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Parikh vectors
1GSP_1 2XZZ_1 6YEY_1 Letter Amino acid
3 13 280 L Leucine
8 15 111 V Valine
7 4 182 A Alanine
9 4 162 N Asparagine
6 4 154 E Glutamic acid
2 5 174 I Isoleucine
2 2 133 K Lycine
0 2 63 M Methionine
4 4 98 F Phenylalanine
6 3 152 D Aspartic acid
4 1 13 C Cysteine
12 9 119 G Glycine
3 1 50 H Histidine
1 5 98 R Arginine
15 8 182 S Serine
6 5 175 T Threonine
1 0 36 W Tryptophan
2 9 131 Q Glutamine
4 7 86 P Proline
9 1 124 Y Tyrosine 

1GSP_1|Chain A|RIBONUCLEASE T1|Aspergillus oryzae (5062)
>2XZZ_1|Chain A|PROTEIN-GLUTAMINE GAMMA-GLUTAMYLTRANSFERASE K|HOMO SAPIENS (9606)
>6YEY_1|Chains A, B, C, D, E|A component of insecticidal toxin complex (Tc)|Xenorhabdus nematophila (628)
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 \(w=f(c)\)
1GSP , Knot 56 104 0.83 38 86 101 
ACDYTCGSNCYSSSDVSTAQAAGYKLHEDGETVGSNSYPHKYNNYEGFDFSVSSPYYEWPILSSGDVYSGGSPGADRVVFNENNQLAGVITHTGASGNNFVECT
2XZZ , Knot 54 102 0.81 38 84 99 
SMYFQSMLSLTLLGAAVVGQECEVQIVFKNPLPVTLTNVVFRLEGSGLQRPKILNVGDIGGNETVTLRQSFVPVRPGPRQLIASLDSPQLSQVHGVIQVDVA
6YEY , Knot 828 2523 0.85 40 367 1944 
MIKVNELLDKINRKRSGDTLLLTNISFMSFSEFRHRTSGTLTWRETDFLYQQAHQESKQNKLEELRILSRANPQLANITNLNITPSTLNNSYNSWFYGRAHRFVKPGSIASIFSPAAYLTELYREAKDFHPDNSQYHLNKRRPDIASLALTQNNMDEEISTLSLSNELLLHNIQTLEKTDYNGVMKMLSTYRQTGMTPYHLPYESARQAILLQDKNLTAFSRNTDVAELMDPTSLLAIKTDISPELYQILVEEITPENSTELMKKNFGTDDVLIFKSYASLARYYDLSYDELSLFVNLSFGKKNTNQQYKNEQLITLVNDGNDTATARLIKRTRKDFYDSHLNYAELIPIKENEYKYNFSVKKTEPDHLDFRLQNGDKEYIYQDKNFVPIANTHYSIPIKLTTEQITNGITLRLWRVKPNPSDAINANAHFKMMEFPGDIFLLKLNKAIRLYKATGISPEDIWQVIESIYDDLTIDSNVLGKLFYVQYYMQHYNISVSDALVLCHSDISQYSTKQQPSHFTMLFNTPLLNGQEFSADNTKLDLTPGESKNHFYLGIMKRAFRVNDTELYTLWKLANGGTNPEFMCSIENLSLLYRVRLLADIHHLTVNELSMLLSVSPYVNTKIALFSDTALTQLISFLFQCTQWLTTQKWSVSDVFLMTTDNYSTVLTPDIENLITTLSNGLSTLSLGDDELIRAAAPLIAASIQMDSAKTAETILLWINQIKPQGLTFDDFMIIAANRDRSENETSNMVAFCQVLGQLSLIVRNIGLSENELTLLVTKPEKFQSETTALQHDLPTLQALTRFHAVIMRCGSYATEILTALELGALTAEQLAVALKFDAQVVTQALQQTDLGVNTFTNWRTIDVTLQWLDVAATLGITPDGVAALIKLKYVGEPETPMPTFDDWQAASTLLQAGLNSQQSDQLQAWLDEATTTAASAYYIKNGAPQQIKSRDELYSYLLIDNQVSAQVKTTRVAEAIASIQLYVNRALNNVEGKVSKPVKTRQFFCDWETYNRRYSTWAGVSELAYYPENYIDPTIRIGQTGMMNNLLQQLSQSQLNIDTVEDSFKNYLTAFEDVANLQVISGYHDSINVNEGLTYLIGYSQTEPRIYYWRNVDHQKCQHGQFAANAWGEWKKIEIPINVWQENIRPVIYKSRLYLLWLEQKELKNESEDGKIDITDYILKLSHIRYDGSWSSPFNFNVTDKIENLINKKASIGMYCSSDYEKDVIIVYFHEKKDNYSFNSLPAREGMTINPDMTLSILTENDLDAIVKSTLSELDTRTEYKVNNQFATDYLAEYKESITTKNKLASFTGNIFDLSYISPGNGHINLTFNPSMEINFSKGNIYNDEVKYLLSMVEDETVILFDYDRHDEMLGKEEEVFHYGTLDFIISIDLKNAEYFRVLMHLRTKEKIPRKSEIGVGINYDYESNDAEFKLDTNIVLDWKDNTGVWHTICESFTNDVSIINNMGNIAALFLREDPCVYLCSIATDIKIASSMIEQIQDKNISFLLKNGSDILVELNAEDHVASKPSHESDPMVYDFNQVKVDIEGYDIPLVSEFIIKQPDGGYNDIVIESPIHIKLKSKDTSNVISLHKMPSGTQYMQIGPYRTRLNTLFSRKLAERANIGIDNVLSMETQNLPEPQLGEGFYATFKLPPYNKEEHGDERWFKIHIGNIDGNSARQPYYEGMLSDIETTVTLFVPYAKGYYIREGVRLGVGYKKIIYDKSWESAFFYFDETKNQFIFINDADHDSGMTQQGIVKNIKKYKGFIHVVVMKNNTEPMDFNGANAIYFWELFYYTPMMVFQRLLQEQNFTESTRWLRYIWNPAGYSVQGEMQDYYWNVRPLEEDTSWNANPLDSVDPDAVAQHDPMHYKVATFMKMLDLLITRGDSAYRQLERDTLNEAKMWYVQALTLLGDEPYFSLDNDWSEPRLEEAASQTMRHHYQHKMLQLRQRAALPTKRTANSLTALFLPQINKKLQGYWQTLTQRLYNLRHNLTIDGQPLSLSLYATPADPSMLLSAAITASQGGGDLPHAVMPMYRFPVILENAKWGVSQLIQFGNTLLSITERQDAEALAEILQTQGSELALQSIKMQDKVMAEIDADKLALQESRHGAQSRFDSFNTLYDEDVNAGEKQAMDLYLSSSVLSTSGTALHMAAAAADLVPNIYGFAVGGSRFGALFNASAIGIEISASATRIAADKISQSEIYRRRRQEWEIQRNNAEAEIKQIDAQLATLAVRREAAVLQKNYLETQQAQTQAQLAFLQSKFSNAALYNWLRGRLSAIYYQFYDLAVSLCLMAEQTYQYELNNAAAHFIKPGAWHGTYAGLLAGETLMLNLAQMEKSYLEKDERALEVTRTVSLAEVYAGLTENSFILKDKVTELVNAGEGSAGTTLNGLNVEGTQLQASLKLSDLNIATDYPDGLGNTRRIKQISVTLPALLGPYQDVRAILSYGGSTMMPRGCKAIAISHGMNDSGQFQMDFNDAKYLPFEGLPVADTGTLTLSFPGISGKQKSLLLSLSDIILHIRYTIRS

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 \(c\).

\(|P_{f(1GSP_1)}(2) \setminus P_{f(2XZZ_1)}(2)|=61\), \(|P_{f(2XZZ_1)}(2) \setminus P_{f(1GSP_1)}(2)|=59\). 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 1:10000010000000010010111001000100110000100000001101010010001111001010011011100111000001111100011010011000
Pair \(Z_2\) Length of longest common subsequence
1GSP_1,2XZZ_1 120 3
1GSP_1,6YEY_1 293 4
2XZZ_1,6YEY_1 289 5

Newick tree

 
[
	6YEY_1:16.40,
	[
		1GSP_1:60,2XZZ_1:60
	]:10.40
]

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 \((0.85)(0.8)(\frac{206 }{\log_{20} 206}-\frac{102}{\log_{20}102})=33.8\)
Status Protein1 Protein2 d d1/2
Query variables 1GSP_1 2XZZ_1 44 43
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]