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Parikh vectors
1EWJ_1 5KJM_1 2MKX_1 Letter Amino acid
10 21 2 D Aspartic acid
3 17 0 C Cysteine
11 22 7 G Glycine
4 13 1 M Methionine
3 20 3 Y Tyrosine
8 27 5 S Serine
6 13 4 T Threonine
11 28 3 A Alanine
6 15 2 Q Glutamine
2 16 6 H Histidine
6 20 4 I Isoleucine
1 35 5 K Lycine
8 17 1 F Phenylalanine
3 25 5 V Valine
7 41 0 E Glutamic acid
8 20 1 R Arginine
2 15 4 N Asparagine
17 41 4 L Leucine
7 19 0 P Proline
3 4 1 W Tryptophan 

1EWJ_1|Chains A, B, C, D, E, F, G, H|BLEOMYCIN RESISTANCE DETERMINANT|Klebsiella pneumoniae (573)
>5KJM_1|Chain A|N-lysine methyltransferase SMYD2|Homo sapiens (9606)
>2MKX_1|Chain A|Autolysin|Enterococcus faecalis (226185)
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 \(w=f(c)\)
1EWJ , Knot 65 126 0.83 40 104 123 
MTDQATPNLPSRDFDSTAAFYERLGFGIVFRDAGWMILQRGDLMLEFFAHPGLDPLASWFSCCLRLDDLAEFYRQCKSVGIQETSSGYPRIHAPELQEWGGTMAALVDPDGTLLRLIQNELLAGIS
5KJM , Knot 179 429 0.84 40 239 406 
GLGGLERFCSPGKGRGLRALQPFQVGDLLFSCPAYAYVLTVNERGNHCEYCFTRKEGLSKCGRCKQAFYCNVECQKEDWPMHKLECSPMVVFGENWNPSETVRLTARILAKQKIHPERTPSEKLLAVKEFESHLDKLDNEKKDLIQSDIAALHHFYSKHLEFPDNDSLVVLFAQVNCNGFTIEDEELSHLGSAIFPDVALMNHSCCPNVIVTYKGTLAEVRAVQEIKPGEEVFTSYIDLLYPTEDRNDRLRDSYFFTCECQECTTKDKDKAKVEIRKLSDPPKAEAIRDMVRYARNVIEEFRRAKHYKSPSELLEICELSQEKMSSVFEDSNVYMLHMMYQAMGVCLYMQDWEGALQYGQKIIKPYSKHYPLYSLNVASMWLKLGRLYMGLEHKAAGEKALKKAIAIMEVAHGKDHPYISEIKQEIESH
2MKX , Knot 32 58 0.74 34 49 52 
MGTNTYYTVKSGDTLNKIAAQYGVSVANLRSWNGISGDLIFVGQKLIVKKGSHHHHHH

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 \(c\).

\(|P_{f(1EWJ_1)}(2) \setminus P_{f(5KJM_1)}(2)|=35\), \(|P_{f(5KJM_1)}(2) \setminus P_{f(1EWJ_1)}(2)|=170\). 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 1:100010101100010001110001111111001111110010111011101110111011000101001101000000111000001010101101001110111110101011011000111110
Pair \(Z_2\) Length of longest common subsequence
1EWJ_1,5KJM_1 205 3
1EWJ_1,2MKX_1 121 3
5KJM_1,2MKX_1 218 3

Newick tree

 
[
	5KJM_1:11.06,
	[
		1EWJ_1:60.5,2MKX_1:60.5
	]:56.56
]

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 \((0.85)(0.8)(\frac{555 }{\log_{20} 555}-\frac{126}{\log_{20}126})=125.\)
Status Protein1 Protein2 d d1/2
Query variables 1EWJ_1 5KJM_1 161 102.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]