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Parikh vectors
5HLP_1 7NKJ_1 6NJN_1 Letter Amino acid
27 27 47 R Arginine
17 10 37 N Asparagine
7 8 10 H Histidine
20 34 58 I Isoleucine
5 7 25 M Methionine
2 3 15 W Tryptophan
34 58 65 A Alanine
18 28 40 Q Glutamine
19 48 55 E Glutamic acid
18 18 50 F Phenylalanine
29 49 61 V Valine
9 4 14 C Cysteine
23 28 52 K Lycine
32 23 33 P Proline
33 39 70 S Serine
30 23 48 T Threonine
24 49 76 G Glycine
37 50 72 L Leucine
18 9 36 Y Tyrosine
22 33 43 D Aspartic acid 

5HLP_1|Chains A, B|Glycogen synthase kinase-3 beta|Homo sapiens (9606)
>7NKJ_1|Chains A, B, C|ATP synthase subunit alpha|Mycolicibacterium smegmatis (strain ATCC 700084 / mc(2)155) (246196)
>6NJN_1|Chain A|Glutamate receptor 1|Rattus norvegicus (10116)
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 \(w=f(c)\)
5HLP , Knot 177 424 0.84 40 232 398 
GSPGMSGRPRTTSFAESCKPVQQPSAFGSMKVSRDKDGSKVTTVVATPGQGPDRPQEVSYTDTKVIGNGSFGVVYQAKLCDSGELVAIKKVLQDKRFKNRELQIMRKLDHCNIVRLRYFFYSSGEKKDEVYLNLVLDYVPETVYRVARHYSRAKQTLPVIYVKLYMYQLFRSLAYIHSFGICHRDIKPQNLLLDPDTAVLKLCDFGSAKQLVRGEPNVSYICSRYYRAPELIFGATDYTSSIDVWSAGCVLAELLLGQPIFPGDSGVDQLVEIIKVLGTPTREQIREMNPNYTEFKFPQIKAHPWTKVFRPRTPPEAIALCSRLLEYTPTARLTPLEACAHSFFDELRDPNVKLPNGRDTPALFNFTTQELSSNPPLATILIPPHARIQAAASTPTNATAASDANTGDRGQTNNAASASASNST
7NKJ , Knot 219 548 0.84 40 225 504 
MAELTISAADIEGAIEDYVSSFSADTEREEIGTVIDAGDGIAHVEGLPSVMTQELLEFPGGVLGVALNLDEHSVGAVILGEFEKIEEGQQVKRTGEVLSVPVGDAFLGRVVNPLGQPIDGQGDIAAETRRALELQAPSVVQRQSVSEPLQTGIKAIDAMTPIGRGQRQLIIGDRKTGKTAVCVDTILNQREAWLTGDPKQQVRCVYVAIGQKGTTIASVKRALEEGGAMEYTTIVAAPASDAAGFKWLAPYTGSAIGQHWMYNGKHVLIVFDDLSKQADAYRAISLLLRRPPGREAFPGDVFYLHSRLLERCAKLSDELGGGSMTGLPIIETKANDISAFIPTNVISITDGQCFLESDLFNQGVRPAINVGVSVSRVGGAAQIKAMKEVAGSLRLDLSQYRELEAFAAFASDLDAASKAQLDRGARLVELLKQPQYSPLAVEEQVVAIFLGTQGHLDSVPVEDVQRFESELLEHVKASHSDIFDGIRETKKLSEEAEEKLVSVINEFKKGFQASDGSSVVVSENAEALDPEDLEKESVKVRKPAPKKA
6NJN , Knot 339 907 0.84 40 317 842 
MPYIFAFFCTGFLGAVVGANFPNNIQIGGLFPNQQSQEHAAFRFALSQLTEPPKLLPQIDIVNISDSFEMTYRFCSQFSKGVYAIFGFYERRTVNMLTSFCGALHVCFITPSFPVDTSNQFVLQLRPELQEALISIIDHYKWQTFVYIYDADRGLSVLQRVLDTAAEKNWQVTAVNILTTTEEGYRMLFQDLEKKKERLVVVDCESERLNAILGQIVKLEKNGIGYHYILANLGFMDIDLNKFKESGANVTGFQLVNYTDTIPARIMQQWRTSDSRDHTRVDWKRPKYTSALTYDGVKVMAEAFQSLRRQRIDISRRGNAGDCLANPAVPWGQGIDIQRALQQVRFEGLTGNVQFNEKGRRTNYTLHVIEMKHDGIRKIGYWNEDDKFVPAATDAQAGGDNSSVQNRTYIVTTILEDPYVMLKKNANQFEGNDRYEGYCVELAAEIAKHVGYSYRLEIVSDGKYGARDPDTKAWNGMVGELVYGRADVAVAPLTITLVREEVIDFSKPFMSLGISIMIKKPQKSKPGVFSFLDPLAYEIWMCIVFAYIGVSVVLFLVSRFSPYEWHSEEFEEGRDQTTSDQSNEFGIFNSLWFSLGAFMQQGCDISPRSLSGRIVGGVWWFFTLIIISSYTANLAAFLTVERMVSPIESAEDLAKQTEIAYGTLEAGSTKEFFRRSKIAVFEKMWTYMKSAEPSVFVRTTEEGMIRVRKSKGKYAYLLESTMNEYIEQRKPCDTMKVGGNLDSKGYGIATPKGSALGAPVNLAVLKLAEQGALDKLKAKWWYDKGECGSKDSGSKDKTSALSLSNVAGVFYILIGGLGLAMLVALIEFCYKSRSESKRMKGFCLIPQQSINEAIRTSTLPRNSGAGASGGGGSGENGRVVSQDFPKSMQSIPCMSHSSGMPLGATGL

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 \(c\).

\(|P_{f(5HLP_1)}(2) \setminus P_{f(7NKJ_1)}(2)|=73\), \(|P_{f(7NKJ_1)}(2) \setminus P_{f(5HLP_1)}(2)|=66\). 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 1:1011101010000110000110010111010100000100100111011011001001000000111010111100101000101111001100001000010110010000110100110001000001010111001100100110000010001111010101001100110100111000010100111010011101001101001101010100100000011011111000000101101101110111101111100110011011011101000010010100001011010101100110100110111100011000101010110101001100100101011010001111010000100011110111110101011100100101100100100100001101010000
Pair \(Z_2\) Length of longest common subsequence
5HLP_1,7NKJ_1 139 5
5HLP_1,6NJN_1 147 5
7NKJ_1,6NJN_1 132 4

Newick tree

 
[
	5HLP_1:73.27,
	[
		7NKJ_1:66,6NJN_1:66
	]:7.27
]

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 \((0.85)(0.8)(\frac{972 }{\log_{20} 972}-\frac{424}{\log_{20}424})=145.\)
Status Protein1 Protein2 d d1/2
Query variables 5HLP_1 7NKJ_1 181 161.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]