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Parikh vectors
6GHB_1 3RWB_1 1JLQ_1 Letter Amino acid
6 16 25 D Aspartic acid
6 6 37 Q Glutamine
6 13 58 K Lycine
6 17 40 T Threonine
3 2 22 Y Tyrosine
7 20 40 V Valine
7 6 37 P Proline
1 3 19 W Tryptophan
4 46 26 A Alanine
15 10 18 R Arginine
3 26 34 G Glycine
2 4 8 H Histidine
14 14 49 L Leucine
3 5 6 M Methionine
6 10 19 N Asparagine
2 0 2 C Cysteine
14 8 47 E Glutamic acid
5 9 13 F Phenylalanine
12 21 41 I Isoleucine
10 11 19 S Serine 

6GHB_1|Chains A, C|Regulatory protein Spx|Bacillus subtilis (strain 168) (224308)
>3RWB_1|Chains A, B, C, D|Pyridoxal 4-dehydrogenase|Mesorhizobium loti (381)
>1JLQ_1|Chain A|HIV-1 RT, A-CHAIN|HIV-1 M:B_HXB2R (11706)
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 \(w=f(c)\)
6GHB , Knot 67 132 0.82 40 102 127 
SMVTLYTSPSCTSCRKARAWLEEHEIPFVERNIFSEPLSIDEIKQILRMTEDGTDEIISTRSKVFQKLNVNVESMPLQDLYRLINEHPGLLRRPIIIDEKRLQVGYNEDEIRRFLPRKVRSFQLREAQRLAN
3RWB , Knot 108 247 0.80 38 142 232 
TERLAGKTALVTGAAQGIGKAIAARLAADGATVIVSDINAEGAKAAAASIGKKARAIAADISDPGSVKALFAEIQALTGGIDILVNNASIVPFVAWDDVDLDHWRKIIDVNLTGTFIVTRAGTDQMRAAGKAGRVISIASNTFFAGTPNMAAYVAAKGGVIGFTRALATELGKYNITANAVTPGLIESDGVKASPHNEAFGFVEMLQAMKGKGQPEHIADVVSFLASDDARWITGQTLNVDAGMVRH
1JLQ , Knot 223 560 0.84 40 255 517 
PISPIETVPVKLKPGMDGPKVKQWPLTEEKIKALVEICTEMEKEGKISKIGPENPYNTPVFAIKKKDSTKWRKLVDFRELNKRTQDFWEVQLGIPHPAGLKKKKSVTVLDVGDAYFSVPLDEDFRKYTAFTIPSINNETPGIRYQYNVLPQGWKGSPAIFQSSMTKILEPFRKQNPDIVIYQYMDDLYVGSDLEIGQHRTKIEELRQHLLRWGLTTPDKKHQKEPPFLWMGYELHPDKWTVQPIVLPEKDSWTVNDIQKLVGKLNWASQIYPGIKVRQLCKLLRGTKALTEVIPLTEEAELELAENREILKEPVHGVYYDPSKDLIAEIQKQGQGQWTYQIYQEPFKNLKTGKYARMRGAHTNDVKQLTEAVQKITTESIVIWGKTPKFKLPIQKETWETWWTEYWQATWIPEWEFVNTPPLVKLWYQLEKEPIVGAETFYVDGAANRETKLGKAGYVTNRGRQKVVTLTDTTNQKTELQAIYLALQDSGLEVNIVTDSQYALGIIQAQPDQSESELVNQIIEQLIKKEKVYLAWVPAHKGIGGNEQVDKLVSAGIRKVL

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 \(c\).

\(|P_{f(6GHB_1)}(2) \setminus P_{f(3RWB_1)}(2)|=56\), \(|P_{f(3RWB_1)}(2) \setminus P_{f(6GHB_1)}(2)|=96\). 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 1:011010001000000010111000011110001100110100100110100010001100000110010101001110010011000111100111100001011000001001110010010100100110
Pair \(Z_2\) Length of longest common subsequence
6GHB_1,3RWB_1 152 4
6GHB_1,1JLQ_1 199 3
3RWB_1,1JLQ_1 179 4

Newick tree

 
[
	1JLQ_1:10.07,
	[
		6GHB_1:76,3RWB_1:76
	]:24.07
]

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 \((0.85)(0.8)(\frac{379 }{\log_{20} 379}-\frac{132}{\log_{20}132})=74.9\)
Status Protein1 Protein2 d d1/2
Query variables 6GHB_1 3RWB_1 92 71.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]