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Parikh vectors
4UZY_1 1XBW_1 6LKB_1 Letter Amino acid
66 6 9 L Leucine
27 1 8 P Proline
27 4 8 R Arginine
13 0 2 C Cysteine
27 5 14 G Glycine
32 6 6 S Serine
29 11 17 T Threonine
32 8 11 D Aspartic acid
32 3 5 Q Glutamine
59 10 8 E Glutamic acid
14 3 8 H Histidine
32 8 8 I Isoleucine
47 12 9 K Lycine
65 5 4 A Alanine
28 4 10 N Asparagine
21 4 5 M Methionine
26 7 9 F Phenylalanine
3 2 2 W Tryptophan
40 4 4 Y Tyrosine
31 6 13 V Valine 

4UZY_1|Chain A|FLAGELLAR ASSOCIATED PROTEIN|CHLAMYDOMONAS REINHARDTII (3055)
>1XBW_1|Chains A, B, C, D|hypothetical protein isdG|Staphylococcus aureus subsp. aureus (196620)
>6LKB_1|Chains A, B|Peptidyl-prolyl cis-trans isomerase CYP71|Arabidopsis thaliana (3702)
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 \(w=f(c)\)
4UZY , Knot 250 651 0.83 40 281 599 
GAASMAFFQQPARPIAEGQYTQTIYTLIKEQKFAEAIQHLQYQLQNVPESRAALSLLGYCYYYTGQYDMASQMYEQLVTLYPSNEDYKLYYAQSLYKGGMYPEASKAVVKVEGHQKAVTTLLVACSYEQDDLTGCRRQLDKCAPEDPDTMVNTGCIMFKEGKFEAARQKFNDAVQALGYQPELLYNIALCYYKTKQFGPALKHLAEIIEKAVREHPELSVGSNTDGMEVRSVGNSQTLKETALIEAFNLKAAIEYTMKNVEAAKEALTDMPPRAEEELDPVTLHNSALINMDSDPTGGFKKLNFLLQSPPFPPETFANLLLLYCKPSHGFYDLAADVLAENPQYAGKLLSPDLYDYLQAAIGRYKSPEEAFRRFDELATRHVEQLRRLTKQIQDARIARDNDAIKRAINEYDEALEAYIPGLMAMASIYWDMELYSNVEKIFRQSAEFCSEHEVWKLNVAHTFFMQDNHYKEAIRYYEPVVKKNADNLLGVTAIVLANLCVSYIMTSQNEEAEELMRKVEKEEERSSMQDPDKPCFHLCIINLVIGTLYCAKGNYEFGVSRIIKSLEPYDKKLETDTWYYAKRCFLALIENLAKHMIVLKDSSFTEIMAFLNEAEKHGKDIRVVFNEGKHQSRTIASEARMLKKMFLKLRD
1XBW , Knot 57 109 0.81 38 93 106 
ETMKFMAENRLTLTKGTAKDIIERFYTRHGIETLEGFDGMFVTQTLEQEDFDEVKILTVWKSKQAFTDWLKSDVFKAAHKHVRSKNEDESSPIINNKVITYDIGYSYMK
6LKB , Knot 79 160 0.83 40 127 156 
SATTSLPENVIMHTTLGDIHMKLYPEECPKTVENFTTHCRNGYYDNHLFHRVIRGFMIQTGDPLGDGTGGQSIWGREFEDEFHKSLRHDRPFTLSMANAGPNTNGSQFFITTVATPWLDNKHTVFGRVVKGMDVVQGIEKVKTDKNDRPYQDVKILNVTV

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 \(c\).

\(|P_{f(4UZY_1)}(2) \setminus P_{f(1XBW_1)}(2)|=203\), \(|P_{f(1XBW_1)}(2) \setminus P_{f(4UZY_1)}(2)|=15\). 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 1:111011110011011101000001001100001101100100010011000111011100000010001100100011010100000010010010011101010011101010001100111100000001010000100011001001100101110010101100010011011100101100111000000011111001101100110001010110000110100110000100011101101011100010010110011001110100010110100011101000101110010111001111100110111100010011001110111001001101101010001011110000100110010011000100100100010010110000110011000001101011111111010101010001001100010100000110101100111000000011000011100010011110111110101001100000010011001000000001001001010101101111010010100011100110010100001000010010001111100110011110000100111110010001001011100100000011001011001110100
Pair \(Z_2\) Length of longest common subsequence
4UZY_1,1XBW_1 218 3
4UZY_1,6LKB_1 218 3
1XBW_1,6LKB_1 146 4

Newick tree

 
[
	4UZY_1:11.59,
	[
		1XBW_1:73,6LKB_1:73
	]:45.59
]

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 \((0.85)(0.8)(\frac{760 }{\log_{20} 760}-\frac{109}{\log_{20}109})=186.\)
Status Protein1 Protein2 d d1/2
Query variables 4UZY_1 1XBW_1 231 133.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]