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Parikh vectors
5PZM_1 5RZT_1 2VAT_1 Letter Amino acid
13 6 13 M Methionine
48 19 39 S Serine
38 9 22 T Threonine
35 16 26 V Valine
57 14 43 A Alanine
42 18 32 R Arginine
12 11 12 H Histidine
59 31 29 L Leucine
15 16 21 F Phenylalanine
18 8 19 Q Glutamine
27 22 16 E Glutamic acid
30 23 13 K Lycine
30 12 29 P Proline
23 13 14 Y Tyrosine
21 8 11 C Cysteine
23 13 26 I Isoleucine
31 15 29 G Glycine
10 4 6 W Tryptophan
16 10 13 N Asparagine
26 18 31 D Aspartic acid

5PZM_1|Chains A, B|RNA-directed RNA polymerase|Hepatitis C virus genotype 1b (isolate Con1) (333284)
>5RZT_1|Chain A|Isoform 2 of Band 4.1-like protein 3|Homo sapiens (9606)
>2VAT_1|Chains A, B, C, D, E, F, G, H, I, J, K, L|ACETYL-COA--DEACETYLCEPHALOSPORIN C ACETYLTRANSFERASE|ACREMONIUM CHRYSOGENUM (5044)
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 \(w=f(c)\)
5PZM , Knot 230 574 0.84 40 278 537
MSMSYTWTGALITPCAAEETKLPINALSNSLLRHHNLVYATTSRSASLRQKKVTFDRLQVLDDHYRDVLKEMKAKASTVKAKLLSVEEACKLTPPHSARSKFGYGAKDVRNLSSKAVNHIRSVWKDLLEDTETPIDTTIMAKNEVFCVQPEKGGRKPARLIVFPDLGVRVCEKMALYDVVSTLPQAVMGSSYGFQYSPGQRVEFLVNAWKAKKCPMGFAYDTRCFDSTVTENDIRVEESIYQCCDLAPEARQAIRSLTERLYIGGPLTNSKGQNCGYRRCRASGVLTTSCGNTLTCYLKAAAACRAAKLQDCTMLVCGDDLVVICESAGTQEDEASLRAFTEAMTRYSAPPGDPPKPEYDLELITSCSSNVSVAHDASGKRVYYLTRDPTTPLARAAWETARHTPVNSWLGNIIMYAPTLWARMILMTHFFSILLAQEQLEKALDCQIYGACYSIEPLDLPQIIQRLHGLSAFSLHSYSPGEINRVASCLRKLGVPPLRVWRHRARSVRARLLSQGGRAATCGKYLFNWAVRTKLKLTPIPAASQLDLSSWFVAGYSGGDIYHSLSRARPRWFM
5RZT , Knot 130 286 0.85 40 190 277
SMPKSMQCKVILLDGSEYTCDVEKRSRGQVLFDKVCEHLNLLEKDYFGLTYRDAENQKNWLDPAKEIKKQVRSGAWHFSFNVKFYPPDPAQLSEDITRYYLCLQLRDDIVSGRLPCSFVTLALLGSYTVQSELGDYDPDECGSDYISEFRFAPNHTKELEDKVIELHKSHRGMTPAEAEMHFLENAKKLSMYGVDLHHAKDSEGVEIMLGVCASGLLIYRDRLRINRFAWPKVLKISYKRNNFYIKIRPGEFEQFESTIGFKLPNHRAAKRLWKVCVEHHTFFRLL
2VAT , Knot 188 444 0.86 40 241 424
MLPSAQVARLKPDPFPPSLSPIPHGAVTFAALAPCHNLPIFSSRQMLRDSLTYSHTSPTMSPQIANRFEASLDAQDIARISLFTLESGVILRDVPVAYKSWGRMNVSRDNCVIVCHTLTSSAHVTSWWPTLFGQGRAFDTSRYFIICLNYLGSPFGSAGPCSPDPDAEGQRPYGAKFPRTTIRDDVRIHRQVLDRLGVRQIAAVVGASMGGMHTLEWAFFGPEYVRKIVPIATSCRQSGWCAAWFETQRQCIYDDPKYLDGEYDVDDQPVRGLETARKIANLTYKSKPAMDERFHMAPGVQAGRNISSQDAKKEINGTDSGNSHRAGQPIEAVSSYLRYQAQKFAASFDANCYIAMTLKFDTHDISRGRAGSIPEALAMITQPALIICARSDGLYSFDEHVEMGRSIPNSRLCVVDTNEGHDFFVMEADKVNDAVRGFLDQSLM

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 \(c\).

\(|P_{f(5PZM_1)}(2) \setminus P_{f(5RZT_1)}(2)|=132\), \(|P_{f(5RZT_1)}(2) \setminus P_{f(5PZM_1)}(2)|=44\). 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
5PZM_1,5RZT_1 176 4
5PZM_1,2VAT_1 151 4
5RZT_1,2VAT_1 173 3

Newick tree

 
[
	5RZT_1:90.83,
	[
		5PZM_1:75.5,2VAT_1:75.5
	]:15.33
]

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 \((0.85)(0.8)(\frac{860 }{\log_{20} 860}-\frac{286}{\log_{20}286})=156.\)
Status Protein1 Protein2 d d1/2
Query variables 5PZM_1 5RZT_1 200 149
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]