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Parikh vectors
1IYI_1 6ZJN_1 6OOW_1 Letter Amino acid
10 31 31 P Proline
6 8 3 W Tryptophan
13 33 44 A Alanine
3 6 6 C Cysteine
12 36 23 E Glutamic acid
7 49 31 G Glycine
15 17 26 D Aspartic acid
14 21 14 I Isoleucine
12 23 13 K Lycine
9 18 9 Y Tyrosine
8 16 19 F Phenylalanine
5 21 19 S Serine
14 22 21 T Threonine
9 23 29 V Valine
10 30 29 R Arginine
9 10 11 Q Glutamine
5 9 8 H Histidine
23 43 41 L Leucine
8 9 11 N Asparagine
6 13 5 M Methionine 

1IYI_1|Chains A, B, C, D|HEMATOPOIETIC PROSTAGLANDIN D SYNTHASE|Homo sapiens (9606)
>6ZJN_1|Chain A[auth 1]|NADH-quinone oxidoreductase subunit 1|Thermus thermophilus (274)
>6OOW_1|Chain A|Cytochrome P450|Rhodopseudomonas palustris (strain HaA2) (316058)
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 \(w=f(c)\)
1IYI , Knot 96 198 0.85 40 150 195 
PNYKLTYFNMRGRAEIIRYIFAYLDIQYEDHRIEQADWPEIKSTLPFGKIPILEVDGLTLHQSLAIARYLTKNTDLAGNTEMEQCHVDAIVDTLDDFMSCFPWAEKKQDVKEQMFNELLTYNAPHLMQDLDTYLGGREWLIGMSVTWADFYWEICSTTLLVFKPDLLDNHPRLVTLRKKVQAIPAVANWIKRRPQTKL
6ZJN , Knot 184 438 0.85 40 230 414 
MTGPILSGLDPRFERTLYAHVGKEGSWTLDYYLRHGGYETAKRVLKEKTPDEVIEEVKRSGLRGRGGAGFPTGLKWSFMPKDDGKQHYLICNADESEPGSFKDRYILEDVPHLLIEGMILAGYAIRATVGYIYVRGEYRRAADRLEQAIKEARARGYLGKNLFGTDFSFDLHVHRGAGAYICGEETALMNSLEGLRANPRLKPPFPAQSGLWGKPTTINNVETLASVVPIMERGADWFAQMGTEQSKGMKLYQISGPVKRPGVYELPMGTTFRELIYEWAGGPLEPIQAIIPGGSSTPPLPFTEEVLDTPMSYEHLQAKGSMLGTGGVILIPERVSMVDAMWNLTRFYAHESCGKCTPCREGVAGFMVNLFAKIGTGQGEEKDVENLEALLPLIEGRSFCPLADAAVWPVKGSLRHFKDQYLALAREKRPVPRPSLWR
6OOW , Knot 166 393 0.84 40 206 372 
TIPHLAIDPFSLDFFDDPYPDQQTLRDAGPVVYLDKWNVYGVARYAEVHAVLNDPTTFCSSRGVGLSDFKKEKPWRPPSLILEADPPAHTRPRAVLSKVLSPATMKTIRDGFAAAADAKVDELLQRGCIDAIADLAEAYPLSVFPDAMGLKQEGREHLLPYAGLVFNAFGPPNELRQTAIERSAPHQAYVNEQCQRPNLAPGGFGACIHAFTDTGEITPDEAPLLVRSLLSAGLDATVNGIGAAVYCLARFPGELQRLRSDPTLARNAFEEAVRFESPVQTFFRTTTREVELGGAVIGEGEKVLMFLGSANRDPRRWSDPDLYDITRKTSGHVGFGSGVHMCVGQLVARLEGEVMLSALARKVAAIDIDGPVKRRFNNTLRGLESLPVKLTPA

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 \(c\).

\(|P_{f(1IYI_1)}(2) \setminus P_{f(6ZJN_1)}(2)|=51\), \(|P_{f(6ZJN_1)}(2) \setminus P_{f(1IYI_1)}(2)|=131\). 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 1:100010010101010110011101010000001001011010001111011110101101000111100100000111000100001011100100110011110000010001100110001101100100011100111110101101010100001111010110001011010001011111101100010001
Pair \(Z_2\) Length of longest common subsequence
1IYI_1,6ZJN_1 182 3
1IYI_1,6OOW_1 176 3
6ZJN_1,6OOW_1 152 5

Newick tree

 
[
	1IYI_1:93.58,
	[
		6OOW_1:76,6ZJN_1:76
	]:17.58
]

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 \((0.85)(0.8)(\frac{636 }{\log_{20} 636}-\frac{198}{\log_{20}198})=124.\)
Status Protein1 Protein2 d d1/2
Query variables 1IYI_1 6ZJN_1 159 115.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]