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Parikh vectors
8OHH_1 4MAP_1 7EDH_1 Letter Amino acid
16 11 82 A Alanine
1 0 29 C Cysteine
18 16 54 E Glutamic acid
4 2 12 M Methionine
3 7 53 Y Tyrosine
8 12 92 V Valine
6 1 44 R Arginine
2 1 65 Q Glutamine
9 6 98 S Serine
5 9 65 D Aspartic acid
3 2 25 H Histidine
14 10 110 L Leucine
7 9 61 P Proline
0 0 12 W Tryptophan
7 6 89 N Asparagine
11 17 89 G Glycine
16 12 74 I Isoleucine
9 18 58 K Lycine
3 5 76 F Phenylalanine
12 13 98 T Threonine 

8OHH_1|Chains A, B|Cyclic di-AMP synthase CdaA|Bacillus subtilis subsp. subtilis str. 168 (224308)
>4MAP_1|Chains A, B|Ara h 8 allergen|Arachis hypogaea (3818)
>7EDH_1|Chains A, B, C|Spike glycoprotein|Severe acute respiratory syndrome coronavirus 2 (2697049)
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 \(w=f(c)\)
8OHH , Knot 72 154 0.78 38 109 147 
GPTPVEEAQQKTIEAITKAINYMAKRRIGALLTIERDTGMGDYIETGIPLNAKVSSELLINIFIPNTPLHDGAVIMKNNEIAAAACYLPLSESPFISKELGTRHRAAVGISEVTDSLTIIVSEETGGVSVAKNGDLHRELTEEALKEMLEAEFK
4MAP , Knot 74 157 0.79 36 107 149 
MGVFTFEDEITSTVPPAKLYNAMKDADSITPKIIDDVKSVEIVEGNGGPGTIKKLTIVEDGETKFILHKVESIDEANYAYNYSVVGGVALPPTAEKITFETKLVEGPNGGSIGKLTLKYHTKGDAKPDEEELKKGKAKGEGLFRAIEGYVLANPTQY
7EDH , Knot 454 1286 0.84 40 331 1111 
METDTLLLWVLLLWVPGSTGDVNLTTRTQLPPAYTNSFTRGVYYPDKVFRSSVLHSTQDLFLPFFSNVTWFHAISGTNGTKRFDNPVLPFNDGVYFASTEKSNIIRGWIFGTTLDSKTQSLLIVNNATNVVIKVCEFQFCNDPFLGVYHKNNKSWMESEFRVYSSANNCTFEYVSQPFLMDLEGKQGNFKNLREFVFKNIDGYFKIYSKHTPINLVRDLPQGFSALEPLVDLPIGINITRFQTLLALHRSYLTPGDSSSGWTAGAAAYYVGYLQPRTFLLKYNENGTITDAVDCALDPLSETKCTLKSFTVEKGIYQTSNFRVQPTESIVRFPNITNLCPFGEVFNATRFASVYAWNRKRISNCVADYSVLYNSASFSTFKCYGVSPTKLNDLCFTNVYADSFVIRGDEVRQIAPGQTGKIADYNYKLPDDFTGCVIAWNSNNLDSKVGGNYNYLYRLFRKSNLKPFERDISTEIYQAGSTPCNGVEGFNCYFPLQSYGFQPTYGVGYQPYRVVVLSFELLHAPATVCGPKKSTNLVKNKCVNFNFNGLTGTGVLTESNKKFLPFQQFGRDIDDTTDAVRDPQTLEILDITPCSFGGVSVITPGTNTSNQVAVLYQGVNCTEVPVAIHADQLTPTWRVYSTGSNVFQTRAGCLIGAEHVNNSYECDIPIGAGICASYQTQTNSHRAAASVASQSIIAYTMSLGAENSVAYSNNSIAIPINFTISVTTEILPVSMTKTSVDCTMYICGDSTECSNLLLQYGSFCTQLNRALTGIAVEQDKNTQEVFAQVKQIYKTPPIKDFGGFNFSQILPDPSKPSKRSFIEDLLFNKVTLADAGFIKQYGDCLGDIAARDLICAQKFNGLTVLPPLLTDEMIAQYTSALLAGTITSGWTFGAGAALQIPFAMQMAYRFNGIGVTQNVLYENQKLIANQFNSAIGKIQDSLSSTASALGKLQDVVNQNAQALNTLVKQLSSNFGAISSVLNDILARLDPPEAEVQIDRLITGRLQSLQTYVTQQLIRAAEIRASANLAATKMSECVLGQSKRVDFCGKGYHLMSFPQSAPHGVVFLHVTYVPAQEKNFTTAPAICHDGKAHFPREGVFVSNGTHWFVTQRNFYEPQIITTHNTFVSGNCDVVIGIVNNTVYDPLQPELDSFKEELDKYFKNHTSPDVDLGDISGINASVVNIQKEIDRLNEVAKNLNESLIDLQELGKYEQEFGSGGYIPEAPRDGQAYVRKDGEWVLLSTFLKGQDNSADIQHSGRPLESRGPFEQKLISEEDLNMHTGHHHHHH

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 \(c\).

\(|P_{f(8OHH_1)}(2) \setminus P_{f(4MAP_1)}(2)|=65\), \(|P_{f(4MAP_1)}(2) \setminus P_{f(8OHH_1)}(2)|=63\). 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 1:1101100100001011001100110001111101000011100100111101010001110111100110011111000011111001110001110001100001111100100010111000011101100101000100011001101010
Pair \(Z_2\) Length of longest common subsequence
8OHH_1,4MAP_1 128 3
8OHH_1,7EDH_1 240 5
4MAP_1,7EDH_1 228 3

Newick tree

 
[
	7EDH_1:12.99,
	[
		8OHH_1:64,4MAP_1:64
	]:65.99
]

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 \((0.85)(0.8)(\frac{311 }{\log_{20} 311}-\frac{154}{\log_{20}154})=48.0\)
Status Protein1 Protein2 d d1/2
Query variables 8OHH_1 4MAP_1 61 60
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]