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Parikh vectors
6VET_1 4FOZ_1 5YQR_1 Letter Amino acid
0 34 21 A Alanine
2 20 14 E Glutamic acid
0 21 19 K Lycine
2 25 12 Y Tyrosine
0 28 23 T Threonine
1 19 20 R Arginine
4 0 2 C Cysteine
2 17 9 Q Glutamine
1 52 15 G Glycine
0 12 7 P Proline
2 21 22 N Asparagine
2 15 17 I Isoleucine
2 33 25 L Leucine
0 5 6 M Methionine
0 23 14 F Phenylalanine
0 34 12 D Aspartic acid
1 13 3 H Histidine
1 31 15 S Serine
0 5 5 W Tryptophan
1 18 15 V Valine 

6VET_1|Chains A, C, E|Insulin A chain|Homo sapiens (9606)
>4FOZ_1|Chain A|Porin D|Pseudomonas aeruginosa (208964)
>5YQR_1|Chain A|Endolysin/Membrane-anchored lipid-binding protein LAM6 fusion protein|Enterobacteria phage T4 (10665)
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 \(w=f(c)\)
6VET , Knot 16 21 0.77 24 20 19 
GIVEQCCHRICSLYQLENYCN
4FOZ , Knot 175 426 0.83 38 223 404 
HHHHHHDAFVSDQAEAKGFIEDSSLDLLLRNYYFNRDGKSGSGDRVDWTQGFLTTYESGFTQGTVGFGVDAFGYLGLKLDGTSDKTGTGNLPVMNDGKPRDDYSRAGGAVKVRISKTMLKWGEMQPTAPVFAAGGSRLFPQTATGFQLQSSEFEGLDLEAGHFTEGKEPTTVKSRGELYATYAGETAKSADFIGGRYAITDNLSASLYGAELEDIYRQYYLNSNYTIPLASDQSLGFDFNIYRTNDEGKAKAGDISNTTWSLAVAYTLDAHTFTLAYQKVHGDQPFDRIGFGRNGSGAGGDSIFLANSVQYSHFNGPGEKSWQARYDLNLASYGVPGLTFMVRYINGKDIDGTKMSDNNVGYKNYGYGEDGKHHETNLEAKYVVQSGPAKDLSFRIRQAWHRANADQGEGDQNEFRLIVDYPLSIL
5YQR , Knot 121 276 0.82 40 180 264 
GSNIFEMLRIDEGLRLKIYKNTEGYYTIGIGHLLTKSPSLNAAKSELDKAIGRNTNGVITKDEAEKLFNQDVDAAVRGILRNAKLKPVYDSLDAVRRAALINMVFQMGETGVAGFTNSLRMLQQKRWDEAAVNLAKSRWYNQTPNRAKRVITTFRTGTWDAYGSEANKKFRQMFKPLAPNTRLITDYFCYFHREFPYQGRIYLSNTHLCFNSTVLNWMAKLQIPLNEIKYLDKVTTNSSAISVETVTNRYTFSGFIARDEVFQLITRVWSKENLTN

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 \(c\).

\(|P_{f(6VET_1)}(2) \setminus P_{f(4FOZ_1)}(2)|=11\), \(|P_{f(4FOZ_1)}(2) \setminus P_{f(6VET_1)}(2)|=214\). 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 1:111000000100100100000
Pair \(Z_2\) Length of longest common subsequence
6VET_1,4FOZ_1 225 3
6VET_1,5YQR_1 184 2
4FOZ_1,5YQR_1 151 3

Newick tree

 
[
	6VET_1:11.36,
	[
		5YQR_1:75.5,4FOZ_1:75.5
	]:34.86
]

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 \((0.85)(0.8)(\frac{447 }{\log_{20} 447}-\frac{21}{\log_{20}21})=135.\)
Status Protein1 Protein2 d d1/2
Query variables 6VET_1 4FOZ_1 169 89
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]

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