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Parikh vectors
5LED_1 4JRI_1 6HHU_1 Letter Amino acid
65 3 73 A Alanine
0 0 0 C Cysteine
12 4 12 P Proline
2 7 30 S Serine
5 2 12 Y Tyrosine
50 3 33 G Glycine
7 2 14 F Phenylalanine
24 2 60 T Threonine
8 1 1 W Tryptophan
37 9 88 V Valine
30 4 84 K Lycine
16 4 1 R Arginine
31 2 30 D Aspartic acid
29 1 7 H Histidine
19 5 13 I Isoleucine
68 7 40 L Leucine
29 3 31 N Asparagine
7 6 17 Q Glutamine
42 2 56 E Glutamic acid
4 1 4 M Methionine 

5LED_1|Chain A|DDD_D12_12_D12_12_D12|synthetic construct (32630)
>4JRI_1|Chains A, B, C, D|Protein hfq|Escherichia coli (83333)
>6HHU_1|Chain A|S-layer protein sap|Bacillus anthracis (1392)
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 \(w=f(c)\)
5LED , Knot 80 485 0.34 38 110 164 
MRGSHHHHHHGSDLGKKLLEAARAGQDDEVRILLANGADVNTADETGFTPLHLAAWEGHLGIVEVLLKNGADVNANDERGHTPLHLAAYTGHLEIVEVLLKNGAGVNATDVIGTAPLHLAAMWGHLEIVEVLLKNGAAVGAQDKFGKTPKDLARDNGNQWIYELLEKAEKDLRRKLLEAARAGHREEVEKLIKLGADVNTADETGFTPLHLAAWEGHLGIVEVLLKNGADVNANDERGHTPLHLAAYTGHLEIVEVLLKNGAGVNATDVIGTAPLHLAAMWGHLEIVEVLLKNGAAVGAQDKFGKTPKDLARDNGNQWIYELLEKAEKDLRRKLLEAARAGHREEVEKLIKLGADVNTADETGFTPLHLAAWEGHLGIVEVLLKNGADVNANDERGHTPLHLAAYTGHLEIVEVLLKNGAGVNATDVIGTAPLHLAAMWGHLEIVEVLLKNGADVNAQDKFGKTPFDLAIDNGNEDIAEVLQKAA
4JRI , Knot 39 68 0.80 38 60 65 
AKGQSLQDPFLNALRRERVPVSIYLVNGIKLQGQIESWDQFVILLKNTVSQMVYKHAISTVVPSRPVS
6HHU , Knot 213 606 0.75 38 180 468 
MSAKAVTTQKVEVKFSKAVEKLTKEDIKVTNKANNDKVLVKEVTLSEDKKSATVELYSNLAAKQTYTVDVNKVGKTEVAVGSLEAKTIEMADQTVVADEPTALQFTVKDENGTEVVSPEGIEFVTPAAEKINAKGEITLAKGTSTTVKAVYKKDGKVVAESKEVKVSAEGAAVASISNWTVAEQNKADFTSKDFKQNNKVYEGDNAYVQVELKDQFNAVTTGKVEYESLNTEVAVVDKATGKVTVLSAGKAPVKVTVKDSKGKELVSKTVEIEAFAQKAMKEIKLEKTNVALSTKDVTDLKVKAPVLDQYGKEFTAPVTVKVLDKDGKELKEQKLEAKYVNKELVLNAAGQEAGNYTVVLTAKSGEKEAKATLALELKAPGAFSKFEVRGLEKELDKYVTEENQKNAMTVSVLPVDANGLVLKGAEAAELKVTTTNKEGKEVDATDANVTVQNNSVITVGQGAKAGETYKVTVVLDGKLITTHSFKVVDTAPTAKGLAVEFTSTSLKEVAPNADLKAALLNILSVDGVPATTAKATVSNVEFVSADTNVVAENGTVGAKGATSIYVKNLTVVKDGKEQKVEFDKAVQVAVSIKEAKPATKHHHHHH

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 \(c\).

\(|P_{f(5LED_1)}(2) \setminus P_{f(4JRI_1)}(2)|=88\), \(|P_{f(4JRI_1)}(2) \setminus P_{f(5LED_1)}(2)|=38\). 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 1:10100000001001100110110110000101111011010010001101101111010111101110011010100001001101110010101101110011110100111011101111110101101110011111100011001001100010011001100100010001101101100001001101110100100011011011110101111011100110101000010011011100101011011100111101001110111011111101011011100111111000110010011000100110011001000100011011011000010011011101001000110110111101011110111001101010000100110111001010110111001111010011101110111111010110111001101010001100110111001000110110011
Pair \(Z_2\) Length of longest common subsequence
5LED_1,4JRI_1 126 4
5LED_1,6HHU_1 146 6
4JRI_1,6HHU_1 164 3

Newick tree

 
[
	6HHU_1:81.92,
	[
		5LED_1:63,4JRI_1:63
	]:18.92
]

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 \((0.85)(0.8)(\frac{553 }{\log_{20} 553}-\frac{68}{\log_{20}68})=145.\)
Status Protein1 Protein2 d d1/2
Query variables 5LED_1 4JRI_1 71 49
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]