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Parikh vectors
3ZCO_1 8AYC_1 9HQL_1 Letter Amino acid
7 8 17 G Glycine
13 12 8 I Isoleucine
15 12 8 L Leucine
8 3 12 T Threonine
2 0 1 W Tryptophan
11 17 17 V Valine
6 4 4 N Asparagine
5 7 13 D Aspartic acid
3 1 4 Y Tyrosine
8 13 10 A Alanine
15 10 18 E Glutamic acid
7 1 5 M Methionine
5 13 10 F Phenylalanine
5 7 7 P Proline
1 2 8 H Histidine
14 13 16 K Lycine
2 5 7 Q Glutamine
9 9 16 S Serine
3 3 15 R Arginine
2 1 2 C Cysteine 

3ZCO_1|Chain A|REGULATORY PROTEIN SIR3|SACCHAROMYCES CEREVISIAE (4932)
>8AYC_1|Chain A|Beta-hydroxyacyl-(Acyl-carrier-protein) dehydratase|Candidatus Scalindua brodae (237368)
>9HQL_1|Chain A|Heterogeneous nuclear ribonucleoprotein A1, N-terminally processed|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 \(w=f(c)\)
3ZCO , Knot 68 141 0.79 40 107 136 
GPSEAINGFKDETISKKIIGMSLLMRTFLYTLAQETEGTNRHTLALETVLIKMVKMLRDNPGYKASKEIKKVICGAWEPAITIEKLKQFSWISVVNDLVGEKLVVVVLEEPSASIMVELKLPLEINYAFSMDEEFKNMDCI
8AYC , Knot 72 141 0.84 38 106 136 
VNFEEVRELVPQKYPFLFIDKVIELQKEARIVCLKNISGNEPFFAGHFPDFAIMPGVLIVEALAQASGILFKKSFSTEQHKDDVFLLASANVRFSKPVFPGDQLILEIDIEKVISSAAIVKGVAKVGDKVVTKATLSFGVA
9HQL , Knot 96 198 0.85 40 143 192 
GPMGSKSESPKEPEQLRKLFIGGLSFETTDESLRSHFEQWGTLTDCVVMRDPNTKRSRGFGFVTYATVEEVDAAMNARPHKVDGRVVEPKRAVSREDSQRPGAHLTVKKIFVGGIKEDTEEHHLRDYFEQYGKIEVIEIMTDRGSGKKRGFAFVTFDDHDSVDKIVIQKYHTVNGHNCEVRKALSKQEMASASSSQRG

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 \(c\).

\(|P_{f(3ZCO_1)}(2) \setminus P_{f(8AYC_1)}(2)|=67\), \(|P_{f(8AYC_1)}(2) \setminus P_{f(3ZCO_1)}(2)|=66\). 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 1:110011011000010001111011100110011000010000011100111011011000110010001001101110111010010010110110011100111111001010111010111010011010001001001
Pair \(Z_2\) Length of longest common subsequence
3ZCO_1,8AYC_1 133 3
3ZCO_1,9HQL_1 156 3
8AYC_1,9HQL_1 147 3

Newick tree

 
[
	9HQL_1:78.63,
	[
		3ZCO_1:66.5,8AYC_1:66.5
	]:12.13
]

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 \((0.85)(0.8)(\frac{282 }{\log_{20} 282}-\frac{141}{\log_{20}141})=43.7\)
Status Protein1 Protein2 d d1/2
Query variables 3ZCO_1 8AYC_1 55 54
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]