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Parikh vectors
3RHW_1 2OTA_1 6RVC_1 Letter Amino acid
7 1 1 C Cysteine
37 8 22 L Leucine
13 6 12 K Lycine
18 1 13 F Phenylalanine
16 3 15 P Proline
13 4 13 N Asparagine
8 2 5 M Methionine
31 4 15 S Serine
28 3 12 T Threonine
14 2 13 R Arginine
17 4 21 D Aspartic acid
15 1 12 G Glycine
14 6 11 H Histidine
26 5 16 I Isoleucine
16 5 17 A Alanine
8 6 13 E Glutamic acid
8 0 6 W Tryptophan
17 1 20 Y Tyrosine
28 11 12 V Valine
13 3 12 Q Glutamine 

3RHW_1|Chains A, B, C, D, E|Avermectin-sensitive glutamate-gated chloride channel GluCl alpha|Caenorhabditis elegans (6239)
>2OTA_1|Chains A, B|UPF0352 protein CPS_2611|Colwellia psychrerythraea (167879)
>6RVC_1|Chains A, B, C|Protein patched homolog 1|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 \(w=f(c)\)
3RHW , Knot 151 347 0.84 40 209 330 
SDSKILAHLFTSGYDFRVRPPTDNGGPVVVSVNMLLRTISKIDVVNMEYSAQLTLRESWIDKRLSYGVKGDGQPDFVILTVGHQIWMPDTFFPNEKQAYKHTIDKPNVLIRIHNDGTVLYSVRISLVLSCPMYLQYYPMDVQQCSIDLASYAYTTKDIEYLWKEHSPLQLKVGLSSSLPSFQLTNTSTTYCTSVTNTGIYSCLRTTIQLKREFSFYLLQLYIPSCMLVIVSWVSFWFDRTAIPARVTLGVTTLLTMTAQSAGINSQLPPVSYIKAIDVWIGACMTFIFCALLEFALVNHIANAGTTEWNDISKRVDLISRALFPVLFFVFNILYWSRFGHHHHHHHH
2OTA , Knot 41 76 0.77 38 63 71 
MPIVSKYSNERVEKIIQDLLDVLVKEEVTPDLALMCLGNAVTNIIAQVPESKRVAVVDNFTKALKQSVLEHHHHHH
6RVC , Knot 115 261 0.81 40 175 250 
ETGHHHHHHRDGLDLTDIVPRETREYDFIAAQFKYFSFYNMYIVTQKADYPNIQHLLYDLHRSFSNVKYVMLEENKQLPKMWLHYFRDWLQGLQDAFDSDWETGKIMPNNYKNGSDDGVLAYKLLVQTGSRDKPIDISQLTKQRLVDADGIINPSAFYIYLTAWVSNDPVAYAASQANIRPHRPEWVHDKADYMPETRLRIPAAEPIEYAQFPFYLNGLRDTSDFVEAIEKVRTICSNYTSLGLSSYPNGYPFLFWEQYIG

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 \(c\).

\(|P_{f(3RHW_1)}(2) \setminus P_{f(2OTA_1)}(2)|=166\), \(|P_{f(2OTA_1)}(2) \setminus P_{f(3RHW_1)}(2)|=20\). 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 1:00001110110010010101100011111101011100100101101000101010001100010011010101011110110011110011100001000010010111010001011001010111001101000110100001011001000001001100001101011100011010100000000001000110001000101000101011010110011111011011100011110101110011010100111000111100101101111101011101110111100110110001001000101100111111111101101001100000000
Pair \(Z_2\) Length of longest common subsequence
3RHW_1,2OTA_1 186 6
3RHW_1,6RVC_1 168 7
2OTA_1,6RVC_1 170 6

Newick tree

 
[
	2OTA_1:90.72,
	[
		3RHW_1:84,6RVC_1:84
	]:6.72
]

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 \((0.85)(0.8)(\frac{423 }{\log_{20} 423}-\frac{76}{\log_{20}76})=106.\)
Status Protein1 Protein2 d d1/2
Query variables 3RHW_1 2OTA_1 137 82
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]