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Parikh vectors
2BJK_1 6XAM_1 7KQZ_1 Letter Amino acid
39 48 47 L Leucine
26 4 39 K Lycine
22 23 45 T Threonine
56 35 39 E Glutamic acid
22 8 40 I Isoleucine
5 15 34 H Histidine
62 49 63 A Alanine
14 3 20 N Asparagine
47 31 59 G Glycine
6 6 14 M Methionine
8 5 15 W Tryptophan
20 4 25 Y Tyrosine
54 30 60 V Valine
1 6 7 C Cysteine
10 9 14 Q Glutamine
27 17 29 F Phenylalanine
29 27 40 P Proline
16 24 35 S Serine
37 36 34 R Arginine
15 21 47 D Aspartic acid 

2BJK_1|Chains A, B|1-PYRROLINE-5-CARBOXYLATE DEHYDROGENASE|THERMUS THERMOPHILUS (300852)
>6XAM_1|Chain A|NzeB|Streptomyces sp. NRRL F-5053 (1463854)
>7KQZ_1|Chain A|Acetyl-coenzyme A synthetase|Coccidioides immitis (strain RS) (246410)
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 \(w=f(c)\)
2BJK , Knot 201 516 0.81 40 220 465 
MTVEPFRNEPIETFQTEEARRAMREALRRVREEFGRHYPLYIGGEWVDTKERMVSLNPSAPSEVVGTTAKAGKAEAEAALEAAWKAFKTWKDWPQEDRSRLLLKAAALMRRRKRELEATLVYEVGKNWVEASADVAEAIDFIEYYARAALRYRYPAVEVVPYPGEDNESFYVPLGAGVVIAPWNFPVAIFTGMIVGPVAVGNTVIAKPAEDAVVVGAKVFEIFHEAGFPPGVVNFLPGVGEEVGAYLVEHPRIRFINFTGSLEVGLKIYEAAGRLAPGQTWFKRAYVETGGKDAIIVDETADFDLAAEGVVVSAYGFQGQKCSAASRLILTQGAYEPVLERVLKRAERLSVGPAEENPDLGPVVSAEQERKVLSYIEIGKNEGQLVLGGKRLEGEGYFIAPTVFTEVPPKARIAQEEIFGPVLSVIRVKDFAEALEVANDTPYGLTGGVYSRKREHLEWARREFHVGNLYFNRKITGALVGVQPFGGFKLSGTNAKTGALDYLRLFLEMKAVAERF
6XAM , Knot 162 401 0.80 40 192 372 
VTTTATLTYPFHDWSQELSPRYAQLRASDAPVCPVVSEGTGDPLWLVTRYATAVKLLEDSRFSSEAAQASGAPRQEPVELRAPGTRGDAIAMLREAGLRSVLADGLGPRAVRRHQGWINDLAETLMSELASREGTFDLAADFVEPLSSALVSRTLLGELSADERDLLAHCADTGLRFCGVTHEEQVHAFTQMHEFFLEHARRLAGTPGEHLLKLIAEAPVDQGPLSDEALAEAGSLLVVAGFPTSSGFLCGALLTLLRHPDAVQELHAHPERVPSAVEELLRYTPLSTGSVKRMATEDLEIDGVRIKAGEVVMVSLEAVNHDPDAFEDPDVFRPGREGPMHFGFGRGRHFCPGNRLARCVIEATVRAVARRPGLRLAVAPEEISWHEGLFFRRPRAIPATW
7KQZ , Knot 279 706 0.86 40 293 654 
MHHHHHHHHENLYFQGSETPAEPKLPVVVEAHQVDTFDVPGVFYENHPHEPHLSGMNEYNQLYQQSINDPDTFWARMARDLITFEKDFDKTHIGTLEGGDNAWFVGGRLNASFNCVDRHAMRDPNKVAIIYEADEPGHGRSITYAELLKEVSRLAWVMKSQGVRKGDTVAIYLPMIPEAIFALLACARIGAIHSVVFAGFSSDSLRDRTLDARSKFIITTDEGKRGGKVIGTKKIVDEALKQCPDVTNCLVFKRTGADVPWTKGRDLWWHEEVDKYPNYLPAESMDSEDPLFLLYTSGSTGKPKGVMHTTAGYLVGAAATGKYVFDIHPADRFFCGGDVGWITGHTYVVYAPLLLGCTTVVFESTPAYPNFSRYWDVIEKHKVTQFYVAPTALRLLKRAGDHHINHEMKDLRILGSVGEPIAAEVWKWYHEVVGKRQAHIVDTYWQTETGSHVITPLGGITPTKPGSASLPFFGIDPVILDPVTGAEIPGNDVEGILAFRKPWPSMARTVWGDHKRYMDTYLNVYKGFYFTGDGAGRDHEGYYWIRGRVDDVVNVSGHRLSTAEIEAALIEHHCVAEAAVVGVPDPLTGQAVHAFVALKSGNDNREQLQKELIMQVRKSIGPFAAPKVVFVIDDLPKTRSGKIMRRILRKILSGEEDSLGDISTLSDPSVVNKIIDTFHEWKKAMAAAAAAAAAVSATAPPNSTTG

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 \(c\).

\(|P_{f(2BJK_1)}(2) \setminus P_{f(6XAM_1)}(2)|=89\), \(|P_{f(6XAM_1)}(2) \setminus P_{f(2BJK_1)}(2)|=61\). 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 1:101011000110010000100110011001000110001101110110000011010101100111001011010101110111011001001100000011101111100000010101100110011010101101101100010111000011101110110000010111111111111011111101111111111001110110011111101101100111111110111111001110110010101101010101110100111011110011001010011001111000101011101111010110100001100111001100111001100100101111000101111101000001100101100010111110010101011110110011101011000111111011010011011011000101101110000000101100010110101000101111110111110101001001110010111010111001
Pair \(Z_2\) Length of longest common subsequence
2BJK_1,6XAM_1 150 5
2BJK_1,7KQZ_1 143 4
6XAM_1,7KQZ_1 161 4

Newick tree

 
[
	6XAM_1:79.78,
	[
		2BJK_1:71.5,7KQZ_1:71.5
	]:8.28
]

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 \((0.85)(0.8)(\frac{917 }{\log_{20} 917}-\frac{401}{\log_{20}401})=137.\)
Status Protein1 Protein2 d d1/2
Query variables 2BJK_1 6XAM_1 169 149.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]