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Parikh vectors
8HPW_1 4FUL_1 6PDY_1 Letter Amino acid
2 17 2 W Tryptophan
10 54 27 A Alanine
14 60 32 I Isoleucine
11 43 22 P Proline
8 48 33 G Glycine
10 50 19 T Threonine
2 50 10 Q Glutamine
9 64 31 E Glutamic acid
23 108 31 L Leucine
0 21 8 M Methionine
12 43 9 F Phenylalanine
14 32 14 N Asparagine
12 60 30 D Aspartic acid
8 18 2 C Cysteine
18 59 22 S Serine
3 26 11 Y Tyrosine
8 62 22 V Valine
7 44 31 R Arginine
18 36 6 H Histidine
13 71 21 K Lycine 

8HPW_1|Chains A, B, C|Leucine-rich glioma-inactivated protein 1|Mus musculus (10090)
>4FUL_1|Chain A|Phosphatidylinositol 4,5-bisphosphate 3-kinase catalytic subunit gamma isoform|Homo sapiens (9606)
>6PDY_1|Chains A, B, C, D, E, F|Membrane-spanning ATPase-like protein|Chaetomium thermophilum (209285)
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 \(w=f(c)\)
8HPW , Knot 85 202 0.74 38 134 185 
HHHHHHHKPKCPAVCTCSKDNALCENARSIPRTVPPDVISLSFVRSGFTEISEGSFLFTPSLQLLLFTSNSFDVISDDAFIGLPHLEYLFIENNNIKSISRHTFRGLKSLIHLSLANNNLQTLPKDIFKGLDSLTNVDLRGNAFNCDCKLKWLVEWLGHTNATVEDIYCEGPPEYKKRKINSLSPKDFDCIIAAAHHHHHHH
4FUL , Knot 364 966 0.86 40 339 883 
MSEESQAFQRQLTALIGYDVTDVSNVHDDELEFTRRGLVTPRMAEVASRDPKLYAMHPWVTSKPLPEYLWKKIANNCIFIVIHRSTTSQTIKVSPDDTPGAILQSFFTKMAKKKSLMDIPESQSEQDFVLRVCGRDEYLVGETPIKNFQWVRHCLKNGEEIHVVLDTPPDPALDEVRKEEWPLVDDCTGVTGYHEQLTIHGKDHESVFTVSLWDCDRKFRVKIRGIDIPVLPRNTDLTVFVEANIQHGQQVLCQRRTSPKPFTEEVLWNVWLEFSIKIKDLPKGALLNLQIYCGKAPALSSKASAESPSSESKGKVRLLYYVNLLLIDHRFLLRRGEYVLHMWQISGKGEDQGSFNADKLTSATNPDKENSMSISILLDNYCHPIALPKHQPTPDPEGDRVRAEMPNQLRKQLEAIIATDPLNPLTAEDKELLWHFRYESLKHPKAYPKLFSSVKWGQQEIVAKTYQLLARREVWDQSALDVGLTMQLLDCNFSDENVRAIAVQKLESLEDDDVLHYLLQLVQAVKFEPYHDSALARFLLKRGLRNKRIGHFLFWFLRSEIAQSRHYQQRFAVILEAYLRGCGTAMLHDFTQQVQVIEMLQKVTLDIKSLSAEKYDVSSQVISQLKQKLENLQNSQLPESFRVPYDPGLKAGALAIEKCKVMASKKKPLWLEFKCADPTALSNETIGIIFKHGDDLRQDMLILQILRIMESIWETESLDLCLLPYGCISTGDKIGMIEIVKDATTIAKIQQSTVGNTGAFKDEVLNHWLKEKSPTEEKFQAAVERFVYSCAGYCVATFVLGIGDRHNDNIMITETGNLFHIDFGHILGNYKSFLGINKERVPFVLTPDFLFVMGTSGKKTSPHFQKFQDICVKAYLALRHHTNLLIILFSMMLMTGMPQLTSKEDIEYIRDALTVGKNEEDAKKYFLDQIEVCRDKGWTVQFNWFLHLVLGIKQGEKHSAHHHHHH
6PDY , Knot 164 383 0.85 40 210 364 
GSIAPYLVKIIDPDYEKNERTRIKAQENLRRIRRKQIAEKGDNEDGTDDPSRRRKIDDLVLNEYENQVALEVVAPEDIPVGFNDIGGLDDIIEELKETIIYPLTMPHLYKHGGALLAAPSGVLLYGPPGCGKTMLAKAVAHESGASFINLHISTLTEKWYGDSNKIVRAVFSLAKKLQPSIIFIDEIDAVLGTRRSGEHEASGMVKAEFMTLWDGLTSTNASGVPNRIVVLGATNRINDIDEAILRRMPKQFPVPLPGLEQRRRILELVLRGTKRDPDFDLDYIARVTAGMSGSDIKETCRDAAMAPMREYIRQHRASGKPLSEINPDDVRGIRTEDFFGRRGGKILSEIPPRQTGYVVQSKNSSEGGYEEVEDDDEQGTAST

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 \(c\).

\(|P_{f(8HPW_1)}(2) \setminus P_{f(4FUL_1)}(2)|=8\), \(|P_{f(4FUL_1)}(2) \setminus P_{f(8HPW_1)}(2)|=213\). 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 1:0000000010011100000001100010011001110110101100110010010111010101111000010110001111110100111000010010000101100110101100010011001101100100101010110000010111011100010100100011100000010010100100111110000000
Pair \(Z_2\) Length of longest common subsequence
8HPW_1,4FUL_1 221 7
8HPW_1,6PDY_1 164 3
4FUL_1,6PDY_1 159 4

Newick tree

 
[
	8HPW_1:10.54,
	[
		6PDY_1:79.5,4FUL_1:79.5
	]:23.04
]

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 \((0.85)(0.8)(\frac{1168 }{\log_{20} 1168}-\frac{202}{\log_{20}202})=259.\)
Status Protein1 Protein2 d d1/2
Query variables 8HPW_1 4FUL_1 331 195
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]