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Parikh vectors
8VKL_1 6WDN_1 8GTZ_1 Letter Amino acid
41 18 40 R Arginine
91 17 38 G Glycine
28 10 8 H Histidine
105 33 63 L Leucine
63 31 27 K Lycine
11 14 7 M Methionine
78 31 12 F Phenylalanine
59 17 27 D Aspartic acid
52 33 22 E Glutamic acid
67 5 15 P Proline
12 4 8 W Tryptophan
54 8 15 Y Tyrosine
85 10 18 N Asparagine
30 4 1 C Cysteine
64 17 13 Q Glutamine
93 22 14 T Threonine
80 11 51 A Alanine
73 16 36 I Isoleucine
105 21 40 S Serine
93 13 28 V Valine 

8VKL_1|Chains A, B, C|Spike glycoprotein|Severe acute respiratory syndrome coronavirus 2 (2697049)
>6WDN_1|Chain A|Calcium uptake protein 2, mitochondrial|Homo sapiens (9606)
>8GTZ_1|Chain A|Ppx/GppA phosphatase|Zymomonas mobilis subsp. mobilis ZM4 = ATCC 31821 (542)
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 \(w=f(c)\)
8VKL , Knot 454 1284 0.84 40 331 1109 
MFVFLVLLPLVSSQCVNLITRTQSYTNSFTRGVYYPDKVFRSSVLHSTQDLFLPFFSNVTWFHAIHVSGTNGTKRFDNPALPFNDGVYFASTEKSNIIRGWIFGTTLDSKTQSLLIVNNATNVVIKVCEFQFCNDPFLDVYQKNNKSWMESEFRVYSSANNCTFEYVSQPFLMDLEGKEGNFKNLREFVFKNIDGYFKIYSKHTPINLERDLPQGFSALEPLVDLPIGINITRFQTLLALHRSYLTPVDSSSGWTAGAAAYYVGYLQPRTFLLKYNENGTITDAVDCALDPLSETKCTLKSFTVEKGIYQTSNFRVQPTESIVRFPNITNLCPFHEVFNATTFASVYAWNRKRISNCVADYSVIYNFAPFFAFKCYGVSPTKLNDLCFTNVYADSFVIRGNEVSQIAPGQTGNIADYNYKLPDDFTGCVIAWNSNKLDSKPSGNYNYLYRLFRKSKLKPFERDISTEIYQAGNKPCNGVAGPNCYSPLQSYGFRPTYGVGHQPYRVVVLSFELLHAPATVCGPKKSTNLVKNKCVNFNFNGLTGTGVLTESNKKFLPFQQFGRDIADTTDAVRDPQTLEILDITPCSFGGVSVITPGTNTSNQVAVLYQGVNCTEVPVAIHADQLTPTWRVYSTGSNVFQTRAGCLIGAEYVNNSYECDIPIGAGICASYQTQTKSHGSASSVASQSIIAYTMSLGAENSVAYSNNSIAIPTNFTISVTTEILPVSMTKTSVDCTMYICGDSTECSNLLLQYGSFCTQLKRALTGIAVEQDKNTQEVFAQVKQIYKTPPIKYFGGFNFSQILPDPSKPSKRSPIEDLLFNKVTLADAGFIKQYGDCLGDIAARDLICAQKFNGLTVLPPLLTDEMIAQYTSALLAGTITSGWTFGAGPALQIPFPMQMAYRFNGIGVTQNVLYENQKLIANQFNSAIGKIQDSLSSTPSALGKLQDVVNHNAQALNTLVKQLSSKFGAISSVLNDILSRLDPPEAEVQIDRLITGRLQSLQTYVTQQLIRAAEIRASANLAATKMSECVLGQSKRVDFCGKGYHLMSFPQSAPHGVVFLHVTYVPAQEKNFTTAPAICHDGKAHFPREGVFVSNGTHWFVTQRNFYEPQIITTDNTFVSGNCDVVIGIVNNTVYDPLQPELDSFKEELDKYFKNHTSPDVDLGDISGINASVVNIQKEIDRLNEVAKNLNESLIDLQELGKYEQGSGYIPEAPRDGQAYVRKDGEWVLLSTFLGRSLEVLFQGPGHHHHHHHHSAWSHPQFEKGGGSGGGGSGGSAWSHPQFEK
6WDN , Knot 145 335 0.84 40 195 320 
SLRKQRFMQFSSLEHEGEYYMTPRDFLFSVMFEQMERKTSVKKLTKKDIEDTLSGIQTAGCGSTFFRDLGDKGLISYTEYLFLLTILTKPHSGFHVAFKMLDTDGNEMIEKREFFKLQKIISKQDDLMTVKTNETGYQEAIVKEPEINTTLQMRFFGKRGQRKLHYKEFRRFMENLQTEIQEMEFLQFSKGLSFMRKEDFAEWLLFFTNTENKDIYWKNVREKLSAGESISLDEFKSFCHFTTHLEDFAIAMQMFSLAHRPVRLAEFKRAVKVATGQELSNNILDTVFKIFDLDGDECLSHEEFLGVLKNRMHRGLWVPQHQSIQEYWKCVKKES
8GTZ , Knot 197 483 0.84 40 222 451 
GPGSIGIIDIGSNSIRLVVYDQLSRAPRILFNEKISAQLGRNIPVDGRIDEKAIELAISELTRFWKLAQIMELSSLRTVATAAVRDAKNGAFLLGEIAKIGLEVEVLSGEEAGYASGYGVLSAIPDADGIVGDLGGGSLELIRVSKGRVKDRVSLPLGVLRIADIRKKSRNALDNFISEAFKKIDWLADARDLPFYMVGGAWRSLAKLDMHVRHYPIPVLHNYIMSPDRPSKLIRVIQRNNPKKLKNKANISTSRVEQLSDAAALLAVVSRHLHSRALVTSAYGLREGLLYLSLDKATRKLDPLLWSANQRGETAGRFYQQGEALYDWMSTLFAQDPPAYHRLRHAACLLADSAWQANPDFRAEQILSIILHGRWVGLDAYGRALIGQALAVSYDGAMDKKITNNLLSEADTIRAVRWGKAIRLGMRLSGGVTTSLKKSTILYRNNKIILQFSGNYKLKGETVLRRLRSLASSFEASEVVEFL

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 \(c\).

\(|P_{f(8VKL_1)}(2) \setminus P_{f(6WDN_1)}(2)|=152\), \(|P_{f(6WDN_1)}(2) \setminus P_{f(8VKL_1)}(2)|=16\). 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 1:111111111110000101100000000010011001001100011000001111110010110110101001000100111110011011000000110111110010000001111001001110100101000111010000000110001010001000010010011110101001010010011100101010100000110100011011011011101111101001001111000010110000110111110011010100111000001010011001101100000010010100110000010101000110110100101100110100110101100001000110001100111111100011010010010100101001110100100111100101100000110010101111000010001010000100110000101100010001001100100111110000110001101001110010011110101101110101100000110000101010110101110000001111001100110000110010010110101001111011011000000111100110000111110100101010100010011000110111100100000001111111010000000001010011000111001011100011000001111001010100011110100001000101010000000111001010001001101111000000001110100100011100111101001110100100001100111001011011110001001101110011010010110111111000111000011111010011011111110111110110010111100011000001110010011101000100010111010011000101100110010001111001100110010110101010011010100100010001101101010101110010001110000101010100110110011011111010011100001001111000101011001111001001110000100101100000110100011111100010011010100100010001000001010110101101011010001001001100100011010011000010101101100101010001011110011100101110111000000000110010100111011110110110010100
Pair \(Z_2\) Length of longest common subsequence
8VKL_1,6WDN_1 168 4
8VKL_1,8GTZ_1 151 4
6WDN_1,8GTZ_1 171 4

Newick tree

 
[
	6WDN_1:87.62,
	[
		8VKL_1:75.5,8GTZ_1:75.5
	]:12.12
]

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 \((0.85)(0.8)(\frac{1619 }{\log_{20} 1619}-\frac{335}{\log_{20}335})=328.\)
Status Protein1 Protein2 d d1/2
Query variables 8VKL_1 6WDN_1 414 261
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]