTASSDGWFNNCALLEVEPTEVFPRGFPFHARPAHAQARTSVCERPADVRINAGLWLGDPDVDAITRLAVRPNALAHSGGSVVLAEGTWCPVNAQNTAVHRDALPAYYFLRMGQPVDGVPMERFGDIFSGYFVQVCAQHLGHAVRFGDPVVEHPRNEHDLLDDLHKEVPAVRLLDDILDHLRDHPLEGGDYLETYESLSYALQEIAERVNGRAWSPDARAFLHRSAHLMRSWTGALRTVAGT

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 \(c\).

\(|P_{f(8EYL_1)}(2) \setminus P_{f(2MXZ_1)}(2)|=111\), \(|P_{f(2MXZ_1)}(2) \setminus P_{f(8EYL_1)}(2)|=46\). 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 1:10001101000110010001111010000110100001000101011010000100101100101101010000001110111101000110101011010101000010000011010110100001011011001011111111101101011100110110010001001010010101111001001001101001111001011110011010000110100101010101001110010110110000101010
Pair \(Z_2\) Length of longest common subsequence
8EYL_1,2MXZ_1 157 4
8EYL_1,7XPV_1 168 3
2MXZ_1,7XPV_1 187 4

Newick tree

 
[
	7XPV_1:92.07,
	[
		8EYL_1:78.5,2MXZ_1:78.5
	]:13.57
]

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 \((0.85)(0.8)(\frac{397 }{\log_{20} 397}-\frac{137}{\log_{20}137})=78.4\)
Status Protein1 Protein2 d d1/2
Query variables 8EYL_1 2MXZ_1 97 73
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]