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Parikh vectors
4ZCT_1 4BCY_1 1TMK_1 Letter Amino acid
9 0 6 Y Tyrosine
8 12 7 A Alanine
5 4 8 R Arginine
16 12 14 E Glutamic acid
5 7 3 H Histidine
12 9 25 L Leucine
17 11 21 K Lycine
4 4 10 F Phenylalanine
7 3 10 Q Glutamine
8 24 15 G Glycine
5 5 5 P Proline
3 1 3 W Tryptophan
14 14 13 V Valine
10 7 9 N Asparagine
19 11 19 D Aspartic acid
1 2 3 C Cysteine
11 9 16 I Isoleucine
4 0 5 M Methionine
10 10 11 S Serine
12 8 13 T Threonine 

4ZCT_1|Chain A|Cholinephosphate cytidylyltransferase|Plasmodium falciparum (36329)
>4BCY_1|Chain A|SUPEROXIDE DISMUTASE [CU-ZN]|HOMO SAPIENS (9606)
>1TMK_1|Chains A, B|THYMIDYLATE KINASE|Saccharomyces cerevisiae (4932)
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 \(w=f(c)\)
4ZCT , Knot 84 180 0.80 40 133 169 
GHMAVPDDDDDDDNSNDESEYESSQMDSEKNKGSIKNSKNVVIYADGVYDMLHLGHMKQLEQAKKLFENTTLIVGVTSDNETKLFKGQVVQTLEERTETLKHIRWVDEIISPCPWVVTPEFLEKYKIDYVAHDDIPYANNQKEDIYAWLKRAGKFKATQRTEGVSTTDLIVRILKNYEDY
4BCY , Knot 73 153 0.80 36 110 148 
ATKAVAVLKGDGPVQGIINFEQKESNGPVKVWGSIKGLTEGLFGFHVHEEEDNTAGCTSAGPHFNPLSRKHGGPKDEERHVGDLGNVTADKDGVADVSIEDSVISLSGDHAIIGRTLVVHEKADDLGKGGNEESTKTGNAGSRLACGVIGIAQ
1TMK , Knot 98 216 0.81 40 151 208 
MMGRGKLILIEGLDRTGKTTQCNILYKKLQPNCKLLKFPERSTRIGGLINEYLTDDSFQLSDQAIHLLFSANRWEIVDKIKKDLLEGKNIVMDRYVYSGVAYSAAKGTNGMDLDWCLQPDVGLLKPDLTLFLSTQDVDNNAEKSGFGDERYETVKFQEKVKQTFMKLLDKEIRKGDESITIVDVTNKGIQEVEALIWQIVEPVLSTHIDHDKFSFF

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 \(c\).

\(|P_{f(4ZCT_1)}(2) \setminus P_{f(4BCY_1)}(2)|=83\), \(|P_{f(4BCY_1)}(2) \setminus P_{f(4ZCT_1)}(2)|=60\). 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 1:101111000000000000000000010000001010000011101011001101101001001001100001111100000001101011001000000100101100110101111010110000100110001101000000101110011010100000110000111011000000
Pair \(Z_2\) Length of longest common subsequence
4ZCT_1,4BCY_1 143 4
4ZCT_1,1TMK_1 154 3
4BCY_1,1TMK_1 157 3

Newick tree

 
[
	1TMK_1:79.72,
	[
		4ZCT_1:71.5,4BCY_1:71.5
	]:8.22
]

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 \((0.85)(0.8)(\frac{333 }{\log_{20} 333}-\frac{153}{\log_{20}153})=54.8\)
Status Protein1 Protein2 d d1/2
Query variables 4ZCT_1 4BCY_1 68 63
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]