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Parikh vectors
6UEC_1 2MNR_1 1ZGI_1 Letter Amino acid
17 15 21 R Arginine
4 2 8 C Cysteine
19 19 11 T Threonine
27 16 16 S Serine
3 6 9 W Tryptophan
54 47 12 A Alanine
19 17 19 D Aspartic acid
15 9 5 H Histidine
27 42 25 L Leucine
28 31 17 V Valine
14 19 19 E Glutamic acid
46 30 24 G Glycine
26 21 17 I Isoleucine
11 19 15 P Proline
7 8 10 F Phenylalanine
9 9 11 Y Tyrosine
9 7 11 N Asparagine
15 15 8 Q Glutamine
10 14 22 K Lycine
8 11 7 M Methionine 

6UEC_1|Chain A|UDP-3-O-acylglucosamine N-acyltransferase|Pseudomonas aeruginosa (strain ATCC 15692 / DSM 22644 / CIP 104116 / JCM 14847 / LMG 12228 / 1C / PRS 101 / PAO1) (208964)
>2MNR_1|Chain A|MANDELATE RACEMASE|Pseudomonas putida (303)
>1ZGI_1|Chain A|Thrombin|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 \(w=f(c)\)
6UEC , Knot 154 368 0.82 40 197 329 
MHHHHHHSSGRENLYFQGSTLSYTLGQLAAHVGAEVRGDADLPIQGLATLQEAGPAQLSFLANPQYRKYLPESRAGAVLLTAADADGFAGTALVVANPYLAYASLSHLFDRKPKAAAGIHPTAIVAADAEVDPSASVGAYAVIESGARIGAGVSIGAHCVIGARSVIGEGGWLAPRVTLYHDVTIGARVSIQSGAVIGGEGFGFANEKGVWQKIAQIGGVTIGDDVEIGANTTIDRGALSDTLIGNGVKLDNQIMIAHNVQIGDHTAMAACVGISGSAKIGRHCMLAGGVGLVGHIEICDNVFVTGMTMVTRSITEPGSYSSGTAMQPAAEWKKSAARIRQLDDMARRLQQLEKRLAAVTSSGDASSD
2MNR , Knot 145 357 0.79 40 196 331 
EVLITGLRTRAVNVPLAYPVHTAVGTVGTAPLVLIDLATSAGVVGHSYLFAYTPVALKSLKQLLDDMAAMIVNEPLAPVSLEAMLAKRFCLAGYTGLIRMAAAGIDMAAWDALGKVHETPLVKLLGANARPVQAYDSHSLDGVKLATERAVTAAELGFRAVKTKIGYPALDQDLAVVRSIRQAVGDDFGIMVDYNQSLDVPAAIKRSQALQQEGVTWIEEPTLQHDYEGHQRIQSKLNVPVQMGENWLGPEEMFKALSIGACRLAMPDAMKIGGVTGWIRASALAQQFGIPMSSHLFQEISAHLLAATPTAHWLERLDLAGSVIEPTLTFEGGNAVIPDLPGVGIIWREKEIGKYLV
1ZGI , Knot 130 287 0.85 40 195 272 
DCGLRPLFEKKSLEDKTERELLESYIDGRIVEGSDAEIGMSPWQVMLFRKSPQELLCGASLISDRWVLTAAHCLLYPPWDKNFTENDLLVRIGKHSRTRYERNIEKISMLEKIYIHPRYNWRENLDRDIALMKLKKPVAFSDYIHPVCLPDRETAASLLQAGYKGRVTGWGNLKETWTANVGKGQPSVLQVVNLPIVERPVCKDSTRIRITDNMFCAGYKPDEGKRGDACEGDSGGPFVMKSPFNNRWYQMGIVSWGEGCDRDGKYGFYTHVFRLKKWIQKVIDQFG

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 \(c\).

\(|P_{f(6UEC_1)}(2) \setminus P_{f(2MNR_1)}(2)|=71\), \(|P_{f(2MNR_1)}(2) \setminus P_{f(6UEC_1)}(2)|=70\). 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 1:10000000010001010100100011011101110101010111011101001111010111010000011000111111011010111101111101011010100110001011111010111110101010101110111001101111101110011110011101111110101000101110101001111110111110001110011011110110010111000100111000111011010001111001011000111101110101011000111111111101010001110110110001001100001011011101000110100100110010010001111000101000
Pair \(Z_2\) Length of longest common subsequence
6UEC_1,2MNR_1 141 5
6UEC_1,1ZGI_1 180 4
2MNR_1,1ZGI_1 171 4

Newick tree

 
[
	1ZGI_1:92.82,
	[
		6UEC_1:70.5,2MNR_1:70.5
	]:22.32
]

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 \((0.85)(0.8)(\frac{725 }{\log_{20} 725}-\frac{357}{\log_{20}357})=100.\)
Status Protein1 Protein2 d d1/2
Query variables 6UEC_1 2MNR_1 119 116.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]