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Parikh vectors
7KWL_1 6COG_1 5NGR_1 Letter Amino acid
0 19 6 S Serine
0 8 4 Y Tyrosine
0 8 1 N Asparagine
2 3 2 C Cysteine
0 16 5 I Isoleucine
0 19 9 K Lycine
0 5 11 Q Glutamine
0 20 12 E Glutamic acid
0 15 11 F Phenylalanine
0 18 13 V Valine
2 10 8 T Threonine
0 15 12 D Aspartic acid
0 25 16 L Leucine
0 12 5 M Methionine
0 12 7 P Proline
0 4 4 W Tryptophan
2 16 4 A Alanine
0 8 8 R Arginine
4 17 17 G Glycine
0 10 4 H Histidine 

7KWL_1|Chains A, B|DNA (5'-D(*(DCZ)P*GP*UP*GP*AP*AP*TP*TP*CP*GP*CP*G)-3')|synthetic construct (32630)
>6COG_1|Chains A, B|Alpha-hydroxynitrile lyase|Arabidopsis thaliana (3702)
>5NGR_1|Chains A, B|7,8-dihydro-8-oxoguanine triphosphatase|Homo sapiens (9606)
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 \(w=f(c)\)
7KWL , Knot 8 12 0.55 12 10 10 
XGUGAATTCGCG
6COG , Knot 121 260 0.86 40 189 250 
MERKHHFVLVHNACHGAWIWYKLKPLLESAGHRVTAVELAASGIDPRPIQAVETVDEYSKPLIETLKSLPENEEVILVGFSFGGINIALAADIFPAKIKVLVFLNAFLPDTTHVPSHVLDKLMEMFGGWGDTEFSSHETRNGTMSLLKMGPKFMKARLYQNCPIEDYELAKMLHRQGSFFTEDLSKKEKFSEEGYGSVQRVYVMSSEDKIIPCDFIRWMIDNFNVSKVYEIDGGDHMVMLSKPQKLFDSLSAIATDYMGL
5NGR , Knot 75 159 0.79 40 122 154 
GSHMGASRLYTLVLVLQPQRVLLGMKKRGFGAGRWNGFGGKVQEGETIEDGARRELQEESGLTVDALHKVGQIVFEFVGEPELMDVHVFCTDSIQGTPVESDEMRPCWFQLDQIPFKDMWPDDSYWFPLLLQKKKFHGYFKFQGQDTILDYTLREVDTV

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 \(c\).

\(|P_{f(7KWL_1)}(2) \setminus P_{f(6COG_1)}(2)|=6\), \(|P_{f(6COG_1)}(2) \setminus P_{f(7KWL_1)}(2)|=185\). 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 1:010111000101
Pair \(Z_2\) Length of longest common subsequence
7KWL_1,6COG_1 191 2
7KWL_1,5NGR_1 130 2
6COG_1,5NGR_1 169 3

Newick tree

 
[
	6COG_1:97.11,
	[
		7KWL_1:65,5NGR_1:65
	]:32.11
]

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 \((0.85)(0.8)(\frac{272 }{\log_{20} 272}-\frac{12}{\log_{20}12})=89.0\)
Status Protein1 Protein2 d d1/2
Query variables 7KWL_1 6COG_1 118 62
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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