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Parikh vectors
1MJL_1 1IIZ_1 6PST_1 Letter Amino acid
2 4 1 F Phenylalanine
2 4 0 W Tryptophan
8 5 7 A Alanine
1 8 4 C Cysteine
2 3 5 Q Glutamine
7 5 6 I Isoleucine
9 7 3 L Leucine
2 1 1 M Methionine
3 4 4 Y Tyrosine
14 4 7 E Glutamic acid
6 8 4 T Threonine
4 11 3 N Asparagine
5 11 2 D Aspartic acid
3 2 2 H Histidine
9 11 3 K Lycine
7 7 6 R Arginine
4 7 4 G Glycine
6 2 4 P Proline
6 10 2 S Serine
4 6 4 V Valine 

1MJL_1|Chains A, B|METHIONINE REPRESSOR PROTEIN METJ|Escherichia coli (562)
>1IIZ_1|Chain A|LYSOZYME|Antheraea mylitta (34739)
>6PST_1|Chain A[auth N]|Protein TraR|Escherichia coli (562)
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 \(w=f(c)\)
1MJL , Knot 54 104 0.80 40 84 102 
AEWSGEYISPYAEHGKKSEQVKKITVSIPLKVLKILTDERTRRKVNNLRHATNSELLCEAFLHAFTGQPLPDDADLRKERSDEIPEAAKEIMREMGINPETWEY
1IIZ , Knot 61 120 0.81 40 100 118 
KRFTRCGLVNELRKQGFDENLMRDWVCLVENESARYTDKIANVNKNGSRDYGLFQINDKYWCSKGSTPGKDCNVTCSQLLTDDITVASTCAKKIYKRTKFDAWSGWDNHCNHSNPDISSC
6PST , Knot 38 72 0.75 38 58 69 
SDEADEAYSVTEQLTMTGINRIRQKINAHGIPVYLCEACGNPIPEARRKIFPGVTLCVECQAYQERQRKHYA

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 \(c\).

\(|P_{f(1MJL_1)}(2) \setminus P_{f(1IIZ_1)}(2)|=57\), \(|P_{f(1IIZ_1)}(2) \setminus P_{f(1MJL_1)}(2)|=73\). 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 1:10101001010100100000100101011101101100000000100100100001100111011010111001010000000110110011001110100100
Pair \(Z_2\) Length of longest common subsequence
1MJL_1,1IIZ_1 130 3
1MJL_1,6PST_1 102 4
1IIZ_1,6PST_1 134 2

Newick tree

 
[
	1IIZ_1:70.30,
	[
		1MJL_1:51,6PST_1:51
	]:19.30
]

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 \((0.85)(0.8)(\frac{224 }{\log_{20} 224}-\frac{104}{\log_{20}104})=38.7\)
Status Protein1 Protein2 d d1/2
Query variables 1MJL_1 1IIZ_1 50 46.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]