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Parikh vectors
7DIZ_1 7TEI_1 4JLY_1 Letter Amino acid
9 88 9 V Valine
5 53 6 D Aspartic acid
3 30 1 C Cysteine
2 56 6 Q Glutamine
7 72 7 G Glycine
2 75 10 F Phenylalanine
10 40 21 E Glutamic acid
11 67 15 I Isoleucine
3 56 3 P Proline
5 90 8 S Serine
3 90 3 T Threonine
7 75 10 A Alanine
10 77 16 N Asparagine
9 94 19 L Leucine
14 56 7 K Lycine
3 11 3 M Methionine
2 40 12 R Arginine
2 16 9 H Histidine
1 9 1 W Tryptophan
3 49 10 Y Tyrosine 

7DIZ_1|Chain A|Glutaredoxin|Plasmodium falciparum (isolate 3D7) (36329)
>7TEI_1|Chains A, B, C|Spike glycoprotein|Severe acute respiratory syndrome coronavirus 2 (2697049)
>4JLY_1|Chains A, B, C, D, E, F|Spermidine n1-acetyltransferase|Vibrio cholerae O1 biovar eltor (243277)
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 \(w=f(c)\)
7DIZ , Knot 57 111 0.80 40 92 107 
MAGTSEAVKKWVNKIIEENIIAVFAKTECPYCIKAISILKGYNLNSHMHVENIEKNPDMANIQAYLKELTGKSSVPRIFINKDVVGGCDDLVKENDEGKLKERLQKLGLVN
7TEI , Knot 410 1144 0.84 40 322 1012 
MFVFLVLLPLVSSQCVNLTTRTQLPPAYTNSFTRGVYYPDKVFRSSVLHSTQDLFLPFFSNVTWFHVISGTNGTKRFDNPVLPFNDGVYFASIEKSNIIRGWIFGTTLDSKTQSLLIVNNATNVVIKVCEFQFCNDPFLDHKNNKSWMESEFRVYSSANNCTFEYVSQPFLMDLEGKQGNFKNLREFVFKNIDGYFKIYSKHTPIIVREPEDLPQGFSALEPLVDLPIGINITRFQTLLALHRSYLTPGDSSSGWTAGAAAYYVGYLQPRTFLLKYNENGTITDAVDCALDPLSETKCTLKSFTVEKGIYQTSNFRVQPTESIVRFPNITNLCPFDEVFNATRFASVYAWNRKRISNCVADYSVLYNLAPFFTFKCYGVSPTKLNDLCFTNVYADSFVIRGDEVRQIAPGQTGNIADYNYKLPDDFTGCVIAWNSNKLDSKVSGNYNYLYRLFRKSNLKPFERDISTEIYQAGNKPCNGVAGFNCYFPLRSYSFRPTYGVGHQPYRVVVLSFELLHAPATVCGPKKSTNLVKNKCVNFNFNGLKGTGVLTESNKKFLPFQQFGRDIADTTDAVRDPQTLEILDITPCSFGGVSVITPGTNTSNQVAVLYQGVNCTEVPVAIHADQLTPTWRVYSTGSNVFQTRAGCLIGAEYVNNSYECDIPIGAGICASYQTQTKSHGSASSVASQSIIAYTMSLGAENSVAYSNNSIAIPTNFTISVTTEILPVSMTKTSVDCTMYICGDSTECSNLLLQYGSFCTQLKRALTGIAVEQDKNTQEVFAQVKQIYKTPPIKYFGGFNFSQILPDPSKPSKRSFIEDLLFNKVTLADAGFIKQYGDCLGDIAARDLICAQKFKGLTVLPPLLTDEMIAQYTSALLAGTITSGWTFGAGAALQIPFAMQMAYRFNGIGVTQNVLYENQKLIANQFNSAIGKIQDSLSSTASALGKLQDVVNHNAQALNTLVKQLSSKFGAISSVLNDIFSRLDKVEAEVQIDRLITGRLQSLQTYVTQQLIRAAEIRASANLAATKMSECVLGQSKRVDFCGKGYHLMSFPQSAPHGVVFLHVTYVPAQEKNFTTAPAICHDGKAHFPREGVFVSNGTHWFVTQRNFYEPQIITTDNTFVSGNCDVVIGIVNNTVYDPLQPELDS
4JLY , Knot 85 176 0.83 40 127 171 
SNAMNSQLTLRALERGDLRFIHNLNNNRNIMSYWFEEPYESFDELEELYNKHIHDNAERRFVVEDAQKNLIGLVELIEINYIHRSAEFQIIIAPEHQGKGFARTLINRALDYSFTILNLHKIYLHVAVENPKAVHLYEECGFVEEGHLVEEFFINGRYQDVKRMYILQSKYLNRSE

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 \(c\).

\(|P_{f(7DIZ_1)}(2) \setminus P_{f(7TEI_1)}(2)|=11\), \(|P_{f(7TEI_1)}(2) \setminus P_{f(7DIZ_1)}(2)|=241\). 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 1:111000110011001100011111100001001011011010010001010010001011010101001010001101110001111000110000010100010011110
Pair \(Z_2\) Length of longest common subsequence
7DIZ_1,7TEI_1 252 3
7DIZ_1,4JLY_1 137 3
7TEI_1,4JLY_1 215 3

Newick tree

 
[
	7TEI_1:12.32,
	[
		7DIZ_1:68.5,4JLY_1:68.5
	]:60.82
]

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 \((0.85)(0.8)(\frac{1255 }{\log_{20} 1255}-\frac{111}{\log_{20}111})=310.\)
Status Protein1 Protein2 d d1/2
Query variables 7DIZ_1 7TEI_1 394 214.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]