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Parikh vectors
3ZFR_1 7WGV_1 8HNV_1 Letter Amino acid
9 41 76 R Arginine
4 16 24 H Histidine
5 71 49 I Isoleucine
5 30 6 C Cysteine
10 57 100 K Lycine
7 53 30 Y Tyrosine
2 9 13 W Tryptophan
15 90 60 V Valine
4 76 79 A Alanine
4 89 58 N Asparagine
9 59 49 D Aspartic acid
5 63 44 Q Glutamine
3 10 18 M Methionine
4 74 45 F Phenylalanine
7 47 92 E Glutamic acid
15 79 59 G Glycine
12 99 125 L Leucine
13 54 36 P Proline
7 93 50 S Serine
8 94 42 T Threonine 

3ZFR_1|Chain A|N-TERMINAL PROTEASE NPRO|PESTIVIRUS STRAIN D32/00_HOBI (266829)
>7WGV_1|Chains A, B, C|Spike glycoprotein|Severe acute respiratory syndrome coronavirus 2 (2697049)
>8HNV_1|Chain A|CRISPR-associated endonuclease Cas9|Haemophilus parainfluenzae (729)
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 \(w=f(c)\)
3ZFR , Knot 74 148 0.83 40 117 144 
MEPLYDKNGAVLFGEPSDTHPQSTLKLPHPRGEKEVIVGIRDLPRKGDCRTGNRLGPVSGLFVKPGPVFYQDYSGPVYHRAPLEQFKQAPMCEVTKRIGRVTGSDGNLYHMYVCTDGCILVKTAKREGQDVLKWVYNVLDSPIWVTSC
7WGV , Knot 429 1204 0.84 40 321 1045 
QCVNLTTRTQLPPAYTNSFTRGVYYPDKVFRSSVLHSTQDLFLPFFSNVTWFHAIHVSGTNGTKRFDNPVLPFNDGVYFASTEKSNIIRGWIFGTTLDSKTQSLLIVNNATNVVIKVCEFQFCNDPFLGVYYHKNNKSWMESEFRVYSSANNCTFEYVSQPFLMDLEGKQGNFKNLREFVFKNIDGYFKIYSKHTPINLVRDLPQGFSALEPLVDLPIGINITRFQTLLALHRSYLTPGDSSSGWTAGAAAYYVGYLQPRTFLLKYNENGTITDAVDCALDPLSETKCTLKSFTVEKGIYQTSNFRVQPTESIVRFPNITNLCPFGEVFNATRFASVYAWNRKRISNCVADYSVLYNSASFSTFKCYGVSPTKLNDLCFTNVYADSFVIRGDEVRQIAPGQTGKIADYNYKLPDDFTGCVIAWNSNNLDSKVGGNYNYLYRLFRKSNLKPFERDISTEIYQAGSTPCNGVEGFNCYFPLQSYGFQPTNGVGYQPYRVVVLSFELLHAPATVCGPKKSTNLVKNKCVNFNFNGLTGTGVLTESNKKFLPFQQFGRDIADTTDAVRDPQTLEILDITPCSFGGVSVITPGTNTSNQVAVLYQDVNCTEVPVAIHADQLTPTWRVYSTGSNVFQTRAGCLIGAEHVNNSYECDIPIGAGICASYQTQTNSRSVASQSIIAYTMSLGAENSVAYSNNSIAIPTNFTISVTTEILPVSMTKTSVDCTMYICGDSTECSNLLLQYGSFCTQLNRALTGIAVEQDKNTQEVFAQVKQIYKTPPIKDFGGFNFSQILPDPSKPSKRSFIEDLLFNKVTLADAGFIKQYGDCLGDIAARDLICAQKFNGLTVLPPLLTDEMIAQYTSALLAGTITSGWTFGAGAALQIPFAMQMAYRFNGIGVTQNVLYENQKLIANQFNSAIGKIQDSLSSTASALGKLQDVVNQNAQALNTLVKQLSSNFGAISSVLNDILSRLDKVEAEVQIDRLITGRLQSLQTYVTQQLIRAAEIRASANLAATKMSECVLGQSKRVDFCGKGYHLMSFPQSAPHGVVFLHVTYVPAQEKNFTTAPAICHDGKAHFPREGVFVSNGTHWFVTQRNFYEPQIITTDNTFVSGNCDVVIGIVNNTVYDPLQPELDSFKEELDKYFKNHTSPDVDLGDISGINASVVNIQKEIDRLNEVAKNLNESLIDLQELGKYEQYIKGSGRENLYFQ
8HNV , Knot 383 1055 0.84 40 301 927 
SMENKNLNYILGLALGIASVGWAVVEIDEKENPLRLIDVGVRTFERAEVPKTGESLALSRRLARSARRLTQRRVARLKKAKRLLKSENILLSTDERLPHQVWQLRVEGLDHKLERQEWAAVLLHLIKHRGYLSQRKNESKSENKELGALLSGVDNNHKLLQQATYRSPAELAVKKFEVEEGHIRNQQGAYTHTFSRLDLLAEMELLFSRQQHFGNPFASEKLLENLTALLMWQKPALSGEAILKMLGKCTFEDEYKAAKNTYSAERFVWITKLNNLRIQENGLERALNDNERLALMEQPYDKNRLFYSQVRSILKLSDEAIFKGLRYSGEDKKAIETKAVLMEMKAYHQIRKVLEGNNLKAEWAELKANPTLLDEIGTAFSLYKTDEDISAYLAGKLSQPVLNALLENLSFDKFIQLSLKALYKLLPLMQQGLRYDEACREIYGDHYGKKTEENHHFLPQIPADEIRNPVVLRTLTQARKVINGVVRLYGSPARIHIETGREVGKSYKDRRELEKRQEENRKQRENAIKEFKEYFPHFAGEPKAKDILKMRLYKQQNAKCLYSGKPIELHRLLEKGYVEVDAALPFSRTWDDSFNNKVLVLANENQNKGNLTPFEWLDGKHNSERWRAFKALVETSAFPYAKKQRILSQKLDEKGFIERNLNDTRYVARFLCNFIADNMHLTGEGKRKVFASNGQITALLRSRWGLAKSREDNDRHHALDAVVVACSTVAMQQKITRFVRFEAGDVFTGERIDRETGEIIPLHFPTPWQFFKQEVEIRIFSDNPKLELENRLPDRPQANHEFVQPLFVSRMPTRKMTGQGHMETVKSAKRLNEGISVIKMPLTKLKLKDLELMVNREREKDLYDTLKARLEAFNDDPAKAFAEPFIKKGGAIVKSVRVEQIQKSGVLVREGNGVADNASMVRVDVFTKGGKYFLVPIYTWQVAKGILPNKAATQYKDEEDWEVMDNSATFKFSLHPNDLVKLVTKKKTILGYFNGLNRATGNIDIKEHDLDKSKGKQGIFEGVGIKLALSFEKYQVDELGKNIRLCKPSKRQPVR

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 \(c\).

\(|P_{f(3ZFR_1)}(2) \setminus P_{f(7WGV_1)}(2)|=11\), \(|P_{f(7WGV_1)}(2) \setminus P_{f(3ZFR_1)}(2)|=215\). 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 1:1011000011111101000010001011010100011111001100100001001111011110111110000011100011100100111001000110101001010010100010111001000100110110011001111000
Pair \(Z_2\) Length of longest common subsequence
3ZFR_1,7WGV_1 226 4
3ZFR_1,8HNV_1 230 4
7WGV_1,8HNV_1 102 5

Newick tree

 
[
	3ZFR_1:12.30,
	[
		7WGV_1:51,8HNV_1:51
	]:77.30
]

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 \((0.85)(0.8)(\frac{1352 }{\log_{20} 1352}-\frac{148}{\log_{20}148})=321.\)
Status Protein1 Protein2 d d1/2
Query variables 3ZFR_1 7WGV_1 405 227
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]