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Parikh vectors
5ONP_1 2IRX_1 1UPT_1 Letter Amino acid
16 31 18 L Leucine
10 7 4 F Phenylalanine
28 8 6 Y Tyrosine
24 19 9 D Aspartic acid
0 0 2 C Cysteine
18 8 11 I Isoleucine
22 24 13 V Valine
10 23 10 R Arginine
2 6 7 M Methionine
25 26 15 T Threonine
8 6 1 H Histidine
11 9 11 K Lycine
8 21 3 P Proline
28 39 10 A Alanine
14 11 6 Q Glutamine
7 13 13 E Glutamic acid
3 6 4 W Tryptophan
20 4 5 N Asparagine
36 22 12 G Glycine
26 20 11 S Serine 

5ONP_1|Chain A|Thermolysin|Geobacillus stearothermophilus (1422)
>2IRX_1|Chain A|DNA ligase-like protein Rv0938/MT0965|Mycobacterium tuberculosis (83332)
>1UPT_1|Chains A, C, E, G|ADP-RIBOSYLATION FACTOR-LIKE PROTEIN 1|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 \(w=f(c)\)
5ONP , Knot 132 316 0.80 38 179 302 
ITGTSTVGVGRGVLGDQKNINTTYSTYYYLQDNTRGNGIFTYDAKYRTTLPGSLWADADNQFFASYDAPAVDAHYYAGVTYDYYKNVHNRLSYDGNNAAIRSSVHYSQGYNNAFWNGSQMVYGDGDGQTFIPLSGGIDVVAHELTHAVTDYTAGLIYQNESGAINEAISDIFGTLVEFYANKNPDWEIGEDVYTPGISGDSLRSMSDPAKYGDPDHYSKRYTGTQDNGGVHINSGIINKAAYLISQGGTHYGVSVVGIGRDKLGKIFYRALTQYLTPTSNFSQLRAAAVQSATDLYGSTSQEVASVKQAFDAVGVK
2IRX , Knot 128 303 0.80 38 173 282 
GSHMGSASEQRVTLTNADKVLYPATGTTKSDIFDYYAGVAEVMLGHIAGRPATRKRWPNGVDQPAFFEKQLALSAPPWLSRATVAHRSGTTTYPIIDSATGLAWIAQQAALEVHVPQWRFVAEPGSGELNPGPATRLVFDLDPGEGVMMAQLAEVARAVRDLLADIGLVTFPVTSGSKGLHLYTPLDEPVSSRGATVLAKRVAQRLEQAMPALVTSTMTKSLRAGKVFVDWSQNSGSKTTIAPYSLRGRTHPTVAAPRTWAELDDPALRQLSYDEVLTRIARDGDLLERLDADAPVADRLTRY
1UPT , Knot 81 171 0.81 40 129 164 
GSHMTREMRILILGLDGAGKTTILYRLQVGEVVTTIPTIGFNVETVTYKNLKFQVWDLGGLTSIRPYWRCYYSNTDAVIYVVDSCDRDRIGISKSELVAMLEEEELRKAILVVFANKQDMEQAMTSSEMANSLGLPALKDRKWQIFKTSATKGTGLDEAMEWLVETLKSRQ

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 \(c\).

\(|P_{f(5ONP_1)}(2) \setminus P_{f(2IRX_1)}(2)|=83\), \(|P_{f(2IRX_1)}(2) \setminus P_{f(5ONP_1)}(2)|=77\). 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 1:1010001111011110000100000000010000010111000100000111011101000111000111101000111000000010001000100111000100001000111010011010101001111011101110010011000011110000011100110011101101010001010110010011101001001001100101000000001000011101001110011011001100011011111000110110011000101000100101111001001010000011010011011110
Pair \(Z_2\) Length of longest common subsequence
5ONP_1,2IRX_1 160 5
5ONP_1,1UPT_1 164 3
2IRX_1,1UPT_1 158 4

Newick tree

 
[
	5ONP_1:81.66,
	[
		2IRX_1:79,1UPT_1:79
	]:2.66
]

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 \((0.85)(0.8)(\frac{619 }{\log_{20} 619}-\frac{303}{\log_{20}303})=88.1\)
Status Protein1 Protein2 d d1/2
Query variables 5ONP_1 2IRX_1 112 107.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]