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Parikh vectors
1LYH_1 6JRD_1 6INW_1 Letter Amino acid
5 8 28 Q Glutamine
16 24 44 L Leucine
5 7 13 M Methionine
3 6 6 W Tryptophan
6 9 8 Y Tyrosine
13 10 21 R Arginine
8 15 24 E Glutamic acid
3 17 23 P Proline
13 10 14 K Lycine
6 22 21 S Serine
16 28 40 A Alanine
12 10 10 N Asparagine
10 13 20 I Isoleucine
1 14 9 H Histidine
5 7 17 F Phenylalanine
11 22 22 T Threonine
9 25 20 V Valine
10 15 23 D Aspartic acid
0 3 6 C Cysteine
12 19 21 G Glycine 

1LYH_1|Chain A|T4 LYSOZYME|Enterobacteria phage T4 (10665)
>6JRD_1|Chains A, B|Zearalenone hydrolase|Clonostachys rosea (29856)
>6INW_1|Chains A, B|O-methyltransferase lepI|Aspergillus flavus (strain ATCC 200026 / FGSC A1120 / NRRL 3357 / JCM 12722 / SRRC 167) (332952)
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 \(w=f(c)\)
1LYH , Knot 77 164 0.79 38 119 158 
MNIFEMLRIDEGLRLKIYKDTEGYYTIGIGHLLTKSPSLNAAKSELDKAIGRNTNGVIGKDEAEKLFNQDVDAAVRGILRNAKLKPVYDSLDAVRRAALINMVFQMGETGVAGFTNSLRMLQQKRWDEAAVNLAKSRWYNQTPNRAKRVITTFRTGTWDAYKNL
6JRD , Knot 127 284 0.84 40 187 274 
MRTRSTISTPNGITWYYEQEGTGPDIVLVPDGLGECQMFDSSVSQIAAQGFRVTTFDMPGMSRSAKAPAETYTEVTAQKLASYVISILDALDIKHATVWGCSSGASTVVALLLGYPDRIRNAMCHELPTKLLDHLSNTAVLEDEEISNILANVMLNDVSGGSEAWQALGVEVHARLHKNYPVWARGYPRTIPPSAPVQDVEALRGKPLDWTVGAATPTESFFDNIVTATKAGVNIGLLPGMHFPYVSHPDVFAKYVVETTQKHLWNSSSVDKLAAALEHHHHHH
6INW , Knot 163 390 0.83 40 215 373 
GSHMETVAAIKTLIQQLAQSTDQFGRAEINDALRELQYSLETPFDTVMRMSLDTCQVAVARIGSDLGLFKHLSQCASPQSAEELADHLGCGRELMSRLLRYMASVRMVQQTDDIKYISSNITQTLAVPGLEAGMRHAFENLWPVLMALPDFLAERKYPDIVDAKDTAFQKAFNTDQDCFHWLATQPTRIANFKVLLTDERTPNFLSTFPLEKELGSWSAEPEKALFVDIGGGMGHACIRLREKYPNQPGRVILQDLPPVLQAAQATLPLSGIESMPHNFHTPQPVQGAKFYFLRLILRDFPDHQALEILQNIVPAMDAESRIVIDDGVPPEKGARWAETGTDICIMSALGSKERTQRQWEELAAKAGLQLQALYQYTWPVVNAAMVFSLQ

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 \(c\).

\(|P_{f(1LYH_1)}(2) \setminus P_{f(6JRD_1)}(2)|=46\), \(|P_{f(6JRD_1)}(2) \setminus P_{f(1LYH_1)}(2)|=114\). 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 1:10110110100110101000001000111101100010101100010011100001111000100110001011101110010101100010110011110111011001111100010110000100111011000100001001001100100101010001
Pair \(Z_2\) Length of longest common subsequence
1LYH_1,6JRD_1 160 3
1LYH_1,6INW_1 188 3
6JRD_1,6INW_1 182 4

Newick tree

 
[
	6INW_1:96.32,
	[
		1LYH_1:80,6JRD_1:80
	]:16.32
]

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 \((0.85)(0.8)(\frac{448 }{\log_{20} 448}-\frac{164}{\log_{20}164})=83.9\)
Status Protein1 Protein2 d d1/2
Query variables 1LYH_1 6JRD_1 106 82.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]