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Parikh vectors
1ERU_1 9KNF_1 7OUW_1 Letter Amino acid
6 35 13 L Leucine
12 20 15 K Lycine
3 11 11 P Proline
4 8 16 T Threonine
1 8 12 Y Tyrosine
11 17 15 V Valine
7 17 13 D Aspartic acid
5 10 12 G Glycine
3 13 6 M Methionine
7 18 16 S Serine
5 1 3 C Cysteine
5 10 10 Q Glutamine
10 18 8 E Glutamic acid
1 14 13 H Histidine
9 5 13 F Phenylalanine
1 2 2 W Tryptophan
8 20 11 A Alanine
0 6 9 R Arginine
3 5 17 N Asparagine
4 13 6 I Isoleucine 

1ERU_1|Chain A|THIOREDOXIN|Homo sapiens (9606)
>9KNF_1|Chain A|Estrogen-related receptor gamma|Homo sapiens (9606)
>7OUW_1|Chains A, B|Protein mono-ADP-ribosyltransferase PARP15|Homo sapiens (9606)
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 \(w=f(c)\)
1ERU , Knot 56 105 0.82 38 88 103 
MVKQIESKTAFQEALDAAGDKLVVVDFSATWCGPCKMIKPFFHSLSEKYSNVIFLEVDVDDCQDVASECEVKCMPTFQFFKKGQKVGEFSGANKEKLEATINELV
9KNF , Knot 111 251 0.81 40 160 241 
MGSSHHHHHHSSGLVPRGSHMPAKKPYNKIVSHLLVAEPEKIYAMPDPTVPDSDIKALTTLCDLADRELVVIIGWAKHIPGFSTLSLADQMSLLQSAWMEILILGVVYRSLSFEDELVYADDYIMDEDQSKLAGLLDLNNAILQLVKKYKSMKLEKEEFVTLKAIALANSDSMHIEDVEAVQKLQDVLHEALQDYEAGQHMEDPRRAGKMLMTLPLLRQTSTKAVQHFYNIKLEGKVPMHKLFLEMLEAKV
7OUW , Knot 103 221 0.83 40 159 215 
MHHHHHHSSGVDLGTENLYFQSMNLPEHWTDMNHQLFCMVQLEPGQSEYNTIKDKFTRTCSSYAIEKIERIQNAFLWQSYQVKKRQMDIKNDHKNNERLLFHGTDADSVPYVNQHGFNRSCAGKNAVSYGKGTYFAVDASYSAKDTYSKPDSNGRKHMYVVRVLTGVFTKGRAGLVTPPPKNPHNPTDLFDSVTNNTRSPKLFVVFFDNQAYPEYLITFTA

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 \(c\).

\(|P_{f(1ERU_1)}(2) \setminus P_{f(9KNF_1)}(2)|=49\), \(|P_{f(9KNF_1)}(2) \setminus P_{f(1ERU_1)}(2)|=121\). 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 1:110010000110011011100111101010101100110111001000000111101010000011000010011010110010011010110000101010011
Pair \(Z_2\) Length of longest common subsequence
1ERU_1,9KNF_1 170 3
1ERU_1,7OUW_1 175 2
9KNF_1,7OUW_1 185 9

Newick tree

 
[
	7OUW_1:91.65,
	[
		1ERU_1:85,9KNF_1:85
	]:6.65
]

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 \((0.85)(0.8)(\frac{356 }{\log_{20} 356}-\frac{105}{\log_{20}105})=77.4\)
Status Protein1 Protein2 d d1/2
Query variables 1ERU_1 9KNF_1 97 68.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]