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Parikh vectors
3BZC_1 6BZK_1 1KEO_1 Letter Amino acid
18 3 3 H Histidine
53 8 10 K Lycine
38 1 14 S Serine
40 1 7 T Threonine
4 0 1 W Tryptophan
13 1 5 Y Tyrosine
51 6 7 D Aspartic acid
63 11 15 E Glutamic acid
28 1 5 I Isoleucine
87 8 11 L Leucine
25 0 8 F Phenylalanine
62 1 11 V Valine
53 4 9 R Arginine
20 3 7 N Asparagine
3 0 6 C Cysteine
26 3 9 Q Glutamine
58 2 13 G Glycine
92 4 6 A Alanine
13 0 4 M Methionine
38 0 3 P Proline 

3BZC_1|Chain A|Tex|Pseudomonas aeruginosa (287)
>6BZK_1|Chain A|M protein|Streptococcus pyogenes (1314)
>1KEO_1|Chains A, B|cation-dependent mannose-6-phosphate receptor|Bos taurus (9913)
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 \(w=f(c)\)
3BZC , Knot 288 785 0.81 40 266 695 
MDSINTRIAEELSALPSGRVQPQQVAAAVALLDEGSTVPFIARYRKEVTGSLDDTQLRMLEERLRYLRELEERRGAILASIEEQGKLTPELARDIKLADTKTRLEDLYLPYKQKRRTKGQIALEAGLGALADALFDDPTLVPESEAARFVDAEKGFADVKAVLEGAKYILMERFAEDATLLDKLRVFMKNEATLTARVVPGKEQEGAKFSDYFEHDEPLKSAPSHRALAIFRGRNEGVLSASLKVGEEAPGTLHPCEVMIAERFGLSNQGRAADKWLAEVVRWTWKVKLYTHLETDLFGELRDGAEDEAISVFARNLHDLLLAAPAGPRATLGLDPGLRTGVKVAVVDATGKLLDTATVYPHAPKNQWDQTLAVLAALCAKHQVELIAIGNGTASRETDKLAGELIKKYPGMKLTKIMVSEAGASVYSASELAAKEFPELDVSLRGAVSIARRLQDPLAELVKIEPKSIGVGQYQHDVSQLKLARSLDAVVEDCVNAVGVDVNTASAALLARISGLNSTLAQNIVAHRDANGAFRTRDELKKVSRLGEKTFEQAAGFLRVMNGDNPLDASAVHPETYPLVQRIAADTERDIRSLIGDSAFLKRLDPKKFTDETFGLPTVTDILKELDKPGRDPRPEFKTAEFQEGVESLKDLKPGMVLEGVVTNVTNFGAFVDIGVHQDGLVHISALSEKFVKDPYEVVKAGDIVKVKVMEVDIPRNRVGLSMRMSDTPGEKVEGQRGGRPTGSGQPRQERGAPRGQSAPPANNAMAALFANAKQLKKKHHHHHH
6BZK , Knot 30 57 0.71 30 44 54 
GSKTIQEKEQELKNLKDNVELERLKNERHDHDEEAERKALEDKLADKQEHLDGALRY
1KEO , Knot 76 154 0.82 40 120 150 
TEEKTCDLVGEKGKESEKELALLKRLTPLFQKSFESTVGQSPDMYSYVFRVCREAGQHSSGAGLVQIQKSNGKETVVGRFNETQIFQGSNWIMLIYKGGDEYDNHCGREQRRAVVMISCNRHTLADNFNPVSEERGKVQDCFYLFEMDSSLACS

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 \(c\).

\(|P_{f(3BZC_1)}(2) \setminus P_{f(6BZK_1)}(2)|=227\), \(|P_{f(6BZK_1)}(2) \setminus P_{f(3BZC_1)}(2)|=5\). 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 1:10010001100101110101010011111111001001111100000101010000101100010010010000111110100010101011001011000001001011000000001011101111111011100101110001101101001110101110110011100110010110010111000101010111100001101000100001100110001111101000111010101100111010100111100111000101100111011010101010001000111010011000110111001001111111110101110111001101111010101100101010110001000111111101000101111101010000001110110001110100111001110100100111001101010101110110010011101101010011110000010010110010111000101111010010111110101100011001110001011100000100100110001001111101101001101011010001110011100000100111001110010100100001111010011001001100101010010100110010010111110111001001111101110001110101100011001001101101101011010110001110101000110010100110101010100001110100111100111111101001000000000
Pair \(Z_2\) Length of longest common subsequence
3BZC_1,6BZK_1 232 4
3BZC_1,1KEO_1 212 4
6BZK_1,1KEO_1 124 3

Newick tree

 
[
	3BZC_1:12.20,
	[
		1KEO_1:62,6BZK_1:62
	]:61.20
]

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 \((0.85)(0.8)(\frac{842 }{\log_{20} 842}-\frac{57}{\log_{20}57})=225.\)
Status Protein1 Protein2 d d1/2
Query variables 3BZC_1 6BZK_1 275 146.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]