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Parikh vectors
7CWH_1 5NDD_1 4OQQ_1 Letter Amino acid
0 9 1 W Tryptophan
1 25 9 Y Tyrosine
1 47 22 V Valine
0 23 18 E Glutamic acid
0 42 19 I Isoleucine
0 74 24 L Leucine
0 30 11 F Phenylalanine
1 33 14 T Threonine
0 29 18 D Aspartic acid
0 19 11 Q Glutamine
1 32 14 S Serine
0 74 21 A Alanine
0 8 2 C Cysteine
1 21 7 H Histidine
2 41 15 K Lycine
2 25 14 R Arginine
0 30 9 N Asparagine
1 24 21 G Glycine
0 15 6 M Methionine
1 18 8 P Proline 

7CWH_1|Chain A|Peptide from Histone H3.3|Homo sapiens (9606)
>5NDD_1|Chain A|Lysozyme,Proteinase-activated receptor 2,Soluble cytochrome b562,Proteinase-activated receptor 2|Enterobacteria phage T4 (10665)
>4OQQ_1|Chains A, B|Deoxyribonucleoside regulator|Bacillus subtilis subsp. subtilis (224308)
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 \(w=f(c)\)
7CWH , Knot 9 11 0.65 18 10 9 
STGRVKKPHRY
5NDD , Knot 239 619 0.82 40 267 555 
MVSAIVLYVLLAAAAHSAFAAASNIFEMLRIDEGLRLKIYKDTEGYYTIGIGHLLTKSPSLNAAKSELDKAIGRNTNGVITKDEAEKLFNQDVDAAVRGILRNAKLKPVYDSLDAVRRAALINMVFQMGETGVAGFTNSLRMLQQKRWDEAAVNLAKSRWYNQTPNRAKRVITTFRTGTWDAYIYEFFSVDEFSASVLTGKLTTVFLPIVYTIVFVVALPSNGMALWVFLFRTKKKAPAVIYMANLALADLLSVIWFPLKIAYHIHGNNWIYGEALCNVLIGFFYANMYCSILFLTCLSVQRAWEIVNPMGHSRKKANIAIGISLAIWLLILLVTIPLYVVKQTIFIPALQITTCHDVLPEQLLVGDMFNYFLSLAIGVFLFPAFLTASAYVLMIRALADLEDNWETLNDNLKVIEKADNAAQVKDALTKMRAAALDAQKATPPKLEDKSPDSPEMKDFRHGFDILVGQIDDALKLANEGKVKEAQAAAEQLKTTRNAYIQKYLENSEKKRKRAIKLAVTVAAMYLICFTPSNLLLVVHYFLIKSQGQSHVYALYIVALCLSTLNSCIDPFVYYFVSHDFRDHAKNALLCRSVRTVKQMQVSLTSKAAAHHHHHHHHHH
4OQQ , Knot 119 264 0.83 40 173 253 
GIDPFTDPFEDLDALGSILEEKYGLLEAHVVFSPTPDYAGITHDLSRYGAEYMHETVKDGDIVGVSWGTTMYQIAQNMQPKQVKGVEVVQLKGGISHSRVNTYSAETIQLFAEAFQTMPRYLPLPVVFDNADVKRMVEKDRHIERIIEMGKQANIALFTVGTVRDEALLFRLGYFNEEEKALLKKQAVGDICSRFFDAKGNICSSAINDRTIGVELQDLRLKERSILVAGGSRKVSSIHGALTGKYANVLIIDQHTARALVNDL

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 \(c\).

\(|P_{f(7CWH_1)}(2) \setminus P_{f(5NDD_1)}(2)|=3\), \(|P_{f(5NDD_1)}(2) \setminus P_{f(7CWH_1)}(2)|=260\). 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 1:00101001000
Pair \(Z_2\) Length of longest common subsequence
7CWH_1,5NDD_1 263 2
7CWH_1,4OQQ_1 173 2
5NDD_1,4OQQ_1 166 4

Newick tree

 
[
	7CWH_1:11.24,
	[
		4OQQ_1:83,5NDD_1:83
	]:36.24
]

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 \((0.85)(0.8)(\frac{630 }{\log_{20} 630}-\frac{11}{\log_{20}11})=189.\)
Status Protein1 Protein2 d d1/2
Query variables 7CWH_1 5NDD_1 235 119.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]

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