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Parikh vectors
3ETJ_1 6JNL_1 8GXP_1 Letter Amino acid
8 13 15 K Lycine
31 3 17 A Alanine
14 3 7 N Asparagine
14 4 6 D Aspartic acid
4 6 8 C Cysteine
23 2 24 E Glutamic acid
28 6 13 G Glycine
17 3 12 I Isoleucine
27 6 15 V Valine
22 10 18 R Arginine
42 4 37 L Leucine
7 2 9 Y Tyrosine
9 3 2 W Tryptophan
23 3 17 Q Glutamine
10 8 15 H Histidine
7 2 6 M Methionine
12 6 14 F Phenylalanine
24 5 6 P Proline
16 6 21 S Serine
17 6 7 T Threonine 

3ETJ_1|Chains A, B|Phosphoribosylaminoimidazole carboxylase ATPase subunit|Escherichia coli (83333)
>6JNL_1|Chain A|Lysine-specific demethylase REF6|Arabidopsis thaliana (3702)
>8GXP_1|Chain A|Nuclear receptor ROR-gamma|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 \(w=f(c)\)
3ETJ , Knot 153 355 0.84 40 207 347 
MKQVCVLGNGQLGRMLRQAGEPLGIAVWPVGLDAEPAAVPFQQSVITAEIERWPETALTRQLARHPAFVNRDVFPIIADRLTQKQLFDKLHLPTAPWQLLAERSEWPAVFDRLGELAIVKRRTGGYDGRGQWRLRANETEQLPAECYGECIVEQGINFSGEVSLVGARGFDGSTVFYPLTHNLHQDGILRTSVAFPQANAQQQARAEEMLSAIMQELGYVGVMAMECFVTPQGLLINELAPRVHNSGHWTQNGASISQFELHLRAITDLPLPQPVVNNPSVMINLIGSDVNYDWLKLPLVHLHWYDKEVRPGRKVGHLNLTDSDTSRLTATLEALIPLLPPEYASGVIWAQSKFG
6JNL , Knot 51 101 0.77 40 82 96 
LMLHKRNICPIKGCGKNFFSHKYLVQHQRVHSDDRPLKCPWKGCKMTFKWAWSRTEHIRVHTGARPYVCAEPDCGQTFRFVSDFSRHKRKTGHSVKKTNKR
8GXP , Knot 121 269 0.84 40 169 262 
ENLYFQGASLTEIEHLVQSVCKSYRETCQLRLEDLLRQRSNIFSREEVTGYQRKSMWEMWERCAHHLTEAIQYVVEFAKRLSGFMELCQNDQIVLLKAGAMEVVLVRMCRAYNADNRTVFFEGKYGGMELFRALGCSELISSIFDFSHSLSALHFSEDEIALYTALVLINAHRPGLQEKRKVEQLQYNLELAFHHHLCKTHRQSILAKLPPKGKLRSLCSQHVERLQIFQHLHPIVVQAAFPPLYKELFSGGSGKEKHKILHRLLQDSS

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 \(c\).

\(|P_{f(3ETJ_1)}(2) \setminus P_{f(6JNL_1)}(2)|=172\), \(|P_{f(6JNL_1)}(2) \setminus P_{f(3ETJ_1)}(2)|=47\). 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 1:1001011101011011001101111111111101011111100011010100110011000110011110001111110010000110010110111011100001111100110111100001100101010101000001110001001100110101010111101101001101100010001110001111010100010100110111001101111110011010111100111010001010001101001010101100111101110010111011100100011011110101000010110011010100000001010101111111100101111100011
Pair \(Z_2\) Length of longest common subsequence
3ETJ_1,6JNL_1 219 3
3ETJ_1,8GXP_1 176 3
6JNL_1,8GXP_1 181 3

Newick tree

 
[
	6JNL_1:10.27,
	[
		3ETJ_1:88,8GXP_1:88
	]:16.27
]

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 \((0.85)(0.8)(\frac{456 }{\log_{20} 456}-\frac{101}{\log_{20}101})=107.\)
Status Protein1 Protein2 d d1/2
Query variables 3ETJ_1 6JNL_1 143 90
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]