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Parikh vectors
7DZX_1 4ICN_1 7UYM_1 Letter Amino acid
91 24 24 V Valine
79 37 13 A Alanine
72 23 5 I Isoleucine
102 30 15 L Leucine
59 14 11 P Proline
96 15 34 S Serine
95 22 18 T Threonine
10 2 6 W Tryptophan
60 17 8 D Aspartic acid
64 8 8 Q Glutamine
24 6 3 H Histidine
57 15 12 K Lycine
90 27 21 G Glycine
75 7 8 F Phenylalanine
42 7 6 R Arginine
89 13 10 N Asparagine
30 3 5 C Cysteine
49 16 7 E Glutamic acid
10 7 2 M Methionine
55 7 10 Y Tyrosine 

7DZX_1|Chains A, B, C|Spike glycoprotein|Severe acute respiratory syndrome coronavirus 2 (2697049)
>4ICN_1|Chains A, B|DIHYDRODIPICOLINATE SYNTHASE|SHEWANELLA BENTHICA (43661)
>7UYM_1|Chain A[auth H]|850 Fab Heavy Chain|Mus musculus (10090)
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 \(w=f(c)\)
7DZX , Knot 444 1249 0.84 40 325 1080 
QCVNLTTRTQLPPAYTNSFTRGVYYPDKVFRSSVLHSTQDLFLPFFSNVTWFHAIHVSGTNGTKRFDNPVLPFNDGVYFASTEKSNIIRGWIFGTTLDSKTQSLLIVNNATNVVIKVCEFQFCNDPFLGVYYHKNNKSWMESEFRVYSSANNCTFEYVSQPFLMDLEGKQGNFKNLREFVFKNIDGYFKIYSKHTPINLVRDLPQGFSALEPLVDLPIGINITRFQTLLALHRSYLTPGDSSSGWTAGAAAYYVGYLQPRTFLLKYNENGTITDAVDCALDPLSETKCTLKSFTVEKGIYQTSNFRVQPTESIVRFPNITNLCPFGEVFNATRFASVYAWNRKRISNCVADYSVLYNSASFSTFKCYGVSPTKLNDLCFTNVYADSFVIRGDEVRQIAPGQTGKIADYNYKLPDDFTGCVIAWNSNNLDSKVGGNYNYLYRLFRKSNLKPFERDISTEIYQAGSTPCNGVEGFNCYFPLQSYGFQPTNGVGYQPYRVVVLSFELLHAPATVCGPKKSTNLVKNKCVNFNFNGLTGTGVLTESNKKFLPFQQFGRDIADTTDAVRDPQTLEILDITPCSFGGVSVITPGTNTSNQVAVLYQGVNCTEVPVAIHADQLTPTWRVYSTGSNVFQTRAGCLIGAEHVNNSYECDIPIGAGICASYQTQTNSPGSAGSVASQSIIAYTMSLGAENSVAYSNNSIAIPTNFTISVTTEILPVSMTKTSVDCTMYICGDSTECSNLLLQYGSFCTQLNRALTGIAVEQDKNTQEVFAQVKQIYKTPPIKDFGGFNFSQILPDPSKPSKRSFIEDLLFNKVTLADAGFIKQYGDCLGDIAARDLICAQKFNGLTVLPPLLTDEMIAQYTSALLAGTITSGWTFGAGAALQIPFAMQMAYRFNGIGVTQNVLYENQKLIANQFNSAIGKIQDSLSSTASALGKLQDVVNQNAQALNTLVKQLSSNFGAISSVLNDILSRLDPPEAEVQIDRLITGRLQSLQTYVTQQLIRAAEIRASANLAATKMSECVLGQSKRVDFCGKGYHLMSFPQSAPHGVVFLHVTYVPAQEKNFTTAPAICHDGKAHFPREGVFVSNGTHWFVTQRNFYEPQIITTDNTFVSGNCDVVIGIVNNTVYDPLQPELDSFKEELDKYFKNHTSPDVDLGDISGINASVVNIQKEIDRLNEVAKNLNESLIDLQELGKYEQYIKGSGRENLYFQGGGGSGYIPEAPRDGQAYVRKDGEWVLLSTFLGHHHHHHHH
4ICN , Knot 130 300 0.82 40 175 287 
ESWGSHMINGSIVALITPLNSDGTVDYTSLEKLVEYHITEGTDAIVAVGTTGESATLPISEHIAVVGQTVKFASGRIPVIGGNGANATAEAIELTKAQNKLGVAAMLGVTPYYNKPSPKGLIAHYTAVAASTDIPQILYNVPGRTAVDMLPETIAQLVEVPNIIGVKDATGDVARVKQLRDLCGNDFLLYSGDDATAREFLTLGGDGVISVANNIVPKLFKLMCDAALAGDTQAAMAAEDQIKGLFSALFCEANPIPVKWAAHKMGLISQGDIRLPLTELSTEFHGLLLDAMKNARIEVK
7UYM , Knot 102 226 0.81 40 145 216 
QVQLVESGGGVVQPGRSLRLSCAASGFTFSNFGMHWIRQSPGKGLEWVAIIWYDGSNTYYADSVKGRFTISRDNSKNTLYLQMNSLRAEDTAVYYCAKVWFGESEDNYSVDVWGQGTTVTVSSASTKGPSVFPLAPSSKSTSGGTAALGCLVKDYFPEPVTVSWNSGALTSGVHTFPAVLQSSGLYSLSSVVTVPSSSLGTQTYICNVNHKPSNTKVDKKVEPKSC

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 \(c\).

\(|P_{f(7DZX_1)}(2) \setminus P_{f(4ICN_1)}(2)|=163\), \(|P_{f(4ICN_1)}(2) \setminus P_{f(7DZX_1)}(2)|=13\). 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 1:0010100000111100001001100100110001100000111111001011011010100100010011111001101100000011011111001000000111100100111010010100011111000000001100010100010000100100111101010010100100111001010101000001101100110110110111011111010010011110000101100001101111100110101001110000010100110011011000000100101001100000101010001101101001011101101001101011000010001100011000101001000110100100101001010011101001001111001011000001100101011110000100011100001001100001011000100010011001001101100011100011010011100100111101011011101011000001100001010101101011100000011110011001100001100100101101010011110110110000001111001100001111101001010101000100110001101111001000000011111110100000000110110110001110010111000110000011110010101000111101000010001010100000001110010100010011011110000000011101001000111001111010011101001000011001110010110111100010011011100110100101101111110001110000111110100110111111101111101100101111000110000011100100111010001000101110100110001011001100100011110011001100101101010100110101001000100011011010101011100100011100001010101001101100110111110100111000010011110001010110011110010011100001001011000001101000111111000100110101001000100010000010101101011010110100010010011001000110100110000010101000101011110101101100101010001011110011100000000
Pair \(Z_2\) Length of longest common subsequence
7DZX_1,4ICN_1 176 4
7DZX_1,7UYM_1 200 4
4ICN_1,7UYM_1 166 4

Newick tree

 
[
	7DZX_1:97.63,
	[
		4ICN_1:83,7UYM_1:83
	]:14.63
]

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 \((0.85)(0.8)(\frac{1549 }{\log_{20} 1549}-\frac{300}{\log_{20}300})=322.\)
Status Protein1 Protein2 d d1/2
Query variables 7DZX_1 4ICN_1 408 252.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]