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Parikh vectors
5PZK_1 2VSI_1 9KGZ_1 Letter Amino acid
31 15 5 G Glycine
38 18 13 T Threonine
23 7 0 Y Tyrosine
48 10 4 S Serine
57 14 7 A Alanine
27 15 23 E Glutamic acid
15 7 6 F Phenylalanine
13 5 0 M Methionine
30 9 3 P Proline
10 1 1 W Tryptophan
35 17 4 V Valine
42 9 0 R Arginine
12 7 0 H Histidine
59 19 16 L Leucine
18 8 0 Q Glutamine
23 28 12 I Isoleucine
30 17 24 K Lycine
16 8 4 N Asparagine
26 20 3 D Aspartic acid
21 2 0 C Cysteine 

5PZK_1|Chains A, B|RNA-directed RNA polymerase|Hepatitis C virus genotype 1b (isolate Con1) (333284)
>2VSI_1|Chains A, B|2-C-METHYL-D-ERYTHRITOL 4-PHOSPHATE CYTIDYLYLTRANSFERASE|STREPTOCOCCUS PNEUMONIAE (1313)
>9KGZ_1|Chain A|De novo designed monomeric mainly-beta protein B10|synthetic construct (32630)
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 \(w=f(c)\)
5PZK , Knot 230 574 0.84 40 278 537 
MSMSYTWTGALITPCAAEETKLPINALSNSLLRHHNLVYATTSRSASLRQKKVTFDRLQVLDDHYRDVLKEMKAKASTVKAKLLSVEEACKLTPPHSARSKFGYGAKDVRNLSSKAVNHIRSVWKDLLEDTETPIDTTIMAKNEVFCVQPEKGGRKPARLIVFPDLGVRVCEKMALYDVVSTLPQAVMGSSYGFQYSPGQRVEFLVNAWKAKKCPMGFAYDTRCFDSTVTENDIRVEESIYQCCDLAPEARQAIRSLTERLYIGGPLTNSKGQNCGYRRCRASGVLTTSCGNTLTCYLKAAAACRAAKLQDCTMLVCGDDLVVICESAGTQEDEASLRAFTEAMTRYSAPPGDPPKPEYDLELITSCSSNVSVAHDASGKRVYYLTRDPTTPLARAAWETARHTPVNSWLGNIIMYAPTLWARMILMTHFFSILLAQEQLEKALDCQIYGACYSIEPLDLPQIIQRLHGLSAFSLHSYSPGEINRVASCLRKLGVPPLRVWRHRARSVRARLLSQGGRAATCGKYLFNWAVRTKLKLTPIPAASQLDLSSWFVAGYSGGDIYHSLSRARPRWFM
2VSI , Knot 106 236 0.81 40 158 226 
HMIYAGILAGGTGTRMGISNLPKQFLELGDRPILIHTIEKFVLEPSIEKIVVGVHGDWVSHAEDLVDKYLPLYKERIIITKGGADRNTSIKNIIEAIDAYRPLTPEDIVVTHDSVRPFITLRMIQDNIQLAQNHDAVDTVVEAVDTIVESTNGQFITDIPNRAHLYQGQTPQTFRCKDFMDLYGSLSDEEKEILTDACKIFVIKGKDVALAKGEYSNLKITTVTDLKIAKSMIEKD
9KGZ , Knot 52 125 0.67 28 68 112 
KKIKLNIKEFKATAEGLSPEEKELWDKFAEKLKAELNNKIINLGEKIEIEEELKTPTKSIKITFSLELVSEDTFKATLKLEIKGKETIVEEETVEFKAGETVKLTIKLPDGKTFTLELKLEATKI

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 \(c\).

\(|P_{f(5PZK_1)}(2) \setminus P_{f(2VSI_1)}(2)|=155\), \(|P_{f(2VSI_1)}(2) \setminus P_{f(5PZK_1)}(2)|=35\). 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 1:1010001011110101100001110110001100001101000001010000101001011000000110010101001010110100100101100100011011001001000110010011001100000110001110001101010011001101111101110100011100110011011110001100011001011101101000111110000010001000010100010000011101001100100010111110000100010000010111000010010001011110011010000111010011110001100000101011001100001111011010001011000000101100101001001000100111011100100011001110111011011101111001101111000100110001011000101101101100101101101000011010011001001111110110001001010110011011001001101110001010111110010100111110011010001001010111
Pair \(Z_2\) Length of longest common subsequence
5PZK_1,2VSI_1 190 3
5PZK_1,9KGZ_1 236 3
2VSI_1,9KGZ_1 142 4

Newick tree

 
[
	5PZK_1:11.70,
	[
		2VSI_1:71,9KGZ_1:71
	]:45.70
]

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 \((0.85)(0.8)(\frac{810 }{\log_{20} 810}-\frac{236}{\log_{20}236})=158.\)
Status Protein1 Protein2 d d1/2
Query variables 5PZK_1 2VSI_1 205 141.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]