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Parikh vectors
5ACB_1 9JOH_1 1IQY_1 Letter Amino acid
17 20 43 D Aspartic acid
15 38 46 E Glutamic acid
27 39 45 L Leucine
11 24 19 Y Tyrosine
16 20 53 V Valine
4 35 23 N Asparagine
13 25 35 I Isoleucine
10 33 40 S Serine
15 25 40 T Threonine
7 6 12 W Tryptophan
20 26 13 K Lycine
15 17 30 F Phenylalanine
11 11 39 P Proline
14 27 51 A Alanine
12 10 48 R Arginine
7 1 4 C Cysteine
18 16 17 Q Glutamine
8 10 16 H Histidine
12 28 57 G Glycine
7 12 7 M Methionine 

5ACB_1|Chains A, B|CYCLIN-K|HOMO SAPIENS (9606)
>9JOH_1|Chains A, B|endo-1.3-fucanase|Wenyingzhuangia fucanilytica (1790137)
>1IQY_1|Chains A, B|AMINE OXIDASE|Arthrobacter globiformis (1665)
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 \(w=f(c)\)
5ACB , Knot 120 259 0.85 40 178 253 
SMSVTSANLDHTKPCWYWDKKDLAHTPSQLEGLDPATEARYRREGARFIFDVGTRLGLHYDTLATGIIYFHRFYMFHSFKQFPRYVTGACCLFLAGKVEETPKKCKDIIKTARSLLNDVQFGQFGDDPKEEVMVLERILLQTIKFDLQVEHPYQFLLKYAKQLKGDKNKIQKLVQMAWTFVNDSLCTTLSLQWEPEIIAVAVMYLAGRLCKFEIQEWTSKPMYRRWWEQFVQDVPVDVLEDICHQILDLYSQGKQQMPH
9JOH , Knot 172 423 0.82 40 215 398 
MGSSHHHHHHSSGLVPRGSHMASMTGGQQMGRGSSTKNGIIELEQETEEELEQNQEIDYSNYPEFSWDTMPLYMHVRKNTAYTDEEINYLASFPLITLEKSQAQNTYGSTEEGTLATASAIKLKNNKAKVLYYRNVVINWGNYKNDDEFISKNPSALLKNQNNELVYMPNGSTPFFDITKSFVQEYWLKSVEDMVATPNIDGTFIDANIKVLVPSFFSSKVGVNKQAEIENSYFSMMSRLKESLSNNLILANIIRVRPEFEENGLEYLGYFNGSYLEGFDSEAFGMSNAEYLVEGIEATQKAAQSGKIITMTLGLGEAIDNNTGIDDQREDVDLNDEELNKRVDYLLAIFLICAEKYSYVYLHDGYLATNSAVWLHQFDQYKKALGAPLGKAIKNGYIYTRKFENLDVWLNLETQTATLTWKE
1IQY , Knot 247 638 0.83 40 270 590 
MTPSTIQTASPFRLASAGEISEVQGILRTAGLLGPEKRIAYLGVLDPARGAGSEAEDRRFRVFIHDVSGARPQEVTVSVTNGTVISAVELDTAATGELPVLEEEFEVVEQLLATDERWLKALAARNLDVSKVRVAPLSAGVFEYAEERGRRILRGLAFVQDFPEDSAWAHPVDGLVAYVDVVSKEVTRVIDTGVFPVPAEHGNYTDPELTGPLRTTQKPISITQPEGPSFTVTGGNHIEWEKWSLDVGFDVREGVVLHNIAFRDGDRLRPIINRASIAEMVVPYGDPSPIRSWQNYFDTGEYLVGQYANSLELGCDCLGDITYLSPVISDAFGNPREIRNGICMHEEDWGILAKHSDLWSGINYTRRNRRMVISFFTTIGNYDYGFYWYLYLDGTIEFEAKATGVVFTSAFPEGGSDNISQLAPGLGAPFHQHIFSARLDMAIDGFTNRVEEEDVVRQTMGPGNERGNAFSRKRTVLTRESEAVREADARTGRTWIISNPESKNRLNEPVGYKLHAHNQPTLLADPGSSIARRAAFATKDLWVTRYADDERYPTGDFVNQHSGGAGLPSYIAQDRDIDGQDIVVWHTFGLTHFPRVEDWPIMPVDTVGFKLRPEGFFDRSPVLDVPANPSQSGSHCHG

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 \(c\).

\(|P_{f(5ACB_1)}(2) \setminus P_{f(9JOH_1)}(2)|=70\), \(|P_{f(9JOH_1)}(2) \setminus P_{f(5ACB_1)}(2)|=107\). 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 1:0101001010000101010000110010010110110010000011011101100111000011011101001011001001100101100111110100010000011001001100101101100100011110011100101010100100111001001010000100110111011000100010101010111111101110100101001000110001100110011101100100011010001000110
Pair \(Z_2\) Length of longest common subsequence
5ACB_1,9JOH_1 177 4
5ACB_1,1IQY_1 192 4
9JOH_1,1IQY_1 141 4

Newick tree

 
[
	5ACB_1:98.53,
	[
		9JOH_1:70.5,1IQY_1:70.5
	]:28.03
]

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 \((0.85)(0.8)(\frac{682 }{\log_{20} 682}-\frac{259}{\log_{20}259})=117.\)
Status Protein1 Protein2 d d1/2
Query variables 5ACB_1 9JOH_1 150 121
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]