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Parikh vectors
1IQW_1 3FKN_1 3VIV_1 Letter Amino acid
7 20 8 R Arginine
10 22 30 I Isoleucine
13 43 15 L Leucine
12 19 12 K Lycine
2 12 5 M Methionine
10 15 14 Y Tyrosine
10 22 12 V Valine
12 14 11 N Asparagine
12 23 13 E Glutamic acid
14 17 14 G Glycine
5 4 1 C Cysteine
11 15 9 Q Glutamine
3 18 7 H Histidine
12 17 12 P Proline
18 19 14 T Threonine
3 5 0 W Tryptophan
14 25 24 A Alanine
14 27 9 D Aspartic acid
7 13 5 F Phenylalanine
29 22 15 S Serine 

1IQW_1|Chain A[auth L]|ANTIBODY M-HFE7A, LIGHT CHAIN|Mus musculus (10090)
>3FKN_1|Chain A|Mitogen-activated protein kinase 14|Homo sapiens (9606)
>3VIV_1|Chains A, B|441aa long hypothetical nfeD protein|Pyrococcus horikoshii (70601)
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 \(w=f(c)\)
1IQW , Knot 103 218 0.84 40 154 212 
DIVLTQSPASLAVSLGQRATISCKASQSVDYDGDSYMNWYQQKPGQPPKLLIYAASNLESGIPARFSGSGSGTDFTLNIHPVEEEDAATYYCQQSNEDPRTFGGGTKLEIKRADAAPTVSIFPPSSEQLTSGGASVVCFLNNFYPKDINVKWKIDGSERQNGVLNSWTDQDSKDSTYSMSSTLTLTKDEYERHNSYTCEATHKTSTSPIVKSFNRNEC
3FKN , Knot 162 372 0.86 40 229 355 
MRGSHHHHHHGSMSQERPTFYRQELNKTIWEVPERYQNLSPVGSGAYGSVCAAFDTKTGLRVAVKKLSRPFQSIIHAKRTYRELRLLKHMKHENVIGLLDVFTPARSLEEFNDVYLVTHLMGADLNNIVKCQKLTDDHVQFLIYQILRGLKYIHSADIIHRDLKPSNLAVNEDCELKILDFGLARHTDDEMTGYVATRWYRAPEIMLNWMHYNQTVDIWSVGCIMAELLTGRTLFPGTDHIDQLKLILRLVGTPGAELLKKISSESARNYIQSLTQMPKMNFANVFIGANPLAVDLLEKMLVLDSDKRITAAQALAHAYFAQYHDPDDEPVADPYDQSFESRDLLIDEWKSLTYDEVISFVPPPLDQEEMES
3VIV , Knot 103 230 0.81 38 149 216 
MSPILAKNIVYVAQIKGQITSYTYDQFDRYITIAEQDNAEAIIIELDTPGGRADAMMNIVQRIQQSKIPVIIYVYPPGASAASAGTYIALGSHLIAMAPGTSIGACRPILGYSQNGSIIEAPPAITNYFIAYIKSLAQESGRNATIAEEFITKDLSLTPEEALKYGVIEVVARDINELLKKSNGMKTKIPVNGRYVTLNFTNVEVRYLAPSFKDKLISYITDLEHHHHHH

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 \(c\).

\(|P_{f(1IQW_1)}(2) \setminus P_{f(3FKN_1)}(2)|=56\), \(|P_{f(3FKN_1)}(2) \setminus P_{f(1IQW_1)}(2)|=131\). 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 1:01110001101110110010100010001000100010100001101101110110010011110101010100101010110000110000000000100111100101001011101011110000100111011011001010010101010100000111001000000000001000101000000000000001000000011100100000
Pair \(Z_2\) Length of longest common subsequence
1IQW_1,3FKN_1 187 3
1IQW_1,3VIV_1 145 4
3FKN_1,3VIV_1 166 6

Newick tree

 
[
	3FKN_1:93.10,
	[
		1IQW_1:72.5,3VIV_1:72.5
	]:20.60
]

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 \((0.85)(0.8)(\frac{590 }{\log_{20} 590}-\frac{218}{\log_{20}218})=105.\)
Status Protein1 Protein2 d d1/2
Query variables 1IQW_1 3FKN_1 133 105.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]