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Parikh vectors
7PJM_1 3REI_1 8ELU_1 Letter Amino acid
6 1 6 C Cysteine
18 8 11 Q Glutamine
30 7 14 E Glutamic acid
31 19 11 A Alanine
14 18 11 R Arginine
33 6 14 G Glycine
15 2 3 H Histidine
28 12 14 L Leucine
37 13 11 K Lycine
14 2 5 M Methionine
18 4 5 F Phenylalanine
12 6 6 P Proline
21 6 11 S Serine
25 6 16 V Valine
17 1 5 N Asparagine
25 4 15 D Aspartic acid
24 10 11 T Threonine
17 3 9 Y Tyrosine
30 7 11 I Isoleucine
7 0 0 W Tryptophan 

7PJM_1|Chains A, B, C|Isocitrate dehydrogenase [NADP] cytoplasmic|Homo sapiens (9606)
>3REI_1|Chains A, E|Histone H3.2|Xenopus laevis (8355)
>8ELU_1|Chain A|GTPase HRas|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 \(w=f(c)\)
7PJM , Knot 178 422 0.85 40 235 405 
MSKKISGGSVVEMQGDEMTRIIWELIKEKLIFPYVELDLHSYDLGIENRDATNDQVTKDAAEAIKKHNVGVKCATITPDEKRVEEFKLKQMWKSPNGTIRNILGGTVFREAIICKNIPRLVSGWVKPIIIGCHAYGDQYRATDFVVPGPGKVEITYTPSDGTQKVTYLVHNFEEGGGVAMGMYNQDKSIEDFAHSSFQMALSKGWPLYLSTKNTILKKYDGRFKDIFQEIYDKQYKSQFEAQKIWYEHRLIDDMVAQAMKSEGGFIWACKNYDGDVQSDFVAQGYGSLGMMTSVLVCPDGKTVEAEAAHGTVTRHYRMYQKGQETSTNPIASIFAWTRGLAHRAKLDNNKELAFFANALEEVSIETIEAGFMTKDLAACIKGLPNVQRSDYLNTFEFMDKLGENLKIKLAQAKLLEHHHHHH
3REI , Knot 65 135 0.78 38 102 126 
ARTKQTARKSTGGKAPRKQLATKAARKSAPATGGVKKPHRYRPGTVALREIRRYQKSTELLIRKLPFQRLVREIAQDFKTDLRFQSSAVMALQEASEAYLVALFEDTNLCAIHAKRVTIMPKDIQLARRIRGERA
8ELU , Knot 90 189 0.83 38 145 186 
MTEYKLVVVGAGGVGKSALTIQLIQNHFVDEYDPTIEDSYRKQVVIDGETCLLDILDTAGQEEYSAMRDQYMRTGEGFLCVFAINNTKSFEDIHQYGEQIKRVKDSDDVPMVLVGNKCDLAARTVESRQAQDLARSYGIPYIETSAKTRQGVEDAFYTLVREIRQHKLRKLNPPDESGPGCMSCKCVLS

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 \(c\).

\(|P_{f(7PJM_1)}(2) \setminus P_{f(3REI_1)}(2)|=164\), \(|P_{f(3REI_1)}(2) \setminus P_{f(7PJM_1)}(2)|=31\). 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 1:10001011011010100100111011000111101010100001110000100001000110110000111001010100001001010011001010100111101100111000110110111011111001010000100111111101010001001000100110010011111111000000100110001011100111101000001100001010011001000000001010011000011001110110001111110000010100011101010111100111010100101011010100000100010000001110111100111001010000011111011001010010111100011101011101000001001011001100101011010110000000
Pair \(Z_2\) Length of longest common subsequence
7PJM_1,3REI_1 195 3
7PJM_1,8ELU_1 190 4
3REI_1,8ELU_1 149 5

Newick tree

 
[
	7PJM_1:10.48,
	[
		8ELU_1:74.5,3REI_1:74.5
	]:27.98
]

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 \((0.85)(0.8)(\frac{557 }{\log_{20} 557}-\frac{135}{\log_{20}135})=123.\)
Status Protein1 Protein2 d d1/2
Query variables 7PJM_1 3REI_1 160 103.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]