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Parikh vectors
1CJR_1 7UGH_1 6VWM_1 Letter Amino acid
0 44 41 G Glycine
1 15 0 F Phenylalanine
1 23 0 D Aspartic acid
0 50 0 L Leucine
1 7 0 M Methionine
1 23 0 T Threonine
0 15 0 Y Tyrosine
3 49 24 A Alanine
1 22 0 R Arginine
0 3 36 C Cysteine
0 24 0 I Isoleucine
0 2 0 W Tryptophan
0 33 0 V Valine
0 29 0 N Asparagine
1 15 0 Q Glutamine
2 43 0 E Glutamic acid
1 15 0 H Histidine
2 33 0 K Lycine
0 11 0 P Proline
1 21 0 S Serine 

1CJR_1|Chain A|PROTEIN (RIBONUCLEASE S)|synthetic construct (32630)
>7UGH_1|Chains A, B|Phosphopyruvate hydratase|Naegleria fowleri (5763)
>6VWM_1|Chain A[auth 3]|5S ribosomal RNA|Escherichia coli (562)
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 \(w=f(c)\)
1CJR , Knot 12 15 0.72 22 13 13 
KETAAAKFERQHMDS
7UGH , Knot 192 477 0.82 40 230 447 
MAHHHHHHQKLVEDALNECYNANASDPVGFLGHFFLNRGKKGAVNRVDKLVGREILDSRGNPTVEVDVYANGQKRPVATASAPSGASTGSNEAHELRDGDKSRYLGKGVLKAVKNVNDVLGKAVEGKSLENLTELDQALIDADGDELKSNLGGNAITACSFALATAGAAVRNEELFLYLARAFHGADKFENLKFRLPTPMVNILNGGKHAGGRLQIQEFMILPKENQPFREKVRCVAEVYQHLGKILAERAGPSAKNVGDEGGFAPNLETADEALNYIEEAIGKAGYKVGEDVFLALDAASSEFYNSDTKKYEITQQKEFLTSEEMVEYYVQLVNRHPAIISIEDGLEEKDYEGWKLLTERLGSKIMLVGDDLYTTNTRLIKQGIEEKWANALLLKVNQIGTITEAMNAARMIFNVGQKVIVSHRSGETATTLISDLVVGIGATHIKTGATARGERVSKYNRLLQIEEYLEQHGLLA
6VWM , Knot 34 120 0.45 8 15 47 
UGCCUGGCGGCCGUAGCGCGGUGGUCCCACCUGACCCCAUGCCGAACUCAGAAGUGAAACGCCGUAGCGCCGAUGGUAGUGUGGGGUCUCCCCAUGCGAGAGUAGGGAACUGCCAGGCAA

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 \(c\).

\(|P_{f(1CJR_1)}(2) \setminus P_{f(7UGH_1)}(2)|=3\), \(|P_{f(7UGH_1)}(2) \setminus P_{f(1CJR_1)}(2)|=220\). 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 1:000111010000100
Pair \(Z_2\) Length of longest common subsequence
1CJR_1,7UGH_1 223 3
1CJR_1,6VWM_1 26 3
7UGH_1,6VWM_1 235 4

Newick tree

 
[
	7UGH_1:13.04,
	[
		1CJR_1:13,6VWM_1:13
	]:11.04
]

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 \((0.85)(0.8)(\frac{492 }{\log_{20} 492}-\frac{15}{\log_{20}15})=150.\)
Status Protein1 Protein2 d d1/2
Query variables 1CJR_1 7UGH_1 186 95.5
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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