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Parikh vectors
3CQF_1 6YED_1 4IRN_1 Letter Amino acid
13 11 14 H Histidine
29 32 41 L Leucine
12 7 21 Q Glutamine
27 19 33 E Glutamic acid
19 21 10 P Proline
20 13 15 Y Tyrosine
1 2 6 C Cysteine
8 5 15 M Methionine
27 28 13 D Aspartic acid
29 13 29 I Isoleucine
45 15 20 T Threonine
6 7 8 W Tryptophan
44 36 39 A Alanine
42 21 15 N Asparagine
38 30 26 K Lycine
16 16 17 F Phenylalanine
39 18 27 S Serine
41 30 13 V Valine
11 10 17 R Arginine
22 18 37 G Glycine 

3CQF_1|Chains A, B|Thiol-activated cytolysin|Bacillus anthracis (1392)
>6YED_1|Chains A, B|Putrescine-binding periplasmic protein|Escherichia coli K12 (83333)
>4IRN_1|Chains A, B, C, D, E, F, G, H|Prolyl-ACP dehydrogenase|Oscillatoria sp. (272129)
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 \(w=f(c)\)
3CQF , Knot 195 489 0.82 40 230 461 
MHHHHHHAAAMETQAGNATGAIKNASDINTGIANLKYDSRDILAVNGDKVESFIPKESINSNGKFVVVEREKKSLTTSPVDILIIDSVVNRTYPGAVQLANKAFADNQPSLLVAKRKPLNISIDLPGMRKENTITVQNPTYGNVAGAVDDLVSTWNEKYSTTHTLPARMQYTESMVYSKSQIASALNVNAKYLDNSLNIDFNAVANGEKKVMVAAYKQIFYTVSAELPNNPSDLFDNSVTFDELTRKGVSNSAPPVMVSNVAYGRTVYVKLETTSKSKDVQAAFKALLKNNSVETSGQYKDIFEESTFTAVVLGGDAKEHNKVVTKDFNEIRNIIKDNAELSFKNPAYPISYTSTFLKDNATAAVHNNTDYIETTTTEYSSAKMTLDHYGAYVAQFDVSWDEFTFDQNGKEVLTHKTWEGSGKDKTAHYSTVIPLPPNSKNIKIVARECTGLAWEWWRTIINEQNVPLTNEIKVSIGGTTLYPTATISH
6YED , Knot 150 352 0.83 40 200 330 
AEQKTLHIYNWSDYIAPDTVANFEKETGIKVVYDVFDSNEVLEGKLMAGSTGFDLVVPSASFLERQLTAGVFQPLDKSKLPEWKNLDPELLKLVAKHDPDNKFAMPYMWATTGIGYNVDKVKAVLGENAPVDSWDLILKPENLEKLKSCGVSFLDAPEEVFATVLNYLGKDPNSTKADDYTGPATDLLLKLRPNIRYFHSSQYINDLANGDICVAIGWAGDVWQASNRAKEAKNGVNVSFSIPKEGAMAFFDVFAMPADAKNKDEAYQFLNYLLRPDVVAHISDHVFYANANKAATPLVSAEVRENPGIYPPADVRAKLFTLKVQDPKIDRVRTRAWTKVKSGKLEHHHHHH
4IRN , Knot 171 416 0.82 40 221 392 
MGSSHHHHHHSSGLVPRGSHMASMTGGQQMGRGSEFDFAWNSQQIQFRKKVIQFAQQSLISDLIKNDKEEIFNRDAWQKCSEFGVHGWPIPARYGGQELDILTTAYALQGLGYGCKDNGLIFAMNAHIWACEMPLLTFGTEEQKEKYLPLLCRGGWIASHAATEPQAGSDIYSLKTTAQKDGDKYILNGYKHYVTNGTIADLFIIFATIDPSLGKEGLTTFMIEKDTPGLILSKPISKMGMRTAEVPELRLENCEVSAANRLGEEGTGLAIFNHSMEWERGFILAAAVGTMERLLEQSIRYARSHKQFGQAIGKFQLVANKLVEMKLRLENAKAYLYKVAWMKENKQMALLEASMANLYISEAWVQSCLEAIEIHGAYGYLTNTELERELRDAIASKFYSGTSEIQRVVIAKFLGL

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 \(c\).

\(|P_{f(3CQF_1)}(2) \setminus P_{f(6YED_1)}(2)|=86\), \(|P_{f(6YED_1)}(2) \setminus P_{f(3CQF_1)}(2)|=56\). 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 1:100000011110001101011100100100111010000001111010010011100010001011110000001000110111100110000111101100111000101111000110101011110000010100100101111100110010000000001110100000110000011011010100100010101011101000111110001100101011001001100010100100011000111111001101001010100000000101110111000010001000011000010111111010000011000100100110001010100110110000011000101110000001000000000101010001101101010100101000100110000101010000100001111110000101110000111101100110000111000101011100101010100
Pair \(Z_2\) Length of longest common subsequence
3CQF_1,6YED_1 142 6
3CQF_1,4IRN_1 147 6
6YED_1,4IRN_1 173 6

Newick tree

 
[
	4IRN_1:83.12,
	[
		3CQF_1:71,6YED_1:71
	]:12.12
]

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 \((0.85)(0.8)(\frac{841 }{\log_{20} 841}-\frac{352}{\log_{20}352})=132.\)
Status Protein1 Protein2 d d1/2
Query variables 3CQF_1 6YED_1 163 139.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]