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Parikh vectors
8FVQ_1 4BCD_1 4ZWI_1 Letter Amino acid
4 41 8 Y Tyrosine
9 48 17 V Valine
14 36 13 A Alanine
11 25 7 R Arginine
11 57 22 G Glycine
6 23 11 H Histidine
2 39 9 I Isoleucine
8 31 17 P Proline
14 42 13 E Glutamic acid
4 36 12 F Phenylalanine
3 12 7 W Tryptophan
8 52 19 D Aspartic acid
5 47 24 K Lycine
9 38 17 S Serine
0 27 10 N Asparagine
1 16 1 C Cysteine
5 27 11 Q Glutamine
28 63 26 L Leucine
2 12 1 M Methionine
8 38 12 T Threonine 

8FVQ_1|Chain A|Proprotein convertase subtilisin/kexin type 9|Homo sapiens (9606)
>4BCD_1|Chain A|PROLYL ENDOPEPTIDASE|SUS SCROFA (9823)
>4ZWI_1|Chain A|Carbonic anhydrase 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 \(w=f(c)\)
8FVQ , Knot 68 152 0.75 38 101 139 
MGTVSSRRSWWPLPLLLLLLLLLGPAGARAQEDEDGDYEELVLALRSEEDGLAEAPEHGTTATFHRCAKDPWRLPGTYVVVLKEETHLSQSERTARRLQAQAARRGYLTKILHVFHGLLPGFLVKMSGDLLELALKLPHVDYIEEDSSVFAQ
4BCD , Knot 278 710 0.85 40 302 663 
MLSFQYPDVYRDETAIQDYHGHKVCDPYAWLEDPDSEQTKAFVEAQNKITVPFLEQCPIRGLYKERMTELYDYPKYSCHFKKGKRYFYFYNTGLQNQRVLYVQDSLEGEARVFLDPNILSDDGTVALRGYAFSEDGEYFAYGLSASGSDWVTIKFMKVDGAKELPDVLERVKFSCMAWTHDGKGMFYNAYPQQDGKSDGTETSTNLHQKLYYHVLGTDQSEDILCAEFPDEPKWMGGAELSDDGRYVLLSIREGCDPVNRLWYCDLQQESNGITGILKWVKLIDNFEGEYDYVTNEGTVFTFKTNRHSPNYRLINIDFTDPEESKWKVLVPEHEKDVLEWVACVRSNFLVLCYLHDVKNTLQLHDLATGALLKIFPLEVGSVVGYSGQKKDTEIFYQFTSFLSPGIIYHCDLTKEELEPRVFREVTVKGIDASDYQTVQIFYPSKDGTKIPMFIVHKKGIKLDGSHPAFLYGYGGFNISITPNYSVSRLIFVRHMGGVLAVANIRGGGEYGETWHKGGILANKQNCFDDFQCAAEYLIKEGYTSPKRLTINGGSNGGLLVATCANQRPDLFGCVIAQVGVMDMLKFHKYTIGHAWTTDYGCSDSKQHFEWLIKYSPLHNVKLPEADDIQYPSMLLLTADHDDRVVPLHSLKFIATLQYIVGRSRKQNNPLLIHVDTKAGHGAGKPTAKVIEEVSDMFAFIARCLNIDWIP
4ZWI , Knot 110 257 0.79 40 173 246 
HWGYGKHNGPEHWHKDFPIAKGERQSPVDIDTHTAKYDPSLKPLSVSYDQATSLRILNNGHAFNVEFDDSQDKAVLKGGPLDGTYRLIQFHFHWGSLDGQGSEHTVDKKKYAAELHLVHWNTKYGDFGKAVQQPDGLAVLGIFLKVGSAKPGLQKVVDVLDSIKTKGKSADFTNFDPRGLLPESLDYWTYPGSLTTPPLLECVTWIVLKEPISVSSEQVLKFRKLNFNGEGEPEELMVDNWRPAQPLKNRQIKASFK

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 \(c\).

\(|P_{f(8FVQ_1)}(2) \setminus P_{f(4BCD_1)}(2)|=21\), \(|P_{f(4BCD_1)}(2) \setminus P_{f(8FVQ_1)}(2)|=222\). 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 1:11010000011111111111111111110100000100001111100000111011001001010001001101110011110000010000001001010110010100110110111111110101011011101101001000001110
Pair \(Z_2\) Length of longest common subsequence
8FVQ_1,4BCD_1 243 3
8FVQ_1,4ZWI_1 154 4
4BCD_1,4ZWI_1 179 4

Newick tree

 
[
	4BCD_1:11.91,
	[
		8FVQ_1:77,4ZWI_1:77
	]:37.91
]

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 \((0.85)(0.8)(\frac{862 }{\log_{20} 862}-\frac{152}{\log_{20}152})=198.\)
Status Protein1 Protein2 d d1/2
Query variables 8FVQ_1 4BCD_1 256 152
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]