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Parikh vectors
8ZND_1 5XMZ_1 2CWV_1 Letter Amino acid
5 2 16 H Histidine
29 8 35 I Isoleucine
10 9 30 F Phenylalanine
20 10 39 P Proline
12 10 40 S Serine
40 21 52 A Alanine
24 10 42 D Aspartic acid
37 17 57 G Glycine
20 3 48 R Arginine
34 8 46 E Glutamic acid
16 6 19 Y Tyrosine
23 8 40 T Threonine
38 5 53 V Valine
1 4 4 C Cysteine
24 14 45 L Leucine
34 5 13 K Lycine
11 0 12 W Tryptophan
15 7 23 N Asparagine
10 6 17 Q Glutamine
13 2 7 M Methionine 

8ZND_1|Chain A|Glutamate dehydrogenase|Thermococcus profundus (49899)
>5XMZ_1|Chain A|Effector protein PevD1|Verticillium dahliae (27337)
>2CWV_1|Chains A, B|Phenylethylamine 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 \(w=f(c)\)
8ZND , Knot 173 416 0.83 40 213 398 
IDPFEMAVKQLERAAQYMDISEEALEWLKKPMRIVEVSVPIEMDDGSVKVFTGFRVQHNWARGPTKGGIRWHPAETLSTVKALATFMTWKVAVVDLPYGGGKGGIIVNPKELSEREQERLARAYIRAVYDVIGPWTDIPAPDVYTNPKIMGWMMDEYETIMRRKGPAFGVITGKPLSIGGSLGRGTATAQGAIFTIREAAKALGIDLKGKKIAVQGYGNAGYYTAKLAKEQLGMTVVAVSDSRGGIYNPDGLDPDEVLKWKREHGSVKDFPGATNITNEELLELEVDVLAPAAIEEVITEKNADNIKAKIVAEVANGPVTPEADDILREKGILQIPDFLCNAGGVTVSYFEWVQNINGYYWTEEEVREKLDKKMTKAFWEVYNTHKDKNIHMRDAAYVVAVSRVYQAMKDRGWVKK
5XMZ , Knot 71 155 0.77 38 108 146 
MQFTLAAAAALFGASALAAPASPGSTGAPPDPNMYENIDIADFNVRKGEDGTIKYVNFKLSGDDADGLLCEAQNPGLPSNVITCGESKYRFALSSGKQYEFALSLYHELGLAVGFYGTGEIFTHCRAGGLGDFICQQQNPTTIVIDSLPDAPAEA
2CWV , Knot 246 638 0.83 40 270 589 
MTPSTIQTASPFRLASAGEISEVQGILRTAGLLGPEKRIAYLGVLDPARGAGSEAEDRRFRVFIHDVSGARPQEVTVSVTNGTVISAVELDTAATGELPVLEEEFEVVEQLLATDERWLKALAARNLDVSKVRVAPLSAGVFEYAEERGRRILRGLAFVQDFPEDSAWAHPVDGLVAYVDVVSKEVTRVIDTGVFPVPAEHGNYTDPELTGPLRTTQKPISITQPEGPSFTVTGGNHIEWEKWSLDVGFDVREGVVLHNIAFRDGDRLRPIINRASIAEMVVPYGDPSPIRSWQNYFATGEYLVGQYANSLELGCDCLGDITYLSPVISDAFGNPREIRNGICMHEEDWGILAKHSDLWSGINYTRRNRRMVISFFTTIGNYDYGFYWYLYLDGTIEFEAKATGVVFTSAFPEGGSDNISQLAPGLGAPFHQHIFSARLDMAIDGFTNRVEEEDVVRQTMGPGNERGNAFSRKRTVLTRESEAVREADARTGRTWIISNPESKNRLNEPVGYKLHAHNQPTLLADPGSSIARRAAFATKDLWVTRYADDERYPTGDFVNQHSGGAGLPSYIAQDRDIDGQDIVVWHTFGLTHFPRVEDWPIMPVDTVGFKLRPEGFFDRSPVLDVPANPSQSGSHCHG

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 \(c\).

\(|P_{f(8ZND_1)}(2) \setminus P_{f(5XMZ_1)}(2)|=150\), \(|P_{f(5XMZ_1)}(2) \setminus P_{f(8ZND_1)}(2)|=45\). 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 1:10110111001001100101000110110011011010111010010101101101000110110011101011001001011101101011110110111011111010010000000110101011001111100111101000101111110000011000111111101011011101101010101111010011011110101001110101011000101100011101111000011100101101001101000010100111100100001101010111111100110000100101011101101110101001100011101101100111101001011001010010000100010001001110100000000101001101111001001100011100
Pair \(Z_2\) Length of longest common subsequence
8ZND_1,5XMZ_1 195 3
8ZND_1,2CWV_1 147 4
5XMZ_1,2CWV_1 216 4

Newick tree

 
[
	5XMZ_1:11.96,
	[
		8ZND_1:73.5,2CWV_1:73.5
	]:37.46
]

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 \((0.85)(0.8)(\frac{571 }{\log_{20} 571}-\frac{155}{\log_{20}155})=120.\)
Status Protein1 Protein2 d d1/2
Query variables 8ZND_1 5XMZ_1 155 104.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]