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Parikh vectors
3CUE_1 9JFF_1 6JGA_1 Letter Amino acid
18 11 30 S Serine
11 19 39 D Aspartic acid
11 35 63 G Glycine
11 11 21 F Phenylalanine
9 6 21 Y Tyrosine
9 16 24 E Glutamic acid
15 13 33 I Isoleucine
8 3 18 M Methionine
1 5 7 W Tryptophan
9 24 52 V Valine
19 44 48 L Leucine
16 5 33 K Lycine
18 14 42 T Threonine
2 2 8 C Cysteine
13 8 19 Q Glutamine
2 13 12 H Histidine
10 23 32 P Proline
9 56 56 A Alanine
8 25 28 R Arginine
20 8 23 N Asparagine 

3CUE_1|Chains A, G, M, S|Transport protein particle 23 kDa subunit|Saccharomyces cerevisiae (4932)
>9JFF_1|Chain A|threonine-phosphate decarboxylase|Rhizobium meliloti (382)
>6JGA_1|Chain A|BETA-D-GLUCAN GLUCOHYDROLASE ISOENZYME EXO1|Hordeum vulgare subsp. vulgare (112509)
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 \(w=f(c)\)
3CUE , Knot 102 219 0.83 40 151 210 
MAIETILVINKSGGLIYQRNFTNDEQKLNSNEYLILASTLHGVFAIASQLTPKALQLTQQTNIENTIPYIPYVGMSSNRSDTRNGGGNNNKHTNNEKLGSFKGDDFFKEPFTNWNKSGLRQLCTDQFTMFIYQTLTGLKFVAISSSVMPQRQPTIATTDKPDRPKSTSNLAIQIADNFLRKVYCLYSDYVMKDPSYSMEMPIRSNLFDEKVKKMVENLQ
9JFF , Knot 142 341 0.81 40 164 312 
MSAPIVHGGGITEAAARYGGRPEDWLDLSTGINPCPVALPAVPERAWHRLPDRQTVDDARSAAADYYRTNGVLPLPVPGTQSVIQLLPRLAPANRHVAIFGPTYGEYARVLEAAGFAVDRVADADALTAEHGLVIVVNPNNPTGRALAPAELLAIAARQKASGGLLLVDEAFGDLEPQLSVAGHASGQGNLIVFRSFGKFFGLAGLRLGFVVATEPVLASFADWLGPWAVSGPALTISKALMQGDTKAIAAGILERRAGLDAALDGAGLNRIGGTGLFVLVEHPRAALLQERLCEAHILTRKFDYAPTWLRVGLAPDAAGDRRLADALARMELLEHHHHHH
6JGA , Knot 242 609 0.85 40 272 568 
HHAADYVLYKDATKPVEDRVADLLGRMTLAEKIGQMTQIERLVATPDVLRDNFIGSLLSGGGSVPRKGATAKEWQDMVDGFQKACMSTRLGIPMIYGIDAVHGQNNVYGATIFPHNVGLGATRDPYLVKRIGEATALEVRATGIQYAFAPCIAVCRDPRWGRCYESYSEDRRIVQSMTELIPGLQGDVPKDFTSGMPFVAGKNKVAACAKHFVGDGGTVDGINENNTIINREGLMNIHMPAYKNAMDKGVSTVMISYSSWNGVKMHANQDLVTGYLKDTLKFKGFVISDFEGIDRITTPAGSDYSYSVKASILAGLDMIMVPNKYQQFISILTGHVNGGVIPMSRIDDAVTRILRVKFTMGLFENPYADPAMAEQLGKQEHRDLAREAARKSLVLLKNGKTSTDAPLLPLPKKAPKILVAGSHADNLGYQCGGWTIEWQGDTGRTTVGTTILEAVKAAVDPSTVVVFAENPDAEFVKSGGFSYAIVAVGEHPYTETKGDNLNLTIPEPGLSTVQAVCGGVRCATVLISGRPVVVQPLLAASDALVAAWLPGSEGQGVTDALFGDFGFTGRLPRTWFKSVDQLPMNVGDAHYDPLFRLGYGLTTNATKKY

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 \(c\).

\(|P_{f(3CUE_1)}(2) \setminus P_{f(9JFF_1)}(2)|=81\), \(|P_{f(9JFF_1)}(2) \setminus P_{f(3CUE_1)}(2)|=94\). 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 1:111001111000111100001000000100000111100101111110010101101000001000110110111000000000111000000000011010100110011001000110010000101110001011011110001110001011000010010000011101100110010010000110010001011100011000100110010
Pair \(Z_2\) Length of longest common subsequence
3CUE_1,9JFF_1 175 4
3CUE_1,6JGA_1 187 4
9JFF_1,6JGA_1 168 5

Newick tree

 
[
	3CUE_1:92.63,
	[
		9JFF_1:84,6JGA_1:84
	]:8.63
]

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 \((0.85)(0.8)(\frac{560 }{\log_{20} 560}-\frac{219}{\log_{20}219})=97.4\)
Status Protein1 Protein2 d d1/2
Query variables 3CUE_1 9JFF_1 124 102.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]