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Parikh vectors
1OYU_1 6LKS_1 6EXB_1 Letter Amino acid
11 19 6 I Isoleucine
5 10 8 F Phenylalanine
11 11 14 D Aspartic acid
15 24 15 G Glycine
1 11 2 H Histidine
9 22 18 E Glutamic acid
3 21 16 P Proline
7 32 10 S Serine
11 17 7 T Threonine
19 18 25 A Alanine
12 13 19 R Arginine
0 9 2 C Cysteine
17 28 21 L Leucine
3 6 5 W Tryptophan
5 2 11 M Methionine
6 14 7 Y Tyrosine
9 21 21 V Valine
11 26 8 N Asparagine
5 8 6 Q Glutamine
15 18 14 K Lycine 

1OYU_1|Chains A, B|Lysozyme|Enterobacteria phage T4 (10665)
>6LKS_1|Chains A, C, E, G, I, K|Hemagglutinin HA1 chain|Influenza A virus (A/Thailand/CU44/2006(H1N1)) (457457)
>6EXB_1|Chains A, B|IcmP (DotM)|Legionella pneumophila subsp. pneumophila (strain Philadelphia 1 / ATCC 33152 / DSM 7513) (272624)
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 \(w=f(c)\)
1OYU , Knot 77 175 0.75 38 116 156 
MNIFEMLRIDEGLRLKIYKDTEGYYTIGIGHLLTKSPSLNAAKSELDKAINAAKSELDKAIGGGGGGVITKDEAEKLFNQDVDAAVRGILRNAKLKPVYDSLDAVRRAALINMVFQMGETGVAGFTNSLRMLQQKRWDEAAVNLAKSRWYNQTPNRAKRVITTFRTGTWDAYKNL
6LKS , Knot 142 330 0.83 40 200 309 
ADPGDTICIGYHANNSTDTVDTVLEKNVTVTHSVNLLEDSHNGKLCLLKGIAPLQLGNCSVAGWILGNPECELLISKESWSYIVEKPNPENGTCYPGHFADYEELREQLSSVSSFERFEIFPKESSWPNHTVTGVSASCSHNGKSSFYKNLLWLTGKNGLYPNLSKSYANNKEKEVLVLWGVHHPPNIGDQRALYHTENAYVSVVSSHYSRKFTPEIAKRPKVRDQEGRINYYWTLLEPGDTIIFEANGNLIAPRYAFALSRGFGSGIINSNAPMDECDAKCQTPQGAINSSLPFQNVHPVTIGECPKYVRSAKLRMVTGLRNIPSIQSR
6EXB , Knot 111 235 0.86 40 164 232 
GPSGGGADVNKGPWAMALTPMEFARKYNLLRKDDALLDNPVPGEEMTAGIRRGDAKRVFTMQLGPYWDGFERCSPQAYALSAVFMARMNRDRDAANNILKVLDKTFVDGKPDFSVARPVMKKYQNSELVQEVVAKHAYVLTVIASLLEAAREDGVVPSSEFLWLKPVDRRLWYMLNCVGRQTPYSEVAGPFAHWKAEKEMGRRSLVPMIDEAIRALEIAVKEVRLTPRQMEELEP

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 \(c\).

\(|P_{f(1OYU_1)}(2) \setminus P_{f(6LKS_1)}(2)|=51\), \(|P_{f(6LKS_1)}(2) \setminus P_{f(1OYU_1)}(2)|=135\). 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 1:1011011010011010100000100011110110001010110001001101100010011111111110000100110001011101110010101100010110011110111011001111100010110000100111011000100001001001100100101010001
Pair \(Z_2\) Length of longest common subsequence
1OYU_1,6LKS_1 186 4
1OYU_1,6EXB_1 148 4
6LKS_1,6EXB_1 178 3

Newick tree

 
[
	6LKS_1:96.02,
	[
		1OYU_1:74,6EXB_1:74
	]:22.02
]

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 \((0.85)(0.8)(\frac{505 }{\log_{20} 505}-\frac{175}{\log_{20}175})=96.2\)
Status Protein1 Protein2 d d1/2
Query variables 1OYU_1 6LKS_1 122 90
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]