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Parikh vectors
4WYY_1 6SUM_1 6JBQ_1 Letter Amino acid
23 20 10 Q Glutamine
14 25 36 E Glutamic acid
33 17 20 G Glycine
13 21 16 K Lycine
18 13 17 S Serine
18 22 5 Y Tyrosine
9 6 9 N Asparagine
23 24 21 D Aspartic acid
11 19 24 I Isoleucine
44 35 38 L Leucine
26 11 16 P Proline
16 18 19 T Threonine
5 6 1 W Tryptophan
24 17 23 R Arginine
0 1 4 C Cysteine
5 9 5 M Methionine
11 16 4 F Phenylalanine
17 16 30 V Valine
44 23 23 A Alanine
6 16 8 H Histidine 

4WYY_1|Chain A|Beta-lactamase|Pseudomonas aeruginosa (strain ATCC 15692 / PAO1 / 1C / PRS 101 / LMG 12228) (208964)
>6SUM_1|Chains A, B|AMICOUMACIN KINASE|Bacillus altitudinis (293387)
>6JBQ_1|Chains A, B|DNA-directed RNA polymerase subunit alpha|Escherichia coli (strain K12) (83333)
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 \(w=f(c)\)
4WYY , Knot 151 360 0.82 38 199 341 
APADRLKALVDAAVQPVMKANDIPGLAVAISLKGEPHYFSYGLASKEDGRRVTPETLFEIGSVSKTFTATLAGYALTQDKMRLDDRASQHWPALQGSRFDGISLLDLATYTAGGLPLQFPDSVQKDQAQIRDYYRQWQPTYAPGSQRLYSNPSIGLFGYLAARSLGQPFERLMEQQVFPALGLEQTHLDVPEAALAQYAQGYGKDDRPLRVGPGPLDAEGYGVKTSAADLLRFVDANLHPERLDRPWAQALDATHRGYYKVGDMTQGLGWEAYDWPISLKRLQAGNSTPMALQPHRIARLPAPQALEGQRLLNKTGSTNGFGAYVAFVPGRDLGLVILANRNYPNAERVKIAYAILSGLE
6SUM , Knot 146 335 0.84 40 204 319 
MHKEVKARYDEQMVLKKAALLYHFQPEQVTFLADAENYVYEYTDEKGSSYILKITHTIRRSSTYILGEMDWIRYLSQHGISVAKPVLSARGKDVEAIPDQAGGAFLLRVYEKAPGRKVTEADWNGQLFQALGAYTGRMHQTTKQYQVKDPRYKRQEWYEEEQLKLETYIPSDQTVVLQRKDELMQKLHQLRISKDVYGLVHADLHHGNFHYHQGEIIAFDFDDCGYHWFINDISILLYNVLWYPVIPYEDKAEFTGEFMSHFLKGYRQENELDDAWLATIPDFLMLRHMLIYGLLHQAFDLRTLSADESAMLARFRKEIEDKTPITPFDFHQLTT
6JBQ , Knot 146 329 0.85 40 180 315 
MQGSVTEFLKPRLVDIEQVSSTHAKVTLEPLERGFGHTLGNALRRILLSSMPGCAVTEVEIDGVLHEYSTKEGVQEDILEILLNLKGLAVRVQGKDEVILTLNKSGIGPVTAADITHDGDVEIVKPQHVICHLTDENASISMRIKVQRGRGYVPASTRIHSEEDERPIGRLLVDACYSPVERIAYNVEAARVEQRTDLDKLVIEMETNGTIDPEEAIRRAATILAEQLEAFVDLRDVRQPEVKEEKPEFDPILLRPVDDLELTVRSANCLKAEAIHYIGDLVQRTEVELLKTPNLGKKSLTEIKDVLASRGLSLGMRLENWPPASIADE

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 \(c\).

\(|P_{f(4WYY_1)}(2) \setminus P_{f(6SUM_1)}(2)|=81\), \(|P_{f(6SUM_1)}(2) \setminus P_{f(4WYY_1)}(2)|=86\). 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 1:111001011101110111010011111111010101001001110000100101001101101000101011101100001010001000111101001011011011000111111011001000010100000010100111000100010111110111001101100110001111111000010110111100101010000110111111010101100011011011010101001001110110100010001101001111010011101001011000111101001101111011010011000100011110111111001111111000010100101101110110
Pair \(Z_2\) Length of longest common subsequence
4WYY_1,6SUM_1 167 3
4WYY_1,6JBQ_1 145 4
6SUM_1,6JBQ_1 160 3

Newick tree

 
[
	6SUM_1:84.63,
	[
		4WYY_1:72.5,6JBQ_1:72.5
	]:12.13
]

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 \((0.85)(0.8)(\frac{695 }{\log_{20} 695}-\frac{335}{\log_{20}335})=98.9\)
Status Protein1 Protein2 d d1/2
Query variables 4WYY_1 6SUM_1 122 120.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]