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Parikh vectors
1NOC_1 1NYQ_1 5AVY_1 Letter Amino acid
35 50 91 L Leucine
14 48 57 K Lycine
11 22 24 M Methionine
23 25 44 P Proline
17 50 61 D Aspartic acid
30 61 65 E Glutamic acid
23 39 86 I Isoleucine
14 29 67 S Serine
12 6 12 W Tryptophan
25 35 83 A Alanine
22 35 48 R Arginine
17 26 41 N Asparagine
20 30 31 Q Glutamine
17 19 14 H Histidine
16 27 25 Y Tyrosine
7 2 23 C Cysteine
23 40 76 G Glycine
18 27 46 F Phenylalanine
24 29 64 T Threonine
20 45 70 V Valine 

1NOC_1|Chain A|INDUCIBLE NITRIC OXIDE SYNTHASE|Mus musculus (10090)
>1NYQ_1|Chains A, B|threonyl-tRNA synthetase 1|Staphylococcus aureus (1280)
>5AVY_1|Chain A|Na, K-ATPase alpha subunit|Squalus acanthias (7797)
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 \(w=f(c)\)
1NOC , Knot 171 388 0.87 40 237 375 
NPKSLTRGPRDKPTPLEELLPHAIEFINQYYGSFKEAKIEEHLARLEAVTKEIETTGTYQLTLDELIFATKMAWRNAPRCIGRIQWSNLQVFDARNCSTAQEMFQHICRHILYATNNGNIRSAITVFPQRSDGKHDFRLWNSQLIRYAGYQMPDGTIRGDAATLEFTQLCIDLGWKPRYGRFDVLPLVLQADGQDPEVFEIPPDLVLEVTMEHPKYEWFQELGLKWYALPAVANMLLEVGGLEFPACPFNGWYMGTEIGVRDFCDTQRYNILEEVGRRMGLETHTLASLWKDRAVTEINVAVLHSFQKQNVTIMDHHTASESFMKHMQNEYRARGGCPADWIWLVPPVSGSITPVFHQEMLNYVLSPFYYYQIEPWKTHIWQNEHHHH
1NYQ , Knot 252 645 0.84 40 273 595 
MEQINIQFPDGNKKAFDKGTTTEDIAQSISPGLRKKAVAGKFNGQLVDLTKPLETDGSIEIVTPGSEEALEVLRHSTAHLMAHAIKRLYGNVKFGVGPVIEGGFYYDFDIDQNISSDDFEQIEKTMKQIVNENMKIERKVVSRDEAKELFSNDEYKLELIDAIPEDENVTLYSQGDFTDLCRGVHVPSTAKIKEFKLLSTAGAYWRGDSNNKMLQRIYGTAFFDKKELKAHLQMLEERKERDHRKIGKELELFTNSQLVGAGLPLWLPNGATIRREIERYIVDKEVSMGYDHVYTPVLANVDLYKTSGHWDHYQEDMFPPMQLDETESMVLRPMNCPHHMMIYANKPHSYRELPIRIAELGTMHRYEASGAVSGLQRVRGMTLNDSHIFVRPDQIKEEFKRVVNMIIDVYKDFGFEDYSFRLSYRDPEDKEKYFDDDDMWNKAENMLKEAADELGLSYEEAIGEAAFYGPKLDVQVKTAMGKEETLSTAQLDFLLPERFDLTYIGQDGEHHRPVVIHRGVVSTMERFVAFLTEETKGAFPTWLAPKQVQIIPVNVDLHYDYARQLQDELKSQGVRVSIDDRNEKMGYKIREAQMQKIPYQIVVGDKEVENNQVNVRQYGSQDQETVEKDEFIWNLVDEIRLKKHR
5AVY , Knot 379 1028 0.85 40 311 914 
MGKGTASDKYEPAATSENATKSKKKGKKDKIDKKRDLDELKKEVSMDDHKLSLDELHNKYGTDLTRGLTNARAKEILARDGPNSLTPPPTTPEWIKFCRQLFGGFSILLWIGAILCFLAYGIQAATEDEPANDNLYLGVVLSTVVIVTGCFSYYQEAKSSRIMDSFKNMVPQQALVIRDGEKSTINAEFVVAGDLVEVKGGDRIPADLRIISAHGCKVDNSSLTGESEPQTRSPEFSSENPLETRNIAFFSTNCVEGTARGVVVYTGDRTVMGRIATLASGLEVGRTPIAIEIEHFIHIITGVAVFLGVSFFILSLILGYSWLEAVIFLIGIIVANVPEGLLATVTVCLTLTAKRMARKNCLVKNLEAVETLGSTSTICSDKTGTLTQNRMTVAHMWFDNQIHEADTTENQSGAAFDKTSATWSALSRIAALCNRAVFQAGQDNVPILKRSVAGDASESALLKCIELCCGSVQGMRDRNPKIVEIPFNSTNKYQLSIHENEKSSESRYLLVMKGAPERILDRCSTILLNGAEEPLKEDMKEAFQNAYLELGGLGERVLGFCHFALPEDKYNEGYPFDADEPNFPTTDLCFVGLMAMIDPPRAAVPDAVGKCRSAGIKVIMVTGDHPITAKAIAKGVGIISEGNETIEDIAARLNIPIGQVNPRDAKACVVHGSDLKDLSTEVLDDILHYHTEIVFARTSPQQKLIIVEGCQRQGAIVAVTGDGVNDSPALKKADIGVAMGISGSDVSKQAADMILLDDNFASIVTGVEEGRLIFDNLKKSIAYTLTSNIPEITPFLVFIIGNVPLPLGTVTILCIDLGTDMVPAISLAYEQAESDIMKRQPRNPKTDKLVNERLISMAYGQIGMIQALGGFFSYFVILAENGFLPMDLIGKRVRWDDRWISDVEDSFGQQWTYEQRKIVEFTCHTSFFISIVVVQWADLIICKTRRNSIFQQGMKNKILIFGLFEETALAAFLSYCPGTDVALRMYPLKPSWWFCAFPYSLIIFLYDEMRRFIIRRSPGGWVEQETYY

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 \(c\).

\(|P_{f(1NOC_1)}(2) \setminus P_{f(1NYQ_1)}(2)|=60\), \(|P_{f(1NYQ_1)}(2) \setminus P_{f(1NOC_1)}(2)|=96\). 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 1:0100100110001011001110110110000101001010001101011000100010001010011110011100110011010100101101000001001100100011010001010011011100001000101100011001100110101010110101001010111010010101111110101001011011101110101001000110011101011111101110111101110110110110011100100000001100110011100001101100011001011110010000101100001000110010000010110110111111110101011100011001101100001011000110000000
Pair \(Z_2\) Length of longest common subsequence
1NOC_1,1NYQ_1 156 4
1NOC_1,5AVY_1 168 6
1NYQ_1,5AVY_1 130 5

Newick tree

 
[
	1NOC_1:85.74,
	[
		1NYQ_1:65,5AVY_1:65
	]:20.74
]

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 \((0.85)(0.8)(\frac{1033 }{\log_{20} 1033}-\frac{388}{\log_{20}388})=170.\)
Status Protein1 Protein2 d d1/2
Query variables 1NOC_1 1NYQ_1 221 175
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]