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Parikh vectors
7NPZ_1 4NQL_1 7NLZ_1 Letter Amino acid
3 9 7 Q Glutamine
4 14 32 G Glycine
3 14 15 I Isoleucine
10 24 31 L Leucine
12 21 8 K Lycine
1 19 22 T Threonine
7 4 3 Y Tyrosine
6 8 17 R Arginine
11 9 21 D Aspartic acid
2 6 3 C Cysteine
5 8 12 E Glutamic acid
6 10 8 H Histidine
6 5 8 M Methionine
6 10 8 S Serine
6 14 39 V Valine
4 7 7 N Asparagine
6 16 12 P Proline
1 1 2 W Tryptophan
5 10 7 F Phenylalanine
11 12 35 A Alanine 

7NPZ_1|Chain A[auth AAA]|Bromodomain-containing protein 2|Homo sapiens (9606)
>4NQL_1|Chain A|AMSH-like protease sst2|Schizosaccharomyces pombe (284812)
>7NLZ_1|Chain A|Acetylglutamate kinase|Mycobacterium tuberculosis H37Rv (83332)
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 \(w=f(c)\)
7NPZ , Knot 59 115 0.81 40 94 113 
GSMGKLSEQLKHCNGILKELLSKKHAAYAWPFYKPVDASALGLHDYHDIIKHPMDLSTVKRKMENRDYRDAQEFAADVRLMFSNCYKYNPPDHDVVAMARKLQDVFEFRYAKMPD
4NQL , Knot 103 221 0.83 40 156 217 
GPLGSMDDNKDIQFIKKPIAVRTSKPRPKPAGTFKIHAYTEGGKPLRTIYLPKLLKKVFLDVVKPNTKKNLATCGILCGKLRQNAFFITHLVIPLQEATSDTCGTTDEASLFEFQDKHNLLTLGWIHTHPTQTCFMSSVALHTHCSYQLMLPEAIAIVMAPSKNTSGIFRLLDPEGLQTIVKCRKPGLFHPHEGKVYTMVAQPGHVREINSKLQVVDLRVK
7NLZ , Knot 124 297 0.79 40 164 272 
GSMVSRIEALPTHIKAQVLAEALPWLKQLHGKVVVVKYGGNAMTDDTLRRAFAADMAFLRNCGIHPVVVHGGGPQITAMLRRLGIEGDFKGGFRVTTPEVLDVARMVLFGQVGRELVNLINAHGPYAVGITGEDAQLFTAVRRSVTVDGVATDIGLVGDVDQVNTAAMLDLVAAGRIPVVSTLAPDADGVVHNINADTAAAAVAEALGAEKLLMLTDIDGLYTRWPDRDSLVSEIDTGTLAQLLPTLESGMVPKVEACLRAVIGGVPSAHIIDGRVTHCVLVELFTDAGTGTKVVRG

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 \(c\).

\(|P_{f(7NPZ_1)}(2) \setminus P_{f(4NQL_1)}(2)|=50\), \(|P_{f(4NQL_1)}(2) \setminus P_{f(7NPZ_1)}(2)|=112\). 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 1:1011010001000011100110000110111100110101111000001100110100100010000000100111010111000000011000111110010011010010110
Pair \(Z_2\) Length of longest common subsequence
7NPZ_1,4NQL_1 162 3
7NPZ_1,7NLZ_1 164 4
4NQL_1,7NLZ_1 162 3

Newick tree

 
[
	7NLZ_1:81.66,
	[
		7NPZ_1:81,4NQL_1:81
	]:0.66
]

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 \((0.85)(0.8)(\frac{336 }{\log_{20} 336}-\frac{115}{\log_{20}115})=68.2\)
Status Protein1 Protein2 d d1/2
Query variables 7NPZ_1 4NQL_1 86 66
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]