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Parikh vectors
7NQG_1 8JZP_1 5PKR_1 Letter Amino acid
16 32 18 S Serine
18 28 20 T Threonine
9 9 8 W Tryptophan
26 19 14 D Aspartic acid
12 9 15 Q Glutamine
11 25 20 I Isoleucine
25 56 24 L Leucine
23 19 27 G Glycine
13 27 19 F Phenylalanine
21 18 18 P Proline
11 13 19 Y Tyrosine
33 33 25 A Alanine
8 20 18 R Arginine
8 14 20 N Asparagine
31 15 23 K Lycine
8 5 11 M Methionine
24 40 14 V Valine
2 10 7 C Cysteine
14 9 26 E Glutamic acid
14 5 18 H Histidine 

7NQG_1|Chain A[auth AAA]|TrapT family, dctP subunit, C4-dicarboxylate periplasmic binding protein|Rhodopseudomonas palustris (strain ATCC BAA-98 / CGA009) (258594)
>8JZP_1|Chain A|C5a anaphylatoxin chemotactic receptor 1|Homo sapiens (9606)
>5PKR_1|Chain A|Lysine-specific demethylase 4D|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 \(w=f(c)\)
7NQG , Knot 143 327 0.84 40 198 312 
MAMDQDKTVNWKVSLWVPPAHPLVPATKAWAEDIQKASGGSIRMTVFPSEQLGKAFDHYDMARDGIADVTYVNPGYQPGRFPIVSAGQLPFVFKDGKKGTLALNEWYHKYAPTEMKDTKLCFAFIHDPGALHGKKKVLLPSDLSGLKVRPAQSTIGEMVKLFGGTNVQASAPESRDALERGVADEITFPWGSVFLFGIDKVVKYHMDVPLYTTVFTYNIGLKAYNALSDAQKKIIDDHCTPEWASKVTDPWTDFEANGRVKMKALQDHEVYPLTDAQLAEWKKATKPLRDSWAEQVKKSGGDPAAVESDLQNALKKYDAGLHHHHHH
8JZP , Knot 165 406 0.81 40 218 372 
MGKTIIALSYIFCLVFADYKDDDDAANFTPVNGSSGNQSVRLVTSSSLEVLFQGPGSDSFNYTTPDYGHYDDKDTLDLNTPVDKTSNTLRVPDILALVIFAVVFLVGVLGNALVVWVTAFEAKRTINAIWFLNLAVADFLSCLALPILFTSIVQHHHWPFGGAACSILPSLILLNMYASILLLATISADRFLLVFKPIWCQNFRGAGLAWIACAVAWGLALLLTIPSFLYRVVREEYFPPKVLCGVDYSHDKRRERAVAIVRLVLGFLWPLLTLTICYTFILLRTWSRRATRSTKTLKVVVAVVASFFIFWLPYQVTGIMMSFLEPSSPTFLLLKKLDSLCVSFAYINCCINPIIYVVAGQGFQGRLRKSLPSLLRNVLTEESVVRESKSFTRSTVDTMAQKTQAV
5PKR , Knot 160 364 0.86 40 229 347 
MHHHHHHSSGVDLGTENLYFQSMETMKSKANCAQNPNCNIMIFHPTKEEFNDFDKYIAYMESQGAHRAGLAKIIPPKEWKARETYDNISEILIATPLQQVASGRAGVFTQYHKKKKAMTVGEYRHLANSKKYQTPPHQNFEDLERKYWKNRIYNSPIYGADISGSLFDENTKQWNLGHLGTIQDLLEKECGVVIEGVNTPYLYFGMWKTTFAWHTEDMDLYSINYLHLGEPKTWYVVPPEHGQRLERLARELFPGSSRGCGAFLRHKVALISPTVLKENGIPFNRITQEAGEFMVTFPYGYHAGFNHGFNCAEAINFATPRWIDYGKMASQCSCGEARVTFSMDAFVRILQPERYDLWKRGQDR

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 \(c\).

\(|P_{f(7NQG_1)}(2) \setminus P_{f(8JZP_1)}(2)|=75\), \(|P_{f(8JZP_1)}(2) \setminus P_{f(7NQG_1)}(2)|=95\). 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 1:111000001010101111110111110011100100101101010111000110110000110011101001011001101111011011111001001011100100001100100001011110011110100011110010110101100011011011110010101100001100111001011110111111001100010111000110001110100110010001100000101100100110010101010101100001011001011010010011000110010001101111000100110000111000000
Pair \(Z_2\) Length of longest common subsequence
7NQG_1,8JZP_1 170 3
7NQG_1,5PKR_1 177 6
8JZP_1,5PKR_1 177 4

Newick tree

 
[
	5PKR_1:89.63,
	[
		7NQG_1:85,8JZP_1:85
	]:4.63
]

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 \((0.85)(0.8)(\frac{733 }{\log_{20} 733}-\frac{327}{\log_{20}327})=111.\)
Status Protein1 Protein2 d d1/2
Query variables 7NQG_1 8JZP_1 138 126
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]