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Parikh vectors
3EUS_1 1VLQ_1 3WJT_1 Letter Amino acid
9 19 10 R Arginine
1 12 1 H Histidine
4 21 11 K Lycine
2 7 5 M Methionine
2 24 3 F Phenylalanine
2 22 10 P Proline
1 19 10 Y Tyrosine
0 7 11 N Asparagine
6 10 19 Q Glutamine
3 14 7 I Isoleucine
9 25 16 L Leucine
8 26 6 E Glutamic acid
3 28 15 G Glycine
6 13 11 T Threonine
13 24 9 A Alanine
5 19 15 D Aspartic acid
2 5 0 C Cysteine
2 14 7 S Serine
1 5 5 W Tryptophan
7 23 7 V Valine 

3EUS_1|Chains A, B|DNA-binding protein|Silicibacter pomeroyi (89184)
>1VLQ_1|Chains A, B, C, D, E, F, G, H, I, J, K, L|acetyl xylan esterase|Thermotoga maritima (243274)
>3WJT_1|Chain A|Outer-membrane lipoprotein LolB|Escherichia coli (536056)
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 \(w=f(c)\)
3EUS , Knot 46 86 0.79 38 68 81 
QAMTKTLRTPEHVYLCQRLRQARLDAGLTQADLAERLDKPQSFVAKVETRERRLDVIEFAKWMAACEGLDVVSEIVATIAEGRAQA
1VLQ , Knot 142 337 0.81 40 194 317 
MGSDKIHHHHHHMAFFDLPLEELKKYRPERYEEKDFDEFWEETLAESEKFPLDPVFERMESHLKTVEAYDVTFSGYRGQRIKGWLLVPKLEEEKLPCVVQYIGYNGGRGFPHDWLFWPSMGYICFVMDTRGQGSGWLKGDTPDYPEGPVDPQYPGFMTRGILDPRTYYYRRVFTDAVRAVEAAASFPQVDQERIVIAGGSQGGGIALAVSALSKKAKALLCDVPFLCHFRRAVQLVDTHPYAEITNFLKTHRDKEEIVFRTLSYFDGVNFAARAKIPALFSVGLMDNICPPSTVFAAYNYYAGPKEIRIYPYNNHEGGGSFQAVEQVKFLKKLFEKG
3WJT , Knot 87 178 0.84 38 134 174 
PGKSPDSPQWRQHQQDVRNLNQYQTRGAFAYISDQQKVYARFFWQQTGQDRYRLLLTNPDGSTELELNAQPGNVQLVDNKGQRYTADDAEEMIGKLTGMPIPLNSLRQWILGLPGDATDYKLDDQYRLSEITYSQNGKNWKVVYGGYDTKTQPAMPANMELTDGGQRIKLKMDNWIVK

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 \(c\).

\(|P_{f(3EUS_1)}(2) \setminus P_{f(1VLQ_1)}(2)|=26\), \(|P_{f(1VLQ_1)}(2) \setminus P_{f(3EUS_1)}(2)|=152\). 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 1:01100010010010100010010101110010110010010011101000000101101101111001101100111011010101
Pair \(Z_2\) Length of longest common subsequence
3EUS_1,1VLQ_1 178 3
3EUS_1,3WJT_1 148 3
1VLQ_1,3WJT_1 188 5

Newick tree

 
[
	1VLQ_1:96.67,
	[
		3EUS_1:74,3WJT_1:74
	]:22.67
]

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 \((0.85)(0.8)(\frac{423 }{\log_{20} 423}-\frac{86}{\log_{20}86})=103.\)
Status Protein1 Protein2 d d1/2
Query variables 3EUS_1 1VLQ_1 129 80.5
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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