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Parikh vectors
3NPI_1 3KNS_1 5JYQ_1 Letter Amino acid
18 16 2 E Glutamic acid
7 23 2 K Lycine
8 5 1 P Proline
0 4 0 W Tryptophan
19 23 2 V Valine
9 4 1 F Phenylalanine
24 11 1 A Alanine
19 12 0 D Aspartic acid
2 0 4 C Cysteine
6 5 0 Q Glutamine
11 13 0 I Isoleucine
7 16 1 N Asparagine
8 7 2 H Histidine
26 25 0 L Leucine
24 13 1 S Serine
15 5 1 R Arginine
14 20 1 G Glycine
11 2 0 M Methionine
15 18 0 T Threonine
8 5 1 Y Tyrosine 

3NPI_1|Chains A, B|TetR family regulatory protein|Corynebacterium diphtheriae (1717)
>3KNS_1|Chains A, B, C, D|Beta-lactamase 2|Bacillus cereus (1396)
>5JYQ_1|Chain A|Insulin 1|Conus geographus (6491)
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 \(w=f(c)\)
3NPI , Knot 114 251 0.83 38 162 243 
GMNESSTHDSVADDPTEVSTDTVLDIALSLFSELGFSDAKLEAIAKKSGMSKRMIHYHFGDKRGLYICCLEEAVRRLRPTAEEMYLASAVPVEGVRTIVEAVFHRYVQHPEAVRMLQMENLHHYGKVAEASPLSDQSAITLQLDRLLMLGQDAGAFRPGISAQDVFTLIASIAVFRINSRSTTLNLYGIDMMNGDNTDGMRRMAVDTVLAFLTSNLKSADEDSYLSRPLLSTVTEHVDEEGSYEVAADPFS
3KNS , Knot 101 227 0.80 38 146 216 
SQKVEKTVIKNETGTISISQLNKNVWVHTELGSFNGEAVPSNGLVLNTSKGLVLVDSSWDDKLTKELIEMVEKKFQKRVTDVIITHAHADRIGGIKTLKERGIKAHSTALTAELAKKNGYEEPLGDLQTVTNLKFGNMKVETFYPGKGHTEDNIVVWLPQYNILVGGDLVKSTSAKDLGNVADAYVNEWSTSIENVLKRYRNINAVVPGHGEVGDKGLLLHTLDLLK
5JYQ , Knot 15 20 0.75 26 19 18 
GVVEHCCHRPCSNAEFKKYC

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 \(c\).

\(|P_{f(3NPI_1)}(2) \setminus P_{f(3KNS_1)}(2)|=86\), \(|P_{f(3KNS_1)}(2) \setminus P_{f(3NPI_1)}(2)|=70\). 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 1:11000000001100100100001101110110011100101011100011000110001100011010010011001010100101101111011001101110001001011011010010001011010110000110101001111100111101110100110111011110100000010101101101000011001110011111000100100000100111001000100010001110110
Pair \(Z_2\) Length of longest common subsequence
3NPI_1,3KNS_1 156 3
3NPI_1,5JYQ_1 163 2
3KNS_1,5JYQ_1 153 2

Newick tree

 
[
	3NPI_1:80.82,
	[
		3KNS_1:76.5,5JYQ_1:76.5
	]:4.32
]

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 \((0.85)(0.8)(\frac{478 }{\log_{20} 478}-\frac{227}{\log_{20}227})=72.5\)
Status Protein1 Protein2 d d1/2
Query variables 3NPI_1 3KNS_1 93 88
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]