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Parikh vectors
6LTN_1 9NLE_1 5TVK_1 Letter Amino acid
10 6 5 M Methionine
15 12 20 T Threonine
17 3 11 Y Tyrosine
27 30 10 A Alanine
11 0 1 C Cysteine
19 8 3 Q Glutamine
29 28 9 V Valine
62 15 33 L Leucine
24 4 15 F Phenylalanine
18 7 6 P Proline
6 0 1 W Tryptophan
19 11 21 N Asparagine
18 19 9 G Glycine
27 11 16 E Glutamic acid
5 2 9 H Histidine
30 11 31 I Isoleucine
29 22 27 K Lycine
39 6 15 S Serine
12 11 7 R Arginine
19 14 18 D Aspartic acid 

6LTN_1|Chains A, B, C, D, E, F, G|Pannexin-1|Homo sapiens (9606)
>9NLE_1|Chain A[auth 1]|Large ribosomal subunit protein uL1|Escherichia coli (562)
>5TVK_1|Chains A, B|PF-32 protein|Borrelia burgdorferi (strain ATCC 35210 / B31 / CIP 102532 / DSM 4680) (224326)
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 \(w=f(c)\)
6LTN , Knot 175 436 0.81 40 224 411 
MAIAQLATEYVFSDFLLKEPTEPKFKGLRLELAVDKMVTCIAVGLPLLLISLAFAQEISIGTQISCFSPSSFSWRQAAFVDSYCWAAVQQKNSLQSESGNLPLWLHKFFPYILLLFAILLYLPPLFWRFAAAPHICSDLKFIMEELDKVYNRAIKAAKSARDLDMRDGACSVPGVTENLGQSLWEVSESHFKYPIVEQYLKTKKNSNNLIIKYISCRLLTLIIILLACIYLGYYFSLSSLSDEFVCSIKSGILRNDSTVPDQFQCKLIAVGIFQLLSVINLVVYVLLAPVVVYTLFVPFRQKTDVLKVYEILPTFDVLHFKSEGYNDLSLYNLFLEENISEVKSYKCLKVLENIKSSGQGIDPMLLLTNLGMIKMDVVDGKTPMSAEMREEQGNQTAELQGMNIDSETKANNGEKNARQRLLDSSCLEWSHPQFEK
9NLE , Knot 100 220 0.81 36 137 208 
KRMRVIREKVDATKQYDINEAIALLKELATAKFVESVDVAVNLGIDARKSDQNVRGATVLPHGTGRSVRVAVFTQGANAEAAKAAGAELVGMEDLADQIKKGEMNFDVVIASPDAMRVVGQLGQVLGPRGLMPNPKVGTVTPNVAEAVKNAKAGQVRYRNDKNGIIHTTIGKVDFDADKLKENLEALLVALKKAKPTQAKGVYIKKVSISTTMGAGVAVD
5TVK , Knot 114 267 0.79 40 153 248 
MAHHHHHHMGTLEAQTQGPGSMDNKKPKIITIASIKGGVGKSTSAIILATLLSKNNKVLLIDMDTQASITSYFYEKIEKLGINFTKFNIYEILKENVDIDSTIINVDNNLDLIPSYLTLHNFSEDKIEHKDFLLKTSLGTLYYKYDYIVIDTNPSLDVTLKNALLCSDYVIIPMTAEKWAVESLDLFNFFVRKLNLFLPIFLIITRFKKNRTHKTLFEILKTKDRFLGTISEREDLNRRIAENNNFDLNKDYIKEYENILEIFLKKI

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 \(c\).

\(|P_{f(6LTN_1)}(2) \setminus P_{f(9NLE_1)}(2)|=132\), \(|P_{f(9NLE_1)}(2) \setminus P_{f(6LTN_1)}(2)|=45\). 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 1:1111011000110011100100101011010111001100111111111101111001011001001010010100111100001111000001000010111110011101111111110111111011111010001011100100100011011001001010011001111000110011010000100111000100000000111001000110111111101011001010010001100100111000001100100011111110110110111011111111001111100000110100111010110100010001010011100010010000010110010001011011111001111010110100110101000010001010110100000100100010001100001010010100
Pair \(Z_2\) Length of longest common subsequence
6LTN_1,9NLE_1 177 4
6LTN_1,5TVK_1 167 5
9NLE_1,5TVK_1 146 4

Newick tree

 
[
	6LTN_1:89.96,
	[
		5TVK_1:73,9NLE_1:73
	]:16.96
]

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 \((0.85)(0.8)(\frac{656 }{\log_{20} 656}-\frac{220}{\log_{20}220})=122.\)
Status Protein1 Protein2 d d1/2
Query variables 6LTN_1 9NLE_1 155 114
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]