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Parikh vectors
4RJV_1 2HHB_1 6UPS_1 Letter Amino acid
4 7 5 F Phenylalanine
1 7 6 P Proline
2 11 16 S Serine
0 1 3 C Cysteine
4 11 16 K Lycine
3 2 8 M Methionine
0 3 5 Y Tyrosine
8 21 15 A Alanine
8 18 33 L Leucine
0 1 1 W Tryptophan
6 10 7 H Histidine
5 9 13 T Threonine
7 8 19 D Aspartic acid
2 1 22 Q Glutamine
6 7 10 G Glycine
5 0 21 I Isoleucine
7 13 17 V Valine
9 3 6 R Arginine
1 4 25 N Asparagine
9 4 12 E Glutamic acid 

4RJV_1|Chains A, B, C, D|OR461|synthetic construct (32630)
>2HHB_1|Chains A, C|HEMOGLOBIN (DEOXY) (ALPHA CHAIN)|Homo sapiens (9606)
>6UPS_1|Chain A|ULP_PROTEASE domain-containing protein|Orientia tsutsugamushi (strain Ikeda) (334380)
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 \(w=f(c)\)
4RJV , Knot 43 87 0.73 34 62 80 
MEMDIRFRGDDDEALARALLKMVVRAVSFGAQVRFTDDGNDLEIRITGVPEQVLKELAKEAERLAKEFGITVTRTIRGSLEHHHHHH
2HHB , Knot 69 141 0.80 38 99 137 
VLSPADKTNVKAAWGKVGAHAGEYGAEALERMFLSFPTTKTYFPHFDLSHGSAQVKGHGKKVADALTNAVAHVDDMPNALSALSDLHAHKLRVDPVNFKLLSHCLLVTLAAHLPAEFTPAVHASLDKFLASVSTVLTSKYR
6UPS , Knot 118 260 0.84 40 163 243 
GMANDQDLSNPEYLYTEDDINQLLKHYLGLDDRISIIQHVALNESLLLKQTLHQVLSDIFSGMQEKAVIPLHTGNNHWVAMAIKKGMNDDIVISYNDPMGVSIDDKVTLINCIKELCPGAKINDLQTVQQTNVYDCGPFVVDNLIKMSQGQPILSTEEAKQQAQNIRQSQVNFLSENRMITSAAAALADTLLKNNNRITEGVLVDRIFDNKILSVQEKQQLLNNLLDNHIKENKSLTKESLTRMLASTHFVQQQANVLLN

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 \(c\).

\(|P_{f(4RJV_1)}(2) \setminus P_{f(2HHB_1)}(2)|=40\), \(|P_{f(2HHB_1)}(2) \setminus P_{f(4RJV_1)}(2)|=77\). 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 1:101010101000011101110111011011101010001001010101110011001100100110011101000101010000000
Pair \(Z_2\) Length of longest common subsequence
4RJV_1,2HHB_1 117 3
4RJV_1,6UPS_1 163 3
2HHB_1,6UPS_1 166 3

Newick tree

 
[
	6UPS_1:88.76,
	[
		4RJV_1:58.5,2HHB_1:58.5
	]:30.26
]

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 \((0.85)(0.8)(\frac{228 }{\log_{20} 228}-\frac{87}{\log_{20}87})=45.8\)
Status Protein1 Protein2 d d1/2
Query variables 4RJV_1 2HHB_1 59 46.5
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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