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Parikh vectors
3UTS_1 5ZGG_1 5DEA_1 Letter Amino acid
24 4 4 A Alanine
23 2 0 R Arginine
2 2 0 N Asparagine
16 5 0 V Valine
5 0 0 M Methionine
23 3 0 T Threonine
18 1 0 D Aspartic acid
22 0 0 E Glutamic acid
20 2 18 G Glycine
4 3 0 I Isoleucine
14 2 0 Y Tyrosine
4 1 6 C Cysteine
17 0 0 Q Glutamine
13 0 0 H Histidine
8 1 0 F Phenylalanine
10 1 0 W Tryptophan
17 3 0 L Leucine
10 1 0 K Lycine
12 1 0 P Proline
14 2 0 S Serine 

3UTS_1|Chains A, F|HLA class I histocompatibility antigen, A-2 alpha chain|Homo sapiens (9606)
>5ZGG_1|Chains A, B|Tumor necrosis factor receptor superfamily member 16|Homo sapiens (9606)
>5DEA_1|Chains A, C|sc1|synthetic construct (32630)
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 \(w=f(c)\)
3UTS , Knot 122 276 0.82 40 182 264 
GSHSMRYFFTSVSRPGRGEPRFIAVGYVDDTQFVRFDSDAASQRMEPRAPWIEQEGPEYWDGETRKVKAHSQTHRVDLGTLRGYYNQSEAGSHTVQRMYGCDVGSDWRFLRGYHQYAYDGKDYIALKEDLRSWTAADMAAQTTKHKWEAAHVAEQLRAYLEGTCVEWLRRYLENGKETLQRTDAPKTHMTHHAVSDHEATLRCWALSFYPAEITLTWQRDGEDQTQDTELVETRPAGDGTFQKWAAVVVPSGQEQRYTCHVQHEGLPKPLTLRWEP
5ZGG , Knot 24 34 0.83 32 32 32 
TRGTTDNLIPVYCSILAAVVVGLVAYIAFKRWNS
5DEA , Knot 14 35 0.47 8 11 23 
GCUGCGGUGUGGAAGGAGUGGCUGGGUUGCGCAGC

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 \(c\).

\(|P_{f(3UTS_1)}(2) \setminus P_{f(5ZGG_1)}(2)|=166\), \(|P_{f(5ZGG_1)}(2) \setminus P_{f(3UTS_1)}(2)|=16\). 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 1:100010011001001101010111110100001101000110001010111100011001010000101000000101101010000001100010010100110010110100001001000111000100101101110000001011011001010101001011000100100010000110001000110000101001110101101010100010000000011000111010100111111101000000001000111011010101
Pair \(Z_2\) Length of longest common subsequence
3UTS_1,5ZGG_1 182 5
3UTS_1,5DEA_1 187 2
5ZGG_1,5DEA_1 41 2

Newick tree

 
[
	3UTS_1:10.87,
	[
		5ZGG_1:20.5,5DEA_1:20.5
	]:85.37
]

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 \((0.85)(0.8)(\frac{310 }{\log_{20} 310}-\frac{34}{\log_{20}34})=90.4\)
Status Protein1 Protein2 d d1/2
Query variables 3UTS_1 5ZGG_1 114 63.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]