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Parikh vectors
9DAS_1 6ZVP_1 8AXK_1 Letter Amino acid
1 3 1 W Tryptophan
22 51 17 A Alanine
30 3 9 N Asparagine
18 27 10 D Aspartic acid
9 19 6 Q Glutamine
36 24 10 G Glycine
9 3 11 M Methionine
6 28 9 P Proline
6 33 4 R Arginine
7 15 1 H Histidine
14 14 20 I Isoleucine
7 20 11 K Lycine
20 49 32 L Leucine
9 25 15 F Phenylalanine
37 34 18 S Serine
24 29 14 V Valine
0 7 1 C Cysteine
4 41 10 E Glutamic acid
27 17 7 T Threonine
15 16 10 Y Tyrosine 

9DAS_1|Chains A, B|Flagellin hook IN motif family|Aeromonas hydrophila (644)
>6ZVP_1|Chains A, B, C, D|Tyrosine 3-monooxygenase|Homo sapiens (9606)
>8AXK_1|Chains A, B, C, D, E|Surface presentation of antigens protein SpaP|Shigella flexneri (623)
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 \(w=f(c)\)
9DAS , Knot 128 301 0.81 38 162 278 
MASSHHHHHHSSGLVPRGSHMPGQNYNIAFMVDSSGSIGSSAMNTIKSQLAQVFSSLKASAGTDGAGTVNIFLVDFDTLANKSVSVNLKDSNALSQLQAILDSMDGSTNSGGGTNYEDVFKTTANWFNSSTVQGNTNAKNLTYFITDGQPTFYLENEGSDPMIRDRSGTVNDVYLSNVIGTSYTYGQTYSVNGRTVVNGNGVVYDYNGNQIGYMRPDGAGRYVYATLAGTGNSTSSTTIANSVAGFGLLTGMGVTVEAIGLGANISYNDLKSYDSDGVIMTGVNASDLANAILGTSVNNLP
6ZVP , Knot 180 458 0.80 40 212 416 
SLIEDARKEREAAVAAAAAAVPSEPGDPLEAVAFEEKEGKAMLNLLFSPRATKPSALSRAVKVFETFEAKIHHLETRPAQRPRAGGPHLEYFVRLEVRRGDLAALLSGVRQVSEDVRSPAGPKVPWFPRKVSELDKCHHLVTKFDPDLDLDHPGFSDQVYRQRRKLIAEIAFQYRHGDPIPRVEYTAEEIATWKEVYTTLKGLYATHACGEHLEAFALLERFSGYREDNIPQLEDVSRFLKERTGFQLRPVAGLLSARDFLASLAFRVFQCTQYIRHASSPMHSPEPDCCHELLGHVPMLADRTFAQFSQDIGLASLGASDEEIEKLSTLYWFTVEFGLCKQNGEVKAYGAGLLSSYGELLHCLSEEPEIRAFDPEAAAVQPYQDQTYQSVYFVSESFSDAKDKLRSYASRIQRPFSVKFDPYTLAIDVLDSPQAVRRSLEGVQDELDTLAHALSAIG
8AXK , Knot 97 216 0.80 40 140 205 
MLSDMSLIATLSFFTLLPFLVAAGTCYIKFSIVFVMVRNALGLQQVPSNMTLNGIALIMALFVMKPIIEAGYENYLNGPQKFDTISDIVRFSDSGLMEYKQYLKKHTDLELARFFQRSEEENADLKSAENNDYSLFSLLPAYALSEIKDAFKIGFYLYLPFVVVDLVISSILLALGMMMMSPITISVPIKLVLFVALDGWGILSKALIEQYINIPA

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 \(c\).

\(|P_{f(9DAS_1)}(2) \setminus P_{f(6ZVP_1)}(2)|=71\), \(|P_{f(6ZVP_1)}(2) \setminus P_{f(9DAS_1)}(2)|=121\). 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 1:1100000000001111010011100001111100010110011001000110110010101100111010111101001100010101000011001011100101000011100000110001011000010100010010011001010101000100111000010100101001110000010000101001101011100001001101010111001010111010000000110011111110111101011111101000010000001111011010011011110010011
Pair \(Z_2\) Length of longest common subsequence
9DAS_1,6ZVP_1 192 4
9DAS_1,8AXK_1 156 4
6ZVP_1,8AXK_1 188 3

Newick tree

 
[
	6ZVP_1:10.03,
	[
		9DAS_1:78,8AXK_1:78
	]:22.03
]

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 \((0.85)(0.8)(\frac{759 }{\log_{20} 759}-\frac{301}{\log_{20}301})=125.\)
Status Protein1 Protein2 d d1/2
Query variables 9DAS_1 6ZVP_1 159 131
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]