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Parikh vectors
6VYB_1 4OQE_1 5CHM_1 Letter Amino acid
31 2 0 C Cysteine
110 23 31 L Leucine
13 6 11 M Methionine
75 11 13 F Phenylalanine
97 16 21 T Threonine
81 33 47 A Alanine
61 13 12 D Aspartic acid
93 28 37 G Glycine
24 21 10 H Histidine
73 6 13 I Isoleucine
57 4 17 K Lycine
62 11 20 P Proline
10 2 3 W Tryptophan
63 4 17 Q Glutamine
51 20 18 E Glutamic acid
55 6 20 Y Tyrosine
42 20 11 R Arginine
89 2 12 N Asparagine
100 17 25 S Serine
94 18 24 V Valine 

6VYB_1|Chains A, B, C|Spike glycoprotein|Severe acute respiratory syndrome coronavirus 2 (2697049)
>4OQE_1|Chains A, B|dTDP-3-amino-3,6-dideoxy-alpha-D-glucopyranose N,N-dimethyltransferase|Streptomyces fradiae (1906)
>5CHM_1|Chain A|Beta-lactamase|Escherichia coli (562)
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 \(w=f(c)\)
6VYB , Knot 457 1281 0.85 40 328 1098 
MGILPSPGMPALLSLVSLLSVLLMGCVAETGTQCVNLTTRTQLPPAYTNSFTRGVYYPDKVFRSSVLHSTQDLFLPFFSNVTWFHAIHVSGTNGTKRFDNPVLPFNDGVYFASTEKSNIIRGWIFGTTLDSKTQSLLIVNNATNVVIKVCEFQFCNDPFLGVYYHKNNKSWMESEFRVYSSANNCTFEYVSQPFLMDLEGKQGNFKNLREFVFKNIDGYFKIYSKHTPINLVRDLPQGFSALEPLVDLPIGINITRFQTLLALHRSYLTPGDSSSGWTAGAAAYYVGYLQPRTFLLKYNENGTITDAVDCALDPLSETKCTLKSFTVEKGIYQTSNFRVQPTESIVRFPNITNLCPFGEVFNATRFASVYAWNRKRISNCVADYSVLYNSASFSTFKCYGVSPTKLNDLCFTNVYADSFVIRGDEVRQIAPGQTGKIADYNYKLPDDFTGCVIAWNSNNLDSKVGGNYNYLYRLFRKSNLKPFERDISTEIYQAGSTPCNGVEGFNCYFPLQSYGFQPTNGVGYQPYRVVVLSFELLHAPATVCGPKKSTNLVKNKCVNFNFNGLTGTGVLTESNKKFLPFQQFGRDIADTTDAVRDPQTLEILDITPCSFGGVSVITPGTNTSNEVAVLYQDVNCTEVPVAIHADQLTPTWRVYSTGSNVFQTRAGCLIGAEHVNNSYECDIPIGAGICASYQTQTNSPSGAGSVASQSIIAYTMSLGAENSVAYSNNSIAIPTNFTISVTTEILPVSMTKTSVDCTMYICGDSTECSNLLLQYGSFCTQLNRALTGIAVEQDKNTQEVFAQVKQIYKTPPIKDFGGFNFSQILPDPSKPSKRSFIEDLLFNKVTLADAGFIKQYGDCLGDIAARDLICAQKFNGLTVLPPLLTDEMIAQYTSALLAGTITSGWTFGAGAALQIPFAMQMAYRFNGIGVTQNVLYENQKLIANQFNSAIGKIQDSLSSTASALGKLQDVVNQNAQALNTLVKQLSSNFGAISSVLNDILSRLDPPEAEVQIDRLITGRLQSLQTYVTQQLIRAAEIRASANLAATKMSECVLGQSKRVDFCGKGYHLMSFPQSAPHGVVFLHVTYVPAQEKNFTTAPAICHDGKAHFPREGVFVSNGTHWFVTQRNFYEPQIITTDNTFVSGNCDVVIGIVNNTVYDPLQPELDSFKEELDKYFKNHTSPDVDLGDISGINASVVNIQKEIDRLNEVAKNLNESLIDLQELGKYEQYIKGSGRENLYFQGGGGSGYIPEAPRDGQAYVRKDGEWVLLSTFLGHHHHHHHH
4OQE , Knot 114 263 0.80 40 152 248 
MAHSSATAGPQADYSGEIAELYDLVHQGKGKDYHREAADLAALVRRHSPKAASLLDVACGTGMHLRHLADSFGTVEGLELSADMLAIARRRNPDAVLHHGDMRDFSLGRRFSAVTCMFSSIGHLAGQAELDAALERFAAHVLPDGVVVVEPWWFPENFTPGYVAAGTVEAGGTTVTRVSHSSREGEATRIEVHYLVAGPDRGITHHEESHRITLFTREQYERAFTAAGLSVEFMPGGPSGRGLFTGLPGAKGETRLEHHHHHH
5CHM , Knot 153 362 0.83 38 195 348 
GHMSGEAPLTATVDGIIQPMLKAYRIPGMAVAVLKDGKAHYFNYGVANRESGQRVSEQTLFEIGSVSKTLTATLGAYAAVKGGFELDDKVSQHAPWLKGSAFDGVTMAELATYSAGGLPLQFPDEVDSNDKMQTYYRSWSPVYPAGTHRQYSNPSIGLFGHLAANSLGQPFEQLMSQTLLPKLGLHHTYIQVPESAMANYAYGYSKEDKPIRATPGVLAAEAYGIKTGSADLLKFVEANMGYQGDAALKSAIALTHTGFHSVGEMTQGLGWESYDYPVTEQVLLAGNSPAVSFQANPVTRFAVPKAMGEQRLYNKTGSTGGFGAYVAFVPARGIAIVMLANRNYPIEARVKAAHAILSQLAE

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 \(c\).

\(|P_{f(6VYB_1)}(2) \setminus P_{f(4OQE_1)}(2)|=195\), \(|P_{f(4OQE_1)}(2) \setminus P_{f(6VYB_1)}(2)|=19\). 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 1:111110111111101101101111101100100010100000111100001001100100110001100000111111001011011010100100010011111001101100000011011111001000000111100100111010010100011111000000001100010100010000100100111101010010100100111001010101000001101100110110110111011111010010011110000101100001101111100110101001110000010100110011011000000100101001100000101010001101101001011101101001101011000010001100011000101001000110100100101001010011101001001111001011000001100101011110000100011100001001100001011000100010011001001101100011100011010011100100111101011011101011000001100001010101101011100000011110011001100001100100101101010011110110110000001111000100001111101001010101000100110001101111001000000011111110100000000101110110001110010111000110000011110010101000111101000010001010100000001110010100010011011110000000011101001000111001111010011101001000011001110010110111100010011011100110100101101111110001110000111110100110111111101111101100101111000110000011100100111010001000101110100110001011001100100011110011001100101101010100110101001000100011011010101011100100011100001010101001101100110111110100111000010011110001010110011110010011100001001011000001101000111111000100110101001000100010000010101101011010110100010010011001000110100110000010101000101011110101101100101010001011110011100000000
Pair \(Z_2\) Length of longest common subsequence
6VYB_1,4OQE_1 214 6
6VYB_1,5CHM_1 157 5
4OQE_1,5CHM_1 147 4

Newick tree

 
[
	6VYB_1:99.69,
	[
		5CHM_1:73.5,4OQE_1:73.5
	]:26.19
]

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 \((0.85)(0.8)(\frac{1544 }{\log_{20} 1544}-\frac{263}{\log_{20}263})=332.\)
Status Protein1 Protein2 d d1/2
Query variables 6VYB_1 4OQE_1 426 254
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]