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Parikh vectors
1BZA_1 5AFY_1 4OJD_1 Letter Amino acid
14 11 22 N Asparagine
18 22 35 G Glycine
5 7 12 M Methionine
15 17 24 V Valine
35 12 34 A Alanine
14 16 34 D Aspartic acid
1 7 14 C Cysteine
12 14 34 E Glutamic acid
5 5 23 H Histidine
28 20 32 L Leucine
16 19 26 K Lycine
15 14 49 S Serine
13 18 27 R Arginine
3 9 5 W Tryptophan
5 10 15 Y Tyrosine
5 9 28 F Phenylalanine
11 16 32 I Isoleucine
10 14 21 P Proline
25 10 35 T Threonine
12 8 24 Q Glutamine 

1BZA_1|Chain A|BETA-LACTAMASE|Escherichia coli (562)
>5AFY_1|Chain A[auth H]|Prothrombin|Homo sapiens (9606)
>4OJD_1|Chain A[auth H]|EFF-1A|Caenorhabditis elegans (6239)
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 \(w=f(c)\)
1BZA , Knot 118 262 0.83 40 168 251 
ANSVQQQLEALEKSSGGRLGVALINTADNSQILYRADERFAMCSTSKVMAAAAVLKQSESDKHLLNQRVEIKKSDLVNYNPIAEKHVNGTMTLAELGAAALQYSDNTAMNKLIAHLGGPDKVTAFARSLGDETFRLDRTAPTLNTAIPGDPRDTTTPLAMAQTLKNLTLGKALAETQRAQLVTWLKGNTTGSASIRAGLPKSWVVGDKTGSGDYGTTNDIAVIWPENHAPLVLVTYFTQPEQKAERRRDILAAAAKIVTHGF
5AFY , Knot 119 258 0.85 40 186 244 
IVEGSDAEIGMSPWQVMLFRKSPQELLCGASLISDRWVLTAAHCLLYPPWDKNFTENDLLVRIGKHSRTRYERNIEKISMLEKIYIHPRYNWRENLDRDIALMKLKKPVAFSDYIHPVCLPDRETAASLLQAGYKGRVTGWGNLKETWTANVGKGQPSVLQVVNLPIVERPVCKDSTRIRITDNMFCAGYKPDEGKRGDACEGDSGGPFVMKSPFNNRWYQMGIVSWGEGCDRDGKYGFYTHVFRLKKWIQKVIDQFG
4OJD , Knot 214 526 0.85 40 272 489 
RSFPLEEKFDGLFRAEPPHCSKTPIVRAQTSQNAMSSIARGMQMQFSIGLHTAVCFRLYEDTQLASQEINDDENAGNQTSLLHTIRLEKLEHHHPITQRYTFGIPEVHASCICECDATSSTCTAESHQFTACPESDKSDETSSCYRTFFPNQTPIGCSEDDIPKLCCDVRFKPYKNMTFLAVKLEQPTTYATFVYAAYDFVNGYWVEKDKTKIRSQLDGGTQDRHLDQKRRISLAVTAGARASHQLETGMYFSRTSNGGETEELRMQPLNEITDNNFDRLGWYRMDDSGHFHVNNGVVKMEEIHKAKVKNCKEQTYKSILSANHYMPGHFNLTRPLEVIKPWIQSARIFDSSLRQAVVTHAEGTNLQISIHLDDEVESQNLVFFHNASRIRDFSGSIIVDSKSNRLFNLTVYEASGKIDGSVKMSTGFGSDTIHTFTAYVSDLHASNRSMIIPLPAIVGQGARAICLRADSMADIDKICHVIEYFESPLFEDDDDKAGWSHPQFEKGGGSGGGSGGGSWSHPQFEK

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 \(c\).

\(|P_{f(1BZA_1)}(2) \setminus P_{f(5AFY_1)}(2)|=81\), \(|P_{f(5AFY_1)}(2) \setminus P_{f(1BZA_1)}(2)|=99\). 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 1:1001000101100001101111110010000110010001110000011111111000000001100010100001100011100010101011011111100000011001110111100101110011000101000110100111101000001111100100101101110000101101101000101010111100111100010100100001111110001111110010010001000001111110110011
Pair \(Z_2\) Length of longest common subsequence
1BZA_1,5AFY_1 180 3
1BZA_1,4OJD_1 188 4
5AFY_1,4OJD_1 192 4

Newick tree

 
[
	4OJD_1:96.61,
	[
		1BZA_1:90,5AFY_1:90
	]:6.61
]

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 \((0.85)(0.8)(\frac{520 }{\log_{20} 520}-\frac{258}{\log_{20}258})=74.7\)
Status Protein1 Protein2 d d1/2
Query variables 1BZA_1 5AFY_1 97 95.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]