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Parikh vectors
2BNM_1 5KIN_1 7ROO_1 Letter Amino acid
1 0 0 C Cysteine
3 10 27 Q Glutamine
6 8 5 H Histidine
7 12 37 F Phenylalanine
8 10 48 S Serine
14 15 54 T Threonine
16 22 28 V Valine
7 5 14 R Arginine
10 8 57 N Asparagine
18 23 86 G Glycine
11 14 38 K Lycine
13 15 61 D Aspartic acid
7 23 34 I Isoleucine
21 26 55 L Leucine
20 26 45 A Alanine
15 19 29 E Glutamic acid
5 3 4 M Methionine
11 13 18 P Proline
2 0 20 W Tryptophan
3 6 26 Y Tyrosine 

2BNM_1|Chains A, B|EPOXIDASE|STREPTOMYCES WEDMORENSIS (43759)
>5KIN_1|Chains A, C|Tryptophan synthase alpha chain|Streptococcus pneumoniae serotype 4 (strain ATCC BAA-334 / TIGR4) (170187)
>7ROO_1|Chain A|CylK|Cylindrospermum licheniforme UTEX B 2014 (379530)
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 \(w=f(c)\)
2BNM , Knot 96 198 0.85 40 145 195 
MSNTKTASTGFAELLKDRREQVKMDHAALASLLGETPETVAAWENGEGGELTLTQLGRIAHVLGTSIGALTPPAGNDLDDGVIIQMPDERPILKGVRDNVDYYVYNCLVRTKRAPSLVPLVVDVLTDNPDDAKFNSGHAGNEFLFVLEGEIHMKWGDKENPKEALLPTGASMFVEEHVPHAFTAAKGTGSAKLIAVNF
5KIN , Knot 111 258 0.79 36 153 247 
MPKTLTEKLNAIKAAGKGIFVPYIMAGDHEKGLDGLAETIHFLEDLGVSAIEVGIPFSDPVADGPVIEEAGLRSLAHGTSTQALVETLKTIETEIPLVIMTYFNPLFQYGVENFVKDLADTAVKGLIIPDLPHEHANFVEPFLANTDIALIPLVSLTTGIERQKELIEGAEGFIYAVAINGVTGKSGNYRADLDKHLAQLHQVADIPVLTGFGVSSQADLERFNAVSDGVIVGSKIVKALHQGEPIQDFIRQAVAYQK
7ROO , Knot 246 686 0.78 38 243 574 
MKKNKKTTKSLLSADEKITESLRSTLSDVLPDQLQTYIRTVLQFSGRPEGANLLTGPNTEIEFFSQDPNKNFPNIFAKYSNVLTVSSDPNFITSEDEEVKIIWGRHGSDSLIGFDPGADLVGKRRIDIFLGDFIDEQFNPIPGALNAGKSWSDRFILGDWQKPYYFEDDETLGLNQSAMILDFNPNEDVIQLHGDRQDYELVNISLGTAIFWREKKGYDLIGVLGGVSDLSLKGDYFEFKGNTAPKTVLKTAEHIGTAANDYIFSSTVDAKGNFYVGGGTGGSLGGRNIGARDAWLAKYDSNGNQRWSRQFGSTGTESLWGMASDGSNIYVAGNTTGQLENNTVKGGNDAYLAKYDSDGNQVWIKQNGTYTLEESYKITVDSSGNIYTAGHTFGSLGGPNQNLEQGEVFELPSTDGYVAKFDSNGNQLWVAQFGTITLDDNWGVAADNNGNVFAGGNTKGSFGAKNTGTAGEYDAWLVKLNKDGQTDWVRQFGTPNYDFMWDIETDSLGDIYATGWTLGDLGGKNAGSYDVWLAKYNTNGNQLWIKQFGTSEDDAPFLDGIDIDANDNIFLTGNTNGNLGGANAGSYDAWAAKFDKDGNQLWLKQFGTPDYDTATTVTAVNFGKLYVSGITEGSLGTTNAGSYDSWALKLDADNGEIQDFNSSTNTFGQTGFLNLGSAWSHPQFEK

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 \(c\).

\(|P_{f(2BNM_1)}(2) \setminus P_{f(5KIN_1)}(2)|=66\), \(|P_{f(5KIN_1)}(2) \setminus P_{f(2BNM_1)}(2)|=74\). 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 1:100000100111011000000101001111011100100111100101101010011011011100111101111001001111011000111011000100010001100001101111110110001001010010110011111010101011000010011110110111000110110110101010111101
Pair \(Z_2\) Length of longest common subsequence
2BNM_1,5KIN_1 140 4
2BNM_1,7ROO_1 172 4
5KIN_1,7ROO_1 162 4

Newick tree

 
[
	7ROO_1:87.58,
	[
		2BNM_1:70,5KIN_1:70
	]: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 \((0.85)(0.8)(\frac{456 }{\log_{20} 456}-\frac{198}{\log_{20}198})=75.4\)
Status Protein1 Protein2 d d1/2
Query variables 2BNM_1 5KIN_1 94 83
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]