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Parikh vectors
5KXA_1 3OIR_1 7AWX_1 Letter Amino acid
53 5 6 T Threonine
13 1 1 W Tryptophan
49 9 3 R Arginine
21 1 2 Q Glutamine
37 8 9 I Isoleucine
69 15 9 L Leucine
55 7 16 K Lycine
63 5 8 S Serine
42 3 4 Y Tyrosine
53 12 9 V Valine
18 3 2 M Methionine
34 11 8 A Alanine
36 3 4 N Asparagine
53 10 7 D Aspartic acid
52 14 10 E Glutamic acid
51 12 15 G Glycine
31 2 3 H Histidine
33 1 0 C Cysteine
43 8 6 F Phenylalanine
57 5 6 P Proline 

5KXA_1|Chain A|Ectonucleotide pyrophosphatase/phosphodiesterase family member 2|Homo sapiens (9606)
>3OIR_1|Chains A, B|SULFATE TRANSPORTER SULFATE TRANSPORTER FAMILY PROTEIN|Wolinella succinogenes (844)
>7AWX_1|Chains A, B|Peptidyl-prolyl cis-trans isomerase FKBP5|Homo sapiens (9606)
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 \(w=f(c)\)
5KXA , Knot 334 863 0.87 40 328 799 
MARRSSFQSCQIISLFTFAVGVNICLGFTAHRIKRAEGWEEGPPTVLSDSPWTAISGSCKGRCFELQEAGPPDCRCDNLCKSYTSCCHDFDELCLKTARGWECTKDRCGEVRNEENACHCSEDCLARGDCCTNYQVVCKGESHWVDDDCEEIKAAECPAGFVRPPLIIFSVDGFRASYMKKGSKVMPNIEKLRSCGTHSPYMRPVYPTKTFPNLYTLATGLYPESHGIVGNSMYDPVFDATFHLRGREKFNHRWWGGQPLWITATKQGVKAGTFFWSVVIPHERRILTILQWLTLPDHERPSVYAFYSEQPDFSGHKYGPFGPEMTNPLREIDKIVGQLMDGLKQLKLHRCVNVIFVGDHGMEDVTCDRTEFLSNYLTNVDDITLVPGTLGRIRSKFSNNAKYDPKAIIAALTCKKPDQHFKPYLKQHLPKRLHYANNRRIEDIHLLVERRWHVARKPLDVYKKPSGKCFFQGDHGFDNKVNSMQTVFVGYGSTFKYKTKVPPFENIELYNVMCDLLGLKPAPNNGTHGSLNHLLRTNTFRPTMPEEVTRPNYPGIMYLQSDFDLGCTCDDKVEPKNKLDELNKRLHTKGSTEERHLLYGRPAVLYRTRYDILYHTDFESGYSEIFLMPLWTSYTVSKQAEVSSVPDHLTSCVRPDVRVSPSFSQNCLAYKNDKQMSYGFLFPPYLSSSPEAKYDAFLVTNMVPMYPAFKRVWNYFQRVLVKKYASERNGVNVISGPIFDYDYDGLHDTEDKIKQYVEGSSIPVPTHYYSIITSCLDFTQPADKCDGPLSVSSFILPHRPDNEESCNSSEDESKWVEELMKMHTARVRDIEHLTSLDFFRKTSRSYPEILTLKTYLHTYESEI
3OIR , Knot 68 135 0.82 40 102 130 
SNADGLEGMDDPDATSKKVVPLGVEIYEINGPFFFGVADRLKGVLDVIEETPKVFILRMRRVPVIDATGMHALWEFQESCEKRGTILLLSGVSDRLYGALNRFGFIEALGEERVFDHIDKALAYAKLLVETAEER
7AWX , Knot 64 128 0.80 38 103 125 
GAPATVTEQGEDITSKKDRGVLKIVKRVGNGEETPMIGDKVYVHYKGKLSNGKKFDSSHDRNEPFVFSLGKGQVIKAWDIGVATMKKGEIAHLLIKPEYAYGSAGSLPKIPSNATLFFEIELLDFKGE

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 \(c\).

\(|P_{f(5KXA_1)}(2) \setminus P_{f(3OIR_1)}(2)|=236\), \(|P_{f(3OIR_1)}(2) \setminus P_{f(5KXA_1)}(2)|=10\). 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 1:11000010000110110111110101110100100101100111011000110110100010010100111100000010000000000100101001011000000010100000100000001101000000011001000110000001011001111101111110101101001001001110100100010001010110100011010011011010001111001001110101010100010001111011110100011011011101111000011011011011000010101100001010100011111010011001001110110110010100010111110011001000000110001001001011110110100010001000101111110000100010101000110010010000100101110001011001101000101001101001100010010011110100100000111100101001100111101110010010100110000101011001001001111010001011000000101000100100010001000000110101111000000110000100100011111110000100010100110010001010101010100001100000010011111101000101000111100111101110011001001110001000011011011110000011000000100010100111100000110001010011000011101001111001000000000000001100110100101001001001011000000010110100010000001
Pair \(Z_2\) Length of longest common subsequence
5KXA_1,3OIR_1 246 4
5KXA_1,7AWX_1 245 4
3OIR_1,7AWX_1 125 4

Newick tree

 
[
	5KXA_1:13.06,
	[
		7AWX_1:62.5,3OIR_1:62.5
	]:74.56
]

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 \((0.85)(0.8)(\frac{998 }{\log_{20} 998}-\frac{135}{\log_{20}135})=238.\)
Status Protein1 Protein2 d d1/2
Query variables 5KXA_1 3OIR_1 309 177
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]