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Parikh vectors
7QUR_1 4JVT_1 1LHI_1 Letter Amino acid
30 3 8 C Cysteine
58 5 5 K Lycine
11 6 2 M Methionine
11 2 5 W Tryptophan
96 19 9 V Valine
88 5 10 N Asparagine
84 19 12 G Glycine
61 13 1 P Proline
98 12 6 S Serine
54 2 6 Y Tyrosine
45 23 14 R Arginine
63 14 6 Q Glutamine
48 17 3 E Glutamic acid
22 9 1 H Histidine
106 38 8 L Leucine
95 22 5 T Threonine
80 52 14 A Alanine
72 14 5 I Isoleucine
76 6 2 F Phenylalanine
61 16 8 D Aspartic acid 

7QUR_1|Chains A, B, C|Spike glycoprotein,Fibritin|Severe acute respiratory syndrome coronavirus 2 (2697049)
>4JVT_1|Chain A|Enoyl-CoA hydratase|Thermobifida fusca (269800)
>1LHI_1|Chain A|HUMAN LYSOZYME|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 \(w=f(c)\)
7QUR , Knot 446 1259 0.84 40 329 1087 
MFVFLVLLPLVSSQCVNLTTRTQLPPAYTNSFTRGVYYPDKVFRSSVLHSTQDLFLPFFSNVTWFHAIHVSGTNGTKRFDNPVLPFNDGVYFASTEKSNIIRGWIFGTTLDSKTQSLLIVNNATNVVIKVCEFQFCNDPFLGVYYHKNNKSWMESEFRVYSSANNCTFEYVSQPFLMDLEGKQGNFKNLREFVFKNIDGYFKIYSKHTPINLVRDLPQGFSALEPLVDLPIGINITRFQTLLALHRSYLTPGDSSSGWTAGAAAYYVGYLQPRTFLLKYNENGTITDAVDCALDPLSETKCTLKSFTVEKGIYQTSNFRVQPTESIVRFPNITNLCPFGEVFNATRFASVYAWNRKRISNCVADYSVLYNSASFSTFKCYGVSPTKLNDLCFTNVYADSFVIRGDEVRQIAPGQTGKIADYNYKLPDDFTGCVIAWNSNNLDSKVGGNYNYLYRLFRKSNLKPFERDISTEIYQAGSTPCNGVEGFNCYFPLQSYGFQPTNGVGYQPYRVVVLSFELLHAPATVCGPKKSTNLVKNKCVNFNFNGLTGTGVLTESNKKFLPFQQFGRDIADTTDAVRDPQTLEILDITPCSFGGVSVITPGTNTSNQVAVLYQDVNCTEVPVAIHADQLTPTWRVYSTGSNVFQTRAGCLIGAEHVNNSYECDIPIGAGICASYQTQTNSPRRAASVASQSIIAYTMSLGAENSVAYSNNSIAIPTNFTISVTTEILPVSMTKTSVDCTMYICGDSTECSNLLLQYGSFCTQLNRALTGIAVEQDKNTQEVFAQVKQIYKTPPIKDFGGFNFSQILPDPSKPSKRSFIEDLLFNKVTLADAGFIKQYGDCLGDIAARDLICAQKFNGLTVLPPLLTDEMIAQYTSALLAGTITSGWTFGAGAALQIPFAMQMAYRFNGIGVTQNVLYENQKLIANQFNSAIGKIQDSLSSTASALGKLQDVVNQNAQALNTLVKQLSSNFGAISSVLNDILSRLDKVEAEVQIDRLITGRLQSLQTYVTQQLIRAAEIRASANLAATKMSECVLGQSKRVDFCGKGYHLMSFPQSAPHGVVFLHVTYVPAQEKNFTTAPAICHDGKAHFPREGVFVSNGTHWFVTQRNFYEPQIITTDNTFVSGNCDVVIGIVNNTVYDPLQPELDSFKEELDKYFKNHTSPDVDLGDISGINASVVNIQKEIDRLNEVAKNLNESLIDLQELGKYEQYIKWPSGRLVPRGSPGSGYIPEAPRDGQAYVRKDGEWVLLSTFLGHHHHHH
4JVT , Knot 123 297 0.78 40 148 269 
MHHHHHHSSGVDLGTENLYFQSMAQTKAPTAEELAAAGLTLTIDGEVARITLSRPHRRNAMTGRMWTELARIGHTLPQAVRIVVITGEGPTFSSGIDLDMFQAGKVDGEPTPFTLLARDPNSTAALDQVIASYQEGFLWLRRADIVSIAAVRGHAIGAGFQLALSCDIRILSDTAQLCMKEPALGLVPDLTGTQPLVELVGVNRAIELCLTARTIDAAEAAQLRLAERVVADAELDAAVDALVAQLLAVPAAAARATKELLLQAGRNDLATQARVERTAQLARLAELAKASAAPRRP
1LHI , Knot 68 130 0.84 40 109 128 
KVFERCELARTLKRLGMDGYRGISLANWMCLAKWESGYNTRATNYNAGDRSTDYGIFQINSRYWCNDGKTGGAVNACHLSCSALLQDNIADAVACAKRVVRDPQGIRAWVAWRNRCQNRDVRQYVQGCGV

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 \(c\).

\(|P_{f(7QUR_1)}(2) \setminus P_{f(4JVT_1)}(2)|=192\), \(|P_{f(4JVT_1)}(2) \setminus P_{f(7QUR_1)}(2)|=11\). 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 1:11111111111000010100000111100001001100100110001100000111111001011011010100100010011111001101100000011011111001000000111100100111010010100011111000000001100010100010000100100111101010010100100111001010101000001101100110110110111011111010010011110000101100001101111100110101001110000010100110011011000000100101001100000101010001101101001011101101001101011000010001100011000101001000110100100101001010011101001001111001011000001100101011110000100011100001001100001011000100010011001001101100011100011010011100100111101011011101011000001100001010101101011100000011110011001100001100100101101010011110110110000001111000100001111101001010101000100110001101111001000000011111110100000000100110110001110010111000110000011110010101000111101000010001010100000001110010100010011011110000000011101001000111001111010011101001000011001110010110111100010011011100110100101101111110001110000111110100110111111101111101100101111000110000011100100111010001000101110100110001011001100100011110011001100100101010100110101001000100011011010101011100100011100001010101001101100110111110100111000010011110001010110011110010011100001001011000001101000111111000100110101001000100010000010101101011010110100010010011001000110100110000010110101110101101011011001010100010111100111000000
Pair \(Z_2\) Length of longest common subsequence
7QUR_1,4JVT_1 203 6
7QUR_1,1LHI_1 242 5
4JVT_1,1LHI_1 159 4

Newick tree

 
[
	7QUR_1:12.50,
	[
		4JVT_1:79.5,1LHI_1:79.5
	]:41.00
]

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 \((0.85)(0.8)(\frac{1556 }{\log_{20} 1556}-\frac{297}{\log_{20}297})=325.\)
Status Protein1 Protein2 d d1/2
Query variables 7QUR_1 4JVT_1 414 250.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]