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Parikh vectors
3AVH_1 3SLB_1 7MTE_1 Letter Amino acid
3 8 75 F Phenylalanine
6 13 62 P Proline
5 13 42 R Arginine
7 14 61 D Aspartic acid
10 6 64 Q Glutamine
15 22 74 I Isoleucine
12 16 57 K Lycine
11 16 97 T Threonine
4 5 55 Y Tyrosine
2 3 31 C Cysteine
11 21 50 E Glutamic acid
19 22 93 G Glycine
13 22 100 S Serine
4 7 10 W Tryptophan
15 24 94 V Valine
14 13 80 A Alanine
14 5 24 H Histidine
9 17 110 L Leucine
5 12 89 N Asparagine
4 9 13 M Methionine 

3AVH_1|Chains A, B|Integrase|Human immunodeficiency virus type 1 (11698)
>3SLB_1|Chains A, B, C, D|Aminoglycoside N3-acetyltransferase|Bacillus anthracis (1392)
>7MTE_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 \(w=f(c)\)
3AVH , Knot 82 183 0.77 40 136 177 
MGSSHHHHHHSSGLVPRGSHMHGQVDSSPGIWQLDCTHLEGKVILVAVHVASGYIEAEVIPAETGQETAYFLLKLAGRWPVKTVHTDNGSNFTSTTVKAACWWAGIKQEDGIPYNPQSQGVIESMNKELKKIIGQVRDQAEHLKTAVQMAVFIHNHKRKGGIGGYSAGERIVDIIATDIQTKE
3SLB , Knot 122 268 0.84 40 177 258 
SNAMNDIVASTQLPNTIKTITNDLRKLGLKKGMTVIVHSSLSSIGWISGGAVAVVEALMEVITEEGTIIMPTQSSDLSDPKHWSRPPVPEEWWQIIRDNVPAFEPHITPTRAMGKVVECFRTYPNVVRSNHPLGSFAAWGRHAEEITVNQSLSMSLGEESPLRKIYDLDGYILLIGVGYDSNTSVGLSEVRSGACELIKVGAPIIENGERVWKEFVDMDYDSDKFVEIGVEFEQKGTVTMGKIGNAKCRLMKQRDIVDFGTEWFRKKN
7MTE , Knot 455 1281 0.84 40 328 1099 
MGILPSPGMPALLSLVSLLSVLLMGCVAETGTQCVNLTTRTQLPPAYTNSFTRGVYYPDKVFRSSVLHSTQDLFLPFFSNVTWFHAIHVSGTNGTKRFDNPVLPFNDGVYFASTEKSNIIRGWIFGTTLDSKTQSLLIVNNATNVVIKVCEFQFCNDPFLGVYYHKNNKSWMESEFRVYSSANNCTFEYVSQPFLMDLEGKQGNFKNLREFVFKNIDGYFKIYSKHTPINLVRDLPQGFSALEPLVDLPIGINITRFQTLLALHRSYLTPGDSSSGWTAGAAAYYVGYLQPRTFLLKYNENGTITDAVDCALDPLSETKCTLKSFTVEKGIYQTSNFRVQPTESIVRFPNITNLCPFGEVFNATRFASVYAWNRKRISNCVADYSVLYNSASFSTFKCYGVSPTKLNDLCFTNVYADSFVIRGDEVRQIAPGQTGKIADYNYKLPDDFTGCVIAWNSNNLDSKVGGNYNYLYRLFRKSNLKPFERDISTEIYQAGSTPCNGVEGFNCYFPLQSYGFQPTNGVGYQPYRVVVLSFELLHAPATVCGPKKSTNLVKNKCVNFNFNGLTGTGVLTESNKKFLPFQQFGRDIADTTDAVRDPQTLEILDITPCSFGGVSVITPGTNTSNQVAVLYQDVNCTEVPVAIHADQLTPTWRVYSTGSNVFQTRAGCLIGAEHVNNSYECDIPIGAGICASYQTQTNSPSGAGSVASQSIIAYTMSLGAENSVAYSNNSIAIPTNFTISVTTEILPVSMTKTSVDCTMYICGDSTECSNLLLQYGSFCTQLNRALTGIAVEQDKNTQEVFAQVKQIYKTPPIKDFGGFNFSQILPDPSKPSKRSFIEDLLFNKVTLIDAGFIKQYGDCLGDIAARDLICAQKFNGLTVLPPLLTDEMIAQYTSALLAGTITSGWTFGAGAALQIPFAMQMAYRFNGIGVTQNVLYENQKLIANQFNSAIGKIQDSLSSTASALGKLQDVVNQNAQALNTLVKQLSSNFGAISSVLNDILSRLDPPEAEVQIDRLITGRLQSLQTYVTQQLIRAAEIRASANLAATKMSECVLGQSKRVDFCGKGYHLMSFPQSAPHGVVFLHVTYVPAQEKNFTTAPAICHDGKAHFPREGVFVSNGTHWFVTQRNFYEPQIITTDNTFVSGNCDVVIGIVNNTVYDPLQPELDSFKEELDKYFKNHTSPDVDLGDISGINASVVNIQKEIDRLNEVAKNLNESLIDLQELGKYEQYIKGSGRENLYFQGGGGSGYIPEAPRDGQAYVRKDGEWVLLSTFLGHHHHHHHH

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 \(c\).

\(|P_{f(3AVH_1)}(2) \setminus P_{f(3SLB_1)}(2)|=68\), \(|P_{f(3SLB_1)}(2) \setminus P_{f(3AVH_1)}(2)|=109\). 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 1:110000000000111101001010100011110100001010111111011010101011110010001011101110111001000010010000101101111100001110010001110010001001110100010010011011111000000111110011001101110010000
Pair \(Z_2\) Length of longest common subsequence
3AVH_1,3SLB_1 177 3
3AVH_1,7MTE_1 220 6
3SLB_1,7MTE_1 189 4

Newick tree

 
[
	7MTE_1:10.81,
	[
		3AVH_1:88.5,3SLB_1:88.5
	]:18.31
]

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 \((0.85)(0.8)(\frac{451 }{\log_{20} 451}-\frac{183}{\log_{20}183})=78.7\)
Status Protein1 Protein2 d d1/2
Query variables 3AVH_1 3SLB_1 100 83.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]