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Parikh vectors
1MXK_1 8DFR_1 5BQY_1 Letter Amino acid
2 7 22 T Threonine
0 3 8 W Tryptophan
0 8 20 R Arginine
0 8 26 N Asparagine
0 12 21 I Isoleucine
0 12 24 S Serine
2 11 17 A Alanine
0 8 12 Y Tyrosine
0 14 19 V Valine
0 11 10 D Aspartic acid
0 14 23 E Glutamic acid
0 4 4 H Histidine
0 8 11 F Phenylalanine
0 18 18 K Lycine
0 5 3 M Methionine
0 11 18 P Proline
3 1 9 C Cysteine
0 7 7 Q Glutamine
3 10 26 G Glycine
0 17 26 L Leucine 

1MXK_1|Chains A, B|5'-D(*GP*GP*AP*AP*GP*CP*TP*TP*CP*C)-3'|
>8DFR_1|Chain A|DIHYDROFOLATE REDUCTASE|Gallus gallus (9031)
>5BQY_1|Chains A, C, E|HEMAGGLUTININ HA1 CHAIN|Influenza A virus (A/chicken/Guangdong/S1312/2010(H6N2)) (1454273)
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 \(w=f(c)\)
1MXK , Knot 7 10 0.53 8 9 8 
GGAAGCTTCC
8DFR , Knot 92 189 0.85 40 148 186 
VRSLNSIVAVCQNMGIGKDGNLPWPPLRNEYKYFQRMTSTSHVEGKQNAVIMGKKTWFSIPEKNRPLKDRINIVLSRELKEAPKGAHYLSKSLDDALALLDSPELKSKVDMVWIVGGTAVYKAAMEKPINHRLFVTRILHEFESDTFFPEIDYKDFKLLTEYPGVPADIQEEDGIQYKFEVYQKSVLAQ
5BQY , Knot 143 324 0.85 40 204 311 
DKICIGYHANNSTTKVDTILEKNVTVTHSVELLENQKEERFCKISNKAPLDLRDCTLEGWILGNPRCGILLADQSWSYIVERPNARNGICYPGTLNEAEELKALIGSGERVERFEMFPKSTWTGVNTESGVSSACPLGNGPSFYRNLLWIIKLKSSEYPVIRGTFNNTGDKSILYFWGVHHPPVTTEQNALYGSGDRYVRMGTESMNFARSPEIAARPAVNGQRGRIDYFWSILKPGETLNVESNGNLIAPWYAYRFVNKDSKGAIFRSNLPIENCDATCQTTEGVIRTNKTFQNVSPLWIGECPKYVKSKSLRLATGLRNVPQ

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 \(c\).

\(|P_{f(1MXK_1)}(2) \setminus P_{f(8DFR_1)}(2)|=6\), \(|P_{f(8DFR_1)}(2) \setminus P_{f(1MXK_1)}(2)|=145\). 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 1:1111100000
Pair \(Z_2\) Length of longest common subsequence
1MXK_1,8DFR_1 151 2
1MXK_1,5BQY_1 203 2
8DFR_1,5BQY_1 172 4

Newick tree

 
[
	5BQY_1:99.49,
	[
		1MXK_1:75.5,8DFR_1:75.5
	]:23.99
]

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 \((0.85)(0.8)(\frac{199 }{\log_{20} 199}-\frac{10}{\log_{20}10})=67.7\)
Status Protein1 Protein2 d d1/2
Query variables 1MXK_1 8DFR_1 90 47.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]

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