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Parikh vectors
1EYV_1 2EEF_1 2MTM_1 Letter Amino acid
11 13 0 D Aspartic acid
9 9 0 E Glutamic acid
2 9 2 F Phenylalanine
1 9 0 Y Tyrosine
5 7 0 Q Glutamine
6 16 1 S Serine
6 10 0 T Threonine
1 6 0 N Asparagine
0 4 4 C Cysteine
16 9 1 L Leucine
4 11 1 K Lycine
20 8 3 V Valine
8 5 3 P Proline
2 2 0 W Tryptophan
28 9 2 A Alanine
15 8 1 R Arginine
9 10 0 G Glycine
6 2 0 H Histidine
5 6 0 I Isoleucine
2 3 1 M Methionine 

1EYV_1|Chains A, B|N-UTILIZING SUBSTANCE PROTEIN B HOMOLOG|Mycobacterium tuberculosis (1773)
>2EEF_1|Chain A|Protein phosphatase 1, regulatory (Inhibitor) subunit 3B|Homo sapiens (9606)
>2MTM_1|Chain A|Putative uncharacterized protein|Ricinus communis (3988)
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 \(w=f(c)\)
1EYV , Knot 72 156 0.77 38 102 147 
MSDRKPVRGRHQARKRAVALLFEAEVRGISAAEVVDTRAALAEAKPDIARLHPYTAAVARGVSEHAAHIDDLITAHLRGWTLDRLPAVDRAILRVSVWELLHAADVPEPVVVDEAVQLAKELSTDDSPGFVNGVLGQVMLVTPQLRAAAQAVRGGA
2EEF , Knot 74 156 0.79 40 121 150 
GSSGSSGAESESFVLDFSQPSADYLDFRNRLQADHVCLENCVLKDKAIAGTVKVQNLAFEKTVKIRMTFDTWKSYTDFPCQYVKDTYAGSDRDTFSFDISLPEKIQSYERMEFAVYYECNGQTYWDSNRGKNYRIIRAELKSTQGMTKPHSGPDLG
2MTM , Knot 13 19 0.67 20 18 17 
ARCCLVMPVPPFACVKFCS

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 \(c\).

\(|P_{f(1EYV_1)}(2) \setminus P_{f(2EEF_1)}(2)|=66\), \(|P_{f(2EEF_1)}(2) \setminus P_{f(1EYV_1)}(2)|=85\). 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 1:100001101000100011111101010110110110001111010101101010011110110001101001101010110100111100111010110110110110111100110110010000011110111101111010101110110111
Pair \(Z_2\) Length of longest common subsequence
1EYV_1,2EEF_1 151 3
1EYV_1,2MTM_1 110 2
2EEF_1,2MTM_1 131 2

Newick tree

 
[
	2EEF_1:75.17,
	[
		1EYV_1:55,2MTM_1:55
	]:20.17
]

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 \((0.85)(0.8)(\frac{312 }{\log_{20} 312}-\frac{156}{\log_{20}156})=47.7\)
Status Protein1 Protein2 d d1/2
Query variables 1EYV_1 2EEF_1 61 61
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]