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Parikh vectors
6NNQ_1 3HCQ_1 6MMU_1 Letter Amino acid
12 23 36 D Aspartic acid
16 14 52 I Isoleucine
9 7 29 M Methionine
9 13 63 S Serine
12 22 48 T Threonine
12 6 45 R Arginine
15 15 42 N Asparagine
5 2 11 C Cysteine
9 29 54 G Glycine
6 9 21 H Histidine
11 5 27 Q Glutamine
14 24 54 E Glutamic acid
19 25 72 L Leucine
6 13 36 P Proline
12 20 63 V Valine
9 29 62 A Alanine
22 20 43 K Lycine
14 9 40 F Phenylalanine
5 5 14 W Tryptophan
7 8 26 Y Tyrosine 

6NNQ_1|Chain A|Sentrin-specific protease 1|Homo sapiens (9606)
>3HCQ_1|Chains A, B|Putative choline ABC transporter, periplasmic solute-binding component|Sinorhizobium meliloti (382)
>6MMU_1|Chains A, C|Glutamate receptor ionotropic, NMDA 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 \(w=f(c)\)
6NNQ , Knot 106 224 0.85 40 164 219 
APEITEEMEKEIKNVFRNGNQDEVLSEAFRLTITRKDIQTLNHLNWLNDEIINFYMNMLMERSKEKGLPSVHAFNTFFFTKLKTAGYQAVKRWTKKVDVFSVDILLVPIHLGVHWCLAVVDFRKKNITYYDSMGGINNEACRILLQYLKQESIDKKRKEFDTNGWQLFSKKSQIPQQMNGSDCGMFACKYADCITKDRPINFTQQHMPYFRKRMVWEILHRKLL
3HCQ , Knot 125 298 0.79 40 175 276 
AEPESCGTVRFSDVGWTDITATTATATTILEALGYETDVKVLSVPVTYTSLKNKDIDVFLGNWMPTMEADIAPYREDKSVETVRENLAGAKYTLATNAKGAELGIKDFKDIAAHKDELDGKIYGIEPGNDGNRLIIDMVEKGTFDLKGFEVVESSEQGMLAQVARAEKSGDPIVFLGWEPHPMNANFKLTYLSGGDDVFGPNYGGATVHTNVRAGYTTECPNVDKLLQNLSFSLQMENEIMGKILNDGEDPEKAAAAWLKDNPQSIEPWLSGVATKDGGDGLAAVKAALGLEHHHHHH
6MMU , Knot 323 838 0.86 40 323 780 
MSTMHLLTFALLFSCSFARAACDPKIVNIGAVLSTRKHEQMFREAVNQANKRHGSWKIQLNATSVTHKPNAIQMALSVCEDLISSQVYAILVSHPPTPNDHFTPTPVSYTAGFYRIPVLGLTTRMSIYSDKSIHLSFLRTVPPYSHQSSVWFEMMRVYNWNHIILLVSDDHEGRAAQKRLETLLEERESKAEKVLQFDPGTKNVTALLMEARELEARVIILSASEDDAATVYRAAAMLNMTGSGYVWLVGEREISGNALRYAPDGIIGLQLINGKNESAHISDAVGVVAQAVHELLEKENITDPPRGCVGNTNIWKTGPLFKRVLMSSKYADGVTGRVEFNEDGDRKFANYSIMNLQNRKLVQVGIYNGTHVIPNDRKIIWPGGETEKPRGYQMSTRLKIVTIHQEPFVYVKPTMSDGTCKEEFTVNGDPVKKVICTGPNDTSPGSPRHTVPQCCYGFCIDLLIKLARTMNFTYEVHLVADGKFGTQERVNNSNKKEWNGMMGELLSGQADMIVAPLTINNERAQYIEFSKPFKYQGLTILVKKEIPRSTLDSFMQPFQSTLWLLVGLSVHVVAVMLYLLDRFSPFGRFKVNSEEEEEDALTLSSAMWFSWGVLLNSGIGEGAPRSFSARILGMVWAGFAMIIVASYTANLAAFLVLDRPEERITGINDPRLRNPSDKFIYATVKQSSVDIYFRRQVELSTMYRHMEKHNYESAAEAIQAVRDNKLHAFIWDSAVLEFEASQKCDLVTTGELFFRSGFGIGMRKDSPWKQNVSLSILKSHENGFMEDLDKTWVRYQECDSRSNAPATLTFENMAGVFMLVAGGIVAGIFLIFIEIAYK

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 \(c\).

\(|P_{f(6NNQ_1)}(2) \setminus P_{f(3HCQ_1)}(2)|=79\), \(|P_{f(3HCQ_1)}(2) \setminus P_{f(6NNQ_1)}(2)|=90\). 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 1:11010001000100110010000110011010100001001001011000110101011100000011101011001110010011001100100010110101111110111010111101000010000011110001001110010000100000010001101100000110010100011110001001000011010000110100011101100011
Pair \(Z_2\) Length of longest common subsequence
6NNQ_1,3HCQ_1 169 3
6NNQ_1,6MMU_1 215 4
3HCQ_1,6MMU_1 184 3

Newick tree

 
[
	6MMU_1:10.72,
	[
		6NNQ_1:84.5,3HCQ_1:84.5
	]:20.22
]

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 \((0.85)(0.8)(\frac{522 }{\log_{20} 522}-\frac{224}{\log_{20}224})=85.6\)
Status Protein1 Protein2 d d1/2
Query variables 6NNQ_1 3HCQ_1 106 93
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]