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Parikh vectors
6LNM_1 3EPK_1 8REK_1 Letter Amino acid
19 29 30 D Aspartic acid
12 11 12 H Histidine
22 42 39 K Lycine
29 21 28 V Valine
25 15 27 A Alanine
8 6 16 C Cysteine
29 28 44 E Glutamic acid
23 20 30 S Serine
3 9 4 W Tryptophan
13 18 17 Y Tyrosine
8 20 39 N Asparagine
17 31 25 I Isoleucine
33 32 25 L Leucine
18 12 23 F Phenylalanine
6 21 16 Q Glutamine
27 18 21 G Glycine
10 8 12 M Methionine
19 15 21 P Proline
9 28 14 T Threonine
21 25 26 R Arginine 

6LNM_1|Chains A, C, E|Peripheral plasma membrane protein CASK|Rattus norvegicus (10116)
>3EPK_1|Chains A, C[auth B]|tRNA isopentenyltransferase|Saccharomyces cerevisiae (4932)
>8REK_1|Chain A|Apical membrane antigen 1 (Fragment)|Plasmodium vivax Sal-1 (126793)
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 \(w=f(c)\)
6LNM , Knot 147 351 0.81 40 199 332 
GPGSEFMADDDVLFEDVYELCEVIGKGPFSVVRRCINRETGQQFAVKIVDVAKFTSSPGLSTEDLKREASICHMLKHPHIVELLETYSSDGMLYMVFEFMDGADLCFEIVKRADAGFVYSEAVASHYMRQILEALRYCHDNNIIHRDVKPHCVLLASKENSAPVKLGGFGVAIQLGESGLVAGGRVGTPHFMAPEVVKREPYGKPVDVWGCGVILFILLSGCLPFYGTKERLFEGIIKGKYKMNPRQWSHISESAKDLVRRMLMLDPAERITVYEALNHPWLKERDRYAYKIHLPETVEQLRKFNARRKLKGAVLAAVSSHKFNSFYGDPPEELPDFSEDPTSSGLLAAER
3EPK , Knot 167 409 0.81 40 224 389 
SKKVIVIAGTTGVGKSQLSIQLAQKFNGEVINSDSMQVYKDIPIITNKHPLQEREGIPHHVMNHVDWSEEYYSHRFETECMNAIEDIHRRGKIPIVVGGTHYYLQTLFNKRVDTKSSERKLTRKQLDILESTDPDVIYNTLVKCDPDIATKYHPNDYRRVQRMLEIYYKTGKKPSETFNEQKITLKFDTLFLWLYSKPEPLFQRLDDRVDDMLERGALQEIKQLYEYYSQNKFTPEQCENGVWQVIGFKEFLPWLTGKTDDNTVKLEDCIERMKTRTRQYAKRQVKWIKKMLIPDIKGDIYLLDATDLSQWDTNASQRAIAISNDFISNRPIKQERAPKALEELLSKGETTMKKLDDWTHYTCNVCRNADGKNVVAIGEKYWKIHLGSRRHKSNLKRNTRQADFEKWKI
8REK , Knot 194 469 0.84 40 239 443 
SIPTVERSTRMGNPWKAFMEKYDIERTHSSGVRVDLGEDAEVENAKYRIPAGRCPVFGKGIVIENSDVSFLRPVATGDQKLKDGGFAFPNANDHISPMTLANLKERYKDNVEMMKLNDIALCRTHAASFVMAGDQNSNYRHPAVYDEKEKTCHMLYLSAQENMGPRYCSPDAQNRDAVFCFKPDKDESFENLVYLSKNVRNDWDKKCPRKNLGNAKFGLWVDGNCEEIPYVKEVEAEDLRECNRIVFGASASDQPTQYEEEMTDYQKIQQGFRQNNREMIKSAFLPVGAFNSDNFKSKGRGFNWANFDSVKKKCYIFNTKPTCLINDKNFIATTALSHPQEVDLEFPCSIYKDEIEREIKKQSRNMNLYSVDGERIVLPRIFISNDKESIKCPCEPERISQSTCNFYVCNCVEKRAEIKENNQVVIKEEFRNYYENGEEKSNKQMLLLEQKLISEEDLNSAVDHHHHHH

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 \(c\).

\(|P_{f(6LNM_1)}(2) \setminus P_{f(3EPK_1)}(2)|=71\), \(|P_{f(3EPK_1)}(2) \setminus P_{f(6LNM_1)}(2)|=96\). 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 1:111001110001110010010011101110110001000010011101101101000111000010001010011001011011000000111011101101101010110010111100011100010011011000000011000101001111000001110111111110110011111101101011110110001010110111011111111010111010000110111010001010010010001001100111101100101001100111000000100101100100100101000101111111000010010101100110100010001111100
Pair \(Z_2\) Length of longest common subsequence
6LNM_1,3EPK_1 167 4
6LNM_1,8REK_1 170 4
3EPK_1,8REK_1 163 4

Newick tree

 
[
	6LNM_1:85.15,
	[
		3EPK_1:81.5,8REK_1:81.5
	]:3.65
]

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 \((0.85)(0.8)(\frac{760 }{\log_{20} 760}-\frac{351}{\log_{20}351})=111.\)
Status Protein1 Protein2 d d1/2
Query variables 6LNM_1 3EPK_1 144 133.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]