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Parikh vectors
7CZR_1 8YKA_1 3IXQ_1 Letter Amino acid
40 12 1 C Cysteine
62 24 2 Q Glutamine
82 50 21 G Glycine
109 68 20 L Leucine
60 42 7 P Proline
99 35 6 S Serine
97 30 12 T Threonine
79 62 13 A Alanine
96 52 26 V Valine
55 11 4 Y Tyrosine
17 10 1 H Histidine
62 46 21 K Lycine
14 19 7 M Methionine
67 55 15 D Aspartic acid
88 30 10 N Asparagine
49 67 23 E Glutamic acid
76 59 20 I Isoleucine
42 58 8 R Arginine
12 3 0 W Tryptophan
77 31 9 F Phenylalanine 

7CZR_1|Chains A, B, C|Spike glycoprotein|Severe acute respiratory syndrome coronavirus 2 (2697049)
>8YKA_1|Chains A, B, C, D, E, F|Transitional endoplasmic reticulum ATPase|Homo sapiens (9606)
>3IXQ_1|Chains A, B, C, D|Ribose-5-phosphate isomerase A|Methanocaldococcus jannaschii (2190)
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 \(w=f(c)\)
7CZR , Knot 456 1283 0.84 40 338 1112 
MFVFLVLLPLVSSQCVNLTTRTQLPPAYTNSFTRGVYYPDKVFRSSVLHSTQDLFLPFFSNVTWFHAIHVSGTNGTKRFDNPVLPFNDGVYFASTEKSNIIRGWIFGTTLDSKTQSLLIVNNATNVVIKVCEFQFCNDPFLGVYYHKNNKSWMESEFRVYSSANNCTFEYVSQPFLMDLEGKQGNFKNLREFVFKNIDGYFKIYSKHTPINLVRDLPQGFSALEPLVDLPIGINITRFQTLLALHRSYLTPGDSSSGWTAGAAAYYVGYLQPRTFLLKYNENGTITDAVDCALDPLSETKCTLKSFTVEKGIYQTSNFRVQPTESIVRFPNITNLCPFGEVFNATRFASVYAWNRKRISNCVADYSVLYNSASFSTFKCYGVSPTKLNDLCFTNVYADSFVIRGDEVRQIAPGQTGKIADYNYKLPDDFTGCVIAWNSNNLDSKVGGNYNYLYRLFRKSNLKPFERDISTEIYQAGSTPCNGVEGFNCYFPLQSYGFQPTNGVGYQPYRVVVLSFELLHAPATVCGPKKSTNLVKNKCVNFNFNGLTGTGVLTESNKKFLPFQQFGRDIADTTDAVRDPQTLEILDITPCSFGGVSVITPGTNTSNQVAVLYQDVNCTEVPVAIHADQLTPTWRVYSTGSNVFQTRAGCLIGAEHVNNSYECDIPIGAGICASYQTQTNSPRRARSVASQSIIAYTMSLGAENSVAYSNNSIAIPTNFTISVTTEILPVSMTKTSVDCTMYICGDSTECSNLLLQYGSFCTQLNRALTGIAVEQDKNTQEVFAQVKQIYKTPPIKDFGGFNFSQILPDPSKPSKRSFIEDLLFNKVTLADAGFIKQYGDCLGDIAARDLICAQKFNGLTVLPPLLTDEMIAQYTSALLAGTITSGWTFGAGAALQIPFAMQMAYRFNGIGVTQNVLYENQKLIANQFNSAIGKIQDSLSSTASALGKLQDVVNQNAQALNTLVKQLSSNFGAISSVLNDILSRLDPPEAEVQIDRLITGRLQSLQTYVTQQLIRAAEIRASANLAATKMSECVLGQSKRVDFCGKGYHLMSFPQSAPHGVVFLHVTYVPAQEKNFTTAPAICHDGKAHFPREGVFVSNGTHWFVTQRNFYEPQIITTDNTFVSGNCDVVIGIVNNTVYDPLQPELDSFKEELDKYFKNHTSPDVDLGDISGINASVVNIQKEIDRLNEVAKNLNESLIDLQELGKYEQYIKWPWYIWLGFIAGLIAIVMVTIMLCCMTSCCSCLKGCCSCGSCCKFDEDDSEPVLKGVKLHYTLEDYKDDDDK
8YKA , Knot 282 764 0.81 40 276 674 
LSTAILKQKNRPNRLIVDEAINEDNSVVSLSQPKMDELQLFRGDTVLLKGKKRREAVCIVLSDDTCSDEKIRMNRVVRNNLRVRLGDVISIQPCPDVKYGKRIHVLPIDDTVEGITGNLFEVYLKPYFLEAYRPIRKGDIFLVRGGMRAVEFKVVETDPSPYCIVAPDTVIHCEGEPIKREDEEESLNEVGYDDIGGCRKQLAQIKEMVELPLRHPALFKAIGVKPPRGILLYGPPGTGKTLIARAVANETGAFFFLINGPEIMSKLAGESESNLRKAFEEAEKNAPAIIFIDELDAIAPKREKTHGEVERRIVSQLLTLMDGLKQRAHVIVMAATNRPNSIDPALRRFGRFDREVDIGIPDATGRLEILQIHTKNMKLADDVDLEQVANETHGHVGADLAALCSEAALQAIRKKMDLIDLEDETIDAEVMNSLAVTMDDFRWALSQSNPSALRETVVEVPQVTWEDIGGLEDVKRELQELVQYPVEHPDKFLKFGMTPSKGVLFYGPPGCGKTLLAKAIANECQANFISIKGPELLTMWFGESEANVREIFDKARQAAPCVLFFDELDSIAKARGGNIGDGGGAADRVINQILTEMDGMSTKKNVFIIGATNRPDIIDPAILRPGRLDQLIYIPLPDEKSRVAILKANLRKSPVAKDVDLEFLAKMTNGFSGADLTEICQRACKLAIRESIESEIRRERERQTNPSAMEVEEDDPVPEIRRDHFEEAMRFARRSVSDNDIRKYEMFAQTLQQSRGFGSFRFPS
3IXQ , Knot 103 226 0.82 38 149 217 
MSNEDLKLKVAKEAVKLVKDGMVIGLGTGSTAALFIRELGNRIREEELTVFGIPTSFEAKMLAMQYEIPLVTLDEYDVDIAFDGADEVEETTLFLIKGGGGCHTQEKIVDYNANEFVVLVDESKLVKKLGEKFPIPVEVIPSAYRVVIRALSEMGGEAVIRLGDRKRGPVITDNGNMIIDVFMNIDDAIELEKEINNIPGVVENGIFTKVDKVLVGTKKGVKTLKK

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 \(c\).

\(|P_{f(7CZR_1)}(2) \setminus P_{f(8YKA_1)}(2)|=86\), \(|P_{f(8YKA_1)}(2) \setminus P_{f(7CZR_1)}(2)|=24\). 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 1:11111111111000010100000111100001001100100110001100000111111001011011010100100010011111001101100000011011111001000000111100100111010010100011111000000001100010100010000100100111101010010100100111001010101000001101100110110110111011111010010011110000101100001101111100110101001110000010100110011011000000100101001100000101010001101101001011101101001101011000010001100011000101001000110100100101001010011101001001111001011000001100101011110000100011100001001100001011000100010011001001101100011100011010011100100111101011011101011000001100001010101101011100000011110011001100001100100101101010011110110110000001111000100001111101001010101000100110001101111001000000011111110100000000100100110001110010111000110000011110010101000111101000010001010100000001110010100010011011110000000011101001000111001111010011101001000011001110010110111100010011011100110100101101111110001110000111110100110111111101111101100101111000110000011100100111010001000101110100110001011001100100011110011001100101101010100110101001000100011011010101011100100011100001010101001101100110111110100111000010011110001010110011110010011100001001011000001101000111111000100110101001000100010000010101101011010110100010010011001000110100110000010111011111111111111101110010000001010000100001000000111011010001000000000
Pair \(Z_2\) Length of longest common subsequence
7CZR_1,8YKA_1 110 4
7CZR_1,3IXQ_1 201 4
8YKA_1,3IXQ_1 159 4

Newick tree

 
[
	3IXQ_1:99.69,
	[
		7CZR_1:55,8YKA_1:55
	]:44.69
]

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 \((0.85)(0.8)(\frac{2047 }{\log_{20} 2047}-\frac{764}{\log_{20}764})=312.\)
Status Protein1 Protein2 d d1/2
Query variables 7CZR_1 8YKA_1 401 315
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]