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Parikh vectors
3PEE_1 7BHJ_1 3LXY_1 Letter Amino acid
10 11 39 A Alanine
26 8 14 E Glutamic acid
13 4 6 Y Tyrosine
15 6 17 N Asparagine
32 9 24 I Isoleucine
28 8 11 S Serine
1 3 1 W Tryptophan
7 5 19 Q Glutamine
12 22 27 G Glycine
3 4 13 H Histidine
20 18 41 L Leucine
9 8 21 P Proline
9 10 9 F Phenylalanine
6 8 23 T Threonine
10 12 25 V Valine
6 10 9 R Arginine
20 10 16 D Aspartic acid
2 3 4 C Cysteine
23 7 9 K Lycine
2 5 6 M Methionine 

3PEE_1|Chains A, B|Toxin B|Clostridium difficile (272563)
>7BHJ_1|Chains A, B, C, D, E|3-hydroxydecanoyl-[acyl-carrier-protein] dehydratase|Pseudomonas aeruginosa PAO1 (208964)
>3LXY_1|Chain A|4-hydroxythreonine-4-phosphate dehydrogenase|Yersinia pestis (632)
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 \(w=f(c)\)
3PEE , Knot 113 254 0.82 40 155 242 
GEDDNLDFSQNIVVDKEYLLEKISSLARSSERGYIHYIVQLQGDKISYEAACNLFAKTPYDSVLFQKNIEDSEIAYYYNPGDGEIQEIDKYKIPSIISDRPKIKLTFIGHGKDEFNTDIFAGFDVDSLSTEIEAAIDLAKEDISPKSIEINLLGCNMFSYSINVEETYPGKLLLKVKDKISELMPSISQDSIIVSANQYEVRINSEGRRELLDHSGEWINKEESIIKDISSKEYISFNPKENKITVKSKNLPEL
7BHJ , Knot 85 171 0.85 40 133 168 
MTKQHAFTREDLLRCSRGELFGPGNAQLPAPNMLMIDRIVHISDVGGKYGKGELVAELDINPDLWFFACHFEGDPVMPGCLGLDAMWQLVGFYLGWQGNPGRGRALGSGEVKFFGQVLPTAKKVTYNIHIKRTINRSLVLAIADGTVSVDGREIYSAEGLRVGLFTSTDSF
3LXY , Knot 144 334 0.83 40 182 309 
SNAMHNHNNRLVITPGEPAGVGPDLAITLAQQDWPVELVVCADPALLLARASQLNLPLQLREYQADQPAIAQQAGSLTILPVKTAVNVVPGKLDVGNSHYVVETLAKACDGAISGEFAALVTGPVQKSIINDAGIPFIGHTEFFADRSHCQRVVMMLATEELRVALATTHLPLLAVPGAITQASLHEVITILDNDLKTKFGITQPQIYVCGLNPHAGEGGHMGHEEIDTIIPALNTLRQQGINLIGPLPADTLFQPKYLQHADAVLAMYHDQGLPVLKYQGFGRAVNITLGLPFIRTSVDHGTALELAATGTADVGSFITALNLAIKMINNSNE

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 \(c\).

\(|P_{f(3PEE_1)}(2) \setminus P_{f(7BHJ_1)}(2)|=90\), \(|P_{f(7BHJ_1)}(2) \setminus P_{f(3PEE_1)}(2)|=68\). 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 1:10000101000111000011001001100000101001101010010001100111001000111000100001100001101010010000110110001010101110100010001111101001000101110110001010010101110011000101000011011101000100111010000111010000101000100011000101100000110010000010101000010100001101
Pair \(Z_2\) Length of longest common subsequence
3PEE_1,7BHJ_1 158 3
3PEE_1,3LXY_1 173 4
7BHJ_1,3LXY_1 153 3

Newick tree

 
[
	3PEE_1:84.84,
	[
		7BHJ_1:76.5,3LXY_1:76.5
	]:8.34
]

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 \((0.85)(0.8)(\frac{425 }{\log_{20} 425}-\frac{171}{\log_{20}171})=75.3\)
Status Protein1 Protein2 d d1/2
Query variables 3PEE_1 7BHJ_1 94 79.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]