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Parikh vectors
6KGM_1 6MMP_1 3PDC_1 Letter Amino acid
30 36 12 D Aspartic acid
9 11 8 C Cysteine
40 63 29 S Serine
33 48 9 T Threonine
9 14 10 W Tryptophan
31 52 12 I Isoleucine
41 43 16 K Lycine
12 29 17 M Methionine
37 36 24 P Proline
17 26 12 Y Tyrosine
32 45 16 R Arginine
27 42 12 N Asparagine
43 54 26 E Glutamic acid
73 72 30 L Leucine
27 40 15 F Phenylalanine
55 62 21 A Alanine
37 27 13 Q Glutamine
47 54 22 G Glycine
16 21 16 H Histidine
53 63 24 V Valine 

6KGM_1|Chain A|Lysine-specific histone demethylase 1A|Homo sapiens (9606)
>6MMP_1|Chains A, C|Glutamate receptor ionotropic, NMDA 1|Rattus norvegicus (10116)
>3PDC_1|Chains A, B|Epoxide hydrolase 2|Homo sapiens (9606)
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 \(w=f(c)\)
6KGM , Knot 261 669 0.84 40 279 617 
GPLGSHMSGVEGAAFQSRLPHDRMTSQEAACFPDIISGPQQTQKVFLFIRNRTLQLWLDNPKIQLTFEATLQQLEAPYNSDTVLVHRVHSYLERHGLINFGIYKRIKPLPTKKTGKVIIIGSGVSGLAAARQLQSFGMDVTLLEARDRVGGRVATFRKGNYVADLGAMVVTGLGGNPMAVVSKQVNMELAKIKQKCPLYEANGQAVPKEKDEMVEQEFNRLLEATSYLSHQLDFNVLNNKPVSLGQALEVVIQLQEKHVKDEQIEHWKKIVKTQEELKELLNKMVNLKEKIKELHQQYKEASEVKPPRDITAEFLVKSKHRDLTALCKEYDELAETQGKLEEKLQELEANPPSDVYLSSRDRQILDWHFANLEFANATPLSTLSLKHWDQDDDFEFTGSHLTVRNGYSCVPVALAEGLDIKLNTAVRQVRYTASGCEVIAVNTRSTSQTFIYKCDAVLCTLPLGVLKQQPPAVQFVPPLPEWKTSAVQRMGFGNLNKVVLCFDRVFWDPSVNLFGHVGSTTASRGELFLFWNLYKAPILLALVAGEAAGIMENISDDVIVGRCLAILKGIFGSSAVPQPKETVVSRWRADPWARGSYSYVAAGSSGNDYDLMAQPITPGPSIPGAPQPIPRLFFAGEHTIRNYPATVHGALLSGLREAGRIADQFLGAM
6MMP , Knot 323 838 0.86 40 323 780 
MSTMHLLTFALLFSCSFARAACDPKIVNIGAVLSTRKHEQMFREAVNQANKRHGSWKIQLNATSVTHKPNAIQMALSVCEDLISSQVYAILVSHPPTPNDHFTPTPVSYTAGFYRIPVLGLTTRMSIYSDKSIHLSFLRTVPPYSHQSSVWFEMMRVYNWNHIILLVSDDHEGRAAQKRLETLLEERESKAEKVLQFDPGTKNVTALLMEARELEARVIILSASEDDAATVYRAAAMLNMTGSGYVWLVGEREISGNALRYAPDGIIGLQLINGKNESAHISDAVGVVAQAVHELLEKENITDPPRGCVGNTNIWKTGPLFKRVLMSSKYADGVTGRVEFNEDGDRKFANYSIMNLQNRKLVQVGIYNGTHVIPNDRKIIWPGGETEKPRGYQMSTRLKIVTIHQEPFVYVKPTMSDGTCKEEFTVNGDPVKKVICTGPNDTSPGSPRHTVPQCCYGFCIDLLIKLARTMNFTYEVHLVADGKFGTQERVNNSNKKEWNGMMGELLSGQADMIVAPLTINNERAQYIEFSKPFKYQGLTILVKKEIPRSTLDSFMQPFQSTLWLLVGLSVHVVAVMLYLLDRFSPFGRFKVNSEEEEEDALTLSSAMWFSWGVLLNSGIGEGAPRSFSARILGMVWAGFAMIIVASYTANLAAFLVLDRPEERITGINDPRLRNPSDKFIYATVKQSSVDIYFRRQVELSTMYRHMEKHNYESAAEAIQAVRDNKLHAFIWDSAVLEFEASQKCDLVTTGELFFRSGFGIGMRKDSPWKQNVSLSILKSHENGFMEDLDKTWVRYQECDSRSNAPATLTFENMAGVFMLVAGGIVAGIFLIFIEIAYK
3PDC , Knot 147 344 0.83 40 200 328 
MGSSHHHHHHSSGLVPRGSHMAPLPTSCNPSDMSHGYVTVKPRVRLHFVELGSGPAVCLCHGFPESWYSWRYQIPALAQAGYRVLAMDMKGYGESSAPPEIEEYCMEVLCKEMVTFLDKLGLSQAVFIGHDWGGMLVWYMALFYPERVRAVASLNTPFIPANPNMSPLESIKANPVFDYQLYFQEPGVAEAELEQNLSRTFKSLFRASDESVLSMHKVCEAGGLFVNSPEEPSLSRMVTEEEIQFYVQQFKKSGFRGPLNWYRNMERNWKWACKSLGRKILIPALMVTAEKDFVLVPQMSQHMEDWIPHLKRGHIEDCGHWTQMDKPTEVNQILIKWLDSDARN

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 \(c\).

\(|P_{f(6KGM_1)}(2) \setminus P_{f(6MMP_1)}(2)|=39\), \(|P_{f(6MMP_1)}(2) \setminus P_{f(6KGM_1)}(2)|=83\). 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 1:111100101101111000110001000011011011011000001111100001011100101010101010010110000011100100010001110111000101110000101111101101111100100111010110100011101101001001101111110111101111100010101101000011001010111000001100010011010001000101011000110110110111010000100001001001100000100110011010001001000000100101100101011100000010110000001100010100010010101100101000000110101101011010110010100100000101010010100100011111101101010011001000101001111000000001100001110011111100011110111111010001100111101001110100111010101110110001001011111010011111111110111110010001111001111011110011101000110010101110100001111001000011101101110111110111011111000100011010111101100110110011111
Pair \(Z_2\) Length of longest common subsequence
6KGM_1,6MMP_1 122 4
6KGM_1,3PDC_1 167 4
6MMP_1,3PDC_1 193 4

Newick tree

 
[
	3PDC_1:98.06,
	[
		6KGM_1:61,6MMP_1:61
	]:37.06
]

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 \((0.85)(0.8)(\frac{1507 }{\log_{20} 1507}-\frac{669}{\log_{20}669})=210.\)
Status Protein1 Protein2 d d1/2
Query variables 6KGM_1 6MMP_1 269 236
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]