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Parikh vectors
5AGQ_1 4ESU_1 7KTL_1 Letter Amino acid
15 12 27 R Arginine
4 4 15 H Histidine
5 23 11 K Lycine
8 7 22 S Serine
4 6 5 W Tryptophan
3 9 5 Y Tyrosine
2 9 14 D Aspartic acid
1 0 7 C Cysteine
4 15 19 T Threonine
1 11 3 N Asparagine
6 9 30 Q Glutamine
9 10 27 G Glycine
3 5 4 M Methionine
5 6 18 F Phenylalanine
8 12 21 P Proline
4 15 28 A Alanine
12 26 27 E Glutamic acid
6 9 6 I Isoleucine
5 16 45 L Leucine
8 6 22 V Valine 

5AGQ_1|Chain A|BROMODOMAIN ADJACENT TO ZINC FINGER DOMAIN PROTEIN 2A|HOMO SAPIENS (9606)
>4ESU_1|Chain A|Ryanodine receptor 1|Oryctolagus cuniculus (9986)
>7KTL_1|Chain A|DNA-directed DNA/RNA polymerase mu|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 \(w=f(c)\)
5AGQ , Knot 57 113 0.79 40 89 108 
GAMERRRIATPEEVRLPLQHGWRREVRIKKGSHRWQGETWYYGPCGKRMKQFPEVIKYLSRNVVHSVRREHFSFSPRMPVGDFFEERDTPEGLQWVQLSAEEIPSRIQAITGS
4ESU , Knot 97 210 0.82 38 148 200 
SNANFDPRPVETLNVIIPEKLDSFINKFAEYTHEKWAFDKIQNNWMYGENVDEELKTHPMLRPYKTFSEKDKEIYRWPIKESLKAMIAWEWTIEKAREGEEERTEKKKTRKISQTAQTYDPREGYNPQPPDLSGVTLSRELQAMAEQLAENYHNTWGRKKKQELEAKGGGTHPLLVPYDTLTAKEKARDREKAQELLKFLQMNGYAVTRG
7KTL , Knot 148 356 0.81 40 197 345 
GSAAAPLSPAWMPAYACQRPTPLTHHNTGLSEALEILAEAAGFEGSEGRLLTFCRAASVLKALPSPVTTLSQLQGLPHFGEHSSRVVQELLEHGVCEEVERVRRSERYQTMKLFTQIFGVGVKTADRWYREGLRTLDDLREQPQKLTQQQKAGLQHHQDLSTPVLRSDVDALQQVVEEAVGQALPGATVTLTGGFRRGKLQGHDVDFLITHPKEGQEAGLLPRVMCRLQDQGLILYHQHQHSCCESPTRLAQQSHMDAFERSFCIFRLPQPGSWKAVRVDLVVAPVSQFPFALLGWTGSDLFQRELRRFSRKEKGLWLNSHGLFDPEQKTFFQAASEEDIFRHLGLEYLPPEQRNA

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 \(c\).

\(|P_{f(5AGQ_1)}(2) \setminus P_{f(4ESU_1)}(2)|=55\), \(|P_{f(4ESU_1)}(2) \setminus P_{f(5AGQ_1)}(2)|=114\). 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 1:11100001101001011100110001010010001010010011010010011011001000110010000101010111101100000101101101010011001011010
Pair \(Z_2\) Length of longest common subsequence
5AGQ_1,4ESU_1 169 3
5AGQ_1,7KTL_1 170 3
4ESU_1,7KTL_1 185 4

Newick tree

 
[
	7KTL_1:90.22,
	[
		5AGQ_1:84.5,4ESU_1:84.5
	]:5.72
]

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 \((0.85)(0.8)(\frac{323 }{\log_{20} 323}-\frac{113}{\log_{20}113})=65.1\)
Status Protein1 Protein2 d d1/2
Query variables 5AGQ_1 4ESU_1 87 66
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]