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Parikh vectors
1NMN_1 4YPY_1 3DKK_1 Letter Amino acid
10 16 22 R Arginine
15 25 43 G Glycine
13 30 48 L Leucine
7 9 35 S Serine
8 16 31 V Valine
4 4 31 N Asparagine
7 14 31 K Lycine
6 14 30 P Proline
7 14 21 D Aspartic acid
6 11 23 Q Glutamine
10 22 32 E Glutamic acid
10 13 26 I Isoleucine
2 7 8 M Methionine
8 13 33 T Threonine
2 5 14 W Tryptophan
12 18 32 A Alanine
0 3 7 C Cysteine
2 8 7 H Histidine
6 7 37 F Phenylalanine
3 5 18 Y Tyrosine 

1NMN_1|Chains A, B|Hypothetical protein yqgF|Escherichia coli (562)
>4YPY_1|Chain A|tRNA (guanine-N(1)-)-methyltransferase|Haemophilus influenzae (strain ATCC 51907 / DSM 11121 / KW20 / Rd) (71421)
>3DKK_1|Chain A|Cholinesterase|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 \(w=f(c)\)
1NMN , Knot 65 138 0.77 38 101 131 
MSGTLLAFDFGTKSIGVAVGQRITGTARPLPAIKAQDGTPDWNIIERLLKEWQPDEIIVGLPLNMDGTEQPLTARARKFANRIHGRFGVEVKLHDERLSTVEARSGLFEQGGYRALNKGKVDSASAVIILESYFEQGY
4YPY , Knot 116 254 0.84 40 175 245 
GLVPRGSHMWIGVISLFPEMFKAITEFGVTGRAVKHNLLKVECWNPRDFTFDKHKTVDDRPYGGGPGMLMMVQPLRDAIHTAKAAAGEGAKVIYLSPQGRKLDQGGVTELAQNQKLILVCGRYEGIDERLIQTEIDEEWSIGDYVLTGGELPAMTLIDAVARFIPGVLGKQASAEEDSFADGLLDCPHYTRPEVLEGLTVPPVLMSGHHEEIRKWRLKQSLQRTWLRRPELLEGLALTDEQRKLLKEAQAEHNS
3DKK , Knot 218 529 0.86 40 265 503 
EDDIIIATKNGKVRGMQLTVFGGTVTAFLGIPYAQPPLGRLRFKKPQSLTKWSDIWNATKYANSCCQNIDQSFPGFHGSEMWNPNTDLSEDCLYLNVWIPAPKPKNATVLIWIYGGGFQTGTSSLHVYDGKFLARVERVIVVSMNYRVGALGFLALPGNPEAPGNMGLFDQQLALQWVQKNIAAFGGNPKSVTLFGESAGAASVSLHLLSPGSHSLFTRAILQSGSFNAPWAVTSLYEARNRTLNLAKLTGCSRENETEIIKCLRNKDPQEILLNEAFVVPYGTPLSVNFGPTVDGDFLTDMPDILLELGQFKKTQILVGVNKDEGTAFLVYGAPGFSKDNNSIITRKEFQEGLKIFFPGVSEFGKESILFHYTDWVDDQRPENYREALGDVVGDYNFICPALEFTKKFSEWGNNAFFYYFEHRSSKLPWPEWMGVMHGYEIEFVFGLPLERRDQYTKAEEILSRSIVKRWANFAKYGNPQETQNQSTSWPVFKSTEQKYLTLNTESTRIMTKLRAQQCRFWTSFFPKV

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 \(c\).

\(|P_{f(1NMN_1)}(2) \setminus P_{f(4YPY_1)}(2)|=36\), \(|P_{f(4YPY_1)}(2) \setminus P_{f(1NMN_1)}(2)|=110\). 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 1:101011110110001111110010101011111010010101011001100101001111111010100011010100110010101110101000010010100111001100110010100101111100010010
Pair \(Z_2\) Length of longest common subsequence
1NMN_1,4YPY_1 146 4
1NMN_1,3DKK_1 190 4
4YPY_1,3DKK_1 176 4

Newick tree

 
[
	3DKK_1:96.96,
	[
		1NMN_1:73,4YPY_1:73
	]:23.96
]

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 \((0.85)(0.8)(\frac{392 }{\log_{20} 392}-\frac{138}{\log_{20}138})=76.6\)
Status Protein1 Protein2 d d1/2
Query variables 1NMN_1 4YPY_1 98 72.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]