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Parikh vectors
1VAK_1 5ZKM_1 5GRR_1 Letter Amino acid
19 5 30 A Alanine
7 5 20 N Asparagine
2 8 9 H Histidine
5 8 19 K Lycine
8 2 13 P Proline
29 4 18 S Serine
15 5 24 L Leucine
2 6 16 F Phenylalanine
20 6 25 T Threonine
5 2 3 W Tryptophan
7 2 11 R Arginine
4 0 6 C Cysteine
4 1 11 Q Glutamine
16 5 19 G Glycine
4 4 10 M Methionine
11 0 14 Y Tyrosine
8 3 30 D Aspartic acid
4 10 12 E Glutamic acid
12 4 11 I Isoleucine
21 2 24 V Valine 

1VAK_1|Chain A|coat protein|Sesbania mosaic virus (12558)
>5ZKM_1|Chain A|SP_0782|Streptococcus pneumoniae TIGR4 (170187)
>5GRR_1|Chain A|Probable phosphatidylethanolamine transferase Mcr-1|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 \(w=f(c)\)
1VAK , Knot 92 203 0.80 40 139 192 
AVSSSRGGITVLTHSELSAEIGVTDSIVVSSELVMPYTVGTWLRGVAANWSKYSWLSVRYTYIPSCPSSTAGSIHMGFQYDMADTVPVSVNQLSNLRGYVSGQVWSGSAGLCFINGTRCSDTSTAISTTLDVSKLGKKWYPYKTSADYATAVGVDVNIATPLVPARLVIALLDGSSSTAVAAGRIYCTYTIQMIEPTASALNN
5ZKM , Knot 44 82 0.78 36 66 76 
MKKMAEFTFEIEEHLLTLSENEKGWTKEINRVSFNGAPAKFDIRAWSPDHTKMGKGITLSNEEFQTMVDAFKGNLEHHHHHH
5GRR , Knot 145 325 0.86 40 205 315 
GSTIYHAKDAVQATKPDMRKPRLVVFVVGETARADHVSFNGYERDTFPQLAKIDGVTNFSNVTSCGTSTAYSVPCMFSYLGADEYDVDTAKYQENVLDTLDRLGVSILWRDNNSDSKGVMDKLPKAQFADYKSATNNAICNTNPYNECRDVGMLVGLDDFVAANNGKDMLIMLHQMGNHGPAYFKRYDEKFAKFTPVCEGNELAKCEHQSLINAYDNALLATDDFIAQSIQWLQTHSNAYDVSMLYVSDHGESLGENGVYLHGMPNAFAPKEQRSVPAFFWTDKQTGITPMATDTVLTHDAITPTLLKLFDVTADKVKDRTAFIR

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 \(c\).

\(|P_{f(1VAK_1)}(2) \setminus P_{f(5ZKM_1)}(2)|=116\), \(|P_{f(5ZKM_1)}(2) \setminus P_{f(1VAK_1)}(2)|=43\). 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 1:11000011101100001010111000111000111100110110111101000011010000110010001101011100011001110100100101010101101011101101000000001100010100110010100001001011110101101111101111110100001111101000001011010101100
Pair \(Z_2\) Length of longest common subsequence
1VAK_1,5ZKM_1 159 3
1VAK_1,5GRR_1 186 5
5ZKM_1,5GRR_1 191 5

Newick tree

 
[
	5GRR_1:98.68,
	[
		1VAK_1:79.5,5ZKM_1:79.5
	]:19.18
]

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 \((0.85)(0.8)(\frac{285 }{\log_{20} 285}-\frac{82}{\log_{20}82})=64.8\)
Status Protein1 Protein2 d d1/2
Query variables 1VAK_1 5ZKM_1 81 56.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]